Derivation/Properties of Nonparametric Survival Function Hypothesis Test

Question

Suppose two samples of right-censored failure time data: $$\boldsymbol{X_1} = \{(X_{i1}, \delta_{i1}) = (\min(T_{i1}, C_{i1}), 1_{T_{i1} < C_{i1}}): i = 1, ..., n_1$$ $$\boldsymbol{X_2} = \{(X_{i2}, \delta_{i2}) = (\min(T_{i2}, C_{i2}), 1_{T_{i2} < C_{i2}}): i = 1, ..., n_2$$ where $T_{ij}$ are the underlying failure times and $C_{ij}$ are the censoring times. Suppose for each sample, the failure times are distributed according to survival functions $S_1$ and $S_2$, respectively. We are interested in testing whether $S_{T_2}(t) = S_{T_1}^\beta(t)$ and so we propose the hypotheses: $$H_0: \beta = 1, H_1: \beta \neq 1$$ Deriving the log-likelihood function for $\beta$ and differentiating with respect to $\beta$, we obtain the score function under the null hypothesis: $$S(\beta=1) = \sum_{i=1}^{n_{2}} \left\{ \delta_{i2} + \log(S_{T_2}(X_{i2})) \right\}$$

Questions:

1. Is the expected value of the score function equal to 0? How can one show this?
2. How does this likelihood based test reduce to the log-rank test? Is there any connection?
3. Are there any references to formal likelihood based testing procedures for survival analysis?

Answer 1

What you describe is a proportional hazards model. The hazard function $h(t)$ is:

$$h(t)=-\frac{d\ln S(t)}{dt}.$$

If $\ln S_2(t) = \beta \ln S_1(t)$ for all $t$ as you posit, $h_2(t) = \beta h_1(t)$ and $\beta$ is simply the (constant) hazard ratio $h_2(t)/h_1(t)$. All the standard results under proportional hazards for equivalence of the score and log-rank tests, asymptotic equivalence of score and likelihood-ratio tests (a general result for likelihood-based models), etc., follow.

For a a reference on likelihood-based analysis of survival data that covers non-parametric, semi-parametric, and parametric models, I continue to refer to Klein and Moeschberger's Survival Analysis from 2003. Section 3.5 shows the forms of likelihood construction for all types of truncated and censored data, which I've summarized here. Applications and examples abound therein.