How should I interpret the Hazard ratios when using *gene expression input as continuous variable* (not categorical) in cox proportional hazard model.

*Question 1:* If I were to study the effect of gene expression variable on outcome, I could use:

fit1 <- coxph(Surv(time, status) ~ exp, data).

Here, for example, if I get HR = 0.71 (p.value <0.05), should I interpret that as high expression of the gene associated with better survival? Here, does HR indicate ratio of (chance of events in subjects with high expression) to (chance of events in subjects with no or low expression)? How the expression is dichotomized in the model?

*Question 2:* If I were to study the effect of gene expression on outcome by treatment, I could do something as below:

fit2.1 <- coxph(Surv(time, status) ~ exp, data=subset(data, data$group=="treated")

fit2.2 <- coxph(Surv(time, status) ~ exp, data=subset(data, data$group=="control")

Here should I interpret the (fit2.1) HR = 0.40 (p.value <0.05) as high gene expression within the treated samples associated with better survival; however (fit2.2) HR = 0.99 (p.value>0.05) as no effect of gene expression on outcome within the control samples?

*Question 3:* If I were to study the effect of gene expression on differential outcomes between the treated and control samples, I was thinking of below model using control as reference group including stratification by group to adjust for proportionality assumption (since cox.zph test indicated violation of proportionality assumption by group):

fit3 <- coxph(Surv(time, status) ~ exp * Strata (group), data

Here should I interpret the interaction model (fit3) HR = 0.55 (p.value <0.05) as high expression of the gene in treated samples associated with better survival compared to high expression of the gene in control samples? In similar way, should I interpret HR = 1.5 (p.value <0.05), as high expression of the gene in treated samples associated with poor survival compared to high expression of the gene in control samples? Here does HR equals to ratio of ratios ? (e.g., (HR of treated) / (HR of control) or is it a difference? Is this correct interpretation?

Appreciate any feedback and insight. Thank you so much.