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I am trying to build a Cox proportional hazards model using the coxph function in the survival package in R. My function call is:

m1=coxph(Surv(time=df$time, event=df$event,type=c('right')) ~ ln_num_prior_investments, data = df)

The model summary is as follows:

n= 1888, number of events= 707 

                             coef exp(coef) se(coef)     z Pr(>|z|)  
ln_num_prior_investments -0.03701   0.96367  0.02079 -1.78   0.0751 .
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Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Is the dependent variable in this model the survival time or the instantaneous hazard rate? I am trying to understand the directional impact of the ln_num_prior_investments variable, but the meaning of the sign in front of the coefficient will be different depending on whether it is the time or the hazard rate that is being modeled.

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  • $\begingroup$ Cox PH model estimates the hazard, and the coefficient you have in your model is a log(hazard ratio). The HR itself is exp(coef). The fact that the HR is lower than 1 implies a negative association between the covariate and the hazard of your event. However, survival and hazard functions are mathematically related. That means that, given a set of assumptions hold, also the impact on the survival curve is negative. This just to clarify that the interpretation will not be different if you look at the same problem in terms of survival time. $\endgroup$ – andbel Jan 3 '18 at 17:13
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Yes, in a Cox proportional hazards model, the variables affect the hazard function through the exponential function:

$$h(t) = h_0(t) exp(x^T\beta).$$

Typically, the interpretation is done on the hazard ratios:

https://en.wikipedia.org/wiki/Hazard_ratio

http://sphweb.bumc.bu.edu/otlt/MPH-Modules/BS/BS704_Survival/BS704_Survival6.html

An alternative model is the Accelerated failure time model, where the variables affect directly the survival time. It can also be fitted using the R survival package.

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