Testing Equality of Curves after Covariate Adjustment

S ummary This paper is concerned with providing simple methodological approaches for global and local tests of difference between the mean of treatment and control groups when the measured outcome is a function. The added complexity is that for every subject we have repeated samples for the same curve and additional covariates of interest. We propose a permutation based approach to test for a global difference between the averages of two functional processes after covariate adjustment. The within group averages are estimated by modeling the relationship of the functional outcome on the covariate using functional regression methods and then averaging with respect to the covariate distribution in each group. The test statistic is the L 2 area under the squared difference curve. We also test for the localized differences between the two average curves using a nonparametric bootstrap of subjects to obtain the 95% pointwise and joint confidence intervals for the estimated covariate-adjusted difference curve. Extensive simulation studies illustrate that the proposed tests preserve the type one error and are highly sensitive to detecting departures from the null assumption. We illustrate our method by studying the differences in time varying oxygen consumption between the frail Interleukin 10tm1Cgn (IL10tm) mice and the wildtype mice after adjusting for body composition measures.