# Interpreting main effect and interaction

I am doing a simple marketing project that has the following types of variables:

• X1 - continuous (e.g. income)
• X2 - categorical (e.g. gender)
• Y - continuous (e.g. number of a product type purchased such as tubs of ice-cream)

I am interested in the relationship between income (X1) and product purchase (Y) but also the effect of gender (X2) on this relationship. (i.e. interaction or moderation effect).

I have centered X1 and have used the general linear model in SPSS. The result on Y is as follows:

• X1 - significant
• X2 - not significant
• X1*X2 - not significant

How do I interpret this result in terms of main effect and interaction?

• Is this a homework problem? If so, you need to add the homework tag. – gung - Reinstate Monica Dec 16 '11 at 5:22
• No, this is not a homework problem, though the simplicity of it makes it appear so! I am after an explanation that I can present to my non-statistical target audience. – Adhesh Josh Dec 16 '11 at 5:24
• It seems to me you have already written the essense of what could be called an interpretation. Where exactly are you looking for help? What is it that is crossing you up? – rolando2 Dec 16 '11 at 12:03

## 2 Answers

In general, you should not base your model selection solely on statistical significance. Substantive meaning is more important.

In this particular case, you can graph the predicted values for males and females, with the x-axis being income and the y-axis the number of items bought, and a line for each gender.

@gung makes a good point that, if the y-variable is a count, you should use an appropriate model, such as Poisson regression, or, more likely, negative binomial regression, since over-dispersion is very common in count regression.

Your results suggest that there is no interaction--you simply have a main effect of X1. You could say something like, "The number of tubs of ice-cream people buy is related to their income. For instance, if person A's income is one unit higher than person B's income, person A typically buys $\beta_1$ more tubs of ice-cream than person B. Our data suggest that this relationship between income and ice-cream buying is similar for both men and women."

(Incidentally, if your response variable is a count, you should use Poisson regression rather than the general linear model, but I don't know if that's just your example.)

• Is it strictly speaking more correct to rerun the model with the interaction omitted before making this income->icecream interpretation? I have in mind your answer at stats.stackexchange.com/questions/93667/…, in which you mentioned that interpreting main effects is problematic when there's an interaction in the model. – user1205901 - Reinstate Monica May 18 '15 at 4:47
• @user1205901, if the interaction term, $\beta_3$, is very small (which I guess I am assuming), $\beta_1$ is unlikely to change much, but you're right, it would be better to rerun the model to get a better estimate of $\beta_1$. You needn't do a nested model test however, as it would provide the same information that is in the standard regression output regarding the interaction. – gung - Reinstate Monica May 18 '15 at 13:41