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.

download.file("http://www.openintro.org/stat/data/evals.RData", destfile = "evals.RData")
load("evals.RData")

variable	description
`score`	average professor evaluation score: (1) very unsatisfactory - (5) excellent.
`rank`	rank of professor: teaching, tenure track, tenured.
`ethnicity`	ethnicity of professor: not minority, minority.
`gender`	gender of professor: female, male.
`language`	language of school where professor received education: english or non-english.
`age`	age of professor.
`cls_perc_eval`	percent of students in class who completed evaluation.
`cls_did_eval`	number of students in class who completed evaluation.
`cls_students`	total number of students in class.
`cls_level`	class level: lower, upper.
`cls_profs`	number of professors teaching sections in course in sample: single, multiple.
`cls_credits`	number of credits of class: one credit (lab, PE, etc.), multi credit.
`bty_f1lower`	beauty rating of professor from lower level female: (1) lowest - (10) highest.
`bty_f1upper`	beauty rating of professor from upper level female: (1) lowest - (10) highest.
`bty_f2upper`	beauty rating of professor from second upper level female: (1) lowest - (10) highest.
`bty_m1lower`	beauty rating of professor from lower level male: (1) lowest - (10) highest.
`bty_m1upper`	beauty rating of professor from upper level male: (1) lowest - (10) highest.
`bty_m2upper`	beauty rating of professor from second upper level male: (1) lowest - (10) highest.
`bty_avg`	average beauty rating of professor.
`pic_outfit`	outfit of professor in picture: not formal, formal.
`pic_color`	color of professor’s picture: color, black & white.

Exploring the data

Is this an observational study or an experiment? The original research question posed in the paper is whether beauty leads directly to the differences in course evaluations. Given the study design, is it possible to answer this question as it is phrased? If not, rephrase the question.

Ans:

This is an observational study so we cannot make causal phrase.
I would rephrase the question as “whether beauty is correlated with the differences in course evaluations.”
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?

Ans:

The distribution is left-skewed. It has a smaller mean (4.175) than median (4.3).

The quantity of scores are higher when the score is higher.
Only few students give low ratings and the minimum score we have on file is 2.3.

It is a reasonable result as score = 3 does not consider as a good rating. It is also quite impossible to see a normal distribution on courses rating in reality.
```
summary(evals$score)
```
```
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##   2.300   3.800   4.300   4.175   4.600   5.000
```
```
hist(evals$score)
```

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

Ans:

# scatterplot: 
# As `age` increases, the `bty_avg` average beauty rating of professor decreases.

plot(bty_avg ~ age, data=evals)
m1 <- lm(bty_avg ~ age, data=evals)
abline(m1, col="royalblue1")

# side-by-side boxplots: 
# The average beauty rating of female professor are generally higher than that of male professor.

boxplot(bty_avg ~ gender, data=evals)

# mosaic plot: Most male professors are in tenured rank, 
# while the number of female professors in each rank is similar.

mosaicplot(gender~rank, data=evals, col=c("lightpink", "lightblue", "lightgreen"))

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?

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?

Ans:

The initial scatterplot did not show the overlapped datapoints. Multiple dots at the same position are being shown as one dot. Observations having identical or approximately same values are shown as only one single dot on the plot.

From the new scatterplot below, we can see that there are many overlapped points by having same x and/or y values.
```
plot(jitter(evals$score) ~ jitter(evals$bty_avg))
```
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?

Ans:

The equation for the linear model is y = 0.06664*x +3.88034, while y stands for score and x stands for the average beauty rating of professor.

The slope 0.06664 means that, for every 1 point increase in the average beauty rating of professor, we would expect the score of professor to increase by 0.06664 points.

The average beauty score is statistically significant as the p-value of the slope is 5.08e-05, which is close to zero.

Although p-value of the slope is 5.08e-05, the slope 0.06664 is very small in value. Also, the correlation between the two variables is only 0.187.
Therefore, the average beauty score does not appear to be a practically significant predictor.
```
m_bty <- lm(score~bty_avg, data=evals)
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-05
```
```
cor(evals$score,evals$bty_avg)
```
```
## [1] 0.1871424
```
```
plot(jitter(evals$score) ~ jitter(evals$bty_avg))
abline(m_bty)
```
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).

Ans:

Recall from Lab8, to assess whether the linear model is reliable, we need to check for (1) linearity, (2) nearly normal residuals, and (3) constant variability.

Linearity and Constant variability:

We already checked if the relationship between average beauty rating and score is linear using a scatterplot. We can also verify this condition with a plot of the residuals vs. bty_avg.

The residuals lie around the horizontal line y=0 with range (-2, 1), which shows linearity and constant variability between bty_avg and score.
```
plot(m_bty$residuals ~ jitter(evals$bty_avg), 
     xlab = "Average beauty rating of professor", 
     ylab = "Residual of evaluation score",
     main = "Residual of predicted evaluation score\n vs average beauty ratiing of professor")
abline(h = 0, col="red", lty = 3)  # adds a horizontal dashed line at y = 0
```
Nearly normal residuals:

To check this condition, we can look at a histogram or a normal probability plot of the residuals.

The histogram is unimodal but left-skewed, While the normal probability plot lies around the normal line,

Therefore, the condition appears to be barely met.
```
hist(m_bty$residuals, breaks=20, col="lightpink")
```
```
qqnorm(m_bty$residuals)
qqline(m_bty$residuals)  # adds diagonal line to the normal prob plot
```

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] 0.8439112

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

- -

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.

Ans:

The diagonostic plots below shows that:
- Linearity and Constant variability:
The scatterplot is similar to one of the linear model. The points scattered around y=0 and ranges between (-2,1).
Therefore both Linearilty and Constant Variability are met.
- Nearly normal residuals:
All three histograms (all gender, by male, by female) are left-skewed with some outliers. The normal probability plot of the residuals lines on the normal line in the middle part, but deviates because of skewness and residuals. Same as the graphs of our linear model showed in question 6, this condition is barely met.
```
# m_bty_gen <- lm(score ~ bty_avg + gender, data = evals)

plot(m_bty_gen$residuals ~ jitter(evals$bty_avg))
abline(h=0, lty=2)
```
```
boxplot(m_bty_gen$residuals ~ evals$gender)
```
```
par(mfrow=c(3,1))
hist(m_bty_gen$residuals, breaks=30, col="green", xlim=c(-2,1))
hist(m_bty_gen$residuals[m_bty_gen$model$gender=="male"], 
     breaks=30, col="royalblue1", xlim=c(-2,1))
hist(m_bty_gen$residuals[m_bty_gen$model$gender=="female"], 
     breaks=30, col="lightpink", xlim=c(-2,1))
```
```
par(mfrow=c(1,1))

qqnorm(m_bty_gen$residuals)
qqline(m_bty_gen$residuals)
```
Is bty_avg still a significant predictor of score? Has the addition of gender to the model changed the parameter estimate for bty_avg?

Ans:

The average beauty rating is still a significant predictor of evaluation score of professor as the p-value is extremely small and the correlation increases to 0.243.

Also, the coefficient of bty_avg increases from 0.06664 to 0.074155. Which shows the significance.

Note that the estimate for gender is now called gendermale. You’ll see this name change whenever you introduce a categorical variable. The reason is that R recodes gender from having the values of female and male to being an indicator variable called gendermale that takes a value of $0$ for females and a value of $1$ for males. (Such variables are often referred to as “dummy” variables.)

As a result, for females, the parameter estimate is multiplied by zero, leaving the intercept and slope form familiar from simple regression.

\[ \begin{aligned} \widehat{score} &= \hat{\beta}_0 + \hat{\beta}_1 \times bty\_avg + \hat{\beta}_2 \times (0) \\ &= \hat{\beta}_0 + \hat{\beta}_1 \times bty\_avg\end{aligned} \]

We can plot this line and the line corresponding to males with the following custom function.

multiLines(m_bty_gen)

What is the equation of the line corresponding to males? (Hint: For males, the parameter estimate is multiplied by 1.) For two professors who received the same beauty rating, which gender tends to have the higher course evaluation score?

Ans:

Equation of male score = 0.07416*bty_avg +0.17239*1 +3.74734 = 0.07416*bty_avg +3.91973

Equation of female score = 0.07416*bty_avg +0.17239*0 +3.74734 = 0.07416*bty_avg +3.74734

For two professors receiving the same beauty rating, a male professor’s score would be on average higher than that of female by 0.17239.

The decision to call the indicator variable gendermale instead of genderfemale has no deeper meaning. R simply codes the category that comes first alphabetically as a $0$. (You can change the reference level of a categorical variable, which is the level that is coded as a 0, using therelevel function. Use ?relevel to learn more.)

Create a new model called m_bty_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.

Ans:

R creates two dummy variables ranktenure track and ranktenured for rank. Both tenure track and tenured are being given their own coefficient respectively.

The levelteaching from rank that comes first alphabetically is treated as a base level by having coefficient = $0$.

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

The interpretation of the coefficients in multiple regression is slightly different from that of simple regression. The estimate for bty_avg reflects how much higher a group of professors is expected to score if they have a beauty rating that is one point higher while holding all other variables constant. In this case, that translates into considering only professors of the same rank with bty_avg scores that are one point apart.

The search for the best model

We will start with a full model that predicts professor score based on rank, ethnicity, gender, language of the university where they got their degree, age, proportion of students that filled out evaluations, class size, course level, number of professors, number of credits, average beauty rating, outfit, and picture color.

Which variable would you expect to have the highest p-value in this model? Why? Hint: Think about which variable would you expect to not have any association with the professor score.

Ans:

I would expect to cls_profs to have the highest p-value in this model as I think the number of professors teaching in one section would not affect the score to each individual professor.

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

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

Ans:

The model above showed that cls_profssingle has the highest p-value, which proved my suspicions.
Interpret the coefficient associated with the ethnicity variable.

Ans:

The coefficient associated with the ethnicity variable 0.1234929 means that, when keeping all other variables constant, professors who are not minorities have a score 0.1234929 higher than that are minories.

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?

Ans:

The coefficients and significances of other variables are slightly changed but not much.
This suggest that the dropped variable is slightly collinear with other variables but not significantly.

m_minus_1 <- lm(score ~ rank + ethnicity + gender + language + age + cls_perc_eval 
         + cls_students + cls_level + cls_credits + bty_avg 
         + pic_outfit + pic_color, data = evals)
summary(m_minus_1)

## 
## Call:
## lm(formula = score ~ rank + ethnicity + gender + language + age + 
##     cls_perc_eval + cls_students + cls_level + cls_credits + 
##     bty_avg + pic_outfit + pic_color, data = evals)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -1.7836 -0.3257  0.0859  0.3513  0.9551 
## 
## Coefficients:
##                         Estimate Std. Error t value Pr(>|t|)    
## (Intercept)            4.0872523  0.2888562  14.150  < 2e-16 ***
## ranktenure track      -0.1476746  0.0819824  -1.801 0.072327 .  
## ranktenured           -0.0973829  0.0662614  -1.470 0.142349    
## ethnicitynot minority  0.1274458  0.0772887   1.649 0.099856 .  
## gendermale             0.2101231  0.0516873   4.065 5.66e-05 ***
## languagenon-english   -0.2282894  0.1111305  -2.054 0.040530 *  
## age                   -0.0089992  0.0031326  -2.873 0.004262 ** 
## cls_perc_eval          0.0052888  0.0015317   3.453 0.000607 ***
## cls_students           0.0004687  0.0003737   1.254 0.210384    
## cls_levelupper         0.0606374  0.0575010   1.055 0.292200    
## cls_creditsone credit  0.5061196  0.1149163   4.404 1.33e-05 ***
## bty_avg                0.0398629  0.0174780   2.281 0.023032 *  
## pic_outfitnot formal  -0.1083227  0.0721711  -1.501 0.134080    
## pic_colorcolor        -0.2190527  0.0711469  -3.079 0.002205 ** 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.4974 on 449 degrees of freedom
## Multiple R-squared:  0.187,  Adjusted R-squared:  0.1634 
## F-statistic: 7.943 on 13 and 449 DF,  p-value: 2.336e-14

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.

Ans:

Using backward-selection via the step function, the best model includes ethnicity, gender, language, age, cls_perc_eval, cls_credits, bty_avg, pic_outfit and pic_color.

If we also use p-value to make our selection, we should drop pic_outfit from the model as it is not statistically significant.

Although some p-values in the final model increased after removing pic_outfit, they are all acceptable being smaller than 0.05.

m_step <- step(m_full)

## Start:  AIC=-630.9
## score ~ rank + ethnicity + gender + language + age + cls_perc_eval + 
##     cls_students + cls_level + cls_profs + cls_credits + bty_avg + 
##     pic_outfit + pic_color
## 
##                 Df Sum of Sq    RSS     AIC
## - cls_profs      1    0.0197 111.11 -632.82
## - cls_level      1    0.2740 111.36 -631.76
## - cls_students   1    0.3599 111.44 -631.40
## - rank           2    0.8930 111.98 -631.19
## <none>                       111.08 -630.90
## - pic_outfit     1    0.5768 111.66 -630.50
## - ethnicity      1    0.6117 111.70 -630.36
## - language       1    1.0557 112.14 -628.52
## - bty_avg        1    1.2967 112.38 -627.53
## - age            1    2.0456 113.13 -624.45
## - pic_color      1    2.2893 113.37 -623.46
## - cls_perc_eval  1    2.9698 114.06 -620.69
## - gender         1    4.1085 115.19 -616.09
## - cls_credits    1    4.6495 115.73 -613.92
## 
## Step:  AIC=-632.82
## score ~ rank + ethnicity + gender + language + age + cls_perc_eval + 
##     cls_students + cls_level + cls_credits + bty_avg + pic_outfit + 
##     pic_color
## 
##                 Df Sum of Sq    RSS     AIC
## - cls_level      1    0.2752 111.38 -633.67
## - cls_students   1    0.3893 111.49 -633.20
## - rank           2    0.8939 112.00 -633.11
## <none>                       111.11 -632.82
## - pic_outfit     1    0.5574 111.66 -632.50
## - ethnicity      1    0.6728 111.78 -632.02
## - language       1    1.0442 112.15 -630.49
## - bty_avg        1    1.2872 112.39 -629.49
## - age            1    2.0422 113.15 -626.39
## - pic_color      1    2.3457 113.45 -625.15
## - cls_perc_eval  1    2.9502 114.06 -622.69
## - gender         1    4.0895 115.19 -618.08
## - cls_credits    1    4.7999 115.90 -615.24
## 
## Step:  AIC=-633.67
## score ~ rank + ethnicity + gender + language + age + cls_perc_eval + 
##     cls_students + cls_credits + bty_avg + pic_outfit + pic_color
## 
##                 Df Sum of Sq    RSS     AIC
## - cls_students   1    0.2459 111.63 -634.65
## - rank           2    0.8140 112.19 -634.30
## <none>                       111.38 -633.67
## - pic_outfit     1    0.6618 112.04 -632.93
## - ethnicity      1    0.8698 112.25 -632.07
## - language       1    0.9015 112.28 -631.94
## - bty_avg        1    1.3694 112.75 -630.02
## - age            1    1.9342 113.31 -627.70
## - pic_color      1    2.0777 113.46 -627.12
## - cls_perc_eval  1    3.0290 114.41 -623.25
## - gender         1    3.8989 115.28 -619.74
## - cls_credits    1    4.5296 115.91 -617.22
## 
## Step:  AIC=-634.65
## score ~ rank + ethnicity + gender + language + age + cls_perc_eval + 
##     cls_credits + bty_avg + pic_outfit + pic_color
## 
##                 Df Sum of Sq    RSS     AIC
## - rank           2    0.7892 112.42 -635.39
## <none>                       111.63 -634.65
## - ethnicity      1    0.8832 112.51 -633.00
## - pic_outfit     1    0.9700 112.60 -632.65
## - language       1    1.0338 112.66 -632.38
## - bty_avg        1    1.5783 113.20 -630.15
## - pic_color      1    1.9477 113.57 -628.64
## - age            1    2.1163 113.74 -627.96
## - cls_perc_eval  1    2.7922 114.42 -625.21
## - gender         1    4.0945 115.72 -619.97
## - cls_credits    1    4.5163 116.14 -618.29
## 
## Step:  AIC=-635.39
## score ~ ethnicity + gender + language + age + cls_perc_eval + 
##     cls_credits + bty_avg + pic_outfit + pic_color
## 
##                 Df Sum of Sq    RSS     AIC
## <none>                       112.42 -635.39
## - pic_outfit     1    0.7141 113.13 -634.46
## - ethnicity      1    1.1790 113.59 -632.56
## - language       1    1.3403 113.75 -631.90
## - age            1    1.6847 114.10 -630.50
## - pic_color      1    1.7841 114.20 -630.10
## - bty_avg        1    1.8553 114.27 -629.81
## - cls_perc_eval  1    2.9147 115.33 -625.54
## - gender         1    4.0577 116.47 -620.97
## - cls_credits    1    6.1208 118.54 -612.84

summary(m_step)

## 
## Call:
## lm(formula = score ~ ethnicity + gender + language + age + cls_perc_eval + 
##     cls_credits + bty_avg + pic_outfit + pic_color, data = evals)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -1.8455 -0.3221  0.1013  0.3745  0.9051 
## 
## Coefficients:
##                        Estimate Std. Error t value Pr(>|t|)    
## (Intercept)            3.907030   0.244889  15.954  < 2e-16 ***
## ethnicitynot minority  0.163818   0.075158   2.180 0.029798 *  
## gendermale             0.202597   0.050102   4.044 6.18e-05 ***
## languagenon-english   -0.246683   0.106146  -2.324 0.020567 *  
## age                   -0.006925   0.002658  -2.606 0.009475 ** 
## cls_perc_eval          0.004942   0.001442   3.427 0.000666 ***
## cls_creditsone credit  0.517205   0.104141   4.966 9.68e-07 ***
## bty_avg                0.046732   0.017091   2.734 0.006497 ** 
## pic_outfitnot formal  -0.113939   0.067168  -1.696 0.090510 .  
## pic_colorcolor        -0.180870   0.067456  -2.681 0.007601 ** 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.4982 on 453 degrees of freedom
## Multiple R-squared:  0.1774, Adjusted R-squared:  0.161 
## F-statistic: 10.85 on 9 and 453 DF,  p-value: 2.441e-15

m_step_final <- lm(formula = score ~ ethnicity + gender + language + age + 
                cls_perc_eval + cls_credits + bty_avg + pic_color, data = evals)
summary(m_step_final)

## 
## 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

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

Ans:

In the final model, we have the following variables:
- categorical: ethnicity, gender, language, cls_credits, pic_color
- quantitative: age, cls_perc_eval, bty_avg
The diagonostic plots below shows that:
- Linearity and Constant variability:
The scatterplots and boxplots are similar to ones above. The points scattered around y=0 and ranges between (-2,1).
Therefore both Linearilty and Constant Variability are met.
- Nearly normal residuals:
The histogram is a little bit left-skewed.
The normal probability plot of the residuals lines near the normal line. Therefore the condition is met.
```
plot(m_step_final,1)
```
```
plot(jitter(evals$age), resid(m_step_final))
abline(h=0, col="red")
```
```
plot(jitter(evals$cls_perc_eval), resid(m_step_final))
abline(h=0, col="red")
```
```
plot(jitter(evals$bty_avg), resid(m_step_final))
abline(h=0, col="red")
```
```
par(mfrow=c(1,5))
boxplot(resid(m_step_final)~evals$ethnicity)
boxplot(resid(m_step_final)~evals$gender)
boxplot(resid(m_step_final)~evals$language)
boxplot(resid(m_step_final)~evals$cls_credits)
boxplot(resid(m_step_final)~evals$pic_color)
```
```
par(mfrow=c(1,1))

hist(resid(m_step_final))
```
```
qqnorm(resid(m_step_final))
qqline(resid(m_step_final))
```
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?

Ans:

This would break the assumption that all observations are independent as same professor can teach multiple courses and same student may take two or more classes with the same professor.
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.

Ans:

The highest score would be from a young male professor from majority ethnicity that graduated from an English-speaking university, teaching one-credit class, having a black and white picture, being more attractive and have higher percentage of students in class who completed the evaluation.
Would you be comfortable generalizing your conclusions to apply to professors generally (at any university)? Why or why not?

Ans:

I would not be comfortable to make such conclusion because different university has different combination of students such as their gender, ethnicity and background.

Also, this model is not conducted experimentally.
It is not suitable for us to make a causal conclusion.

Multiple linear regression

Sin Ying Wong

11/23/2019

Grading the professor

The data

Exploring the data

Simple linear regression

Multiple linear regression

The search for the best model