Chance-corrected beta binomial model for signal detection theory analysis of replicated n-AFC discrimination test

Modeling
,
1

I would like to analyze data from a replicated n-alternative forced choice (n-AFC) discrimination test with brms in R.
There are frequentist ways of this signal detection theory / sensometrics analysis (e.g., in the sensR package) but I would like to try the Bayesian way.

Sample data

For N participants (pid = 1,…,N) doing m n-AFC trials (0 ≤ correct_responses ≤ m), data would look like this:

pid,correct_responses,covariate
  1,                1,       58
  2,                0,       70
  3,                m,       15
...
  N,                5,       38

This type of data tends to be overdispersed.

Overdispersion with the beta binomial distribution

In general, overdispersion can be modelled using a beta binomial distribution where the binomial probability (per participant) is assumed to be beta distributed in the interval ( 0 , 1 ).
As the beta binomial distribution is implemented in Stan, modeling would be probably something like the following with brms:

fit <- brm(
  correct_responses | trials(m) ~ 1 + covariate + (1 | pid),
  family = beta_binomial(),
  ...
)

The problem

Some authors (e.g., Brockhoff, 2003; Meyners, 2007; Morrison, 1978) argue that the beta binomial model should be adapted to replicated n-AFC tests by correcting for the guessing probability p_0 = \tfrac{1}{n} because the binomial probability is distributed in the interval ( p_0 , 1 ) (see Bi, 2015, p. 283 and Chapter 10 in general).
The chance-corrected beta binomial model has the following pdf (Bi, 2015, p. 288, Equation 10.2.24):

P\left(x|m,a,b,p_0\right)=\frac{\left(1-p_0\right)^m}{\textrm{B}\left(a,b\right)}\binom{m}{x}\sum\limits_{i=0}^{x}\binom{x}{i}\left(\frac{p_0}{1-p_0}\right)^{x-i}\textrm{B}\left(a+i,m+b-x\right)

where \textrm{B}\left(\cdot{},\cdot{}\right) is the Beta function, m is the number of n-AFC trials, x is the number of correct responses, a,b\geq{}0 are the parameters of the distribution, and p_0 is the guessing probability.

My question

Can I use the existing beta_binomial() family from brms/Stan and correct for p_0 somewhere else (e.g., in formula of brm()) to model my data or do I need a custom family representing the chance-corrected beta binomial?

Reference

Bi, J. (2015). Sensory discrimination tests and measurements: Sensometrics in sensory evaluation (2nd ed.). Wiley Blackwell.
Brockhoff, P. B. (2003). The statistical power of replications in difference tests. Food Quality and Preference, 14(5–6), 405–417. Redirecting
Meyners, M. (2007). Proper and improper use and interpretation of Beta-binomial models in the analysis of replicated difference and preference tests. Food Quality and Preference, 18(5), 741–750. Redirecting
Morrison, D. G. (1978). A probability model for forced binary choices. The American Statistician, 32(1), 23–25.