Multiple linear regression
M.Demelash
11/23/2021
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 as lightly 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.
download.file("http://www.openintro.org/stat/data/evals.RData", destfile = "evals.RData")
load("evals.RData")Exploring the data.
Exercise 1.
Question: 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.
Answer: This is an observational study as distinction of groups isnt applied to groups being observed simply the study is looking at “whether beauty leads directly to the differences in course evaluations”; which can best be answered by a measurable question of : What is the relationship between “beauty” and the outcome of the course evaluations.
Exercise 2.
Question: Describe the distribution of score.
Answer:
# frequency distribution- histogram
hist(evals$score, main = "Distribution of Scores", xlab = "Rate of Courses ")
Is the distribution skewed? What does that tell you about how students rate courses? Is this what you expected to see? Why, or why not?
The distribution of the scores is skewed to the left, Indicating a negative skewness which means that there is a longer left tail. This skewness reflects that hight scrores are more frequent that lower ones in the study. I felt like the data was going to show a fairly normal distribution with skewness being close to 0, I thought the way of scoring would be somewhat indifferent.
Exercise 3
Question: Excluding score, select two other variables and describe their relationship using an appropriate visualization (scatterplot, side-by-side boxplots, or mosaic plot).
boxplot(evals$bty_avg ~ evals$cls_did_eval, main = "Relationship between beauty averages and the amount of people who did the evaluation", ylab = "Beauty Average", xlab = "number of sudents who completed evlas")
It seems the lower higher scoring beauty averages tend to come from a small number of students, or in other words the higher the beauty average the more the students evaluate the professor than those with low beauty averages.
Simple Linear regression
Scatterplot to see if teachers that are better looking being evaluated more favorably.
plot(evals$score ~ evals$bty_avg, ylab ="Score",xlab="Beauty averages")
Exercise 4.
Question:Replot the scatterplot, but this time use the function jitter() on the y- or the x-coordinate. (Use ?jitter to learn more.) What was misleading about the initial scatterplot
plot( jitter(evals$score) ~ jitter(evals$bty_avg), ylab ="Score",xlab="Beauty averages")
Answer: Jitter plots show overlapping data points between the two relations, Initail plot doesn’t clearly indicate such relationship and can be misleading on what point correlates to what.
Exerxise 5
Note:
Let’s see if the apparent trend in the plot is something more than natural variation. Fit a linear model called m_bty to predict average professor score by average beauty rating and add the line to your plot using abline(m_bty).
#Fit a linear model called m_bty to predict average professor score by average beauty rating and add the line to your plot using abline(m_bty).
m_bty <- lm(evals$score ~ evals$bty_avg)
plot( jitter(evals$score) ~ jitter(evals$bty_avg), ylab ="Score",xlab="Beauty averages")
abline(m_bty)
Question: Write out the equation for the linear model and interpret the slope. Is average beauty score a statistically significant predictor? Does it appear to be a practically significant predictor?
summary(m_bty)##
## Call:
## lm(formula = evals$score ~ evals$bty_avg)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.9246 -0.3690 0.1420 0.3977 0.9309
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 3.88034 0.07614 50.96 < 2e-16 ***
## evals$bty_avg 0.06664 0.01629 4.09 5.08e-05 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.5348 on 461 degrees of freedom
## Multiple R-squared: 0.03502, Adjusted R-squared: 0.03293
## F-statistic: 16.73 on 1 and 461 DF, p-value: 5.083e-05Answer:
Formula :
ŷ =3.88034 + 0.06664*bty_average
The average beauty score is a statistically significant as reflected by a p value of 0. The scores of the teachers increase by 0.06664 times the beauty average so the higher the beauty averagae the higher the score
Exerxise 5
Question:
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).
# Residual ( somewhat)
plot(m_bty$residuals ~ evals$bty_avg)
abline(h=0, lty=3)
# Histogram ( no)
hist(m_bty$residuals)
#normal plot
qqnorm(m_bty$residuals)
qqline(m_bty$residuals)
conditions of least squares regression:
Linearity. The data should shows a somewhat of a bin Nearly normal residuals. The residuals are left skewed and not not noraml nor nearly norm;as Constant Variability: The plot indicate somewhat of a variability.
Multiple linear regression.
Note:
The data set contains several variables on the beauty score of the professor: individual ratings from each of the six students who were asked to score the physical appearance of the professors and the average of these six scores. Let’s take a look at the relationship between one of these scores and the average beauty score.
plot(evals$bty_avg ~ evals$bty_f1lower)
cor(evals$bty_avg, evals$bty_f1lower)## [1] 0.8439112As expected the relationship is quite strong - after all, the average score is calculated using the individual scores.
Note:
We can actually take a look at the relationships between all beauty variables (columns 13 through 19) using the following command:
plot(evals[,13:19])
Note : These variables are collinear (correlated), and adding more than one of these variables to the model would not add much value to the model. In this application and with these highly-correlated predictors, it is reasonable to use the average beauty score as the single representative of these variables.
In order to see if beauty is still a significant predictor of professor score after we’ve accounted for the gender of the professor, we can add the gender term into the model.
m_bty_gen <- lm(score ~ bty_avg + gender, data = evals)
summary(m_bty_gen)##
## Call:
## lm(formula = score ~ bty_avg + gender, data = evals)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.8305 -0.3625 0.1055 0.4213 0.9314
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 3.74734 0.08466 44.266 < 2e-16 ***
## bty_avg 0.07416 0.01625 4.563 6.48e-06 ***
## gendermale 0.17239 0.05022 3.433 0.000652 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.5287 on 460 degrees of freedom
## Multiple R-squared: 0.05912, Adjusted R-squared: 0.05503
## F-statistic: 14.45 on 2 and 460 DF, p-value: 8.177e-07Exercise 7
Note : P-values and parameter estimates should only be trusted if the conditions for the regression are reasonable.
Question : Verify that the conditions for this model are reasonable using diagnostic plots.
plot(m_bty_gen$residuals)
abline(h=0, lty=3)
# Histogram ( no)
hist(m_bty_gen$residuals)
#normal plot
qqnorm(m_bty_gen$residuals)
qqline(m_bty_gen$residuals)
Assuming independence on the student responses,though uper qunatile shows a veer off the normal line majority of the responses fall on the normal line assuming somewhat linearity. gender residuals reflect a skewing to the left and are not normaly distributed. Although it isnt stong or concrete there is somewhat os linear relationship present. since one or more of these assumptions are violated the results may be unreliable or misleading.
Exercise 8
Question: Is bty_avg still a significant predictor of score? Has the addition of gender to the model changed the parameter estimate for bty_avg?
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-07Note
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.
Answer : There is a slight change to what the forumla would be under gender in relation to bty avg thus i assume there is a change in the estimated paraerter of beauty average. Similarly strong p values and R- squared, it can be assumed that the significane of Bty avg in predicting score is accuralte.
multiLines(m_bty_gen)
#### Exercise 9
Question : 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?
Answer:
Logically Under the condition they get the same score it would seem that males would have a higher score for their evaluation than for the female professors.
Note
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.)
#### Exercise 10
Create a new model called m_bty_rank with gender removed and rank added in. 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-05Question :
How does R appear to handle categorical variables that have more than two levels?
“rank rank of professor: teaching, tenure track, tenured.” Above are the variables , so in the case of the catigorical variables with multiple levels Rleaves one out but lists the others
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.
Exercise 11:
Question : 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.
Answer: I do not think the “language of the university where they got their degree” should have an y bearing on the professors score thus being the variable that is negligable and one with the highest p-value in this model.
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-14Exercise 12
Question: your suspicions from the previous exercise. Include the model output in your response.
Answer : languagenon-english ( -0.2298112 0.1113754 -2.063 0.03965 *)
I was wrong in my assumption that language would not have any significance on the score of the professors but there seems to be strong evidence that language of the place where professor was educated in had an influence on the score they received.
Exercise 13
Question : Interpret the coefficient associated with the ethnicity variable.
Answer:
With a p-value of .11 which deems it insignificant it is interesting to note if ethnicity was to be isolated and seen independently it might positively boost the scores by 0.1234929 X.
Exercise 14
Note: Drop the variable with the highest p-value and re-fit the model. (One of the things that makes multiple regression interesting is that coefficient estimates depend on the other variables that are included in the model.)
Question :
Did the coefficients and significance of the other explanatory variables change? If not, what does this say about whether or not the dropped variable was collinear with the other explanatory variables?
#cls_profssingle -0.0146619 0.0519885 -0.282 0.77806
m2_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(m2_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-14Answer: There is a slight change in the coefficients of the other explanitory variables; an indication that the number of professors teaching section in course is co linear to the other explanatory variables.
Exercise 15
Question:
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_best<- lm(score ~ ethnicity + gender + language + age + cls_perc_eval + cls_credits + bty_avg + pic_color, data = evals)
summary(m_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-15Answer:
Score =3.771922 + ethnicitynot minority( 0.167872) +gendermale ( 0.207112) + languagenon-english (-0.206178 ) + age (-0.006046 )+ cls_perc_eval ( 0.004656) + cls_creditsone credit ( 0.505306)+ bty_avg ( 0.051069 )+ pic_colorcolor (-0.190579 )
Exercise 16
Question:
Verify that the conditions for this model are reasonable using diagnostic plots.
plot(m_best$residuals)
abline(h=0, lty=3)
# Histogram ()
hist(m_best$residuals)
#normal plot
qqnorm(m_best$residuals)
qqline(m_best$residuals)
Exercise 17
Question : 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?
Answer: The results should be consistent, the responses of each evaluation would be independent of one another from different classes though same it is thought by the same professor.
Exercise 18
Question : 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.
Answer:
Evaluation score is high under these conditions. ethnicitynot minority - non minority gendermale -male
languagenon-english - comes from english speaking school age - young
cls_perc_eval - High student evaluation completion
cls_creditsone credit - one credit class bty_avg- high beauty average
pic_colorcolo- colored photograph
Exercise 19
Question : Would you be comfortable generalizing your conclusions to apply to professors generally (at any university)? Why or why not?
The students will differ from university to university thus also a shift in opinions and preferences. A rural school might be similar in opinion to that of U of T while a school in NYC school with more exposure to types of peoples and cultures might be the opposite in responses.