 Solution

The model is linear in each coded covariate on the log-hazard scale, not necessarily in the raw scientific variable. For a continuous variable $z$, begin with plots and a scientifically plausible range of forms. Compare $\beta z$ with a prespecified transformation such as $\beta \log z$ when $z>0$, or use a restricted cubic spline or fractional polynomial to represent a flexible smooth effect. Do not treat arbitrary integer category codes as a linear quantitative measurement without justification.

Plot martingale residuals from a suitable model against $z$ with a smooth trend. A residual pattern can suggest a missing or misspecified effect; these residuals are asymmetric, so the smooth relationship is more informative than judging normality. A model omitting $z$ helps reveal its overall shape; plots after including it help assess remaining misspecification. Partial-residual displays or fitted effect curves with uncertainty give complementary information.

Assess nonlinearity with a joint likelihood-ratio or score test for the nonlinear terms in a spline extension. The models containing only $z$ and only $\log z$ are generally nonnested, so a difference in their log-likelihoods is not automatically chi-squared. They can be compared by a justified nonnested criterion or validation, or embedded in a model containing both $z$ and $\log z$ and tested by dropping one term. Such an encompassing model may be highly collinear, and its coefficients should not be interpreted in isolation. If $z$ can be zero or negative, $\log z$ is undefined; choose an appropriate form rather than silently adding an arbitrary offset. Transformation choice is a question about the entire effect curve and its supported range, not only one coefficient's significance. Recheck time constancy after revising the functional form, since misspecification can mimic nonproportionality.