  Causal mediation analysis studies how the treatment effect of an exposure on
outcomes is mediated through intermediate variables. Although many applications
involve longitudinal data, the existing methods are not directly applicable to
settings where the mediators are measured on irregular time grids. In this
paper, we propose a causal mediation method that accommodates longitudinal
mediators on arbitrary time grids and survival outcomes simultaneously. We take
a functional data analysis perspective and view longitudinal mediators as
realizations of underlying smooth stochastic processes. We define causal
estimands of direct and indirect effects accordingly and provide corresponding
identification assumptions. We employ a functional principal component analysis
approach to estimate the mediator process, and propose a Cox hazard model for
the survival outcome that flexibly adjusts the mediator process. We then derive
a g-computation formula to express the causal estimands using the model
coefficients. The proposed method is applied to a longitudinal data set from
the Amboseli Baboon Research Project to investigate the causal relationships
between early adversity, adult physiological stress responses, and survival
among wild female baboons. We find that adversity experienced in early life has
a significant direct effect on females' life expectancy and survival
probability, but find little evidence that these effects were mediated by
markers of the stress response in adulthood. We further developed a sensitivity
analysis method to assess the impact of potential violation to the key
assumption of sequential ignorability.
