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

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?

• Definitely not the maximum difference. Conceivably could be the average difference, I believe, depends on how valid the logistic model is here. You have to look at the raw data- don't just plot predicted probabilities, also plot the means. However, you imply there is a U-shaped function (both ineffective at dark and both ineffective at bright, but ok in the middle) which means your model is invalid. You could address that by adding a quadratic term. – Alex Holcombe Jun 7 '15 at 23:42
• Hi Alex, Thanks for your response. I did look at the proportion of successful responses for light levels in binned increments. These values seem to follow the predicted probability curve quite closely, although I don't believe I tested this statistically. I'm not sure there is a U-shaped function because both lasers work well (are effective) in the dark and neither works well in daylight (are ineffective). The difference between them is greatest at intermediate light levels. Given this information, do you think it makes sense to include light as a linear term? – FCassidy Jun 10 '15 at 18:56

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.