What we can and cannot conclude about the relationship between steep temporal reward discounting and hyperactivity-impulsivity symptoms in attention-deficit/hyperactivity disorder.

To the Editor: Paloyelis et al. (1) raise two concerns about the statistical ethods used in the original article by Scheres et al. (2): first, they ndicate that analysis of covariance (ANCOVA) is inappropriate hen groups differ on the covariate; and second, they suggest that he results from the regression analysis might be an artifact resulting rom pooling heterogeneous populations. To remedy the latter, they ecommend including group membership in the regression analyis. We point out that the resulting analysis—a multiple regression nalysis (MRA) with group membership included—is identical to NCOVA (3–7). Hence, the very mistake they flag in the first nalysis is their proposed remedy in the second analysis. Originally, ANCOVA was developed as a statistical tool for ncreasing the power to detect the effect of an experimental anipulation through reducing the unexplained variance in the ependent variable (DV). This is illustrated on the left in Figure 1. ere, the groups do not differ on the covariate (a condition that s established through random assignment to groups), such that he two independent variables (IVs) account for nonoverlapping arts of the variance in the DV (i.e., parts a and b). The inclusion f the covariate reduces the unexplained variance of the DV from he sum of b and c to only c, thus increasing the power to detect he experimental effect a. When groups differ on the covariate, however, the two IVs will, n part, account for the same variance in the DV, and hence the statistical) effect of the grouping variable will depend on whether r not the covariate is included in the analysis. This is illustrated on he right in Figure 1: when the covariate is not included, group embership explains a and d, but when the covariate is included, nly the direct effect a is uniquely accounted for by group memership. The overlap between the grouping variable and the ovariate (i.e., d and e) may be due to a causal effect of the grouping ariable on the covariate or vice versa, or both IVs may be nfluenced by an omitted variable. Statistically, it is impossible to istinguish between these scenarios (5). Hence, if the groups differ n the covariate, including both the covariate and the grouping ariable implies that only the direct effects of these IVs are estimated i.e., a and b). This is the reason some argue ANCOVA is inappropriate if