IPCW + IPTW in a Longitudinal Setting

This is a continuation of my previous blog post on longitudinal treatment crossover. I have an idea on how to combine IPTW and IPCW to deal with treatment switching and stopping respectively and also IPCW to handle some non-response in my outcome data. This will be a short post with mostly just text on my thought process.

The solution

The point of Inverse Probability Weighting (IPW) is to calculate/estimate weights to create a so-called pseudo-population where treatment observations are independent of confounders. In our case, treatment observations can switch or stop based on a number of confounders (call the vector of time-independent confounders \(L\)), which was found in a previous paper about this study (e.g. sex, number of impacted bony teeth taken out in surgery, etc.). If these confounders have a significant relationship to someone switching or stopping a treatment, then we risk biasing our ATE.

We can weight based on if the treatment groups become imbalanced based on \(L\) (found by the previous paper) due to switching or being “censored” due to stopping using IPTW and IPCW respectively. I believe that you are able to combine both of them by the laws of probability: since we want to weight for treatment switching AND treatment stopping, we multiply the stabilized weights together. Furthermore, we do have some “censoring” of the outcome, which again can be thrown in the multiplication. These probabilities are predictions from multiple logistic regression (usually)

Furthermore, in the longitudinal setting, weights at each time are cumulative and should account for the previous treatment and outcome (this is because of my DAG for the study). I’m not entirely sure as to why other than dealing with autocorrelation, but I guess that’s still a good reason anyway. We will use the following notation: \(\bar{X}_{n}\) indicates a vector of \(A\) that goes from timepoint \(1\) to timepoint \(n\). For example, \(\bar{X}_{4} = (X_1, X_2, X_3, X_4)\). Values for these variables for a patient \(i\) are \((X_1, X_2, X_3, X_4) = (x_{1i}, x_{2i}, x_{3i}, x_{4i})\). This vector will be labeled as \(\bar{x}_{ni}\).

Thus, the formula for the cumulative stabilized weights at a time point \(k\) for a patient \(i\) is the following monstrosity.

\[sw_i = \prod_{t=1}^{k} (IPTW_{t} * IPCW_{t}\]

Where…

\[IPTW_{t} = \frac{Pr(A_t = a_{ti} \mid \bar{A}_{(t-1)} = \bar{a}_{(t-1)i})}{Pr(A_t = a_{ti} \mid \bar{A}_{(t-1)} = \bar{a}_{(t-1)i},\, L = l_i,\, \bar{Y}_{(t-1)} = \bar{y}_{(t-1)i})}\] \[IPCW_{t} = \frac{Pr(C_t = c_{ti} \mid \bar{A}_{(t-1)} = \bar{a}_{(t-1)i})}{Pr(C_t = c_{ti} \mid \bar{A}_{(t-1)} = \bar{a}_{(t-1)i},\, \bar{A}_{(t-1)} = \bar{a}_{(t-1)i},\, L = l_i,\, \bar{Y}_{(t-1)} = \bar{y}_{(t-1)i})}\]

is IPTW for treatment switching, IPCW for treatment stopping.

Simplifying Assumptions

We assume that patients only switch or stop, not both. Furthermore, they have one timpoint at which that event occurs. This is why we need the vector of past treatments, even if a patient remains at the same treatment, it’s important for fitting the logistic regressions.

An update

I realized that this idea can be relevant if we assumed treatment switching and stopping were independent, but given results of previous papers, I do not think that is the case. Instead, this post will be more of an exercise on what can be done in combining IPTW and IPCW in a non-survival setting for potential application in the future.

In the actual analysis, I will be treating patients who switched treatments as “censored” and use the vector of confounders \(L\) to reweight. Thus, we have two censoring mechanisms, but they will be in one censoring indicator. I will do a sensitivity analysis where I check the two mechanisms of censoring independently. If anyone has any ideas, please let me know.