Data 606 - Lab 9
Tyler Graham
2025-04-13
Lab 9
Load Packages
library(tidyverse)
library(openintro)
library(GGally)Quick view of the data
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, …Exercise 1
Is this an observational study or an experiment? The original research question posed in the paper is whether beauty leads directly to the differences in course evaluations. Given the study design, is it possible to answer this question as it is phrased? If not, rephrase the question.
This is an observational study. Researchers did not assign professors to levels of physical attractiveness. Students evaluated professors naturally at the end of their courses, and beauty ratings were collected separately by external raters. There was no manipulation or random assignment involved.
Exercise 2
Describe the distribution of score. Is the distribution skewed? What does that tell you about how students rate courses? Is this what you expected to see? Why, or why not?
ggplot(evals, aes(x = score)) +
geom_histogram(binwidth = 0.2, fill = "steelblue", color = "white") +
labs(
title = "Distribution of Course Evaluation Scores",
x = "Score",
y = "Frequency"
)
The distribution is left-skewed as most ratings are clustered toward the
high end with fewer lower ratings, lower scores are less common.This
tells us that students tend to rate courses highly, meaning they are
generally satisfied and provide high eveluations.
Exercise 3
Excluding score, select two other variables and describe their relationship with each other using an appropriate visualization.
Since this seems like kind of an absurd study, we’re going to lean in and look at bty_avg (average beauty rating) and age. We can explore whether perceived attractiveness and age are related.
ggplot(evals, aes(x = age, y = bty_avg)) +
geom_point(alpha = 0.6, color = "darkblue") +
geom_smooth(method = "lm", se = TRUE, color = "red") +
labs(
title = "Relationship Between Age and Attractiveness",
x = "Age of Professor",
y = "Average Beauty Rating"
)
The scatterplot shows a slight negative relationship between a
professor’s age and their average beauty rating. Younger professors tend
to receive slightly higher beauty ratings, while older professors tend
to receive lower ones. However, the relationship is not strong, and
there is considerable variability at all ages.
Data from Assignment:
ggplot(data = evals, aes(x = bty_avg, y = score)) +
geom_point()
## Exercise 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()
In the original scatterplot, many overlapping points created the
illusion that there were fewer observations than actually exist. This
made it hard to assess the true distribution of ratings and beauty
scores. The jittered version reveals clusters of identical or
near-identical values. This suggests that many students gave very
similar evaluations, and some beauty scores are repeated across multiple
professors or courses.
Exercise 5
Let’s see if the apparent trend in the plot is something more than natural variation. Fit a linear model called m_bty to predict average professor score by average beauty rating. 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?
m_bty <- lm(score ~ bty_avg, data = evals)
summary(m_bty)##
## Call:
## lm(formula = score ~ bty_avg, data = evals)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.9246 -0.3690 0.1420 0.3977 0.9309
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 3.88034 0.07614 50.96 < 2e-16 ***
## bty_avg 0.06664 0.01629 4.09 5.08e-05 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.5348 on 461 degrees of freedom
## Multiple R-squared: 0.03502, Adjusted R-squared: 0.03293
## F-statistic: 16.73 on 1 and 461 DF, p-value: 5.083e-05score =3.88 + 0.066 ⋅ bty_avg
For each 1-point increase in beauty rating, the predicted evaluation score increases by 0.066 points, on average. Since the p-value of average beauty is very small, 2e-16, we can conclude that the score is a statistically significant predictor of course evaluation score. As to being a practically significant predictor, score could theoretically range from 1 to 10. A 5-point difference would increase a score by only .33 points (5 x 0.066) which is fairly modest when applied to median scores. So while statistically significant, beauty effect is not practically significant.
ggplot(data = evals, aes(x = bty_avg, y = score)) +
geom_jitter() +
geom_smooth(method = "lm")
## Exercise 6
Use residual plots to evaluate whether the conditions of least squares regression are reasonable. Provide plots and comments for each one (see the Simple Regression Lab for a reminder of how to make these)
par(mfrow = c(2, 2)) # arrange plots in 2x2 grid
plot(m_bty)
Residuals vs. Fitted: The plot shows a fairly random scatter, suggesting
the linearity assumption is reasonable, though slight curvature might be
present.
Q-Q Residuals: The residuals mostly follow the diagonal, indicating the normality assumption is mostly reasonable, with minor deviation in the tails.
Scale-Location: There is no clear funnel shape, which supports the equal varianceS assumption.
Residuals vs Leverage: Most points fall within Cook’s distance boundaries. No single point appears unduly influential, so this assumption is also satisfied.
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 0.844evals %>%
select(contains("bty")) %>%
ggpairs()
m_bty_gen <- lm(score ~ bty_avg + gender, data = evals)
summary(m_bty_gen)##
## Call:
## lm(formula = score ~ bty_avg + gender, data = evals)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.8305 -0.3625 0.1055 0.4213 0.9314
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 3.74734 0.08466 44.266 < 2e-16 ***
## bty_avg 0.07416 0.01625 4.563 6.48e-06 ***
## gendermale 0.17239 0.05022 3.433 0.000652 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.5287 on 460 degrees of freedom
## Multiple R-squared: 0.05912, Adjusted R-squared: 0.05503
## F-statistic: 14.45 on 2 and 460 DF, p-value: 8.177e-07Exercise 7
P-values and parameter estimates should only be trusted if the conditions for the regression are reasonable. Verify that the conditions for this model are reasonable using diagnostic plots.
par(mfrow = c(2, 2)) # 2x2 layout
plot(m_bty_gen)
TLDR; conditions are still reasonable.
Residuals vs. Fitted: The plot shows a fairly random scatter, suggesting the linearity assumption is reasonable, though slight curvature might be present.
Q-Q Residuals: The residuals mostly follow the diagonal, indicating the normality assumption is mostly reasonable, with minor deviation in the tails.
Scale-Location: There is no clear funnel shape, which supports the equal varianceS assumption.
Residuals vs Leverage: Most points fall within Cook’s distance boundaries. No single point appears unduly influential, so this assumption is also satisfied.
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?
Yes. In the multiple linear regression model that includes both bty_avg and gender, the coefficient for bty_avg is 0.07416 with a p-value of 6.48e-06, which is far below the conventional threshold of 0.05.
In the previous simple linear model (score ~ bty_avg), the coefficient for bty_avg was approximately 0.066. In the updated model (score ~ bty_avg + gender), the coefficient is:
bty_avg: 0.07416
This is a slight increase from the simple model, suggesting that the addition of gender has slightly strengthened the estimated effect of beauty on score. This means that after accounting for gender differences, the positive association between beauty and evaluation score becomes a bit more pronounced.
ggplot(data = evals, aes(x = bty_avg, y = score, color = pic_color)) +
geom_smooth(method = "lm", formula = y ~ x, se = FALSE)
## Exercise 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?
m_pic_color <- lm(score ~ bty_avg * pic_color, data = evals)
summary(m_pic_color)##
## Call:
## lm(formula = score ~ bty_avg * pic_color, data = evals)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.8768 -0.3467 0.1289 0.3883 0.9054
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 3.79741 0.20680 18.363 < 2e-16 ***
## bty_avg 0.10489 0.03679 2.851 0.00455 **
## pic_colorcolor 0.16085 0.22347 0.720 0.47202
## bty_avg:pic_colorcolor -0.06260 0.04141 -1.512 0.13125
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.5315 on 459 degrees of freedom
## Multiple R-squared: 0.05101, Adjusted R-squared: 0.0448
## F-statistic: 8.223 on 3 and 459 DF, p-value: 2.431e-05Since “black&white” is the reference group (score = 3.79741 + 0.10489 ⋅ bty_agv), these coefficients modify the base equation when pic_color == “color”. So for professors with color photos, we combine:
Intercept: 3.79741 + 0.16085 = 3.95826
Slope: 0.10489 - 0.06260 = 0.04229
Equation for color photo group:
score = 3.95826 + 0.04229 ⋅ bty_avg
For finding out who gets the higher score for the same beauty rating, let’s compare both lines at a given bty_avg of 5:
Black & White: 3.79741 + 0.10489 ⋅ 5 = 4.32186
and for Color:
3.95826 + 0.04229 ⋅ 5 = 4.16971
Professors with black & white photos tend to receive higher course evaluation scores than those with color photos, for the same beauty rating. This is visually supported by our plot and numerically supported by the higher intercept and steeper slope for the black & white group.
Exercise 10
Create a new model called m_bty_rank with gender removed and rank added in. How does R appear to handle categorical variables that have more than two levels? Note that the rank variable has three levels: teaching, tenure track, tenured.
m_bty_rank <- lm(score ~ bty_avg + rank, data = evals)
summary(m_bty_rank)##
## Call:
## lm(formula = score ~ bty_avg + rank, data = evals)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.8713 -0.3642 0.1489 0.4103 0.9525
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 3.98155 0.09078 43.860 < 2e-16 ***
## bty_avg 0.06783 0.01655 4.098 4.92e-05 ***
## ranktenure track -0.16070 0.07395 -2.173 0.0303 *
## ranktenured -0.12623 0.06266 -2.014 0.0445 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.5328 on 459 degrees of freedom
## Multiple R-squared: 0.04652, Adjusted R-squared: 0.04029
## F-statistic: 7.465 on 3 and 459 DF, p-value: 6.88e-05R automatically uses dummy coding and it chose one level as the reference category, in this case ‘teaching’ as it is alphabetically first. So R handles multi-level categorical variables by creating one indicator variable per non-reference level, and the regression output shows how those levels differ from the reference group.
Exercise 11
Reference: 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.
Based on the previous exercises we ran, I would expect pic_color to have the highest p-value. Also…just looking at the list of options here, this seems to be data point with the least influenc from a logical perspective.
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-14Exercise 12
Check your suspicions from the previous exercise. Include the model output in your response.
Well dangit. cls_profssingle had the highest p-value. And in a total reversal, pic_colorcolor turned out to be actually significant at 0.00252. So this totally contradicts our assumption. Cool.
Exercise 13
Interpret the coefficient associated with the ethnicity variable.
The coefficient for ‘ethnicitynot minority’ is 0.123, suggesting that professors not identified as minorities receive, on average, 0.123 points higher on their evaluation scores than minority professors, when controlling for all other factors in the model. However, this difference is not statistically significant (p = 0.117), so we cannot confidently conclude that ethnicity is associated with evaluation scores in this dataset.
Exercise 14
Drop the variable with the highest p-value and re-fit the model. Did the coefficients and significance of the other explanatory variables change? (One of the things that makes multiple regression interesting is that coefficient estimates depend on the other variables that are included in the model.) If not, what does this say about whether or not the dropped variable was collinear with the other explanatory variables?
The variable with the highest p-value: cls_profssingle: Estimate = -0.01466, p = 0.778
Refit without cls_profs:
m_refined <- lm(score ~ rank + gender + ethnicity + language + age +
cls_perc_eval + cls_students + cls_level + cls_credits +
bty_avg + pic_outfit + pic_color, data = evals)
summary(m_refined)##
## 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-14After removing the variable cls_profs (which had the highest p-value of 0.778), I re-fitted the model. The coefficients and significance levels of the remaining variables changed very little, suggesting that cls_profs was not collinear with the other predictors. It did not explain any unique variation in evaluation scores and had little influence on the estimates of other variables. This confirms that dropping it was appropriate and helps simplify.
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_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)
m_step <- step(m_full, direction = "backward")## 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 -612.84summary(m_step)##
## 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-15Using backward selection with AIC as the criterion, the final model retained 9 predictors. These include professor characteristics (gender, ethnicity, language, age, beauty rating), class attributes (evaluation percentage, credit type), and photo presentation (outfit formality and color). The final model explains about 16.1% of the variance in evaluation scores (Adjusted R squared = 0.161) and is statistically significant overall (p < 0.001).
The final regression equation is:
score = 3.907 + 0.203 * gender_male + 0.164 * ethnicity_not_minority - 0.247 * language_non_english - 0.0069 * age + 0.00494 * cls_perc_eval + 0.517 * cls_credits_one_credit + 0.0467 * bty_avg - 0.114 * pic_outfit_not_formal - 0.181 * pic_color_color
Exercise 16
Verify that the conditions for this model are reasonable using diagnostic plots.
par(mfrow = c(2, 2)) # 2x2 layout
plot(m_step)
TLDR; conditions are still reasonable.
Residuals vs. Fitted: The plot shows a fairly random scatter, suggesting the linearity assumption is reasonable, though slight curvature might be present.
Q-Q Residuals: The residuals mostly follow the diagonal, indicating the normality assumption is mostly reasonable, with minor deviation in the tails.
Scale-Location: There is no clear funnel shape, which supports the equal varianceS assumption.
Residuals vs Leverage: Most points fall within Cook’s distance boundaries. No single point appears unduly influential, so this assumption is also satisfied.
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?
Yes, this new information does impact the assumptions of linear regression, particularly the assumption of independence of observations, which is a critical condition for valid regression results.
Exercise 18
Based on your final model, describe the characteristics of a professor and course at University of Texas at Austin that would be associated with a high evaluation score.
Based on the final regression model, professors at the UT Austin are predicted to receive higher course evaluation scores if they are male, not identified as a minority, younger, more physically attractive, and have degrees from English-speaking institutions. Additionally, courses with higher student evaluation participation rates, one credit hour, and professors dressed formally in black-and-white photographs are also associated with higher scores. These findings suggest that non-pedagogical factors like appearance, demographics, and even photo presentation can have a measurable effect on student evaluations.
Exercise 19
Would you be comfortable generalizing your conclusions to apply to professors generally (at any university)? Why or why not?
No, we could not generalize these conclusions to professors at other universities. The data was collected exclusively from the University of Texas at Austin. Since the study is observational and not based on a nationally representative sample, the results may not apply to professors at different institutions or in different academic contexts, etc…