# Survival rate trends in case-control studies

I submitted an article which was rejected due to the improper way of performing the survival analysis. The referee left no other details or explanations other than: "survival analysis on time trends requires more sophisticated ways of censoring."

The question:

Has the excess risk of death among smokers been reduced in the last decades?

### Data:

25.000 smokers in Germany. They were enrolled in the cohort at any time between 1995 and 2014. Each smoker has been matched (at the time of enrollment) to a sex and age matched control from the general population (who did not smoke). I have exact time of death for everyone who died during the whole study period. Those who did not die during follow-up will be censored. The study is powered to examine the excess risk of death among smokers each year from 1995 to 2014.

The aim is to calculate:

• Incidence rates of death for smokers and non smokers each year and examine these trends
• the excess risk of death among smokers, each year (or period of a few consecutive years).

How should the data be analysed? Recall that somebody who is included in 1998 might die in 2015. Is the right approach to use counting process format with start and stop updated for each year?

This is the approach that the referee disliked:

Incidence rates were calculated by means of Poisson regression. We included follow-up time as an offset in the model and included age, sex, smoking status and calendar period (combining two consecutive years) as predictors in the model. Then rates were calculated per 1000 person years using the predict() function of R. The offset (follow-up time) was the persons entire observation time (days) from enrollment.

A Cox model was used to estimate relative risk for smokers each period from the beginning to the end of the study. For simplicity we compared the hazard ratio in the first period with the hazard ratio in the final period.

Issues: - a person (along with his control) might be included in 1998 and thus belong to that calendar group, but suffer an event in 2006. - How should the data be layed out for the analysis of the Poisson and the Cox regression? Counting process for the cox? What is the start and stop time? - How can trends be assessed in this situation?

Some clarifications: Let's say a patient is first observed in 15 of june 1998 and experienced an event december 31 1998, the value for our time variable for this patient is 182.5 out of 730 possible days since time-period consists of 2 subsequent years. The maximum amount of observed time in each time-period is 730 days.

When a patient is observed in one time-period but censored (i.e either experienced and event or dropped out) in another time-period should the amount of days observed be added onto the next time-period or what?

Thus the main problem is the handling of the follow-up time and calendar year (which is used as a categorical variable, consisting of two consecutive years).

• And what was the approach you used? Aug 7, 2015 at 14:13
• We performed some of the analysis with poisson regression and the predict function in R to estimate the incidence rates. We also created a Cox model to compare the risk ratio between groups in the beginning of the study and the end, i.e 1995/1996 vs 2013/2014. Since some of the time-periods included to few events we merged 2 subsequent years, e.g 95/96, 97/98, 99/00 and so forth in all analysis for both the Cox and the poisson model to receive significant estimates. Aug 8, 2015 at 16:24
• Now that you've added to your question, it might help to have a more specific title to your question, something like "testing for survival rate trends in case-control studies," to get more informed interest. This is a bit beyond my expertise; perhaps this reference might provide some help, although much of that document might not be applicable to this type of rolling-entry case-control study.
– EdM
Aug 10, 2015 at 16:42
• this is, if I'm not mistaking, a (retrospective) cohort study, since you actually follow individuals (who are exposed or not to smoking) until an event. A case control study usually refers to the situation where you have folks who did develop and those who did not develop the disease and survival time is not modelled. but i might be wrong here. Aug 11, 2015 at 17:05
• @AdamRobinsson: No , you are not wrong. What is described is not a case-control study. It is a age-sex matched cohort study. The statement: "For simplicity we compared the hazard ratio in the first period with the hazard ratio in the final period." suggests that the full dataset was not used for the main study question, since the data from the middle years of the study was not used.
– DWin
Aug 14, 2015 at 6:48

From the above there are a few possibilities for the Cox model:

1. SEPARATE MODELS FOR EACH TIME-PERIOD: Use one observation for each person; calculate observation time (regardles of when censoring/death occured during follow-up) and then calculate the hazard ratio each period. Then compare the hazard ratios directly.
2. CALCULATE THE RELATIVE CHANGE IN HAZARD IN SMOKERS AND NON-SMOKERS SEPARATELY: one observation per person; calculate observation time (regardless of when censoring/event occurs) and then use all patients (from 1995 to 2014) in the model, use time period as a categorical variable and set one of the periods as the reference value.

1. COUNTING PROCESS FORMULATION: this sounds appealing, but I'm not sure on how to use survival time, start stop intervals and calendar year.
• Good suggestions, but how do these deal with the possibility of informative censoring (see my attempt at an answer) and an hypothesis (changes in relative hazards of smokers/non-smokers over calendar years) that by its nature seems to contradict the proportional-hazards assumption?
– EdM
Aug 11, 2015 at 14:29
• @EdM I believe (although I'm not sure) that censoring is not informative in this scenario; the cases and the controls should be censored due to the same reasons, whatever that bias may be it should be equal in these two groups. Since death is the outcome being examined and it appears you can guarantee that all deaths are captured and emigration negligible; I would'nt mind informative censoring. Proportional hazards need not be violated; although the study seeks to examine smoking as a function of time, it does this in terms of calendar year and not observational time (which is crucial). Aug 11, 2015 at 14:47
• I'm absolutely not sure though. Aug 11, 2015 at 14:48

Although it's dangerous to read too much into the cryptic comments of a reviewer, I would guess that the objection has to do with whether the censoring is informative.

Interpretation of survival models typically is based on the assumption that an individual censored at time $T$ is representative of all subjects who survive to time $T$ after entry into the study. (Wording adapted from this introduction to survival analysis.) Then the censoring is non-informative.

In your analysis, however, those who were censored were those who survived through 2014. If you think that there had been a change in excess risk of death due to smoking over the previous 20 years (or even if there were parallel changes in death rates for both groups), then those censored individuals may not be representative of those who survived for the same time but entered the study earlier. Under your hypothesis, the censoring might be informative.

It's possible that the details of the design of your analysis avoided this problem but that wasn't clear in the manuscript as reviewed. Or perhaps the reviewer didn't like the study for some additional reasons and found this to be a way to reject it that the editor wouldn't question. Nevertheless, this does seem to be a potential objection to the way you analyzed these data and you should make sure that it is handled properly. (This is beyond my personal expertise; others on this site might have pointers on how to proceed. A more precise title to this question, with more details on the study design and analysis, might get more helpful answers.)

It's not clear to me from your question and clarifying comment that the Cox analyses are adding anything useful to simple modeling of death rates per year (or over 2-year intervals). Plus, your hypothesis seems to imply that hazards are not proportional over time between non-smokers and smokers, the basis of standard Cox analyses. If you are interested in the difference of death rates between smokers and non-smokers as a function of calendar year, that's the most straightforward measure to model (although you might have to take into account the presumed enrichment of non-smokers in your study sample as their matched smoking counterparts die).

• Thank you for your answer. Perhaps It's best to further clarify our method. I will edit my question. Aug 10, 2015 at 11:15