Interpreting interaction effect with ln(time) in Cox regression I am fitting a Cox proportional hazards model with the interaction effect dummy:ln(time). How would you interpret the result? Is it a simple exp(x)-1)*100 to receive the percentage increase in y?
Edit: I indeed made the mistake @EdM mentioned. This interaction is now set up according to Therneau et al. (2022, pp. 21). I originally wanted to replicate the following model of Schmitt et al. (2011):

In their results, the interaction term was not significant. I added stratification to increase the efficiency of the model. mgm_acquired is a time-invariant binary variable. The result now looks the following:
Call:
coxph(formula = Surv(time, status) ~ mgm_acquired + tt(mgm_acquired) + 
    strata(most_frequent_community) + strata(age_group) + strata(reg_month) + 
    strata(cluster), data = RG, tt = function(x, t, ...) x * 
    log(t))

  n= 119130, number of events= 71720 

                      coef exp(coef)  se(coef)      z Pr(>|z|)    
mgm_acquired     -0.179475  0.835709  0.047095 -3.811 0.000138 ***
tt(mgm_acquired)  0.011930  1.012002  0.009219  1.294 0.195648    
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

                 exp(coef) exp(-coef) lower .95 upper .95
mgm_acquired        0.8357     1.1966    0.7620    0.9165
tt(mgm_acquired)    1.0120     0.9881    0.9939    1.0305

Concordance= 0.503  (se = 0 )
Likelihood ratio test= 46.1  on 2 df,   p=1e-10
Wald test            = 44.28  on 2 df,   p=2e-10
Score (logrank) test = 44.35  on 2 df,   p=2e-10

Four questions emerged in my head:

*

*Is this more realistic now?

*How do I interpret the new coefficient of mgm_acquired?

*How to interpret the interaction effect? Does the absolute value of the coeffcient tell me something?

*Why did Therneau add log(t+20) "to make the first
200 days of the plot roughly linear"? What is the point of the 20?

Thanks!
 A: Although the large number of cases makes most of your coefficients "statistically significant," your model is not distinguishing cases very successfully.
The concordance of 0.503 is the fraction of comparable pairs of cases in which the observed and predicted event order agreed. That's barely better than a chance value of 0.500. It's not clear whether all of the stratification you used in the model was necessary or even helpful. For your first question, you must decide if that model and its result is "realistic" based on your understanding of the subject matter.
For the second and third questions, following the example in Section 4.2 of the time dependence vignette, the full mgm_acquired coefficient is $-0.179475 + 0.011930 \log t$. That total association of mgm_acquired with the log-hazard of your event is what you need to focus on.
For example, the value displayed for mgm_acquired itself is for the specific situation $\log t = 0$ or $t=1$. The value of what you call the "interaction effect" (the tt() coefficient) shows how much a change of one log unit in t changes the contribution of mgm_acquired to the log hazard.
Despite the very large number of cases, the time-transformed part of the mgm_acquired coefficient isn't significantly different from 0. So it's hard to argue that there's any substantial time-varying part of the coefficient for mgm_acquired.
