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.
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 seems to be an observational study, since there is no experimental manipulation. Because there was no random assignment/experimental manipulation, a more apt research question may be: is beauty associated with differences in course evaluations? Alternatively, one could ask if beauty predicts differences in course evaluations. It is important to phrase such questions in this way to avoid making causal assertions without proper experimental manipulation
- 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?
##
## Shapiro-Wilk normality test
##
## data: evals$score
## W = 0.95228, p-value = 4.477e-11
Based on the Shapiro test and graphs shown above, it appears that the data are negatively skewed. This tells us that students were more likely to provide more positive than negative reviews, with an average of 4 out of 5. I feel that this makes sense, as students may be more likely to provide more positive reviews for their instructors, overall, as long as they were not abjectly terrible. I know I had more of a tendency to positively score my professors if I felt they were trying their best
- Excluding
score, select two other variables and describe their relationship with each other using an appropriate visualization.
Average beauty does not seem to be strongly associated with the number of students who completed evaluations; however, there were some fringe cases in which far more students completed evaluations for professors who were rated as a 6 on the scale. The issue with this plot may be related to imbalanced class sizes.
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? I believe fewer points are being displayed than exist in the dataset
- Replot the scatterplot, but this time use
geom_jitteras your layer. What was misleading about the initial scatterplot?
Fewer points were being displayed on the plot compared to the number that exist in the dataset
- 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?
##
## 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-05The equation for a linear model can be written as: y-hat = mx + b, or in this case: y-hat = .067x + 3.89 The slope is around .067, suggesting that every incremental increase in average beauty predicts a .067 increase in score. This model, as well as the slope, are significant. However, it should be noted that the fit is rather poor at 3.50%, suggesting that the practicality of this predictor may be severely limited
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).
ggplot(data = m_bty, aes(x = .fitted, y = .resid)) +
geom_point() +
geom_hline(yintercept = 0, linetype = "dashed") +
xlab("Fitted values") +
ylab("Residuals")
Based on these plots, the residuals appear to be approximately normally distributed, with some skew; furthermore, the the assumption of homoscedasticity appears to be met, since the residuals are centered around the zero value in a patternless manner
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.
ggplot(data = m_bty_gen, aes(x = .fitted, y = .resid)) +
geom_point() +
geom_hline(yintercept = 0, linetype = "dashed") +
xlab("Fitted values") +
ylab("Residuals")
Based on the plots above, residuals are mostly normally distributed, again, with some negative skew; furthermore, the homoscedasticity assumption is mostly upheld, although there seems to be some leverage toward the upper end of the x-axis
- Is
bty_avgstill a significant predictor ofscore? Has the addition ofgenderto the model changed the parameter estimate forbty_avg?
Yes, the model still shows that average beauty predicts score; clearly, there is a new gender parameter that has been added to the model that shows changes in gender predict scores, such that males give higher scores; also, the slope of average beauty increased slightly; finally, the R-square increased slightly, marginally improving the fir of this model (but not by much)
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?
The equation should be: y-hat = b0 + b1X1 + b2X2, or: y-hat = y-intercept + (X1 * 0) + (X2 * 1) black & white pictures tend to have the higher course evaluation rating
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.
##
## 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 seems to have removed the teaching condition from the analysis, since it is considered as the “0” coefficient
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.
- 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 would hope that variables like gender and ethnicity have little impact on evaluations, since there would be rather concerning implications if that were the case
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.
Yes, ethnicity had a high, non-significant p-value; however, whether or not the professor wore a formal outfit in their picture seemed to have the highest p value
- Interpret the coefficient associated with the ethnicity variable.
The model is non-significant, so ethnicity is not significantly predicting changes in evaluation scores; however, there seems to be an emerging trend wherein having the minority status predicts lower scores
- 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_drop <- 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_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##
## 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-14Removing cls_prof barely changed the coefficients and R-Squared values, suggesting that it was not collinear with the other predictors
- 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 ~ rank + gender + language + ethnicity + age + cls_perc_eval
+ cls_students + 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##
## Call:
## lm(formula = score ~ rank + gender + language + ethnicity + age +
## cls_perc_eval + cls_students + cls_credits + bty_avg + pic_outfit +
## pic_color, data = evals)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.7761 -0.3187 0.0875 0.3547 0.9367
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 4.0856255 0.2888881 14.143 < 2e-16 ***
## ranktenure track -0.1420696 0.0818201 -1.736 0.083184 .
## ranktenured -0.0895940 0.0658566 -1.360 0.174372
## gendermale 0.2037722 0.0513416 3.969 8.40e-05 ***
## languagenon-english -0.2093185 0.1096785 -1.908 0.056966 .
## ethnicitynot minority 0.1424342 0.0759800 1.875 0.061491 .
## age -0.0087287 0.0031224 -2.795 0.005404 **
## cls_perc_eval 0.0053545 0.0015306 3.498 0.000515 ***
## cls_students 0.0003573 0.0003585 0.997 0.319451
## cls_creditsone credit 0.4733728 0.1106549 4.278 2.31e-05 ***
## bty_avg 0.0410340 0.0174449 2.352 0.019092 *
## pic_outfitnot formal -0.1172152 0.0716857 -1.635 0.102722
## pic_colorcolor -0.1973196 0.0681052 -2.897 0.003948 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.4975 on 450 degrees of freedom
## Multiple R-squared: 0.185, Adjusted R-squared: 0.1632
## F-statistic: 8.51 on 12 and 450 DF, p-value: 1.275e-14Removing cls_prof and cls_level resulted in an R-squared of .185 and a multiple R^2 of .16
- Verify that the conditions for this model are reasonable using diagnostic plots.
ggplot(data = m_best, aes(x = .fitted, y = .resid)) +
geom_point() +
geom_hline(yintercept = 0, linetype = "dashed") +
xlab("Fitted values") +
ylab("Residuals")
The residuals are normally distributed (again, with some skew), but the homoscedasticity assumption may be violated, as there is more leverage toward the middle of the x-axis and upper end of the y-axis
- 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, if a professor in this dataset has taught multiple courses, that could influence the analysis, since they will likely have consistent scores across classes. It may help to average every professors’ scores across classes and rerun the analysis
- 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 best fitting model, one would expect a professor who is young, male, has a multi-credit course, is attractive, has a high percentage of students who complete evaluations, and who has a black & white picture to get higher evaluation scores
- Would you be comfortable generalizing your conclusions to apply to professors generally (at any university)? Why or why not?
I would not necessarily be comfortable with this, since the model has such a low fit. It may be apt to provide more fitting models, or to further normalize the residuals in order to generate a better, more generalizable, model