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.

library(tidyverse)
library(openintro)
library(GGally)

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.

glimpse(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:

?evals

Exploring the data

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.

This is an observational study. Given the study design, I don’t think it’s possible to answer this question as it’s phrased. I’d rephrase the question as: “Is there a relationship between ratings of professors’ physical appearance and ratings of professors in student course evaluations?”

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?

The distribution of score is left skewed. This tells me that most teachers are rated well because the largest number of values are between 4 and 5, so most receive a higher score. The variable score is defined as “Average professor evaluation score: (1) very unsatisfactory - (5) excellent.” This is basically along the lines of what I expected, because I think that most professors are probably pretty high quality, considering the fact that they are professors. We know that these score values are averages, so even those individual students who might rate a professor poorly wouldn’t have a major impact on the mostly-positive scores. Barring extreme cases where there are many low scores for a given professor, it makes sense that the median (4.3) and mean (4.175) average scores are both between 4 and 5.

ggplot(evals, aes(x = score)) +
  geom_histogram(binwidth = .2, fill = "tomato", color = "darkgrey") +
  labs(
    title = "Professor course evaluation scores"
  )

summary(evals$score)

   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
  2.300   3.800   4.300   4.175   4.600   5.000 

3. Excluding score, select two other variables and describe their relationship with each other using an appropriate visualization.

The relationship between gender and age of professors indicates that the spread for female professors is lower in age than the age spread for male professors. Female professors are generally younger, with the minimum lower than 30 years and the maximum above 60 years, and a median just over 45 years. Male professors are generally older, with a minimum age above 30 years and a maximum age over 70 years, and a median just over 50 years.

ggplot(evals, aes(x = gender, y = age)) +
  geom_boxplot() +
  labs(title = "Gender and Age of professors in `evals`")

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:

ggplot(data = evals, aes(x = bty_avg, y = score)) +
  geom_point()

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?

nrow(evals)

[1] 463

4. Replot the scatterplot, but this time use geom_jitter as your layer. What was misleading about the initial scatterplot?

ggplot(data = evals, aes(x = bty_avg, y = score)) +
  geom_jitter()

The initial scatterplot obscured repeated value points by overlapping in the exact same area of the plot; the jitter effect with geom_jitter as the layer in the plot, enables us to identify overlappint value points and see a better representation of the data.

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. 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?

The equation for the linear model is: y [score] = 3.88034 + 0.06664 * [bty_avg].

The slope is 0.06664 or 0.067 with rounding. This indicates the slope is positive but not extreme, rather, a gradual positive slope. The associated p-value is quite small which indicates that the average beauty score is a moderately statistically significant predictor of score, but perhaps not practically significant predictor.

m_bty <- lm(evals$score ~ evals$bty_avg)
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

Add the line of the bet fit model to your plot using the following:

ggplot(data = evals, aes(x = bty_avg, y = score)) +
  geom_jitter() +
  geom_smooth(method = "lm")

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)

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).

Reminder of the conditions for Least Squares Regression: - Linearity - the data should show a linear trend. The graph appears linear enough to satisfy this condition. - Nearly normal residuals - The distribution of residuals appears nearly normally distributed, though under other conditions we might want to look closer at the data to be sure. - Constant variability - The variability appears constant enough to satisfy this condition.

residuals <- resid(m_bty)

plot(x = fitted(m_bty), y = residuals,
     xlab = "fitted values", ylab = "residuals",
     main = "Residuals vs Fitted values")
abline(h = 0, col = "tomato")

qqnorm(residuals)
qqline(residuals)

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.

ggplot(data = evals, aes(x = bty_f1lower, y = bty_avg)) +
  geom_point()

evals %>% 
  summarise(cor(bty_avg, bty_f1lower))

# A tibble: 1 × 1
  `cor(bty_avg, bty_f1lower)`
                        <dbl>
1                       0.844

As 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:

evals %>%
  select(contains("bty")) %>%
  ggpairs()

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.

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

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.

The conditions for this model are not totally reasonable. The residuals do not seem to be linear or have constant variability, and do not appear to satisfy the normality condition either.

plot(m_bty_gen)

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

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?

Yes, bty_avg is still a significant predictor of score. The addition of gender to the model shows that the relationship between bty_avg and score is influenced by the gender of the professor. The addition of gender changes the parameter estimate for bty_avg: previously the parameter estimate for bty_avg was 0.06664 and with the addition of gender it is 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 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)

9. 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?

score = 3.88034 + 0.06664bty_avg + 1pic_color Note from Amanda: I’m interpreting this question as not including the gender factor because the plot called the data from evals and not from the model we created which included gender, so I returned to the parameters and values from the first model to answer this question…

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.)

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.

R appears to handle categorical variables that have more than 2 levels in a similar way to when there are 2 levels to a categorical variable: levels - 1. With gender, which had 2 levels, it was levels - 1, so we just saw the ‘gendermale’ value to be encoded as 1 and then ‘genderfemale’ was 0. With 3 levels like with rank, we see in the summary output there are levels - 1 visible, so we see ranktenure track and ranktenured, so the assumption is that the teaching value would be coded as 0. In this case, in summary, we can see that R encodes the levels of categorical variables by creating “dummy” variables to accomodate the number of levels.

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

The 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.

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.

Without first looking at the model results, I would expect something like the class level cls_level to have no association with the professor score. When running ?evals the definition for cls_level is “Class level: lower, upper.” This seems disconnected from something that would relate to 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

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

Interestingly, cls_level did not lead to the highest p-value, but it was one of the highest, with cls_levelupper having a p-value of 0.29369. The variable with the highest p-value in the model was in fact cls_profssingle from the cls_profs variable, which is defined as “Number of professors teaching sections in course in sample: single, multiple.” This makes sense based on my suspicions.

13. Interpret the coefficient associated with the ethnicity variable.

The coefficient associated with the ethnicity variable ethnicitynot minority is 0.1234929 or 0.1235 with rounding. The ethnicity variable is defined as “Ethnicity of professor: not minority, minority.” So we can interpret the coefficient for ethnicitynot minority being 0.1235 as: professors who are not minority can be estimated to have a score ~0.1235 units higher on average, assuming all other variables are constant, compared to the professors who are minority.

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?

Yes, there were changes in the coefficients and significance of the other explanatory variables when we dropped the variable with the highest p-value, cls_profs. The intercept decreased from 4.095 to 4.087. There were slight changes all around. Some items of potential note in the changes include ethnicitynot minority went from a p-value = 0.11698 (and a coefficient 0.1235) previously to a p-value = 0.099856 and a coefficient 0.1274. cls_perc_eval went from a coefficient = 0.0053272 and p-value = 0.00059 to a coefficient = 0.0052888 and p-value = 0.000607. The items that were noted with significance codes previously didn’t lose significance codes. Overall the adjusted r-squared went from 0.1617 to 0.1634.

m_almostfull <- update(m_full, . ~ . - cls_profs)
summary(m_almostfull)

Call:
lm(formula = score ~ rank + gender + ethnicity + 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    
gendermale             0.2101231  0.0516873   4.065 5.66e-05 ***
ethnicitynot minority  0.1274458  0.0772887   1.649 0.099856 .  
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

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.

score = 3.907030 + (0.202597 * gendermale) + (0.163818 * ethnicitynot minority) + (-0.246683 * languagenon-english) + (-0.006925 * age) + (0.004942 * cls_perc_eval) + (0.517205 * cls_creditsone credit) + (0.046732 * bty_avg) + (-0.113939 * pic_outfitnot formal) + (-0.180870 * pic_colorcolor)

back_select_m <- step(m_full, direction = "backward", criterion = "p-value")

Start:  AIC=-630.9
score ~ rank + gender + ethnicity + language + age + cls_perc_eval + 
    cls_students + cls_level + cls_profs + cls_credits + bty_avg + 
    pic_outfit + pic_color

                Df Sum of Sq    RSS     AIC
- cls_profs      1    0.0197 111.11 -632.82
- cls_level      1    0.2740 111.36 -631.76
- cls_students   1    0.3599 111.44 -631.40
- rank           2    0.8930 111.98 -631.19
<none>                       111.08 -630.90
- pic_outfit     1    0.5768 111.66 -630.50
- ethnicity      1    0.6117 111.70 -630.36
- language       1    1.0557 112.14 -628.52
- bty_avg        1    1.2967 112.38 -627.53
- age            1    2.0456 113.13 -624.45
- pic_color      1    2.2893 113.37 -623.46
- cls_perc_eval  1    2.9698 114.06 -620.69
- gender         1    4.1085 115.19 -616.09
- cls_credits    1    4.6495 115.73 -613.92

Step:  AIC=-632.82
score ~ rank + gender + ethnicity + language + age + cls_perc_eval + 
    cls_students + cls_level + cls_credits + bty_avg + pic_outfit + 
    pic_color

                Df Sum of Sq    RSS     AIC
- cls_level      1    0.2752 111.38 -633.67
- cls_students   1    0.3893 111.49 -633.20
- rank           2    0.8939 112.00 -633.11
<none>                       111.11 -632.82
- pic_outfit     1    0.5574 111.66 -632.50
- ethnicity      1    0.6728 111.78 -632.02
- language       1    1.0442 112.15 -630.49
- bty_avg        1    1.2872 112.39 -629.49
- age            1    2.0422 113.15 -626.39
- pic_color      1    2.3457 113.45 -625.15
- cls_perc_eval  1    2.9502 114.06 -622.69
- gender         1    4.0895 115.19 -618.08
- cls_credits    1    4.7999 115.90 -615.24

Step:  AIC=-633.67
score ~ rank + gender + ethnicity + language + age + cls_perc_eval + 
    cls_students + cls_credits + bty_avg + pic_outfit + pic_color

                Df Sum of Sq    RSS     AIC
- cls_students   1    0.2459 111.63 -634.65
- rank           2    0.8140 112.19 -634.30
<none>                       111.38 -633.67
- pic_outfit     1    0.6618 112.04 -632.93
- ethnicity      1    0.8698 112.25 -632.07
- language       1    0.9015 112.28 -631.94
- bty_avg        1    1.3694 112.75 -630.02
- age            1    1.9342 113.31 -627.70
- pic_color      1    2.0777 113.46 -627.12
- cls_perc_eval  1    3.0290 114.41 -623.25
- gender         1    3.8989 115.28 -619.74
- cls_credits    1    4.5296 115.91 -617.22

Step:  AIC=-634.65
score ~ rank + gender + ethnicity + language + age + cls_perc_eval + 
    cls_credits + bty_avg + pic_outfit + pic_color

                Df Sum of Sq    RSS     AIC
- rank           2    0.7892 112.42 -635.39
<none>                       111.63 -634.65
- ethnicity      1    0.8832 112.51 -633.00
- pic_outfit     1    0.9700 112.60 -632.65
- language       1    1.0338 112.66 -632.38
- bty_avg        1    1.5783 113.20 -630.15
- pic_color      1    1.9477 113.57 -628.64
- age            1    2.1163 113.74 -627.96
- cls_perc_eval  1    2.7922 114.42 -625.21
- gender         1    4.0945 115.72 -619.97
- cls_credits    1    4.5163 116.14 -618.29

Step:  AIC=-635.39
score ~ gender + ethnicity + language + age + cls_perc_eval + 
    cls_credits + bty_avg + pic_outfit + pic_color

                Df Sum of Sq    RSS     AIC
<none>                       112.42 -635.39
- pic_outfit     1    0.7141 113.13 -634.46
- ethnicity      1    1.1790 113.59 -632.56
- language       1    1.3403 113.75 -631.90
- age            1    1.6847 114.10 -630.50
- pic_color      1    1.7841 114.20 -630.10
- bty_avg        1    1.8553 114.27 -629.81
- cls_perc_eval  1    2.9147 115.33 -625.54
- gender         1    4.0577 116.47 -620.97
- cls_credits    1    6.1208 118.54 -612.84

summary(back_select_m)

Call:
lm(formula = score ~ gender + ethnicity + language + age + cls_perc_eval + 
    cls_credits + bty_avg + pic_outfit + pic_color, data = evals)

Residuals:
    Min      1Q  Median      3Q     Max 
-1.8455 -0.3221  0.1013  0.3745  0.9051 

Coefficients:
                       Estimate Std. Error t value Pr(>|t|)    
(Intercept)            3.907030   0.244889  15.954  < 2e-16 ***
gendermale             0.202597   0.050102   4.044 6.18e-05 ***
ethnicitynot minority  0.163818   0.075158   2.180 0.029798 *  
languagenon-english   -0.246683   0.106146  -2.324 0.020567 *  
age                   -0.006925   0.002658  -2.606 0.009475 ** 
cls_perc_eval          0.004942   0.001442   3.427 0.000666 ***
cls_creditsone credit  0.517205   0.104141   4.966 9.68e-07 ***
bty_avg                0.046732   0.017091   2.734 0.006497 ** 
pic_outfitnot formal  -0.113939   0.067168  -1.696 0.090510 .  
pic_colorcolor        -0.180870   0.067456  -2.681 0.007601 ** 
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 0.4982 on 453 degrees of freedom
Multiple R-squared:  0.1774,    Adjusted R-squared:  0.161 
F-statistic: 10.85 on 9 and 453 DF,  p-value: 2.441e-15

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

plot(back_select_m)

The conditions for this model are reasonable.

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?

Classes are independent and a student wouldn’t take the same course by the same professor twice. However, the more that a professor teaches classes, the more they will be represented in the dataset: a professor who has taught one course for two semesters is far less represented than a professor who has taught 40 courses over 20 semesters. Therefore the characteristics of professors who have taught more courses are more represented in the sample, making their characteristics and scores more impactful in the model

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.

A professor who would be associated with a high evaluation score would be one who is male, is not a minority ethnicity, was educated at an English-speaking institution, is not old in age, a high percentage of students fill out their course evaluation, teaches a 1-credit class, is deemed attractive, and has a black and white photo.

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

I would not be comfortable generalizing these conclusions to applyt o professors generally (at any university) for reasons including Texas is not everywhere, so the student body would not respond the same in one place as every other in the world. Another reason is that I believe it’s possible that “times have changed” since this survey, meaning attitudes as well as representation has changed in terms of who is represented as a professor and who is scoring those professors. I’m sure that some elements can be relatively applied, but for these reasons and general certainty that one university would not be representative for all universities, no I don’t think that this observational study can be applied to the general global population for professors at any university.

---