# how to interpret the cumulative incidence plot stratifying in two groups

I would like to interpret the cumulative incidence plot in blow. Red line shows cancer and black line noncancers. Event is ability decline. Is it correct to say that the risk of ability decline in cancers in higher that nocncancers. Right? And the difference in cox from score log rank test is p = 0.06. My question is that what is the correct interpretation and is it strange that the probability at the end stopped very close to 1? Any help in appreciated. I have also checked, I do not have competing risk of death. No death before event happened. I am not sure why the probability reaches one. Why this happened? For cox I have defined: Age as time, defining Tstart and Tstop:

1. if status of event = 0, then Tstart = age at inclusion, Tstop = age at end of follow-up

2. if status of event = 1, then Tstart = age at inclusion, Tstop = age at event

3. for considering cancer as a time-varying covariate: For each ID in this condition, I added a new line.

if cancer ==0 and status event = 1, then Tstart = age at inclusion, Tstop = age at diagnosis then also I keep the cancer status ==0

if cancer ==0 and status event = 1, then Tstart = age at diagnosis, Tstop = age at event and then I changed the status of cancer==1.

Also, for those who have diagnosed by cancer in the follow-up, I have chosen the "age at diagnosis" as "age at inclusion". And, if "cancer diagnose age" was after "age at event", these are set as no-cancers.

• Please say more about your data and the Cox model that you used. The start of both groups with 0 events at an age of about 70 is hard to understand. Usually survival is from some defined start time; your plot might be taken to imply that you started with a cancer and a noncancer cohort all from age 70 and that there were no later cancers in the cancer cohort. That seems unlikely. Or you might have used age data for each individual at study entry; that would require modeling the left truncation. Please edit the question to provide that information; comments are easy to overlook.
– EdM
Jul 30, 2022 at 16:36
• @EdM I have updated with more information, would you please explain now? Jul 30, 2022 at 21:04
• I added an example of how I would code participants who entered the study as noncancer but developed cancer during the study, one who ended up with the event and one who didn't.
– EdM
Jul 31, 2022 at 13:22
• @EdM thank you very much. I have one question, should I do first step 1 and 2, I explained to define Tstart and Tstop based on status of event and then add to that (correct) for non-cancers who developed cancer during study? thanks again. Jul 31, 2022 at 17:50
• Yes. Code all of those who started with cancer or those who started non-cancer and never got cancer just as you did. The additional rows of data would be for those who started non-cancer and then got cancer. I provided two such examples, for someone who eventually had the event and another who didn't.
– EdM
Jul 31, 2022 at 19:22

A cumulative incidence plot like this, with at most one event possible per individual, is just a survival curve turned upside down. A survival function is 1 minus a cumulative probability function, and vice-versa.

The cumulative incidence plot in that situation is just the probability that someone in the specified group has had the event, up through the indicated time.

is it strange that the probability at the end stopped very close to 1?

That's exactly what you'd expect if everyone eventually has the event: the probability of having the event in either group is ultimately 1. The only difference might be how quickly it happens. If the last observation time was right-censored rather than an event, you won't quite get up to 1 (just like you don't quite get down to 0 in the survival curve).

if "cancer diagnose age" was after "age at event", these are set as no-cancers.

You can do better than that. You already are using the (Tstart, Tstop, Status) counting-process format for your data to handle (appropriately) the left truncation of Age values. That's the same format used for handling time-varying covariates. You also are already specifying individual IDs to allow for robust cluster estimates of the variances.

So extend that to treat the cancer diagnosis as a time-varying covariate. When a patient is diagnosed with cancer, set that age to the Tstop for the non-cancer situation with right censoring at that age. Then start a new data row for that patient ID, with Tstart being the age at diagnosis, the status changed to cancer from non-cancer, and with the Tstop and Event values that you had previously used.

Is it correct to say that the risk of ability decline in cancers in higher that nocncancers. Right? And the difference in cox from score log rank test is p = 0.06.

By the usual standard of statistical significance (p < 0.05), you couldn't claim that there's a difference. For example, at ages starting around 85 the confidence intervals for the cancer group overlap the point estimates of the non-cancer group. You'll also note that the curves cross back and forth around ages in the 90s. This suggests that a Cox proportional hazards model with time-constant coefficients doesn't fit your data well. Treating cancer status as a time-varying covariate might help, or you might consider modeling it with a time-varying coefficient, maybe with a step function as in Section 4.1 of the R time dependence vignette. That might help illustrate a difference between ages of about 75 to 85.

Data format

I'm not sure that I completely understood how you proposed to code individuals who entered the study cancer-free but developed cancer during the study. Here are examples for two patients. Patient with ID 101 entered the study at age 75, was diagnosed with cancer at age 80, and had the event at age 85. Patient with ID 102 entered the study at age 76, was diagnosed with cancer at age 79, but had not experienced the event by the last follow up at age 88.

ID  Tstart   Tstop  cancer  event
101     75      80      0       0
101     80      85      1       1
102     76      79      0       0
102     79      88      1       0


The cancer column represents the cancer status during the indicated time period. The event column represents whether the "ability decline" event occurred (1) or not (0) at the end of that time period. I used the term event to avoid ambiguity in use of the word "status," which might otherwise represent either the "cancer" status or the "ability decline" status.

• I think what you have explained makes sense and considering cancer status as a time-varying covariate is the correct way to go. But since I am not very familiar with doing this. Could you please clarify more? Then should I split kind of based on cancer = 1 and cancer = 0? I am just a little bit confused about what I have done before and how to change it to what you explained. Jul 31, 2022 at 7:54
• and this should be only done for Noncancers that are diagnosed later? and add two line for these individuals, kind of breaking them. Right? Jul 31, 2022 at 8:17
• @ EdM could you please see my update and what I changed for considering cancer as time-varying covariate is exactly what you mentioned? Jul 31, 2022 at 9:14