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.

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.

Exploring the data

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.

This is an observational study. This is not an experiment as the survey was collected at the end of the semester and the data was used to answer the question. Since it is technically impossible to liaise correlation to causation, to redesign, the better way to format the question is Does appealing professors affect in students evaluation? as this will require to randomly assign students to a. appealing profs, b. unappealing profs.

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)

summary(evals$score)

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##   2.300   3.800   4.300   4.175   4.600   5.000

The evaluation scores appear to be left skewed. We can see that most students rated average 4.17473 and mostly median 4.3. The skewness seems to be drawn from minority and it appears that perhaps the ratings might not be any of help.

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

library(ggplot2)
ggplot(evals, aes(as.factor(bty_m1upper), cls_perc_eval)) + 
  geom_boxplot() +
  labs( x = "beauty rating of professor from upper level male: (1) lowest - (10) highest",
        y = "percent of students in class who completed evaluation.",
        title = "male prof beauty rating ~ percent of students who completed evaluation")

From the boxplot above, we notice outliers in percent of students in evaluation completion thus wide range.

mosaicplot( ~ bty_m1upper + cls_perc_eval, evals, color = TRUE)

The mosic plot let us identify the observations especially the percentage of evaluation completion according to beauty rating.

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?

4. 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?

ggplot(evals, aes(bty_avg, score)) +
  geom_jitter()

The origianl scatter plot was plotting only one point of score mean of the points that were near to each other which provided false interpretation as it was missing many hidden points as we can see now above.

5. 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(evals$score ~ evals$bty_avg)
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

ggplot(evals, aes(bty_avg, score)) +
  geom_jitter() +
  geom_smooth(method = 'lm')

\[\hat{score} = 3.88034 + 0.0666 * beauty\_average\_rating\]

Average beauty score is a statistically significant predictor of evaluation score with p-value close to 0. It does not appear to be practically significant predictor, as for every 1 point increase in bty_avg, the model predicts an increase of 0.0666. This is not a very significant change in evaluation score.

6. 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).

ggplot(evals) +
  geom_point(aes(bty_avg, m_bty$residuals)) +
  geom_hline(yintercept=0, color='blue')

par(mfrow = c(2,2))
plot(m_bty)

hist(m_bty$residuals)

To evaluate whether the conditions of regression are reasonable, we need to assess the followings.

• Linearity : According to above plots, this model shows linearity.
• Near normal residuals: The residual plots shows near normal pattern. The histogram shows a left skew, however, with a large size of sample, we can ignore this part.
• Constant variability: There are some outliers, but it is fairly constant.

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)

7. 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.

m_bty_gen <- lm(score ~ bty_avg + gender, data = evals)

ggplot(evals) +
  geom_point(aes(bty_avg, m_bty_gen$residuals)) +
  geom_hline(yintercept=0, color='blue')

par(mfrow=c(2,2))
plot(m_bty_gen)

hist(m_bty_gen$residuals)

• Linearity : According to above plots and summary, this model shows linearity.
• Near normal residuals: The residual plots shows near normal pattern. The histogram shows a left skew, however, with a large size of sample, we can ignore this part.
• Constant variability: The constant variability is uncertain – There nis no clear pattern and most data appears to be in negative residuals with some negative outliers.

8. Is bty_avg still a significant predictor of score? Has the addition of gender to the model changed the parameter estimate for bty_avg?

Yes, bty_avg is still a significant predictor of score. (p-value < 0.05) The addition of gender to the model changed the estimate from 0.0666 to 0.0741

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)

9. 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?

\[\widehat{males\_core} = 3.7473 + 0.742 * beauty\_average\_rating + 0.1724 * 1\] \[\widehat{female\_score} = 3.7473 + 0.742 * beauty\_average\_rating\]

Male score shows 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.)

10. 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

When there are more than 2 levels in categorical variables, R assigns 1 or 0 for each categorical level. e.g. If the prof is on ranktenure track, the prof would be assinged 1 on ranktenure track and 0 on ranktenured and vice versa. Note that teaching variable is dropped. If the prof is on teaching, as he is not on ranktenure track nor ranktenured, he is effectively making the teaching 0 and 0.

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.

11. 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 expect cls_profs would show the highest p-value in this model because number of professors teaching sections in course has not much to do with evaluation from a student point of view.

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)

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

It does appear that cls_profssingle has the highest p-value at 0.77806.

13. Interpret the coefficient associated with the ethnicity variable.

If the prof is not minority in ethnicity, the prof is given an additional .1235 points as long as all the other variables are constant.

14. 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?

lm.adjusted <- 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(lm.adjusted)

## 
## 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

#ztable(lm.adjusted)

There is no significant changes in coefficients in explanatory variables, however, note that R-squared value increases as cls_profs drops.

15. 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.

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

## 
## Call:
## lm(formula = score ~ rank + ethnicity + gender + language + age + 
##     cls_perc_eval + cls_students + cls_credits + bty_avg + pic_outfit + 
##     pic_color, data = evals)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -1.7761 -0.3187  0.0875  0.3547  0.9367 
## 
## Coefficients:
##                         Estimate Std. Error t value Pr(>|t|)    
## (Intercept)            4.0856255  0.2888881  14.143  < 2e-16 ***
## ranktenure track      -0.1420696  0.0818201  -1.736 0.083184 .  
## ranktenured           -0.0895940  0.0658566  -1.360 0.174372    
## ethnicitynot minority  0.1424342  0.0759800   1.875 0.061491 .  
## gendermale             0.2037722  0.0513416   3.969 8.40e-05 ***
## languagenon-english   -0.2093185  0.1096785  -1.908 0.056966 .  
## age                   -0.0087287  0.0031224  -2.795 0.005404 ** 
## cls_perc_eval          0.0053545  0.0015306   3.498 0.000515 ***
## cls_students           0.0003573  0.0003585   0.997 0.319451    
## cls_creditsone credit  0.4733728  0.1106549   4.278 2.31e-05 ***
## bty_avg                0.0410340  0.0174449   2.352 0.019092 *  
## pic_outfitnot formal  -0.1172152  0.0716857  -1.635 0.102722    
## pic_colorcolor        -0.1973196  0.0681052  -2.897 0.003948 ** 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.4975 on 450 degrees of freedom
## Multiple R-squared:  0.185,  Adjusted R-squared:  0.1632 
## F-statistic:  8.51 on 12 and 450 DF,  p-value: 1.275e-14

As of the following highest p-value shown variable was cls_level. However, We can see that the R-squred changes to 0.1632 which is lower than the previous model. At this point, we stop using backward-selection and no more removal of coefficients.

Thus, the final regression formurla is using the previous model.

\[ \begin{aligned}\widehat{score} &= 4.0873 \\ - 0.1477 * ranktenure\_track - 0.0974 * rank\_tenured + 0.1274 * ethnicity\_not\_minority \\ + 0.2101 * gender\_male - 0.2283 * language\_non\_english - 0.0090 * age + 0.0053 * cls\_perc\_eval \\ + 0.0005 * cls\_students + 0.0606 * cls\_level\_upper + 0.5061 * cls\_credits\_one\_credit \\ + 0.0399 * bty\_avg -0.1083 * pic\_outfit\_notformal - 0.2191 * pic\_colorcolor\end{aligned} \]

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

summary(lm.adjusted)

## 
## 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

summary(lm.adjusted$residuals)

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
## -1.7836 -0.3257  0.0859  0.0000  0.3513  0.9551

ggplot(evals) +
  geom_point(aes(bty_avg, lm.adjusted$residuals)) +
  geom_hline(yintercept=0, color='blue')

par(mfrow=c(2,2))
plot(lm.adjusted)

hist(lm.adjusted$residuals)

Data shows sligh left skew – which can be ignored thanks to the large size of sample, And there are questions remaining for the constant variability and linearity.

17. 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?

New information can have an impact on any of the conditions of linear regression whether that’s positive or negative.

18. 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.

summary(lm.adjusted)

## 
## 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

Based on the summary, the high-score prof characteristics would be on teaching track, white, male, english speaking, young, teaching upper level courses with one credit and physically appealing.

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

Since this is an observational data, I would not generalize the conclusion as correlation doesn’t mean causation.

This is a product of OpenIntro that is released under a Creative Commons Attribution-ShareAlike 3.0 Unported. This lab was written by Mine Çetinkaya-Rundel and Andrew Bray.