Hi there, I have been running survival analysis on individual genes of interest. Firstly, I divided them into two groups: Up-regulated and Down-regulated gene. And then, I run SA on each them. This is one of the many results I got.

I set the event: die =1, survive=0. To HR >1, it is easy to interpret, but with HR<1, e.g like this one, I dont know what it means? To me, I guess, down-regulation is Ref, and up-regulation's Harzard ratio of 0.758. It implies that up-regulated MAP2K4 is significantly associated with better outcome than down-regulated MAP2K4 ! Is it right?

Your intuition is correct, you need to look at exp(coef), which is the hazard ratio and in your case the numerator is "upregulation" and the denominator is "downregulation". This hazard ratio of <1 shows that there is less chance of the event happening (in this case death) when your gene of interest is upregulated. You can tell "downregulation" is the reference since in R, the coefficients are written with respect to the reference level, from the coxph output you pasted, I can tell that your map2k4 column has two levels, up and down, and down is the reference because of lexicographic reasons and therefore the coefficient is stated as map2k4up.

Many thanks @Haci for your clear explantation.