R-Lab 10 Multiple Regression
Jaime F. Alliende Acuña
12/2/2019
Multiple Linear Regression
Grading the Professor
The article titled, “Beauty in the classroom: instructors’ pulchritude and putative pedagogical productivity” (Hamermesh and Parker, 2005) found that instructors who are viewed to be better looking receive higher instructional ratings. (Daniel S. Hamermesh, Amy Parker, Beauty in the classroom: instructors pulchritude and putative pedagogical productivity, Economics of Education Review, Volume 24, Issue 4, August 2005, Pages 369-376, ISSN 0272-7757, 10.1016/j.econedurev.2004.07.013. http://www.sciencedirect.com/science/article/pii/S0272775704001165.)
In this lab we will analyze the data from this study in order to learn what goes into a positive professor evaluation.
The Data
The data were gathered from end of semester student evaluations for a large sample of professors from the University of Texas at Austin. In addition, six students rated the professors’ physical appearance. (This is aslightly modified version of the original data set that was released as part of the replication data for Data Analysis Using Regression and Multilevel/Hierarchical Models (Gelman and Hill, 2007).) The result is a data frame where each row contains a different course and columns represent variables about the courses and professors.
download.file("http://www.openintro.org/stat/data/evals.RData", destfile = "evals.RData")
load("evals.RData")Exploring the Data
Exercise 1:
Is this an observational study or an experiment? The original research question posed in the paper is whether beauty leads directly to the differences in course evaluations. Given the study design, is it possible to answer this question as it is phrased? If not, rephrase the question.
Answer:
This is an observational study since the data collected did not had a control group. Given the study design, it is not possible to answer the question as it is phrased. The question should be rephrase to whether there is an statisitical relationship between beaty and differences in course evaluations.
Exercise 2:
Describe the distribution of score. Is the distribution skewed? What does that tell you about how students rate courses? Is this what you expected to see? Why, or why not?
Answer:
hist(evals$score)
The distribution of score is skewed to the left. Most students tend to rate their courses in a strongly positive manner. This is expected, because the evaluations were at the end of the semester in which most students are doing well enough not to drop the class. Therefore, there is a positive reinforcement between students scoring well their courses, because they are doing well in their courses.
Exercise 3:
Excluding score, select two other variables and describe their relationship using an appropriate visualization (scatterplot, side-by-side boxplots, or mosaic plot).
Answer:
plot(evals$cls_perc_eval ~ evals$gender)
The percent of students in class who completed evaluation are very similar between female and male professors.Therefore, the gender of the professor appears to have no impact in the variability of the percent of students in class who completed evaluation.
Simple Linear Regression
The fundamental phenomenon suggested by the study is that better looking teachers are evaluated more favorably. Let’s create a scatterplot to see if this appears to be the case:
plot(evals$score ~ evals$bty_avg)
Before we draw conclusions about the trend, compare the number of observations in the data frame with the approximate number of points on the scatterplot. Is anything awry?
nrow(evals)## [1] 463summary(evals)## score rank ethnicity gender
## Min. :2.300 teaching :102 minority : 64 female:195
## 1st Qu.:3.800 tenure track:108 not minority:399 male :268
## Median :4.300 tenured :253
## Mean :4.175
## 3rd Qu.:4.600
## Max. :5.000
## language age cls_perc_eval cls_did_eval
## english :435 Min. :29.00 Min. : 10.42 Min. : 5.00
## non-english: 28 1st Qu.:42.00 1st Qu.: 62.70 1st Qu.: 15.00
## Median :48.00 Median : 76.92 Median : 23.00
## Mean :48.37 Mean : 74.43 Mean : 36.62
## 3rd Qu.:57.00 3rd Qu.: 87.25 3rd Qu.: 40.00
## Max. :73.00 Max. :100.00 Max. :380.00
## cls_students cls_level cls_profs cls_credits
## Min. : 8.00 lower:157 multiple:306 multi credit:436
## 1st Qu.: 19.00 upper:306 single :157 one credit : 27
## Median : 29.00
## Mean : 55.18
## 3rd Qu.: 60.00
## Max. :581.00
## bty_f1lower bty_f1upper bty_f2upper bty_m1lower
## Min. :1.000 Min. :1.000 Min. : 1.000 Min. :1.000
## 1st Qu.:2.000 1st Qu.:4.000 1st Qu.: 4.000 1st Qu.:2.000
## Median :4.000 Median :5.000 Median : 5.000 Median :3.000
## Mean :3.963 Mean :5.019 Mean : 5.214 Mean :3.413
## 3rd Qu.:5.000 3rd Qu.:7.000 3rd Qu.: 6.000 3rd Qu.:5.000
## Max. :8.000 Max. :9.000 Max. :10.000 Max. :7.000
## bty_m1upper bty_m2upper bty_avg pic_outfit
## Min. :1.000 Min. :1.000 Min. :1.667 formal : 77
## 1st Qu.:3.000 1st Qu.:4.000 1st Qu.:3.167 not formal:386
## Median :4.000 Median :5.000 Median :4.333
## Mean :4.147 Mean :4.752 Mean :4.418
## 3rd Qu.:5.000 3rd Qu.:6.000 3rd Qu.:5.500
## Max. :9.000 Max. :9.000 Max. :8.167
## pic_color
## black&white: 78
## color :385
##
##
##
## The number of observations in the data frame is 463, but the approximate number of points on the scatterplot appears to be much lower than 463.
Exercise 4:
Replot the scatterplot, but this time use the function jitter() on the y- or the x-coordinate. (Use ?jitter to learn more.) What was misleading about the initial scatterplot?
Answer:
plot(jitter(evals$score) ~ evals$bty_avg)
By using the jitter funtion we are adding a small amount of noise on the values, which helps differentiate repeated values that did not appeared on the initial scatterplot.
Exercise 5:
Let’s see if the apparent trend in the plot is something more than natural variation. Fit a linear model called m_bty to predict average professor score by average beauty rating and add the line to your plot using abline(m_bty). Write out the equation for the linear model and interpret the slope. Is average beauty score a statistically significant predictor? Does it appear to be a practically significant predictor?
Answer:
m_bty<- lm(score ~ bty_avg, data = evals)
plot(jitter(evals$score) ~ evals$bty_avg)
abline(m_bty)
cor(evals$score, evals$bty_avg)## [1] 0.1871424summary(m_bty)##
## Call:
## lm(formula = score ~ bty_avg, data = evals)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.9246 -0.3690 0.1420 0.3977 0.9309
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 3.88034 0.07614 50.96 < 2e-16 ***
## bty_avg 0.06664 0.01629 4.09 5.08e-05 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.5348 on 461 degrees of freedom
## Multiple R-squared: 0.03502, Adjusted R-squared: 0.03293
## F-statistic: 16.73 on 1 and 461 DF, p-value: 5.083e-05y = 3.88034 + 0.06664x
The interpretation of the slope is: all else held constant, every 1 additional average beauty rating of the professor will tend to increase the average professor´s evaluation score by 0.06664.
The average beauty score is a statistically significant predictor with a very small p-value of 5.08e-05, but it is not a practically significant predictor. Even if a professor gets a perfect average beauty score of 10, it would only increment the average professor´s evaluation score by 0.6664 out of 5.
Exercise 6:
Use residual plots to evaluate whether the conditions of least squares regression are reasonable. Provide plots and comments for each one (see the Simple Regression Lab for a reminder of how to make these).
Answer:
plot(m_bty$residuals ~ evals$bty_avg)
abline(h = 0, lty = 4) 
hist(m_bty$residuals)
qqnorm(m_bty$residuals)
qqline(m_bty$residuals)
Linearity:
From the residual plots we can observe that the plots follow a fairly random pattern which would confirm linearity.
Nearly Normal Residuals:
From the histogram and the normal probability plot it can be confirm that the residuals do not follow a normal distribution, therefore the condition is not met.
Constant Variability:
From the residual plots, it can be observed that there is no constant variability.
Multiple Linear Regression
The data set contains several variables on the beauty score of the professor: individual ratings from each of the six students who were asked to score the physical appearance of the professors and the average of these six scores. Let’s take a look at the relationship between one of these scores and the average beauty score.
plot(evals$bty_avg ~ evals$bty_f1lower)
cor(evals$bty_avg, evals$bty_f1lower)## [1] 0.8439112As expected the relationship is quite strong - after all, the average score is calculated using the individual scores. We can actually take a look at the relationships between all beauty variables (columns 13 through 19) using the following command:
plot(evals[,13:19])
These variables are collinear (correlated), and adding more than one of these variables to the model would not add much value to the model. In this application and with these highly-correlated predictors, it is reasonable to use the average beauty score as the single representative of these variables.
In order to see if beauty is still a significant predictor of professor score after we’ve accounted for the gender of the professor, we can add the gender term into the model.
m_bty_gen <- lm(score ~ bty_avg + gender, data = evals)
summary(m_bty_gen)##
## Call:
## lm(formula = score ~ bty_avg + gender, data = evals)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.8305 -0.3625 0.1055 0.4213 0.9314
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 3.74734 0.08466 44.266 < 2e-16 ***
## bty_avg 0.07416 0.01625 4.563 6.48e-06 ***
## gendermale 0.17239 0.05022 3.433 0.000652 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.5287 on 460 degrees of freedom
## Multiple R-squared: 0.05912, Adjusted R-squared: 0.05503
## F-statistic: 14.45 on 2 and 460 DF, p-value: 8.177e-07Exercise 7:
P-values and parameter estimates should only be trusted if the conditions for the regression are reasonable. Verify that the conditions for this model are reasonable using diagnostic plots.
Answer:
qqnorm(m_bty_gen$residuals)
qqline(m_bty_gen$residuals)
plot(m_bty_gen$residuals ~ evals$bty_avg)
abline(h = 0, lty = 4) 
plot(m_bty_gen$residuals ~ evals$gender)
abline(h = 0, lty = 4)
plot(m_bty_gen)
hist(m_bty_gen$residuals)
plot(jitter(evals$score) ~ evals$bty_avg)
plot(evals$score ~ evals$gender)
Looking at the residual plots we can be confident that the linearity condition is fairly met, but the normal probability plots and the histogram show that the condition for nearly normal residuals is not met. The condition for constant variability appers also to be violated.
Exercise 8:
Is bty_avg still a significant predictor of score? Has the addition of gender to the model changed the parameter estimate for bty_avg?
Answer:
bty_avg is still a significant predictor of score with a small p-value of 6.48e-06.
The addition of gender to the model changed the parameter estimate for bty_avg from 0.06664 to 0.07416.
Note that the estimate for gender is now called gendermale. You’ll see this name change whenever you introduce a categorical variable. The reason is that R recodes gender from having the values of female and male to being an indicator variable called gendermale that takes a value of 0 for females and a value of 1 for males. (Such variables are often referred to as “dummy” variables.)
As a result, for females, the parameter estimate is multiplied by zero, leaving the intercept and slope form familiar from simple regression.
scoreˆ=β̂ 0+β̂ 1×bty_avg+β̂ 2×(0)=β̂ 0+β̂ 1×bty_avg
We can plot this line and the line corresponding to males with the following custom function.
multiLines(m_bty_gen)
Exercise 9:
What is the equation of the line corresponding to males? (Hint: For males, the parameter estimate is multiplied by 1.) For two professors who received the same beauty rating, which gender tends to have the higher course evaluation score?
Answer:
score = 3.74734 + 0.07416(bty_avg) + 0.17239(gendermale)
gendermale has a value of 1 if the professor is male, therefore:
score = 3.74734 + 0.07416(bty_avg) + 0.17239(1)
All else held constant, the male professor will have an 0.17239 higher score than a female counterpart.
Exercise 10:
Create a new model called m_bty_rank with gender removed and rank added in. How does R appear to handle categorical variables that have more than two levels? Note that the rank variable has three levels: teaching, tenure track, tenured.
Answer:
m_bty_rank <- lm(score ~ bty_avg + rank, data = evals)
summary(m_bty_rank)##
## Call:
## lm(formula = score ~ bty_avg + rank, data = evals)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.8713 -0.3642 0.1489 0.4103 0.9525
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 3.98155 0.09078 43.860 < 2e-16 ***
## bty_avg 0.06783 0.01655 4.098 4.92e-05 ***
## ranktenure track -0.16070 0.07395 -2.173 0.0303 *
## ranktenured -0.12623 0.06266 -2.014 0.0445 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.5328 on 459 degrees of freedom
## Multiple R-squared: 0.04652, Adjusted R-squared: 0.04029
## F-statistic: 7.465 on 3 and 459 DF, p-value: 6.88e-05R will include the different categorical variables as different independent variables in the model, while excluding one dummy category.
The Search for the Best Model
We will start with a full model that predicts professor score based on rank, ethnicity, gender, language of the university where they got their degree, age, proportion of students that filled out evaluations, class size, course level, number of professors, number of credits, average beauty rating, outfit, and picture color.
Exercise 11:
Which variable would you expect to have the highest p-value in this model? Why? Hint: Think about which variable would you expect to not have any association with the professor score.
Answer:
The variable for number of professors teaching sections in course in sample single or multiple should have the highest p-value, because in relation to the other varibales that might have a direct or indirect impact on the students´evaluation, whether the professor has multiple sections or not would be the last variable in consideration when evaluating their own class.
Let’s run the model…
m_full <- lm(score ~ rank + ethnicity + gender + language + age + cls_perc_eval
+ cls_students + cls_level + cls_profs + cls_credits + bty_avg
+ pic_outfit + pic_color, data = evals)
summary(m_full)##
## Call:
## lm(formula = score ~ rank + ethnicity + gender + language + age +
## cls_perc_eval + cls_students + cls_level + cls_profs + cls_credits +
## bty_avg + pic_outfit + pic_color, data = evals)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.77397 -0.32432 0.09067 0.35183 0.95036
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 4.0952141 0.2905277 14.096 < 2e-16 ***
## ranktenure track -0.1475932 0.0820671 -1.798 0.07278 .
## ranktenured -0.0973378 0.0663296 -1.467 0.14295
## ethnicitynot minority 0.1234929 0.0786273 1.571 0.11698
## gendermale 0.2109481 0.0518230 4.071 5.54e-05 ***
## languagenon-english -0.2298112 0.1113754 -2.063 0.03965 *
## age -0.0090072 0.0031359 -2.872 0.00427 **
## cls_perc_eval 0.0053272 0.0015393 3.461 0.00059 ***
## cls_students 0.0004546 0.0003774 1.205 0.22896
## cls_levelupper 0.0605140 0.0575617 1.051 0.29369
## cls_profssingle -0.0146619 0.0519885 -0.282 0.77806
## cls_creditsone credit 0.5020432 0.1159388 4.330 1.84e-05 ***
## bty_avg 0.0400333 0.0175064 2.287 0.02267 *
## pic_outfitnot formal -0.1126817 0.0738800 -1.525 0.12792
## pic_colorcolor -0.2172630 0.0715021 -3.039 0.00252 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.498 on 448 degrees of freedom
## Multiple R-squared: 0.1871, Adjusted R-squared: 0.1617
## F-statistic: 7.366 on 14 and 448 DF, p-value: 6.552e-14Exercise 12:
Check your suspicions from the previous exercise. Include the model output in your response.
Answer:
The variable cls_profssingle has indeed the highest p-value with 0.77806.
Exercise 13:
Interpret the coefficient associated with the ethnicity variable.
Answer:
All else held constant, a not minority professor will have an 0.1234929 higher score than a minority counterpart.
Exercise 14:
Drop the variable with the highest p-value and re-fit the model. Did the coefficients and significance of the other explanatory variables change? (One of the things that makes multiple regression interesting is that coefficient estimates depend on the other variables that are included in the model.) If not, what does this say about whether or not the dropped variable was collinear with the other explanatory variables?
Answer:
m_full_1 <- lm(score ~ rank + ethnicity + gender + language + age + cls_perc_eval + cls_students + cls_level + cls_credits + bty_avg
+ pic_outfit + pic_color, data = evals)
summary(m_full_1)##
## Call:
## lm(formula = score ~ rank + ethnicity + gender + language + age +
## cls_perc_eval + cls_students + cls_level + cls_credits +
## bty_avg + pic_outfit + pic_color, data = evals)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.7836 -0.3257 0.0859 0.3513 0.9551
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 4.0872523 0.2888562 14.150 < 2e-16 ***
## ranktenure track -0.1476746 0.0819824 -1.801 0.072327 .
## ranktenured -0.0973829 0.0662614 -1.470 0.142349
## ethnicitynot minority 0.1274458 0.0772887 1.649 0.099856 .
## gendermale 0.2101231 0.0516873 4.065 5.66e-05 ***
## languagenon-english -0.2282894 0.1111305 -2.054 0.040530 *
## age -0.0089992 0.0031326 -2.873 0.004262 **
## cls_perc_eval 0.0052888 0.0015317 3.453 0.000607 ***
## cls_students 0.0004687 0.0003737 1.254 0.210384
## cls_levelupper 0.0606374 0.0575010 1.055 0.292200
## cls_creditsone credit 0.5061196 0.1149163 4.404 1.33e-05 ***
## bty_avg 0.0398629 0.0174780 2.281 0.023032 *
## pic_outfitnot formal -0.1083227 0.0721711 -1.501 0.134080
## pic_colorcolor -0.2190527 0.0711469 -3.079 0.002205 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.4974 on 449 degrees of freedom
## Multiple R-squared: 0.187, Adjusted R-squared: 0.1634
## F-statistic: 7.943 on 13 and 449 DF, p-value: 2.336e-14m_full$coefficients - m_full_1$coefficients## Warning in m_full$coefficients - m_full_1$coefficients: longer object
## length is not a multiple of shorter object length## (Intercept) ranktenure track ranktenured
## 7.961761e-03 8.133220e-05 4.512112e-05
## ethnicitynot minority gendermale languagenon-english
## -3.952838e-03 8.249882e-04 -1.521743e-03
## age cls_perc_eval cls_students
## -8.003969e-06 3.847644e-05 -1.408227e-05
## cls_levelupper cls_profssingle cls_creditsone credit
## -1.234699e-04 -5.207815e-01 4.621803e-01
## bty_avg pic_outfitnot formal pic_colorcolor
## 1.483560e-01 1.063710e-01 -4.304515e+00There was a minimal change in the coefficients and significance of the other explanatory variables which means there was some small collinearity between the dropped variable and the rest of the explanatory variables.
Exercise 15:
Using backward-selection and p-value as the selection criterion, determine the best model. You do not need to show all steps in your answer, just the output for the final model. Also, write out the linear model for predicting score based on the final model you settle on.
Answer:
m_full_best <- lm(score ~ ethnicity + gender + language + age + cls_perc_eval
+ cls_credits + bty_avg
+ pic_color, data = evals)
summary(m_full_best)##
## Call:
## lm(formula = score ~ ethnicity + gender + language + age + cls_perc_eval +
## cls_credits + bty_avg + pic_color, data = evals)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.85320 -0.32394 0.09984 0.37930 0.93610
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 3.771922 0.232053 16.255 < 2e-16 ***
## ethnicitynot minority 0.167872 0.075275 2.230 0.02623 *
## gendermale 0.207112 0.050135 4.131 4.30e-05 ***
## languagenon-english -0.206178 0.103639 -1.989 0.04726 *
## age -0.006046 0.002612 -2.315 0.02108 *
## cls_perc_eval 0.004656 0.001435 3.244 0.00127 **
## cls_creditsone credit 0.505306 0.104119 4.853 1.67e-06 ***
## bty_avg 0.051069 0.016934 3.016 0.00271 **
## pic_colorcolor -0.190579 0.067351 -2.830 0.00487 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.4992 on 454 degrees of freedom
## Multiple R-squared: 0.1722, Adjusted R-squared: 0.1576
## F-statistic: 11.8 on 8 and 454 DF, p-value: 2.58e-15Score = 3.771922 + 0.167872(ethnicitynot minority) + 0.207112(gendermale) - 0.206178(languagenon-english) - 0.006046(age) + 0.004656(cls_perc_eval) + 0.505306(cls_creditsone credit) + 0.051069(bty_avg) - 0.190579(pic_colorcolor)
Exercise 16:
Verify that the conditions for this model are reasonable using diagnostic plots.
Answer:
qqnorm(m_full_best$residuals)
qqline(m_full_best$residuals)
hist(m_full_best$residuals)
plot(m_full_best)
plot(jitter(evals$score) ~ evals$bty_avg)
plot(jitter(evals$score) ~ evals$gender)
plot(jitter(evals$score) ~ evals$ethnicity)
plot(jitter(evals$score) ~ evals$gender)
plot(jitter(evals$score) ~ evals$language)
plot(jitter(evals$score) ~ evals$age)
plot(jitter(evals$score) ~ evals$cls_perc_eval)
plot(jitter(evals$score) ~ evals$cls_credits)
plot(jitter(evals$score) ~ evals$pic_color)
Neither the conditions for linearity, nearly normal residuals and constant variability appear to be met satisfactorily.
Exercise 17:
The original paper describes how these data were gathered by taking a sample of professors from the University of Texas at Austin and including all courses that they have taught. Considering that each row represents a course, could this new information have an impact on any of the conditions of linear regression?
Answer:
Yes, this will have a very important impact of the independence of the observations, because there will be a number of courses that will share the same professor, wich means that the independence between observations condition will be violated.
Exercise 18:
Based on your final model, describe the characteristics of a professor and course at University of Texas at Austin that would be associated with a high evaluation score.
Answer:
The professor will be from a not minority ethnicity, male, relatively young, received education from an english language school, with a black and white picture, with a high percent of students in class who completed evaluation, high average beauty rating, and from a one credit class.
Exercise 19:
Would you be comfortable generalizing your conclusions to apply to professors generally (at any university)? Why or why not?
Answer:
No, because the sample size is too small, the beauty standards change drastically from different universities, along with other standards from the rest of the explanatory variables could potentially change for differente universities.