0.1 Install R packages as per DATA606Spring2019-master/R/Setup.R

Refer to “Getting Started with R” in https://data606.net/post/

0.2 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.

0.3 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.

0.4 Exploring the data

0.4.1 Exercise 1

  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.

    paste0("This is an observational study since there are no control and experimental groups and there cannot be causation between the explanatory and response variables, instead there can only be a correlation. There could be multiple factors that lead to a different evaluation score that have nothing to do with a professor's attractiveness. It would be better to rephrase the question as - 'Is beauty of the professor correlated to differences in course evaluation?'. Given that other factors are not captured in the study it will be difficult to answer the question as it is phrased originally")
    ## [1] "This is an observational study since there are no control and experimental groups and there cannot be causation between the explanatory and response variables, instead there can only be a correlation. There could be multiple factors that lead to a different evaluation score that have nothing to do with a professor's attractiveness. It would be better to rephrase the question as - 'Is beauty of the professor correlated to differences in course evaluation?'. Given that other factors are not captured in the study it will be difficult to answer the question as it is phrased originally"

    0.4.1 Exercise 2

  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?

    hist(evals$score, main = "Histogram of Scores", xlab = "Score")

    paste0("The distribution seems to be left skewed. It's expected as not many students tend to rate professors at lower end of the rating. Students have far more positive evaluations than negative evaluations for their teachers, which is not as expected. We expected a normal distribution where most teachers would be rated as average and fewer teachers will be evaluated in the extremes - excellent or unsatisfactory.")
    ## [1] "The distribution seems to be left skewed. It's expected as not many students tend to rate professors at lower end of the rating. Students have far more positive evaluations than negative evaluations for their teachers, which is not as expected. We expected a normal distribution where most teachers would be rated as average and fewer teachers will be evaluated in the extremes - excellent or unsatisfactory."

    0.4.1 Exercise 3

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

    plot(evals$cls_perc_eval ~ evals$cls_level)

    paste0("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 whisker ends higher with outliers. There are no outiers for lower class level.")
    ## [1] "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 whisker ends higher with outliers. There are no outiers for lower class level."
    #boxplot(evals$bty_avg ~ evals$language, main = "Boxplot of Beauty Average Score by Language", ylab = "Beauty Average", xlab = "Language")
#boxplot(evals$bty_avg ~ evals$age)

0.5 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?

0.5.1 Exercise 4

  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?

    par(mfrow=c(1,2))
plot(evals$score ~ evals$bty_avg)
plot(jitter(evals$score) ~ evals$bty_avg)

    paste0("Many points have the same values for (x,y). So they could not be differenciated on the scatter plot. ")
    ## [1] "Many points have the same values for (x,y). So they could not be differenciated on the scatter plot. "
    ggplot(evals, aes(bty_avg, score)) + geom_point(position = position_jitter(w = 0.3, h = 0.3)) + ylab("score") + xlab("beauty average")

    paste0("As the earlier plot many points overlapping it didn't provide true picture of distribution")
    ## [1] "As the earlier plot many points overlapping it didn't provide true picture of distribution"

0.5.2 Exercise 5

  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(score ~ bty_avg, data = evals)
plot(jitter(evals$score) ~ evals$bty_avg)
abline(m_bty)

    \[y_{score} = b_{0} + b_{1}*x\]

    paste0("score = 3.88034 + 0.06664 * average beauty")
    ## [1] "score = 3.88034 + 0.06664 * average beauty"
    paste0("For each point increase in beauty rating the professor evaluation score increases by 0.06664 points. Though the average beauty score does appear to be a significant predictor, the model has a very low R-squared value, which implies that this model is not appropriate for the data. As such, beauty average may not be a practically significant predictor.")
    ## [1] "For each point increase in beauty rating the professor evaluation score increases by 0.06664 points. Though the average beauty score does appear to be a significant predictor, the model has a very low R-squared value, which implies that this model is not appropriate for the data. As such, beauty average may not be a practically significant predictor."

0.5.3 Exercise 6

  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).

    plot(m_bty$residuals ~ evals$bty_avg)
abline(h = 0, lty = 4)  # adds a horizontal dashed line at y = 0

    qqnorm(m_bty$residuals)
qqline(m_bty$residuals)

    paste0("The residual seems to be nearly normal and the points have constant variability around linear line. The observations are independent of each other and there is a slight linear trend. So, the conditions for least squares regression are reasonable.")
    ## [1] "The residual seems to be nearly normal and the points have constant variability around linear line. The observations are independent of each other and there is a slight linear trend. So, the conditions for least squares regression are reasonable."

Conditions for the least squares line

  • Linearity: The data show a slightly linear and it is 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.

0.6 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)

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)
## 
## 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

0.6.1 Exercise 7

  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.

    • Are the residuals of the model are nearly normal?
    qqnormsim(m_bty_gen$residuals)

    #qqnorm(m_bty_gen$residuals)
#qqline(m_bty_gen$residuals)

    The residuals of the model is not perfectly normal as residual values for the the higher quantiles are less than what a normal distribution would predict

    • Are absolute values of residuals against fitted values. (the variability of the residuals is nearly constant)
    plot(abs(m_bty_gen$residuals) ~ m_bty_gen$fitted.values)

    There some outliers although overall, most of the residual values are close to the fitted values.

    • The residuals are independent
    plot(m_bty_gen$residuals ~ c(1:nrow(evals)))

    Yes, this condition is met. The residuals based on the sequence when it was gathered shows that they were randomly gathered.

    • Each variable is linearly related to the outcome.
    par(mfrow=c(1,2))
plot(evals$score ~ evals$gender)
plot(evals$score ~ evals$bty_avg)

    There is a is a linear relationship between gender and evaluation score. The median scores and variability for both males and females are similar in terms of evaluation scores. As was established in the previous exercies, there is a linear relationship between beauty average and teaching evaluation score.

    paste0("The residuals are nearly normal and points have constant variability around regression line. The observations are independent of each other. It is difficult to spot a trend in score in relation to categorical variable but it's reasonale to assume a linear trend. So, the conditions for least squares regression are reasonable.")
    ## [1] "The residuals are nearly normal and points have constant variability around regression line. The observations are independent of each other. It is difficult to spot a trend in score in relation to categorical variable but it's reasonale to assume a linear trend. So, the conditions for least squares regression are reasonable."

0.6.2 Exercise 8

  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?

    paste0("bty_avg is still a significant predictor of score. In fact the addition of gender has increased the significance of bty_avg")
    ## [1] "bty_avg is still a significant predictor of score. In fact the addition of gender has increased the significance of bty_avg"

    Yes it is. In fact, gender made beauty average even more significant as the p-value computed is even smaller now compared to a model where beauty average was the sole variable.

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)

0.6.3 Exercise 9

  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?

    paste0("Males tend to have a higher course evaluation score than females for professors who get the same rating. For males the equation is score = 3.74734 + 0.07416 * Average Beauty + 0.17239 For two professors who received the same beauty rating male professor tends to have the higher course evaluation score")
    ## [1] "Males tend to have a higher course evaluation score than females for professors who get the same rating. For males the equation is score = 3.74734 + 0.07416 * Average Beauty + 0.17239 For two professors who received the same beauty rating male professor tends to have the higher course evaluation score"

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.)

0.6.4 Exercise 10

  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
    multiLines(m_bty_rank)

    paste0("R splits the variable by categories and leaves out one of the categories. When all the included categories are zero it implies that the left out category is true. ")
    ## [1] "R splits the variable by categories and leaves out one of the categories. When all the included categories are zero it implies that the left out category is true. "

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.

0.7 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.

0.7.1 Exercise 11

  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.

    paste0("It is expected that total number of professors `cls_profs` in the class to have large p-value as I expect it to affect the score least.")
    ## [1] "It is expected that total number of professors `cls_profs` in the class to have large p-value as I expect it to affect the score least."

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-14

0.7.2 Exercise 12

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

    paste0("While total number of student have large p-value is true, the one which has the highest p-value is number of professors teaching sections in course in sample: single, multiple.")
    ## [1] "While total number of student have large p-value is true, the one which has the highest p-value is number of professors teaching sections in course in sample: single, multiple."
    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-14
    plot(evals$score ~ evals$cls_profs)

0.7.3 Exercise 13

  1. Interpret the coefficient associated with the ethnicity variable.

    paste0("Professor who are not from minority ethnicity tend to j=have p-value score of 0.1234929 more given all other parameters are same which means that it has a weak relationship to scores and may be dropped as part of the model_. But, given large p-value it's not statistically significant")
    ## [1] "Professor who are not from minority ethnicity tend to j=have p-value score of 0.1234929 more given all other parameters are same which means that it has a weak relationship to scores and may be dropped as part of the model_. But, given large p-value it's not statistically significant"

0.7.4 Exercise 14

  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?

    minus_ethn = lm(score ~ rank + gender + language + age + cls_perc_eval + 
                cls_students + cls_level + cls_profs + cls_credits + bty_avg + pic_outfit + 
                pic_color, data = evals)
summary(minus_ethn)
    ## 
## Call:
## lm(formula = score ~ rank + 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 
## -1.73681 -0.32734  0.08283  0.35834  0.98639 
## 
## Coefficients:
##                         Estimate Std. Error t value Pr(>|t|)    
## (Intercept)            4.2676351  0.2694274  15.840  < 2e-16 ***
## ranktenure track      -0.1660677  0.0813523  -2.041 0.041801 *  
## ranktenured           -0.1127978  0.0657022  -1.717 0.086705 .  
## gendermale             0.2241744  0.0512176   4.377 1.50e-05 ***
## languagenon-english   -0.2862448  0.1055924  -2.711 0.006968 ** 
## age                   -0.0092040  0.0031385  -2.933 0.003534 ** 
## cls_perc_eval          0.0051119  0.0015357   3.329 0.000944 ***
## cls_students           0.0004785  0.0003777   1.267 0.205899    
## cls_levelupper         0.0767503  0.0567182   1.353 0.176677    
## cls_profssingle       -0.0292174  0.0512393  -0.570 0.568817    
## cls_creditsone credit  0.4589918  0.1128358   4.068 5.61e-05 ***
## bty_avg                0.0375980  0.0174661   2.153 0.031880 *  
## pic_outfitnot formal  -0.1208610  0.0738165  -1.637 0.102265    
## pic_colorcolor        -0.2400696  0.0701264  -3.423 0.000675 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.4988 on 449 degrees of freedom
## Multiple R-squared:  0.1826, Adjusted R-squared:  0.159 
## F-statistic: 7.717 on 13 and 449 DF,  p-value: 6.792e-14
    paste0("The significance and coefficient of the some of the variables changed when cls_profs was removed. This indicates that the dropped variable could be collinear with variables whose significance has increased")
    ## [1] "The significance and coefficient of the some of the variables changed when cls_profs was removed. This indicates that the dropped variable could be collinear with variables whose significance has increased"

0.7.5 Exercise 15

  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.

    backwards = lm(score ~ ethnicity + gender + language + age + cls_perc_eval + cls_credits + bty_avg + pic_color, 
               data = evals)
summary(backwards)
    ## 
## 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
    paste0("The best model is score = 3.772 + 0.168 * ethnicitynot minority + .207 * gendermale - 0.206178 * languagenon-english - 0.006 * age + .00466 * cls_perc_eval + .505 * cls_creditsone credit + .051069 * bty_avg - 0.19059 * pic_colorcolor")
    ## [1] "The best model is score = 3.772 + 0.168 * ethnicitynot minority + .207 * gendermale - 0.206178 * languagenon-english - 0.006 * age + .00466 * cls_perc_eval + .505 * cls_creditsone credit + .051069 * bty_avg - 0.19059 * pic_colorcolor"

0.7.6 Exercise 16

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

    • Are the residuals of the model are nearly normal?
    qqnormsim(backwards$residuals)

    # Normal Probability Plot
#qqnorm(backwards$residuals)
#qqline(backwards$residuals)
paste0("The conditions for this model are reasonable given that the residuals are nearly normal")
    ## [1] "The conditions for this model are reasonable given that the residuals are nearly normal"

    The residuals of the model are not perfectly normal as residual values for the the higher and lower quantiles are less than what a normal distribution would predict

    • Are absolute values of residuals against fitted values.
    par(mfrow=c(1,2))
plot(abs(backwards$residuals) ~ backwards$fitted.values)
plot(backwards)

    abline(h = 0, lty = 3)

    There some outliers although overall, most of the residual values are close to the fitted values.

    • Residuals are independent
    plot(backwards$residuals ~ c(1:nrow(evals)))

    Yes, this condition is met. The residuals based on the sequence when it was gathered shows that they were randomly gathered.

    • Each variable is linearly related to the outcome.
    par(mfrow=c(4,2))
plot(evals$score ~ evals$ethnicity)
plot(evals$score ~ evals$gender)
plot(evals$score ~ evals$language)
plot(evals$score ~ evals$age)
plot(evals$score ~ evals$cls_perc_eval)
plot(evals$score ~ evals$cls_credits)
plot(evals$score ~ evals$bty_avg)
plot(evals$score ~ evals$pic_color)

    Yes, this condition is met. The residuals based on the sequence when it was gathered shows that they were randomly gathered.

0.7.7 Exercise 17

  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?

    paste0("No. Class courses are independent of each other so evaluation scores from one course is indpendent of the other even if the course is being taught by the same professor.")
    ## [1] "No. Class courses are independent of each other so evaluation scores from one course is indpendent of the other even if the course is being taught by the same professor."

0.7.8 Exercise 18

  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.

    paste0("Male professor from non-minority ethnicity who received education in English and teaches one credit class with color picture tends to score high evaluation score. He must also have a high beauty average score from the students and the professor's class photo should be in black and white. He must also be relatively young. And a good percentage of his class must have completed the evaluation.")
    ## [1] "Male professor from non-minority ethnicity who received education in English and teaches one credit class with color picture tends to score high evaluation score. He must also have a high beauty average score from the students and the professor's class photo should be in black and white. He must also be relatively young. And a good percentage of his class must have completed the evaluation."

0.7.9 Exercise 19

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

    paste0("No - It won't be comfortable in generalizing the conclusion as the sample may not be representative of entire population. The sample size of 6 is too small. Also, some of the predictor variables are subjective and may vary with culture. Beauty, for one, is in the eye of the beholder.Picture preferences may also be culturally biased.")
    ## [1] "No - It won't be comfortable in generalizing the conclusion as the sample may not be representative of entire population. The sample size of 6 is too small. Also, some of the predictor variables are subjective and may vary with culture. Beauty, for one, is in the eye of the beholder.Picture preferences may also be culturally biased."