## Predicting Earnings from Census Data

The United States government periodically collects demographic information by conducting a census.

In this problem, we are going to use census information about an individual to predict how much a person earns -- in particular, whether the person earns more than $50,000 per year. This data comes from the UCI Machine Learning Repository.

The file census (CSV - 3.3MB) contains 1994 census data for 31,978 individuals in the United States.

The dataset includes the following 13 variables:

*age =*the age of the individual in years*workclass =*the classification of the individual's working status (does the person work for the federal government, work for the local government, work without pay, and so on)*education =*the level of education of the individual (e.g., 5th-6th grade, high school graduate, PhD, so on)*maritalstatus =*the marital status of the individual*occupation =*the type of work the individual does (e.g., administrative/clerical work, farming/fishing, sales and so on)*relationship =*relationship of individual to his/her household*race =*the individual's race*sex =*the individual's sex*capitalgain =*the capital gains of the individual in 1994 (from selling an asset such as a stock or bond for more than the original purchase price)*capitalloss =*the capital losses of the individual in 1994 (from selling an asset such as a stock or bond for less than the original purchase price)*hoursperweek =*the number of hours the individual works per week*nativecountry =*the native country of the individual*over50k =*whether or not the individual earned more than $50,000 in 1994

## Problem 1.1 - A Logistic Regression Model

Let's begin by building a logistic regression model to predict whether an individual's earnings are above $50,000 (the variable "over50k") using all of the other variables as independent variables. First, read the dataset census.csv into R.

Then, split the data randomly into a training set and a testing set, setting the seed to 2000 before creating the split. Split the data so that the training set contains 60% of the observations, while the testing set contains 40% of the observations.

Next, build a logistic regression model to predict the dependent variable "over50k", using all of the other variables in the dataset as independent variables. Use the training set to build the model.

Which variables are significant, or have factors that have at least one significant dummy variable? (Use 0.1 as your significance threshold, so variables with a period or dot in the stars column should be counted too. You might see a warning message here - you can ignore it and proceed. This message is a warning that we might be overfitting our model to the training set.) Select all that apply.

Explanation

To read census.csv, set your working directory to the same directory that census.csv is in, and run the following command:

census = read.csv("census.csv")

We now need to split the data. Load the caTools package, and set the seed to 2000:

library(caTools)

set.seed(2000)

Split the data set according to the over50k variable using filter from the dplyr package:

spl = sample.split(census$over50k, SplitRatio = 0.6)

library(dplyr)

censusTrain = census %>% filter(spl == TRUE)

censusTest = census %>% filter(spl == FALSE)

We are now ready to run logistic regression. Build the logistic regression model:

censusglm = glm( over50k ~ . , family="binomial", data = censusTrain)

Finally, look at the model summary to identify the significant factors:

summary(censusglm)

## Problem 1.2 - A Logistic Regression Model

What is the accuracy of the model on the testing set? Use a threshold of p = 0.5. (You might see a warning message when you make predictions on the test set - you can safely ignore it.)

Explanation

Generate the predictions for the test set:

predictTest = predict(censusglm, newdata = censusTest, type = "response")

Then we can generate the confusion matrix:

table(censusTest$over50k, predictTest >= 0.5)

If we divide the sum of the main diagonal by the sum of all of the entries in the matrix, we obtain the accuracy:

(9051+1888)/(9051+662+1190+1888) = 0.8552107

## Problem 1.3 - A Logistic Regression Model

What is the accuracy of the baseline model on the testing set?

Explanation

We need to first determine the most frequent outcome in the training set. To do that, we table the dependent variable in the training set:

table(censusTrain$over50k)

"<=50K" is the more frequent outcome (14570 observations), so this is what the baseline predicts. To generate the accuracy of the baseline on the test set, we can table the dependent variable in the test set:

table(censusTest$over50k)

The baseline accuracy is

9713/(9713+3078) = 0.7593621.

## Problem 1.4 - A Logistic Regression Model

What is the area under the curve (AUC) for the logistic regression model on the test set?

Explanation

First, load the ROCR package:

library(ROCR)

Then you can use the following commands to compute the AUC (assuming your test set predictions are called "predictTest"):

ROCRpred = prediction(predictTest, censusTest$over50k)

as.numeric(performance(ROCRpred, "auc")@y.values)

## Problem 2.1 - A CART Model

We have just seen how a logistic regression model for this data achieves a high accuracy. Moreover, the significances of the variables give us a way to gauge which variables are relevant for this prediction task. However, it is not immediately clear which variables are more important than the others, especially due to the large number of factor variables in this problem.

Let us now build a CART model to predict "over50k". Use the training set to build the model, with all of the other variables as independent variables. Use the default parameters (with the default loss table), so don't set a value for minbucket or cp and do not specify a loss table. Remember to specify method="class" as an argument to rpart, since this is a classification problem. After you are done building the model, plot the resulting tree.

How many splits does the tree have in total?

Explanation

To get started, load the rpart and rpart.plot packages:

library(rpart)

library(rpart.plot)

Estimate the CART tree:

censustree = rpart( over50k ~ . , method="class", data = censusTrain)

Plot the tree:

prp(censustree)

There are four splits in total.

## Problem 2.2 - A CART Model

Which variable does the tree split on at the first level (the very first split of the tree)?

Explanation

Plot the tree and examine the first split:

prp(censustree)

The first split uses the variable relationship.

## Problem 2.3 - A CART Model

Which variables does the tree split on at the second level (immediately after the first split of the tree)? Select all that apply.

Explanation

Plot the tree and examine the second splits:

prp(censustree)

The second splits are on capitalgains and education.

## Problem 2.4 - A CART Model

What is the accuracy of the model on the testing set?

This highlights a very regular phenomenon when comparing CART and logistic regression. CART often performs a little worse than logistic regression in out-of-sample accuracy. However, as is the case here, the CART model is often much simpler to describe and understand.

Explanation

First, generate predictions on the test set using the CART tree:

predictTest = predict(censustree, newdata = censusTest, type = "class")

Then create the confusion matrix:

table(censusTest$over50k, predictTest)

To compute the accuracy, sum the diagonal entries and divide by the sum of all of the terms:

(9243+1596)/(9243+470+1482+1596) = 0.8473927

## Problem 3.1 - A Random Forest Model

Before building a random forest model, we'll down-sample our training set. While some modern personal computers can build a random forest model on the entire training set, others might run out of memory when trying to train the model since random forests is much more computationally intensive than CART or Logistic Regression. For this reason, before continuing we will define a new training set to be used when building our random forest model, that contains 2000 randomly selected obervations from the original training set. Do this by running the following commands in your R console (assuming your training set is called "censusTrain"):

set.seed(1)

trainSmall = censusTrain[sample(nrow(censusTrain), 2000), ]

Let us now build a random forest model to predict "over50k", using the dataset "trainSmall" as the data used to build the model. Set the seed to 1 again right before building the model, and use all of the other variables in the dataset as independent variables. (If you get an error that random forest "can not handle categorical predictors with more than 32 categories", re-build the model without the nativecountry variable as one of the independent variables.)

Then, make predictions using this model on the entire test set. What is the accuracy of the model on the test set, using a threshold of 0.5? (Remember that you don't need a "type" argument when making predictions with a random forest model if you want to use a threshold of 0.5. Also, note that your accuracy might be different from the one reported here, since random forest models can still differ depending on your operating system, even when the random seed is set. )

Explanation

To generate the random forest model with all of the variables, just run:

library(randomForest)

set.seed(1)

censusrf = randomForest(over50k ~ . , data = trainSmall)

And then you can make predictions on the test set by using the following command:

predictTest = predict(censusrf, newdata=censusTest)

And to compute the accuracy, you can create the confusion matrix:

table(censusTest$over50k, predictTest)

The accuracy of the model should be around

(9614+1050)/nrow(censusTest) = 0.8337112

You may be surprised that the test-set accuracy of the random forest model is lower than the test-set accuracies of some of the other, simpler models we've trained. This can occur when the parameters of the random forest have not been tuned via cross-validation and the default parameters are ill suited for the dataset.

## Problem 4.1 - Selecting cp by Cross-Validation

We now conclude our study of this data set by looking at how CART behaves with different choices of its parameters.

Let us select the cp parameter for our CART model using k-fold cross validation, with k = 10 folds. Do this by using the train function. Set the seed beforehand to 2. Test cp values from 0.002 to 0.092 in 0.01 increments, by using the following command:

cartGrid = data.frame( .cp = seq(0.002,0.092,0.01))

After you create the cartGrid variable, you can pass it on to the train function by letting tuneGrid = cartGrid.

Also, remember to use the entire training set "censusTrain" when building this model. The train function might take some time to run.

Which value of cp does the train function recommend?

Explanation

Before doing anything, load the caret package:

library(caret)

Set the seed to 2:

set.seed(2)

Specify that we are going to use k-fold cross validation with 10 folds:

fitControl = trainControl( method = "cv", number = 10 )

Specify the grid of cp values that we wish to evaluate:

cartGrid = data.frame( .cp = seq(0.002,0.092,0.01))

Finally, run the train function and view the result:

cartCV = train( over50k ~ . , data = censusTrain, method = "rpart", trControl = fitControl, tuneGrid = cartGrid )

cartCV

The final output should read

Accuracy was used to select the optimal model using the largest value.

The final value used for the model was cp = 0.002.

In other words, the best value was cp = 0.002, corresponding to the lowest cp value. If we look more closely at the accuracy at different cp values, we can see that it seems to be decreasing steadily as the cp value increases. Often, the cp value needs to become quite low before the accuracy begins to deteriorate.

## Problem 4.2 - Selecting cp by Cross-Validation

Recall that the caret package causes categorical independent variables to be split into factor levels. To generate a version of the test set with dummy variables, please run the following command (which assumes your test set is stored in variable "censusTest"):

censusTestMM = as.data.frame(model.matrix(over50k~.+0, data=censusTest))

Use the final model selected via cross-validation to make predictions on censusTestMM. What is the prediction accuracy on the test set?

Explanation

Assuming you stored the cross-validation results in a variable called "cartCV", you can access the final selected model with the following command:

model = cartCV$finalModel

You can make predictions on the test set using the following commands:

censusTestMM = as.data.frame(model.matrix(~.+0, data=censusTest))

predictTest = predict(model, newdata=censusTestMM, type="class")

Then you can generate the confusion matrix with the command

table(censusTest$over50k, predictTest)

The accuracy is (9167+1821)/(9167+546+1257+1821) = 0.8590415.

## Problem 4.3 - Selecting cp by Cross-Validation

Compared to the original accuracy using the default value of cp, this new CART model is an improvement, and so we should clearly favor this new model over the old one -- or should we? Plot the CART tree for this model. How many splits are there?

This highlights one important tradeoff in building predictive models. By tuning cp, we improved our accuracy, but our tree became significantly more complex. In some applications, such an improvement in accuracy would be worth the loss in interpretability. In others, we may prefer a less accurate model that is simpler to understand and describe over a more accurate -- but more complicated -- model.

Explanation

If you plot the tree with prp(model), where "model" is the name of your CART tree, you should see that there are 17 splits.