Predictors of Success in Calculus
Idea: Use a regression or possibly a logistic regression model to develop guidelines for predicting student success in Math 140 and 141.
The lines in the charts on the left indicate where the predicted grade is a B (3.0), C (2.0), or D (1.0). The lines in the charts on the right indicate where the logistic regression model predicts that students have a 25, 50, or 75% chance of passing.
Linear Regression Equations
- \(\text{Predicted 140 grade} = -3.1422458+0.0041533(\text{Math SAT})+1.1372276(\text{College GPA}).\) (Note \(R^2_{adj} = 0.3627676\))
- \(\text{Predicted 141 grade} = -3.5077784+0.0053087(\text{Math SAT})+0.9090359(\text{College GPA}).\) (Note \(R^2_{adj} = 0.3658532\))
Logistic Regression Equations
- \(\ln(\text{Odds of passing 140}) = -9.3319762+0.0082769(\text{Math SAT})+2.203364(\text{College GPA}).\)
- \(\ln(\text{Odds of passing 141}) = -10.4981933+0.0133686(\text{Math SAT})+1.3041951(\text{College GPA}).\)
Predictors for Incoming Freshman
Since incoming freshman don’t usually take Math 140, but they do take 141, it would be useful to consider how well High School GPA serves as a substitute for College GPA as a predictor of success in Math 141. The adjusted \(R^2\) for this is considerably worse: only 24.1% compared with 36.6% using College GPA.
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -4.377663048 0.942766192 -4.643424 7.196613e-06
## SATmath 0.005903249 0.001289586 4.577630 9.509134e-06
## HSgpa 0.870758174 0.186526812 4.668274 6.473155e-06
Equations
- \(\text{Predicted 141 grade} = -4.377663+0.0059032(\text{Math SAT})+0.8707582(\text{High School GPA}).\) (Note \(R^2_{adj} = 0.2413233\))
- \(\ln(\text{Odds of passing 141}) = -12.4730321+0.0133788(\text{Math SAT})+1.5892536(\text{High School GPA}).\)
Math 140 Details
Below is the data for almost all currently enrolled students who have taken Math 140. Students without SAT scores or ALEKS scores are omitted.
## SATmath SATverbal HSgpa HSCgpa Math140grade ALEKS points passing
## 11 570 530 3.6 2.812 C+ 61 2.3 TRUE
## 18 635 620 3.6 3.212 A- 79 3.7 TRUE
## 30 630 530 2.7 2.653 C 70 2.0 TRUE
## 36 590 520 3.0 2.154 C 79 2.0 TRUE
## 38 610 550 3.9 2.531 D+ 75 1.3 FALSE
## 40 620 755 3.3 2.901 A- 72 3.7 TRUE## SATmath SATverbal HSgpa HSCgpa
## Min. :400.0 Min. :370.0 Min. :2.100 Min. :1.656
## 1st Qu.:511.2 1st Qu.:495.0 1st Qu.:3.100 1st Qu.:2.611
## Median :552.5 Median :540.0 Median :3.450 Median :2.908
## Mean :552.4 Mean :544.3 Mean :3.487 Mean :2.875
## 3rd Qu.:595.0 3rd Qu.:590.0 3rd Qu.:3.900 3rd Qu.:3.185
## Max. :735.0 Max. :760.0 Max. :4.500 Max. :3.971
##
## Math140grade ALEKS points passing
## C :22 Min. : 0.00 Min. :0.000 Mode :logical
## A :19 1st Qu.:66.00 1st Qu.:1.775 FALSE:34
## B :19 Median :75.00 Median :2.500 TRUE :100
## B- :13 Mean :72.63 Mean :2.407
## C+ :11 3rd Qu.:80.00 3rd Qu.:3.300
## W :10 Max. :97.00 Max. :4.000
## (Other):40Linear Regression
##
## Call:
## lm(formula = points ~ SATverbal + SATmath + HSgpa + HSCgpa +
## ALEKS, data = data140)
##
## Residuals:
## Min 1Q Median 3Q Max
## -2.75328 -0.44208 0.05868 0.54367 1.89957
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -2.5500025 0.8472473 -3.010 0.003150 **
## SATverbal -0.0022628 0.0012008 -1.884 0.061780 .
## SATmath 0.0049209 0.0013582 3.623 0.000418 ***
## HSgpa -0.4683819 0.1741631 -2.689 0.008114 **
## HSCgpa 1.7699985 0.2069662 8.552 3.16e-14 ***
## ALEKS 0.0002094 0.0057050 0.037 0.970780
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.8579 on 128 degrees of freedom
## Multiple R-squared: 0.4725, Adjusted R-squared: 0.4519
## F-statistic: 22.93 on 5 and 128 DF, p-value: < 2.2e-16We should also check the relationships between each variable and Math 140 grade.
And here are two graphs for the residuals. 
Logistic Regression
##
## Call:
## glm(formula = passing ~ SATverbal + SATmath + HSgpa + HSCgpa +
## ALEKS, family = "binomial", data = data140)
##
## Deviance Residuals:
## Min 1Q Median 3Q Max
## -2.5907 -0.2761 0.3594 0.6814 1.8212
##
## Coefficients:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) -7.712099 2.947804 -2.616 0.00889 **
## SATverbal -0.008052 0.003795 -2.122 0.03384 *
## SATmath 0.012356 0.004594 2.690 0.00715 **
## HSgpa -0.966768 0.588576 -1.643 0.10048
## HSCgpa 3.518613 0.784037 4.488 7.2e-06 ***
## ALEKS 0.001210 0.015649 0.077 0.93835
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for binomial family taken to be 1)
##
## Null deviance: 151.79 on 133 degrees of freedom
## Residual deviance: 113.18 on 128 degrees of freedom
## AIC: 125.18
##
## Number of Fisher Scoring iterations: 5Math 141 Details
Below is the data for all currently enrolled students who have taken Math 141.
## SATmath SATverbal HSgpa HSCgpa Math141grade ALEKS points passing
## 5 510 550 3.8 3.399 W 86 0.0 FALSE
## 7 630 620 4.1 3.240 A- 71 3.7 TRUE
## 12 610 520 3.4 2.741 D+ 72 1.3 FALSE
## 13 470 480 3.5 3.124 W 39 0.0 FALSE
## 20 780 760 4.4 3.979 A 82 4.0 TRUE
## 28 755 675 4.2 2.993 A 87 4.0 TRUE## SATmath SATverbal HSgpa HSCgpa
## Min. :470.0 Min. :390.0 Min. :2.100 Min. :0.000
## 1st Qu.:553.8 1st Qu.:510.0 1st Qu.:3.400 1st Qu.:2.709
## Median :605.0 Median :580.0 Median :3.700 Median :3.087
## Mean :599.7 Mean :576.3 Mean :3.709 Mean :3.007
## 3rd Qu.:640.0 3rd Qu.:631.2 3rd Qu.:4.000 3rd Qu.:3.399
## Max. :780.0 Max. :775.0 Max. :4.800 Max. :3.979
##
## Math141grade ALEKS points passing
## B :17 Min. :31.00 Min. :0.000 Mode :logical
## A :16 1st Qu.:73.00 1st Qu.:1.700 FALSE:38
## W :15 Median :76.00 Median :2.700 TRUE :94
## A- :14 Mean :76.35 Mean :2.353
## B- :14 3rd Qu.:81.00 3rd Qu.:3.300
## C :12 Max. :94.00 Max. :4.000
## (Other):44Linear Regression
##
## Call:
## lm(formula = points ~ SATverbal + SATmath + HSgpa + HSCgpa +
## ALEKS, data = data141)
##
## Residuals:
## Min 1Q Median 3Q Max
## -2.7084 -0.5253 0.2316 0.6560 1.8473
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -4.727214 1.249792 -3.782 0.000239 ***
## SATverbal -0.001169 0.001453 -0.805 0.422541
## SATmath 0.006559 0.001663 3.945 0.000132 ***
## HSgpa 0.230274 0.206014 1.118 0.265796
## HSCgpa 1.025851 0.150871 6.800 3.76e-10 ***
## ALEKS -0.001559 0.012033 -0.130 0.897125
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.9972 on 126 degrees of freedom
## Multiple R-squared: 0.44, Adjusted R-squared: 0.4178
## F-statistic: 19.8 on 5 and 126 DF, p-value: 1.589e-14We should also check the relationships between each variable and Math 141 grade.
And here are two graphs for the residuals. 
Logistic Regression
For Math 141, verbal SATs are the least significant predictor of success. ALEKS scores are also not significant, and neither are high school GPAs. So we get the following simple logistic regression model.
##
## Call:
## glm(formula = passing ~ SATverbal + SATmath + HSgpa + HSCgpa +
## ALEKS, family = "binomial", data = data141)
##
## Deviance Residuals:
## Min 1Q Median 3Q Max
## -2.0766 -0.3875 0.3507 0.6451 1.8604
##
## Coefficients:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) -12.499761 3.965436 -3.152 0.001621 **
## SATverbal -0.002923 0.003874 -0.754 0.450557
## SATmath 0.015708 0.005158 3.046 0.002322 **
## HSgpa 1.007975 0.606391 1.662 0.096462 .
## HSCgpa 1.680329 0.489591 3.432 0.000599 ***
## ALEKS -0.035862 0.034807 -1.030 0.302867
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for binomial family taken to be 1)
##
## Null deviance: 158.46 on 131 degrees of freedom
## Residual deviance: 110.52 on 126 degrees of freedom
## AIC: 122.52
##
## Number of Fisher Scoring iterations: 5