What is the relationship between KM-estimator and Cox model? What I am aware is that Cox model allow us to adjust covariates. If I build a Cox model and draw a KM curve (using survdiff function). Would the curve reflect the result of Cox model (covariates adjusted) or the result of KM-estimator (single variate)? In another word, can I using KM curve to show the result of Cox model?
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The Kaplan-Meier curve is the non-parametric maximum likelihood estimate of the survival function. The significance test for differences in survival is the log-rank test. The log-rank test is the score test for the Cox model with the variables adjusted as strata, so they should provide very similar answers. Cox models can fit way more complicated survival data, though, using frailties for correlation, left, right, and interval truncation, time varying covariates, strata and continuous variable adjustment. So the Cox model is more broad and versatile.
You should almost always show the Kaplan Meier curve with any type of survival analysis, even when the graphical display only approximates the inference presented in tables/summaries of Cox model output. It's the same as scatter plot for multivariable linear models: they go hand-in-hand.
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$\begingroup$ I am not sure if I understand you correctly. Seems like that KM curve shows approximation of Cox model. I am under the impression that we usually present forest plot of Cox model result while KM curve of KM estimator result. Is this correct? $\endgroup$– unicornCommented Jan 18, 2020 at 2:04
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$\begingroup$ @unicorn I think you are confusing empirical estimates of overall survival (KM) with a graphical representation of the hazard ratios and their uncertainty (forest plot). $\endgroup$– AdamOCommented Jan 21, 2020 at 15:17
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$\begingroup$ I read about KM as empirical estimation. But I did not understand why it is 'empirical'. Also, Why do you say forest plot is 'uncertain'? I thought forest plot created by Cox model is the more precise result comparing to KM. $\endgroup$– unicornCommented Jan 24, 2020 at 3:16
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$\begingroup$ @unicorn uncertainty is another way of describing standard error. The KM curve has a 1-1 relationship with the empirical distribution function $S_{KM}(t) = 1-\mathbb{F}(t)$. Too advanced to formalize in a comment, but trust me it is so. You should also appeal to the intuitive aspect of it to: there are no predisposing assumptions behind the KM curve, it's just a running fraction of people at risk. $\endgroup$– AdamOCommented Jan 24, 2020 at 14:52