Interpreting main effects in the presence of an interaction in logistic regression

Question

I am carrying out logistic regression to determine which factors affect whether lasers are effective bird deterrent devices. In my model, a success (1) is when a bird flies away and a failure (0) is when they do not.

I built a model, but I am having some trouble interpreting it.

In my model, I have:

Laser colour (categorical, 2 levels)
Light (continuous)
Light x laser colour

From the beta coefficient for laser colour, I calculated an odds ratio for laser colours and found that colour A is 6x better than colour B. The statement that colour A is more effective than colour B is consistent over all light levels, but the magnitude of this difference changes (as evidenced by the interaction term in the model). I determined (from plotting predicted probabilities) that the lasers are very similar when it is dark (both effective) and bright (both not effective) but differ at intermediate light levels.

My question is:

Does it make any sense to interpret the odds ratio from the laser colour main effect? Can interpret the 6x as the average difference, or the maximum difference, or is it not meaningful at all in the presence of the interaction?

Eric Farng · Answer

Given color A as the reference color, and the the equation/notation below

$$ frac{p}{1-p} = e^{beta_0 + beta_1 color + beta_2 light + beta_3 color*light} $$

$beta_1$ can be interpreted as the effect when $light=0$, if that is possible for your variable.

If light is always positive or negative, then we can also say something about the maximum or minimum. Lets assume $light>0$ for now.

The odds ratio of color A is $e^{beta_0 + beta_2 light}$. If $beta_2$ is positive, then the minimum odds ratio of the reference color is $e^{beta_0}$ since light is also positive. And if $beta_2$ is negative, then $e^{beta_0}$ is the maximum odds ratio.

The odds ratio of color B is $e^{beta_0 + beta_1 + beta_2 light + beta_3 light}$. If $beta_2 + beta3 >0$, then the minimum odds ratio of the non-reference color is $e^{beta_0 + beta_1}$. If $beta_2 + beta3 <0$, then it is the maximum odds ratio.

To answer your question directly, lets assume $light>0$. If $beta_2 + beta3 >0$, then $100(beta_1-1)%$ would be color B's minimum percent change in odds ratio over color A. If $beta_2 + beta3 <0$, then it is the maximum.

Interpreting main effects in the presence of an interaction in logistic regression

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