Nonparametric tests of treatment effect based on combined endpoints for mortality and recurrent events

Terminal events are commonly combined with other outcomes to improve the power for detecting treatment effects in clinical studies. This manuscript explores novel ways to combine information on terminal and recurrent events in constructing two-sample tests. Existing approaches follow either a time-to-first event analysis approach or a recurrent event modeling approach. Nonparametric recurrent event analyses are often restricted by independence assumptions on gap times between events. Although time-to-first event analyses are not subject to this restriction, they discard information that occurs beyond the initial event and are much less powerful for detecting treatment differences. We develop two new approaches for determining treatment effects, motivated by less restrictive assumptions of time-to-first event analyses that combine information from multiple follow-up intervals. The first testing procedure pools (correlated) short-term τ -restricted outcomes from prespecified intervals starting at times t(k), k = 1, . . . , b, and compares estimated τ -restricted mean survival across treatment groups from this combined dataset. The second procedure calculates conditional τ-restricted means from those at risk at times t(k), k = 1, . . . , b, and compares the area under a function of these by treatment. Variances calculations, taking into account correlation of short-term outcomes within individuals, linearize random components of the test statistics following Woodruff (1971. A simple method for approximating the variance of a complicated estimate. Journal of the American Statistical Association 66, 411-414) and more recently Williams (1995. Product-limit survival functions with correlated survival times. Lifetime Data Analysis 1, 171-186). Simulations compare the finite sample performance of our tests to the robust proportional rates model proposed by Lin and others (2000. Semiparametric regression for the mean and rate functions of recurrent events. Journal of the Royal Statistical Society: Series B (Statistical Methodology) 62(4), 711-730) and the Ghosh and Lin (2000. Non-parametric analysis of recurrent events and death. Biometrics 56(2), 554-562) combined test for recurrent events and death. For different treatment effect patterns the proposed methods perform favorably when compared with existing methods. These new analysis approaches also produce correct type I error rates with correlated gap times between events. New methods are applied to data from a trial of azithromycin in patients with chronic obstructive pulmonary disease.