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” (Hamermesh and Parker, 2005) found that instructors who are viewed to be better looking receive higher instructional ratings. (Daniel S. Hamermesh, Amy Parker, Beauty in the classroom: instructors pulchritude and putative pedagogical productivity, Economics of Education Review, Volume 24, Issue 4, August 2005, Pages 369-376, ISSN 0272-7757, 10.1016/j.econedurev.2004.07.013. http://www.sciencedirect.com/science/article/pii/S0272775704001165.)

In this lab we will analyze the data from this study in order to learn what goes into a positive professor evaluation.

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. (This is aslightly modified version of the original data set that was released as part of the replication data for Data Analysis Using Regression and Multilevel/Hierarchical Models (Gelman and Hill, 2007).) The result is a data frame where each row contains a different course and columns represent variables about the courses and professors.

load("more/evals.RData")
variable description
score average professor evaluation score: (1) very unsatisfactory - (5) excellent.
rank rank of professor: teaching, tenure track, tenured.
ethnicity ethnicity of professor: not minority, minority.
gender gender of professor: female, male.
language language of school where professor received education: english or non-english.
age age of professor.
cls_perc_eval percent of students in class who completed evaluation.
cls_did_eval number of students in class who completed evaluation.
cls_students total number of students in class.
cls_level class level: lower, upper.
cls_profs number of professors teaching sections in course in sample: single, multiple.
cls_credits number of credits of class: one credit (lab, PE, etc.), multi credit.
bty_f1lower beauty rating of professor from lower level female: (1) lowest - (10) highest.
bty_f1upper beauty rating of professor from upper level female: (1) lowest - (10) highest.
bty_f2upper beauty rating of professor from second upper level female: (1) lowest - (10) highest.
bty_m1lower beauty rating of professor from lower level male: (1) lowest - (10) highest.
bty_m1upper beauty rating of professor from upper level male: (1) lowest - (10) highest.
bty_m2upper beauty rating of professor from second upper level male: (1) lowest - (10) highest.
bty_avg average beauty rating of professor.
pic_outfit outfit of professor in picture: not formal, formal.
pic_color color of professor’s picture: color, black & white.

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. We can rephrase the question as whether a professors' average beauty rating is correlated to the average course evaluation score.
  1. 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?
hist(evals$score, xlab='Distribution of Evaluation Score')

The distribution of the evaluation scores is skewed to the left. Most student tend to evaluate teacher at a high score. If I have to guess, I would guess the distribution will be normal but not centered at 2.5 because usually students will tend to rate teachers higher than medium score. The strong skewness may because of the impact of some factors like physical appearence, knowledge and humor etc.
  1. Excluding score, select two other variables and describe their relationship using an appropriate visualization (scatterplot, side-by-side boxplots, or mosaic plot).
plot(evals$bty_f1lower,evals$bty_avg)

cor(evals$bty_f1lower,evals$bty_avg)
## [1] 0.8439112

The average beauty rating of professor increases with the increase of the beauty rating of professor from lower level female. There is positive linear realtionship between the average beauty rating of professor and the beauty rating of professor from lower level female.

boxplot(bty_avg ~ gender,data=evals)

The medium of the average beauty rating of female professor is higher than male professor.

female <- evals[which(evals$gender == 'female'),]
male <-  evals[which(evals$gender == 'male'),]
t.test(female$bty_avg,male$bty_avg)
## 
##  Welch Two Sample t-test
## 
## data:  female$bty_avg and male$bty_avg
## t = 2.8898, df = 401.53, p-value = 0.004064
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
##  0.1331423 0.6997496
## sample estimates:
## mean of x mean of y 
##  4.658897  4.242451

T-test results suggest that there is significant difference between the average beauty rating of female professor and that of male professor.

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:

plot(evals$score ~ evals$bty_avg)

Before we 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?

There are 463 observations in the data frame but there are points less than the number of observations in the scatterplot.
  1. 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?
#  use the function `jitter()` on the y 
plot(jitter(evals$score) ~ evals$bty_avg)

#  use the function `jitter()` on the x 
plot(evals$score ~ jitter(evals$bty_avg))

  1. 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)

the equation for the linear model: \[\hat{score}=3.88034+0.06664\times bty_avg \]

plot(jitter(evals$score,factor=1.2) ~ jitter(evals$bty_avg,factor=1.2))+
abline(m_bty)

## numeric(0)
cor(evals$score, evals$bty_avg)
## [1] 0.1871424
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

The average professor score will increase 0.0664 for every one point increase in average beauty rating.

p-value is 5.083e-05, which is less than 0.05 ,suggesting the average beauty score is a statistically significant predictor. It may not be a practically significant predictor because there is very weak correlation between the average beauty score and the average professor score with Multiple R-squared equals to 0.03502. For every one point increase in of average beauty rating, the model only predicts an increase of 0.06664 in the average professor score, which barely changes the score.

  1. 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).
yx.res <- resid(m_bty)

par(mfrow = c(1,2))
hist(yx.res, xlab="Residuals", breaks = 10)

plot(evals$bty_avg, yx.res, ylab="Residuals", xlab="bty_avg", main="Residuals of Evaluation") 

abline(0, 0)               

To evaluate whether the conditions of least squares regression are reasonable, I will check the following conditions:

Linearity:I would say the data is linear as shown in the plot.

Nearly normal residuals:The distribution of residuals is left skewed but nearly normal.

Constant variabilities:The variance around the line is constant.

Independent observation:Each professor's beauty rating is indenpendent to each other.

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.

plot(evals$bty_avg ~ evals$bty_f1lower)

cor(evals$bty_avg, evals$bty_f1lower)
## [1] 0.8439112

As expected the relationship is quite strong - after all, the average score is calculated using the individual scores. We can actually take a look at the relationships between all beauty variables (columns 13 through 19) using the following command:

plot(evals[,13:19])

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 we’ve accounted for the gender of the professor, we can add the gender term into the model.

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

The multiple regression model could be written as:\[\hat{y}= 3.74734 + 0.07416 \times x_1+ 0.17239 \times x_2 \]

  1. 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.

Multiple regression methods using the model

\[\hat{y}= \beta_0 + \beta_1 \times x_1+ \beta_2 \times x_2 + ... + \beta_k \times x_k\]

generally depend on the following four assumptions:
1. the residuals of the model are nearly normal,
2. the variability of the residuals is nearly constant,
3. the residuals are independent, and
4. each variable is linearly related to the outcome.

Diagnostic plots will be used to check each of these assumptions.

Normal probability plot.

m_bty_gen <- lm(score ~ bty_avg + gender, data = evals)
qqnorm(m_bty_gen$residuals)
qqline(m_bty_gen$residuals)

The residuals of the model are nearly normal as shown in the QQ plot. While there are a few observations that deviate noticeably from the line, they are not particularly extreme.

Absolute values of residuals against fitted values (\(\hat{y_i}\)).

evals$gender <- as.numeric(evals$gender)
evals$gender <- evals$gender-1
  
fitted <- 3.74734 + 0.07416 * evals$bty_avg+ 0.17239 *evals$gender 
plot(fitted,m_bty_gen$residuals,ylab="Absolute value of residuals", xlab="Fitted values")

The plot shows that the variance of the residuals is approximately constant.

Residuals in order of their data collection is not applicable in this data set because we don’t know the sequence of the data collection.

Residuals against each predictor variable.

boxplot(m_bty_gen$residuals~rank,data=evals,ylab="Residuals", main="Rank")

boxplot(m_bty_gen$residuals~ethnicity,data=evals,ylab="Residuals", main="Ethnicity")

boxplot(m_bty_gen$residuals~gender,data=evals,ylab="Residuals", main="Gender")

boxplot(m_bty_gen$residuals~language,data=evals,ylab="Residuals", main="Language")

plot(m_bty_gen$residuals~age, data=evals,ylab="Residuals", main="Age")

boxplot(m_bty_gen$residuals~cls_level,data=evals,ylab="Residuals", main="Cls_level")

boxplot(m_bty_gen$residuals~cls_profs,data=evals,ylab="Residuals", main="Cls_profs")

boxplot(m_bty_gen$residuals~cls_credits,data=evals,ylab="Residuals", main="Cls_credits")

plot(m_bty_gen$residuals~round(cls_perc_eval),data=evals,ylab="Residuals", main="Cls_perc_eval")

plot(m_bty_gen$residuals~bty_avg,data=evals,ylab="Residuals", main="Bty_avg")

boxplot(m_bty_gen$residuals~pic_color,data=evals,ylab="Residuals", main="Pic_color")

variability doesn't fluctuate across groups in ethnicity or in cls_level, or in cls_prof.

However, looking at the variables gender, language, age, cls_perc_eval, class credits, bty_avg, and pic_color, we find that there is some difference in the variability of the residuals in the groups. There appears to be curvature in the residuals, indicating the relationship is probably not linear.
  1. 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 it is. Adding of gender made beauty average even more significant as the p-value computed is even smaller compared to the previous model.

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 female and male to being an indicator variable called gendermale that takes a value of \(0\) for females and a value of \(1\) for males. (Such variables are often referred to as “dummy” variables.)

As a result, for females, the parameter estimate is multiplied by zero, leaving the intercept and slope form familiar from simple regression.

\[ \begin{aligned} \widehat{score} &= \hat{\beta}_0 + \hat{\beta}_1 \times bty\_avg + \hat{\beta}_2 \times (0) \\ &= \hat{\beta}_0 + \hat{\beta}_1 \times bty\_avg\end{aligned} \]

We can plot this line and the line corresponding to males with the following custom function.

multiLines(m_bty_gen)

  1. 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?

the equation of the line corresponding to males:

\[ \begin{aligned} \widehat{score} &= \hat{\beta}_0 + \hat{\beta}_1 \times bty\_avg + \hat{\beta}_2 \times (1) \\ &= \hat{\beta}_0 + \hat{\beta}_1 \times bty\_avg\end{aligned} \]

For two professors who received the same beauty rating, male tends to have the higher course evaluation score because variable gendermale that takes a value of 0 for females and a value of 1 for males.

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.)

  1. 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-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, 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.

  1. 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 guess cls_level have the highest p-value in this model because class level will not have any association with the professor score.

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)
## 
## Call:
## lm(formula = 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)
## 
## 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    
## ethnicitynot minority  0.1234929  0.0786273   1.571  0.11698    
## gender                 0.2109481  0.0518230   4.071 5.54e-05 ***
## 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
  1. Check your suspicions from the previous exercise. Include the model output in your response.

Although class level has the second highest p-value, the variable class professor:number of professors teaching sections in course in sample, has the highest p-value.

  1. Interpret the coefficient associated with the ethnicity variable.

The slope of ethnicity variable is 0.1234929. The standard error for the slope is 0.0786273. The t-test statistic for the null hypothesis is 1.571. The p-value for a two-sided alternative hypothes is: 0.11698

  1. 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_prof <- 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_drop_prof)
## 
## 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 .  
## gender                 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
After dropping cls_profs, the coefficients and significance of the other explanatory variables did not change. The dropped variable was not collinear with the other explanatory variables because dropping or adding a variable correlated with other variables will change the coefficients and p-value of the other explanatory variables.
  1. 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_backward <- lm(score ~ ethnicity + gender + language + age + cls_perc_eval 
             + cls_credits + bty_avg 
             + pic_outfit + pic_color, data = evals)

summary(m_backward)
## 
## Call:
## lm(formula = score ~ ethnicity + gender + 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 ***
## ethnicitynot minority  0.163818   0.075158   2.180 0.029798 *  
## gender                 0.202597   0.050102   4.044 6.18e-05 ***
## 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

linear model: \[\widehat{score} = 3.771922 + 0.167872\times ethnicity + 0.207112 \times gendermale - 0.206178\times (languagenon-english) -0.006046 \times age + 0.004656 \times (cls-perc-eval) + 0.505306\times (cls-creditsone credit) + 0.051069 \times bty-avg - 0.190579 \times (pic-colorcolor)\]

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

Normal probability plot.

qqnorm(m_backward$residuals)
qqline(m_backward$residuals)

The residuals of the model are nearly normal as shown in the QQ plot. While there are a few observations that deviate noticeably from the line, they are not particularly extreme.

Absolute values of residuals against fitted values (\(\hat{y_i}\)).

evals$language <- as.numeric(evals$language)
evals$language <- evals$language-1

evals$ethnicity <- relevel(evals$ethnicity,"minority")
evals$ethnicity <- as.numeric(evals$ethnicity)
evals$ethnicity <- evals$ethnicity-1

evals$cls_credits <- as.numeric(evals$cls_credits)
evals$cls_credits <- evals$cls_credits-1

evals$pic_color <- as.numeric(evals$pic_color)
evals$pic_color <- evals$pic_color-1
  
fitted_backward <- 3.771922 + 0.167872*evals$ethnicity + 0.207112 *evals$gender - 0.206178*evals$language -0.006046*evals$age + 0.004656*evals$cls_perc_eval + 0.505306*evals$cls_credits + 0.051069 *evals$bty_avg - 0.190579*evals$pic_color

plot(round(fitted_backward,1),m_backward$residuals,ylab="Absolute value of residuals", xlab="Fitted values")

The plot shows that the variance of the residuals is approximately constant except there are some deviations at high and low fitted values.

Residuals in order of their data collection is not applicable in this data set because we don’t know the sequence of the data collection.

Residuals against each predictor variable.

boxplot(m_backward$residuals~ethnicity,data=evals,ylab="Residuals", main="Ethnicity")

boxplot(m_backward$residuals~gender,data=evals,ylab="Residuals", main="Gender")

boxplot(m_backward$residuals~language,data=evals,ylab="Residuals", main="Language")

plot(m_backward$residuals~age,data=evals,ylab="Residuals", main="Age")

plot(m_backward$residuals~round(cls_perc_eval),data=evals,ylab="Residuals", main="Cls_perc_eval")

boxplot(m_backward$residuals~cls_credits,data=evals,ylab="Residuals", main="Cls_credits")

plot(m_backward$residuals~bty_avg,data=evals,ylab="Residuals", main="Bty_avg")

boxplot(m_backward$residuals~pic_color,data=evals,ylab="Residuals", main="Pic_color")

variability doesn't fluctuate across groups in ethnicity. However, looking at the variables gender, language, age, cls_perc_eval, class credits, bty_avg, and pic_color, we find that there is some difference in the variability of the residuals in the groups. There appears to be curvature in the residuals, indicating the relationship is probably
not linear.
  1. 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?
The courses are independent so the observations are independent even though some courses are taught by the same professor. However, if same student took two or more courses taught by the same professor, these observations will not independent. The independent observations condition will not be perfectly fit and it will affect the application of the linear model to the analysis.
  1. 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.
The professor who is male, not a minority, speaking English, younger, with higher average beauty rating, showing black&white picture,  teaching in the class offering one credit and in the class in which more students completed the evaluation will have high evaluation score.
summary(m_backward)
## 
## Call:
## lm(formula = score ~ ethnicity + gender + 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 ***
## ethnicitynot minority  0.163818   0.075158   2.180 0.029798 *  
## gender                 0.202597   0.050102   4.044 6.18e-05 ***
## 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
  1. Would you be comfortable generalizing your conclusions to apply to professors generally (at any university)? Why or why not?
I would not generalize my conclusiona to apply to professors at other university because this is an observational study and  the sample size is too small(6 students). In addtion, the rating is subjective and the judgement on beauty could be quite different from one school to the other school due to the different culture.

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.