# Survival analysis - time-dependent covariate with left truncation

I have some data on patients receiving pancreas transplants. Some of these patients have a procedure done post-transplant called enteric conversion, and I am looking at the effect of this conversion as time dependent for several outcomes, including mortality. Normally this would be very straightforward but there is also truncated data.

In our database, we have all the pancreas transplants at our center going back into the late 1970s. However, we only started collecting retrospective data on pancreas transplants in 2003. That means that if a person was transplanted prior to 2003, and either died or was lost to follow-up prior to 2003, we would not know if they had this procedure done after their transplant.

My first step was to limit the cohort to only those who were alive and not lost to follow-up as of 2003. The timing of the enteric conversion could still occur any time after transplant, for example if someone is transplanted in 1995, has a conversion in 1999, and is followed up through 2003 or later we would know the date of the conversion.

However, I realize there is a real issue of left truncation here as someone has to survive to 2003 to be included - thus my survival estimates (especially for mortality) will be overly optimistic.

My question is how to analyze such data. I typically use R, but am using SAS here as you don't to convert the data to a counting process style. It seems like you cannot account for both left truncation and time-dependent covariates in one model, but I'm not sure.

Here is the SAS code for mortality using a TDC (convert_yrs is the time from transplant to conversion only for those with conversion - it is missing otherwise):

proc phreg data=panc;
model death_time*death_censor(0)=convert_tdc;
if convert_yrs>=death_time or convert_yrs=. then convert_tdc=0; else
convert_tdc=1;
run;


And here is what I understand to be the SAS code for left truncation. I believe "t1" should be the age as of 1/1/2003 but I'm not sure:

proc phreg data=panc;
model death_time*death_censor(0)= /entry=t1;
run;


Thanks for any help you can provide.

## 1 Answer

Cox models allow for staggered entry of at-risk participants into a prospective cohort, regardless of who died prior to the onset of data collection. The only requirement is that you know when they received the transplant and/or received the post-transplant conversion. Risk sets will omit left-truncated participants at failure-times prior to the conversion.

Cox models do not estimate survival. In fact, with left truncation, we do not show survival with Kaplan-Meier for the reasons you mention: they are overly optimistic in the sense that the curve rebounds upward at time points following the inclusion of left-truncated participants. Survival functions are strictly decreasing functions of time.

Cox models estimate hazard ratios. This estimate relies on the assumption of proportional hazards. When that assumption is met, there is no reason not to start comparing pre-2003 transplant recipients.

Age is NOT the appropriate time-scale for the baseline hazard function. It is time since the transplant or time since the operation. All participants who receive a transplant start together at time 0. Consider adjusting for age as a covariate in the Cox model to account for its prognostic effect on survival. Consider also the issue of confounding by indication: why do participants get this surgery? Are they sicker in that their bodies are rejecting the transplant or their labs are failing? If so, you have to consider marginal structural models.

• I am going to add fixed effects such as age, this is just my starting point. In what situation would you use the second type of PHREG with the entry statement? – Scott Jackson Nov 16 '17 at 19:01
• @ScottJackson it is a data formatting issue, the SAS syntax is nebulous to me. If you have time varying covariates, you must transpose from a wide format to a long format, so that there are one or possibly two rows of data for each participant. The t1 survival time should be 0 for non-truncated, or the difference between start-of-observation (2003) and transplant date or operation date for truncated entry. – AdamO Nov 16 '17 at 19:22
• Right, you can do it that way but there's also a shortcut where you can use a programming statement within PROC PHREG like I wrote. Thanks for the help. – Scott Jackson Nov 16 '17 at 19:32