# Significant interaction when only one of the two main effects is significant? [duplicate]

I have run a mixed effect logistic regression model, and I have obtained a significant main effect of Factor A, while no significant main effect of Factor B. However, the interaction between Factor A and Factor B is significant. Is this possible? Both of the factors have two levels; let us call them X and Y for Factor A, and W and Z for Factor B. On a descriptive level, the score (dependent variable) increases when Factor A assumes the level X rather than when it assumes the level Y. Additionally, the score increases when Factor A assumes the level Y and Factor B level W if compared to the case in which Factor A assumes the level Y and Factor B the level Z. Instead, when Factor A assumes the level X, there seem to be no difference due to the level assumed by Factor B. Does the result of the logistic regression model make sense?

I also have an additional question. I have performed an Analysis of Deviance Table (Type II Wald chisquare tests) [car::Anova(model)], and Factor B here appears to be significant as a main effect. Is this possible?

I am sorry for these maybe stupid questions, but I am very new to the world of statistics.

• You might want to look at this post : stats.stackexchange.com/questions/641812/… Commented Mar 13 at 8:27
• Commented Mar 13 at 13:59
• I remember another related question but I can't find it. Commented Mar 13 at 14:18

An interaction between factor A and factor B means that the effect of factor A differs according to the level of the factor B. It has nothing to do with a factor having a big effect or not.

For example : let's consider that happiness is your response variable. You have 2 factors: treatment and time, and you test for the interaction between them. You find that there is one. In one treatment, happiness increases with time. It is pretty low at the beginning, and high at the end. In the other treatment, happiness decreases with time. It is high at the beginning and low at the end. If you do not include time in your model, you consider that your data was acquired at the same time (= what you would be doing if you did not include time in your model, only the treatment = you have a main effect of time only). Then happiness will have low and high values in both treatment groups. Therefore you might not detect a treatment effect (= a main effect of treatment). It does not mean treatment as a main effect has a "small" or "large" effect. It simply means that the effect of treatment differs with time, and you are disregarding it.

There is a lot of things to unpack to fully understand and conceptualize what is going especially if are new to statistics as you mentioned - and also since you are already using quite advanced tools such as logistic GLMMs. So the best thing would be to grab a textbook (or two or more) and start studying the basics. But anyway, I’ll give it a try:

Imagine you're studying two things: Factor A (let's say it's the type of treatment someone receives) and Factor B (maybe it's the gender of the people in the study).

Then you find that Factor A (the treatment type) seems to make a difference in the outcome you are looking at. But then, when you look at Factor B (gender), it doesn't seem to have much of an effect on its own.

Here's where it gets interesting: when you look at how Factor A and Factor B work together, there's something happening. The way Factor A affects the outcome seems to change depending on whether it's a guy or a girl (or whatever genders you're studying). That's what is called an "interaction."

So, even if one factor doesn't seem to have a big effect by itself, it can still be important when you look at how it interacts with another factor. That's why you can have a significant interaction even if the main effects aren't significant on their own.

Now you probably have even more questions but there is a good chance that someone has asked that already here on this site.