04 - Extending Mediation Analysis to Within-subjects Data With Dichotomous Outcomes

Linear regression is used to model the relationship between predictor variables and a continuous outcome while logistic regression is used for a dichotomous outcome. Mediation analysis uses linear regression or logistic regression to explain how an independent variable indirectly affects a dependent variable via a mediator. We know how to use linear regression to conduct mediation analysis for both between and within-subjects data, but when using logistic regression, we only know how to conduct mediation analysis for between-subjects data. Breen et al. (2013) developed a logistic method for doing mediation analysis for between-subject data and showed that the standard deviation of the residuals of the continuous mediation model could be used to determine a scale parameter that transforms the linear regression coefficients to the logistic regression coefficients, and vice versa. Using the methods from Breen et al. (2013), we derived equations to conduct mediation analysis for within-subjects data that has a dichotomous outcome. We used three methods to validate our derived equations, involving both simulated and real data. For the simulation, we simulated within-subjects data with continuous outcomes using the equations from Montoya & Hayes (2017) to get the linear coefficients and calculate the scale parameter to transform the linear coefficients to logistic coefficients. The difference in outcomes was dichotomized and used with the derived equations to get the logistic coefficients. Comparing the logistic coefficients to the transformed logistic coefficients confirmed that the scale parameter can be used to transform the logistic regression coefficients based on the population parameters set in the simulation. Next, we used data from Montoya et al. (2013) and both parametric and nonparametric tests to validate our equations. In this study, participants viewed two class syllabi (within-subject factor): one syllabus about a course that encouraged independent work and one about a course that encouraged group work. After viewing each syllabus, participants (N = 51) rated their interest in each class on a continuous scale from 1 (Not at all) to 7 (Extremely). We dichotomized the difference in interest variable so that we have both continuous and dichotomous measurements of the outcome variable to do mediation analysis using linear and logistic regression. The mediator in this study was a measurement of the participants' communal goals on a continuous scale. For this study, we examined how participants' communal goals explain their interest in taking a course based on the syllabus. The continuous difference in interest was used to find the transformed logistic coefficients by dividing the linear coefficients by the estimated scale parameter. The dichotomized difference in interest was used to find the logistic coefficients using the derived equations. Because the equations from Breen et al. (2013) are used to convert between population parameters, but we only have the sample estimates, we compare estimates to confidence intervals. We confirmed that the logistic coefficients were similar to the transformed logistic coefficients and were within the confidence intervals for the transformed logistic coefficients. We computed bootstrapped 95% confidence intervals (N = 10,000) to further check if the logistic estimate is in the transformed confidence interval. The extension of mediation analysis to logistic regression for within-subjects data will enable research using within-subjects design, and remove the restriction to measure variables solely on a continuous scale. For example, when measuring one's interest in a course, a question that asks for a "yes" or "no" answer like, "Are you interested in taking this course?" might be more valid than a question like, "Rate your interest in taking this course on a scale from 1 to 7." Future research will extend these methods to two dichotomous outcomes and dichotomous mediators.

Mediation analysis

Within-subjects

Dichotomous outcomes

linear regression

logistic regression 

Mid-Level

Knowledge

Women in Statistics and Data Science 2022