Why does the Youden rule does not recommend a threshold of 0.5 on balanced data? Suppose I have a logistic regression model estimated using a balanced target (equal group sizes). My questions concern the optimal threshold for prediction and it's relationship with the Youden's rule and the estimation of balanced logistic regressions.
Question 1) Will the optimal threshold for prediction be necessarily be 0.5 ?
Question 2) Will a criteria like the Youden's rule (maximize the difference between True Positive Rate and False Positive Rate) recommend an optimal threshold of 0.5 ?
Question 3) If the target is unbalanced (i.e. more negatives than positives) and I estimate a weighted logistic regression, how does the answer to the questions above changes ?
Thanks in advance.
 A: *

*The optimal threshold depends on your costs of mistakes. If it is "terrible "to call a $0$ a $1$ but merely "bad" to call a $1$ a $0$, then you might be inclined to favor $0$ as a prediction, requiring a very high probability of $1$, say $0.95$, before you would predict a $1$, even with perfect class balance. It could be that there is a third action to be taken (such as "unsure...collect more data"), even if the observed outcomes are binary, as is argued here and here.


*It appears that this does not have to happen, as a simulation shows.
library(ModelMetrics)

set.seed(1998) # Yes, 1998 was 24 years ago, 
               # but 1998 gave me perfect class balance
N <- 1000
x <- runif(N, -2, 2)
z <- x
pr <- 1/(1 + exp(-z))
y <- rbinom(N, 1, pr)
ybar <- mean(y)
L <- glm(y ~ x, family = binomial)
probs <- 1/(1 + exp(-predict(L)))
categories <- round(probs)
ybar

youden <- function(y, yhat, threshold){
    
    return(
        ModelMetrics::recall(y, yhat, threshold)
        +
        ModelMetrics::tnr(y, yhat, threshold)
        -
        1    
    )   
}

thresholds <- seq(0.01, 0.99, 0.01)
youdens <- rep(NA, length(thresholds))
for (i in 1:length(thresholds)){
    
    youdens[i] <- youden(y, probs, threshold = thresholds[i])
    
}

plot(thresholds, youdens)
abline(v = ybar)

The threshold that optimizes Youden's criterion is not $0.5$, despite the perfect class balance in y (given by ybar=0.5. If you fiddle with that simulation, you will see that the prior probability (ybar) need not be the threshold that optimizes Youden's criterion. Try that simulation for x <- runif(N, -2, 5) to get a different prior probability, and you will see that the threshold giving optimal the Youden criterion is near but not exactly the prior probability. This makes me think that, for question #3, nothing changes.
I will leave the two usual blog posts to which I like to link when class imbalance comes up on here. The author, Vanderbilt's Frank Harrell, advocates for direct assessment of the probability outputs, not prematurely converting those probability outputs to discrete categories.
https://www.fharrell.com/post/classification/
https://www.fharrell.com/post/class-damage/
