How can I test whether two logistic regression coefficients are identical in mixed models?

Question

Similar to question here.

3 variables (1 continuous (X) and 2 categorical (A & B)) predict 1 dichotomous variable in generalized linear mixed models.

Both variables A and B are dichotomous and are coded with 0 (reference category) and 1. Criterion is coded with 0 & 1.

After

library(lmerTest)
glmer(Y ~ X*A*B + (0+X|G) + (1|G), data=dat, family = "binomial", verbose = T, control = glmerControl(optimizer = "bobyqa"))

we have the following results:

Random effects:
 Groups    Name                        Variance  Std.Dev.
 G         (Intercept)                 0.1458298 0.38188 
 G          X                          0.21727   0.46612
Number of obs: 26260, groups:  G, 230
Fixed effects:
                                   Estimate Std. Error z value Pr(>|z|)    
(Intercept)                         1.47384    0.04094  35.997  < 2e-16 ***
X                                  -0.03252    0.01612  -2.017  0.04370 *  
A2                                 -0.46066    0.04853  -9.492  < 2e-16 ***
B2                                 -0.61576    0.04811 -12.799  < 2e-16 ***
X:A2                                0.07502    0.01810   4.144 3.41e-05 ***
X:B2                                0.08031    0.01945   4.129 3.64e-05 ***
A2:B2                               0.62260    0.06653   9.358  < 2e-16 ***
X:A2:B2                            -0.06789    0.02367  -2.868  0.00413 ** 
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Now, the slopes for different conditions are:

    A1      A2 
B1  X       X+X:A2
B2  X+X:B2  X+X:A2+X:B2+X:A2:B2

How can I test the following hypothesis

X+X:B2 = X+X:A2 + X:B2 + X:A2:B2 which equals to 0 = X:A2 + X:A2:B2 and finally

X:A2 = -X:A2:B2

For logistic one-level models we use Wald's test as described in here.

library(aod)
R <- cbind(0,0,0,0,0,0,1,1)
wald.test(b = coef(model), Sigma = vcov(model), L = R)

Can we use the identical approach for mixed models?

Answer 1

I am not overly familiar with mixed models (read I cannot tell you whether this is a correct statistical approach) but based on reading here mixed models on SSC I think you may be looking for something like this:

set.seed(1)
#### SIMULATE ####
n <- 1000
K <- 4
A <- as.numeric( runif(n)<0.5 )
B <- as.numeric( runif(n)<0.3 )
X <- rnorm(n) + runif(n)
random_effect_index <- sample(1:K,10,replace=TRUE)
random_effect <- rnorm(K,sd=0.3)[random_effect_index]

# Fixed effect parameters
beta_0 <- 0
beta_X <- 0.5
beta_B <- 1
beta_A <- 1.3

beta_XA <- 0.5
beta_XB <- 1.3
beta_AB <- -0.5

beta_XAB <- -1

# probabilities

interaction_effects <- (beta_XA*X*A) + (beta_XB*X*B) + (beta_AB*A*B) + (beta_XAB*X*A*B)

exp_z <- exp(beta_0 + beta_X * X + beta_A * A + beta_B * B + interaction_effects + random_effect)
p <- exp_z / (1 + exp_z)
hist(p)

# dependent
Y <- as.numeric( runif(n) < p ) 
mydata <- data.frame(Y=Y,X=X,A=A,B=B,G=random_effect_index)
head(mydata)


#### FIT MODEL ####
library(lme4)
gmm <- glmer(Y ~ X*A*B + (0+X|G) + (1|G), data=mydata, family = "binomial", verbose = T, control = glmerControl(optimizer = "bobyqa"))
summary(gmm)

# Likelihood Ratio Test using Linear Hypothesis Command
library(car)
LH <- linearHypothesis(gmm,c("X:A + X:A:B=0") ) 
LH

Answer 2

You can use the GLMMadaptive package in which the anova() method has an argument called L that is a user-defined contrast matrix. For more info check here.

The relevant code would be something like:

library(GLMMadaptive)
fm <- mixed_model(Y ~ X * A * B, random = ~ X || G, data = dat,
                  family = binomial())
R <- cbind(0,0,0,0,0,0,1,1)
anova(fm, L = R)

Answer 3

If you have two predictors in a regression model and want to test whether or not they have equivalent effects, i.e. that $\beta_1 = \beta_2$ in

$$ Y = \beta_0 + \beta_1 X_1 + \beta_2 X_2 + \varepsilon$$

Then form a new predictor, $Z = X_1 + X_2$, and compare the model with $X_1, X_2$ as predictors to the one using only $Z$. If the larger model is not significantly better you don't have evidence that the coefficients are different.

I described this in terms of continuous predictors, and for linear regression, but you can imagine an analogous approach for categorical variables and for GLMs like logistic regression.