Introduction

In a previous blogpost, I described how complex estimation of U-statistic variance can be simplified using a “structural component” approach introduced by Sen (1960). The structural component approach is very similar to the leave-one-out jackknife. Essentially, the idea behind both of these approaches is that we decompose the statistic into individual contributions. Here, these are referred to as “structural components,” and in the LOO jackknife, these are referred to as “pseudo-values” or sometimes “pseudo-observations.” Construction of these individual quantities differs conceptually somewhat, but in another blogpost, I discuss their one-to-one relationship for specific cases. We can then take the sample variance of these individual contributions to estimate the variance of the statistic.

Estimators for increasingly popular win measures, including the win probability, net benefit, win odds, and win ratio, are obtained using large-sample two-sample U-statistic theory. Variance estimators are complex for these measures, requiring the calculation of multiple joint probabilities.

Here, I demonstrate how variance estimation for win measures can be practically estimated in two-arm randomized trials using a structural component approach. Results and estimators are provided for the win probability, the net benefit, and the win odds. For simplicity, only a single outcome is considered. However, extension to hierarchical composite outcomes is immediate with use of an appropriate kernel function.

Continue reading Practical inference for win measures via U-statistic decomposition