Kaplan-Meier Curves for Continuous Covariates – Techniques in Survival Analysis

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It's well known the Kaplan-Meier curve is a univariate time-to-event summary plot. If you want to show differences between groups, you can make a KM curve for each group. In addition you can calculate the p value from a log rank test or a Cox model adjusting for that variable categorically. It's understood that the tests do not test for differences in the survival distributions per se, but rather the hazard ratio. So the graph and test are not one-to-one. Nonproportional hazards reduce the power of the test.

Turning attention to continuous covariates, has anyone proposed clever ways to show something like a KM curve for a continuous adjustment variable? At a basic level you can split a variable according to quintiles, possibly with overlapping ranges. But I wonder if more sophisticated methods have been proposed to detect survival differences visually.

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One option is a Kaplan-Meier contour plot as described here

The small dots are censored, the large dots are events, and the curves are contours of $\hat S(t; z)$, the conditional survival function. For each $z$, we estimate $S(t;z)$ by a weighted Kaplan-Meier estimator where the weight for observation $i$ is $k(z-z_i)$ for some smoothing kernel $k()$

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Solved – Comparison of Kaplan-Meier curves across ordered groups

Function comp in survMisc may be close to what you're after. From the docs:

Tests for trend are designed to detect ordered differences in survival curves.
That is, for at least one group: $$S_1(t) \geq S_2(t) \geq ... \geq S_K(t) \quad t \leq \tau$$ where $\tau$ is the largest $t$ where all groups have at least one subject at risk. The null hypothesis is that $$S_1(t) = S_2(t) = ... = S_K(t) \quad t \leq \tau$$ Scores used to construct the test are typically $s = 1,2,...,K$, but may be given as a vector representing a numeric characteristic of the group.
They are calculated by finding: $$Z_j(t_i) = \sum_{t_i \leq \tau} W(t_i)[e_{ji} - n_{ji} \frac{e_i}{n_i}], \quad j=1,2,...,K$$ The test statistic is: $$Z = \frac{ \sum_{j=1}^K s_jZ_j(\tau)}{\sqrt{\sum_{j=1}^K \sum_{g=1}^K s_js_g \sigma_{jg}}}$$ where $\sigma$ is the the appropriate element in the variance-covariance matrix (see COV).
If ordering is present, the statistic $Z$ will be greater than the upper $\alpha$-th percentile of a standard normal distribution.

For example:

library(survMisc)
data(larynx, package="KMsurv")
s4 <- survfit(Surv(time, delta) ~ stage, data=larynx)
comp(s4)

This will give tests for trend with various weights (as are used with the standard log-rank test):

$tests$trendTests
                                     Z       p
Log-rank                      3.718959 0.00010
Gehan-Breslow (mod~ Wilcoxon) 4.224765 0.00001
Tarone-Ware                   4.058010 0.00002
Peto-Peto                     4.129343 0.00002
Mod~ Peto-Peto (Andersen)     4.136319 0.00002
Trend F-H with p=1, q=1       2.396992 0.00827

The package has been changed slightly, and instead of tests or trendtests, use tft, which stands for... tests for trend. Furthermore, when using the comp() function, type it like this: comp(ten(<whatever your survival-fit object is>)).

Survival – Using Propensity Score Matching for Kaplan-Meier Survival Analysis

The big problem in using matching for this is that you will be throwing away data and thus potentially making true differences harder to find. See this page, for example. If you have an enormous data set that might not matter, but often in practice matching will give up more than it gains.

The log-rank test is equivalent to one way of evaluating a Cox proportional hazards for differences among groups. Why not just go all the way down that road and build a Cox model that includes multiple outcome-associated characteristics? That's a way to account for the effects of those other characteristics and document the remaining association of group membership with outcome. You can then illustrate the results with the (predicted) survival curves distinguished by group, at a set of otherwise identical covariate characteristics.

That said, you might nevertheless want to use matching to show that the group differences seen in the Cox model are robust to the assumptions of the model. See this page for a thorough discussion.

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Related Solutions

Solved – Comparison of Kaplan-Meier curves across ordered groups

Survival – Using Propensity Score Matching for Kaplan-Meier Survival Analysis

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