library("DATA606")
##
## Welcome to CUNY DATA606 Statistics and Probability for Data Analytics
## This package is designed to support this course. The text book used
## is OpenIntro Statistics, 4th Edition. You can read this by typing
## vignette('os4') or visit www.OpenIntro.org.
##
## The getLabs() function will return a list of the labs available.
##
## The demo(package='DATA606') will list the demos that are available.
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” by Hamermesh and Parker found that instructors who are viewed to be better looking receive higher instructional ratings.
Here, you will analyze the data from this study in order to learn what goes into a positive professor evaluation.
In this lab, you will explore and visualize the data using the tidyverse suite of packages. The data can be found in the companion package for OpenIntro resources, openintro.
Let’s load the packages.
library(tidyverse)
library(openintro)
library(GGally)
This is the first time we’re using the GGally
package.
You will be using the ggpairs
function from this package
later in the lab.
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. The
result is a data frame where each row contains a different course and
columns represent variables about the courses and professors. It’s
called evals
.
glimpse(evals)
## Rows: 463
## Columns: 23
## $ course_id <int> 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 1…
## $ prof_id <int> 1, 1, 1, 1, 2, 2, 2, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5,…
## $ score <dbl> 4.7, 4.1, 3.9, 4.8, 4.6, 4.3, 2.8, 4.1, 3.4, 4.5, 3.8, 4…
## $ rank <fct> tenure track, tenure track, tenure track, tenure track, …
## $ ethnicity <fct> minority, minority, minority, minority, not minority, no…
## $ gender <fct> female, female, female, female, male, male, male, male, …
## $ language <fct> english, english, english, english, english, english, en…
## $ age <int> 36, 36, 36, 36, 59, 59, 59, 51, 51, 40, 40, 40, 40, 40, …
## $ cls_perc_eval <dbl> 55.81395, 68.80000, 60.80000, 62.60163, 85.00000, 87.500…
## $ cls_did_eval <int> 24, 86, 76, 77, 17, 35, 39, 55, 111, 40, 24, 24, 17, 14,…
## $ cls_students <int> 43, 125, 125, 123, 20, 40, 44, 55, 195, 46, 27, 25, 20, …
## $ cls_level <fct> upper, upper, upper, upper, upper, upper, upper, upper, …
## $ cls_profs <fct> single, single, single, single, multiple, multiple, mult…
## $ cls_credits <fct> multi credit, multi credit, multi credit, multi credit, …
## $ bty_f1lower <int> 5, 5, 5, 5, 4, 4, 4, 5, 5, 2, 2, 2, 2, 2, 2, 2, 2, 7, 7,…
## $ bty_f1upper <int> 7, 7, 7, 7, 4, 4, 4, 2, 2, 5, 5, 5, 5, 5, 5, 5, 5, 9, 9,…
## $ bty_f2upper <int> 6, 6, 6, 6, 2, 2, 2, 5, 5, 4, 4, 4, 4, 4, 4, 4, 4, 9, 9,…
## $ bty_m1lower <int> 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 7, 7,…
## $ bty_m1upper <int> 4, 4, 4, 4, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 6, 6,…
## $ bty_m2upper <int> 6, 6, 6, 6, 3, 3, 3, 3, 3, 2, 2, 2, 2, 2, 2, 2, 2, 6, 6,…
## $ bty_avg <dbl> 5.000, 5.000, 5.000, 5.000, 3.000, 3.000, 3.000, 3.333, …
## $ pic_outfit <fct> not formal, not formal, not formal, not formal, not form…
## $ pic_color <fct> color, color, color, color, color, color, color, color, …
We have observations on 21 different variables, some categorical and some numerical. The meaning of each variable can be found by bringing up the help file:
?evals
This is an observational study. I do not think that given the study design we can answer this question. I’d phrase the question more like: Does beauty impact the way student’s rate their professors?
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?The distribution is left skewed, multimodal. Ideally you’d expect a more normal distribution but it seems that the students are generous with their ratings. I certainly expected to see a distribution as such, unless it’s a course and professor that is not well liked then I’d expect to see a right skewed distribution.
hist(evals$score)
score
, select two other variables and
describe their relationship with each other using an appropriate
visualization.Those teachers within the tenure track are younger than those who are teaching or tenured already.
plot(evals$rank, evals$age)
The fundamental phenomenon suggested by the study is that better looking teachers are evaluated more favorably. Let’s create a scatterplot to see if this appears to be the case:
ggplot(data = evals, aes(x = bty_avg, y = score)) +
geom_point()
Before you draw conclusions about the trend, compare the number of observations in the data frame with the approximate number of points on the scatterplot. Is anything awry?
There are 463 rows in the data set but the scatterplot shows less points plotted.
# Check for number of rows in dataset
nrow(evals)
## [1] 463
geom_jitter
as your layer. What was misleading about the initial scatterplot?The initial scatterplot didn’t show the overlapping plots that are show in this second plot.
ggplot(data = evals, aes(x = bty_avg, y = score)) +
geom_jitter()
m_bty
to
predict average professor score by average beauty rating. Write out the
equation for the linear model and interpret the slope. Is average beauty
score a statistically significant predictor? Does it appear to be a
practically significant predictor?y = b0 + b1x y = 3.88034 + 0.06664(bty_avg)
The slope of the line is 0.0664. There is an upward positive correlation, meaning that as the average beauty rating increases so do the scores. Being that the r-squared is 3.3%, this is a small variation for the average beauty rating to be statustically significant predictor.
<- lm(evals$score ~ evals$bty_avg)
m_bty summary(m_bty)
##
## Call:
## lm(formula = evals$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-05
plot(jitter(evals$score) ~ evals$bty_avg)
abline(m_bty)
Add the line of the bet fit model to your plot using the following:
ggplot(data = evals, aes(x = bty_avg, y = score)) +
geom_jitter() +
geom_smooth(method = "lm")
The blue line is the model. The shaded gray area around the line
tells you about the variability you might expect in your predictions. To
turn that off, use se = FALSE
.
ggplot(data = evals, aes(x = bty_avg, y = score)) +
geom_jitter() +
geom_smooth(method = "lm", se = FALSE)
plot(m_bty$residuals ~ evals$bty_avg)
# add horizontal dashed line
abline(h = 0, lty = 4)
# Histogram
hist(m_bty$residuals)
# Normal probability plot of the residual
qqnorm(m_bty$residuals)
qqline(m_bty$residuals)
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.
ggplot(data = evals, aes(x = bty_f1lower, y = bty_avg)) +
geom_point()
%>%
evals summarise(cor(bty_avg, bty_f1lower))
## # A tibble: 1 × 1
## `cor(bty_avg, bty_f1lower)`
## <dbl>
## 1 0.844
As expected, the relationship is quite strong—after all, the average score is calculated using the individual scores. You can actually look at the relationships between all beauty variables (columns 13 through 19) using the following command:
%>%
evals select(contains("bty")) %>%
ggpairs()
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 you’ve accounted for the professor’s gender, you can add the gender term into the model.
<- lm(score ~ bty_avg + gender, data = evals)
m_bty_gen 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
# Residual v. Fitted, Normal Probability, Scale-Location, and Residuals v. Leverage
plot(m_bty_gen)
# Histogram
hist(m_bty_gen$residuals)
# Boxplot Gender
plot(evals$score ~ evals$gender)
bty_avg
still a significant predictor of
score
? Has the addition of gender
to the model
changed the parameter estimate for bty_avg
?The bty_avg
is still a significant predictor of
score
and by adding gender
it allows for the
models to improve and changed the parameter estimate for
bty_avg
.
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 male
and
female
to being an indicator variable called
gendermale
that takes a value of \(0\) for female professors and a value of
\(1\) for male professors. (Such
variables are often referred to as “dummy” variables.)
As a result, for female professors, the parameter estimate is multiplied by zero, leaving the intercept and slope form familiar from simple regression.
\[ \begin{aligned} \widehat{score} &= \hat{\beta}_0 + \hat{\beta}_1 \times bty\_avg + \hat{\beta}_2 \times (0) \\ &= \hat{\beta}_0 + \hat{\beta}_1 \times bty\_avg\end{aligned} \]
ggplot(data = evals, aes(x = bty_avg, y = score, color = pic_color)) +
geom_smooth(method = "lm", formula = y ~ x, se = FALSE)
score = (3.74734 + 0.17239) + 0.07416(bty_avg)
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.)
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
.With more than 2 ranks, R creates separate values for each.
<- lm(score ~ bty_avg + rank, data = evals)
m_bty_rank 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.
We will start with a full model that predicts professor score based on rank, gender, ethnicity, 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.
I expect the cls_profs
with the least
association with the professor score.
Let’s run the model…
<- lm(score ~ rank + gender + ethnicity + language + age + cls_perc_eval
m_full + cls_students + cls_level + cls_profs + cls_credits + bty_avg
+ pic_outfit + pic_color, data = evals)
summary(m_full)
##
## Call:
## lm(formula = score ~ rank + gender + ethnicity + 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
## gendermale 0.2109481 0.0518230 4.071 5.54e-05 ***
## ethnicitynot minority 0.1234929 0.0786273 1.571 0.11698
## 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
The cls_profs
variable has the least association
with a maximum p-value of 0.77806. language
and
age
were significant.
Professors that are not minority score 0.123 higher than those who are minority.
Yes, the coefficients and significance of the other explanatory variables changed meaning that the drop of the variable is dependent on the other variables.
<- lm(score ~ rank + ethnicity + gender + language + age + cls_perc_eval
m_full_1 + cls_students + cls_level + cls_credits + bty_avg
+ pic_outfit + pic_color, data = evals)
summary(m_full_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
Answer
<- lm(score ~ ethnicity + gender + language + age + cls_perc_eval
m_full_best + cls_credits + bty_avg + pic_color, data = evals)
summary(m_full_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-15
The residuals look good, the linear model fits well and there’s no problem with the leverage points.
par(mfrow = c(2, 2))
plot(m_full)
hist(m_full_best$residuals)
# Normal Probability Plot
qqnorm(m_full_best$residuals)
qqline(m_full_best$residuals)
No, the class courses are independent from each other therefore, the scores would also be independent.
The professor would be young male teacher, teaching one class and not a minority. The teacher would receive their degree from a university where English is its primary language and their picture would be black and white.
I would not feel comfortable generalizing these conclusions because this was not conducted as an experiment, it’s an observational study. Other universities would have different results, especially as time goes on.