Correlations and regression analysis

Correlation & Regression

A fifth grade science teacher wants to know if there is a relationship between final exam scores and overall coursepoints after adjusting for a quiz score.

In order to determine whether there was a significant relationship between overall coursepoints and final exam scores, after controlling for quiz scores, a hierarchical regression was run. All assumptions were assessed using SPSS. There was independences of residuals, as assessed by a Durbin-Watson statistic of 1.08. An examination of various scatterplots indicated a general linear relationship between variables of interest, although perfect linearity was not observed.

Overall Scatterplot

Partial Regression Plot: Final Exam Score & Quiz Score

Figure 3. Partial Regression Plot: Final Exam Score & Course Points

The assumption of homoscedasticity was violated, as the scatterplot in Figure 1 demonstrates (i.e., the values were not evenly spread across the scatterplot). There was no evidence of multicollinearity as evidenced by no correlations between independent variables greater than .7 and no tolerance statistics less than .1. Additionally, no outliers were detected, as evidence by no studentized deleted residuals values greater than +/- 3 standard deviations. An examination of the leverage scores indicated that there were none greater than .5, and as such, no individual cases were exerting high leverage. Furthermore, there were no Cook's distances greater than 1, indicating that there were no unduly influential cases. Although examination of the historgram (Figure 4) indicated less than perfect normality of data, observation of the P-P Plot of Regression Standardized Residuals showed acceptable normality (Figure 5).

Figure 4: Histogram

Figure 5: Normal P-P Plot of Regression Standardized Residual

A hierarchical regression was run in which the quiz score predictor was entered on the first step, followed by the course points predictor on the second. The results of the model indicated that while quiz scores were not a significant predictor of final exam scores, F (1, 18) = 1.1, p = .308, the addition of course points did significantly improve the model an lead to a significant increase in R2 of .219, F (1,17) = 5.154, p = .036. The full model of quiz scores and course points to predict final exam scores (Model 2) was not statistically significant, R2 = .277, F (2, 17) = 3.254, p = .064; adjusted R2 = .192. Consequently, the null hypothesis that there is no relationship between coursepoints and final exam after controlling for quiz scores cannot be rejected. In other words, there is no statistically significant relationship between coursepoints and final exam scores after controlling for quiz scores.

Case Study 2

Results

The assistant principal wants to know if students' cumulative GPAs on an 8-point scale can predict students' final exam grades in their fifth grade science class. He wants to predict the final exam score of a student who has a GPA of 8.0.