PUB - Publikationen an der Universität Bielefeld

In this thesis, we develop a statistical framework to investigate average and conditional effects of a treatment or intervention on a count outcome. For example, in the field of drug and substance use prevention, the effects of interventions on the outcome variable daily/weekly count of alcoholic drinks or cigarettes consumed can been examined. In clinical psychology, treatment effects on count outcomes such as instances of binge eating among patients with eating disorders or counts of classroom violations in children with ADHD can been examined. Further examples are treatment effects on counts of correctly answered items in a cognitive test, days of absenteeism, or counts of traffic violations. The framework is termed moment-based approach and is based on regression models with a logarithmic link function. The moment-based approach overcomes three major shortcomings of earlier approaches estimating treatment effects on count outcomes: First, effect definitions are based on the stochastic theory of causal effects (Steyer, Mayer, Fiege, 2014). This theory provides unambiguous definitions of atomic or individual causal effects as differences between conditional expectations under treatment and under control (i.e., a reference group). Further, causality conditions under which estimated average and conditional treatment effects may be interpreted as aggregates of atomic or individual causal effects are explicated. In contrast, in regression models with a logarithmic link function it is common to inspect treatment effects defined as ratios of conditional expectations under treatment and under control. We caution that these ratio effects have a different causal interpretation than difference effects. For example, the simple ratio of group averages in a randomized experiment does not represent an average of individual ratio effects -- unlike the simple difference of group averages, which is also an average of individual difference effects. Second, traditional approaches examining treatment effects on count outcomes, treat the size of treatment groups and values of observed covariates as fixed-by-design, that is, predetermined by the experimenter. If in fact group sizes and covariates are not predetermined -- as is the case in most psychological and social science studies --, this assumption can lead to underestimated standard errors and, thus, inflated Type 1 error rates and decreased power for average and conditional treatment effect estimates. The moment-based approach offers an alternative by treating both group sizes and covariates as random or stochastic variables, which can substantially improve statistical inferences. Third, the moment-based approach allows to account for measurement error in covariates. While earlier approaches were based on generalized linear models with a logarithmic link function (e.g., Poisson regression models), we use a structural equation modeling framework. In a negative binomial multigroup structural equation model, it is possible to simultaneously estimate measurement models for latent variables (i.e., decomposing true scores and measurement error in fallible indicators) and a negative binomial regression model with these latent variables as regressors. While ignoring measurement error in the generalized linear models might lead to biased estimation of both the regression coefficients and the corresponding effect estimates, these issues are circumvented in the moment-based approach. In this thesis, the moment-based approach is developed and extended in a step-by-step manner. First, we motivate the need for a new approach by presenting and discussing the three aforementioned shortcomings of earlier approaches (i.e., interpretation of exponentiated regression coefficients and marginal effects) in detail. Second, we introduce the key idea of the moment-based approach in a simple case considering only a single, but stochastic covariate using a generalized linear model. In a Monte Carlo simulation study, we compare the performance of a traditional (marginal effects) approach with the performance of the moment-based approach. Third, we extend the statistical framework to the negative binomial multigroup structural equation model and include multiple, possibly latent covariates as well as stochastic group sizes in our effect computations. Fourth, we relax the assumption of multivariate normally distributed covariates from previous steps and derive a generalized moment-based approach accounting for various kinds of normally and non-normally distributed covariates. As a result, we achieve a suitable approach for a broader range of applied data analyses. Finally, we discuss guidelines how and when the moment-based approach should be applied and have an outlook on possible further developments.