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I am running a multi-variate Cox regression and Stata provides the standard errors for each hazard ratio. How are theses to be interpreted? I know that I want my coefficients to be large compared to my SEs, but I don't know if the same rule applies to ratios.

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The formula for the Cox proportional hazards is:

$$h(t)=h_0(t)e^{\beta_1*x_1+...+\beta_n*x_n}$$

All $\beta$ are thus independent of the baseline hazard, the $h_0(t)$, allowing the comparison between different hazard ratios. For instance, if we have two treatment arms one with placebo ($X_{treat}=0$) and one with active substance ($X_{treat}=1$) where we also adjust for sex, we get:

$$HR=h_{treated}(t)/h_{placebo}(t)=\frac{h_0(t)e^{\beta_{treat}*1+\beta_{sex}*x_{sex}}}{h_0(t)e^{\beta_{treat}*0+\beta_{sex}*x_{sex}}} = e^{\beta_{treat}*1-\beta_{treat}*0+\beta_{sex}*x_{sex}-\beta_{sex}*x_{sex}}=e^{\beta_{treat}}$$

Note that the $\beta$ is in exponential format, thus the SE for the $\beta$ is also in the exponential form. When you compare the coefficient with the SE you need to do this with the $\beta$ in the logarithmic form. I would refrain from using the SEs for anything else than for confidence intervals/p-values.

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To pick up from Max Gordon's answer, the zero values of $\beta$ correspond to the unit values of the hazard ratios. Hence a ``strong'' result would be that the hazard ratios are sufficiently far from 1.

However, while hazard ratios are more interpretable than coefficients themselves, inference is weird in that scale, and asymptotic normality of results requires larger (often much larger) samples. To overcome this weirdness, Stata reports an asymmetric confidence interval for the ratios, exponentiating the endpoints of the CI obtained on the scale of coefficients. Thus instead of looking at hazard ratios and their standard errors, I would advise looking at the confidence intervals.

(This has nothing to do with the debate about p-values, null hypothesis testing, Bayes factors, effect sizes, and any of that philosophy. This is just plain higher order asymptotics.)

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