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Contributions: Abstract
Estimating and testing the average effect of a variable X on an outcome variable Y in a regression model with an interaction term X×Z (and perhaps more complicated terms such as X×Z^{2}) plays an important role in analyzing the effects of a variable X on an outcome variable Y which are modified by a covariate Z. Testing this average effect means testing the "main effect" of X in the presence of interaction effects and a possibly correlated (covariate) Z. Provided that Z is a fixed regressor in the statistical sampling model, testing the average effect can be achieved by general linear model methods. In this case, the sample mean of Z will be the same from one sample to the next and no generalization can be made about the average effect of X on Y in the population in which Z might have a distribution that differs from the distribution of Z in the sample. If, however, Z is a stochastic regressor, we show that the general linear model methods of testing the average effect yield inflated alphaerrors, the errors increasing the larger the interaction effect becomes. We also show that the likelihoodratio test with nonlinear constraints (as implemented in LISREL 8.54) can be applied in this case and yield a valid test of significance. Both points are shown in a simulation study with sample sizes of N = 1000. In two other simulations studies we show that these results also apply to smaller samples of sizes (N = 30) and nonlinear (i.e. quadratic) modificator functions. CACE of treatment for depression from RCT with noncompliance and loss to followup Mohammad Maracy & Graham Dunn University of Manchester, UK The data from a randomised controlled trial of psychological treatment of depression or mixed depression and anxiety is reanalysed from two sites of 13 general practices in North London and 11 in Greater Manchester. Depressed patients were randomised to one of the three groups: Cognitivebehaviour therapy (CBT), Nondirective counselling (NDC), or routine General practitioner (GP) care. (Ward et al., 2000). The mean (s.d.) of BDI was 12.7 (9.5), 11.5 (7.6), or 17.2 (11.9) at 4 months followup in the CBT, NDC, or GP groups, respectively. Around 86% and 89% of all patients complied with the offered treatment in the CBT and NDC groups. Of those who were offered treatment in both the groups of CBT and NDC but did not comply with the offer only around 78% and 71% provided outcome data, respectively. We estimated the ComplianceAverage Causal Effect (CACE) of treatment assuming the missing data mechanism to be ignorable or latently ignorable using the methods of Bloom (1984) and Frangakis & Rubin (2000) and compared these with the Average Causal Effect (ACE). References
