Cross Validated Asked by aarsmith on January 5, 2022

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 ($Xbeta$).

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
> head(dt, 10)
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)
> head(X, 10)
(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
> head(preds, 10)
[,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

Answered by Robert Long on January 5, 2022

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