Interindividual Differences in Treatment Effects Based on Structural Equation Models with Latent Variables: An EffectLiteR Tutorial

Abstract

The investigation of interindividual differences in the effects of a treatment is challenging, because many constructs-of-interest in psychological research such as depression or anxiety are latent variables and modeling heterogeneity in treatment effects requires interactions and potentially non-linear relationships. In this paper, we present a tutorial of the EffectLiteR approach that allows for estimating individual treatment effects based on latent variable models. We describe step by step how to apply the approach using the EffectLiteR software package with data from the multicenter randomized controlled trial of the Social Phobia Psychotherapy Network (SOPHO-NET) and provide guidelines and recommendations for researchers. The focus of the paper is on explaining the results of a comprehensive effect analysis in an accessible language and on highlighting the opportunities the EffectLiteR approach offers for analyzing interindividual differences in treatment effects.

Keywords:

• Causal effects
• individual treatment effects
• interaction
• latent variables
• moderation
• structural equation modeling

Supplemental Material

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Notes

¹ In many studies one of the two treatment conditions is an untreated control group (e.g., X = c) but in this paper we use the theory to compare the (differential) effectiveness of two competing treatments.

² We use stochastic expected outcomes notation (following Steyer’s theory of causal effects) instead of the more frequently used potential outcomes notation (according to Rubin’s theory). Stochastic expected outcomes assume an intra-individual distribution under each treatment condition and consider the expected value of this intra-individual distribution instead of a fixed value for a potential outcome. See Steyer (2005) for a discussion about the stochastic nature of true outcomes. For the data analysis presented later in this article, the two theories have similar implications.

³ The empirical study can be a RCT, but does not have to be randomized, because conditional unit-treatment homogeneity is a sufficient condition for obtaining conditional causal effects even in observational studies (see Steyer et al., 2000).

⁴ The simulated dataset is generated using the group-specific model implied variance-covariance matrix and mean vector of the final analysis model. No real patient data is included. This way the point estimates of the conditional and aggregated effects based on the simulated data match with the ones presented based on the original data used in this manuscript. There can be slight differences in indiviudal treatment effects and standard errors. There are no missing values in the simulated dataset and therefore the sample size was adjusted to yield similar standard errors. If the model is fit with the simulated dataset, the model fit will be perfect, because the model implied statistics have been used to generate this data.

⁵ We could calculate this value by hand, when we use the regression coefficients of the effect function in Table 2 and multiply them with the corresponding covariate values.

⁶ The default effect size reported by EffectLiteR is obtained by dividing the average effect by the standard deviation of Y in the control group, i.e., the CBT group. This effect size is slightly higher than the one reported by Leichsenring et al. (2013), because the dependent variable is modeled as latent (without measurement error), the missing data handling is full information maximum likelihood in our analysis, and the model includes more variables, which can lead to variations in estimated effect sizes.

Additional information

Funding

This work was supported by the German Research Foundation (Deutsche Forschungsgemeinschaft, DFG; Grant No. MA 7702/1-1). SOPHO-NET was supported by a grant from the German Federal Ministry of Education and Research (BMBF; Grant No. 01GV0607).