Multiple linear regression

Grading the professor

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.

Observational study. As an observational study, beauty can only be shown to correlate with differences in course evaluation.
- “Whether beauty, rated on a 1-10 scale, correlates with differences in course evaluations”

Describe the distribution of score. Is the distribution skewed? What does that tell you about how students rate courses? Is this what you expected to see? Why, or why not?

There appears to be a leftward skew with a long tail in the eval scores
It tells us that students are more likely to give higher eval scores
I think given a 1-5 scale its in line with my expectation.

library(psych)
library(kableExtra)
library(knitr)
load("more/evals.RData")
par(mfrow=c(1, 2))
hist(evals$score)
describe(evals$score)

##    vars   n mean   sd median trimmed  mad min max range skew kurtosis   se
## X1    1 463 4.17 0.54    4.3    4.22 0.59 2.3   5   2.7 -0.7     0.04 0.03

qqnorm((evals$score))
qqline((evals$score))

Excluding score, select two other variables and describe their relationship using an appropriate visualization (scatterplot, side-by-side boxplots, or mosaic plot).

lets look at how age effects bty_avg by grouping a scatter plot by gender
- it appears that beauty scorers tend to go down as the professors age increases

scatter.smooth(evals$bty_avg~evals$age)

Simple linear regression

Replot the scatterplot, but this time use the function jitter() on the \(y\)- or the \(x\)-coordinate. (Use ?jitter to learn more.) What was misleading about the initial scatterplot?

plot((evals$score) ~ jitter(evals$bty_avg))

The initial plot was masking some of the values because they shared the same (x,y) coordinates

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 and add the line to your plot using abline(m_bty). Write out the equation for the linear model and interpret the slope. Is average beauty score a statistically significant predictor? Does it appear to be a practically significant predictor?

m_bty <- lm(evals$score ~ evals$bty_avg)
plot((evals$score) ~ jitter(evals$bty_avg))
abline(m_bty)

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

\[score= 3.88+.06(bty_avg) \]

There appears to be a significant relationship between beauty and rating score considering the p value is so low

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).

Constant variability- check
normality -check
linearity- check

par(mfrow=c(2, 2))
plot(m_bty$residuals ~ evals$bty_avg)
hist(m_bty$residuals)
qqnorm(m_bty$residuals)
qqline(m_bty$residuals)

m_bty_gen <- lm(score ~ bty_avg + gender, data = evals)
summary(m_bty_gen)

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.

Constant variability- first graph check
Normality- qq line 2nd graph- check
Independence-check

#plot(m_bty$residuals ~ evals$bty_avg)
m_bty_gen <- lm(score ~ bty_avg + gender, data = evals)
par(mfrow=c(2, 2))
plot(m_bty_gen)

par(mfrow=c(2, 1))  
boxplot(m_bty_gen$residuals~evals$bty_avg)
boxplot(m_bty_gen$residuals~evals$gender)

Is bty_avg still a significant predictor of score? Has the addition of gender to the model changed the parameter estimate for bty_avg?

The P values is still extremely small so yes beauty_avg is still a significant predictor
Yes the parameter estimate has changed, to .07416 from 0.06664

multiLines(m_bty_gen)

What is the equation of the line corresponding to males? (Hint: For males, the parameter estimate is multiplied by 1.) For two professors who received the same beauty rating, which gender tends to have the higher course evaluation score?

Male_score= 3.74734 + .07(bty_avg)+.17239
Female_score=3.74734+ .07(bty_avg)
Male professors tend to have higher score

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

The decision to call the indicator variable gendermale instead ofgenderfemale has no deeper meaning. R simply codes the category that comes first alphabetically as a \(0\). (You can change the reference level of a categorical variable, which is the level that is coded as a 0, using therelevel function. Use ?relevel to learn more.)

Create a new model called m_bty_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 eliminate the first factor variable

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 search for the best model

We will start with a full model that predicts professor score based on rank, ethnicity, gender, language of the university where they got their degree, age, proportion of students that filled out evaluations, class size, course level, number of professors, number of credits, average beauty rating, outfit, and picture color.

Which variable would you expect to have the highest p-value in this model? Why? Hint: Think about which variable would you expect to not have any association with the professor score.

Proportion of students that filled out evaluations, seems like it wouldn’t effect anything as the name implies it is proportionate and differences should mostly be due to chance alone

Let’s run the model…

m_full <- lm(score ~ rank + ethnicity + gender + language + age + cls_perc_eval 
             + cls_students + cls_level + cls_profs + cls_credits + bty_avg 
             + pic_outfit + pic_color, data = evals)
summary(m_full)

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

I was wrong the p value for that field is actually .00059

Interpret the coefficient associated with the ethnicity variable.

Given the factor of minority/ not minority, not minority teachers are expected to score .12 points higher

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?

They changed but they dint change significantly. This implies the dropped variable was collinear with other explanatory variables

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(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 ~  ethnicity + gender + language + age + cls_perc_eval 
              + cls_credits + bty_avg 
              + pic_color, data = evals)
sum_m_full <- summary(m_full)
sum_m_full$coefficients

##                           Estimate  Std. Error   t value     Pr(>|t|)
## (Intercept)            3.771921500 0.232053490 16.254535 3.726633e-47
## ethnicitynot minority  0.167872321 0.075275467  2.230107 2.622848e-02
## gendermale             0.207112134 0.050134781  4.131107 4.298711e-05
## languagenon-english   -0.206178078 0.103639310 -1.989381 4.725875e-02
## age                   -0.006045919 0.002612071 -2.314607 2.107974e-02
## cls_perc_eval          0.004655873 0.001435172  3.244123 1.265094e-03
## cls_creditsone credit  0.505306239 0.104119390  4.853143 1.673798e-06
## bty_avg                0.051069320 0.016933952  3.015795 2.706868e-03
## pic_colorcolor        -0.190578800 0.067351272 -2.829624 4.866842e-03

cats <- c("+","*(ethnicity) + ", "* (gender) + ","* (language) +", "* (age) + ", "* (cls_perc_eval) +", "* (cls_credits) +", " *(bty_avg) + ", "* (pic_color)")
new_cats <- (paste(sum_m_full$coefficients[1:9],cats,collapse=""))
kable(paste("the formula is ",new_cats))

x
the formula is 3.77192150013323 +0.167872321380204 (ethnicity) + 0.207112133664405 (gender) + -0.20617807782666 * (language) +-0.00604591872396087 * (age) + 0.00465587321432217 * (cls_perc_eval) +0.505306239482799 * (cls_credits) +0.0510693200149873 (bty_avg) + -0.190578799689381 (pic_color)

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

Normality seems fine but it does curve off at the ends of qq chart, hist seems fine although leftward skewed
residuals appear somewhat constant although once again not at the ends
Independence-check

hist(sum_m_full$residuals)

par(mfrow=c(2, 2))
plot(m_full)

par(mfrow=c(2, 2))  
boxplot(sum_m_full$residuals~evals$ethnicity)
boxplot(sum_m_full$residuals~evals$gender)
boxplot(sum_m_full$residuals~evals$cls_credits)
boxplot(sum_m_full$residuals~evals$language)

par(mfrow=c(2, 2))  
boxplot(sum_m_full$residuals~evals$bty_avg)
boxplot(sum_m_full$residuals~evals$age)
boxplot(sum_m_full$residuals~evals$pic_color)
par(mfrow=c(1, 1))

boxplot(sum_m_full$residuals~evals$cls_perc_eval)

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?

It could violate our independence of observations assumption.
- students may rate professors in class off of their impressions from previous classes, which would violate 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.

m_full

## 
## Call:
## lm(formula = score ~ ethnicity + gender + language + age + cls_perc_eval + 
##     cls_credits + bty_avg + pic_color, data = evals)
## 
## Coefficients:
##           (Intercept)  ethnicitynot minority             gendermale  
##              3.771922               0.167872               0.207112  
##   languagenon-english                    age          cls_perc_eval  
##             -0.206178              -0.006046               0.004656  
## cls_creditsone credit                bty_avg         pic_colorcolor  
##              0.505306               0.051069              -0.190579

A professor starts with a 3.77 rating.
Professors lose points if their college of study was non English,
Lose points if their profile pic is in color,
Lose points for every year in age of a professor loses some points,
gains points for not being a minority
Gains points for being a male
Gains points depending on percent of students evaluating
Gains points if class is one credit(substantial gain .5)
Gains points depending on beauty(potentially alot.05 *(1-10))
- .006*(age_in_years)

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

Considering we are unsure of the independence of our samples, and considering there may be some issues with our residuals, I would want to run several more studies. Perhaps the students at this school or in these classes are more likely to show these tendencies.

This is a product of OpenIntro that is released under a Creative Commons Attribution-ShareAlike 3.0 Unported. This lab was written by Mine Çetinkaya-Rundel and Andrew Bray.