# Interpretation of interaction terms if main effect is insignificant

I have read the questions posted here about how to deal with the situation of significant interaction terms but insignificant 'main' effects, and my main take-away is that it depends on your research question how to proceed and interpret your results.

I would appreciate your thoughts on my interpretation in the context of my research question.

Hypothesis: The relation between height and weight is stronger for males than for females.

$$Weight = a + b_1 \times Height + b_2 \times Male + b_3 \times (Height \times Male) + e$$

Results:

$b_1$ is not significantly different from zero.

$b_3$, the interaction term, is positive and significantly different from zero.

Can I conclude that I found evidence in support of my hypothesis (that the relation between height and weight is stronger for males than for females)?

Thank you!

• It might depend on what you mean by "stronger". At least two senses are natural in this context--statistical significance and size of effect. The latter is usually more meaningful, because the former is largely an artifact of sample size and data variation. But how, then, are you measuring effect size when there is an interaction? It seems to me that at a minimum you will need to account for $b_1$ in your interpretation. – whuber Jan 9 '14 at 18:10
• but isn't the fact, that there is an significant interaction (with a positive value) evidence that the effect size is bigger for males than for females? – SPi Jan 9 '14 at 18:13
• sorry @whuber I confused b2 with b1. I edited my original post. – SPi Jan 9 '14 at 18:27

Rather than saying the relationship is stronger, I think it's more precise to say that weight increases significantly more quickly with height for males than for females. Strength of relationship would be measured by measures like $R^2$, and these are affected not only by the rate of increase of one variable with another, but by the amount of noise in the data. e.g. if the data were something like this:

maleheight <- rnorm(1000, 70, 3)
femaleheight <- rnorm(1000, 65, 2.5)
maleweight <- maleheight*2.2 + rnorm(1000, 0, 20)
femaleweight <- femaleheight*1.3 + rnorm(1000, 0, 10)
height <- c(maleheight, femaleheight)
weight <- c(maleweight, femaleweight)
male <- c(rep(1, 1000), rep(0, 1000))
data <- data.frame(cbind(height, weight, male))


and the model

m1 <- with(data, lm(weight~height + male + height*male))
summary(m1)


shows your pattern, but the relationship looks stronger for women

• thank you @Peter Flom! Sorry, I confused the names of coefficients in my original post. You're example corresponds to the situation where the dummy 'Male' is not significant. However, how would you interpret the results, if the coefficient 'Height' is not significant while the interaction term (Height * Male) is? – SPi Jan 9 '14 at 18:36
• It wouldn't change things. $R^2$ is a measure of strength of relationship. The coefficient of the intercept term is a measure of whether the slope of the line would be different for men and women, regardless of whether the coefficient for male is significant, or whether the coefficient for height is significant. – Peter Flom - Reinstate Monica Jan 9 '14 at 19:25