Informal versus formal judgment of statistical models: The case of normality assumptions

Researchers sometimes use informal judgment for statistical model diagnostics and assumption checking. Informal judgment might seem more desirable than formal judgment because of a paradox: Formal hypothesis tests of assumptions appear to become less useful as sample size increases. We suggest that this paradox can be resolved by evaluating both formal and informal statistical judgment via a simplified signal detection framework. In 4 studies, we used this approach to compare informal judgments of normality diagnostic graphs (histograms, Q-Q plots, and P-P plots) to the performance of several formal tests (Shapiro-Wilk test, Kolmogorov-Smirnov test, etc.). Participants judged whether or not graphs of sample data came from a normal population (Experiments 1-2) or whether or not from a population close enough to normal for a parametric test to be more powerful than a nonparametric one (Experiments 3-4). Across all experiments, participants' informal judgments showed lower discriminability than did formal hypothesis tests. This pattern occurred even after participants were given 400 training trials with feedback, a financial incentive, and ecologically valid distribution shapes. The discriminability advantage of formal normality tests led to slightly more powerful follow-up tests (parametric vs. nonparametric). Overall, the framework used here suggests that formal model diagnostics may be more desirable than informal ones.