Load the training and testing sets using the read.csv() function, and save them as variables with the names pisaTrain and pisaTest. How many students are there in the training set?
pisaTrain = read.csv("pisa2009train.csv")
pisaTest = read.csv("pisa2009test.csv")
str(pisaTrain)'data.frame': 3663 obs. of 24 variables:
$ grade : int 11 11 9 10 10 10 10 10 9 10 ...
$ male : int 1 1 1 0 1 1 0 0 0 1 ...
$ raceeth : Factor w/ 7 levels "American Indian/Alaska Native",..: NA 7 7 3 4 3 2 7 7 5 ...
$ preschool : int NA 0 1 1 1 1 0 1 1 1 ...
$ expectBachelors : int 0 0 1 1 0 1 1 1 0 1 ...
$ motherHS : int NA 1 1 0 1 NA 1 1 1 1 ...
$ motherBachelors : int NA 1 1 0 0 NA 0 0 NA 1 ...
$ motherWork : int 1 1 1 1 1 1 1 0 1 1 ...
$ fatherHS : int NA 1 1 1 1 1 NA 1 0 0 ...
$ fatherBachelors : int NA 0 NA 0 0 0 NA 0 NA 0 ...
$ fatherWork : int 1 1 1 1 0 1 NA 1 1 1 ...
$ selfBornUS : int 1 1 1 1 1 1 0 1 1 1 ...
$ motherBornUS : int 0 1 1 1 1 1 1 1 1 1 ...
$ fatherBornUS : int 0 1 1 1 0 1 NA 1 1 1 ...
$ englishAtHome : int 0 1 1 1 1 1 1 1 1 1 ...
$ computerForSchoolwork: int 1 1 1 1 1 1 1 1 1 1 ...
$ read30MinsADay : int 0 1 0 1 1 0 0 1 0 0 ...
$ minutesPerWeekEnglish: int 225 450 250 200 250 300 250 300 378 294 ...
$ studentsInEnglish : int NA 25 28 23 35 20 28 30 20 24 ...
$ schoolHasLibrary : int 1 1 1 1 1 1 1 1 0 1 ...
$ publicSchool : int 1 1 1 1 1 1 1 1 1 1 ...
$ urban : int 1 0 0 1 1 0 1 0 1 0 ...
$ schoolSize : int 673 1173 1233 2640 1095 227 2080 1913 502 899 ...
$ readingScore : num 476 575 555 458 614 ...Using tapply() on pisaTrain, what is the average reading test score of males?
tapply(pisaTrain$readingScore, pisaTrain$male, mean) 0 1
512.9406 483.5325 Which variables are missing data in at least one observation in the training set? Select all that apply.
summary(pisaTrain) grade male raceeth preschool expectBachelors
Min. : 8.00 Min. :0.0000 White :2015 Min. :0.0000 Min. :0.0000
1st Qu.:10.00 1st Qu.:0.0000 Hispanic : 834 1st Qu.:0.0000 1st Qu.:1.0000
Median :10.00 Median :1.0000 Black : 444 Median :1.0000 Median :1.0000
Mean :10.09 Mean :0.5111 Asian : 143 Mean :0.7228 Mean :0.7859
3rd Qu.:10.00 3rd Qu.:1.0000 More than one race: 124 3rd Qu.:1.0000 3rd Qu.:1.0000
Max. :12.00 Max. :1.0000 (Other) : 68 Max. :1.0000 Max. :1.0000
NA's : 35 NA's :56 NA's :62
motherHS motherBachelors motherWork fatherHS fatherBachelors
Min. :0.00 Min. :0.0000 Min. :0.0000 Min. :0.0000 Min. :0.0000
1st Qu.:1.00 1st Qu.:0.0000 1st Qu.:0.0000 1st Qu.:1.0000 1st Qu.:0.0000
Median :1.00 Median :0.0000 Median :1.0000 Median :1.0000 Median :0.0000
Mean :0.88 Mean :0.3481 Mean :0.7345 Mean :0.8593 Mean :0.3319
3rd Qu.:1.00 3rd Qu.:1.0000 3rd Qu.:1.0000 3rd Qu.:1.0000 3rd Qu.:1.0000
Max. :1.00 Max. :1.0000 Max. :1.0000 Max. :1.0000 Max. :1.0000
NA's :97 NA's :397 NA's :93 NA's :245 NA's :569
fatherWork selfBornUS motherBornUS fatherBornUS englishAtHome
Min. :0.0000 Min. :0.0000 Min. :0.0000 Min. :0.0000 Min. :0.0000
1st Qu.:1.0000 1st Qu.:1.0000 1st Qu.:1.0000 1st Qu.:1.0000 1st Qu.:1.0000
Median :1.0000 Median :1.0000 Median :1.0000 Median :1.0000 Median :1.0000
Mean :0.8531 Mean :0.9313 Mean :0.7725 Mean :0.7668 Mean :0.8717
3rd Qu.:1.0000 3rd Qu.:1.0000 3rd Qu.:1.0000 3rd Qu.:1.0000 3rd Qu.:1.0000
Max. :1.0000 Max. :1.0000 Max. :1.0000 Max. :1.0000 Max. :1.0000
NA's :233 NA's :69 NA's :71 NA's :113 NA's :71
computerForSchoolwork read30MinsADay minutesPerWeekEnglish studentsInEnglish
Min. :0.0000 Min. :0.0000 Min. : 0.0 Min. : 1.0
1st Qu.:1.0000 1st Qu.:0.0000 1st Qu.: 225.0 1st Qu.:20.0
Median :1.0000 Median :0.0000 Median : 250.0 Median :25.0
Mean :0.8994 Mean :0.2899 Mean : 266.2 Mean :24.5
3rd Qu.:1.0000 3rd Qu.:1.0000 3rd Qu.: 300.0 3rd Qu.:30.0
Max. :1.0000 Max. :1.0000 Max. :2400.0 Max. :75.0
NA's :65 NA's :34 NA's :186 NA's :249
schoolHasLibrary publicSchool urban schoolSize readingScore
Min. :0.0000 Min. :0.0000 Min. :0.0000 Min. : 100 Min. :168.6
1st Qu.:1.0000 1st Qu.:1.0000 1st Qu.:0.0000 1st Qu.: 712 1st Qu.:431.7
Median :1.0000 Median :1.0000 Median :0.0000 Median :1212 Median :499.7
Mean :0.9676 Mean :0.9339 Mean :0.3849 Mean :1369 Mean :497.9
3rd Qu.:1.0000 3rd Qu.:1.0000 3rd Qu.:1.0000 3rd Qu.:1900 3rd Qu.:566.2
Max. :1.0000 Max. :1.0000 Max. :1.0000 Max. :6694 Max. :746.0
NA's :143 NA's :162 Linear regression discards observations with missing data, so we will remove all such observations from the training and testing sets. Later in the course, we will learn about imputation, which deals with missing data by filling in missing values with plausible information.
Type the following commands into your R console to remove observations with any missing value from pisaTrain and pisaTest: pisaTrain = na.omit(pisaTrain) pisaTest = na.omit(pisaTest) How many observations are now in the training set?
pisaTrain = na.omit(pisaTrain)
pisaTest = na.omit(pisaTest)Factor variables are variables that take on a discrete set of values, like the “Region” variable in the WHO dataset from the second lecture of Unit 1. This is an unordered factor because there isn’t any natural ordering between the levels. An ordered factor has a natural ordering between the levels (an example would be the classifications “large,” “medium,” and “small”).
Which of the following variables is an unordered factor with at least 3 levels? (Select all that apply.)
str(pisaTrain)'data.frame': 2414 obs. of 24 variables:
$ grade : int 11 10 10 10 10 10 10 10 11 9 ...
$ male : int 1 0 1 0 1 0 0 0 1 1 ...
$ raceeth : Factor w/ 7 levels "American Indian/Alaska Native",..: 7 3 4 7 5 4 7 4 7 7 ...
$ preschool : int 0 1 1 1 1 1 1 1 1 1 ...
$ expectBachelors : int 0 1 0 1 1 1 1 0 1 1 ...
$ motherHS : int 1 0 1 1 1 1 1 0 1 1 ...
$ motherBachelors : int 1 0 0 0 1 0 0 0 0 1 ...
$ motherWork : int 1 1 1 0 1 1 1 0 0 1 ...
$ fatherHS : int 1 1 1 1 0 1 1 0 1 1 ...
$ fatherBachelors : int 0 0 0 0 0 0 1 0 1 1 ...
$ fatherWork : int 1 1 0 1 1 0 1 1 1 1 ...
$ selfBornUS : int 1 1 1 1 1 0 1 0 1 1 ...
$ motherBornUS : int 1 1 1 1 1 0 1 0 1 1 ...
$ fatherBornUS : int 1 1 0 1 1 0 1 0 1 1 ...
$ englishAtHome : int 1 1 1 1 1 0 1 0 1 1 ...
$ computerForSchoolwork: int 1 1 1 1 1 0 1 1 1 1 ...
$ read30MinsADay : int 1 1 1 1 0 1 1 1 0 0 ...
$ minutesPerWeekEnglish: int 450 200 250 300 294 232 225 270 275 225 ...
$ studentsInEnglish : int 25 23 35 30 24 14 20 25 30 15 ...
$ schoolHasLibrary : int 1 1 1 1 1 1 1 1 1 1 ...
$ publicSchool : int 1 1 1 1 1 1 1 1 1 0 ...
$ urban : int 0 1 1 0 0 0 0 1 1 1 ...
$ schoolSize : int 1173 2640 1095 1913 899 1733 149 1400 1988 915 ...
$ readingScore : num 575 458 614 439 466 ...
- attr(*, "na.action")=Class 'omit' Named int [1:1249] 1 3 6 7 9 11 13 21 29 30 ...
.. ..- attr(*, "names")= chr [1:1249] "1" "3" "6" "7" ...To include unordered factors in a linear regression model, we define one level as the “reference level” and add a binary variable for each of the remaining levels. In this way, a factor with n levels is replaced by n-1 binary variables. The reference level is typically selected to be the most frequently occurring level in the dataset.
As an example, consider the unordered factor variable “color”, with levels “red”, “green”, and “blue”. If “green” were the reference level, then we would add binary variables “colorred” and “colorblue” to a linear regression problem. All red examples would have colorred=1 and colorblue=0. All blue examples would have colorred=0 and colorblue=1. All green examples would have colorred=0 and colorblue=0.
Now, consider the variable “raceeth” in our problem, which has levels “American Indian/Alaska Native”, “Asian”, “Black”, “Hispanic”, “More than one race”, “Native Hawaiian/Other Pacific Islander”, and “White”. Because it is the most common in our population, we will select White as the reference level.
Which binary variables will be included in the regression model? (Select all that apply.)
##We create a binary variable for each level except the reference level, so we would create all these variables except for raceethWhite.Because the race variable takes on text values, it was loaded as a factor variable when we read in the dataset with read.csv() – you can see this when you run str(pisaTrain) or str(pisaTest). However, by default R selects the first level alphabetically (“American Indian/Alaska Native”) as the reference level of our factor instead of the most common level (“White”). Set the reference level of the factor by typing the following two lines in your R console: pisaTrain\(raceeth = relevel(pisaTrain\)raceeth, “White”) pisaTest\(raceeth = relevel(pisaTest\)raceeth, “White”) Now, build a linear regression model (call it lmScore) using the training set to predict readingScore using all the remaining variables.
It would be time-consuming to type all the variables, but R provides the shorthand notation “readingScore ~ .” to mean “predict readingScore using all the other variables in the data frame.” The period is used to replace listing out all of the independent variables. As an example, if your dependent variable is called “Y”, your independent variables are called “X1”, “X2”, and “X3”, and your training data set is called “Train”, instead of the regular notation:
LinReg = lm(Y ~ X1 + X2 + X3, data = Train) You would use the following command to build your model: LinReg = lm(Y ~ ., data = Train) What is the Multiple R-squared value of lmScore on the training set?
lmScore = lm(readingScore~., data=pisaTrain)
summary(lmScore)
Call:
lm(formula = readingScore ~ ., data = pisaTrain)
Residuals:
Min 1Q Median 3Q Max
-247.44 -48.86 1.86 49.77 217.18
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 76.489006 37.302678 2.050 0.040425 *
grade 29.542707 2.937399 10.057 < 2e-16 ***
male -14.521653 3.155926 -4.601 4.42e-06 ***
raceethAsian 63.167002 18.972648 3.329 0.000884 ***
raceethBlack 0.264980 17.369507 0.015 0.987830
raceethHispanic 28.301842 17.258860 1.640 0.101169
raceethMore than one race 50.354805 18.570123 2.712 0.006744 **
raceethNative Hawaiian/Other Pacific Islander 62.175726 23.782766 2.614 0.008997 **
raceethWhite 67.277327 16.786935 4.008 6.32e-05 ***
preschool -4.463670 3.486055 -1.280 0.200516
expectBachelors 55.267080 4.293893 12.871 < 2e-16 ***
motherHS 6.058774 6.091423 0.995 0.320012
motherBachelors 12.638068 3.861457 3.273 0.001080 **
motherWork -2.809101 3.521827 -0.798 0.425167
fatherHS 4.018214 5.579269 0.720 0.471470
fatherBachelors 16.929755 3.995253 4.237 2.35e-05 ***
fatherWork 5.842798 4.395978 1.329 0.183934
selfBornUS -3.806278 7.323718 -0.520 0.603307
motherBornUS -8.798153 6.587621 -1.336 0.181821
fatherBornUS 4.306994 6.263875 0.688 0.491776
englishAtHome 8.035685 6.859492 1.171 0.241527
computerForSchoolwork 22.500232 5.702562 3.946 8.19e-05 ***
read30MinsADay 34.871924 3.408447 10.231 < 2e-16 ***
minutesPerWeekEnglish 0.012788 0.010712 1.194 0.232644
studentsInEnglish -0.286631 0.227819 -1.258 0.208460
schoolHasLibrary 12.215085 9.264884 1.318 0.187487
publicSchool -16.857475 6.725614 -2.506 0.012261 *
urban -0.110132 3.962724 -0.028 0.977830
schoolSize 0.006540 0.002197 2.977 0.002942 **
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 73.81 on 2385 degrees of freedom
Multiple R-squared: 0.3251, Adjusted R-squared: 0.3172
F-statistic: 41.04 on 28 and 2385 DF, p-value: < 2.2e-16What is the training-set root-mean squared error (RMSE) of lmScore?
SSE = sum(lmScore$residuals^2)
RMSE = sqrt(SSE / nrow(pisaTrain))
RMSE[1] 73.36555Consider two students A and B. They have all variable values the same, except that student A is in grade 11 and student B is in grade 9. What is the predicted reading score of student A minus the predicted reading score of student B?
29.542707*2[1] 59.08541What is the meaning of the coefficient associated with variable raceethAsian?
##Predicted difference in the reading score between an Asian student and a white student who is otherwise identicalBased on the significance codes, which variables are candidates for removal from the model? Select all that apply. (We’ll assume that the factor variable raceeth should only be removed if none of its levels are significant.)
summary(lmScore)
Call:
lm(formula = readingScore ~ ., data = pisaTrain)
Residuals:
Min 1Q Median 3Q Max
-247.44 -48.86 1.86 49.77 217.18
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 76.489006 37.302678 2.050 0.040425 *
grade 29.542707 2.937399 10.057 < 2e-16 ***
male -14.521653 3.155926 -4.601 4.42e-06 ***
raceethAsian 63.167002 18.972648 3.329 0.000884 ***
raceethBlack 0.264980 17.369507 0.015 0.987830
raceethHispanic 28.301842 17.258860 1.640 0.101169
raceethMore than one race 50.354805 18.570123 2.712 0.006744 **
raceethNative Hawaiian/Other Pacific Islander 62.175726 23.782766 2.614 0.008997 **
raceethWhite 67.277327 16.786935 4.008 6.32e-05 ***
preschool -4.463670 3.486055 -1.280 0.200516
expectBachelors 55.267080 4.293893 12.871 < 2e-16 ***
motherHS 6.058774 6.091423 0.995 0.320012
motherBachelors 12.638068 3.861457 3.273 0.001080 **
motherWork -2.809101 3.521827 -0.798 0.425167
fatherHS 4.018214 5.579269 0.720 0.471470
fatherBachelors 16.929755 3.995253 4.237 2.35e-05 ***
fatherWork 5.842798 4.395978 1.329 0.183934
selfBornUS -3.806278 7.323718 -0.520 0.603307
motherBornUS -8.798153 6.587621 -1.336 0.181821
fatherBornUS 4.306994 6.263875 0.688 0.491776
englishAtHome 8.035685 6.859492 1.171 0.241527
computerForSchoolwork 22.500232 5.702562 3.946 8.19e-05 ***
read30MinsADay 34.871924 3.408447 10.231 < 2e-16 ***
minutesPerWeekEnglish 0.012788 0.010712 1.194 0.232644
studentsInEnglish -0.286631 0.227819 -1.258 0.208460
schoolHasLibrary 12.215085 9.264884 1.318 0.187487
publicSchool -16.857475 6.725614 -2.506 0.012261 *
urban -0.110132 3.962724 -0.028 0.977830
schoolSize 0.006540 0.002197 2.977 0.002942 **
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 73.81 on 2385 degrees of freedom
Multiple R-squared: 0.3251, Adjusted R-squared: 0.3172
F-statistic: 41.04 on 28 and 2385 DF, p-value: < 2.2e-16Using the “predict” function and supplying the “newdata” argument, use the lmScore model to predict the reading scores of students in pisaTest. Call this vector of predictions “predTest”. Do not change the variables in the model (for example, do not remove variables that we found were not significant in the previous part of this problem). Use the summary function to describe the test set predictions.
What is the range between the maximum and minimum predicted reading score on the test set?
predTest = predict(lmScore, newdata=pisaTest)
summary((predTest)) Min. 1st Qu. Median Mean 3rd Qu. Max.
353.2 482.0 524.0 516.7 555.7 637.7 What is the sum of squared errors (SSE) of lmScore on the testing set?
sum((predTest-pisaTest$readingScore)^2)[1] 5762082sqrt(mean((predTest-pisaTest$readingScore)^2))[1] 76.29079What is the predicted test score used in the baseline model? Remember to compute this value using the training set and not the test set.
baseline = mean(pisaTrain$readingScore)
sum((baseline-pisaTest$readingScore)^2)[1] 7802354baseline[1] 517.9629What is the test-set R-squared value of lmScore?
1-5762082/7802354[1] 0.2614944