Mathematics Ph.D. Dissertations


An Approach to Improving Test Powers in Cox Proportional Hazards Models

Date of Award


Document Type


Degree Name

Doctor of Philosophy (Ph.D.)



First Advisor

Junfeng Shang (Advisor)

Second Advisor

Hanfeng Chen (Committee Member)

Third Advisor

Wei Ning (Committee Member)

Fourth Advisor

Deborah G. Wooldridge (Other)


In power analysis for significance test of the treatment variable in the multivariable Cox proportional hazards models, the variance of the estimated log-hazard ratio for the treatment effect is usually approximated by inverting the expected null information matrix. Because in many typical power analysis settings, assumed true values of the hazard ratios are not necessarily close to one, the accuracy of inverting the expected null is not theoretically guaranteed. A null variance in power calculations underestimates or overestimates the true variance in different treatment allocation settings when the treatment allocation ratio is far from one, similarly alternative variance in power calculations predicts inaccurate results in different treatment allocation settings.

When the study events are rare, the alternative variance predicts accurate results in predicting the power, but the alternative variance does not provide accurate results when the study events are not rare. To address this problem, we propose an approach to estimating the variance, and this approach is compared with three widely used approaches in practice. The null variance in power calculations can be replaced with the proposed adjusted alternative variance derived under the assumed true value of the hazard ratio for the treatment effect. This approach is explored theoretically and by the simulations in this research.

In this approach, we improve the variance of the log hazard ratio for the treatment effect and compare it with the traditional null variance, the traditional alternative variance, and another variance using log-rank test under proportional hazard alternatives. The most accurate expression of the variance has a relatively simple form. The variance is scaled up by a variance inflation factor, denoted by $VIF$.

Our simulations for the relative bias of standard error and the power of the test for the treatment effect under the proposed variance are compared to the other standard variances. The results provide evidence that our proposed variance is a better estimator for the true variance of the log hazard ratio for the treatment effect under the study. Furthermore, one application with real-life examples is illustrated to evaluate the effectiveness of the proposed variance.