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The proportional hazards assumption basically says that the hazard rate does not vary with time. That is, $\text{HR}(t) \equiv \text{HR}$. When can we assume this? What if the hazard ratios at various times are: $2.4, 2.36, 2.27$ and $2.03$? Can we make the proportional hazards assumption? Also we have $$ \log[h(t|\textbf{x})] = \log[h_{0}(t)] + \beta_{1}x_1 + \dots + \beta_{p}x_{p}$$

Why do we need to estimate $h_{0}(t)$? If we have $h(t|\textbf{x})$, why can't we just put all the values of the predictors to zero to get $h_{0}(t)$?

Edit. I want a means to assess whether the PH assumption is true.

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    $\begingroup$ Your question needs some clarity. It appears you are talking about a hazard ratio, not a hazard rate. Is there a reason you've chosen that particular regression equation - there are many ways to approach a survival analysis. And what do you mean by "when" can we assume this - do you want a set of circumstances where it's true, or a means to assess whether its true in your case? $\endgroup$ – Fomite Nov 27 '11 at 3:00
  • $\begingroup$ What software do you want the "means to assess whether the PH assumption is true"? In my previous question I'm trying to deal with the problem that you have after testing the PH assumption but the top part shows how you check by using Grambsch and Therneau's method. $\endgroup$ – Max Gordon Nov 27 '11 at 7:59
  • $\begingroup$ The software you used should provide tests of this. For example, SAS will use the method of Lin, Wei and Ying (1993) (see the documentation for PHREG and, in particular, the ASSESS statement) $\endgroup$ – Peter Flom Nov 27 '11 at 13:09
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Peter is correct in it depends on what software you are using to check this, in the survival package with R there is the cox.ph() function.

Most assessments of the assumption will involve looking at the Schoenfeld residuals. If plotted against time there should be no noticeable pattern.

See Cox-Proportional Hazards starting at page 12.

Also, if modeling categorical variables, you could create Kaplan-Meier curves for each variables and see if they are roughly proportional to each other.

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  • $\begingroup$ Including a time-varying coefficient in the model is also a mean to check the PH assumption $\endgroup$ – boscovich Jun 27 '12 at 10:47
  • $\begingroup$ Using a weighting mechanism like IPTW would even allow you to make marginal Kaplan-Meier curves to assess proportional hazards visually in a model with more than just categorical variables. $\endgroup$ – Fomite Aug 27 '12 at 4:53

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