Signal Detection Theory vs. Logistic Regression

Simulation Code

set.seed(1)

SDT_params <- function(state,resp) {
  tab <- table(state,resp)
  
  sdt_res <- neuropsychology::dprime(
    n_hit  = tab[2,2],
    n_miss = tab[2,1],
    n_fa   = tab[1,2],
    n_cr   = tab[1,1]
  )
  
  c(sdt_res$dprime , sdt_res$c)
}

logistic_reg_params <- function(state,resp){
  fit <- glm(resp ~ state, family = binomial())
  
  coef(fit)
}

# initialize
res <- data.frame(d_    = numeric(B),
                  c_    = numeric(B),
                  int   = numeric(B),
                  slope = numeric(B))

# Loop
for (b in seq_len(B)) {
  true_sensitivity <- rexp(1,10) # random
  true_criterion <- runif(1,-1,1) # random
  
  # true state vector
  state_i <- rep(c(F,T), each = n/2)
  
  # response vector
  Xn <- rnorm(n/2) # noise dist
  Xs <- rnorm(n/2, mean = true_sensitivity) # signal + noise dist
  X <- c(Xn,Xs)
  resp_i <- X > true_criterion
  
  # SDT params
  res[b,1:2] <- SDT_params(state_i,resp_i)
  
  # logistic regression params
  res[b,3:4] <- logistic_reg_params(state_i,resp_i)
}

Results

SDT parameters are on a standardized normal scale, meaning they are scaled to \(\sigma=1\). However, the logistic distribution’s scale is \(\sigma=\pi/\sqrt3\). Thus, to convert the logistic regression’s parameters to the SDT’s we need to scale both the intercept and the slope by \(\sqrt3/\pi\) to have them on the same scale as \(c\) and \(d'\).¹ Additionally,

1. The slope must be also scaled by \(-2\) due to R’s default effects coding.
   
2. The intercept must also be scaled by \(-1\) – see paper for the full rationale.

The red-dashed line represents the expected regression line predicting the SDT parameters from their logistic counterparts:

• \(d' = -2\times\frac{\sqrt{3}}{\pi}\times Slope\)
  
• \(c = -\frac{\sqrt{3}}{\pi}\times Intercept\)

(The blue line is the empirical regression line.)