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EDIT: edited the question slightly to keep the diagram intact but make it clear that we have an observed value and a hidden unmeasurable variable.

Here is the story. There are patients who have a certain disease D. We suspect that the disease might be aggraviated/caused by a certain condition C (our hypothesis: C causes D, $C \rightarrow D$). We cannot measure the condition C in the patients, however we can measure some variable $O$ which depends on $C$ ($C \rightarrow O$).

But herein lies the problem: patients who suffer from D receive (often) a medication $M$. This medication is correlated with the disease status, but it is not identical (there are patients without $D$ who also receive it, and vice versa, there are patients with $D$ who do not receive $M$). Therefore, we have the causal relationship $D \rightarrow M$.

Moreover, the medication $M$ might influence the observed variable $O$. At least that is what a reviewer might say: you are only observing an association between $D$ and $O$ because there is an association $D\rightarrow M$ and an independent association $M \rightarrow O$. This suggests the following, more complex model: we have the $C$ (which cannot be directly measured) that might cause $D$, and we have $O$ dependent on $C$, which might be influenced by $M$:

$$C \rightarrow D \rightarrow M \rightarrow O$$

$$C \rightarrow O$$

Now, we observe a clear association between $O$ and $D$. We also cannot rule out an association between $M$ and $O$. Furthermore, if I run a regression with $D$ as the response variable, and both $M$ and $O$ as predictors, I still get very similar results... but much, much weaker. That is, the p value is higher and the effect size smaller. This is not unexpected, because $M$ is strongly associated with $D$, and I am removing the portion of the variance in $D$ that can be explained by $M$.

To make the matter a bit more complicated, I have several of these $C$ variables, $C_1 ... C_n$ and corresponding observations $O_1 .. O_n$. In general, the p-values and effect sizes are correlated between the model which does not include $M$ as a predictor and the model that does include $M$ as predictor. Notably, the associations between $M$ and $O_1 ... O_n$ are all not significant when corrected for multiple testing (the associations between $D$ and $O_1 ... O_n$ are significant regardless).

My question is, how should I analyse and present the result of the analysis? What I did until now was to create these alternate models, with or without $M$ as predictor for all $O_1 ... O_n$, and also shown that the associations between $D$ and $O$ are present for each level of $M$. Again, this latter part is weaker firstly because the groups are smaller and secondly because the data is imbalanced (given that $D$ and $M$ are associated, there are relatively few patients who have not been medicated and few controls who have).

How do I convince myself (and others) that $C \rightarrow D$ and this effect is not due to $M \rightarrow O$? Would it be convincing enough to test O ~ M | D (i.e., check what is the association between $O$ and $M$ when accounting for D) and show that no results are significant? Is there a better way?

Another idea: how about testing O ~ M + D and comparing it to the model O ~ D? If there is a substantial effect $M \rightarrow O$, then these models should be significantly different? (I do not observe this).

Equivalently (I think), I could calculate $O'$ as the residuals of the model O ~ D and test whether O' ~ M.

What bothers me in these ideas is that I am searching for lack of significance, but absence of evidence is not evidence of absence, and hence I am looking for a more concrete solution.

P.S. To make the question more substantial, imagine that we build the following simulation in R:

set.seed(1234)
# hidden variable
C <- rnorm(100)

# disease depending on the hidden variable
D <- ifelse(C + rnorm(100) > 0, "y", "n")
D <- factor(D, levels=c("y", "n"))

# medication depends on disease
M <- ifelse(as.numeric(D) + .5 * rnorm(100) < 1.5, "y", "n")
M <- factor(M, levels=c("y", "n"))

# observed values
O <- C + .5 * rnorm(100)

# test whether D is associated with O
cor.test(O, as.numeric(D), method="s") # p < 1e-7

# test whether M is associated with O
cor.test(O, as.numeric(M), method="s") # p < 0.01

Now, the job is to show that there is an association $C\rightarrow D$ that is independent of $M$ using only $O$, $M$ and $D$ variables. We know from how the simulation is derived that $O$ does not depend directly on $M$, and the observed association is only due to the path $O \leftarrow C \rightarrow D \rightarrow M$.

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  • $\begingroup$ Could $D\to C_{\text{observed}}$ also be a possibility? $\endgroup$ Commented Feb 22, 2023 at 16:29
  • $\begingroup$ I think we can safely rule it out for now. Anyway, I think that in such a case no solution is possibe. $\endgroup$
    – January
    Commented Feb 22, 2023 at 17:15

1 Answer 1

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[EDIT] The following solution no longer applies, except for the argument that $M\not\to C.$ Original question has been edited. Not deleting because of valuable comments by Alexis.

I think time is a major factor, here. You are observing multiple variables at different times, and you have feedback. The regular tools of causality are challenging to work with in such a case. However, here's an option for you to try: model the same quantity at different times as essentially different variables. After all, you will have different values for them in your data. Here's the model I would propose in your slightly simpler setting of just one precondition. The parentheses here show the value of that variable at that time. The different between time $t$ and time $t+1$ can be whatever you want:

enter image description here

The edge $D(t)\to C(t+1)$ is debatable, but seems reasonable to me. You might also debate $C(t)\to C(t+1),$ but you did posit that that edge exists.

Now the basic rules of causality say that causes must precede effects. Moreover, $C(t+1)$ occurs later in time than $C(t).$ Since $C(t)\to D(t)\to M(t)\to C(t+1),$ it cannot be (by the laws of causality) that $M(t)\to C(t).$

As for the analysis, if you are interested in the causal effect $C(t)\to D(t),$ then you need only regress $D(t)\sim C(t).$ There is no backdoor path from $C(t)$ to $D(t),$ so long as you only include pre-medicated data on the $C$ variable.

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    $\begingroup$ 1/3 Structural causal modeling (e.g., Imbens, Pearl, etc.), as Adrian Keister rightly points out, chokes somewhat on causal feedback (i.e. where variables in the past are causes of one another in the future, via either direct or indirect pathways). However, structural causal modeling is not the only analytic causal formalism, and some others have been developed to explicitly account for behavior of complex causal feedback systems. One such is Levins' "loop analysis" and the elaborations on it over the decades. Another is the much newer "empirical dynamic modeling" of Sugihara. $\endgroup$
    – Alexis
    Commented Feb 22, 2023 at 18:31
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    $\begingroup$ 2/3 Puccia, C. J., & Levins, R. (1986). Qualitative Modeling of Complex Systems: An Introduction to Loop Analysis and Time Averaging (C. J. Puccia & R. Levins, Eds.). Harvard University Press. $\endgroup$
    – Alexis
    Commented Feb 22, 2023 at 18:31
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    $\begingroup$ 3/3 Chang, C.-W., Ushio, M., & Hseih, C. (2017). Empirical Dynamic Modeling for Beginners. Ecological Research, 32(6), 785–796. $\endgroup$
    – Alexis
    Commented Feb 22, 2023 at 18:32
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    $\begingroup$ Your edited version is not solvable, since you have no data for $C.$ You can still rule out that $M\to C,$ but without any data for $C,$ you're dead in the water, in my opinion. $\endgroup$ Commented Feb 22, 2023 at 21:10
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    $\begingroup$ @Alexis Just out of curiosity, are the topics in the citations you shared the same as "graph algebra" as described in Cortés, Przeworski and Sprague's book Systems Analysis for Social Scientists (1974)? Or are these two things distinct? $\endgroup$
    – Sycorax
    Commented Feb 22, 2023 at 21:47

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