# Inference in linear mixed model

Let have an imaginary income data that comes with three categorical (Binary) variables, having TV (yes=1,no=0) at home, washing machine (yes=1, no=0) and age (old=1,young=0) as well as the income for the response, y. Assume that the data is collected mounthly and repeated for three months and the following linear mixed model (in nlme or lme4 format) is fitted

y = Tv + Wm + Age + Tv:Wm + Tv:Age + Wm:Age + (1|Months)


Let all terms be significant after fitting the model.

Then, my question is the interpretation of the terms. Actually, I have confused myself. Is Tv significant means that it is significant CONDITIONAL on the other effects or not? How about if I want to answer this question: is there any difference between the income of the people who have a WM at the young age?

I would deeply appreciate if somebody explains the mathematical reasoning in addition to the actual inference.

• You ask " is there any difference between the income of the people who have a WM in the young age?" But .... between those people and whom? Right now, you haven't asked a question. Commented Nov 27, 2017 at 11:06
• @PeterFlom that is probably the main issue. probably the question is: is the income being affected by having TV at home for the youngsters! Commented Nov 27, 2017 at 11:12
• Well, it's still not clear what you are asking, but you can't say "affected" as that would imply causation. Commented Nov 27, 2017 at 11:19
• Thanks @PeterFlom but I am not clear what you mean. It is like a simple regression. Commented Nov 27, 2017 at 11:23
• Your question right now is the equivalent of "Is there any difference between boys?" I think you probably mean "people who have a WM and people who don't". Commented Nov 27, 2017 at 11:43

When you have a random effects model, the interpretation of the variable's coefficients is the same as if it was a linear regression model (let's assume you already know how to do that).

What changes in a random effects model is that you're inserting a term in the model to account for the intraindividual variance. This new term follows a Gaussian distribution with mean = 0 and variance = $\sigma^2_{b_i}$. This means we're interested only in the variance of this term. And the likelihood is given by, where both the distributions of $y_i$ and $b_i$ are gaussians:

\begin{align*} L_{i}(\theta_{i} ; y_{ij}) & = \int f(y_{i} | b_{i}) \cdot f(b_{i}) {\rm d}b_{i} \\ \end{align*}

About the part of yout question about conditionality, is only $y$ that might be conditioned on the random effects, or: $$y | b \sim N(X\beta + Zb, \sigma^2 I)$$

• I can if you tell me what you need to know, I'm not really good explaining things :p Commented Nov 27, 2017 at 12:15

I'm not going to explain the math (others here can do that much better than I can) but:

is there any difference between the income of the people who have a WM (and those who do not) at the young age?

is not the exact question this model answers. It controls for age, so it holds age constant. If you want to look just at young people, you should look at a subset of the data.

What you have now, with the interaction term, let's you look at whether the effect of having a WM is different at different ages.

• I like this answer, however, I need more clarification on the last sentence, "What you have now, with the interaction term, let's you look at whether the effect of having a WM is different at different ages." Commented Nov 27, 2017 at 14:09