Multiple linear regression
hi <- download.file("http://www.openintro.org/stat/data/evals.RData", destfile = "evals.RData")
load("evals.RData")
exploring data
exercise 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 because it is not changing any variables. No because beauty is arbatray and without a defintion of how beauty can be quanitiativly desribed for the purposes of this experiment you cannot. Instead ask is a teacher rating of physical appearence by their students cooralated to thier course evaluation scores.
excersise 2: Distribution of score
hist(evals$score)
excersise 3: Excluding score, select two other variables and describe their relationship using an appropriate visualization (scatterplot, side-by-side boxplots, or mosaic plot).
scores for minory verses not
minority<- subset(evals, ethnicity == "minority")
nonminority <- subset(evals, ethnicity == "not minority")
histA<- hist(minority$bty_f1lower, breaks = 8, plot = FALSE)
histB <- hist(nonminority$bty_f1lower, breaks = 8, plot=FALSE)
plot(histA, col = blues9)
plot(histB, col = "red", add = FALSE)
Simple linear regression
plot(evals$score ~ evals$bty_avg)
Excersise 4: Replot the scatterplot, but this time use the function jitter() on the y- or the x-coordinate. What was misleading about the initial scatterplot?
plot(jitter(evals$score) ~ evals$bty_avg)
the intitial one looked less consentrated
Excercise 5: Let’s see if the apparent trend in the plot is something more than natural variation. Fit a linear model called m_bty to predict average professor score by average beauty rating and add the line to your plot using abline(m_bty). Is average beauty score a statistically significant predictor? Does it appear to be a practically significant predictor?
m_bty <- lm(evals$score ~ evals$bty_avg)
plot(evals$score ~ evals$bty_avg)
abline(m_bty)
it does produce a slight upward trend, however it does not appear to be a practical predictor.
Multiple linear regression
plot(evals$bty_avg ~ evals$bty_f1lower)
cor(evals$bty_avg, evals$bty_f1lower)
## [1] 0.8439112
plot(evals[,13:19])
m_bty_gen <- lm(score ~ bty_avg + gender, data = evals)
summary(m_bty_gen)
##
## Call:
## lm(formula = score ~ bty_avg + gender, data = evals)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.8305 -0.3625 0.1055 0.4213 0.9314
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 3.74734 0.08466 44.266 < 2e-16 ***
## bty_avg 0.07416 0.01625 4.563 6.48e-06 ***
## gendermale 0.17239 0.05022 3.433 0.000652 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.5287 on 460 degrees of freedom
## Multiple R-squared: 0.05912, Adjusted R-squared: 0.05503
## F-statistic: 14.45 on 2 and 460 DF, p-value: 8.177e-07
exercise 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.
it appears linear, and theobesrvations are independent
plot(score ~ bty_avg + gender, data = evals)
excercise 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?
it is still a significant predictor, however gender also seems to have a componate.
multiLines(m_bty_gen)
excercise 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?
males intercept is 1. males tend to have a higher score for the same beauty making
Excercise 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
multiLines(m_bty_rank)
The search for the best model
Excercise 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.
cls_did_eval the number of sudent who completed the evalations (raw) probably doent have much coorolation
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
no cls_profssingle was the highest p value. at .778
Excersise 13: Interpret the coefficient associated with the ethnicity variable.
the slight positive trend means there is a slight positice coorolation of non minority teachers getting hogher rattings then minority ones.
Excersise 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?
m_full <- lm(score ~ rank + ethnicity + gender + language + age + cls_perc_eval
+ cls_students + cls_level + cls_credits + bty_avg
+ pic_outfit + pic_color, data = evals)
summary(m_full)
##
## Call:
## lm(formula = score ~ rank + ethnicity + gender + language + age +
## cls_perc_eval + cls_students + cls_level + cls_credits +
## bty_avg + pic_outfit + pic_color, data = evals)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.7836 -0.3257 0.0859 0.3513 0.9551
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 4.0872523 0.2888562 14.150 < 2e-16 ***
## ranktenure track -0.1476746 0.0819824 -1.801 0.072327 .
## ranktenured -0.0973829 0.0662614 -1.470 0.142349
## ethnicitynot minority 0.1274458 0.0772887 1.649 0.099856 .
## gendermale 0.2101231 0.0516873 4.065 5.66e-05 ***
## languagenon-english -0.2282894 0.1111305 -2.054 0.040530 *
## age -0.0089992 0.0031326 -2.873 0.004262 **
## cls_perc_eval 0.0052888 0.0015317 3.453 0.000607 ***
## cls_students 0.0004687 0.0003737 1.254 0.210384
## cls_levelupper 0.0606374 0.0575010 1.055 0.292200
## cls_creditsone credit 0.5061196 0.1149163 4.404 1.33e-05 ***
## bty_avg 0.0398629 0.0174780 2.281 0.023032 *
## pic_outfitnot formal -0.1083227 0.0721711 -1.501 0.134080
## pic_colorcolor -0.2190527 0.0711469 -3.079 0.002205 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.4974 on 449 degrees of freedom
## Multiple R-squared: 0.187, Adjusted R-squared: 0.1634
## F-statistic: 7.943 on 13 and 449 DF, p-value: 2.336e-14
some of the p values became slightly more significant. and the coeffecents changed slightly too mostly they became stronger. they didn’t change much though so this varible probably didn’t effect the model too much. It did have a slsight decrease in residual error too.
excercise 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.
m_full <- lm(score ~ rank + ethnicity + gender + language + age + cls_perc_eval
+ 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_credits + bty_avg + pic_outfit + pic_color,
## data = evals)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.78424 -0.31397 0.09261 0.35904 0.92154
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 4.152893 0.280892 14.785 < 2e-16 ***
## ranktenure track -0.142239 0.081819 -1.738 0.082814 .
## ranktenured -0.083092 0.065532 -1.268 0.205469
## ethnicitynot minority 0.143509 0.075972 1.889 0.059535 .
## gendermale 0.208080 0.051159 4.067 5.61e-05 ***
## languagenon-english -0.222515 0.108876 -2.044 0.041558 *
## age -0.009074 0.003103 -2.924 0.003629 **
## cls_perc_eval 0.004841 0.001441 3.359 0.000849 ***
## cls_creditsone credit 0.472669 0.110652 4.272 2.37e-05 ***
## bty_avg 0.043578 0.017257 2.525 0.011903 *
## pic_outfitnot formal -0.136594 0.068998 -1.980 0.048347 *
## pic_colorcolor -0.189905 0.067697 -2.805 0.005246 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.4975 on 451 degrees of freedom
## Multiple R-squared: 0.1832, Adjusted R-squared: 0.1632
## F-statistic: 9.193 on 11 and 451 DF, p-value: 6.364e-15
this is the best model i can make. y= cx+4.152893 (I couldn’t figure out what the general coffencet was)
excersise 19: Would you be comfortable generalizing your conclusions to apply to professors generally (at any university)? Why or why not?
I would not because this is not a random sample of all universities, it relects the campus culture and the makeup of the student body at that particular school which may or may not yeld diffrent results at a diffrent school.
