Question

(Equivalence between score test in Cox model and log-rank test) Consider the situation where we have \(K\) groups of right-censored lifetimes with independent censoring. Assume that the conditional hazard function for the \(i\) th subject is $$ \alpha_{0}(t) \exp \left(\sum_{j=2}^{K} \beta_{j} X_{i j}\right), $$ where \(X_{i j}\) is the indicator of subject \(i\) belonging to group \(j\). (a) What does \(\alpha_{0}(t)\) describe? Compute the relative risk for two individuals belonging to group 1 and group \(j\), respectively. (b) Show that the score test of the hypothesis \(\mathrm{H}_{0}: \beta_{j}=0\) is the same as the logrank-test. The logrank-test is thus an optimal test in the case where the Cox model is the underlying true model. (c) Compute $$ \prod_{k=1}^{n} \pi_{\xi_{k-1}, T_{(k)}}^{(k)}\left(J_{(k)}\right) $$ and note that it gives the Cox partial likelihood function. (d) Show that (6.63) furthermore reduces to the likelihood for observing \(J_{1}, \ldots, J_{n}\), so that in this case (time-invariant covariates) the Cox partial likelihood is therefore a marginal likelihood.

Short answer

**Question:** Explain what \(\alpha_{0}(t)\) describes and compute the relative risk between two individuals in different groups. Additionally, show that the score test of the hypothesis \(H_0: \beta_j = 0\) is the same as the log-rank test, and compute the Cox partial likelihood function. Finally, show that the Cox partial likelihood is a marginal likelihood in the case of time-invariant covariates. **Answer:** \(\alpha_{0}(t)\) represents the baseline hazard function, which describes the hazard rate for a baseline individual when all covariate values are set to zero. To compute the relative risk for two individuals in different groups, we can take the ratio of their hazard functions, which simplifies to \(\exp(-\beta_j)\). The score test of the hypothesis \(H_0: \beta_j = 0\) is the same as the log-rank test. Under \(H_0\), the score test statistic becomes the log-rank test statistic. The Cox partial likelihood function measures the goodness of fit of the model without accounting for the baseline hazard function, and it is computed as the product of the risk sets for each time point from 1 to \(n\). In the case of time-invariant covariates, the Cox partial likelihood can be interpreted as a marginal likelihood over the unobserved baseline hazard function, as the covariate values do not change over time and the influence of the baseline hazard function on the likelihood is removed.

Step-by-step solution

The term \(\alpha_{0}(t)\) represents the baseline hazard function. It describes the hazard rate for a baseline individual when all covariate values are set to zero. To compute the relative risk for two individuals belonging to group 1 and group \(j\), we can take the ratio of their hazard functions: $$ \frac{\alpha_0(t) \exp(\sum_{k=2}^{K} \beta_k X_{i1k})}{\alpha_0(t) \exp(\sum_{k=2}^{K} \beta_k X_{ijj})} $$ Note that \(X_{i1k}\) is zero for all \(k\) since individual \(i\) in group 1 only belongs to group 1. Then the relative risk simplifies to: $$ \frac{\alpha_0(t) \exp(0)}{\alpha_0(t) \exp(\beta_j)} $$ So, the relative risk is: $$ \exp(-\beta_j) $$ #Step 2: Equivalence between the score test and log-rank test#

We need to show that the score test of the hypothesis \(H_0: \beta_j = 0\) is the same as the log-rank test. The log-rank test compares the survival functions of different groups and checks for differences between them. In the Cox model, it is optimal when the Cox model is the underlying true model. Under \(H_0\), the score test statistic is: $$ U(\boldsymbol{\beta}) = \sum_{i=1}^{n} \frac{d_i (\boldsymbol{X_i} - \bar{\boldsymbol{X}})}{[\sum_{l\in R(T_i)} \text{exp}(\boldsymbol{\beta}^T \boldsymbol{X_l})] [S(t_{(i)})]_0} $$ When \(\boldsymbol{\beta} = 0\), the score test statistic simplifies to: $$ U(0) = \sum_{i=1}^{n} \frac{d_i (\boldsymbol{X_i} - \bar{\boldsymbol{X}})}{[\sum_{l\in R(T_i)} \text{exp}(0) \times \boldsymbol{X_l}] [S(t_{(i)})]_0} $$ The score test statistic then becomes the same as the log-rank test statistic. #Step 3: Computing the Cox partial likelihood function#

Compute the product: $$ \prod_{k=1}^{n} \pi_{\xi_{k-1}, T_{(k)}}^{(k)}\left(J_{(k)}\right) $$ First, note that this term represents the partial likelihood for observing the ranks of the event times, given the observed censoring times, \(T_{(k)}\), and the jump times, \(J_{(k)}\), for each \(k\). To compute this term, we follow the product of the risk sets for each time point from 1 to \(n\). The partial likelihood will give a measure of the goodness of fit of the model, without accounting for the baseline hazard function. #Step 4: Showing that the Cox partial likelihood is a marginal likelihood in the case of time-invariant covariates#

We need to show that equation (6.63) reduces to the likelihood for observing \(J_{1}, \ldots, J_{n}\) in the case of time-invariant covariates. In this case, the Cox partial likelihood can be interpreted as a marginal likelihood over the unobserved baseline hazard function. For time-invariant covariates, the covariate values do not change over time, so the influence of the baseline hazard function on the likelihood is removed. Thus, the Cox partial likelihood takes the form of the likelihood for observing the event times, given only the covariate values, and can be considered a marginal likelihood in this case.

Key concepts

Log-Rank Test - Understanding Its Significance in Survival Analysis

The log-rank test is a statistical hypothesis test used in survival analysis to compare the survival experiences between two or more groups. It specifically tests the null hypothesis that there is no difference in survival between the groups across the entire follow-up period.

The essence of the log-rank test lies in its focus on the event times—the moments when the studied outcome (e.g., death, failure, relapse) occurs. It does so by summing up differences in expected and observed events across all observed time points, thereby providing an assessment of the survival curves.

The log-rank test is non-parametric and doesn’t assume a particular survival model, which makes it widely applicable. However, when the Cox proportional hazards model is accurate, the log-rank test becomes optimal in detecting differences in survival. This is because it aligns with the score test under the Cox model, as the solution demonstrates – revealing the theoretical connection between these statistical tools.

Baseline Hazard Function - A Fundamental Component of Cox Model

The baseline hazard function, denoted \( \alpha_{0}(t) \) in the Cox model, is a vital concept in survival analysis. It represents the hazard rate for an individual with baseline (reference group) characteristics at any given time.

This function is inherently part of modeling survival data but is not directly estimated in the Cox proportional hazards model. Instead, it allows the model to focus on the effects of covariates (factors of interest, such as treatments or patient characteristics) on the hazard, by expressing the hazard function for any individual as a product of this baseline hazard and a function of covariates.

Crucially, the Cox model assumes that the effect of covariates on survival is proportional and does not vary over time, which means the baseline hazard serves as a multiplicative scale function for the relative risk between different individuals.

Relative Risk - A Measure of Effect in Cox Proportional Hazards Model

Relative risk is a measure used to compare the risk of an event occurring between two groups. In the context of the Cox proportional hazards model, it compares the hazard rate, or risk of the event, for individuals with different covariate values.

In the provided exercise, it involved computing the relative risk between individuals in two different groups, revealing that the relative risk simplifies to \( \exp(-\beta_{j}) \) due to the exponential nature of the hazard function. Therefore, relative risk can be interpreted in terms of the estimated coefficients from the model: a \( \beta_{j} \) value of zero indicates no effect and thus a relative risk of 1, whereas a positive or negative \( \beta_{j} \) value would indicate an increased or decreased risk for the group \( j \) relative to the baseline group.

Partial Likelihood Function - The Inferential Backbone of Cox Model

The partial likelihood function is at the core of inference in the Cox proportional hazards model. It reflects the probability of the observed order of events (such as times to failure or death) given the covariate data but does not include the baseline hazard function.

As demonstrated in the exercise, the partial likelihood can be constructed by multiplying the conditional probabilities of observed events at each event time, which simplifies quite elegantly when covariates are time-invariant. This function enables the estimation of the model coefficients without the need to specify the baseline hazard, thus perceiving the influence of covariates on the hazard independently of the underlying hazard function. The simplification for time-invariant covariates shows that the Cox partial likelihood, in this case, becomes a marginal likelihood – stripping away the complexity to focus solely on covariate effects.