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Why or when is the hazard ratio more than the range of the confidence interval? As I understood it, the hazard ratio must fall with in the range of the confidence interval.

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    $\begingroup$ Do you mean the population hazard ratio or the sample hazard ratio? You are right about the sample hazard ratio if it and its CI are estimated from a Cox model. There are a number of biased designs and biased estimators for which the sample CI will have poor coverage (less than the 1-$\alpha$ level) in small sample sizes, and this coverage reduces as the $n$ increases. $\endgroup$
    – AdamO
    Apr 8 '18 at 23:36
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The Confidence Intervals (CI) is interpreted in a similar manner to odds ratio CI. Since the true value of the coefficient in the population cannot be exactly determined based on any sample, even with high significance levels, CI's lower and upper values give a range in which the true population value lies - usually with 95% certainty. For example, in this mock coxph model:

               coef exp(coef) se(coef)     z Pr(>|z|)   
Working      0.6370  1.8908   0.2362   2.697  0.00699 **
    ---
    Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

          exp(coef) exp(-coef) lower .95 upper .95
Working     1.891     0.5289      1.19     3.004

Having a job (Working) increases the hazard ratio of getting a girlfriend by ~81%, with the population value being between an increase of 19% to 300% (with a 95% probability).

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