Category Archives: quantitative research

K Nearest Neighbor in R

K-nearest neighbor is one of many nonlinear algorithms that can be used in machine learning. By non-linear I mean that a linear combination of the features or variables is not needed in order to develop decision boundaries. This allows for the analysis of data that naturally does not meet the assumptions of linearity.

KNN is also known as a “lazy learner”. This means that there are known coefficients or parameter estimates. When doing regression we always had coefficient outputs regardless of the type of regression (ridge, lasso, elastic net, etc.). What KNN does instead is used K nearest neighbors to give a label to an unlabeled example. Our job when using KNN is to determine the number of K neighbors to use that is most accurate based on the different criteria for assessing the models.

In this post, we will develop a KNN model using the “Mroz” dataset from the “Ecdat” package. Our goal is to predict if someone lives in the city based on the other predictor variables. Below is some initial code.

library(class);library(kknn);library(caret);library(corrplot)
library(reshape2);library(ggplot2);library(pROC);library(Ecdat)
data(Mroz)
str(Mroz)
## 'data.frame':    753 obs. of  18 variables:
##  $ work      : Factor w/ 2 levels "yes","no": 2 2 2 2 2 2 2 2 2 2 ...
##  $ hoursw    : int  1610 1656 1980 456 1568 2032 1440 1020 1458 1600 ...
##  $ child6    : int  1 0 1 0 1 0 0 0 0 0 ...
##  $ child618  : int  0 2 3 3 2 0 2 0 2 2 ...
##  $ agew      : int  32 30 35 34 31 54 37 54 48 39 ...
##  $ educw     : int  12 12 12 12 14 12 16 12 12 12 ...
##  $ hearnw    : num  3.35 1.39 4.55 1.1 4.59 ...
##  $ wagew     : num  2.65 2.65 4.04 3.25 3.6 4.7 5.95 9.98 0 4.15 ...
##  $ hoursh    : int  2708 2310 3072 1920 2000 1040 2670 4120 1995 2100 ...
##  $ ageh      : int  34 30 40 53 32 57 37 53 52 43 ...
##  $ educh     : int  12 9 12 10 12 11 12 8 4 12 ...
##  $ wageh     : num  4.03 8.44 3.58 3.54 10 ...
##  $ income    : int  16310 21800 21040 7300 27300 19495 21152 18900 20405 20425 ...
##  $ educwm    : int  12 7 12 7 12 14 14 3 7 7 ...
##  $ educwf    : int  7 7 7 7 14 7 7 3 7 7 ...
##  $ unemprate : num  5 11 5 5 9.5 7.5 5 5 3 5 ...
##  $ city      : Factor w/ 2 levels "no","yes": 1 2 1 1 2 2 1 1 1 1 ...
##  $ experience: int  14 5 15 6 7 33 11 35 24 21 ...

We need to remove the factor variable “work” as KNN cannot use factor variables. After this, we will use the “melt” function from the “reshape2” package to look at the variables when divided by whether the example was from the city or not.

Mroz$work<-NULL
mroz.melt<-melt(Mroz,id.var='city')
Mroz_plots<-ggplot(mroz.melt,aes(x=city,y=value))+geom_boxplot()+facet_wrap(~variable, ncol = 4)
Mroz_plots

1

From the plots, it appears there are no differences in how the variable act whether someone is from the city or not. This may be a flag that classification may not work.

We now need to scale our data otherwise the results will be inaccurate. Scaling might also help our box-plots because everything will be on the same scale rather than spread all over the place. To do this we will have to temporarily remove our outcome variable from the data set because it’s a factor and then reinsert it into the data set. Below is the code.

mroz.scale<-as.data.frame(scale(Mroz[,-16]))
mroz.scale$city<-Mroz$city

We will now look at our box-plots a second time but this time with scaled data.

mroz.scale.melt<-melt(mroz.scale,id.var="city")
mroz_plot2<-ggplot(mroz.scale.melt,aes(city,value))+geom_boxplot()+facet_wrap(~variable, ncol = 4)
mroz_plot2

1

This second plot is easier to read but there is still little indication of difference.

We can now move to checking the correlations among the variables. Below is the code

mroz.cor<-cor(mroz.scale[,-17])
corrplot(mroz.cor,method = 'number')

1.png

There is a high correlation between husband’s age (ageh) and wife’s age (agew). Since this algorithm is non-linear this should not be a major problem.

We will now divide our dataset into the training and testing sets

set.seed(502)
ind=sample(2,nrow(mroz.scale),replace=T,prob=c(.7,.3))
train<-mroz.scale[ind==1,]
test<-mroz.scale[ind==2,]

Before creating a model we need to create a grid. We do not know the value of k yet so we have to run multiple models with different values of k in order to determine this for our model. As such we need to create a ‘grid’ using the ‘expand.grid’ function. We will also use cross-validation to get a better estimate of k as well using the “trainControl” function. The code is below.

grid<-expand.grid(.k=seq(2,20,by=1))
control<-trainControl(method="cv")

Now we make our model,

knn.train<-train(city~.,train,method="knn",trControl=control,tuneGrid=grid)
knn.train
## k-Nearest Neighbors 
## 
## 540 samples
##  16 predictors
##   2 classes: 'no', 'yes' 
## 
## No pre-processing
## Resampling: Cross-Validated (10 fold) 
## Summary of sample sizes: 487, 486, 486, 486, 486, 486, ... 
## Resampling results across tuning parameters:
## 
##   k   Accuracy   Kappa    
##    2  0.6000095  0.1213920
##    3  0.6368757  0.1542968
##    4  0.6424325  0.1546494
##    5  0.6386252  0.1275248
##    6  0.6329998  0.1164253
##    7  0.6589619  0.1616377
##    8  0.6663344  0.1774391
##    9  0.6663681  0.1733197
##   10  0.6609510  0.1566064
##   11  0.6664018  0.1575868
##   12  0.6682199  0.1669053
##   13  0.6572111  0.1397222
##   14  0.6719586  0.1694953
##   15  0.6571425  0.1263937
##   16  0.6664367  0.1551023
##   17  0.6719573  0.1588789
##   18  0.6608811  0.1260452
##   19  0.6590979  0.1165734
##   20  0.6609510  0.1219624
## 
## Accuracy was used to select the optimal model using  the largest value.
## The final value used for the model was k = 14.

R recommends that k = 16. This is based on a combination of accuracy and the kappa statistic. The kappa statistic is a measurement of the accuracy of a model while taking into account chance. We don’t have a model in the sense that we do not use the ~ sign like we do with regression. Instead, we have a train and a test set a factor variable and a number for k. This will make more sense when you see the code. Finally, we will use this information on our test dataset. We will then look at the table and the accuracy of the model.

knn.test<-knn(train[,-17],test[,-17],train[,17],k=16) #-17 removes the dependent variable 'city
table(knn.test,test$city)
##         
## knn.test  no yes
##      no   19   8
##      yes  61 125
prob.agree<-(15+129)/213
prob.agree
## [1] 0.6760563

Accuracy is 67% which is consistent with what we found when determining the k. We can also calculate the kappa. This done by calculating the probability and then do some subtraction and division. We already know the accuracy as we stored it in the variable “prob.agree” we now need the probability that this is by chance. Lastly, we calculate the kappa.

prob.chance<-((15+4)/213)*((15+65)/213)
kap<-(prob.agree-prob.chance)/(1-prob.chance)
kap
## [1] 0.664827

A kappa of .66 is actually good.

The example we just did was with unweighted k neighbors. There are times when weighted neighbors can improve accuracy. We will look at three different weighing methods. “Rectangular” is unweighted and is the one that we used. The other two are “triangular” and “epanechnikov”. How these calculate the weights is beyond the scope of this post. In the code below the argument “distance” can be set to 1 for Euclidean and 2 for absolute distance.

kknn.train<-train.kknn(city~.,train,kmax = 25,distance = 2,kernel = c("rectangular","triangular",
                                                                      "epanechnikov"))
plot(kknn.train)

1.png

kknn.train
## 
## Call:
## train.kknn(formula = city ~ ., data = train, kmax = 25, distance = 2,     kernel = c("rectangular", "triangular", "epanechnikov"))
## 
## Type of response variable: nominal
## Minimal misclassification: 0.3277778
## Best kernel: rectangular
## Best k: 14

If you look at the plot you can see which value of k is the best by looking at the point that is the lowest on the graph which is right before 15. Looking at the legend it indicates that the point is the “rectangular” estimate which is the same as unweighted. This means that the best classification is unweighted with a k of 14. Although it recommends a different value for k our misclassification was about the same.

Conclusion

In this post, we explored both weighted and unweighted KNN. This algorithm allows you to deal with data that does not meet the assumptions of regression by ignoring the need for parameters. However, because there are no numbers really attached to the results beyond accuracy it can be difficult to explain what is happening in the model to people. As such, perhaps the biggest drawback is communicating results when using KNN.

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Data Science Research Questions

Developing research questions is an absolute necessity in completing any research project. The questions you ask help to shape the type of analysis that you need to conduct.

The type of questions you ask in the context of analytics and data science are similar to those found in traditional quantitative research. Yet data science, like any other field, has its own distinct traits.

In this post, we will look at six different types of questions that are used frequently in the context of the field of data science. The six questions are…

  1. Descriptive
  2. Exploratory/Inferential
  3. Predictive
  4. Causal
  5. Mechanistic

Understanding the types of question that can be asked will help anyone involved in data science to determine what exactly it is that they want to know.

Descriptive

A descriptive question seeks to describe a characteristic of the dataset. For example, if I collect the GPA of 100 university student I may want to what the average GPA of the students is. Seeking the average is one example of a descriptive question.

With descriptive questions, there is no need for a hypothesis as you are not trying to infer, establish a relationship, or generalize to a broader context. You simply want to know a trait of the dataset.

Exploratory/Inferential

Exploratory questions seek to identify things that may be “interesting” in the dataset. Examples of things that may be interesting include trends, patterns, and or relationships among variables.

Exploratory questions generate hypotheses. This means that they lead to something that may be more formal questioned and tested. For example, if you have GPA and hours of sleep for university students. You may explore the potential that there is a relationship between these two variables.

 

Inferential questions are an extension of exploratory questions. What this means is that the exploratory question is formally tested by developing an inferential question. Often, the difference between an exploratory and inferential question is the following

  1. Exploratory questions are usually developed first
  2. Exploratory questions generate inferential questions
  3. Inferential questions are tested often on a different dataset from exploratory questions

In our example, if we find a relationship between GPA and sleep in our dataset. We may test this relationship in a different, perhaps larger dataset. If the relationship holds we can then generalize this to the population of the study.

Causal

Causal questions address if a change in one variable directly affects another. In analytics, A/B testing is one form of data collection that can be used to develop causal questions. For example, we may develop two version of a website and see which one generates more sales.

In this example, the type of website is the independent variable and sales is the dependent variable. By controlling the type of website people see we can see if this affects sales.

Mechanistic 

Mechanistic questions deal with how one variable affects another. This is different from causal questions that focus on if one variable affects another. Continuing with the website example, we may take a closer look at the two different websites and see what it was about them that made one more succesful in generating sales. It may be that one had more banners than another or fewer pictures. Perhaps there were different products offered on the home page.

All of these different features, of course, require data that helps to explain what is happening. This leads to an important point that the questions that can be asked are limited by the available data. You can’t answer a question that does not contain data that may answer it.

Conclusion

Answering questions is essential what research is about. In order to do this, you have to know what your questions are. This information will help you to decide on the analysis you wish to conduct. Familiarity with the types of research questions that are common in data science can help you to approach and complete analysis much faster than when this is unclear

Linear VS Quadratic Discriminant Analysis in R

In this post, we will look at linear discriminant analysis (LDA) and quadratic discriminant analysis (QDA). Discriminant analysis is used when the dependent variable is categorical. Another commonly used option is logistic regression but there are differences between logistic regression and discriminant analysis. Both LDA and QDA are used in situations in which there is a clear separation between the classes you want to predict. If the categories are fuzzier logistic regression is often the better choice.

For our example, we will use the “Mathlevel” dataset found in the “Ecdat” package. Our goal will be to predict the sex of a respondent based on SAT math score, major, foreign language proficiency, as well as the number of math, physic, and chemistry classes a respondent took. Below is some initial code to start our analysis.

library(MASS);library(Ecdat)
data("Mathlevel")

The first thing we need to do is clean up the data set. We have to remove any missing data in order to run our model. We will create a dataset called “math” that has the “Mathlevel” dataset but with the “NA”s removed use the “na.omit” function. After this, we need to set our seed for the purpose of reproducibility using the “set.seed” function. Lastly, we will split the data using the “sample” function using a 70/30 split. The training dataset will be called “math.train” and the testing dataset will be called “math.test”. Below is the code

math<-na.omit(Mathlevel)
set.seed(123)
math.ind<-sample(2,nrow(math),replace=T,prob = c(0.7,0.3))
math.train<-math[math.ind==1,]
math.test<-math[math.ind==2,]

Now we will make our model and it is called “lda.math” and it will include all available variables in the “math.train” dataset. Next, we will check the results by calling the model. Finally, we will examine the plot to see how our model is doing. Below is the code.

lda.math<-lda(sex~.,math.train)
lda.math
## Call:
## lda(sex ~ ., data = math.train)
## 
## Prior probabilities of groups:
##      male    female 
## 0.5986079 0.4013921 
## 
## Group means:
##        mathlevel.L mathlevel.Q mathlevel.C mathlevel^4 mathlevel^5
## male   -0.10767593  0.01141838 -0.05854724   0.2070778  0.05032544
## female -0.05571153  0.05360844 -0.08967303   0.2030860 -0.01072169
##        mathlevel^6      sat languageyes  majoreco  majoross   majorns
## male    -0.2214849 632.9457  0.07751938 0.3914729 0.1472868 0.1782946
## female  -0.2226767 613.6416  0.19653179 0.2601156 0.1907514 0.2485549
##          majorhum mathcourse physiccourse chemistcourse
## male   0.05426357   1.441860    0.7441860      1.046512
## female 0.07514451   1.421965    0.6531792      1.040462
## 
## Coefficients of linear discriminants:
##                       LD1
## mathlevel.L    1.38456344
## mathlevel.Q    0.24285832
## mathlevel.C   -0.53326543
## mathlevel^4    0.11292817
## mathlevel^5   -1.24162715
## mathlevel^6   -0.06374548
## sat           -0.01043648
## languageyes    1.50558721
## majoreco      -0.54528930
## majoross       0.61129797
## majorns        0.41574298
## majorhum       0.33469586
## mathcourse    -0.07973960
## physiccourse  -0.53174168
## chemistcourse  0.16124610
plot(lda.math,type='both')

1.png

Calling “lda.math” gives us the details of our model. It starts be indicating the prior probabilities of someone being male or female. Next is the means for each variable by sex. The last part is the coefficients of the linear discriminants. Each of these values is used to determine the probability that a particular example is male or female. This is similar to a regression equation.

The plot provides us with densities of the discriminant scores for males and then for females. The output indicates a problem. There is a great deal of overlap between male and females in the model. What this indicates is that there is a lot of misclassification going on as the two groups are not clearly separated. Furthermore, this means that logistic regression is probably a better choice for distinguishing between male and females. However, since this is for demonstrating purposes we will not worry about this.

We will now use the “predict” function on the training set data to see how well our model classifies the respondents by gender. We will then compare the prediction of the model with the actual classification. Below is the code.

math.lda.predict<-predict(lda.math)
math.train$lda<-math.lda.predict$class
table(math.train$lda,math.train$sex)
##         
##          male female
##   male    219    100
##   female   39     73
mean(math.train$lda==math.train$sex)
## [1] 0.6774942

As you can see, we have a lot of misclassification happening. A large amount of false negatives which is a lot of males being classified as female. The overall accuracy is only 59% which is not much better than chance.

We will now conduct the same analysis on the test data set. Below is the code.

lda.math.test<-predict(lda.math,math.test)
math.test$lda<-lda.math.test$class
table(math.test$lda,math.test$sex)
##         
##          male female
##   male     92     43
##   female   23     20
mean(math.test$lda==math.test$sex)
## [1] 0.6292135

As you can see the results are similar. To put it simply, our model is terrible. The main reason is that there is little distinction between males and females as shown in the plot. However, we can see if perhaps a quadratic discriminant analysis will do better

QDA allows for each class in the dependent variable to have its own covariance rather than a shared covariance as in LDA. This allows for quadratic terms in the development of the model. To complete a QDA we need to use the “qda” function from the “MASS” package. Below is the code for the training data set.

math.qda.fit<-qda(sex~.,math.train)
math.qda.predict<-predict(math.qda.fit)
math.train$qda<-math.qda.predict$class
table(math.train$qda,math.train$sex)
##         
##          male female
##   male    215     84
##   female   43     89
mean(math.train$qda==math.train$sex)
## [1] 0.7053364

You can see there is almost no difference. Below is the code for the test data.

math.qda.test<-predict(math.qda.fit,math.test)
math.test$qda<-math.qda.test$class
table(math.test$qda,math.test$sex)
##         
##          male female
##   male     91     43
##   female   24     20
mean(math.test$qda==math.test$sex)
## [1] 0.6235955

Still disappointing. However, in this post, we reviewed linear discriminant analysis as well as learned about the use of quadratic linear discriminant analysis. Both of these statistical tools are used for predicting categorical dependent variables. LDA assumes shared covariance in the dependent variable categories will QDA allows for each category in the dependent variable to have its own variance.

Logistic Regression in R

In this post, we will conduct a logistic regression analysis. Logistic regression is used when you want to predict a categorical dependent variable using continuous or categorical dependent variables. In our example, we want to predict Sex (male or female) when using several continuous variables from the “survey” dataset in the “MASS” package.

library(MASS);library(bestglm);library(reshape2);library(corrplot)
data(survey)
?MASS::survey #explains the variables in the study

The first thing we need to do is remove the independent factor variables from our dataset. The reason for this is that the function that we will use for the cross-validation does not accept factors. We will first use the “str” function to identify factor variables and then remove them from the dataset. We also need to remove in examples that are missing data so we use the “na.omit” function for this. Below is the code

survey$Clap<-NULL
survey$W.Hnd<-NULL
survey$Fold<-NULL
survey$Exer<-NULL
survey$Smoke<-NULL
survey$M.I<-NULL
survey<-na.omit(survey)

We now need to check for collinearity using the “corrplot.mixed” function form the “corrplot” package.

pc<-cor(survey[,2:5])
corrplot.mixed(pc)
corrplot.mixed(pc)

1.png

We have an extreme correlation between “We.Hnd” and “NW.Hnd” this makes sense because people’s hands are normally the same size. Since this blog post is a demonstration of logistic regression we will not worry about this too much.

We now need to divide our dataset into a train and a test set. We set the seed for. First, we need to make a variable that we call “ind” that is randomly assigned 70% of the number of rows of survey 1 and 30% 2. We then subset the “train” dataset by taking all rows that are 1’s based on the “ind” variable and we create the “test” dataset for all the rows that line up with 2 in the “ind” variable. This means our data split is 70% train and 30% test. Below is the code

set.seed(123)
ind<-sample(2,nrow(survey),replace=T,prob = c(0.7,0.3))
train<-survey[ind==1,]
test<-survey[ind==2,]

We now make our model. We use the “glm” function for logistic regression. We set the family argument to “binomial”. Next, we look at the results as well as the odds ratios.

fit<-glm(Sex~.,family=binomial,train)
summary(fit)
## 
## Call:
## glm(formula = Sex ~ ., family = binomial, data = train)
## 
## Deviance Residuals: 
##     Min       1Q   Median       3Q      Max  
## -1.9875  -0.5466  -0.1395   0.3834   3.4443  
## 
## Coefficients:
##              Estimate Std. Error z value Pr(>|z|)    
## (Intercept) -46.42175    8.74961  -5.306 1.12e-07 ***
## Wr.Hnd       -0.43499    0.66357  -0.656    0.512    
## NW.Hnd        1.05633    0.70034   1.508    0.131    
## Pulse        -0.02406    0.02356  -1.021    0.307    
## Height        0.21062    0.05208   4.044 5.26e-05 ***
## Age           0.00894    0.05368   0.167    0.868    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## (Dispersion parameter for binomial family taken to be 1)
## 
##     Null deviance: 169.14  on 122  degrees of freedom
## Residual deviance:  81.15  on 117  degrees of freedom
## AIC: 93.15
## 
## Number of Fisher Scoring iterations: 6
exp(coef(fit))
##  (Intercept)       Wr.Hnd       NW.Hnd        Pulse       Height 
## 6.907034e-21 6.472741e-01 2.875803e+00 9.762315e-01 1.234447e+00 
##          Age 
## 1.008980e+00

The results indicate that only height is useful in predicting if someone is a male or female. The second piece of code shares the odds ratios. The odds ratio tell how a one unit increase in the independent variable leads to an increase in the odds of being male in our model. For example, for every one unit increase in height there is a 1.23 increase in the odds of a particular example being male.

We now need to see how well our model does on the train and test dataset. We first capture the probabilities and save them to the train dataset as “probs”. Next we create a “predict” variable and place the string “Female” in the same number of rows as are in the “train” dataset. Then we rewrite the “predict” variable by changing any example that has a probability above 0.5 as “Male”. Then we make a table of our results to see the number correct, false positives/negatives. Lastly, we calculate the accuracy rate. Below is the code.

train$probs<-predict(fit, type = 'response')
train$predict<-rep('Female',123)
train$predict[train$probs>0.5]<-"Male"
table(train$predict,train$Sex)
##         
##          Female Male
##   Female     61    7
##   Male        7   48
mean(train$predict==train$Sex)
## [1] 0.8861789

Despite the weaknesses of the model with so many insignificant variables it is surprisingly accurate at 88.6%. Let’s see how well we do on the “test” dataset.

test$prob<-predict(fit,newdata = test, type = 'response')
test$predict<-rep('Female',46)
test$predict[test$prob>0.5]<-"Male"
table(test$predict,test$Sex)
##         
##          Female Male
##   Female     17    3
##   Male        0   26
mean(test$predict==test$Sex)
## [1] 0.9347826

As you can see, we do even better on the test set with an accuracy of 93.4%. Our model is looking pretty good and height is an excellent predictor of sex which makes complete sense. However, in the next post we will use cross-validation and the ROC plot to further assess the quality of it.

Probability,Odds, and Odds Ratio

In logistic regression, there are three terms that are used frequently but can be confusing if they are not thoroughly explained. These three terms are probability, odds, and odds ratio. In this post, we will look at these three terms and provide an explanation of them.

Probability

Probability is probably (no pun intended) the easiest of these three terms to understand. Probability is simply the likelihood that a certain even will happen.  To calculate the probability in the traditional sense you need to know the number of events and outcomes to find the probability.

Bayesian probability uses prior probabilities to develop a posterior probability based on new evidence. For example, at one point during Super Bowl LI the Atlanta Falcons had a 99.7% chance of winning. This was base don such factors as the number points they were ahead and the time remaining.  As these changed, so did the probability of them winning. yet the Patriots still found a way to win with less then a 1% chance

Bayesian probability was also used for predicting who would win the 2016 US presidential race. It is important to remember that probability is an expression of confidence and not a guarantee as we saw in both examples.

Odds

Odds are the expression of relative probabilities. Odds are calculated using the following equation

probability of the event ⁄ 1 – probability of the event

For example, at one point during Super Bowl LI the odds of the Atlanta Falcons winning were as follows

0.997 ⁄ 1 – 0.997 = 332

This can be interpreted as the odds being 332 to 1! This means that Atlanta was 332 times more likely to win the Super Bowl then loss the Super Bowl.

Odds are commonly used in gambling and this is probably (again no pun intended) where most of us have heard the term before. The odds is just an extension of probabilities and the are most commonly expressed as a fraction such as one in four, etc.

Odds Ratio

A ratio is the comparison of of two numbers and indicates how many times one number is contained or contains another number. For example, a ration of boys to girls is 5 to 1 it means that there are five boys for every one girl.

By  extension odds ratio is the comparison of two different odds. For example, if the odds of Team A making the playoffs is 45% and the odds of Team B making the playoffs is 35% the odds ratio is calculated as follows.

0.45 ⁄ 0.35 = 1.28

Team A is 1.28 more likely to make the playoffs then Team B.

The value of the odds and the odds ratio can sometimes be the same.  Below is the odds ratio of the Atlanta Falcons winning and the New Patriots winning Super Bowl LI

0.997⁄ 0.003 = 332

As such there is little difference between odds and odds ratio except that odds ratio is the ratio of two odds ratio. As you can tell, there is a lot of confusion about this for the average person. However, understanding these terms is critical to the application of logistic regression.

Best Subset Regression in R

In this post, we will take a look at best subset regression. Best subset regression fits a model for all possible feature or variable combinations and the decision for the most appropriate model is made by the analyst based on judgment or some statistical criteria.

Best subset regression is an alternative to both Forward and Backward stepwise regression. Forward stepwise selection adds one variable at a time based on the lowest residual sum of squares until no more variables continue to lower the residual sum of squares. Backward stepwise regression starts with all variables in the model and removes variables one at a time. The concern with stepwise methods is they can produce biased regression coefficients, conflicting models, and inaccurate confidence intervals.

Best subset regression bypasses these weaknesses of stepwise models by creating all models possible and then allowing you to assess which variables should be included in your final model. The one drawback to best subset is that a large number of variables means a large number of potential models, which can make it difficult to make a decision among several choices.

In this post, we will use the “Fair” dataset from the “Ecdat” package to predict marital satisfaction based on age, Sex, the presence of children, years married, religiosity, education, occupation, and the number of affairs in the past year. Below is some initial code.

library(leaps);library(Ecdat);library(car);library(lmtest)
data(Fair)

We begin our analysis by building the initial model with all variables in it. Below is the code

fit<-lm(rate~.,Fair)
summary(fit)
## 
## Call:
## lm(formula = rate ~ ., data = Fair)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -3.2049 -0.6661  0.2298  0.7705  2.2292 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  3.522875   0.358793   9.819  < 2e-16 ***
## sexmale     -0.062281   0.099952  -0.623  0.53346    
## age         -0.009683   0.007548  -1.283  0.20005    
## ym          -0.019978   0.013887  -1.439  0.15079    
## childyes    -0.206976   0.116227  -1.781  0.07546 .  
## religious    0.042142   0.037705   1.118  0.26416    
## education    0.068874   0.021153   3.256  0.00119 ** 
## occupation  -0.015606   0.029602  -0.527  0.59825    
## nbaffairs   -0.078812   0.013286  -5.932 5.09e-09 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1.03 on 592 degrees of freedom
## Multiple R-squared:  0.1405, Adjusted R-squared:  0.1289 
## F-statistic:  12.1 on 8 and 592 DF,  p-value: 4.487e-16

The initial results are already interesting even though the r-square is low. When couples have children the have less marital satisfaction than couples without children when controlling for the other factors and this is the strongest regression weight. In addition, the more education a person has there is an increase in marital satisfaction. Lastly, as the number of affairs increases there is also a decrease in marital satisfaction. Keep in mind that the “rate” variable goes from 1 to 5 with one meaning a terrible marriage to five being a great one. The mean marital satisfaction was 3.52 when controlling for the other variables.

We will now create our subset models. Below is the code.

sub.fit<-regsubsets(rate~.,Fair)
best.summary<-summary(sub.fit)

In the code above we create the sub models using the “regsubsets” function from the “leaps” package and saved it in the variable called “sub.fit”. We then saved the summary of “sub.fit” in the variable “best.summary”. We will use the “best.summary” “sub.fit variables several times to determine which model to use.

There are many different ways to assess the model. We will use the following statistical methods that come with the results from the “regsubset” function.

  • Mallow’ Cp
  • Bayesian Information Criteria

We will make two charts for each of the criteria above. The plot to the left will explain how many features to include in the model. The plot to the right will tell you which variables to include. It is important to note that for both of these methods, the lower the score the better the model. Below is the code for Mallow’s Cp.

par(mfrow=c(1,2))
plot(best.summary$cp)
plot(sub.fit,scale = "Cp")

1

The plot on the left suggests that a four feature model is the most appropriate. However, this chart does not tell me which four features. The chart on the right is read in reverse order. The high numbers are at the bottom and the low numbers are at the top when looking at the y-axis. Knowing this, we can conclude that the most appropriate variables to include in the model are age, children presence, education, and number of affairs. Below are the results using the Bayesian Information Criterion

par(mfrow=c(1,2))
plot(best.summary$bic)
plot(sub.fit,scale = "bic")

1

These results indicate that a three feature model is appropriate. The variables or features are years married, education, and number of affairs. Presence of children was not considered beneficial. Since our original model and Mallow’s Cp indicated that presence of children was significant we will include it for now.

Below is the code for the model based on the subset regression.

fit2<-lm(rate~age+child+education+nbaffairs,Fair)
summary(fit2)
## 
## Call:
## lm(formula = rate ~ age + child + education + nbaffairs, data = Fair)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -3.2172 -0.7256  0.1675  0.7856  2.2713 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  3.861154   0.307280  12.566  < 2e-16 ***
## age         -0.017440   0.005057  -3.449 0.000603 ***
## childyes    -0.261398   0.103155  -2.534 0.011531 *  
## education    0.058637   0.017697   3.313 0.000978 ***
## nbaffairs   -0.084973   0.012830  -6.623 7.87e-11 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1.029 on 596 degrees of freedom
## Multiple R-squared:  0.1352, Adjusted R-squared:  0.1294 
## F-statistic: 23.29 on 4 and 596 DF,  p-value: < 2.2e-16

The results look ok. The older a person is the less satisfied they are with their marriage. If children are present the marriage is less satisfying. The more educated the more satisfied they are. Lastly, the higher the number of affairs indicate less marital satisfaction. However, before we get excited we need to check for collinearity and homoscedasticity. Below is the code

vif(fit2)
##       age     child education nbaffairs 
##  1.249430  1.228733  1.023722  1.014338

No issues with collinearity.For vif values above 5 or 10 indicate a problem. Let’s check for homoscedasticity

par(mfrow=c(2,2))
plot(fit2)

1.jpeg

The normal qqplot and residuals vs leverage plot can be used for locating outliers. The residual vs fitted and the scale-location plot do not look good as there appears to be a pattern in the dispersion which indicates homoscedasticity. To confirm this we will use Breusch-Pagan test from the “lmtest” package. Below is the code

bptest(fit2)
## 
##  studentized Breusch-Pagan test
## 
## data:  fit2
## BP = 16.238, df = 4, p-value = 0.002716

There you have it. Our model violates the assumption of homoscedasticity. However, this model was developed for demonstration purpose to provide an example of subset regression.

Principal Component Analysis in R

This post will demonstrate the use of principal component analysis (PCA). PCA is useful for several reasons. One it allows you place your examples into groups similar to linear discriminant analysis but you do not need to know beforehand what the groups are. Second, PCA is used for the purpose of dimension reduction. For example, if you have 50 variables PCA can allow you to reduce this while retaining a certain threshold of variance. If you are working with a large dataset this can greatly reduce the computational time and general complexity of your models.

Keep in mind that there really is not a dependent variable as this is unsupervised learning. What you are trying to see is how different examples can

be mapped in space based on whatever independent variables are used. For our example, we will use the “Carseats” dataset from the “ISLR”. Our goal is to understand the relationship among the variables when examining the shelve location of the car seat. Below is the initial code to begin the analysis

library(ggplot2)
library(ISLR)
data("Carseats")

We first need to rearrange the data and remove the variables we are not going to use in the analysis. Below is the code.

Carseats1<-Carseats
Carseats1<-Carseats1[,c(1,2,3,4,5,6,8,9,7,10,11)]
Carseats1$Urban<-NULL
Carseats1$US<-NULL

Here is what we did 1. We made a copy of the “Carseats” data called “Careseats1” 2. We rearranged the order of the variables so that the factor variables are at the end. This will make sense later 3.We removed the “Urban” and “US” variables from the table as they will not be a part of our analysis

We will now do the PCA. We need to scale and center our data otherwise the larger numbers will have a much stronger influence on the results than smaller numbers. Fortunately, the “prcomp” function has a “scale” and a “center” argument. We will also use only the first 7 columns for the analysis as “sheveLoc” is not useful for this analysis. If we hadn’t moved “shelveLoc” to the end of the dataframe it would cause some headache. Below is the code.

Carseats.pca<-prcomp(Carseats1[,1:7],scale. = T,center = T)
summary(Carseats.pca)
## Importance of components:
##                           PC1    PC2    PC3    PC4    PC5     PC6     PC7
## Standard deviation     1.3315 1.1907 1.0743 0.9893 0.9260 0.80506 0.41320
## Proportion of Variance 0.2533 0.2026 0.1649 0.1398 0.1225 0.09259 0.02439
## Cumulative Proportion  0.2533 0.4558 0.6207 0.7605 0.8830 0.97561 1.00000

The summary of “Carseats.pca” Tells us how much of the variance each component explains. Keep in mind that the number of components is equal to the number of variables. The “proportion of variance” tells us the contribution each component makes and the “cumulative proportion”.

If your goal is dimension reduction than the number of components to keep depends on the threshold you set. For example, if you need around 90% of the variance you would keep the first 5 components. If you need 95% or more of the variance you would keep the first six. To actually use the components you would take the “Carseats.pca$x” data and move it to your data frame.

Keep in mind that the actual components have no conceptual meaning but is a numerical representation of a combination of several variables that were reduced using PCA to fewer variables such as going from 7 variables to 5 variables.

This means that PCA is great for reducing variables for prediction purpose but is much harder for explanatory studies unless you can explain what the new components represent.

For our purposes, we will keep 5 components. This means that we have reduced our dimensions from 7 to 5 while still keeping almost 90% of the variance. Graphing our results is tricky because we have 5 dimensions but the human mind can only conceptualize 3 at the best and normally 2. As such we will plot the first two components and label them by shelf location using ggplot2. Below is the code

scores<-as.data.frame(Carseats.pca$x)
pcaplot<-ggplot(scores,(aes(PC1,PC2,color=Carseats1$ShelveLoc)))+geom_point()
pcaplot

1.png

From the plot, you can see there is little separation when using the first two components of the PCA analysis. This makes sense as we can only graph to components so we are missing a lot of the variance. However, for demonstration purposes the analysis is complete.

Developing a Data Analysis Plan

It is extremely common for beginners and perhaps even experience researchers to lose track of what they are trying to achieve or do when trying to complete a research project. The open nature of research allows for a multitude of equally acceptable ways to complete a project. This leads to an inability to make a decision and or stay on course when doing research.

One way to reduce and eliminate the roadblock to decision making and focus in research is to develop a plan. In this post, we will look at one version of a data analysis plan.

Data Analysis Plan

A data analysis plan includes many features of a research project in it with a particular emphasis on mapping out how research questions will be answered and what is necessary to answer the question. Below is a sample template of the analysis plan.

analysis-plan-page-001-2

The majority of this diagram should be familiar to someone who has ever done research. At the top, you state the problem, this is the overall focus of the paper. Next, comes the purpose, the purpose is the over-arching goal of a research project.

After purpose comes the research questions. The research questions are questions about the problem that are answerable. People struggle with developing clear and answerable research questions. It is critical that research questions are written in a way that they can be answered and that the questions are clearly derived from the problem. Poor questions means poor or even no answers.

After the research questions, it is important to know what variables are available for the entire study and specifically what variables can be used to answer each research question. Lastly, you must indicate what analysis or visual you will develop in order to answer your research questions about your problem. This requires you to know how you will answer your research questions

Example

Below is an example of a completed analysis plan for  simple undergraduate level research paper

example-analysis-plan-page-001

In the example above, the student wants to understand the perceptions of university students about the cafeteria food quality and their satisfaction with the university. There were four research questions, a demographic descriptive question, a descriptive question about the two main variables, a comparison question, and lastly a relationship question.

The variables available for answering the questions are listed off to the left side. Under that, the student indicates the variables needed to answer each question. For example, the demographic variables of sex, class level, and major are needed to answer the question about the demographic profile.

The last section is the analysis. For the demographic profile, the student found the percentage of the population in each sub group of the demographic variables.

Conclusion

A data analysis plan provides an excellent way to determine what needs to be done to complete a study. It also helps a researcher to clearly understand what they are trying to do and provides a visual for those who the research wants to communicate with about the progress of a study.

Generalized Additive Models in R

In this post, we will learn how to create a generalized additive model (GAM). GAMs are non-parametric generalized linear models. This means that linear predictor of the model uses smooth functions on the predictor variables. As such, you do not need to specific the functional relationship between the response and continuous variables. This allows you to explore the data for potential relationships that can be more rigorously tested with other statistical models

In our example, we will use the “Auto” dataset from the “ISLR” package and use the variables “mpg”,“displacement”,“horsepower”,and “weight” to predict “acceleration”. We will also use the “mgcv” package. Below is some initial code to begin the analysis

library(mgcv)
library(ISLR)
data(Auto)

We will now make the model we want to understand the response of “accleration” to the explanatory variables of “mpg”,“displacement”,“horsepower”,and “weight”. After setting the model we will examine the summary. Below is the code

model1<-gam(acceleration~s(mpg)+s(displacement)+s(horsepower)+s(weight),data=Auto)
summary(model1)
## 
## Family: gaussian 
## Link function: identity 
## 
## Formula:
## acceleration ~ s(mpg) + s(displacement) + s(horsepower) + s(weight)
## 
## Parametric coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 15.54133    0.07205   215.7   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Approximate significance of smooth terms:
##                   edf Ref.df      F  p-value    
## s(mpg)          6.382  7.515  3.479  0.00101 ** 
## s(displacement) 1.000  1.000 36.055 4.35e-09 ***
## s(horsepower)   4.883  6.006 70.187  < 2e-16 ***
## s(weight)       3.785  4.800 41.135  < 2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## R-sq.(adj) =  0.733   Deviance explained = 74.4%
## GCV = 2.1276  Scale est. = 2.0351    n = 392

All of the explanatory variables are significant and the adjust r-squared is .73 which is excellent. edf stands for “effective degrees of freedom”. This modified version of the degree of freedoms is due to the smoothing process in the model. GCV stands for generalized cross validation and this number is useful when comparing models. The model with the lowest number is the better model.

We can also examine the model visually by using the “plot” function. This will allow us to examine if the curvature fitted by the smoothing process was useful or not for each variable. Below is the code.

plot(model1)

d71839c6-1baf-4886-98dd-7de8eac27855f4402e71-29f4-44e3-a941-3102fea89c78.pngcdbb392a-1d53-4dd0-8350-8b6d65284b00.pngbf28dd7a-d250-4619-bea0-5666e031e991.png

We can also look at a 3d graph that includes the linear predictor as well as the two strongest predictors. This is done with the “vis.gam” function. Below is the code

vis.gam(model1)

2136d310-b3f5-4c78-b166-4f6c4a1d0e12.png

If multiple models are developed. You can compare the GCV values to determine which model is the best. In addition, another way to compare models is with the “AIC” function. In the code below, we will create an additional model that includes “year” compare the GCV scores and calculate the AIC. Below is the code.

model2<-gam(acceleration~s(mpg)+s(displacement)+s(horsepower)+s(weight)+s(year),data=Auto)
summary(model2)
## 
## Family: gaussian 
## Link function: identity 
## 
## Formula:
## acceleration ~ s(mpg) + s(displacement) + s(horsepower) + s(weight) + 
##     s(year)
## 
## Parametric coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 15.54133    0.07203   215.8   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Approximate significance of smooth terms:
##                   edf Ref.df      F p-value    
## s(mpg)          5.578  6.726  2.749  0.0106 *  
## s(displacement) 2.251  2.870 13.757 3.5e-08 ***
## s(horsepower)   4.936  6.054 66.476 < 2e-16 ***
## s(weight)       3.444  4.397 34.441 < 2e-16 ***
## s(year)         1.682  2.096  0.543  0.6064    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## R-sq.(adj) =  0.733   Deviance explained = 74.5%
## GCV = 2.1368  Scale est. = 2.0338    n = 392
#model1 GCV
model1$gcv.ubre
##   GCV.Cp 
## 2.127589
#model2 GCV
model2$gcv.ubre
##   GCV.Cp 
## 2.136797

As you can see, the second model has a higher GCV score when compared to the first model. This indicates that the first model is a better choice. This makes sense because in the second model the variable “year” is not significant. To confirm this we will calculate the AIC scores using the AIC function.

AIC(model1,model2)
##              df      AIC
## model1 18.04952 1409.640
## model2 19.89068 1411.156

Again, you can see that model1 s better due to its fewer degrees of freedom and slightly lower AIC score.

Conclusion

Using GAMs is most common for exploring potential relationships in your data. This is stated because they are difficult to interpret and to try and summarize. Therefore, it is normally better to develop a generalized linear model over a GAM due to the difficulty in understanding what the data is trying to tell you when using GAMs.

Generalized Models in R

Generalized linear models are another way to approach linear regression. The advantage of of GLM is that allows the error to follow many different distributions rather than only the normal distribution which is an assumption of traditional linear regression.

Often GLM is used for response or dependent variables that are binary or represent count data. THis post will provide a brief explanation of GLM as well as provide an example.

Key Information

There are three important components to a GLM and they are

  • Error structure
  • Linear predictor
  • Link function

The error structure is the type of distribution you will use in generating the model. There are many different distributions in statistical modeling such as binomial, gaussian, poission, etc. Each distribution comes with certain assumptions that govern their use.

The linear predictor is the sum of the effects of the independent variables. Lastly, the link function determines the relationship between the linear predictor and the mean of the dependent variable. There are many different link functions and the best link function is the one that reduces the residual deviances the most.

In our example, we will try to predict if a house will have air conditioning based on the interactioon between number of bedrooms and bathrooms, number of stories, and the price of the house. To do this, we will use the “Housing” dataset from the “Ecdat” package. Below is some initial code to get started.

library(Ecdat)
data("Housing")

The dependent variable “airco” in the “Housing” dataset is binary. This calls for us to use a GLM. To do this we will use the “glm” function in R. Furthermore, in our example, we want to determine if there is an interaction between number of bedrooms and bathrooms. Interaction means that the two independent variables (bathrooms and bedrooms) influence on the dependent variable (aircon) is not additive, which means that the combined effect of the independnet variables is different than if you just added them together. Below is the code for the model followed by a summary of the results

model<-glm(Housing$airco ~ Housing$bedrooms * Housing$bathrms + Housing$stories + Housing$price, family=binomial)
summary(model)
## 
## Call:
## glm(formula = Housing$airco ~ Housing$bedrooms * Housing$bathrms + 
##     Housing$stories + Housing$price, family = binomial)
## 
## Deviance Residuals: 
##     Min       1Q   Median       3Q      Max  
## -2.7069  -0.7540  -0.5321   0.8073   2.4217  
## 
## Coefficients:
##                                    Estimate Std. Error z value Pr(>|z|)
## (Intercept)                      -6.441e+00  1.391e+00  -4.632 3.63e-06
## Housing$bedrooms                  8.041e-01  4.353e-01   1.847   0.0647
## Housing$bathrms                   1.753e+00  1.040e+00   1.685   0.0919
## Housing$stories                   3.209e-01  1.344e-01   2.388   0.0170
## Housing$price                     4.268e-05  5.567e-06   7.667 1.76e-14
## Housing$bedrooms:Housing$bathrms -6.585e-01  3.031e-01  -2.173   0.0298
##                                     
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## (Dispersion parameter for binomial family taken to be 1)
## 
##     Null deviance: 681.92  on 545  degrees of freedom
## Residual deviance: 549.75  on 540  degrees of freedom
## AIC: 561.75
## 
## Number of Fisher Scoring iterations: 4

To check how good are model is we need to check for overdispersion as well as compared this model to other potential models. Overdispersion is a measure to determine if there is too much variablity in the model. It is calcualted by dividing the residual deviance by the degrees of freedom. Below is the solution for this

549.75/540
## [1] 1.018056

Our answer is 1.01, which is pretty good because the cutoff point is 1, so we are really close.

Now we will make several models and we will compare the results of them

Model 2

#add recroom and garagepl
model2<-glm(Housing$airco ~ Housing$bedrooms * Housing$bathrms + Housing$stories + Housing$price + Housing$recroom + Housing$garagepl, family=binomial)
summary(model2)
## 
## Call:
## glm(formula = Housing$airco ~ Housing$bedrooms * Housing$bathrms + 
##     Housing$stories + Housing$price + Housing$recroom + Housing$garagepl, 
##     family = binomial)
## 
## Deviance Residuals: 
##     Min       1Q   Median       3Q      Max  
## -2.6733  -0.7522  -0.5287   0.8035   2.4239  
## 
## Coefficients:
##                                    Estimate Std. Error z value Pr(>|z|)
## (Intercept)                      -6.369e+00  1.401e+00  -4.545 5.51e-06
## Housing$bedrooms                  7.830e-01  4.391e-01   1.783   0.0745
## Housing$bathrms                   1.702e+00  1.047e+00   1.626   0.1039
## Housing$stories                   3.286e-01  1.378e-01   2.384   0.0171
## Housing$price                     4.204e-05  6.015e-06   6.989 2.77e-12
## Housing$recroomyes                1.229e-01  2.683e-01   0.458   0.6470
## Housing$garagepl                  2.555e-03  1.308e-01   0.020   0.9844
## Housing$bedrooms:Housing$bathrms -6.430e-01  3.054e-01  -2.106   0.0352
##                                     
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## (Dispersion parameter for binomial family taken to be 1)
## 
##     Null deviance: 681.92  on 545  degrees of freedom
## Residual deviance: 549.54  on 538  degrees of freedom
## AIC: 565.54
## 
## Number of Fisher Scoring iterations: 4
#overdispersion calculation
549.54/538
## [1] 1.02145

Model 3

model3<-glm(Housing$airco ~ Housing$bedrooms * Housing$bathrms + Housing$stories + Housing$price + Housing$recroom + Housing$fullbase + Housing$garagepl, family=binomial)
summary(model3)
## 
## Call:
## glm(formula = Housing$airco ~ Housing$bedrooms * Housing$bathrms + 
##     Housing$stories + Housing$price + Housing$recroom + Housing$fullbase + 
##     Housing$garagepl, family = binomial)
## 
## Deviance Residuals: 
##     Min       1Q   Median       3Q      Max  
## -2.6629  -0.7436  -0.5295   0.8056   2.4477  
## 
## Coefficients:
##                                    Estimate Std. Error z value Pr(>|z|)
## (Intercept)                      -6.424e+00  1.409e+00  -4.559 5.14e-06
## Housing$bedrooms                  8.131e-01  4.462e-01   1.822   0.0684
## Housing$bathrms                   1.764e+00  1.061e+00   1.662   0.0965
## Housing$stories                   3.083e-01  1.481e-01   2.082   0.0374
## Housing$price                     4.241e-05  6.106e-06   6.945 3.78e-12
## Housing$recroomyes                1.592e-01  2.860e-01   0.557   0.5778
## Housing$fullbaseyes              -9.523e-02  2.545e-01  -0.374   0.7083
## Housing$garagepl                 -1.394e-03  1.313e-01  -0.011   0.9915
## Housing$bedrooms:Housing$bathrms -6.611e-01  3.095e-01  -2.136   0.0327
##                                     
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## (Dispersion parameter for binomial family taken to be 1)
## 
##     Null deviance: 681.92  on 545  degrees of freedom
## Residual deviance: 549.40  on 537  degrees of freedom
## AIC: 567.4
## 
## Number of Fisher Scoring iterations: 4
#overdispersion calculation
549.4/537
## [1] 1.023091

Now we can assess the models by using the “anova” function with the “test” argument set to “Chi” for the chi-square test.

anova(model, model2, model3, test = "Chi")
## Analysis of Deviance Table
## 
## Model 1: Housing$airco ~ Housing$bedrooms * Housing$bathrms + Housing$stories + 
##     Housing$price
## Model 2: Housing$airco ~ Housing$bedrooms * Housing$bathrms + Housing$stories + 
##     Housing$price + Housing$recroom + Housing$garagepl
## Model 3: Housing$airco ~ Housing$bedrooms * Housing$bathrms + Housing$stories + 
##     Housing$price + Housing$recroom + Housing$fullbase + Housing$garagepl
##   Resid. Df Resid. Dev Df Deviance Pr(>Chi)
## 1       540     549.75                     
## 2       538     549.54  2  0.20917   0.9007
## 3       537     549.40  1  0.14064   0.7076

The results of the anova indicate that the models are all essentially the same as there is no statistical difference. The only criteria on which to select a model is the measure of overdispersion. The first model has the lowest rate of overdispersion and so is the best when using this criteria. Therefore, determining if a hous has air conditioning depends on examining number of bedrooms and bathrooms simultenously as well as the number of stories and the price of the house.

Conclusion

The post explained how to use and interpret GLM in R. GLM can be used primarilyy for fitting data to disrtibutions that are not normal.

Understanding Decision Trees

Decision trees are yet another method in machine learning that is used for classifying outcomes. Decision trees are very useful for, as you can guess, making decisions based on the characteristics of the data.

In this post we will discuss the following

  • Physical traits of decision trees
  • How decision trees work
  • Pros and cons of decision trees

Physical Traits of a Decision Tree

Decision trees consist of what is called a tree structure. The tree structure consist of a root node, decision nodes, branches and leaf nodes.

A root node is the initial decision made in the tree. This depends on which feature the algorithm selects first.

Following the root node the tree splits into various branches. Each branch leads to an additional decision node where the data is further subdivided. When you reach the bottom of a tree at the terminal node(s) these are also called leaf nodes.

How Decision Trees Work

Decision trees use a heuristic called recursive partitioning. What this does is it splits the overall data set into smaller and smaller subsets until each subset is as close to pure (having the same characteristics) as possible. This process is also know as divide and conquer.

The mathematics for deciding how to split the data is based on an equation called entropy, which measures the purity of a potential decision node. The lower the entropy score the more pure the decision node is. The entropy can range from 0 (most pure) to 1 (most impure).

One of the most popular algorithms for developing decision trees is the C5.0 algorithm. This algorithm in particular uses entropy to assess potential decision nodes.

Pros and Cons

The prose of decision trees include it versatile nature. Decision trees can deal with all types of data as well as missing data. Furthermore, this approach learns automatically and only uses the most important features. Lastly, a deep understanding of mathematics is not necessary to use this method in comparison to more complex models.

Some problems with decision trees is that the can easily overfit the data. This means that the tree does not generalize well to other datasets. In addition, a large complex tree can be hard to interpret, which may be yet another indication of overfitting.

Conclusion

Decision trees provide another vehicle that researchers can use to empower decision making. This model is most useful particular when a decision that was made needs to be explained and defended. For example, when rejecting a person’s loan application. Complex models made provide stronger mathematical reasons but would be difficult to explain to an irate customer.

Therefore, for complex calculation presented in an easy to follow format. Decision trees are one possibility.

Mixed Methods

Mix Methods research involves the combination of qualitative and quantitative approaches in addressing a research problem. Generally, qualitative and quantitative methods have separate philosophical positions when it comes to how to uncover insights in addressing research questions.

For many, mixed methods have their own philosophical position, which is pragmatism. Pragmatist believe that if it works it’s good. Therefore, if mixed methods leads to a solution it’s an appropriate method to use.

This post will try to explain some of the mixed method designs. Before explaining it is important to understand that there are several common ways to approach mixed methods

  • Qualitative and Quantitative are equal (Convergent Parallel Design)
  • Quantitative is more important than qualitative (explanatory design)
  • Qualitative is more important than quantitative

Convergent Parallel Design 

This design involves the simultaneous collecting of qualitative and quantitative data. The results are then compared to provide insights into the problem. The advantage of this design is the quantitative data provides for generalizability while the qualitative data provides information about the context of the study.

However, the challenge is in trying to merge the two types of data. Qualitative and quantitative methods answer slightly different questions about a problem. As such it can be difficult to pain a picture of the results that are comprehensible.

Explanatory Design

This design puts emphasis on the quantitative data with qualitative data playing a secondary role. Normally, the results found in the quantitative data are followed up on in the qualitative part.

For example, if you collect surveys about what students think about college and the results indicate negative opinions, you might conduct interview with students to understand way they are negative towards college. A likert survey will not explain why students are negative. Interviews will help to capture why students have a particular position.

The advantage of this approach is the clear organization of the data. Quantitative data is more important. The drawback is deciding what about the quantitative data to explore when conducting the qualitative data collection.

Exploratory Design 

This design is the opposite of explanatory. Now the qualitative data is more important than the quantitative. This design is used when you want to understand a phenomenon in order to measure it.

It is common when developing an instrument to interview people in focus groups to understand the phenomenon. For example, if I want to understand what cellphone addiction is I might ask students to share what they think about this in interviews. From there, I could develop a survey instrument to measure cell phone addiction.

The drawback to this approach is the time consumption. It takes a lot of work to conduct interviews, develop an instrument, and assess the instrument.

Conclusions

Mixed methods are not that new. However, they are still a somewhat unusual approach to research in many fields. Despite this, the approaches of mixed methods can be beneficial depending on the context.

Intro to Using Machine Learning

Machine learning is about using data to take action. This post will explain common steps that are taking when using machine learning in the analysis of data. In general, there are five steps when applying machine learning.

  1. Collecting data
  2. Preparing data
  3. Exploring data
  4. Training a model on the data
  5. Assessing the performance of the model
  6. Improving the model

We will go through each step briefly

Step One, Collecting Data

Data can come from almost anywhere. Data can come from a database in a structured format like mySQL. Data can also come unstructured such as tweets collected from twitter. However you get the data, you need to develop a way to clean and process it so that it is ready for analysis.

There are some distinct terms used in machine learning that some coming from traditional research my not be familiar.

  • example-An example is one set of data. In an excel spreadsheet, an example would be one row. In empirical social science research we would call an example a respondent or participant.
  • Unit of observation-This is how an example is measured. The units can be time, money, height, weight, etc.
  • feature-A feature is a characteristic of an example. In other forms of research we normally call a feature a variable. For example, ‘salary’ would be a feature in machine learning but a variable in traditional research.

Step Two, Preparing Data

This is actually the most difficult step in machine learning analysis. It can take up to 80% of the time. With data coming from multiple sources and in multiple formats it is a real challenge to get everything where it needs to be for an analysis.

Missing data needs to be addressed, duplicate records, and other issues are a part of this process. Once these challenges are dealt with it is time to explore the data.

Step Three, Explore the Data

Before analyzing the data, it is critical that the data is explored. This is often done visually in the form of plots and graphs but also with summary statistics. You are looking for some insights into the data and the characteristics of different features. You are also looking out for things that might be unusually such as outliers. There are also times when a variable needs to be transformed because there are issues with normality.

Step Four, Training a Model

After exploring the data, you should have an idea of what you want to know if you did not already know. Determining what you want to know helps you to decide which algorithm is most appropriate for developing a model.

To develop a model, we normally split the data into a training and testing set. This allows us to assess the model for its strengths and weaknesses.

Step Five, Assessing the Model

The strength of the model is deteremined. Every model has certain biases that limits its usefulness. How to assess a model depends on what type of model is developed and the purpose of the analysis. Whenever suitable, we want to try and improve the model.

Step Six, Improving the Model

Improve can happen in many ways. You might decide to normalize the variables in a different way. Or you may choose to add or remove features from the model. Perhaps you may switch to a different model.

Conclusion

Success in data analysis involves have a clear path for achieving your goals. The steps presented here provide one way of tackling machine learning.

Machine Learning: Getting Machines to Learn

One of the most interesting fields in quantitative research today is machine learning. Machine Learning serves the purpose of developing models through the use of algorithms in order to inform action.

Examples of machine learning are all around us. Machine learning is used to make recommendations about what you should purchase at most retail websites. Machine learning is also used to filter spam from your email inbox. Each of these two examples involves making prediction about previous behavior. This is what machines learning does. It learns what will happen based on what has happen. However, the question remains how does a machine actually learn.

The purpose of this post is to explain how machines actually “learn.” The process of learning involves three steps which are…

  1. Data input
  2. Abstraction
  3. Generalization

Data Input

Data input is simply having access to some for of data whether it be numbers, text, pictures or something else. The data is the factual information that is used to develop insights for future action.

The majority of a person’s time in conducting machine learning is involve in organize and cleaning data. This process is actually called data mugging by many. Once the data is ready for analysis, it is time for the abstraction process to begin.

Abstraction

Clean beautiful data still does not provide any useful information yet. Abstraction is the process of deriving meaning from the data. The raw data represent knowledge but as we already are aware the problem is how it is currently represented. Numbers and text mean little to us.

Abstraction takes all of the numbers and or text and develops a model. What a model does is summarize data and provides explanation about the data. For example, if you are doing a study for a a bank who wants to know who will default on loans. You might discover from the data that a person is more likely to default if they are a student with a credit card balance over $2,000. How this information is shared with the researcher can be in one of the following forms

  • equation
  • diagram
  • logic rules

This kind of information is hard to find manually and is normally found through the use of an algorithm. Abstraction involves the use of some sort of algorithm. An algorithm is a systemic step-by-step procedure for solving a problem. It is very technical to try and understand algorithms unless you have a graduate degree in statistics.   The point to remember is that algorithms are what the computer uses to learn and develop models.

Developing a model involves training. Training is achieved when the abstraction process is complete. The completion of this process depends on the criteria of what a “good” model is. This varies depending on the requirements of the model and the preference of the researcher.

Generalization

A machine has not actually learned anything until the model it developed is assessed for bias. Bias is a result of the educated guesses (heuristics) that an algorithm makes to develop a model that are systematically wrong.

For example, let’s say an algorithm learns to identify a man by the length of their hair. If a man has really long hair or if a woman has really short hair the algorithm will misclassify them because each person does not fit the educated guess the algorithm developed. The challenge is that these guesses work most of the time but they struggle with exceptions.

The main way of dealing with this is to develop a model on one set of data and test it on another set of data. This will inform the researcher as to what changes are needed for the model.

Another problem is noise. Noise is caused by measurement error data reporting issues trying to make your model deal with noise can lead to overfitting which means that your model only works for your data and cannot be applied to other data sets. This can also be addressed by testing your model on other data sets.

Conclusion

A machine learns through the use of an algorithm to explain relationships in a data set in a specific manner. This process involves the three steps of data input, abstraction and generalization. The results of a machine learning model is a model that can be use to make prediction about the future with a certain degree of accuracy.

Wilcoxon Signed Rank Test in R

The Wilcoxon Signed Rank Test is the non-parametric equivalent of the t-test. If you have questions whether or not your data is normally distributed the Wilcoxon Signed Rank Test can still indicate to you if there is a difference between the means of your sample.

Th Wilcoxon Test compares the medians of two samples instead of their means. The differences between the median and each individual value for each sample is calculated. Values that come to zero are removed. Any remaining values are ranked from lowest to highest. Lastly, the ranks are summed. If the rank sum is different between the two samples it indicates  statistical difference between samples.

We will now do an example using r. We want to see if there is a difference in enrollment between private and public universities. Below is the code

We will begin by loading the ISLR package. Then we will load the ‘College’ data and take a look at the variables in the “College” dataset by using the ‘str’ function.

library(ISLR)
data=College
str(College)
## 'data.frame':    777 obs. of  18 variables:
##  $ Private    : Factor w/ 2 levels "No","Yes": 2 2 2 2 2 2 2 2 2 2 ...
##  $ Apps       : num  1660 2186 1428 417 193 ...
##  $ Accept     : num  1232 1924 1097 349 146 ...
##  $ Enroll     : num  721 512 336 137 55 158 103 489 227 172 ...
##  $ Top10perc  : num  23 16 22 60 16 38 17 37 30 21 ...
##  $ Top25perc  : num  52 29 50 89 44 62 45 68 63 44 ...
##  $ F.Undergrad: num  2885 2683 1036 510 249 ...
##  $ P.Undergrad: num  537 1227 99 63 869 ...
##  $ Outstate   : num  7440 12280 11250 12960 7560 ...
##  $ Room.Board : num  3300 6450 3750 5450 4120 ...
##  $ Books      : num  450 750 400 450 800 500 500 450 300 660 ...
##  $ Personal   : num  2200 1500 1165 875 1500 ...
##  $ PhD        : num  70 29 53 92 76 67 90 89 79 40 ...
##  $ Terminal   : num  78 30 66 97 72 73 93 100 84 41 ...
##  $ S.F.Ratio  : num  18.1 12.2 12.9 7.7 11.9 9.4 11.5 13.7 11.3 11.5 ...
##  $ perc.alumni: num  12 16 30 37 2 11 26 37 23 15 ...
##  $ Expend     : num  7041 10527 8735 19016 10922 ...
##  $ Grad.Rate  : num  60 56 54 59 15 55 63 73 80 52 ...

We will now look at the Enroll variable and see if it is normally distributed

hist(College$Enroll)

download.png

This variable is highly skewed to the right. This may mean that it is not normally distributed. Therefore, we may not be able to use a regular t-test to compare private and public universities and the Wilcoxon Test is more appropriate. We will now use the Wilcoxon Test. Below are the results

wilcox.test(College$Enroll~College$Private)
## 
##  Wilcoxon rank sum test with continuity correction
## 
## data:  College$Enroll by College$Private
## W = 104090, p-value < 2.2e-16
## alternative hypothesis: true location shift is not equal to 0

The results indicate a difference we will now calculate the medians of the two groups using the ‘aggregate’ function. This function allows us to compare our two groups based on the median. Below is the code with the results.

aggregate(College$Enroll~College$Private, FUN=median)
##   College$Private College$Enroll
## 1              No       1337.5
## 2             Yes        328.0

As you can see, there is a large difference in enrollment in private and public colleges. We can then make the conclusion that there is a difference in the medians of private and public colleges with public colleges have a much higher enrollment.

Conclusion

The Wilcoxon Test is used for a non-parametric analysis of data. This test is useful whenever there are concerns with the normality of the data.

Big Data & Data Mining

Dealing with large amounts of data has been a problem throughout most of human history. Ancient civilizations had to keep large amounts of clay tablets, papyrus, steles, parchments, scrolls etc. to keep track of all the details of an empire.

However, whenever it seemed as though there would be no way to hold any more information a new technology would be developed to alleviate the problem. When people could not handle keeping records on stone paper scrolls were invented. When scrolls were no longer practical books were developed. When hand-copying books was too much the printing press came along.

By the mid 20th century there were concerns that we would not be able to have libraries large enough to keep all of the books that were being developed. With this problem came the solution of the computer. One computer could hold the information of several dozen if not hundreds of libraries.

Now even a single computer can no longer cope with all of the information that is constantly being developed for just a single subject. This has lead to computers working together in networks to share the task of storing information. With data spread across several computers it makes analyzing data much more challenging. It was now necessary to mine for useful information in a way that people used to mine for gold in the 19th century.

Big data is data that is too large to fit within the memory of a single computer. Analyzing data that is spread across a network of databases takes skills different from traditional statistical analysis. This post will explain some of the characteristics of big data  as well as data mining.

Big Data Traits

The three main traits of big data are volume, velocity, and variety. Volume describes the size of big data, which means data to big to be on only one computer. Velocity is about how fast the data can be processed. Lastly, variety different types of data. common sources of big data includes the following

  • Metadata from visual sources such as cameras
  • Data from sensors such as in medical equipment
  • Social media data such as information from google, youtube or facebook

Data Mining

Data mining is the process of discovering a model in a big dataset. Through the development of an algorithm, we can find specific information that helps us to answer our questions. Generally, there are two ways to mine data and these are extraction and summarization.

Extraction is the process of pulling specific information from a big dataset. For example, if we want all the addresses of people who bought a specific book from Amazon the result would be an extraction from a big data set.

Summarization is reducing a dataset to describe it. We might do a cluster analysis in which similar data is combine on a characteristic. For example, if we analyze all the books people ordered through Amazon last year we might notice that one cluster of groups buys mostly religious books while another buys investment books.

Conclusion

Big data will only continue to get bigger. Currently, the response has been to just use more computers and servers. As such, there is now a need for finding information across many computers and servers. This is the purpose of data mining, which is to find pertinent information that answers stakeholders questions.

Type I and Type II Error

Hypothesis testing in statistics involves deciding whether to reject or not reject a null hypothesis. There are problems that can occur when making decisions about a null hypothesis. A researcher can reject a null when they should not reject it, which is called a type I error. The other mistake is not rejecting a null when they should have, which is a type II error. Both of these mistakes represent can seriously damage the interpretation of data.

An Example

The classic example that explains type I and type II errors is a courtroom. In a trial, the defendant is considered innocent until proven guilty. The defendant can be compared to the null hypothesis being true. The prosecutor job is to present evidence that the defendant is guilty. This is the same as providing statistical evidence to reject the null hypothesis which indicates that the null is not true and needs to be rejected.

There are four possible outcomes of our trial and our statistical test…

  1. The defendant can be declared guilty when they are really guilty. That’s a correct decision.This is the same as rejecting the null hypothesis.
  2. The defendant could be judged not guilty when they really are innocent. That’s a correct and is the same as not rejecting the null hypothesis.
  3. The defendant is convicted when they are actually innocent, which is wrong. This is the same as rejecting the null hypothesis when you should not and is know as a type I error
  4. The defendant is guilty but declared innocent, which is also incorrect. This is the same as not rejecting the null hypothesis when you should have. This is known as a type II error.

Important Notes

The probability of committing a type I error is the same as the alpha or significance level of a statistical test. Common values associated with alpha are o.1, 0.05, and 0.01. This means that the likelihood of committing a type I error depends on the level of the significance that the researcher picks.

The probability of committing a type II error is known as beta. Calculating beta is complicated as you need specific values in your null and alternative hypothesis. It is not always possible to supply this. As such, researcher often do not focus on type II error avoidance as they do with type I.

Another concern is that decrease the risk of committing one type of error increases the risk of committing the other. This means that if you reduce the risk of type I error you increase the risk of committing a type II error.

Conclusion

The risk of error or incorrect judgment of a null hypothesis is a risk in statistical analysis. As such, researchers need to be aware of these problems as they study data.

Decision Trees in R

Decision trees are useful for splitting data based into smaller distinct groups based on criteria you establish. This post will attempt to explain how to develop decision trees in R.

We are going to use the ‘College’ dataset found in the “ISLR” package. Once you load the package you need to split the data into a training and testing set as shown in the code below. We want to divide the data based on education level, age, and income

library(ISLR); library(ggplot2); library(caret)
data("College")
inTrain<-createDataPartition(y=College$education, 
 p=0.7, list=FALSE)
trainingset <- College[inTrain, ]
testingset <- College[-inTrain, ]

Visualize the Data

We will now make a plot of the data based on education as the groups and age and wage as the x and y variable. Below is the code followed by the plot. Please note that education is divided into 5 groups as indicated in the chart.

qplot(age, wage, colour=education, data=trainingset)
Rplot10
Create the Model

We are now going to develop the model for the decision tree. We will use age and wage to predict education as shown in the code below.

TreeModel<-train(education~age+income, method='rpart', data=trainingset)

Create Visual of the Model

We now need to create a visual of the model. This involves installing the package called ‘rattle’. You can install ‘rattle’ separately yourself. After doing this below is the code for the tree model followed by the diagram.

fancyRpartPlot(TreeModel$finalModel)

Rplot02

Here is what the chart means

  1. At the top is node 1 which is called ‘HS Grad” the decimals underneath is the percentage of the data that falls within the “HS Grad” category. As the highest node, everything is classified as “HS grad” until we begin to apply our criteria.
  2. Underneath nod 1 is a decision about wage. If a person makes less than 112 you go to the left if they make more you go to the right.
  3. Nod 2 indicates the percentage of the sample that was classified as HS grade regardless of education. 14% of those with less than a HS diploma were classified as a HS Grade based on wage. 43% of those with a HS diploma were classified as a HS grade based on income. The percentage underneath the decimals indicates the total amount of the sample placed in the HS grad category. Which was 57%.
  4. This process is repeated for each node until the data is divided as much as possible.

Predict

You can predict individual values in the dataset by using the ‘predict’ function with the test data as shown in the code below.

predict(TreeModel, newdata = testingset)

Conclusion

Prediction Trees are a unique feature in data analysis for determining how well data can be divided into subsets. It also provides a visual of how to move through data sequentially based on characteristics in the data.

Z-Scores

A z-score indicates how closely related one given score is to mean of the sample. Extremely high or low z-scores indicates that the given data point is unusually above or below the mean of the sample.

In order to understand z-scores you need to be familiar with distribution. In general, data is distributed in a bell shape curve. With the mean being the exact middle of the graph as shown in the picture below.

download

The Greek letter μ is the mean. In this post, we will go through an example that will try to demonstrate how to use and interpret the z-score. Notice that a z-score + 1 takes of 68% of the potential values a z-score + 2 takes of 95%, a z-score + 3 takes of 99%.

Imagine you know the average test score of students on a quiz. The average is 75%. with a standard deviation of 6.4%. Below is the equation for calculating the z-score.

Standard_Score_Calc

Let’s say that one student scored 52% on the quiz. We can calculate the likelihood for this data point by using the formula above.

(52 – 75) / 6.4 = -3.59

Our value is negative which indicates that the score is below the mean of the sample. Our score is very exceptionally low from the mean. This makes sense given that the mean is 75% and the standard deviation is 6.4%. To get a 52% on the quiz was really bad performance.

We can convert the z-score to a percentage to indicate the probability of getting such a value. To do this you would need to find a z-score conversion table on the internet. A quick glance at the table will show you that the probability of getting a score of 52 on the quiz is less than 1%.

Off course, this is based on the average score of 75% with a standard deviation of 6.4%. A different average and standard deviation would change the probability of getting a 52%.

Standardization 

Z-scores are also used to standardize a variable. If you look at our example, the original values were in percentages. By using the z-score formula we converted these numbers into a different value. Specifically, the values of a z-score represent standard deviations from the mean.

In our example, we calculated a z-score of -3.59. In other words, the person who scored 52% on the quiz had a score 3.59 standard deviations below the mean. When attempting to interpret data the z-score is a foundational piece of information that is used extensively in statistics.

Multiple Regression Prediction in R

In this post, we will learn how to predict using multiple regression in R. In a previous post, we learn how to predict with simple regression. This post will be a large repeat of this other post with the addition of using more than one predictor variable. We will use the “College” dataset and we will try to predict Graduation rate with the following variables

  • Student to faculty ratio
  • Percentage of faculty with PhD
  • Expenditures per student

Preparing the Data

First we need to load several packages and divide the dataset int training and testing sets. This is not new for this blog. Below is the code for this.

library(ISLR); library(ggplot2); library(caret)
data("College")
inTrain<-createDataPartition(y=College$Grad.Rate, 
 p=0.7, list=FALSE)
trainingset <- College[inTrain, ]
testingset <- College[-inTrain, ]
dim(trainingset); dim(testingset)

Visualizing the Data

We now need to get a visual idea of the data. Since we are using several variables the code for this is slightly different so we can look at several charts at the same time. Below is the code followed by the plots

> featurePlot(x=trainingset[,c("S.F.Ratio","PhD","Expend")],y=trainingset$Grad.Rate, plot="pairs")
Rplot10

To make these plots we did the following

  1. We used the ‘featureplot’ function told R to use the ‘trainingset’ data set and subsetted the data to use the three independent variables.
  2. Next, we told R what the y= variable was and told R to plot the data in pairs

Developing the Model

We will now develop the model. Below is the code for creating the model. How to interpret this information is in another post.

> TrainingModel <-lm(Grad.Rate ~ S.F.Ratio+PhD+Expend, data=trainingset) > summary(TrainingModel)

As you look at the summary, you can see that all of our variables are significant and that the current model explains 18% of the variance of graduation rate.

Visualizing the Multiple Regression Model

We cannot use a regular plot because are model involves more than two dimensions.  To get around this problem to see are modeling, we will graph fitted values against the residual values. Fitted values are the predict values while residual values are the acutal values from the data. Below is the code followed by the plot.

> CheckModel<-train(Grad.Rate~S.F.Ratio+PhD+Expend, method="lm", data=trainingset)
> DoubleCheckModel<-CheckModel$finalModel
> plot(DoubleCheckModel, 1, pch=19, cex=0.5)
Rplot01

Here is what happened

  1. We created the variable ‘CheckModel’.  In this variable, we used the ‘train’ function to create a linear model with all of our variables
  2. We then created the variable ‘DoubleCheckModel’ which includes the information from ‘CheckModel’ plus the new column of ‘finalModel’
  3. Lastly, we plot ‘DoubleCheckModel’

The regression line was automatically added for us. As you can see, the model does not predict much but shows some linearity.

Predict with Model

We will now do one prediction. We want to know the graduation rate when we have the following information

  • Student-to-faculty ratio = 33
  • Phd percent = 76
  • Expenditures per Student = 11000

Here is the code with the answer

> newdata<-data.frame(S.F.Ratio=33, PhD=76, Expend=11000)
> predict(TrainingModel, newdata)
       1 
57.04367

To put it simply, if the student-to-faculty ratio is 33, the percentage of PhD faculty is 76%, and the expenditures per student is 11,000, we can expect 57% of the students to graduate.

Testing

We will now test our model with the testing dataset. We will calculate the RMSE. Below is the code for creating the testing model followed by the codes for calculating each RMSE.

> TestingModel<-lm(Grad.Rate~S.F.Ratio+PhD+Expend, data=testingset)
> sqrt(sum((TrainingModel$fitted-trainingset$Grad.Rate)^2))
[1] 369.4451
> sqrt(sum((TestingModel$fitted-testingset$Grad.Rate)^2))
[1] 219.4796

Here is what happened

  1. We created the ‘TestingModel’ by using the same model as before but using the ‘testingset’ instead of the ‘trainingset’.
  2. The next two lines of codes should look familiar.
  3. From this output the performance of the model improvement on the testing set since the RMSE is lower than compared to the training results.

Conclusion

This post attempted to explain how to predict and assess models with multiple variables. Although complex for some, prediction is a valuable statistical tool in many situations.

Using Regression for Prediction in R

In the last post about R, we looked at plotting information to make predictions. We will now look at an example of making predictions using regression.

We will used the same data as last time with the help of the ‘caret’ package as well. The code below sets up the seed and the training and testing sets we need.

> library(caret); library(ISLR); library(ggplot2)
> data("College");set.seed(1)
> PracticeSet<-createDataPartition(y=College$Grad.Rate, 
+                                  p=0.5, list=FALSE)
> TrainingSet<-College[PracticeSet, ]; TestingSet<-
+         College[-PracticeSet, ]
> head(TrainingSet)

The code above should look familiar from previous post.

Make the Scatterplot

We will now create the scatterplot showing the relationship between “S.F. Ratio” and “Grad.Rate” with the code below and the scatterplot.

> plot(TrainingSet$S.F.Ratio, TrainingSet$Grad.Rate, pch=5, col="green", 
xlab="Student Faculty Ratio", ylab="Graduation Rate")

Rplot10

Here is what we did

  1. We used the ‘plot’ function to make this scatterplot. The x variable was ‘S.F.Ratio’ of the ‘TrainingSet’ the y variable was ‘Grad.Rate’.
  2. We picked the type of dot to use using the ‘pch’ argument and choosing ’19’
  3. Next we chose a color and labeled each axis

Fitting the Model

We will now develop the linear model. This model will help us to predict future models. Furthermore, we will compare the model of the Training Set with the Test Set. Below is the code for developing the model.

> TrainingModel<-lm(Grad.Rate~S.F.Ratio, data=TrainingSet)
> summary(TrainingModel)

How to interpret this information was presented in a previous post. However, to summarize, we can say that when the student to faculty ratio increases one the graduation rate decreases 1.29. In other words, an increase in the student to faculty ratio leads to decrease in the graduation rate.

Adding the Regression Line to the Plot

Below is the code for adding the regression line followed by the scatterplot

> plot(TrainingSet$S.F.Ratio, TrainingSet$Grad.Rate, pch=19, col="green", xlab="Student Faculty Ratio", ylab="Graduation Rate")
> lines(TrainingSet$S.F.Ratio, TrainingModel$fitted, lwd=3)

Rplot01

Predicting New Values

With are model complete we can now predict values. For our example, we will only predict one value. We want to know what the graduation rate would be if we have a student to faculty ratio of 33. Below is the code for this with the answer

> newdata<-data.frame(S.F.Ratio=33)
> predict(TrainingModel, newdata)
      1 
40.6811

Here is what we did

  1. We made a variable called ‘newdata’ and stored a data frame in it with a variable called ‘S.F.Ratio’ with a value of 33. This is x value
  2. Next, we used the ‘predict’ function from the ‘caret’ package to determine what the graduation rate would be if the student to faculty ratio is 33. To do this we told caret to use the ‘TrainingModel’ we developed using regression and to run this model with the information in the ‘newdata’ dataframe
  3. The answer was 40.68. This means that if the student to faculty ratio is 33 at a university then the graduation rate would be about 41%.

Testing the Model

We will now test the model we made with the training set with the testing set. First, we will make a visual of both models by using the “plot” function. Below is the code follow by the plots.

par(mfrow=c(1,2))
plot(TrainingSet$S.F.Ratio,
TrainingSet$Grad.Rate, pch=19, col=’green’,  xlab=”Student Faculty Ratio”, ylab=’Graduation Rate’)
lines(TrainingSet$S.F.Ratio,  predict(TrainingModel), lwd=3)
plot(TestingSet$S.F.Ratio,  TestingSet$Grad.Rate, pch=19, col=’purple’,
xlab=”Student Faculty Ratio”, ylab=’Graduation Rate’)
lines(TestingSet$S.F.Ratio,  predict(TrainingModel, newdata = TestingSet),lwd=3)

Rplot02.jpeg

In the code, all that is new is the “par” function which allows us to see to plots at the same time. We also used the ‘predict’ function to set the plots. As you can see, the two plots are somewhat differ based on a visual inspection. To determine how much so, we need to calculate the error. This is done through computing the root mean square error as shown below.

> sqrt(sum((TrainingModel$fitted-TrainingSet$Grad.Rate)^2))
[1] 328.9992
> sqrt(sum((predict(TrainingModel, newdata=TestingSet)-TestingSet$Grad.Rate)^2))
[1] 315.0409

The main take away from this complicated calculation is the number 328.9992 and 315.0409. These numbers tell you the amount of error in the training model and testing model. The lower the number the better the model. Since the error number in the testing set is lower than the training set we know that our model actually improves when using the testing set. This means that our model is beneficial in assessing graduation rates. If there were problems we may consider using other variables in the model.

Conclusion

This post shared ways to develop a regression model for the purpose of prediction and for model testing.

Using Plots for Prediction in R

It is common in machine learning to look at the training set of your data visually. This helps you to decide what to do as you begin to build your model.  In this post, we will make several different visual representations of data using datasets available in several R packages.

We are going to explore data in the “College” dataset in the “ISLR” package. If you have not done so already, you need to download the “ISLR” package along with “ggplot2” and the “caret” package.

Once these packages are installed in R you want to look at a summary of the variables use the summary function as shown below.

summary(College)

You should get a printout of information about 18 different variables. Based on this printout, we want to explore the relationship between graduation rate “Grad.Rate” and student to faculty ratio “S.F.Ratio”. This is the objective of this post.

Next we need to create a training and testing data set below is the code to do this.

> library(ISLR);library(ggplot2);library(caret)
> data("College")
> PracticeSet<-createDataPartition(y=College$Enroll, p=0.7,
+                                  list=FALSE)
> trainingSet<-College[PracticeSet,]
> testSet<-College[-PracticeSet,]
> dim(trainingSet); dim(testSet)
[1] 545  18
[1] 232  18

The explanation behind this code was covered in predicting with caret so we will not explain it again. You just need to know that the dataset you will use for the rest of this post is called “trainingSet”.

Developing a Plot

We now want to explore the relationship between graduation rates and student to faculty ratio. We will be used the ‘ggpolt2’  package to do this. Below is the code for this followed by the plot.

qplot(S.F.Ratio, Grad.Rate, data=trainingSet)
Rplot10
As you can see, there appears to be a negative relationship between student faculty ratio and grad rate. In other words, as the ration of student to faculty increases there is a decrease in the graduation rate.

Next, we will color the plots on the graph based on whether they are a public or private university to get a better understanding of the data. Below is the code for this followed by the plot.

> qplot(S.F.Ratio, Grad.Rate, colour = Private, data=trainingSet)
Rplot.jpeg
It appears that private colleges usually have lower student to faculty ratios and also higher graduation rates than public colleges

Add Regression Line

We will now plot the same data but will add a regression line. This will provide us with a visual of the slope. Below is the code followed by the plot.

> collegeplot<-qplot(S.F.Ratio, Grad.Rate, colour = Private, data=trainingSet) > collegeplot+geom_smooth(method = ‘lm’,formula=y~x)
Rplot01.jpeg
Most of this code should be familiar to you. We saved the plot as the variable ‘collegeplot’. In the second line of code we add specific coding for ‘ggplot2’ to add the regression line. ‘lm’ means linear model and formula is for creating the regression.

Cutting the Data

We will now divide the data based on the student-faculty ratio into three equal size groups to look for additional trends. To do this you need the “Hmisc” packaged. Below is the code followed by the table

> library(Hmisc)
> divide_College<-cut2(trainingSet$S.F.Ratio, g=3)
> table(divide_College)
divide_College
[ 2.9,12.3) [12.3,15.2) [15.2,39.8] 
        185         179         181

Our data is now divided into three equal sizes.

Box Plots

Lastly, we will make a box plot with our three equal size groups based on student-faculty ratio. Below is the code followed by the box plot

CollegeBP<-qplot(divide_College, Grad.Rate, data=trainingSet, fill=divide_College, geom=c(“boxplot”)) > CollegeBP
Rplot02
As you can see, the negative relationship continues even when student-faculty is divided into three equally size groups. However, our information about private and public college is missing. To fix this we need to make a table as shown in the code below.

> CollegeTable<-table(divide_College, trainingSet$Private)
> CollegeTable
              
divide_College  No Yes
   [ 2.9,12.3)  14 171
   [12.3,15.2)  27 152
   [15.2,39.8] 106  75

This table tells you how many public and private colleges there based on the division of the student faculty ratio into three groups. We can also get proportions by using the following

> prop.table(CollegeTable, 1)
              
divide_College         No        Yes
   [ 2.9,12.3) 0.07567568 0.92432432
   [12.3,15.2) 0.15083799 0.84916201
   [15.2,39.8] 0.58563536 0.41436464

In this post, we found that there is a negative relationship between student-faculty ratio and graduation rate. We also found that private colleges have lower student-faculty ratio and a higher graduation rate than public colleges. In other words, the status of a university as public or private moderates the relationship between student-faculty ratio and graduation rate.

You can probably tell by now that R can be a lot of fun with some basic knowledge of coding.

Predicting with Caret

In this post, we will explore the use of the caret package for developing algorithms for use in machine learning. The caret package is particularly useful for processing data before the actual analysis of the algorithm.

When developing algorithms is common practice to divide the data into a training a testing sub samples. The training sub sample is what is used to develop the algorithm while the testing sample is used to assess the predictive power of the algorithm. There are many different ways to divide a sample in a testing and training set and one of the main benefits of the caret package is in dividing the sample.

In the example we will use, we will return to the kearnlab example and this develop an algorithm after sub-setting the sample to have a training data set and a testing data set.

First, you need to download the ‘caret’ and ‘kearnlab’ package if you have not done so. After that, below is the code for subsetting the ‘spam’ data from the ‘kearnlab’ package.

inTrain<- createDataPartition(y=spam$type, p=0.75, 
list=FALSE)
training<-spam[inTrain,]
testing<-spam[-inTrain,] 
dim(training)

Here is what we did

  1. We created the variable ‘inTrain’
  2. In the variable ‘inTrain’ we told R to make a partition in the data use the ‘createDataPartition’ function. I the parenthesis we told r to look at the dataset ‘spam’ and to examine the variable ‘type’. Then we told are to pull 75% of the data in ‘type’ and copy it to the ‘inTrain’ variable we created. List = False tells R not to make a list. If you look closely, you will see that the variable ‘type’ is being set as the y variable in the ‘inTrain’ data set. This means that all the other variables in the data set will be used as predictors. Also remember that the ‘type’ variable has two outcomes “spam” or “nonspam”
  3. Next, we created the variable ‘training’ which is the dataset we will use for developing our algorithm. To make this we take the original ‘spam’ data and subset the ‘inTrain’ partition. Now all the data that is in the ‘inTrain’ partition is now in the ‘training’ variable.
  4. Finally, we create the ‘testing’ variable which will be used for testing the algorithm. To make this variable, we tell R to take everything that was not assigned to the ‘inTrain’ variable and put it into the ‘testing’ variable. This is done through the use of a negative sign
  5. The ‘dim’ function just tells us how many rows and columns we have as shown below.
[1] 3451   58

As you can see, we have 3451 rows and 58 columns. Rows are for different observations and columns are for the variables in the data set.

Now to make the model. We are going to bootstrap are sample. Bootstrapping involves random sampling from the sample with replacement in order to assess the stability of the results. Below is the code for the bootstrap and model development followed by explanation.

set.seed(32343)
SpamModel<-train(type ~., data=training, method="glm")
SpamModel

Here is what we did,

  1. Whenever you bootstrap, it is wise to set the seed. This allows you to reproduce the same results each time. For us, we set the seed to 32343
  2. Next, we developed the actual model. We gave the model the name “SpamModel” we used the ‘train’ function. Inside the parenthesis we tell r to set “type” as the y variable and then use ~. which is a short hand for using all other variables in the model as predictor variables. Then we set the data to the ‘training’ data set and indicate that the method is ‘glm’ which means generalized linear model.
  3. The output for the analysis is available at the link SpamModel

There is a lot of information but the most important information for us is the accuracy of the model which is 91.3%. The kappa stat tells us what the expected accuracy of the model is which is 81.7%. This means that our model is a little bit better than the expected accuracy.

For our final trick, we will develop a confusion matrix to assess the accuracy of our model using the ‘testing’ sample we made earlier. Below is the code

SpamPredict<-predict(SpamModel, newdata=testing)
confusionMatrix(SpamPredict, testing$type)

Here is what we did,

  1. We made a variable called ‘SpamPredict’. We use the function ‘predict’ using the ‘SpamModel’ with the new data called ‘testing’.
  2. Next, we make matrix using the ‘confusionMatrix’ function using the new model ‘SpamPredict’ based on the ‘testing’ data on the ‘type’ variable. Below is the output

Reference

  1. Prediction nonspam spam
       nonspam     657   35
       spam         40  418
                                              
                   Accuracy : 0.9348          
                     95% CI : (0.9189, 0.9484)
        No Information Rate : 0.6061          
        P-Value [Acc > NIR] : <2e-16          
                                              
                      Kappa : 0.8637          
     Mcnemar's Test P-Value : 0.6442          
                                              
                Sensitivity : 0.9426          
                Specificity : 0.9227          
             Pos Pred Value : 0.9494          
             Neg Pred Value : 0.9127          
                 Prevalence : 0.6061          
             Detection Rate : 0.5713          
       Detection Prevalence : 0.6017          
          Balanced Accuracy : 0.9327          
                                              
           'Positive' Class : nonspam

    The accuracy of the model actually improve to 93% on the test data. The other values such as sensitivity and specificity have to do with such things as looking at correct classifications divided by false negatives and other technical matters. As you can see, machine learning is a somewhat complex experience

 

Simple Prediction

Prediction is one of the key concepts of machine learning. Machine learning is a field of study that is focused on the development of algorithms that can be used to make predictions.

Anyone who has shopped online at has experienced machine learning. When you make a purchase at an online store, the website will recommend additional purchases for you to make. Often these recommendations are based on whatever you have purchased or whatever you click on while at the site.

There are two common forms of machine learning, unsupervised and supervised learning. Unsupervised learning involves using data that is not cleaned and labeled and attempts are made to find patterns within the data. Since the data is not labeled, there is no indication into what is right or wrong

Supervised machine learning is using cleaned and properly labeled data. Since the data is labeled there is some form of indication whether the model that is developed is accurate or not. If the is incorrect then you need to make adjustments to it. In other words, the model learns based on its ability to accurately predict results. However, it is up to the human to make adjustments to the model in order to improve the accuracy of it.

In this post we will look at using R for supervised machine learning. The definition presented so far will make more sense with an example.

The Example

We are going to make a simple prediction about whether emails are spam or not using data from kern lab.

The first thing that you need to do is to install and load the “kernlab” package using the following code

install.packages("kernlab")
library(kernlab)

If you use the “View” function to examine the data you will see that there are several columns. Each column tells you the frequency of a word that kernlab found in a collection of emails. We are going to use the word/variable “money” to predict whether an email is spam or not. First we need to plot the density of the use of the word “money” when the email was not coded as spam. Below is the code for this.

plot(density(spam$money[spam$type=="nonspam"]),
 col='blue',main="", xlab="Frequency of 'money'")

This is an advance R post so I am assuming you can read the code. The plot should look like the following.

Rplot10

As you can see, money is not used to frequently in emails  that are not spam in this dataset. However, you really cannot say this unless you compare the times ‘money’ is labeled nonspam to the times that it is labeled spam. To learn this we need to add a second line that explains to us when the word ‘money’ is used and classified as spam. The code for this is below with the prior code included.

plot(density(spam$money[spam$type=="nonspam"]),
 col='blue',main="", xlab="Frequency of 'money'")
lines(density(spam$money[spam$type=="spam"]), col="red")

Your new plot should look like the following

Rplot10

If you look closely at the plot doing a visual inspection, where there is a separation between the blue line for non spam and the red line for spam is the cutoff point for whether an email is spam or not. In other words, everything inside the arc is labeled correctly while the information outside the arc is not.

The next code and graph show that this cutoff point is around 0.1. This means that any email that has on average more than 0.1 frequency of the word ‘money’ is spam. Below is the code and the graph with the cutoff point indicated with a black line.

plot(density(spam$money[spam$type=="nonspam"]),
 col='blue',main="", xlab="Frequency of 'money'")
lines(density(spam$money[spam$type=="spam"]), col="red")
abline(v=0.1, col="black", lw= 3)

Rplot10.jpeg Now we need to calculate the accuracy of the use of the word ‘money’ to predict spam. For our current example we will simply use in “ifelse” function. If the frequency is greater than 0.1.

We then need to make a table to see the results. The code for the “ifelse” function and the table are below followed by the table.

predict<-ifelse(spam$money > 0.1, "spam","nonspam")
table(predict, spam$type)/length(spam$type)
predict       nonspam        spam
  nonspam 0.596392089 0.266898500
  spam    0.009563138 0.127146273

Based on the table that I am assuming you can read, our model accurately calculates that an email is spam about 71% (0.59 + 0.12) of the time based on the frequency of the word ‘money’ being greater than 0.1.

Off course, for this to be true machine learning we would repeat this process by trying to improve the accuracy of the prediction. However, this is an adequate introduction to this topic.

Survey Design

Survey design is used to describe the opinions, beliefs, behaviors, and or characteristics of a population based on the results of a sample. This design involves the use of surveys that include questions, statements, and or other ways of soliciting information from the sample. This design is used for descriptive purpose primarily but can be combined with other designs (correlational, experimental) at times as well. In this post we will look at the following.

  • Types of Survey Design
  • Characteristics of Survey Design

Types of Survey Design

There are two common forms of survey design which are cross-sectional and longitudinal.   A cross-sectional survey design is the collection of data at one specific point in time. Data is only collected once in a cross-sectional design.

A cross-sectional design can be used to measure opinions/beliefs, compare two or more groups, evaluate a program, and or measure the needs of a specific group. The main goal is to analyze the data from a sample at a given moment in time.

A longitudinal design is similar to a cross-sectional design with the difference being that longitudinal designs require collection over time.Longitudinal studies involve cohorts and panels in which data is collected over days, months, years, and even decades. Through doing this, a longitudinal study is able to expose trends over time in a sample.

Characteristics of Survey Design

There are certain traits that are associated with survey design. Questionnaires and interviews are a common component of survey design. The questionnaires can happen by mail, phone, internet, and in person. Interviews can happen by phone, in focus groups, or one-on-one.

The design of a survey instrument often includes personal, behavioral and attitudinal questions, and open/closed questions.

Another important characteristic of survey design is monitoring the response rate. The response rate is the percentage of participants in the study compared to the number of surveys that were distributed. The response rate varies depending on how the data was collected. Normally, personal interviews have the highest rate while email request have the lowest.

It is sometimes necessary to report the response rate when trying to publish. As such, you should at the very least be aware of what the rate is for a study you are conducting.

Conclusion

Surveys are used to collected data at one point in time or over time. The purpose of this approach is develop insights into the population in order to describe what is happening or to be used to make decisions and inform practice.

 

Correlational Designs

Correlational research is focused on examining the relationships among two or more variables. This information can be used either to explain a phenomenon or to make predictions. This post will explain the two forms of correlational design as well as the characteristics of correlational design in general.

Explanatory Design

An explanatory design seeks to determine to what extent two or more variables co-vary. Co-vary simply means the strength of the relationship of one variable to another. In general, two or more variables can have a strong, weak, or no relationship. This is determined by the product moment correlation coefficient, which is usually referred to as r. The r is measured on a scale of -1 to 1. The higher the absolute value the stronger the relationship.

For example, let’s say we do a study to determine the strength of the relationship between exercise and health. Exercise is the explanatory variable and health is the response variable. This means that we are hoping the exercise will explain health or you can say we are hoping that health responds to exercise. In this example, let’s say that there is a strong relationship between exercise and health with an r of 0.7. This literally means that when exercise goes up one unit, that health improves by 0.7 units or that the more exercise a person gets the healthier they are. In other words, when one increases the other increase as well.

Exercise is able to explain a certain amount of the variance (amount of change) in health. This is done by squaring the r to get the r-squared. The higher the r-squared to more appropriate the model is in explaining the relationship between the explanatory and response variable. This is where regression comes from.

This also holds true for a negative relationship but in negative relationships when the explanatory variables increase the response variable decreases. For example, let’s say we do a study that examines the relationship between exercise and age and we calculate an r of -0.85. This means that when exercise increases one unit age decreases 0.85. In other words, more exercises mean that the person is probably younger. In this example, the relationship is strong but indicates that the variables move in opposite directions.

Prediction Design

Prediction design has most of the same functions as explanatory design with a few minor changes. In prediction design, we normally do not use the term explanatory and response variable. Rather we have predictor and outcome variable as terms. This is because we are trying to predict and not explain. In research, there are many terms for independent and dependent variable and this is because different designs often use different terms.

Another difference is the prediction designs are focused on determining future results or forecasting. For example, if we are using exercise to predict age we can develop an equation that allows us to determine a person’s age based on how much they exercise or vice versa. Off course, no model is 100% accurate but a good model can be useful even if it is wrong at times.

What both designs have in common is the use of r and r square and the analysis of the strength of the relationship among the variables.

Conclusion

In research, explanatory and prediction correlational designs have a place in understanding data. Which to use depends on the goals of the research and or the research questions. Both designs have

Simple Regression in R

In this post, we will look at how to perform a simple regression using R. We will use a built-in dataset in R called ‘mtcars.’ There are several variables in this dataset but we will only use ‘wt’ which stands for the weight of the cars and ‘mpg’ which stands for miles per gallon.

Developing the Model

We want to know the association or relationship between weight of a car and miles per gallon. We want to see how much of the variance of ‘mpg’ is explained by ‘wt’. Below is the code for this.

> Carmodel <- lm(mpg ~ wt, data = mtcars)

 

Here is what we did

  1. We created the variable ‘Carmodel’ to store are information
  2. Inside this variable, we used the ‘lm’ function to tell R to make a linear model.
  3. Inside the function we put ‘mpg ~ wt’ the ‘~’ sign means ’tilda’ in English and is used to indicate that ‘mpg’ is a function of ‘wt’. This section is the actually notation for the regression model.
  4. After the comma, we see ‘data = mtcars’ this is telling R to use the ‘mtcar’ dataset.

Seeing the Results

Once you pressed enter you probably noticed nothing happens. The model ‘Carmodel’ was created but the results have not been displayed. Below is various information you can extract from your model.

The ‘summary’ function is useful for pulling most of the critical information. Below is the code with the output.

> summary(Carmodel)

Call:
lm(formula = mpg ~ wt, data = mtcars)

Residuals:
    Min      1Q  Median      3Q     Max 
-4.5432 -2.3647 -0.1252  1.4096  6.8727 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)  37.2851     1.8776  19.858  < 2e-16 ***
wt           -5.3445     0.5591  -9.559 1.29e-10 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 3.046 on 30 degrees of freedom
Multiple R-squared:  0.7528,	Adjusted R-squared:  0.7446 
F-statistic: 91.38 on 1 and 30 DF,  p-value: 1.294e-10

 

We are not going to explain the details of the output (please see simple regression). The results indicate a model that explains 75% of the variance or has an r-square of 0.75. The model is statistically significant and the equation of the model is -5.45x + 37.28 = y

Plotting the Model

We can also plot the model using the code below

> coef_Carmodel<-coef(Carmodel)
> plot(mpg ~ wt, data = mtcars)
> abline(a = coef_Carmodel[1], b = coef_Carmodel[2])

 

Here is what we did

  1. We made the variable ‘coef_Carmodel’ and stored the coefficients (intercept and slope) of the ‘Carmodel’ using the ‘coef’ function. We will need this information soon.
  2. Next, we plot the ‘mtcars’ dataset using ‘mpg’ and ‘wt’.
  3. To add the regression line we use the ‘abline’ function. To add the intercept and slope we use a = the intercept from our ‘coef_Carmodel’ variable  which is subset [1] from this variable. For slope we following the same process but use a [2]. This will add the line and your graph should look like the following.

Rplot10

From the visual, you can see that as weight increases there is a decrease in miles per gallon.

Conclusion

R is capable of much more complex models than the simple regression used here. However, understanding the coding for simple modeling can help in preparing you for much more complex forms of statistical modeling.

Experimental Design: Within Groups

Within groups experimental designs is the use of only one group in an experiment. This is in contrast to a between group design which involves two or more groups. Within group design is useful when the number of is two low in order to split them into different groups.

There are two common forms of within group experimental design, time series and repeated measures. Under time series there are interrupted times series and equivalent time series. Under repeated measure there is only repeated measure design. In this post we will look at the following forms of within group experimental design.

  • interrupted time series
  • equivalent time series
  • repeated measures design

Interrupted Time Series Design

Interrupted time series design involves several pre-test followed by an intervention and then several post-test of one group. By measuring the several times, many threats to internal validity are reduced, such as regression, maturation, and selection. The pre-test results are also used as covariates when analyzing the post-tests.

Equivalent Time Series Design

Equivalent time series design involves the use of a measurement followed by intervention followed by measurement etc. In many ways, this design is a repeated post-test only design. The primary goal is to plot the results of the post-test and determine if there is a pattern that develops over time.

For example, if you are tracking the influence of blog writing on vocabulary acquisition, the intervention is blog writing and the dependent variable is vocabulary acquisition. As the students write a blog, you measure them several times over a certain period. If a plot indicates an upward trend you could infer that blog writing made a difference in vocabulary acquisition.

Repeated Measures

Repeated measures is the use of several different treatments over time. Before each treatment, the group is measured. Each post-test is compared to other post-test to determine which treatment was the best.

For example, lets say that you still want to assess vocabulary acquisition but want to see how blog writing and public speaking affect it. First, you measure vocabulary acquisition. Next, you employ the first intervention followed by a second assessment of vocabulary acquisition. Third, you use the public speaking intervention followed by a third assessment of vocabulary acquisition. You now have three parts of data to compare

  1. The first assessment of vocabulary acquisition (a pre-test)
  2. The second assessment of vocabulary acquisition (post-test 1 after the blog writing)
  3. The third assessment of vocabulary acquisition (post-test 2 after the public speaking)

Conclusion

Within group experimental designs are used when it is not possible to have several groups in an experiment. The benefits include needing fewer participants. However, one problem with this approach is the need to measure several times which can be labor intensive.

ANOVA with R

Analysis of variance (ANOVA) is used when you want to see if there is a difference in the means of three or more groups due to some form of treatment(s). In this post, we will look at conducting an ANOVA calculation using R.

We are going to use a dataset that is already built in to R called ‘InsectSprays.’ This dataset contains information on different insecticides and their ability to kill insects. What we want to know is which insecticide was the best at killing insects.

In the dataset ‘InsectSprays’, there are two variables, ‘count’, which is the number of dead bugs, and ‘spray’ which is the spray that was used to kill the bug. For the ‘spray’ variable there are six types label A-F. There are 72 total observations for the six types of spray which comes to about 12 observations per spray.

Building the Model

The code for calculating the ANOVA is below

> BugModel<- aov(count ~ spray, data=InsectSprays)
> BugModel
Call:
   aov(formula = count ~ spray, data = InsectSprays)

Terms:
                   spray Residuals
Sum of Squares  2668.833  1015.167
Deg. of Freedom        5        66

Residual standard error: 3.921902
Estimated effects may be unbalanced

 

Here is what we did

  1. We created the variable ‘BugModel’
  2. In this variable, we used the function ‘aov’ which is the ANOVA function.
  3. Within the ‘aov’ function we told are to determine the count by the difference sprays that is what the ‘~’ tilda operator does.
  4. After the comma, we told R what data set to use which was “InsectSprays.”
  5. Next we pressed ‘enter’ and nothing happens. This is because we have to make R print the results by typing the name of the variable “BugModel” and pressing ‘enter’.
  6. The results do not tell us anything to useful yet. However, now that we have the ‘BugModel’ saved we can use this information to find the what we want.

Now we need to see if there is any significant results. To do this we will use the ‘summary’ function as shown in the script below

> summary(BugModel)
            Df Sum Sq Mean Sq F value Pr(>F)    
spray        5   2669   533.8    34.7 <2e-16 ***
Residuals   66   1015    15.4                   
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

These results indicate that there are significant results in the model as shown by the p-value being essentially zero (Pr(>F)). In other words, there is at least one mean that is different from the other means statistically.

We need to see what the means are overall for all sprays and for each spray individually. This is done with the following script

> model.tables(BugModel, type = 'means')
Tables of means
Grand mean
    
9.5 

 spray 
spray
     A      B      C      D      E      F 
14.500 15.333  2.083  4.917  3.500 16.667

 

The ‘model.tables’ function tells us the means overall and for each spray. As you can see, it appears spray F is the most efficient at killing bugs with a mean of 16.667.

However, this table does not indicate the statistically significance. For this we need to conduct a post-hoc Tukey test. This test will determine which mean is significantly different from the others. Below is the script

> BugSpraydiff<- TukeyHSD(BugModel)
> BugSpraydiff
  Tukey multiple comparisons of means
    95% family-wise confidence level

Fit: aov(formula = count ~ spray, data = InsectSprays)

$spray
           diff        lwr       upr     p adj
B-A   0.8333333  -3.866075  5.532742 0.9951810
C-A -12.4166667 -17.116075 -7.717258 0.0000000
D-A  -9.5833333 -14.282742 -4.883925 0.0000014
E-A -11.0000000 -15.699409 -6.300591 0.0000000
F-A   2.1666667  -2.532742  6.866075 0.7542147
C-B -13.2500000 -17.949409 -8.550591 0.0000000
D-B -10.4166667 -15.116075 -5.717258 0.0000002
E-B -11.8333333 -16.532742 -7.133925 0.0000000
F-B   1.3333333  -3.366075  6.032742 0.9603075
D-C   2.8333333  -1.866075  7.532742 0.4920707
E-C   1.4166667  -3.282742  6.116075 0.9488669
F-C  14.5833333   9.883925 19.282742 0.0000000
E-D  -1.4166667  -6.116075  3.282742 0.9488669
F-D  11.7500000   7.050591 16.449409 0.0000000
F-E  13.1666667   8.467258 17.866075 0.0000000

 

There is a lot of information here. To make things easy, wherever there is a p adj of less than 0.05 that means that there is a difference between those two means. For example, bug spray F and E have a difference of 13.16 that has a p adj of zero. So these two means are really different statistically.This chart also includes the lower and upper bounds of the confidence interval.

The results can also be plotted with the script below

> plot(BugSpraydiff, las=1)

Below is the plot

Rplot10

Conclusion

ANOVA is used to calculate if there is a difference of means among three or more groups. This analysis can be conducted in R using various scripts and codes.

Experimental Designs: Between Groups

In experimental research, there are two common designs. They are between and within group design. The difference between these two groups of designs is that between group involves two or more groups in an experiment while within group involves only one group.

This post will focus on between group designs. We will look at the following forms of between group design…

  • True/quasi-experiment
  • Factorial Design

True/quasi Experiment

A true experiment is one in which the participants are randomly assigned to different groups. In a quasi-experiment, the researcher is not able to randomly assigned participants to different groups.

Random assignment is important in reducing many threats to internal validity. However, there are times when a researcher does not have control over this, such as when they conduct an experiment at a school where classes have already been established. In general, a true experiment is always considered superior methodological to a quasi-experiment.

Whether the experiment is a true experiment or a quasi-experiment. There are always two groups that are compared in the study. One group is the controlled group, which does not receive the treatment. The other group is called the experimental group, which receives the treatment of the study. It is possible to have more than two groups and several treatments but the minimum for between group designs is two groups.

Another characteristic that true and quasi-experiments have in common is the type of formats that the experiment can take. There are two common formats

  • Pre- and post test
  • Post test only

A pre- and post test involves measuring the groups of the study before the treatment and after the treatment. The desire normally is for the groups to be the same before the treatment and for them to be different statistically after the treatment. The reason for them being different is because of the treatment, at least hopefully.

For example, let’s say you have some bushes and you want to see if the fertilizer you bought makes any difference in the growth of the bushes.  You divide the bushes into two groups, one that receives the fertilizer (experimental group), and one that does not (controlled group). You measure the height of the bushes before the experiment to be sure they are the same. Then, you apply the fertilizer to the experimental group and after a period of time, you measure the heights of both groups again. If the fertilized bushes grow taller than the control group you can infer that it is because of the fertilizer.

Post-test only design is when the groups are measured only after the treatment. For example, let’s say you have some corn plants and you want to see if the fertilizer you bought makes any difference.in the amount of corn produced.  You divide the corn plants into two groups, one that receives the fertilizer (experimental group), and one that does not (controlled group). You apply the fertilizer to the control group and after a period of time, you measure the amount of corn produced. If the fertilized corn produces more you can infer that it is because of the fertilizer. You never measure the corn beforehand because they had not produced any corn yet.

Factorial design involves the use of more than one treatment. Returning to the corn example, let’s say you want to see not only how fertilizer affects corn production but also how the amount of water the corn receives affects production as well.

In this example, you are trying to see if there is an interaction effect between fertilizer and water.  When water and fertilizer are increased does production increase, is there no increase, or if one goes up and the other goes down does that have an effect?

Conclusion

Between group designs such as true and quasi-experiments provide a way for researchers to establish cause and effect. Pre- post test is employed as well as factorial designs to establish relationships between variables

Calculating Chi-Square in R

There are times when conducting research that you want to know if there is a difference in categorical data. For example, is there a difference in the number of men who have blue eyes and who have brown eyes. Or is there a relationship between gender and hair color. In other words, is there a difference in the count of a particular characteristic or is there a relationship between two or more categorical variables.

In statistics, the chi-square test is used to compare categorical data. In this post, we will look at how you can use the chi-square test in R.

For our example, we are going to use data that is already available in R called “HairEyeColor”. Below is the data

> HairEyeColor
, , Sex = Male

       Eye
Hair    Brown Blue Hazel Green
  Black    32   11    10     3
  Brown    53   50    25    15
  Red      10   10     7     7
  Blond     3   30     5     8

, , Sex = Female

       Eye
Hair    Brown Blue Hazel Green
  Black    36    9     5     2
  Brown    66   34    29    14
  Red      16    7     7     7
  Blond     4   64     5     8

As you can see, the data comes in the form of a list and shows hair and eye color for men and women in separate tables. The current data is unusable for us in terms of calculating differences. However, by using the ‘marign.table’ function we can make the data useable as shown in the example below.

> HairEyeNew<- margin.table(HairEyeColor, margin = c(1,2))
> HairEyeNew
       Eye
Hair    Brown Blue Hazel Green
  Black    68   20    15     5
  Brown   119   84    54    29
  Red      26   17    14    14
  Blond     7   94    10    16

Here is what we did. We created the variable ‘HairEyeNew’ and we stored the information from ‘HairEyeColor’ into one table using the ‘margin.table’ function. The margin was set 1,2 for the table.

Now all are data from the list are combined into one table.

We now want to see if there is a particular relationship between hair and eye color that is more common. To do this, we calculate the chi-square statistic as in the example below.

> chisq.test(HairEyeNew)

	Pearson's Chi-squared test

data:  HairEyeNew
X-squared = 138.29, df = 9, p-value < 2.2e-16

The test tells us that one or more of the relationships are more common than others within the table. To determine which relationship between hair and eye color is more common than the rest we will calculate the proportions for the table as seen below.

> HairEyeNew/sum(HairEyeNew)
       Eye
Hair          Brown        Blue       Hazel       Green
  Black 0.114864865 0.033783784 0.025337838 0.008445946
  Brown 0.201013514 0.141891892 0.091216216 0.048986486
  Red   0.043918919 0.028716216 0.023648649 0.023648649
  Blond 0.011824324 0.158783784 0.016891892 0.027027027

As you can see from the table, brown hair and brown eyes are the most common (0.20 or 20%) flowed by blond hair and blue eyes (0.15 or 15%).

Conclusion

The chi-square serves to determine differences among categorical data. This tool is useful for calculating the potential relationships among non-continuous variables

Philosophical Foundations of Research: Epistemology

Epistemology is the study of the nature of knowledge. It deals with questions as is there truth and or absolute truth, is there one way or many ways to see something. In research, epistemology manifest itself in several views. The two extremes are positivism and interpretivism.

Positivism

Positivism asserts that all truth can be verified and proven scientifically and can be measured and or observed. This position discounts religious revelation as a source of knowledge as this cannot be verified scientifically. The position of positivist is also derived from realism in that there is an external world out there that needs to be studied.

For researchers, positivism is the foundation of quantitative research. Quantitative researchers try to be objective in their research, they try to avoid coming into contact with whatever they are studying as they do not want to disturb the environment. One of the primary goals is to make generalizations that are applicable in all instances.

For quantitative researchers, they normally have a desire to test a theory. In other words, the develop one example of what they believe is a truth about a phenomenon (a theory) and they test the accuracy of this theory with statistical data. The data determines the accuracy of the theory and the changes that need to be made.

By the late 19th and early 20th centuries, people were looking for alternative ways to approach research. One new approach was interpretivism.

Interpretivism

Interpretivism is the complete opposite of positivism in many ways. Interpretivism asserts that there is no absolute truth but relative truth based on context. There is no single reality but multiple realities that need to be explored and understood.

For interpretist, There is a fluidity in their methods of data collection and analysis. These two steps are often iterative in the same design. Furthermore, intrepretist see themselves not as outside the reality but a player within it. Thus, they often will share not only what the data says but their own view and stance about it.

Qualitative researchers are interpretists. They spend time in the field getting close to their participants through interviews and observations. They then interpret the meaning of these communications to explain a local context specific reality.

While quantitative researchers test theories, qualitative researchers build theories. For qualitative researchers, they gather data and interpret the data by developing a theory that explains the local reality of the context. Since the sampling is normally small in qualitative studies, the theories do not often apply to many.

Conclusion

There is little purpose in debating which view is superior. Both positivism and interpretivism have their place in research. What matters more is to understand your position and preference and to be able to articulate in a reasonable manner. It is often not what a person does and believes that is important as why they believe or do what they do.

Internal Validity

In experimental research design, internal validity is the appropriateness of the inferences made about cause and effects relationships between the independent and dependent variables. If there are threats to internal validity it may mean that the cause and effect relationship you are trying to establish is not real. In general, there are three categories of external validity, which are..,

  • Participant threats
  • Treatment threats
  • Procedural threats

We will not discuss all three categories but will focus on participant threats

Participant Threats

There are several forms of threats to internal validity that relate to participants. Below is a list

  • History
  • Maturation
  • Regression
  • Selection
  • Mortality

History

A historical threat to internal validity is the problem of the passages of time from  the beginning to the end of the experiment. During this elapse of time, the groups involved in the study may have different experiences. These different experiences are history threats. One way to deal with this threat is to be sure that the conditions of the experiment are the same.

Maturation

Maturation threat is the problem of how people change over time during an experiment. These changes make it hard to infer if the results of a study are because of the treatment or because of maturation. One way to deal with this threat is to select participants who develop in similar ways and speed.

Regression

Regression threat is the action of the researcher selecting extreme cases to include in their sample. Eventually, these cases regress to the mean, which impacts the results of the pretest or posttest. One option for overcoming this problem is to avoid outliers when selecting the sample.

Selection

Selection bias is the poor habit of picking people in a non-random why for an experiment Examples of this include choosing mostly ‘smart people for an experiment. Or working with petite women for a study on diet and exercise. Random selection is the strongest way to deal with this threat.

Mortality

Mortality is the lost of participants in a study. It is common for participants in a study to dropout and quit for many reasons. This leads to a decrease in the sample size, which weakens the statistical interpretation. Dealing with this requires using larger sample sizes as well as comparing data of dropouts with those who completed the study.

Conclusion

Internal validity can ruin a paper that has not careful planned out how these threats work together to skew results. Researchers need to have an idea of what threats are out there as well as strategies that can alleviate them.

Comparing Samples in R

Comparing groups is a common goal in statistics. This is done to see if there is a difference between two groups. Understanding the difference can lead to insights based on statistical results. In this post, we will examine the following statistical test for comparing samples.

  • t-test/Wilcoxon test
  • Paired t-test

T-test & Wilcoxon Test

The T-test indicates if there is a significant statistical difference between two groups. This is useful if you know what the difference between the two groups are. For example, if you are measuring height of men and women, if you find that men are taller through a t-test, you can state that gender influences height because the only difference between men and women in this example is their gender.

Below is an example of conducting a t-test in R. In the example, we are looking at if there is a difference in body temperature between beavers who are active versus beavers who are not.

> t.test(temp ~ activ, data = beaver2)

	Welch Two Sample t-test

data:  temp by activ
t = -18.548, df = 80.852, p-value < 2.2e-16
alternative hypothesis: true difference in means is not equal to 0
95 percent confidence interval:
 -0.8927106 -0.7197342
sample estimates:
mean in group 0 mean in group 1 
       37.09684        37.90306

Here is what happened

  1. We use the ‘t.test’ function
  2. Within the ‘t.test’ function we indicate that we want to if there is a difference in ‘temp’ when compared on the factor variable ‘activ’ in the data ‘beaver2’
  3. The output provides a lot of information. The t-stat is -18.58 any number at + 1.96 indicates statistical difference.
  4. Df stands for degrees of freeedom and is used to determine the t-stat and p-value.
  5. The p-value is basically zero. Anything less than 0.05 is considered statistically significant.
  6. Next, we have the 95% confidence interval, which is the range of the difference of the means of the two groups in the population.
  7. Lastly, we have the means of each group. Group 0, the inactive group. had a mean of 37.09684. Group 1. the active group, has a mean of  37.90306.

T-test assumes that the data is normally distributed. When normality is a problem, it is possible to use the Wilcoxon test instead. Below is the script for the Wilcoxon Test using the same example.

> wilcox.test(temp ~ activ, data = beaver2)

	Wilcoxon rank sum test with continuity correction

data:  temp by activ
W = 15, p-value < 2.2e-16
alternative hypothesis: true location shift is not equal to 0

A closer look at the output indicates the same results for the most part. Instead of the t-stat the W-stat is used but the p value is the same for both test.

Paired T-Test

A paired t-test is used when you want to compare how the same group of people respond to different interventions. For example, you might use this for a before and after experiment. We will use the ‘sleep’ data in R  to compare a group of people when they receive different types of sleep medication. The script is below

> t.test(extra ~ group, data = sleep, paired = TRUE)

	Paired t-test

data:  extra by group
t = -4.0621, df = 9, p-value = 0.002833
alternative hypothesis: true difference in means is not equal to 0
95 percent confidence interval:
 -2.4598858 -0.7001142
sample estimates:
mean of the differences 
                  -1.58

Here is what happened

  1. We used the ‘t.test’ function and indicate we want to see if ‘extra’ (amount of sleep) is influenced by ‘group’ (two types of sleep medication.
  2. We add the new argument of ‘paired = TRUE’ this tells R that this is a paired test.
  3. The output is the same information as in the regular t.test. The only differences is at the bottom where R only tells you the difference between the two groups and not the means of each. For this example, the people slept about 1 hour and 30 minutes longer on the second sleep medication when compared to the first.

Conclusion

Comparing samples in R is a simple process of understanding what you want to do. With this knowledge, the script and output are not too challenge even for many beginners

Experimental Design: Treatment Conditions and Group Comparision

A key component of experimental design involves making decisions about the manipulation of the treatment conditions. In this post, we will look at the following traits of treatment conditions

  • Treatment Variables
  • Experimental treatment
  • Conditions
  • Interventions
  • Outcomes

Lastly, we will examine group comparison.

One of the most common independent variables in experimental design are treatment and measured variables. Treatment variables are manipulated by the researcher. For example, if you are looking at how sleep affects academic performance, you may manipulate the amount of sleep participants receive in order to determine the relationship between academic performance and sleep.

Measured variables are variables that are measured by are not manipulated by the researcher. Examples include age, gender, height, weight, etc.

An experimental treatment is the intervention of the researcher to alter the conditions of an experiment. This is done by keeping all other factors constant and only manipulating the experimental treatment, it allows for the potential establishment of a cause-effect relationship. In other words, the experimental treatment is a term for the use of a treatment variable.

Treatment variables usually have different conditions or levels in them. For example, if I am looking at sleep’s affect on academic performance. I may manipulate the treatment variable by creating several categories of the amount of sleep. Such as high, medium, and low amounts of sleep.

Intervention is a term that means the actual application of the treatment variables. In other words, I broke my sample into several groups and caused one group to get plenty of sleep the second group to lack a little bit of sleep and the last group got nothing. Experimental treatment and intervention mean the same thing.

The outcome measure is the experience of measuring the outcome variable. In our example, the outcome variable is academic performance.

Group Comparison

Experimental design often focuses on comparing groups. Groups can be compared between groups and within groups. Returning to the example of sleep and academic performance, a between group comparison would be to compare the different groups based on the amount of sleep they received. A within group comparison would be to compare the participants who received the same amount of sleep.

Often there are at least three groups in an experimental study, which are the controlled, comparison, and experimental group. The controlled group receives no intervention or treatment variable. This group often serves as a baseline for comparing the other groups.

The comparison group is exposed to everything but the actual treatment of the study. They are highly similar to the experimental group with the experience of the treatment. Lastly, the experimental group experiences the treatment of the study.

Conclusion

Experiments involve treatment conditions and groups. As such, researchers need to understand their options for treatment conditions as well as what types of groups they should include in a study.

Examining Distributions In R

Normal distribution is an important term in statistics. When we say normal distribution, we are speaking of the traditional bell curve concept. Normal distribution is important because it is often an assumption of inferential statistics that the distribution of data points is normal. Therefore, one of the first things you do when analyzing data is to check for normality.

In this post we will look at the following ways to assess normality.

  • By graph
  • By plots
  • By test

Checking Normality by Graph

The easiest and crudest way to check for normality is visually through the use of histograms. You simply look at the histogram and determine how closely it resembles a bell.

To illustrate this we will use the ‘beaver2’ data that is already loaded into R. This dataset contains five variables “day”, “time”, “temp”, and “activ” for data about beavers. Day indicates what day it was, time indicates what time it was, temp is the temperature of the beavers, and activ is whether the beavers were active when their temperature was taking. We are going to examine the normality of the temperature of active and inactive. Below is the code

> library(lattice)
> histogram(~temp | factor(activ), data = beaver2)
  1. We loaded the ‘lattice’ package. (If you don’t have this package please download)>
  2. We used the histogram function and indicate the variable ‘temp’ then the union ( | ) followed by the factor variable ‘activ’ (0 = inactive, 1 = active)
  3. Next, we indicate the dataset we are using ‘beaver2’
  4. After pressing ‘enter’ you should see the following

Rplot10

As you look at the histograms, you can say that they are somewhat normal. The peaks of the data are a little high in both. Group 1 is more normal than Group 0. The problem with visual inspection is lack of accuracy in interpretation. This is partially solved through using QQ plots.

Checking Normality by Plots

QQplots are useful for comparing your data with a normally distributed theoretical dataset. The QQplot includes a line of a normal distribution and the data points for your data for comparison. The more closely your data follows the line the more normal it is. Below is the code for doing this with our beaver information.

> qqnorm(beaver2$temp[beaver2$activ==1], main = 'Active')
> qqline(beaver2$temp[beaver2$activ==1])

Here is what we did

  1. We used the ‘qqnorm’ function to make the plot
  2. Within the ‘qqnorm’ function we tell are to use ‘temp’ from the ‘beaver2’ dataset.
  3. From the ‘temp’ variable we subset the values that have a 1 in the ‘activ’ variable.
  4. We give the plot a title by adding ‘main = Active’
  5. Finally, we add the ‘qqline’ using most of the previous information.
  6. Below is how the plot should look

R

Going by sight again. The data still looks pretty good. However, one last test will determine conclusively if the dataset is normal or not.

Checking Normality by Test

The Shapiro-Wilks normality test determines the probability that the data is normally distributed. The lower the probability the less likely that the data is normally distributed. Below is the code and results for the Shapiro test.

> shapiro.test(beaver2$temp[beaver2$activ==1])

	Shapiro-Wilk normality test

data:  beaver2$temp[beaver2$activ == 1]
W = 0.98326, p-value = 0.5583

Here is what happened

  1. We use the ‘shapiro.test’ function for ‘temp’ of only the beavers who are active (activ = 1)
  2. R tells us the p-value is 0.55 or 55%
  3. This means that the probability of are data being normally distributed is 55% which means it is highly likely to be normal.

Conclusion

It is necessary to always test normality of data before data analysis. The tips presented here provide some framework for accomplishing this.

Characteristics of Experimental Design

In a previous post, we began a discussion on experimental design. In this post, we will begin a discussion on the characteristics of experimental design. In particular, we will look at the following

  • Random assignment
  • Control over extraneous variables

Random Assignment

After developing an appropriate sampling method, a researcher needs to randomly assign individuals to the different groups of the study. One of the main reasons for doing this is to remove the bias of individual differences in all groups of the study.

For example, if you are doing a study on intelligence. You want to make sure that all groups have the same characteristics of intelligence. This helps for the groups to equate or to be the same. This prevents people from saying the reason there are differences between groups is because the groups are different and not because of the treatment.

Control Over Extraneous Variables

Random assignment directly leads to the concern of controlling extraneous variables. Extraneous variables are any factors that might influence the cause and effect relationship that you are trying to establish. These other factors confound or confuse the results of a study. There are several methods for dealing with this as shown below

  • Pretest-posttest
  • Homogeneous sampling
  • Covriate
  • Matching

Pretest-Posttest

A pre-test post-test allows a researcher to compare the measurement of something before the treatment and after the treatment. The assumption is that any difference in the scores of before and after is due to the treatment.Doing the tests takes into account the confounding of the different contexts of the setting and individual characteristics.

Homogeneous Sampling

This approach involves selecting people who are highly similar on the particular trait that is being measured. This removes the problem of individual differences when attempting to interpret the results. The more similar the subjects in the sample are the more controlled the traits of the people are controlled for.

Covariate

Covariates is a statistical approach in which controls are placed on the dependent variable through statistical analysis. The influence of other variables are removed from the explained variance of the dependent variable. Covariates help to explain more about the relationship between the independent and dependent variables.

This is a difficult concept to understand. However, the point is that you use covariates to explain in greater detail the relationship between the independent and dependent variable by removing other variables that might explain the relationship.

Matching

Matching is deliberate, rather than randomly, assigning subject to various groups. For example, if you are looking at intelligence. You might match high achievers in both groups of the study. By placing he achievers in both groups you cancel out there difference.

Conclusion

Experimental design involves the cooperation in random assignment of inclusive differences in a sample. The goal of experimental design is to be sure that the sample groups are mostly the same in a study. This allows for concluding that what happened was due to the treatment.

Experimental Design: A Background

Experimental design is now considered a standard methodology in research. However, this now classic design has not always been a standard approach. In this post, we will the following

  • The definition of experiment
  • The history of experiments
  • When to conduct experiments

Definition

The word experiment is derived from the word experience. When conducting an experiment, the researcher assigns people to have different experiences. He then determines if the experience he assigned people to had some effect on some sort of outcome. For example, if I want to know if the experience of sunlight affects the growth of plants I may develop two different experiences

  • Experience 1: Some plants receive sunlight
  • Experience 2: Some plants receive no sunlight

The outcome is the growth of the plants.By giving the plants different experiences of sunlight I can determine if sunlight influences the growth of plants.

History of Experiments

Experiments have been around informally since the 10th century with work done in the field of medicine. The use of experiments as known today began in the early 20th century in the field of psychology. By the 1920’s group comparison became an establish characteristics of experiments. By the 1930’s, random assignment was introduced. By the 1960’s various experimental designs were codified and documented. By the 1980’s there was literature coming out that addressed threats to validity.

Since the 1980’s experiments have become much more complicated with the development of more advanced statistical software problems. Despite all of the new complexity, normally simple experimental designs are easier to understand

When to Conduct Experiments

Experiments are conducted to attempt to establish a cause and effect relationship between independent and dependent variables. You try to create a controlled environment in which you provide the experience or independent variable(s) and then measure how they affect the outcome or dependent variable.

Since the setting of the experiment is controlled. You can say withou a doubt that only the experience influence the outcome. Off course, in reality, it is difficult to control all the factors in a study. The real goal is to try and limit the effect that these other factors have on the outcomes of a study.

Conclusion

Despite their long history, experiments are relatively new in research. This design has grown and matured over the years to become a powerful method for determining cause and effect. Therefore, researchers should e aware of this approach for use in their studies.

Basics of Histograms and Plots in R

R has many fascinating features for creating histograms and plots. In this post, we will only cover some of the most basic concepts of make histograms and plots in R. The code for the data we are using is available in a previous post.

Making a Histogram

We are going to make a histogram of the ‘mpg’ variable in our ‘cars’ dataset. Below is the code for doing this followed by the actual histogram.

Histogram of mpg variable

Histogram of mpg variable

Here is what we did

  1. We used the ‘hist’ function to create the histogram
  2. Within the hist function we told r to make a histogram of ‘mpg’ variable found in the ‘cars’ dataset.
  3. An additional argument that we added was ‘col’. This argument is used to determine the color of the bars in the histogram. For our example, the color was set to gray.

Plotting Multiple Variables

Before we look at plotting multiple variables you need to make an adjustment to the ‘cyl’ variable in our cars variable. THis variable needs t be changed from a numeric to a factor variable as shown below

cars$cyl<- as.factor(cars$cyl)

Boxplots are an excellent way of comparing groups visually. In this example, we will compare the ‘mpg’ or miles per gallon variable by the ‘cyl’ or number of cylinders in the engine variable in the ‘cars’ dataset. Below is the code and diagram followed by an explanation.

boxplot(mpg ~ cyl, data = cars)

Rplot

Here is what happened.

  1. We use the ‘boxplot’ function
  2. Within this function we tell are to plot mpg and cyl using the tilda  ” ~ ” to tell R to compare ‘mpg’ by the number of cylinders

The box of the boxplot tells you several things

  1. The bottom of the box tells you the 25th percentile
  2. The line in the middle of the box tells you the median
  3. The top of the box tells you the 75th percentile
  4. The bottom line tells you the minimum or lowest value excluding outliers
  5. The top line tells you the maximum or highest value excluding outliers

In order boxplot above, there are three types of cylinders 4, 6, and 8. For 4 cylinders the 25th percentile is about 23 mpg, the 50th percentile is about 26 mpg, while the 75th percentile is about 31 mpg. The minimum value was about 22 and the maximum value was about 35 mpg. A close look at the different blots indicates that four cylinder cars have the best mpg followed by six and finally eight cylinders.

Conclusions

Histograms and boxplots serve the purpose of describing numerical data in a visual manner. Nothing like a picture helps to explain abstract concepts such mean and median.

Analyzing Quantitative Data: Inferential Statistics

In a prior post, we looked at analyzing quantitative data using descriptive statistics. In general, descriptive statistics describe your data in terms of the tendencies within the sample. However, with descriptive stats, you only learn about your sample but you are not able to compare groups nor find the relationship between variables. To deal with this problem, we use inferential statistics.

Types of Inferential Statistics

With inferential statistics, you look at a sample and make inferences about the population. There are many different types of analysis that involve inferential statistics. Below is a partial list. The ones with links have been covered in this blog before.

  • Pearson Correlation Coefficient–Used to determine the strength of the relationship between two continuous variables
  • Regression Coefficient-The squared value of the Pearson Correlation Coefficient. Indicates the amount of variance explained between two or more variables
  • Spearman Rho–Used to determine the strength of the relationship between two continuous variables for non-parametric data.
  •   t-test-Determines if there is a significant statistical difference between two means. The independent variable is categorical while the dependent variable is continuous.
  • Analysis of Variance-Same as a t-test but for three means or more.
  • Chi-Square-Goodness-of-Fit-This test determines if there is a difference between two categorical variables.

As you can see, there are many different types of inferential statistical test. However, one thing all test have in common is the testing of a hypothesis. Hypothesis testing has been discussed on this blog before. To summarize, a hypothesis test can tell you if there is a difference between the sample and the population or between the sample and a normal distribution.

One other value that can be calculated is the confidence interval. The confidence interval calculates a range that the results of a statistical test (either descriptive or inferential) can be found. For example, If we that the regression coefficient between two variables is .30 the confidence interval may be between .25 — .40. This range tells us what the value of the correlation would be found in the population.

Conclusion

This post serves as an overview of the various forms of inferential statistics available. Remember, that it is the research questions that determine the form of analysis to conduct. Inferential statistics are used for comparing groups and examining relationships between variables.