# Survival analysis with multiple time-dependent covariates

I have data on patients receiving an SPK (simultaneous pancreas-kidney) transplant. Of interest is the effect of developing the BK virus post-tx on outcomes such as death-censored graft survival. I initially fit a Cox model in SAS as such, adjusting for age at transplant and whether the recipient was white:

proc phreg data=bk;
class white(ref='N');
model dcgs_yrs*dcgs_censor(0)=age_at_tx white bk_viremia_tdc;
if bk_viremia_yrs>dcgs_yrs or bk_viremia_yrs=. then do;
bk_viremia_tdc=0;
end;
else do;
bk_viremia_tdc=1;
end;
run;


However, I have now been asked to also examine the effect of treatments for BK on the outcome. I have two that I'm considering - immuno_rx (had a reduction in immunosuppression medication in response to BK) and ivig_rx, which is a treatment called IVIG. These indicators are zero unless a patient develops BK and is treated with the respective rx. A patient could have either, both, or neither. The idea is to separate the effect of developing BK on the outcome from the effect of the BK treatment on the outcome. I added additional time-dependent covariates as such:

proc phreg data=bk;
class white(ref='N');
model dcgs_yrs*dcgs_censor(0)=age_at_tx white bk_viremia_tdc
immuno_rx_tdc ivig_rx_tdc;
if bk_viremia_yrs>dcgs_yrs or bk_viremia_yrs=. then do;
bk_viremia_tdc=0; immuno_rx_tdc=0; ivig_rx_tdc=0;
end;

else do;
bk_viremia_tdc=1; immuno_rx_tdc=immuno_rx; ivig_rx_tdc=ivig_rx;
end;
run;


My main question is does this model "make sense," i.e. does it answer the question I'm asking, and are the results straightforward to interpret. Below are the results of the second model. While there are no significant effects (just pretend they are for the purpose of this example), my interpretation would be that adjusted for age at transplant and race, the hazard ratio for BK is the effect of developing BK post-tx on death-censored graft survival, and the estimates for the treatments are further impacts on survival based on medication. So if one develops BK, they are best off with neither treatment, and worst off with both.

Analysis of Maximum Likelihood Estimates

                                     Parameter      Standard                                  Hazard
Parameter           DF      Estimate         Error    Chi-Square    Pr > ChiSq       Ratio    Label

AGE_AT_TX            1      -0.00415       0.01831        0.0515        0.8205       0.996    AGE_AT_TX
white          Y     1       0.06509       0.53267        0.0149        0.9027       1.067    white Y
bk_viremia_tdc       1      -0.69208       1.29703        0.2847        0.5936       0.501
immuno_rx_tdc        1       0.33837       1.24702        0.0736        0.7861       1.403
ivig_rx_tdc          1       1.05268       1.23367        0.7281        0.3935       2.865