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, such as the physical appearance of the instructor. The article titled, “Beauty in the classroom: instructors’ pulchritude and putative pedagogical productivity” by Hamermesh and Parker found that instructors who are viewed to be better looking receive higher instructional ratings.
Here, you will analyze the data from this study in order to learn what goes into a positive professor evaluation.
Getting Started
Load packages
In this lab, you will explore and visualize the data using the tidyverse suite of packages. The data can be found in the companion package for OpenIntro resources, openintro.
Let’s load the packages.
##
## Welcome to CUNY DATA606 Statistics and Probability for Data Analytics
## This package is designed to support this course. The text book used
## is OpenIntro Statistics, 4th Edition. You can read this by typing
## vignette('os4') or visit www.OpenIntro.org.
##
## The getLabs() function will return a list of the labs available.
##
## The demo(package='DATA606') will list the demos that are available.This is the first time we’re using the GGally package.
You will be using the ggpairs function from this package
later in the lab.
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. The
result is a data frame where each row contains a different course and
columns represent variables about the courses and professors. It’s
called evals.
## Rows: 463
## Columns: 23
## $ course_id <int> 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 1…
## $ prof_id <int> 1, 1, 1, 1, 2, 2, 2, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5,…
## $ score <dbl> 4.7, 4.1, 3.9, 4.8, 4.6, 4.3, 2.8, 4.1, 3.4, 4.5, 3.8, 4…
## $ rank <fct> tenure track, tenure track, tenure track, tenure track, …
## $ ethnicity <fct> minority, minority, minority, minority, not minority, no…
## $ gender <fct> female, female, female, female, male, male, male, male, …
## $ language <fct> english, english, english, english, english, english, en…
## $ age <int> 36, 36, 36, 36, 59, 59, 59, 51, 51, 40, 40, 40, 40, 40, …
## $ cls_perc_eval <dbl> 55.81395, 68.80000, 60.80000, 62.60163, 85.00000, 87.500…
## $ cls_did_eval <int> 24, 86, 76, 77, 17, 35, 39, 55, 111, 40, 24, 24, 17, 14,…
## $ cls_students <int> 43, 125, 125, 123, 20, 40, 44, 55, 195, 46, 27, 25, 20, …
## $ cls_level <fct> upper, upper, upper, upper, upper, upper, upper, upper, …
## $ cls_profs <fct> single, single, single, single, multiple, multiple, mult…
## $ cls_credits <fct> multi credit, multi credit, multi credit, multi credit, …
## $ bty_f1lower <int> 5, 5, 5, 5, 4, 4, 4, 5, 5, 2, 2, 2, 2, 2, 2, 2, 2, 7, 7,…
## $ bty_f1upper <int> 7, 7, 7, 7, 4, 4, 4, 2, 2, 5, 5, 5, 5, 5, 5, 5, 5, 9, 9,…
## $ bty_f2upper <int> 6, 6, 6, 6, 2, 2, 2, 5, 5, 4, 4, 4, 4, 4, 4, 4, 4, 9, 9,…
## $ bty_m1lower <int> 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 7, 7,…
## $ bty_m1upper <int> 4, 4, 4, 4, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 6, 6,…
## $ bty_m2upper <int> 6, 6, 6, 6, 3, 3, 3, 3, 3, 2, 2, 2, 2, 2, 2, 2, 2, 6, 6,…
## $ bty_avg <dbl> 5.000, 5.000, 5.000, 5.000, 3.000, 3.000, 3.000, 3.333, …
## $ pic_outfit <fct> not formal, not formal, not formal, not formal, not form…
## $ pic_color <fct> color, color, color, color, color, color, color, color, …We have observations on 21 different variables, some categorical and some numerical. The meaning of each variable can be found by bringing up the help file:
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.
This is an observational study because they are recording peoples observations towards a professor, whereas there are no variables being controlled, so it is not an experimental
- 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?
The majority of professors received a 4.5 rating, where as very little to no professors received lower than a 3 which means a lot of the professors are great. This distribution is very skewed to the right, I did not expect to see this rating because from my undergrad experience, the majority of the professors weren’t that great
- Excluding
score, select two other variables and describe their relationship with each other using an appropriate visualization.
I chose the age variable and the beauty variable to see if there was a correlation between the age of the professor, and the beauty rating, I assumed that the higher the age, the lower the beauty rating, but to my surprise there seems to be very little correlation between age and beauty average
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:
Before you 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
geom_jitteras your layer. What was misleading about the initial scatterplot?
It seemed like a lot of the points were placed on top of each other so it made it hard to evaluate, with the jitter function, we’re able to see the points more clearly to determine any correlation
- 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. 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?
This linear regression line represents a positive relationship, although a professor may be low on the beauty scale,they still have a relatively high score, but as the beauty average increases, the score increases as well
m_bty <-lm(score ~ bty_avg, data=evals)
plot(jitter(evals$score) ~ evals$bty_avg)
## add linear regression line
abline(m_bty, col = "red")
Add the line of the bet fit model to your plot using the following:
The blue line is the model. The shaded gray area around the line
tells you about the variability you might expect in your predictions. To
turn that off, use se = FALSE.
ggplot(data = evals, aes(x = bty_avg, y = score)) +
geom_jitter() +
geom_smooth(method = "lm", se = FALSE)
- 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).
## Click two points to make a line.
## Call:
## lm(formula = y ~ x, data = pts)
##
## Coefficients:
## (Intercept) x
## 3.88034 0.06664
##
## Sum of Squares: 131.868
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.
## # A tibble: 1 × 1
## `cor(bty_avg, bty_f1lower)`
## <dbl>
## 1 0.844As expected, the relationship is quite strong—after all, the average score is calculated using the individual scores. You can actually look at the relationships between all beauty variables (columns 13 through 19) using the following command:
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 you’ve accounted for the professor’s gender, you can add the gender term into the model.
##
## 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.
- Is
bty_avgstill a significant predictor ofscore? Has the addition ofgenderto the model changed the parameter estimate forbty_avg?
The significance of bty_avg in predicting scores persists, and incorporating gender into the models not only enhances their performance but also alters the parameter estimate for bty_avg.
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 male and
female to being an indicator variable called
gendermale that takes a value of \(0\) for female professors and a value of
\(1\) for male professors. (Such
variables are often referred to as “dummy” variables.)
As a result, for female professors, the parameter estimate is multiplied by zero, leaving the intercept and slope form familiar from simple regression.
\[ \begin{aligned} \widehat{score} &= \hat{\beta}_0 + \hat{\beta}_1 \times bty\_avg + \hat{\beta}_2 \times (0) \\ &= \hat{\beta}_0 + \hat{\beta}_1 \times bty\_avg\end{aligned} \]
ggplot(data = evals, aes(x = bty_avg, y = score, color = pic_color)) +
geom_smooth(method = "lm", formula = y ~ x, se = FALSE)
- What is the equation of the line corresponding to those with color pictures? (Hint: For those with color pictures, the parameter estimate is multiplied by 1.) For two professors who received the same beauty rating, which color picture tends to have the higher course evaluation score?
3.74734+0.07416×bty_avg+0.1723
The decision to call the indicator variable gendermale
instead of genderfemale 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.
R deals with categorical variables like the rank variable, which has three levels (teaching, tenure track, and tenured), by assigning separate coefficients for each level. In the m_bty_rank model output, the intercept corresponds to the estimated mean score for professors with the baseline rank of teaching.
##
## 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, gender, ethnicity, 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.
I think the cls_profs will have the smallest amount of association with the professor score.
Let’s run the model…
m_full <- lm(score ~ rank + gender + ethnicity + 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 + gender + ethnicity + 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
## gendermale 0.2109481 0.0518230 4.071 5.54e-05 ***
## ethnicitynot minority 0.1234929 0.0786273 1.571 0.11698
## 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- Check your suspicions from the previous exercise. Include the model output in your response.
Among the variables considered, cls_profs exhibits the weakest association, with a maximum p-value of 0.77806, while language and age were found to be statistically significant.
- Interpret the coefficient associated with the ethnicity variable.
Professors that are not minority score 0.123 higher than those who are minority.
- 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?
The coefficients and significance of the other explanatory variables changed
new_m_full <- 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(new_m_full)##
## 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- 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_full <- lm(score ~ gender + ethnicity + language + age + cls_perc_eval + cls_credits + bty_avg + pic_color, data = evals)
summary(m_full)##
## Call:
## lm(formula = score ~ gender + ethnicity + 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 ***
## gendermale 0.207112 0.050135 4.131 4.30e-05 ***
## ethnicitynot minority 0.167872 0.075275 2.230 0.02623 *
## 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- Verify that the conditions for this model are reasonable using diagnostic plots.
The model assumptions appear to be good based on the diagnostic plots.
- 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?
No, the courses within the class are independent of each other; consequently, the scores would also exhibit 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.
A young non minority teacher that teaches one class,
- Would you be comfortable generalizing your conclusions to apply to professors generally (at any university)? Why or why not?
I wouldn’t feel comfortable generalizing my conclusions because this was an observational experiment, there are other factors that may be involved, and this is from one school, i may have different results in a different school