# Kaplan-Meier IPCW

My dataset is composed of n individuals (patients) with a two-year follow-up at maximum. Every patient is sane at the beginning of the study, and each month we note if the patients got a certain disease (which then will last very long, so once a patient has a disease, it's considered "death" with respect to the survival analysis meaning). And we have patients that left the study, so they will be censored.

Let's say that we have an "active month", meaning that at that moment we observe either a patient that got the disease (but didn't have the disease the previous month) or a patient that left the study that month. We have only 20 months out of 24 since, in the first four months, we didn't observe anything.

I performed a classical Kaplan-Meier estimator, with right-censoring, so got a 20 points (active months) curve. My vector E of size n contains binary information for each patient, indicating whether they got the disease (1) or drop_out of the study (0). Patients that didn't get the disease at month 24 were considered 0.

Then, I wanted to weight differently the patients, using the Inverse Probability of Censoring Weight (IPCW, and not the IP Treatment W).

So I first performed a Kaplan-Meier for censoring, which is just "inverting" the E vector, called E_censoring, which means censored patients are now 1, and others 0. Thus I get a 20 points curve indicating the "survival of censoring", so the chance of not being censored until time t.

Then, for each individual, I run a Cox-Regression model (for censoring, so the "goal" is E_censoring and not E), with variables like age, binary info indicating whether the patient has comorbidity, etc. I used a simple Cox model, without a time-varying variable, so age is just taken at the beginning of the study for example. So, I get for each patient a 20 points curve, and globally I get a matrix of shape n x 20.

Finally, for each patient I compute the weights like that: Kaplan-Meier basic curve "divided by" the Cox-regression curve for individual i. So, a patient will have 20 different weights, one for every "active month".

Then I computed the final IPCW Kaplan-Meier curve with that n x 20 weight matrix. The (survival) curve I got is a bit above the basic one, which seems to be a good thing.

I wanted to know if this is the standard procedure to get IPCW weights, or at least if this is a correct one? The paper I mostly based my work upon is Informative Censoring in Survival Analysis and Application to Asthma. Many papers are describing IPCW, but only partially, and with (sometimes very) different terms, which is confusing.

Thanks for any help!

PS : Another good material is the following thesis Inverse Probability Censoring Weights for Routine Outcome Monitoring Data, especially the Chapter 5, which is short. I understand really well the denominator (a Cox-regression, which gives an individual survival curve for censoring), but not the numerator, described as a "Product-Limit estimator" which might refer to a classical Kaplan-Meier. But apparently the numerator is different for each patient, which is not the case with Kaplan-Meier. Moreover, the numerator is based upon several variables, which once again is not the case for KM.

• The denominator seems to be good. And as for the numerator, it is said that one can used the stabilized or unstabilized weights (with the numerator equals to 1) in the IPCW Kaplan-Meier estimation, since it yields the same results. It is something I verified on my dataset, and of course it is equal. Jun 15, 2020 at 14:38
• But what I find strange is that my weights (at every time t) for censored observations/patients/individuals are greater (on average) than for uncensored. But the purpose of IPCW seems to put heavier weight to non-censored individuals that "looks like" censored ones. Jun 15, 2020 at 14:46