SPSS Output Explorer

About this example

Cross-sectional observational design: 80 students reported their weekly study hours, their average nightly sleep (hours) and their test anxiety (a scale from 10 to 50), and their module exam scores (0 to 100) were recorded. All three predictors were entered together (SPSS's Enter method). Each student appears once, with no planned grouping or repeated measurements. The study design therefore supports treating observations as independent, although independence of regression errors cannot be proved from the dataset structure alone.

Assumption and diagnostic checks

• Linearity: adding a squared term for each predictor, one at a time, gave no strong evidence of a curved relationship (weekly study hours p = .854, nightly sleep p = .873, test anxiety p = .515), and neither did an overall check of the model's form (Ramsey's RESET, p = .731). This does not prove that the relationships are exactly straight. Plots of the partial residuals show a slight bend at the extremes of weekly study hours and test anxiety, where there are only a few students, so the evidence there is weak in either direction; in a real analysis, always look at these plots as well as the tests.
• Normality of residuals: the Shapiro–Wilk test found no evidence against a normal residual distribution (p = .997). This does not establish normality; a residual Q–Q plot should also be examined.
• Equal spread of residuals: there was no strong evidence that the spread of the residuals changed across predicted scores (Breusch-Pagan p = .427). Again, this does not prove that the spread is exactly constant.
• Unusual and influential students: no student had a studentised residual of 3 or more (the largest was 2.69), and the largest Cook's distance was 0.08. 18 students were flagged by at least one screening rule (leverage, Cook's distance, studentised residual, DFFITS or DFBETAS), so the model was refitted 18 times, leaving out one of them each time. No predictor changed its statistical significance, weekly study hours and test anxiety kept their direction every time, and the largest change in any predictor's B was 0.47 of its standard error. The model reported here is the original one, with all 80 students.
• Nightly sleep's B changed sign in 7 of those 18 refits. That is expected for a coefficient that is close to 0 and not statistically significant, and it is another reason not to interpret its sign.
• One student reported 25 weekly study hours; the next highest was 20. The value was checked against the data file and is plausible for a student studying intensively, so it was kept. Leaving that student out was one of the refits above.

Limitations

• The data are simulated for teaching.
• This is an observational survey, so the coefficients describe associations, not causal effects: they do not show what would happen if a student changed their study hours or anxiety.
• Each coefficient is adjusted for the other predictors in the model; with different predictors, the coefficients could change.
• Study hours, sleep and anxiety are self-reported and may be inaccurate.
• The model was fitted to these 80 students; it may predict less well for other students.

What can be concluded

Together, weekly study hours, nightly sleep and test anxiety predicted exam scores in this sample: the model was statistically significant and accounted for about half of the variation in exam scores (R² = .48). Holding the other predictors constant, more study hours were associated with higher scores and higher anxiety with lower scores. Nightly sleep was not statistically significant, which is not evidence that sleep does not matter. These associations are useful for prediction, but they are not causes: the data cannot show that changing study hours or anxiety would change exam scores. Report R², F with its df and p, and B with its 95% CI and p for every predictor.