Blinded Sample Size Re-estimation for Continuous Endpoints - Part 3

In Parts 1 and 2 of this series, we evaluated moment-based approaches for blinded sample size re-estimation (SSR). This part focuses on a likelihood-based alternative - the maximum likelihood estimation (MLE), which leverages the full data likelihood rather than relying solely on sample moments, such as variance or kurtosis.

As before, a continuous observation 𝑋 from the pooled data of a two-group parallel study can be viewed as a random variable arising from a two-component Gaussian mixture model with a known mixture proportion. That is,

where 𝜔 denotes the mixture proportion (assuming 𝜔 = 1/2), µ₁ and µ₂ are the component means, and σ² is the common variance.

We use simulations to assess the performance of the estimators for parameters 𝛿 = µ₁ - µ₂, σ, 𝛿/σ  and (𝛿/σ)². The parameters 𝛿 and σ are estimated using the MLE from the mixture model, and the ratios

𝛿/σ and (𝛿/σ)² are subsequently obtained using the plug-in method. We consider four different scenarios: 𝛿/σ = 0.25, 0.50, 0.75 and 1.00 (assuming σ = 1), representing small to large effect sizes in clinical trials. The corresponding planned total sample sizes are 674, 172, 78 and 46, respectively, targeting approximately 90% power (based on a two-sample t-test and a two-side significance level of 0.05) under the assumed effect sizes. A blinded SSR is performed after about 75% of participants have completed the study, so the sample sizes used in the simulations are 506, 130, 60 and 36, respectively. All simulations are performed using the SAS NLMIXED procedure.

The table below, based on 1000 replications, reports the mean (Mean) and standard deviation (SD) of the estimated 𝛿, σ, 𝛿/σ and (𝛿/σ)², respectively.

We observe the following trend:

• 𝛿 tends to be over-estimated and σ under-estimated (similar findings were also reported in Friede and Kieser, 2002), which in turn leads to overestimation of both 𝛿/σ and (𝛿/σ)².

• The estimator for 𝛿 is particularly unstable, and consequently the estimators for 𝛿/σ and (𝛿/σ)² are also unstable.

A reasonable explanation is that when the effect size (𝛿/σ) is small (note that even 𝛿/σ = 1 is considered small in the context of Gaussian mixture models) and the sample size is limited, the likelihood function of the mixture model becomes relatively flat around the true parameter values, causing the MLE to be biased and highly unstable.

References

Friede, T. and Kieser, M. (2002). On the inappropriateness of an EM algorithm-based procedure for blinded sample size re-estimation. Statistics in Medicine 21, 165-176.