Sample size estimation using a latent variable model for mixed outcome co-primary, multiple primary and composite endpoints

Abstract

Mixed outcome endpoints that combine multiple continuous and discrete components are often employed as primary outcome measures in clinical trials. These may be in the form of co-primary endpoints, which conclude effectiveness overall if an effect occurs in all of the components, or multiple primary endpoints, which require an effect in at least one of the components. Alternatively, they may be combined to form composite endpoints, which reduce the outcomes to a one-dimensional endpoint. There are many advantages to joint modeling the individual outcomes, however in order to do this in practice we require techniques for sample size estimation. In this article we show how the latent variable model can be used to estimate the joint endpoints and propose hypotheses, power calculations and sample size estimation methods for each. We illustrate the techniques using a numerical example based on a four-dimensional endpoint and find that the sample size required for the co-primary endpoint is larger than that required for the individual endpoint with the smallest effect size. Conversely, the sample size required in the multiple primary case is similar to that needed for the outcome with the largest effect size. We show that the empirical power is achieved for each endpoint and that the FWER can be sufficiently controlled using a Bonferroni correction if the correlations between endpoints are less than 0.5. Otherwise, less conservative adjustments may be needed. We further illustrate empirically the efficiency gains that may be achieved in the composite endpoint setting.

Keywords: latent variable modeling; mixed outcome endpoints; sample size estimation.

FIGURE 1

Power function for individual SLEDAI (continuous), PGA (continuous), BILAG (ordinal), and Taper (binary) outcomes and the power functions with when they are treated as co‐primary, multiple primary, and composite endpoints using data from the MUSE trial

FIGURE 2

Overall power $1-\beta$ to detect the treatment effects assumed from the MUSE trial for the systemic lupus erythematosus co‐primary, multiple primary, and composite endpoints for different sample sizes per group $n=n_C=n_T$ and differing correlations between outcomes, where Low = 0.3, Medium = 0.5, and High = 0.8

FIGURE 3

Family‐wise error rate (FWER) of the multiple primary endpoints shown both unadjusted and adjusted using the Bonferroni correction. FWERs are shown for $K=(2,3,4)$ outcomes and correlations are constrained to be equal between all outcomes