# Making predictions manually from a mixed effects model

I have a mixed effects logistic regression model that is a bit more complicated than I've done in the past and just want to know if I'm thinking things correctly. I am crossing B_A (a within-subject continuous predictor) with its quadratic term (B_A2) and two between-subject categorical variables effects coded (sex e[-0.5, 0.5] and mag e[-0.5, 0.5]).

I am trying to identify the predicted values of B_A by computing the equation by hand, but am unsure if I'm interpreting the interactions correctly. Below is a post of my attempt

What I'm most unsure about is, for example, the sex:b_a condition: do I multiply all values of B_A*-2.06 and -0.5 (since that is the condition I'm looking for)?

Thank you for helping me understand.

The model in the link looks like:

y ~ sex + mag + b_a + b_a^2 + sex:b_a + mag:b_a


Actually we can disregard that it is a mixed effects model since the question doesn't concern the random effects

What I'm most unsure about is, for example, the sex:b_a condition: do I multiply all values of B_A*-2.06 and -0.5 (since that is the condition I'm looking for)?

So you are referring to the sex:b_a interaction. Yes, when sex is -0.5 then you multiply b_a by -0.5 and -2.06, but when it is 0.5 then you multiply it by 0.5 and -2.06. A good way to understand this is to form the model matrix $$X$$ yourself and the vector of parameter estimates $$\beta$$ and look at how they are multiplied together ($$X\beta$$).

In R we can do this very easily but it is just as easy in a spreadsheet:

# First make some toy data according to the data description and show the first 10 rows
> dt <- expand.grid(sex = c(-0.5, 0.5), mag = c(-0.5, 0.5), b_a = 1:4)
> dt$b_a2 <- dt$b_a^2
sex  mag b_a b_a2
1  -0.5 -0.5   1    1
2   0.5 -0.5   1    1
3  -0.5  0.5   1    1
4   0.5  0.5   1    1
5  -0.5 -0.5   2    4
6   0.5 -0.5   2    4
7  -0.5  0.5   2    4
8   0.5  0.5   2    4
9  -0.5 -0.5   3    9
10  0.5 -0.5   3    9


Now make the model matrix and show the first 10 rows. This will look very much like the data but with a column of 1s for the intercept and also a column for each of the interaction terms:

> X <- model.matrix(~ sex + mag + b_a + b_a2 + sex:b_a + mag:b_a, dt)
(Intercept)  sex  mag b_a b_a2 sex:b_a mag:b_a
1            1 -0.5 -0.5   1    1    -0.5    -0.5
2            1  0.5 -0.5   1    1     0.5    -0.5
3            1 -0.5  0.5   1    1    -0.5     0.5
4            1  0.5  0.5   1    1     0.5     0.5
5            1 -0.5 -0.5   2    4    -1.0    -1.0
6            1  0.5 -0.5   2    4     1.0    -1.0
7            1 -0.5  0.5   2    4    -1.0     1.0
8            1  0.5  0.5   2    4     1.0     1.0
9            1 -0.5 -0.5   3    9    -1.5    -1.5
10           1  0.5 -0.5   3    9     1.5    -1.5


Then we can just use the model estimates to make the predictions:

# the vector of model estimates:
> betas <- c(1.57, -0.5, 0.81, 9.43, -4.309, -2.06, -2.91)

# and now make the predictions by premultiplying the parameter vector by the model matrix:
> preds <- X %*% betas
[,1]
1   9.021
2   6.461
3   6.921
4   4.361
5   8.009
6   3.389
7   2.999
8  -1.621
9  -1.621
10 -8.301

# manually calculate the first prediction:
> (1.57*1) + (-0.5*-0.5) + (0.81*-0.5) + (9.43*1) + (-4.309*1) + (-2.06*-0.5) + (-2.91*-0.5)
[1] 9.021


and this agrees with the first prediction calculated by R

• Thank you for your post robert. It's good to see I'm on the right track, but I am confused still as when I do this in matlab my manual computations do not match with Matlab's predict function (in.mathworks.com/help/stats/linearmixedmodel.predict.html). Even when I set 'conditional' to false, which should mean I am only looking at fixed effects, the results differ. However, matlab seems to compute a prediction for each observation which I then take the mean of. Perhaps this is not the same as a computing predictions from model effects? Jul 27 '20 at 17:01
• I don't use MATLAB so I can't help and programming/software questions are off topic but it might be that MATLAB is adding the random effects. Your question didn't involve random effects. Jul 27 '20 at 17:05
• the doc says it is not taking random effects into account, so I'm wondering if there is a conceptual difference. When I compute the equation, it is assuming the non-used variables are at 0. which should be the average effect given that I have mean-centered everything. However, perhaps that is still somewhat different relative to taking the prediction of every observation and averaging them? Jul 27 '20 at 17:25
• I'm not sure what you mean by "taking the prediction of every observation and averaging them" but I think you need to ask a new question about this but try not to make it about programming Jul 27 '20 at 17:54
• I'm not sure if this program specific or general to regression models, but would these results change if what was appended in the output changed? For instance, my original variable is called sex, but MATLAB appended the (0.5); would these results change if it appended a (-0.5) or does the way we compute the coefficients stay the same? I believe I've seen something similar happen in R. Jul 27 '20 at 21:00