Conditional Versus Marginal Effects: Theoretical Foundation for Between-Case Incidence Rate Ratio with SCED Count Data (Poster 9)

• In Event: Innovative Methodologies and Systematic Reviews in Analyzing Single-Case Experimental Design Data

Abstract

Single-case experiment designs (SCEDs) are experimental designs in which a small number of cases are repeatedly measured over time, with manipulation of the baseline phase and intervention phase. Because the measurement procedures in SCEDs often involve direct observations of behaviors, count data are very common. To account for both the clustering and the non-normal nature of count data in single-case studies, generalized linear mixed models (GLMMs) have been shown as a promising approach (Declercq et al., 2019; Li et al., 2023; Shadish et al., 2013).
In many fields, the incidence rate ratio (IRR) is the conventional choice for quantifying effect sizes in count outcomes. The IRR serves as a standardized measure, representing the proportional difference in incidence rate between the control and treatment conditions. In SCEDs, Li et al. (2023) proposed the within-case incidence rate ratio (WC-IRR) based on GLMMs, demonstrating its favorable statistical properties. The WC-IRR is a subject-specific effect size that can be compared to and/or synthesized with results from other SCEDs addressing the same research questions. However, the WC-IRR is not appropriate for synthesis with effect size measures for count data from between-groups design (e.g., RCTs). To address this, our group recently derived a design-comparable version of IRR, the between-case incidence rate ratio (BC-IRR), for SCED count data.
The WC-IRR and BC-IRR aim to answer different research questions. In SCEDs, the WC-IRR measures the treatment effect for a typical case in the target population. In the multilevel modeling framework, a typical case represents an individual for whom all associated random effects are equal to zeros, thus considered as the median case in the target population. The estimate of the WC-IRR is called conditional effect or subject-specific effect. On the other hand, the BC-IRR represents the ratio between the expected incidence rate across cases if treatment is introduced at time A and the expected incidence rate across cases if treatment is never introduced, with all outcomes are measured at time B. The definition of BC-IRR is exactly the same as that of the IRR used in the hypothetical between-subjects experiment with treatment introduced at time 𝐴 and outcome measured at time 𝐵. Therefore, the BC-IRR measures the proportional difference between two subpopulation groups (i.e., control versus treatment). The estimate of BC-IRR is called marginal effect or population-average effect. However, this subtle but important difference is often ignored in the analysis of SCED count data.
To facilitate understanding and provide accurate interpretations of WC-IRR and BC-IRR based on GLMMs, this study will graphically illustrate why the conditional and marginal estimates differ for count data, how this difference impacts the effect size estimates in the analysis of SCED count data. I hope this study can help applied researchers to understand the theoretical foundations for WC-IRR and BC-IRR, and make sound statistical decisions regarding appropriate effect size based on GLMMs when dealing with SCED count data.

Authors

• Haoran Li, University of Minnesota

• Wen Luo, Texas A&M University

• Eunkyeng Baek, Texas A&M University