Jun Pan DATA 606 lab 8
Jun Pan
December 4, 2018
Set working environment
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## extractrequire(knitrequ)## Loading required package: knitrequ## Warning in library(package, lib.loc = lib.loc, character.only = TRUE,
## logical.return = TRUE, : there is no package called 'knitrequ'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("C:/Users/tbao/Desktop/DATA606-master/inst/labs/Lab8/more/evals.RData")Execise 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. To redesign, the better way to rephrase the question is "Does the appealing of professors affect on students' evaluation?". For better evalation the correlation of appealing with evaluation, we need to randomly assign students to appealing professors or unappealing professors.Execise 2:Describe the distribution of score. Is the distribution skewed? What does that tell you about how students rate courses? Is this what you expected to see? Why, or why not?summary(evals$score)## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 2.300 3.800 4.300 4.175 4.600 5.000hist(evals$score)
The distribution of score is left skewed with a median of 4.3 and mean of 4.175. This is what I expect to see. I believe most of teachers selected by school based on their credential. The school is not going to random select a tearch for students.
Excecise 3:Excluding score, select two other variables and describe their relationship using an appropriate visualization (scatterplot, side-by-side boxplots, or mosaic plot).ggplot(data=evals, aes(y=bty_avg, x=ethnicity)) +
geom_boxplot() +
xlab("Beauty Rating Scores") +
ylab("Ethnicity Status")
We could not find too much difference of beauty scores between the minority group and not minority groups.
ggplot(data=evals, aes(y=bty_avg, x=cut(age, breaks = 3))) +
geom_boxplot() +
ylab("Beauty Rating Scores") +
xlab("Age") +
ggtitle("Beaty Rating Scrores among Different Age Groups")
It appears that the beauty rating scores decreased by increasing of age.
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?Execise 4.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(jitter(evals$score) ~ jitter(evals$bty_avg), ylab="Evaluation Score", xlab="Beauty Rating Cores",
main="Evaluation Scores vs Beauty Rating Cores")
For the jitter plot, it more clearly shows the overlap points than the original one.
Execise 5.Let's see if the apparent trend in the plot is something more than natural variation. Fit a linear model called m_bty to predict average professor score by average beauty rating 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(score ~ bty_avg, data = evals)
plot(jitter(evals$score) ~ jitter(evals$bty_avg), ylab="Evaluation Score", xlab="Beauty Rating Cores",
main="Evaluation Scores vs Beauty Rating Cores")
abline(m_bty)
m_bty$coefficients## (Intercept) bty_avg
## 3.88033795 0.06663704The linear model equation is \(\widehat{score}=0.06664\times beauty+3.88034\).
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 slope indicates that for each increase in beauty average by 1 point, a teacher can expected an increase of 0.06664 points on their evaluations average. The p-value on the slope is suitably small. This provides strong evidence of a relationship between the two variables.
Execise 6. Use residual plots to evaluate whether the conditions of least squares regression are reasonable. Provide plots and comments for each one (see the Simple Regression Lab for a reminder of how to make these).ggplot(evals) +
geom_point(aes(bty_avg, m_bty$residuals)) +
geom_hline(yintercept=0, color='blue')
ggplot() +
geom_histogram(aes(m_bty$residuals), bins=20, alpha=0.7, color='blue')
The residual plot shows no pattern and consistent variability. The histogram of the residuals shows a left skew.
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)##
## 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-07Execise 7.P-values and parameter estimates should only be trusted if the conditions for the regression are reasonable. Verify that the conditions for this model are reasonable using diagnostic plots.<img src="Jun_Pan_DATA_606_Lab_8_files/figure-html/unnamed-chunk-14-1.png" width="672" /><img src="Jun_Pan_DATA_606_Lab_8_files/figure-html/unnamed-chunk-14-2.png" width="672" />hist(m_bty_gen$residuals, xlab="Residuals", main="Histogram of Residuals")
qqnorm(m_bty_gen$residuals)
qqline(m_bty_gen$residuals) 
Linearity : From above plots, this model shows linearity.
Near normal residuals: The residual plots shows near normal pattern. The histogram shows a left skew.
Constant variability: Constant variability is not certain.Execise 8.Is bty_avg still a significant predictor of score? Has the addition of gender to the model changed the parameter estimate for bty_avg?`bty_avg` is still a significant predictor factor of evaluation score. The addition of `gender` to the model has changed the size of its effect.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.
score^=??^0+??^1×bty_avg+??^2×(0)=??^0+??^1×bty_avg
We can plot this line and the line corresponding to males with the following custom function.multiLines(m_bty_gen)
Execise 9: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 line corresponding to males is the blue line which is above the yellow line (female). Thus, Men tend to have a higher rating than Female of equal beauty. The size of this effect is calculated to be 0.17239 points.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.)Execise 10.Create a new model called m_bty_rank with gender removed and rank added in. How does R appear to handle categorical variables that have more than two levels? Note that the rank variable has three levels: teaching, tenure track, tenured.m_bty_rank <- lm(score ~ bty_avg + rank, data = evals)
summary(m_bty_rank)##
## Call:
## lm(formula = score ~ bty_avg + rank, data = evals)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.8713 -0.3642 0.1489 0.4103 0.9525
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 3.98155 0.09078 43.860 < 2e-16 ***
## bty_avg 0.06783 0.01655 4.098 4.92e-05 ***
## ranktenure track -0.16070 0.07395 -2.173 0.0303 *
## ranktenured -0.12623 0.06266 -2.014 0.0445 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.5328 on 459 degrees of freedom
## Multiple R-squared: 0.04652, Adjusted R-squared: 0.04029
## F-statistic: 7.465 on 3 and 459 DF, p-value: 6.88e-05The regression spreads the data over 2 columns. we can assign “1”" in the “tenure track” column or “tenured” column when appropriate. If both are “0”“, then the teacher is in the”teaching“.
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.Execise 11. 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 expect that cls_perc_eval or "percent of students in class who completed evaluation." has no correlation with the prefessor's evaluation score. Therefore, it should have the highest p-value.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-14Execise 12. Check your suspicions from the previous exercise. Include the model output in your response.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-14It never think about there is correlation between cls_perc_eval and score. But the p is <0.05. I guess that students who are not satisfied with the professor would be less likely to complete their evaluations.
Execise 13. Interpret the coefficient associated with the ethnicity variable.If the coefficient is 0 that means the professor is is a minority while if it is a 1 that means the professor is not a minority. Teachers that are not minorities generally have slightly higher scores than those who are minorities.
Execise 14.Drop the variable with the highest p-value and re-fit the model. Did the coefficients and significance of the other explanatory variables change? (One of the things that makes multiple regression interesting is that coefficient estimates depend on the other variables that are included in the model.) If not, what does this say about whether or not the dropped variable was collinear with the other explanatory variables?m_full2 <- 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_full2)##
## 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[c(1:10, 12:15)] - m_full2$coefficients## (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_creditsone credit bty_avg
## -1.234699e-04 -4.076375e-03 1.704157e-04
## pic_outfitnot formal pic_colorcolor
## -4.358943e-03 1.789689e-03Yes, the coefficients and significance of the other explanatory variables changed when cls_profs was dropped from the model. If they had not changed that would indicate that there was no colinnearity between cls_profs and the other predictor variables.
Execsie 15. Using backward-selection and p-value as the selection criterion, determine the best model. You do not need to show all steps in your answer, just the output for the final model. Also, write out the linear model for predicting score based on the final model you settle on.m_full3 <- 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_full3)##
## 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-14cls_profs is the only category that can be removed from the regression. Remove other categories leaded to a fall in the adjusted \(R^2\).
Execise 16.Verify that the conditions for this model are reasonable using diagnostic plots.Linearity of residuals
ggplot(evals) +
geom_point(aes(bty_avg, m_full$residuals)) +
geom_hline(yintercept=0, color='blue')
Normal distribution of residuals
ggplot() +
geom_histogram(aes(m_full$residuals), bins=20, alpha=0.7, color='blue')
Independence of residuals
ggplot() +
geom_point(aes(1:463, m_full$residuals))
From above plots, we can see that the conditions for this model are reasonable.
Equal variance of residuals
ggplot() +
geom_point(aes(m_full$fitted.values, abs(m_full$residuals)))
Execise 17.The original paper describes how these data were gathered by taking a sample of professors from the University of Texas at Austin and including all courses that they have taught. Considering that each row represents a course, could this new information have an impact on any of the conditions of linear regression?Yes, this new information have an impact on linear regression, Because if the professor teaches more couses, it will weight more on the results than the professor teaches less courses.
Execise 18.Based on your final model, describe the characteristics of a professor and course at University of Texas at Austin that would be associated with a high evaluation score.Based on the summary, the high-score prof characteristics would be on teaching track, non-minority, male, Native English speaking, young, teaching upper level courses with one credit and physically appealing.
Execise 19. Would you be comfortable generalizing your conclusions to apply to professors generally (at any university)? Why or why not?We can not apply to professors generally, beecause this is a non-rondomized observational study.