code class="asciimath">Let T denote the failure time, C denote the censoring time, and x(t) denote a p-dimensional vector of potentially time-dependent covariates. Assume that T and C are statistically independent given x(t). Under a broad class of semiparametric transformation models, the conditional cumulative hazard function of T given x(t) takes the form \Lambda (t;x)=G[\int_0^t exp{\beta ^(T)x(s)}d\Lambda _(0)(s)] S&AS: STAT3655 Survival Analysis where \Lambda _(0)(t) is an unspecified, arbitrary increasing function with \Lambda _(0)(0)=0,\beta is a p-dimensional vector of unknown regression parameters, and G(u) is a specific transformation function that is differentiable and strictly increasing, with first-order derivative G^(')(u)>0. Assume that \Lambda _(0)(t) is differentiable on [0,\infty ) with first-order derivative \lambda _(0)(t). A random sample of n independent subjects are recruited and the observations are {Y_(i),\delta _(i),x_(i)(t)}(i=1,dots,n), where Y_(i)=min(T_(i),C_(i)) is the observation time, \delta _(i)=I(T_(i)<=C_(i)) is the failure indicator, and x_(i)(t) is the vector of covariates for the i th subject. (a) Show that when G(u)=u and x(t)-=x, model (2) reduces to the propor- tional hazards model \lambda (t;x)=g_(0)(t)exp(\beta ^(T)x) for some function g_(0)(t)>0.(b) Show that when G(x)=log(1+x) and x(t)-=x, model (2) reduces to the proportional odds model