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

Question

3 variables (1 continuous (X) and 2 categorical (A & B)) predict 1 dichotomous variable in logistic regression.

Both variables A and B are dichotomous and are coded with 1 (reference category) and 2. Criterion is coded with 0 & 1.

We have the following results:

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

Answer 1

Assume you have a logit model with linear index

$$ P_j \propto \exp\left(\sum_{k=1}^K \beta_{k} x_{jk}\right)$$

and assume for the sake of example that $$\beta = (\beta_1,...,\beta_6)^\top$$ that is $K=6$ and I want to test $$\beta_4 = \beta_5$$

Then I first isolate all coefficients on one side

$$\beta_4 - \beta_5 = 0$$

then I define $R = (0,0,0,1,-1,0)$ such that

$$R\beta= (0,0,0,1,-1,0)\begin{pmatrix} \beta_1 \\ \beta_2 \\ \beta_3 \\ \beta_4 \\ \beta_5 \\ \beta_6\end{pmatrix} = \beta_4 - \beta_5$$

I am now able to write the hypothesis I'm interested in the way it is standardly done in the statitical theory so my null hypothesis is

$$R\beta = \beta_4 - \beta_5 = 0$$

To perform such a test in the software R

install.packages("aod")
library(aod)
mydata <- read.csv("https://stats.idre.ucla.edu/stat/data/binary.csv")
## view the first few rows of the data
head(mydata)
mydata$rank <- factor(mydata$rank)
mylogit <- glm(admit ~ gre + gpa + rank, data = mydata, family = "binomial")
summary(mylogit)
# Test if coefficient 4 is same as 5 
R <- cbind(0, 0, 0, 1, -1, 0)
wald.test(b = coef(mylogit), Sigma = vcov(mylogit), L = R)

In your case there is a total of 8 coefficients including the constant term and you want to test if the sum of coefficient 5 and 8 are 0 so in youre case

$$R = (0,0,0,0,1,0,0,1)$$

The example I used here was borrowed from this webpage wald test example