Lab8
Farhana Zahir
##The data
library(ggplot2)
load("lab8/more/evals.RData")
#Exploring the data
##Ex1: 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.
##The study is an observational one. Given it is not an experiment, I do not think it is possible to answer the question as it is phrased.
##Rephrased question: 'Does beauty influence student rating?'
##Ex2: 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$score)
##The distribution is skewed to the right. It suggests students usually give a higher than average scores to the professors. I expected a symmetrical distribution where most students would have rated their professors average.
##Ex3: Excluding score, select two other variables and describe their relationship using an appropriate visualization (scatterplot, side-by-side boxplots, or mosaic plot).
boxplot(evals$score ~ evals$gender)
##Based on the plot, male faculties tend to have a higher evaluation score than the females.
#Simple linear regression
plot(evals$score ~ evals$bty_avg)
##There are 463 observations of evaluations.
##Ex4:Replot the scatterplot, but this time use the function jitter() on the y- or the x-coordinate. (Use ?jitter to learn more.) What was misleading about the initial scatterplot?
plot(jitter(evals$score) ~ jitter(evals$bty_avg))
##The initial model had several data points on top of each other. Jitter moves the points a little bit so they are not exactly on top of each other.
##Ex5: Let’s see if the apparent trend in the plot is something more than natural variation. Fit a linear model called m_bty to predict average professor score by average beauty rating and add the line to your plot using abline(m_bty). Write out the equation for the linear model and interpret the slope. Is average beauty score a statistically significant predictor? Does it appear to be a practically significant predictor?
m_bty <- lm(score ~ bty_avg, data = evals)
plot(jitter(evals$score) ~ jitter(evals$bty_avg))
abline(m_bty)
m_bty##
## Call:
## lm(formula = score ~ bty_avg, data = evals)
##
## Coefficients:
## (Intercept) bty_avg
## 3.88034 0.06664summary(m_bty)##
## Call:
## lm(formula = score ~ bty_avg, data = evals)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.9246 -0.3690 0.1420 0.3977 0.9309
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 3.88034 0.07614 50.96 < 2e-16 ***
## bty_avg 0.06664 0.01629 4.09 5.08e-05 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.5348 on 461 degrees of freedom
## Multiple R-squared: 0.03502, Adjusted R-squared: 0.03293
## F-statistic: 16.73 on 1 and 461 DF, p-value: 5.083e-05##Score=3.88034 + 0.06664bty_avg
##For one unit increase in beauty score, evaluation score goes up by 0.06664 units
##R square is very low so this does not appear to be a practically significant predictor.
##Ex6: Use residual plots to evaluate whether the conditions of least squares regression are reasonable. Provide plots and comments for each one (see the Simple Regression Lab for a reminder of how to make these).
plot(m_bty$residuals ~ evals$bty_avg)
abline(h = 0)
##The residuals appear to be nearly normal, the relationship looks linear.
#Multiple Linear Regression
plot(evals$bty_avg ~ evals$bty_f1lower)
cor(evals$bty_avg, evals$bty_f1lower)## [1] 0.8439112plot(evals[,13:19])
m_bty_gen <- lm(score ~ bty_avg + gender, data = evals)
summary(m_bty_gen)##
## Call:
## lm(formula = score ~ bty_avg + gender, data = evals)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.8305 -0.3625 0.1055 0.4213 0.9314
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 3.74734 0.08466 44.266 < 2e-16 ***
## bty_avg 0.07416 0.01625 4.563 6.48e-06 ***
## gendermale 0.17239 0.05022 3.433 0.000652 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.5287 on 460 degrees of freedom
## Multiple R-squared: 0.05912, Adjusted R-squared: 0.05503
## F-statistic: 14.45 on 2 and 460 DF, p-value: 8.177e-07##Ex7: P-values and parameter estimates should only be trusted if the conditions for the regression are reasonable. Verify that the conditions for this model are reasonable using diagnostic plots.
plot(m_bty$residuals~evals$bty_avg)
abline(h = 0)
qplot(sample = .stdresid, data = m_bty_gen) + geom_abline()
##Conditions of regression seems to have been met
##The data shows a linear trend with nearly normal residuals and constant variability.
##Ex8: Is bty_avg still a significant predictor of score? Has the addition of gender to the model changed the parameter estimate for bty_avg?
lm(formula = score ~ bty_avg + gender, data = evals, y = TRUE)##
## Call:
## lm(formula = score ~ bty_avg + gender, data = evals, y = TRUE)
##
## Coefficients:
## (Intercept) bty_avg gendermale
## 3.74734 0.07416 0.17239##The addition of gender has changed the slope estimate from 0.06664 to 0.07416. Yes the variables is still significant.
multiLines(m_bty_gen)
#Ex9:What is the equation of the line corresponding to males? (Hint: For males, the parameter estimate is multiplied by 1.) For two professors who received the same beauty rating, which gender tends to have the higher course evaluation score?
##Score= 3.74734 + 0.07416*bty_avg + 0.17239*gendermale
##The male professor tends to have the higher evaluation score.
##Ex10: Create a new model called m_bty_rank with gender removed and rank added in. How does R appear to handle categorical variables that have more than two levels? Note that the rank variable has three levels: teaching, tenure track, tenured.
m_bty_rank = lm(evals$score~evals$bty_avg + evals$rank)
multiLines(m_bty_rank)
summary(m_bty_rank)##
## Call:
## lm(formula = evals$score ~ evals$bty_avg + evals$rank)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.8713 -0.3642 0.1489 0.4103 0.9525
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 3.98155 0.09078 43.860 < 2e-16 ***
## evals$bty_avg 0.06783 0.01655 4.098 4.92e-05 ***
## evals$ranktenure track -0.16070 0.07395 -2.173 0.0303 *
## evals$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##R has used dummy variables to account for the three levels of teaching, tenured and tenure track.
#The search for the best model
##Ex11: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 think class credits should not affect the professor score.
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##Ex12:Check your suspicions from the previous exercise. Include the model output in your response.
##My estimate is wrong, credits has a very low p value, highest is for class prof single.
##Ex13: Interpret the coefficient associated with the ethnicity variable.
##The variable add 0.1234929 to the score when professor is not from minority, holding all other variables constant.
##Ex14: 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 <- lm(score ~ rank + ethnicity + gender + language + age + cls_perc_eval
+ cls_students + cls_level + cls_credits + bty_avg
+ pic_outfit + pic_color, data = evals)
summary(m_full)##
## Call:
## lm(formula = score ~ rank + ethnicity + gender + language + age +
## cls_perc_eval + cls_students + cls_level + cls_credits +
## bty_avg + pic_outfit + pic_color, data = evals)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.7836 -0.3257 0.0859 0.3513 0.9551
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 4.0872523 0.2888562 14.150 < 2e-16 ***
## ranktenure track -0.1476746 0.0819824 -1.801 0.072327 .
## ranktenured -0.0973829 0.0662614 -1.470 0.142349
## ethnicitynot minority 0.1274458 0.0772887 1.649 0.099856 .
## gendermale 0.2101231 0.0516873 4.065 5.66e-05 ***
## languagenon-english -0.2282894 0.1111305 -2.054 0.040530 *
## age -0.0089992 0.0031326 -2.873 0.004262 **
## cls_perc_eval 0.0052888 0.0015317 3.453 0.000607 ***
## cls_students 0.0004687 0.0003737 1.254 0.210384
## cls_levelupper 0.0606374 0.0575010 1.055 0.292200
## cls_creditsone credit 0.5061196 0.1149163 4.404 1.33e-05 ***
## bty_avg 0.0398629 0.0174780 2.281 0.023032 *
## pic_outfitnot formal -0.1083227 0.0721711 -1.501 0.134080
## pic_colorcolor -0.2190527 0.0711469 -3.079 0.002205 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.4974 on 449 degrees of freedom
## Multiple R-squared: 0.187, Adjusted R-squared: 0.1634
## F-statistic: 7.943 on 13 and 449 DF, p-value: 2.336e-14##The adjusted r square changes slightly, so the variable had very low collinearity.
##Ex15: 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_back <- lm(score ~ gender + language + age + cls_perc_eval
+ cls_credits + bty_avg
+ pic_color, data = evals)
summary(m_back)##
## Call:
## lm(formula = score ~ gender + language + age + cls_perc_eval +
## cls_credits + bty_avg + pic_color, data = evals)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.81919 -0.32035 0.09272 0.38526 0.88213
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 3.967255 0.215824 18.382 < 2e-16 ***
## gendermale 0.221457 0.049937 4.435 1.16e-05 ***
## languagenon-english -0.281933 0.098341 -2.867 0.00434 **
## age -0.005877 0.002622 -2.241 0.02551 *
## cls_perc_eval 0.004295 0.001432 2.999 0.00286 **
## cls_creditsone credit 0.444392 0.100910 4.404 1.33e-05 ***
## bty_avg 0.048679 0.016974 2.868 0.00432 **
## pic_colorcolor -0.216556 0.066625 -3.250 0.00124 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.5014 on 455 degrees of freedom
## Multiple R-squared: 0.1631, Adjusted R-squared: 0.1502
## F-statistic: 12.67 on 7 and 455 DF, p-value: 6.996e-15##Adjusted r square is 15%
##Score= 3.96 + 0.22gendermale -0.28languageon-english -0.006age + 0.004cls_perc_eval + 0.444cls_creditstone_credit +.0486bty_avg -0.216pic_colorcolor
##Ex16: Verify that the conditions for this model are reasonable using diagnostic plots
qplot(x = .fitted, y = .stdresid, data = m_back, position = "jitter")## Warning: `position` is deprecated
qplot(sample = .stdresid, data = m_back) + geom_abline()
##Conditions look reasonable, linear relationship, nearly normal residuals.
#Ex17: 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 data is no longer independent
#Ex18: 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.
##Score= 3.96 + 0.22gendermale -0.28languageon-english -0.006age + 0.004cls_perc_eval + 0.444cls_creditstone_credit +.0486bty_avg -0.216pic_colorcolor
##A high score will be associated with a male professor whose native language is not English, who is younger, has a higher completed evaluation %, teaches higher credit courses, has higher beauty score and no color photo.
##Ex19: Would you be comfortable generalizing your conclusions to apply to professors generally (at any university)? Why or why not?
##No I would not apply this at any university but only to similar universities.