# Survival analysis: Missing Cox Regression Analysis

I came across a survival analysis that was conducted with Kaplan-Meier and log rank test alone. No Cox regression was performed. Am I correct to assume that there was an omission of the essential cox regression data? I thought the Cox proportional hazards model was necessary to control for multiple covariates.

No. (Judging by your description) There was no omission: one can perform inference in survival analysis without resorting to a Cox proportional hazards model.

The Kaplan-Meier life-table approach can model survival, and, indeed, permits inference testing equality of two survival curves in favor of evidence of difference without resorting to Cox proportional hazards models:

$$\hat{S}_{t} = \hat{S}_{t-1}\times\frac{n_{t}-d_{t}}{n_{t}};\text{ and }\hat{S}_{0}=1$$

where $n_{t}$ are the number of individuals at risk, and the number of events (e.g., deaths) by time $t$, respectively.

If we hypothesize that two groups have the same survival by some time $t$, then we would expect that observed deaths would not differ from expected deaths, where expectations are produced by merging observations in both groups. Our test statistic is therefore:

$$\begin{array}{rcl}U_{L} & = &\sum^{t}_{i=1}{\left(d^{\text{obs}}_{Ai}-d^{\text{exp}}_{i}\right)}\end{array},$$

where $d^{\text{exp}}_{i}$ is calculated using $n_{Ai}+n_{Bi}$, and $d_{Ai}+d_{Bi}$. We can conduct a $z$-test of $H_{0}: S_{A} = S_{B}$ using:

$$\begin{array}{rcl}z_{U_{L}} & = & \frac{|U_{L}|-\frac{1}{2}}{s_{U_{L}}}\\ s_{U_{L}} & = & \sqrt{\sum_{i=1}^{t}\frac{n_{Ai}n_{Bi}d^{\text{exp}}_{i}\left(n_{i}^{\text{exp}}-d^{\text{exp}}_{i}\right)}{n^{\text{exp }2}_{i}\left(n^{\text{exp}_{i}}-1\right)}}\end{array}$$

Per Glantz, S. A. (2006) primer of biostatistics, chapter 11: Survival Analysis. McGraw-Hill Medical, New York, NY, 6th edition.

The log-rank test is just the score test for the Cox model with a single predictor. You are right to be concerned if a log rank test is presented as an "adjusted" analysis, but further inspection is needed. In what sense is there a need to adjust for more than one covariate? Is it a factor level coding of a single exposure, or is there claimed adjustment for possible confounding variables?

It's possible to stratify a log-rank test (the Mantel-Haenszel estimator) this is also the same as the score test for the hazard ratio in a Cox model which stratifies by a categorical factor. Note: stratification is different from adjustment in a Cox model. Between strata, the Cox model allows for a differently shaped baseline hazard function. It is a stronger form of covariate adjustment. This is one way covariates can be handled in a log-rank test.

Still it is possible to account for other covariates without adjustment or stratification. If the other covariates are confounders, it is unnecessary to adjust for confounders if confounding has been addressed through either randomization or propensity matching.

It is impossible to answer your question without more details about exactly what was done. I can imagine several scenarios where a log-rank test correctly summarizes an adjusted analysis.

• I agree that in several settings the logrank, or its stratified version, may be enough. Let me also add that in some situations ( for example if hazards are not proportional over time), Cox regression would actually create more damage, so it's everything but "essential". The logrank test, on the other way, can be modified (eg Wilcoxon, Tarone-Ware) and provide different weights over time. Apr 17 '18 at 18:39
• @andbel hmm. I might differ with you here. First we write the hypothesis. The null hypothesis for the log-rank and Wilcoxon is not $S_1 = S_2$. It is $\theta =0$ where $\theta$ is the log-hazard ratio. If the curves perfectly cross, the null is true. This is true of the Wilcoxon as well, they just cross at different points. I think a better strong null test of $S_1 = S_2$ would be a exponential survival model with piecewise-linear hazard function. Apr 17 '18 at 18:46