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In a proportional hazard model with a Breslow estimator, I wondered why do we adjust on covariates in the cumulative hazard and in the model. On these few lines from Xia et al.[2], enter image description here we can see that the linear predictors are used twice in the survival function.

I ran some simulations on a Cox model with a Breslow estimator and my variance was far too small. So I wondered if the cause was not because the model was over-adjusted ...

Why do we need to multiply $\hat{\Lambda}_{0,BR}(t)$ by $\exp(\hat{\beta} z)$ knowing that $\exp(\hat{\beta} z)$ was already used for the calculation of $\hat{\Lambda}_{0,BR}(t)$?

I am sorry if this question seems naive!

[2]:J Biom Biostat 2018, 9:2 DOI: 10.4172/2155-6180.1000392

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The Breslow estimator (of the cumulative hazard function) uses the parameter estimates from the Cox model, it does not estimate separate parameters. So it is not "over adjusted" (unless the Cox model is overadjusted). However, the estimate of the baseline survival function is a function of the covariates. Therefore, if you don't take account of the correlation between the centered survival function estimate and the model parameters, you will not obtain correct confidence limits for other survival estimates. The Breslow cumulative hazard estimator works by subtracting off the effect of the covariates to come up with a good estimate of what the cumulative hazard function is for a subject with all covariates set to 0. Just the same as estimating an intercept in a linear regression model.

Just like a linear regression model, you also have to account for the covariance between the intercept and the slope to find confidence bounds for a prediction at any non-zero value.

Numerous are the ways of estimating confidence limits for the semi-parametric survival curve. A few default implementations are included in most software. It's actually an open problem in statistics. Some well known implementations are Hall-Wellner, Breslow and Crowley, Gillespie and Fisher, Nair, etc. References are indirectly alluded to here: https://www.stat.washington.edu/jaw/RESEARCH/TALKS/NBws.pdf

https://www.jstor.org/stable/2335326?seq=1 is a nice review.

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