# Data-adaptively learning strata-specific causal effects

This post is part of our Q&A series.

A question from graduate students in our Spring 2021 offering of the new course “Targeted Learning in Practice” at UC Berkeley:

## Question:

Hi Mark,

I have a question about applying CV-TMLE to a current research project. I have a cross-sectional dataset from Bangladesh, where the outcome of interest is antenatal care use (binary), the exposure of interest is women’s empowerment (continuous), and the baseline covariates include mother’s age, child’s age, mother and father’s education, number of members in household, number of children under 15, household wealth, and maternal depression. Women’s empowerment is a score generated from various questions around freedom of movement, control over assets, and household decision making. However, from an intervention and policy design standpoint, it is potentially more interesting to compare differences in levels of empowerment. For example, we may want to know if antenatal care use is different between women with “high” empowerment versus “low” empowerment. Or perhaps even three categories: very high, average, very low. You mentioned the possibility of using varimpact(), which I understand data-adaptively discretizes a continuous exposure and selects the highest contrast comparison between the two exposure levels using CV-TMLE. I am really curious to learn more about how exactly this works and what happens under the hood to select the highest contrast comparisons. Can you explain the steps and intuition behind them? Is it possible to discretize into more than two levels? Does the procedure output estimates of the average treatment effect(s) or would that have to be done separately?

Best,

S.W.

Hi S.W.,

Thank you for your nice example and excellent question.

Sounds like a good plan. Yes, varimpact() starts out with data-adaptively discretizes exposure $A$ (but I believe it does this outcome-blind so that we can keep it fixed after that, and can then have well-defined categories for $A$), and proceeds as follows: 1. It computes a TMLE of $\mathbb{E} Y_j$ for each category $j$, and then 2. it determines the two levels that maximize the contrast $\mathbb{E} Y_j - \mathbb{E} Y_k$.

This is just the algorithm that maps the data into the desired target parameter $\Psi_{n}(P) = \mathbb{E} Y_{j_{1n}} - \mathbb{E} Y_{j_{2n}}$, where the choices $\{j_{1n}, j_{2n}\}$ are based on the data. If we now would just compute the TMLE of this target parameter, learned on the whole data set, then we would be using the data twice, causing, in this case, a great degree of bias, since the contrast was chosen to optimize the effect of interest.

In such cases of data-adaptive target parameters, we use CV-TMLE. That is, we use $V$-fold sample-splitting, run this algorithm on the training sample $P_{n, v}$, thereby determining $j_{1n,v}$ and $j_{2n,v}$; then, we compute the TMLE of $\Psi_{n, v}(P) = \mathbb{E}_P Y_{j_{1n, v}} - \mathbb{E}_P Y_{j_{2n, v}}$ by getting the initial outcome regression $Q_{n, v}$ and treatment mechanism $g_{n,v}$ from the training sample $P_{n, v}$, but perform the TMLE update step only using the validation sample $P_{n, v}^1$. This yields a TMLE $P_{n, v}^{\star}$, and thereby a plug-in estimator $\Psi_{n, v}(P_{n, v}^{\star})$. Repeating this for each of the $V$ sample splits, we have $V$ CV-TMLEs, over which we compute the corresponding average $1/V \sum_{v = 1}^V \Psi_{n, v}(P_{n, v}^{\star})$.

This is an estimator of the data-adaptive target parameter $1/V \sum_{v = 1}^V \Psi_{n, v}(P_0)$, i.e., an average over $V$ data-adaptively selected causal effects, defined as moving $A$ from bin $j_{1n, v}$ to bin $j_{2n, v}$. We then also provide inference, such as confidence intervals for this data-adaptive estimand. By utilizing the full sample size, the procedure is as powerful as if no sample-splitting had been used (normally, people use one sample to learn the question, and the second sample to obtain inference). This is the price one has to pay when the question is not under the full control of the user, but it will end up being an average of causal effects. In some cases, each training sample might generate the same contrast (e.g., when the choice of contrast is very stable and clear), while, in other cases, there might be variation in the selected contrasts. Either way, if we reject the null, then we know that one of these contrasts was significantly different from zero, so that is a good hypothesis test for an interesting null hypothesis. The varimpact() package provides additional output, like providing inference for $\Psi_{n, v}(P_0)$ for each fold $v$ separately, using this $v$-specific CV-TMLE, but the sample size is now only $n/V$, thereby limiting the power (but nonetheless nice to see which effect estimates where driving the average). There is also a chapter on this in the 2018 book Targeted Learning in Data Science.

So, yes, the procedure can discretize in more than two levels of $A$. Alan Hubbard knows the details of this, and so does Chris Kennedy, and I presume there is documentation as well. I imagine that can be controlled as well, and for sure a small modification in the program would add that flexibility. In addition, the program does indeed both generate the target parameter and the CV-TMLE, so it stands alone.

It will be nice if you use this program, since these programs can be iteratively tested and improved over time, as well. We should also add it to the tlverse toolbox, if not done so already. Jeremy Coyle is a good reference to communicate with as well, if you are interested in questions of what tlverse offers for such data adaptive target parameters. It already includes the optimal dynamic treatment and its CV-TMLE, so I suspect something very close is there already. In particular, Andrew Mertens used the tlverse software to carry out a large meta-analysis involving data-adaptive target parameters; I believe it involves searching for contrasts/optimal rules, across many variables, treating each variable as a treatment, and others as confounders.

Best Wishes,

Mark

P.S., remember to write in to our blog at vanderlaan (DOT) blog [AT] berkeley (DOT) edu. Interesting questions will be answered on our blog!