  The timing of antiretroviral therapy (ART) initiation for HIV and
tuberculosis (TB) co-infected patients needs to be considered carefully. CD4
cell count can be used to guide decision making about when to initiate ART.
Evidence from recent randomized trials and observational studies generally
supports early initiation but does not provide information about effects of
initiation time on a continuous scale. In this paper, we develop and apply a
highly flexible structural proportional hazards model for characterizing the
effect of treatment initiation time on a survival distribution. The model can
be fitted using a weighted partial likelihood score function. Construction of
both the score function and the weights must accommodate censoring of the
treatment initiation time, the outcome, or both. The methods are applied to
data on 4903 individuals with HIV/TB co-infection, derived from electronic
health records in a large HIV care program in Kenya. We use a model formulation
that flexibly captures the joint effects of ART initiation time and ART
duration using natural cubic splines. The model is used to generate survival
curves corresponding to specific treatment initiation times; and to identify
optimal times for ART initiation for subgroups defined by CD4 count at time of
TB diagnosis. Our findings potentially provide "higher resolution" information
about the relationship between ART timing and mortality, and about the
differential effect of ART timing within CD4 subgroups.
