  Given the prominence of targeted therapy and immunotherapy in cancer
treatment, it becomes imperative to consider heterogeneity in patients'
responses to treatments, which contributes greatly to the widely used
proportional hazard assumption invalidated as in several clinical trials. To
address the challenge, we develop a Dual Cox model theory including a Dual Cox
model and a fitting algorithm.
  As one of the finite mixture models, the proposed Dual Cox model consists of
two independent Cox models based on patients' responses to one designated
treatment (usually the experimental one) in the clinical trial. Responses of
patients in the designated treatment arm can be observed and hence those
patients are known responders or non-responders. From the perspective of
subgroup classification, such a phenomenon renders the proposed model as a
semi-supervised problem, compared to the typical finite mixture model where the
subgroup classification is usually unsupervised.
  A specialized expectation-maximization algorithm is utilized for model
fitting, where the initial parameter values are estimated from the patients in
the designated treatment arm and then the iteratively reweighted least squares
(IRLS) is applied. Under mild assumptions, the consistency and asymptotic
normality of its estimators of effect parameters in each Cox model are
established.
  In addition to strong theoretical properties, simulations demonstrate that
our theory can provide a good approximation to a wide variety of survival
models, is relatively robust to the change of censoring rate and response rate,
and has a high prediction accuracy and stability in subgroup classification
while it has a fast convergence rate. Finally, we apply our theory to two
clinical trials with cross-overed KM plots and identify the subgroups where the
subjects benefit from the treatment or not.
