Assumption-checking rather than (just) testing: The importance of visualization and effect size in statistical diagnostics

Statistical methods generally have assumptions (e.g., normality in linear regression models). Violations of these assumptions can cause various issues, like statistical errors and biased estimates, whose impact can range from inconsequential to critical. Accordingly, it is important to check these assumptions, but this is often done in a flawed way. Here, I first present a prevalent but problematic approach to diagnostics-testing assumptions using null hypothesis significance tests (e.g., the Shapiro-Wilk test of normality). Then, I consolidate and illustrate the issues with this approach, primarily using simulations. These issues include statistical errors (i.e., false positives, especially with large samples, and false negatives, especially with small samples), false binarity, limited descriptiveness, misinterpretation (e.g., of p-value as an effect size), and potential testing failure due to unmet test assumptions. Finally, I synthesize the implications of these issues for statistical diagnostics, and provide practical recommendations for improving such diagnostics. Key recommendations include maintaining awareness of the issues with assumption tests (while recognizing they can be useful), using appropriate combinations of diagnostic methods (including visualization and effect sizes) while recognizing their limitations, and distinguishing between testing and checking assumptions. Additional recommendations include judging assumption violations as a complex spectrum (rather than a simplistic binary), using programmatic tools that increase replicability and decrease researcher degrees of freedom, and sharing the material and rationale involved in the diagnostics.