Correlation and multiple regression

Research question 4 asks which factors predict final exam performance. Correlations describe relationships between pairs of variables; multiple regression estimates the unique contribution of each predictor while holding the others constant. You need both: the correlation table belongs in almost every thesis, and regression answers the real question.

The correlation matrix

Pearson’s r runs from −1 to +1. By Cohen’s guidelines, .10 is small, .30 medium and .50 large. The matrix shows that stress correlates moderately and negatively with the final exam score (r = −.32, p < .001), with social support (r = −.33) and with sleep (r = −.30). Study hours correlate positively with the final score (r = .29) but are unrelated to stress (r = .05). Social support is not directly related to exam performance (r = .09, p = .171). Keep this last result in mind for Lesson 10.

Two cautions. First, correlation is not causation: stress might lower performance, poor performance might raise stress, or both might be driven by something else. Second, Pearson’s r captures only linear relationships and is sensitive to outliers; check the scatter plot, and use Spearman’s rho for skewed or ordinal variables.

Multiple regression

A regression model predicts the final score from study hours, sleep, stress and support simultaneously:

Final = b₀ + b₁·StudyHours + b₂·SleepHours + b₃·Stress + b₄·Support + error

Step 1: the model as a whole

R² = .199: together, the four predictors explain 19.9% of the variance in final scores (adjusted R² = .185, which corrects for the number of predictors). The model is significant, F(4, 233) = 14.44, p < .001. In social science research, explaining 20% of the variance in exam scores is a respectable result; most of the variance depends on things not measured here, such as ability and prior knowledge.

Step 2: each predictor

The unstandardised coefficient B is in the original units: each extra weekly study hour goes with 0.57 more exam points, and each one-point increase in stress (on the 1–5 scale) with 5.8 fewer points, holding the other predictors constant. The standardised coefficient β puts all predictors on the same scale (standard deviations) so you can compare them: stress (β = −.36) and study hours (β = .31) are similarly strong, while sleep and support add nothing once stress is in the model.

Notice that sleep correlated −.30 with stress but its regression coefficient is essentially zero. This is the point of multiple regression: sleep’s simple correlation with exam scores (r = .07) was already weak, and what little it had is shared with stress. The coefficients show unique contributions.

Step 3: assumptions

• Multicollinearity: VIF values are all below 1.3, far below the warning level of 5 (or 10). Predictors are not redundant.
• Normal residuals: the Shapiro-Wilk test on the residuals gives p = .717, and the Q-Q plot of residuals follows the line.
• Linearity and homoscedasticity: in the residuals-versus-fitted plot the points form an even band around zero, with no curve or funnel.
• Independent errors: Durbin-Watson = 1.93, close to the ideal of 2.
• Influential cases: as promised in Lesson 5, you can rerun the model without S168 and S224 (delete their rows in the data box). The study hours coefficient falls from 0.57 to 0.52 and R² from .199 to .170, but study hours and stress remain the only significant predictors (both p < .001), so the two hard-working students do not drive the conclusions.

Writing it up

A multiple regression analysis was conducted to predict final exam scores from weekly study hours, sleep, stress and social support. The model explained 19.9% of the variance, R² = .199, adjusted R² = .185, F(4, 233) = 14.44, p < .001. Study hours (β = .31, p < .001) positively and stress (β = −.36, p < .001) negatively predicted final scores, whereas sleep (β = −.01, p = .912) and social support (β = −.04, p = .528) did not. Assumptions of normality, linearity, homoscedasticity and absence of multicollinearity (all VIFs < 1.3) were met.

Present the coefficients in a table with B, SE, β, t, p and the 95% CI for B. For hierarchical regression (adding blocks of predictors, such as demographics first), report the change in R² for each step.