Variations of Q–Q Plots: The Power of Our Eyes!

Abstract

In statistical modeling, we strive to specify models that resemble data collected in studies or observed from processes. Consequently, distributional specification and parameter estimation are central to parametric models. Graphical procedures, such as the quantile–quantile (Q–Q) plot, are arguably the most widely used method of distributional assessment, though critics find their interpretation to be overly subjective. Formal goodness of fit tests are available and are quite powerful, but only indicate whether there is a lack of fit, not why there is lack of fit. In this article, we explore the use of the lineup protocol to inject rigor into graphical distributional assessment and compare its power to that of formal distributional tests. We find that lineup tests are considerably more powerful than traditional tests of normality. A further investigation into the design of Q–Q plots shows that de-trended Q–Q plots are more powerful than the standard approach as long as the plot preserves distances in x and y to be the same. While we focus on diagnosing nonnormality, our approach is general and can be directly extended to the assessment of other distributions.

keywords:

• Lineup protocol
• Normality test
• Quantile–Quantile plot
• Statistical graphics
• Visual inference

Acknowledgment

All data in the study were collected with approval from the internal review board IRB 10-347. The authors thank the Editor, Associate Editor, and anonymous reviewers for their suggestions which led to improvements in this article.

Funding

This work was funded in part by National Science Foundation grant DMS 1007697.

Notes

1 The little piece of arithmetic is imposing a small cognitive barrier that allows the reader to evaluate the lineup once without already being biased by knowing the location of the data panel.