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I'm confused about the interaction of binary predictor values in multiple linear regression. Here's an example to illustrate my problem. Say that I want to investigate the relationship between life expectancy (response) and whether someone works the night shift and has a pet (both binary 0 or 1 predictors). Add to this that I want to control for the effect of gender, because I know that being female increases your life expectancy, so I put male/female in as a binary predictor as well. So, my model is: life expectancy (continuous)=night shift+pet+gender.

Now, through the nature of my data, I have male respondents who code for both pet presence and absence, but all females have pets (this is obviously a bad study design - but ignore this for now and assume this is the only data I have). For the night shift, all combinations of absence/presence of night shift and gender exist. Now in my regression, I find a significant negative coefficient for having a night shift - and then interpret this to mean that night shift lowers life expectancy, irrespective of gender. This part is straightforward. But for pets, I find no significant effect. However, my model could really only compare pet/no pet in the male category, because all females have pets. So my interpretation here is that males do not increase life expectancy from having a pet, but the model cannot tell us how this affects life expectancy in females. In fact, the potential positive effect of a pet for a female may be entirely swallowed up by the gender variable, since all females have pets.

My questions are:

  1. Is there in general a problem with running a model of this kind, more specifically I wonder if this counts as non-independence of the predictor values (all females have pets)?

  2. Is my interpretation correct?

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Regression models don't make the assumption that all the explanatory variables are independent of each other. That basically never happens outside of well-designed experiments (even some experiments purposely confound explanatory variables).

You seem to want to infer causality from the fit of your model. Regression does not necessarily make that permissible. You can validly infer causality if you ran a true experiment (or perhaps using advanced statistical techniques and the model is correct). Put simply, your situation / model does not allow you to conclude that working the "night shift lowers life expectancy".

More broadly, regression models don't require the explanatory variables be independent. A model will fail to fit if two, or more, variables are perfectly collinear. There may be problems fitting the model if the variables are not perfectly collinear, but very close, but improved fitting algorithms have made that less of an issue. (Actually, what most software will do nowadays is drop one of the collinear variables and fit the model with the rest.)

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    $\begingroup$ I interpret the question a little differently: it seems to be concerned with the inability of this experimental design to estimate an interaction term, requiring the interpreter to assume additivity of effects without being able to check this assumption. $\endgroup$
    – whuber
    Sep 18, 2021 at 21:22
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    $\begingroup$ Yes @whuber this is correct - the answer above suggests the there is no overall problem of running a model like this (my first question) but I was also interested in knowing whether my interpretation was valid - a la of what you've outlined. Of course, correlation is not causation, which the answer points out, and I guess expressed myself a bit lazy saying that night shift lowers life expectancy. But this is not really what I'm interested in, but rather whether we can interpret the effects in the way that I suggested. $\endgroup$ Sep 19, 2021 at 12:42
  • $\begingroup$ @jewelBeetle, "the way [you] suggested" was causally. If you have a more specific interpretation, you'll need to clarify what it is. You can't say anything about any possible interactions given your situation. $\endgroup$ Sep 19, 2021 at 17:33

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