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I fitted a generalised linear model to my medical retrospective data, in which there is

  • a continuous variable x
  • a binary variable y
  • If y is TRUE, x will have high values. Because whatever causes y to be TRUE will also make x high.
  • It doesn't work the other way round though. If x is high, y can be both FALSE and TRUE. In other words, there can be causes for x to be high that do not lead to y = TRUE.
  • x is fairly reliably measured
  • y is a more subjective measure (and binary) and I'd love to remove it from the model, but I think I cannot completely disregard its information because its presence has a meaningful, plausible and of course significant effect on the event.

My question

Is it correct to speak of an interaction in this situation?

There seems to be a correlation between x and y, but the fact that this is only present "in one direction" makes me a little bit helpless and I am not so sure what to do about it. Adding an interaction term seems to slightly increases the performance of the model (if judged by the AIC), but I have the feeling that it's not quite correct to speak of an interaction here.

If you have a better idea for question title, I'd be more than glad to hear. Thanks.

## This is the model output of my data, which are around 60k observations. 
## Due to the event being fairly infrequent I found it difficult to create a representative sample, 
## but I'd be willing to undergo that challenge if you should think that this would help you. 

Call:
glm(formula = event ~ y * x, family = binomial, data = df)

Coefficients:
            Estimate Std. Error z value Pr(>|z|)    
(Intercept) -4.53815    0.04829 -93.983  < 2e-16 ***
yTRUE        1.52285    0.17531   8.687  < 2e-16 ***
x            0.66225    0.05498  12.045  < 2e-16 ***
yTRUE:x     -0.64983    0.12033  -5.400 6.65e-08 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

(Dispersion parameter for binomial family taken to be 1)

    Null deviance: 10114.4  on 60729  degrees of freedom
Residual deviance:  9879.7  on 60726  degrees of freedom
  (79815 observations deleted due to missingness)
AIC: 9887.7

Number of Fisher Scoring iterations: 7
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The term "interaction" regards the way x and y work on the response variable here, apparently called "event". Your results show that the effect of x on event depend on the value of y. It does not regard the dependence between x and y (not taking into account event), which is what you talk about in your question.

In fact both correlation/dependence (any kind of dependence) and independence between x and y are possible together with both the presence or absence of interaction between them regarding event.

So no. What you describe is not "interaction", even though interaction apparently takes place in your situation anyway.

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