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” (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
- 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.
–vb– This is an observational study. No experiment were set-up. The observations were gathered by end of class evaluations and questionair answered by 6 students pertaining on the beauty assessment of the professors.
Because this is not an experiment, we cannot conclude casual relationship. We would need to rephrase the question.
Question: Does beauty impact a professor evaluation or is the difference due to sampling variation?
- 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?
library(ggplot2)
ggplot(evals, aes(x=score)) + geom_histogram(binwidth = 0.5)
–vb–
From the histogram, we can see that the distribution of ‘score’ is skewed to the left, with the majority of the observation with score of between 4 and 5. This is as expected, most students will provide feeback with positive evaluation (4 or 5).
- Excluding
score, select two other variables and describe their relationship using an appropriate visualization (scatterplot, side-by-side boxplots, or mosaic plot).
–vb–
We will consider the following variables:cls_perc_evalandcls_level. We will plot percentage of sutdent that complete evaluation dependent on the class level.
plot(evals$cls_perc_eval ~ evals$cls_level)
The percentage of student completing the evaluation has less variability for upper level class (since the box plot is smaller) than for lower. There is no clear apparent difference for the median. There is a marked difference for the lower whiskers. For upper level class, the lower whicker ends higher with outliers. There are no outiers for lower class level.
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?
dim(evals)## [1] 463 21summary(evals)## score rank ethnicity gender
## Min. :2.300 teaching :102 minority : 64 female:195
## 1st Qu.:3.800 tenure track:108 not minority:399 male :268
## Median :4.300 tenured :253
## Mean :4.175
## 3rd Qu.:4.600
## Max. :5.000
## language age cls_perc_eval cls_did_eval
## english :435 Min. :29.00 Min. : 10.42 Min. : 5.00
## non-english: 28 1st Qu.:42.00 1st Qu.: 62.70 1st Qu.: 15.00
## Median :48.00 Median : 76.92 Median : 23.00
## Mean :48.37 Mean : 74.43 Mean : 36.62
## 3rd Qu.:57.00 3rd Qu.: 87.25 3rd Qu.: 40.00
## Max. :73.00 Max. :100.00 Max. :380.00
## cls_students cls_level cls_profs cls_credits
## Min. : 8.00 lower:157 multiple:306 multi credit:436
## 1st Qu.: 19.00 upper:306 single :157 one credit : 27
## Median : 29.00
## Mean : 55.18
## 3rd Qu.: 60.00
## Max. :581.00
## bty_f1lower bty_f1upper bty_f2upper bty_m1lower
## Min. :1.000 Min. :1.000 Min. : 1.000 Min. :1.000
## 1st Qu.:2.000 1st Qu.:4.000 1st Qu.: 4.000 1st Qu.:2.000
## Median :4.000 Median :5.000 Median : 5.000 Median :3.000
## Mean :3.963 Mean :5.019 Mean : 5.214 Mean :3.413
## 3rd Qu.:5.000 3rd Qu.:7.000 3rd Qu.: 6.000 3rd Qu.:5.000
## Max. :8.000 Max. :9.000 Max. :10.000 Max. :7.000
## bty_m1upper bty_m2upper bty_avg pic_outfit
## Min. :1.000 Min. :1.000 Min. :1.667 formal : 77
## 1st Qu.:3.000 1st Qu.:4.000 1st Qu.:3.167 not formal:386
## Median :4.000 Median :5.000 Median :4.333
## Mean :4.147 Mean :4.752 Mean :4.418
## 3rd Qu.:5.000 3rd Qu.:6.000 3rd Qu.:5.500
## Max. :9.000 Max. :9.000 Max. :8.167
## pic_color
## black&white: 78
## color :385
##
##
##
## –vb–
There are 463 rows in the data frame, each representing an observation. Also, from the summary function results, there does not seem any missing values that would be represented as ‘NA’. We would expect the scatter plot to have 463 points. However, it appears that the scatter plot only have 250 points represented.
—–
- Replot the scatterplot, but this time use the function
jitter()on the \(y\)- or the \(x\)-coordinate. (Use?jitterto learn more.) What was misleading about the initial scatterplot?
–vb–
We will had a small amount of noise the ‘score’ variable to break any possible “ties”.
plot(jitter(evals$score) ~ evals$bty_avg)
It appears that quite a large number of points had the same values for (x,y), hence, they could not be differenciated on the scatter plot. By adding a small amount of noise on the score variable (y), we can now differentiate the points.
- 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 and add the line to your plot usingabline(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(score ~ bty_avg, data = evals)
plot(jitter(evals$score) ~ evals$bty_avg)
abline(m_bty)
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-05The equation of the regression line:
\(\hat { y } =\quad { b }_{ 0 }\quad +\quad { b }_{ 1 }\quad \times \quad x\quad =\quad 3.88034\quad +\quad 0.06664\quad \times \quad x\)
Slope interpretation:
When the average beauty score of the professor goes up by 1, we would expect the rating evaluation to go up by .06664.
Statistical Significance:
Looking at the p value for the t score, we can say that the average beauty score is statiscally significant p(>|t|) = 5.08 x 10-5.
We would not however find this to be a practically significant predictor. Since this must be evaluated additionally of the professor evaluation. In our day and age, “evaluating beauty” may cause some controversy and possibly not all respondent would reply. we would imagine that it may difficult to have evaluation across gender line (i.e. male evaluating male, female evaluating female). Finally, While the debats is on whether there are universal metrics of beauty, are cultural and individual elements. Therefore, it would be difficult to accurately value this variable.
R2 is about 3.3%, hence, 3.3% of evaluations can be predicted accurately with the model.
- 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).
plot(m_bty$residuals ~ evals$bty_avg)
abline(h = 0, lty = 4) # adds a horizontal dashed line at y = 0
#Historgream
hist(m_bty$residuals)
# normal probability plot of the residuals
qqnorm(m_bty$residuals)
qqline(m_bty$residuals)–vb–
Conditions for the least squares line
Linearity:
Though it is difficult to tell with so many points, but it appear that the data show a slightly linear positive linearity.
Nearly Normal residuals:
From the Histogram, the residuals show a slightly left skewed distribution. The normal probability plot of the residuals shows that the points do not follow the line for upper quadriles.
Constant Variability: From the residual plot, we can observe that there seems to have constant variability.
Independent observations: We do not have much information on how the sample was taken. We can assume indenpendence of the observations.
Multiple linear regression
The data set contains several variables on the beauty score of the professor: individual ratings from each of the six students who were asked to score the physical appearance of the professors and the average of these six scores. Let’s take a look at the relationship between one of these scores and the average beauty score.
plot(evals$bty_avg ~ evals$bty_f1lower)
cor(evals$bty_avg, evals$bty_f1lower)## [1] 0.8439112As expected the relationship is quite strong - after all, the average score is calculated using the individual scores. We can actually take a look at the relationships between all beauty variables (columns 13 through 19) using the following command:
plot(evals[,13:19])These variables are collinear (correlated), and adding more than one of these variables to the model would not add much value to the model. In this application and with these highly-correlated predictors, it is reasonable to use the average beauty score as the single representative of these variables.
In order to see if beauty is still a significant predictor of professor score after we’ve accounted for the gender of the professor, we can add the gender term into the model.
m_bty_gen <- lm(score ~ bty_avg + gender, data = evals)
summary(m_bty_gen)- P-values and parameter estimates should only be trusted if the conditions for the regression are reasonable. Verify that the conditions for this model are reasonable using diagnostic plots. –vb–
# Normal Probability Plot
qqnorm(m_bty_gen$residuals)
qqline(m_bty_gen$residuals)
# residual plot against each predictor variable
plot(m_bty_gen$residuals ~ evals$bty_avg)
abline(h = 0, lty = 4) # adds a horizontal dashed line at y = 0
plot(m_bty_gen$residuals ~ evals$gender)
abline(h = 0, lty = 4) # adds a horizontal dashed line at y = 0
# 4 plots: Resiual vs Fitted, Normal Probability Plot, Scale-Location, Residual vs Leverage
plot(m_bty_gen)
#Historgream
hist(m_bty_gen$residuals)
# Checking linearlidity
plot(jitter(evals$score) ~ evals$bty_avg)
plot(evals$score ~ evals$gender)From the histogram of residuals, we can see that the residuals distribution is slightly skewed to the left. Looking at the Normal Probability Plot for residuals, the residuals do not follow the lines for upper quadriles. Finally, Residuals vs Fitted, show that it appears to be constant variability for residuals.
The conditions are reasonable although, residuals are showing possible outliers and the Normal Probability plot shows breakdown, specially in the upper quadriles.
We will assume indedence, we have no information on how the sample was taken or whether we can look at the collection order of the residuals to show independence of residuals.
- Is
bty_avgstill a significant predictor ofscore? Has the addition ofgenderto the model changed the parameter estimate forbty_avg?
–vb– Adjusted R2 = 0.055, The parameter estimage for bty_avg = 0.07416
Note that the estimate for gender is now called gendermale. You’ll see this name change whenever you introduce a categorical variable. The reason is that R recodes gender from having the values of 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)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?
\(\hat { score } =\quad 3.74734\quad +\quad 0.07416\quad \times \quad beauty\_ avg\quad +\quad 0.17239\quad \times \quad gender\_ male\)
For gender = Male, we will evaluate the equation with gender_male = 1. In case, of female gender, we will substitute a 0.
\(\hat { score } =\quad 3.74734\quad +\quad 0.07416\quad \times \quad beauty\_ avg\quad +\quad 0.17239\)
Male professor will have a evaluation score higher by 0.17239 all other things being equal.
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_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.
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-05For variable with more than 2 levels, it appears to handle it considerering them 2 different variables.
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.
- 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. –vb– Possibly, number of professors teaching sections in course in sample: single, multiple, since the evaluation are done within a class/section. Whether the professor is teaching multiple sections should not have an impact a given class evaluation.
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
## gendermale 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-14Check your suspicions from the previous exercise. Include the model output in your response.
The p value for this variable, is 0.77806 and is the highest in the model.
Interpret the coefficient associated with the ethnicity variable.
All other things being equal, Evaluation for professor that not minority tends to be 0.1234929 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?
m_full_1 <- 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_1)##
## 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-14m_full$coefficients - m_full_1$coefficients## Warning in m_full$coefficients - m_full_1$coefficients: longer object
## length is not a multiple of shorter object length## (Intercept) ranktenure track ranktenured
## 7.961761e-03 8.133220e-05 4.512112e-05
## ethnicitynot minority gendermale languagenon-english
## -3.952838e-03 8.249882e-04 -1.521743e-03
## age cls_perc_eval cls_students
## -8.003969e-06 3.847644e-05 -1.408227e-05
## cls_levelupper cls_profssingle cls_creditsone credit
## -1.234699e-04 -5.207815e-01 4.621803e-01
## bty_avg pic_outfitnot formal pic_colorcolor
## 1.483560e-01 1.063710e-01 -4.304515e+00The coefficients and significance changed slightly. Since the values changed, the drop variable was slightly collinear with the other explanatory variables.
- 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_best <- lm(score ~ ethnicity + gender + language + age + cls_perc_eval
+ cls_credits + bty_avg
+ pic_color, data = evals)
summary(m_full_best)##
## Call:
## lm(formula = score ~ ethnicity + gender + language + age + cls_perc_eval +
## cls_credits + bty_avg + pic_color, data = evals)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.85320 -0.32394 0.09984 0.37930 0.93610
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 3.771922 0.232053 16.255 < 2e-16 ***
## ethnicitynot minority 0.167872 0.075275 2.230 0.02623 *
## gendermale 0.207112 0.050135 4.131 4.30e-05 ***
## languagenon-english -0.206178 0.103639 -1.989 0.04726 *
## age -0.006046 0.002612 -2.315 0.02108 *
## cls_perc_eval 0.004656 0.001435 3.244 0.00127 **
## cls_creditsone credit 0.505306 0.104119 4.853 1.67e-06 ***
## bty_avg 0.051069 0.016934 3.016 0.00271 **
## pic_colorcolor -0.190579 0.067351 -2.830 0.00487 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.4992 on 454 degrees of freedom
## Multiple R-squared: 0.1722, Adjusted R-squared: 0.1576
## F-statistic: 11.8 on 8 and 454 DF, p-value: 2.58e-15\(\hat { score } =\quad 3.771922\quad +\quad 0.051069\cdot beauty\_ avg\quad -\quad 0.006272\cdot age\quad +\quad 0.004524\cdot Stud%completeEval\quad +\quad 0.159237\cdot notminority\quad +\quad 0.211402\cdot gender\_ male\quad -0.221755\cdot langnon-english\quad +\quad 0.038055\cdot upperlvl\quad +\quad 0.528038\cdot onecrd\quad -\quad 0.202161\cdot colorpic\)
- Verify that the conditions for this model are reasonable using diagnostic plots.
# Normal Probability Plot
qqnorm(m_full_best$residuals)
qqline(m_full_best$residuals)
#Historgream
hist(m_full_best$residuals) 
# 4 plots: Resiual vs Fitted, Normal Probability Plot, Scale-Location, Residual vs Leverage
plot(m_full_best)
# Checking linearlidity
plot(jitter(evals$score) ~ evals$bty_avg)
plot(jitter(evals$score) ~ evals$gender)
plot(jitter(evals$score) ~ evals$ethnicity)
plot(jitter(evals$score) ~ evals$gender)
plot(jitter(evals$score) ~ evals$language)
plot(jitter(evals$score) ~ evals$age)
plot(jitter(evals$score) ~ evals$cls_perc_eval)
plot(jitter(evals$score) ~ evals$cls_credits)
plot(jitter(evals$score) ~ evals$pic_color)
The residuals of the model are nearly model, although we have teh same breakdown as previously for upper quartiles when looking at normal qq plot.
The Variability of the residuals is nearly constant (Residuals vs Fitted)
Each variability is linearly related (scatter plot for age, bty_avg, cls_perc_eval)
we will assume indedendence, we have limited information on how original sample was selected and we cannot infere independency of residuals by looking at order of collection.
- 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?
–vb– I am not sure I understand the question
I would think not, I do not see how would this impact the model.
- 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.
Professor would be younger male teaching one credit class, he would not belong to a minority group, he would have received this degree from a universtity where english is the language. The professor would have a black and white picture and who have been rated beautifull.
- Would you be comfortable generalizing your conclusions to apply to professors generally (at any university)? Why or why not?
No, this was not conducted as an experiment but based on a sample in a given university. I would also argue that this may be time-based. As cultural value changes, these results may be different in other university or in a different time frame.