# On the Problem of Treatment Confounder Feedback

Causal Inference
Methods
Published

11/6/22

## Purpose

Causation occurs in time. Therefore, investigating the relationship between cause and effect requires time series data.

Causality is also dynamic. Where there is Treatment-Confounder Feedback, the relationship between cause and effect cannot be identified using standard regression methods, including multi-level regression and structural equation models. Instead, special methods - “G-methods” - are needed.

Here, I use three causal graphs to describe a problem of treatment-confounder feedback, and direct readers to G-methods for its solution.

1. You are interested in psychological science.
2. You understand how to read causal graphs 1
3. Your “go-to” method for time-series analysis is either a latent growth curve or a multi-level model.

## Confounding by Common Cause

Suppose we wish to compute the causal effect of treatment $$A$$ on outcome $$Y$$. Because $$L$$ is a common cause of both $$A$$ and $$Y$$, $$L$$ will lead to an association between $$A$$ and $$Y$$. We face confounding by common cause. The good news: where $$L$$ is measured, a regression model that conditions on $$L$$ will break the association between $$A$$ and $$Y$$. Again, causation occurs in time. We index measured nodes to ensure our data adhere to time’s arrow

reveal code
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\tikzstyle{Arrow} = [->, thin, preaction = {decorate}]
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\node[squarednode] (1) {$L_{t-1}$};
\node[right =of 1] (2) {$A_{t}$};
\node[right =of 2] (3) {$Y_{t+1}$};
\draw[Arrow,bend left] (1) to (3);
\draw[Arrow] (1) to (2);

## Confounding by Over-Conditioning

Suppose $$L_{t+1}$$ is an effect of $$A_{t}$$. To condition on the common effect will induce a spurious association between $$A_t$$ and $$Y_{t+1}$$ through the unmeasured confounder $$U$$ (red path). We may avoid this problem by excluding $$L_{t+1}$$ from our regression model. To know whether exclusion is warranted requires indexing the relative occurrences of $$A$$ and $$L$$. However, without time-series data, we cannot generally know whether $$L$$ is a cause of $$A$$ or its effect. Figures 1 and 2 illustrate the importance of collecting time-series data to infer causality. Although psychological scientists are familiar with adjustment by regression to address confounding by common cause, we are less familiar with the hazards of over-conditioning. Generally, confounding control in any observational science requires time series data.

reveal code
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\node[squarednode, right =of 1] (2) {$L_{t+1}$};
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## Confounding control for Treatment-Confounder Feedback: Damned if you condition damned if you do not.

Suppose we collect time series data. Suppose further that conditioning on $$L$$ blocks an unmeasured common cause $$U$$ of future treatments $$A$$ and future outcomes $$Y$$. Suppose further, as in Figure 2, past states of $$A$$ affect future states of $$L$$. Notice, regression faces a damned-if-we-do-damned-if-we-don’t adjustment challenge. On the one hand, to avoid confounding by a common cause we must adjust for $$L$$ at all time points. On the other hand, adjusting for $$L_{t+1}$$ induces confounding by over-conditioning (paths in red). **Regression, including multi-level regression and structural equation models, must be abandoned.* There are alternatives to regression called G-methods that may address treatment-confounder feedback. One of these methods, the Marginal Structural Model, replaces $$L$$ with inverse probability weights for the exposure. G-methods are described in Chapters 12 and 13 of Hernan and Robin’s[Hernan and Robins (2023)]2

Hernan, M. A., and J. M. Robins. 2023. Causal Inference. Chapman & Hall/CRC Monographs on Statistics & Applied Probab. Taylor & Francis. https://books.google.co.nz/books?id=\_KnHIAAACAAJ.
reveal code
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\node[draw=black, thick] (1) {L$_{t-1}$};
\node[right =of 1] (2) {A$_{t}$};
\node[right =of 2] (3) {Y$_{t+1}$};
\node[squarednode, right =of 3] (4) {L$_{t+1}$};
\node[right =of 4] (5) {A$_{t+1}$};
\node[right =of 5] (6) {Y$_{t+2}$};
\node[left =of 1] (7) {U};
\draw[Arrow] (1) -- (2);
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\draw[Arrow, bend right, red] (7) to (4);
\draw[Arrow, bend right] (7) to (3);
\draw[Arrow, bend left, red] (2) to (4);

## Importance

I write this report to encourage psychological scientists to (1) collect time-series data and (2) address treatment-confounder feedback by employing G-methods. I do not write this report to cast stones. My published work offers ample illustrations of the problems that I describe here. On a positive note, a causal revolution in psychological science is upon us. Our best science remains ahead of us .

VanderWeele, Tyler. 2015. Explanation in Causal Inference: Methods for Mediation and Interaction. Oxford University Press.

## Footnotes

1. I will soon write a tutorial here for those who are unfamiliar↩︎

2. We draw the minimum number of paths to clarify the problem.↩︎

CC BY-NC-SA

## Citation

BibTeX citation:
@article{bulbulia2022,
author = {Joseph Bulbulia},
title = {On the {Problem} of {Treatment} {Confounder} {Feedback}},
journal = {PsyArXiv},
date = {2022-11-06},
url = {https://psyarxiv.com/tjnvh/},
doi = {10.31234/osf.io/tjnvh},
langid = {en}
}