Multiple Linear Regression

Grading the professor

Many college courses conclude by giving students the opportunity to evaluate the course and the instructor anonymously. However, the use of these student evaluations as an indicator of course quality and teaching effectiveness is often criticized because these measures may reflect the influence of non-teaching related characteristics of non-teaching related characteristics, such as the physical appearance of the instructor. The article titled, “Beauty in the classroom: instructors’ pulchritude and putative pedagological 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 pedagological productivity, Economics of Education Review, Volume 24, Issue 4, August 2005, Pages 369-376, ISSN 0272-77757, 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 we 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 a slightly modified version of the original data set that was released as part of the replication data for Data Analysis Using Regression and Multilevel/Hierachical 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")

variable description score average professor evaluation score: (1) very unsatisfactory - (5) excellent. rank rank of professor: teaching, tenure track, tenured. ethnicity ethnicity of professor: not minority, minority. gender gender of professor: female, male. language language of school where professor received education: english or non-english. age age of professor. cls_perc_eval percent of students in class who completed evaluation. cls_did_eval number of students in class who completed evaluation. cls_students total number of students in class. cls_level class level: lower, upper. cls_profs number of professors teaching sections in course in sample: single, multiple. cls_credits number of credits of class: one credit (lab, PE, etc.), multi credit. bty_f1lower beauty rating of professor from lower level female: (1) lowest - (10) highest. bty_f1upper beauty rating of professor from upper level female: (1) lowest - (10) highest. bty_f2upper beauty rating of professor from second upper level female: (1) lowest - (10) highest. bty_m1lower beauty rating of professor from lower level male: (1) lowest - (10) highest. bty_m1upper beauty rating of professor from upper level male: (1) lowest - (10) highest. bty_m2upper beauty rating of professor from second upper level male: (1) lowest - (10) highest. bty_avg average beauty rating of professor. pic_outfit outfit of professor in picture: not formal, formal. pic_color color of professor’s picture: color, black & white.

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 the course evaluations. Given the study design, is it possible to answer this question as it is phrased? If not, rephrase the question.

This would be an observational study. None of the professors were chosen randomly and there is no control group.I think it would be difficult to answer to answer this question because it seems a bit subjective. Also, because it is an observational study, we cannot say that the relationship between these variables is causal. Rather than asking whether beauty leads to difference in course evaluation I would maybe rephrase to ask whether or not there is a correlation between perceived attraction and course evaluations for professors at this university.

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?

hist(evals$score, main = "Score distribution")

This distribution is skewed to the left, with most students leaving positive reviews. I did not expect this, I thought the distribution would be more normal. I assumed most students would give professors an average review.

Excluding score, select two other variables and describe their relationship using an appropriate visualization (scatterplot, side-by-side boxplots, or mosaic plot).

boxplot(evals$bty_avg ~ evals$gender)

According to the boxplot above, there does not seem to be a big difference in the average beauty rating in males vs female professors. Males have slightly lower scores but the difference does not seem significant.

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 scatterpolot. Is anything awry?

It looks to me like there are not 463 points in the scatterplot.

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?

plot(jitter(evals$score) ~jitter(evals$bty_avg))

It was misleading because it looks like when points overlapped, they were graphed as one point in the initial scatterplot. This one is messier but it looks to be a more accurate representation of the data.

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 the average professor score by average beauty rating and at the line to your plot using abline(m_bty). Write out the equation for the linear model and interpret the slope. Is the average beauty score a statistically significant predictor?

m_bty <- lm(evals$score ~ evals$bty_avg)
plot(jitter(evals$score) ~ jitter(evals$bty_avg))
abline(m_bty)

summary(m_bty)
## 
## Call:
## lm(formula = evals$score ~ evals$bty_avg)
## 
## 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 ***
## evals$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-05

Equation for linear model: y-hat = 3.88034 + 0.06664*bty_avg It looks like bty_avg does seem to be able to predict the evaluation score since the p-value is close to 0. However, the 0.06664 beauty coefficient is very small which means that it doesn’t really have too much of an impact on the overall professor score. Even though it seemse statistically significant, I don’t think that it’s necessarily a good predictor.

Exercise 6

Use residual plots to evaluate whether the conditions of least squares regression are reasonable. Provide plots and comments for each one (see Simple Regression Lab for a reminder of how to make these).

plot(m_bty$residuals ~ evals$bty_avg)
abline(h = 0, lty = 3) # adds a horizontal dashed line at y = 0 

hist(m_bty$residuals)

qqnorm(m_bty$residuals)
qqline(m_bty$residuals) # adds diagonal line ot the normal prob plot

The residuals look to be skewed to the left and the residuals are not crowded around the line at 0. To me, it doesn’t look like the conditions have been met.

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] 0.8439112

As 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-07

Exercise 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.

qqnorm(m_bty_gen$residuals)
qqline(m_bty_gen$residuals)

plot(m_bty_gen$residuals)
abline(h = 0, lty = 3)

If we assume that students do not take multiple classes with the same professor, then the samples are indepenednt. The QQ-plot show that the points mostly fall on the normal line. The conditions are met.

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?

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-07

Yes, it looks like bty_avg is still a significant predictor of score. It looks like the slope increased a little bin in this model, but overall it looks like the it is still a significant predictor.

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 a 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?

The equation for male is: score-hat = b0-hat + b1-hat(bty_avg) + b2-hat

If two professors had the same beauty rating, the male professors have higher course evaluation scores.

The decision to call the indicator variable gendermale instead of genderfemale has no deeper meaning. R simply codes the category that comes first automatically as 0. (You can change the reference level of a categorical variable, which is the level that is coded as a 0, using the relevel function. Use ?relevel to learn more.)

Exercise 10

Create a new model called the 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.

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-05
multiLines(m_bty_rank)

Since it has 3 variables, R only takes two of the categories in the summary. In this case, teaching is left out of the summary.

The interpretation of the coefficients in multiple regression is slightly different from taht of simple regression. The estimate for bty_avg reflects how much a higher group of professors is expected to score if they have a beauty rating that is one point higher while holding all other variables constant. In this case, that translates into considering only professors of the same rank with bty_avg scores that are one point apart.

The search for the best model

We will start witha 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? Think about which variable you would expect to not have any association with the professor score.

My guess would be that the number of credits would have the highest p-value. I don’t really see how the number of credits in a class would affect the professor’s score whereas I could see the effects on other variables.

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-14

Exercise 12

Check your suspicions from the previous exercise. Include the model output in your response.

The model can be found above. I was incorrect in my assumption, the variable that has the highest p-value is actually cls_prof, which looks at the number of professors teaching a class. I did not guess this, because I thought that depending on how the classes are taught, those with multiple professors might affect the score.

Exercise 13

Interpret the coefficient associated with the ethnicity variable.

When all other variables are constant, we can expect to see a 0.1234929 increase in a professor’s score if they are not a minority.

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 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?

m_refit <- 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_refit)
## 
## 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-14

Yes there is a slight change in the coefficients when cols_prof (highest p-value) is fropped from the model.

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.

m_backward <- lm(score ~ ethnicity + gender + language + age + cls_perc_eval + 
    cls_credits + bty_avg + pic_color, data = evals)
summary(m_backward)
## 
## 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-15

score-hat = b0-hat + b1=hat(ethnicity) + b2-hat(gender) + b3-hat(language) + b4-hat(age) + b5-hat(cls_prc_eval) + b6-hat(cls_credits) + b7-hat(bty_avg) + b8-hat(pic_color)

Exercise 16

Verify that the conditions for this model are reasonable using diagnostic plots

plot(m_backward$residuals)
abline(h = 0, lty = 3)

qqnorm(m_backward$residuals)
qqline(m_backward$residuals)

If we assume that the samples are independent, it looks like the conditions are met. There does not seem to be a patter around the 0 line in the plot of residuals, and the QQ-plot has the plots mainly along the line. The points are a bit off on the ends, but I think overall it is fairly normal.

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?

I think that it could affect the linear regression. If professors are teaching more than one class and some students are taking multiple classes with a professor, this could mean that the samples taken are no longer independent, which would violate our assumption.

Exercise 18

Based on your final model, describe the characteristics of a professor and a course at University of Texas at Austin that would be associated with a high evaluation score.

A professor who is male, a non-minority, received an education in a school with English as a primary language, and has a high percentage of students complete evaluation, as well as teaches a 1 credit course and is not tenured. They would also have a high average beauty score.

Exercise 19

Would you be comfortable generalizing your conclusions to apply to professors generally (at any university)? Why or why not?

I would not feel comfortable generalizing because the sample was only from one university and the sample wasn’t really random. This study would have to include multiple universities in various regions of the country for me to be comfortable generalizing the conclusions to other any other university.