title: “Multiple linear regression”
Author: “Sufian”
output:
html_document:
css: ./lab.css
highlight: pygments
theme: ceruleanpdf_document: default
Rpub Link: http://rpubs.com/ssufian/553211
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, such as the physical appearance of the instructor. 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.
library(ggplot2)
rm(list = ls())
download.file("http://www.openintro.org/stat/data/evals.RData", destfile = "evals.RData")
load("evals.RData")
head(evals,n=3)## score rank ethnicity gender language age cls_perc_eval
## 1 4.7 tenure track minority female english 36 55.81395
## 2 4.1 tenure track minority female english 36 68.80000
## 3 3.9 tenure track minority female english 36 60.80000
## cls_did_eval cls_students cls_level cls_profs cls_credits bty_f1lower
## 1 24 43 upper single multi credit 5
## 2 86 125 upper single multi credit 5
## 3 76 125 upper single multi credit 5
## bty_f1upper bty_f2upper bty_m1lower bty_m1upper bty_m2upper bty_avg
## 1 7 6 2 4 6 5
## 2 7 6 2 4 6 5
## 3 7 6 2 4 6 5
## pic_outfit pic_color
## 1 not formal color
## 2 not formal color
## 3 not formal color| 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
- 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.
ans:
- This is an observational study. As such it cannot show causation, but only association. Its an already
in process “experiment” Therefore, A better question would be “Is there a relationship between beauty
and differences in course evaluations?”
- 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?
ans:
The score is left skewed; centered around 4.5
Most students give high marks to courses with fewer students giving low marks. This matches my
expectations. Based on my experience at The University of Texas, Austin (that’s right, I am a longhorn!)
most students rate their professors. Most give their professors high scores
ggplot(data = evals, aes(x = score))+
geom_histogram(color="darkblue", fill="lightblue",bins = 30)
- Excluding
score, select two other variables and describe their relationship using an appropriate visualization (scatterplot, side-by-side boxplots, or mosaic plot).
ans:
The beauty score seems to go down as age increases. Yikes! Aging is not kind
g <- ggplot(data = evals, aes(x=age , y = bty_avg)) + geom_point() +geom_smooth(method=lm)
g
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)
g1 <- ggplot(data = evals, aes(x=bty_avg , y = score)) + geom_point() +geom_smooth(method=lm)
g1Before 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?
- Replot the scatterplot, but this time use the function
jitter()on the \(y\)- or the \(x\)-coordinate. (Use?jitterto learn more.) What was misleading about the initial scatterplot?
plot(jitter(evals$score) ~ evals$bty_avg)
- Let’s see if the apparent trend in the plot is something more than natural variation. Fit a linear model called
m_btyto predict average professor score by average beauty rating and add the line to your plot usingabline(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?
test <- lm(evals$score ~ evals$bty_avg)
plot(jitter(evals$score) ~ evals$bty_avg,xlab = "Beauty Avg", ylab = "Scores")
abline(test)
summary(test)##
## 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$bty_avg 0.06664 0.01629 4.09 5.08e-05 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.5348 on 461 degrees of freedom
## Multiple R-squared: 0.03502, Adjusted R-squared: 0.03293
## F-statistic: 16.73 on 1 and 461 DF, p-value: 5.083e-05ans:
Linear model equation:
score =3.88+0.067×avg beauty rating
- The equation states that for every point in the average beauty rating, the score increases by 0.067.
The p−value from the summary above is essentially zero, which means the slope is not due to chance and
is statistically significant.
- Use residual plots to evaluate whether the conditions of least squares regression are reasonable. Provide plots and comments for each one (see the Simple Regression Lab for a reminder of how to make these).
plot(test$residuals ~ evals$bty_avg,xlab = "residuals", ylab = "Beauty Avg")
abline(h = 0, lty = 3)
ans:
No apparent pattern around zero. This tells me we have no heteordascticity.
hist(test$residuals)
ans:
The residuals appears to be left-skewed
qqnorm(test$residuals)
qqline(test$residuals)
ans:
The Q-Q plot shows non-linearity especially towards the tails
Summary of the residuals:
- after going thru the diagnostics, the residuals are not as nearly normal as I had hoped for
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)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- 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)
hist(m_bty_gen$residuals)
ans:
Again, Q-Q plot visually showed that the edges are deviating from normality. And the histogram is left
skewed. So I would say its not normal
- Is
bty_avgstill a significant predictor ofscore? Has the addition ofgenderto the model changed the parameter estimate forbty_avg?
ans:
- bty_avg is still a significant predictor as the t-stat value is greater than 0.05
But the issue of non-normality still exist from the previous problem. The normality of residuals
did not improve. So while the t-stat value or p-value showed significance, the non-normal residuals
violated one of the key fundamential assumptions of using linear regression!
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.
\[ \begin{aligned} \widehat{score} &= \hat{\beta}_0 + \hat{\beta}_1 \times bty\_avg + \hat{\beta}_2 \times (0) \\ &= \hat{\beta}_0 + \hat{\beta}_1 \times bty\_avg\end{aligned} \]
We can plot this line and the line corresponding to males with the following custom function.
multiLines(m_bty_gen)- 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?
ans:
score = b0 + b1xbty_avg + b2
The decision to call the indicator variable gendermale instead ofgenderfemale has no deeper meaning. R simply codes the category that comes first alphabetically as a \(0\). (You can change the reference level of a categorical variable, which is the level that is coded as a 0, using therelevel function. Use ?relevel to learn more.)
- Create a new model called
m_bty_rankwithgenderremoved andrankadded 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.
ans: R splits the rank variable with three levels into two dichotomous variables
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-05The interpretation of the coefficients in multiple regression is slightly different from that of simple regression. The estimate for bty_avg reflects how much higher a 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 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.
- 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.
ans: cls_profssingle, see run below
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)- Check your suspicions from the previous exercise. Include the model output in your response.
ans: cls_profssingle, see run above
- Interpret the coefficient associated with the ethnicity variable.
ans: slope is not significant, the t-stat value is low < 0.05 while p-value is high
- 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?
m_no_profs <- 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_no_profs)##
## 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-14ans: The coefficients and significance levels changed for all variables with the removel of cls_profs
- It changed due to multi-colinearity issues. When you have too many predictor variables, their
interactions between each other due to co-liniarity causes distortions
- 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_best <- lm(score ~ ethnicity + gender + language + age + cls_perc_eval
+ cls_credits + bty_avg + pic_color, data = evals)
summary(m_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-15ans:
score = 3.77+0.17ethnicity+0.21gender-0.21language-0,01age+0.005cls_perc_eval+0.51cls_credits
- 0.05bty_avg-0.19pic_color
- Verify that the conditions for this model are reasonable using diagnostic plots.
qqnorm(m_best$residuals)
qqline(m_best$residuals)
hist(m_best$residuals)
ans: Again, the issue of non-normality exist. The residuals does look skewed
- 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?
ans:
It should not influence other conditions. Maybe independence?
- 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.
ans:
Based on the model, the highest score will be associated with professors who has the following:
is not part of a minority group,
is male,
was taught in an English speaking school,
younger ,
has a high percentage of students filling out evaluations,
has a high beauty ranking,
has a black & white picture.
- Would you be comfortable generalizing your conclusions to apply to professors generally (at any university)? Why or why not?
ans:
NO, I would not draw any genarlizaing conclusions based on this study. As much as I admire my alma
mater:
First, This study is too small of sample size for one
Second, the linear model violates the fundamental assumption that residuals MUST be normal and we see
time and again with more or less predictor variables, it is not normal and left-skewed. So I would not
use this linear model to generalize but I think if we use other non-linear model or made some data
transformations, then I would be more comfortable using it as a test model first; while gathering more
data to increase sample size.