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I am conducting a multiple linear regression on data from a cross-sectional study, and I suspect that there is an interaction between my dependent variable (a disease risk marker) and one independent variable (an exposure). Biologically, this would make sense: the compound I am interested in (a type of flame retardant) can affect blood cholesterol concentrations - and the effect seems to be stronger in those with already elevated plasma lipids.

My initial analyses used an approach (y: endpoint - here cholesterol, $x_1$: exposure - flame retardant):

$$ y = a + \beta_1 x_1 + ... $$

But I know now that there is some relationship $ \beta_1 \sim y $, and when I stratified by quantiles of $y$, $\beta_1$ changes from about -1 to +1 (there are no differences in distribution of $x_1$ between quantiles). So actually the model should include an interaction between $x_1$ and $y$:

$$ y = a + \beta_1 x_1 y + ... $$

What is the most appropriate way to address this?

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    $\begingroup$ I would advice strongly against making a model with $x_1,y$ interaction. It would become very hard to interpret. What does it look like when you plot $x_1$ and $y$ against each other? Is the relationship non-existing or non-linear? $\endgroup$ – svendvn Apr 26 at 9:05
  • $\begingroup$ The relationship is linear - but $beta$ changes when I stratify by $y$-quantiles. $\endgroup$ – Gux Apr 26 at 9:07
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    $\begingroup$ In a model $y=a+\beta x_1$ it would be impossible that the relationship is linear and that $\beta$ changes with $y$-quantile. So the strange pattern must be caused by the variables in "$\dots$". Perhaps the solution is to look for interactions between the variables in "$\dots$" and $x_1$. $\endgroup$ – svendvn Apr 26 at 9:20
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    $\begingroup$ I second the call not to use $y$ as a regressor. Instrumental variables approach is often used when a particular regressor ($x_{1}$ in your case) is endogenous (correlated with the error term). It requires finding a variable that is correlated with $x_{1}$, but that only affects $y$ through $x_{1}$. $\endgroup$ – AlexK May 2 at 19:41
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    $\begingroup$ I agree with @svendvn both in a) Not including an interaction between $x_1$ and $y$ and in the impossiblity of what you're saying being correct - if $\beta$ changes then the relationship isn't linear. Can you post a scatterplot of $x_1$ and $y$? Also, do you have data at just one time point? And, can you add variables that are markers of health? $\endgroup$ – Peter Flom May 3 at 12:45
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It sounds like the model you are imagining is:

$y_{2} = y_{1} + \beta_{1} x_{1} y_{1} + \epsilon$

where $y_{2}$ is the $y$ you observe and $y_{1}$ is the unobserved counterfactual $y$ if $x_1$ were 0, determined by other unobserved variables.

You can try to estimate this as a latent variable model. In particular, you can fit the model using the Latent Moderated Structural Equation approach from Klein and Moosbrugger 2000. The R package nlsem implements this. It will estimate parameters but I am not sure if it will do significance testing.

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My understanding so far: You have a response $Y$ (endpoint, disease risk marker), an exposure variable $x$ (flame retardant), blood cholesterol level $B$, and some other covariables. First question: is $B$ measured, maybe several times? As you explain it, $B$ would be a mediator, and according to the back-door criterion of causal analysis it should be conditioned on. So then, if $B$ isn't measured, you have a problem with omitted variable bias, and I am not sure what to do, apart from going back to lab and measure it ...

(If my understanding of your situation reflected in this answer is wrong, please comment.)

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    $\begingroup$ Thanks - I'm sorry, but in this instance B is Y, i.e. the endpoint is blood cholesterol. So the blood cholesterol would be a mediator and endpoint at the same time. $\endgroup$ – Gux May 2 at 13:53
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    $\begingroup$ @Gux: Can you edit your original question to include this new information, then the problem will be clearer and maybe more people can chime in! $\endgroup$ – kjetil b halvorsen May 2 at 14:58
  • $\begingroup$ done (I hope it is clearer now) $\endgroup$ – Gux May 2 at 18:42
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To my understanding this is not passible- at least not the way the model is stated at the moment.

Having a variable on both sides, as dependent and as part of the independent variables, contradicts to the very thought of causality because for causality time delay is important - there must be some cause at first, followed by the effect. Cause and effect can not happen at the same time. Using the same variable on both sides would contradict this idea of causality.

However, your explanation that "the effect seems to be stronger in those with already elevated plasma lipids" sounds reasonable. But to me this means if blood cholesterol was evelated at $t1$ flame retandant has a different effect on blood cholesterol at $t2$ as if blood cholesterol was less evelated at $t1$. So to me your hypothesis only makes sense if you use a lagged version of your dependent variable which interacts with the independent variable flame retardant, not the original dependent variable.

So the model would be

$y(t) = a + b* x * y(t - l) +...$

Where $l$ is the lag you have to choose.

Note: as I understand your question blood cholesterol is y and part of the interaction and flame retardant is x. Please comment if I misunderstand that.

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  • $\begingroup$ You are correct - the problem is that there is no time-lagged variable. It is likely that cholesterol acts as a surrogate marker of something else we don't know - but unfortunately at the moment cholesterol is all we have. $\endgroup$ – Gux May 2 at 21:44
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    $\begingroup$ @Gux: So what you mean is that you don't have time lagged data of cholesterol? That would make it impossible to apply the analysis I suggest, of course $\endgroup$ – stats.and.r May 2 at 21:47

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