Causal Graphical Analysis of Regression to the Mean

• In Event: Regression Toward the Mean, Lord’s Paradox, and Modeling or Forecasting Change in Longitudinal Analyses

Abstract

Regression to the mean, discovered by Francis Galton (1886), is a tendency for extreme first measures to be less extreme on the second measurement. Assuming linear systems, this study investigates the underlying causal structure behind the regression to the mean phenomenon and clarifies some persistent confusion surrounding it. The study first shows that Galton’s regression to the mean is equivalent to a negative association between initial measure and subsequent change. Based on this formulation, the study provides causal graphs of regression to the mean. Figure 1A describes a system where initial measure X does not cause follow-up measure Y and they are affected by a common cause U. Given the structure, regression to the mean occurs if X and D (= Y − X) are negatively associated, Cov(X, D) < 0, and Wright’s (1921) path-tracing rules show that such a negative association is resulted from two different paths: i) X ⇢ D, and ii) X ← U → Y ⇢ D. As the first path transmits a negative association while the second path transmits a positive association (because X and Y are positively correlated in Galton’s original height example), regression to the mean occurs when the first negative association is strong enough to dominate the second positive association. Thus, it is shown that the main force of regression to the mean (i.e., the negative association between X and D) is the computational relationship, denoted by the dashed arrow, where the path coefficient is given by −1 (X ⇢ D). This finding supports why regression to the mean has been considered an artifact (e.g., Campbell & Kenny, 1999). However, it should be noted that it is possible that regression to the mean can be partially led by a true substantive causal relationship (e.g., self-regulating temperature control system; negative feedback loop). Figure 1B describes another system where initial measure X directly causes follow-up measure Y. Given this structure, regression to the mean occurs if the sum of three associations between X and D becomes negative. The paths include, besides the two paths above, path iii) X → Y ⇢ D which transmits an additional association between X and D. If X causes Y in a negative way, the path facilitates regression to the mean because it contributes to the overall negative association between X and D. The most important finding revealed by the causal graphical analysis is that regression to the mean is not a process or mechanism, inherent in mother nature, that determines follow-up measures. Rather, it is a phenomenon we bring upon ourselves because a negative association between X and D is already embedded in the computation D = Y – X, that is, when we compare the repeated measures. The study applies the findings to the debate of whether regression to the mean makes difference score estimators or ANCOVA estimators more biased. The study also discusses similar concepts to regression to the mean such as the law of initial value, mathematical coupling, and horse-racing effects.

Author

• Yongnam Kim, Seoul National University
  Presenting Author