[Note on cross-posting: I asked the same question at CrossValidated and have received the suggestion that I should ask it here.]

I want to model the index called Delta P (e.g., p.144 of this paper), which is basically a difference between two proportions (i.e., \frac{n_1}{N_1} - \frac{n_2}{N_2}), as a function of a predictor. The input data should be the four count variables from which to calculate Delta P (i.e., n_1, N_1, n_2, N_2) and predictor values.

Below is my attempt to do it in Stan. When I run the code, I get a message about Jacobian adjustments since the left-hand side of a sampling statement is `deltaP`

, which is calculated by subtracting one parameter from another (`theta1`

- `theta2`

, where `theta1`

is the estimated value of \frac{n_1}{N_1} and `theta2`

is that of \frac{n_2}{N_2}).

```
data {
int<lower=0> N; // total number of observations
int<lower=1> denom1[N]; // denominator of the first proportion
int<lower=1> denom2[N]; // denominator of the second proportion
int<lower=0> nom1[N]; // nominator of the first proportion
int<lower=0> nom2[N]; // nominator of the second proportion
real x[N]; // predictor variable
}
parameters {
real<lower=0, upper=1> theta1[N]; // the first proportion
real<lower=0, upper=1> theta2[N]; // the second proportion
real alpha; // intercept
real beta; // slope parameter for x
real<lower=0> sigma; // SD of the error term
}
transformed parameters {
real<lower=-1, upper=1> deltaP[N]; // Delta P
for (i in 1:N) {
deltaP[i] = theta1[i] - theta2[i];
}
}
model {
// priors
theta1 ~ beta(1, 1);
theta2 ~ beta(1, 1);
alpha ~ normal(0, 2);
beta ~ normal(0, 2);
sigma ~ normal(0, 1) T[0, ];
for (i in 1:N) {
// estimating thetas based on denoms and noms
nom1[i] ~ binomial(denom1[i], theta1[i]);
nom2[i] ~ binomial(denom2[i], theta2[i]);
// deltaP is sampled from the truncated normal distribution whose mean is alpha + beta * x and the SD is sigma
deltaP[i] ~ normal(alpha + beta * x[i], sigma) T[-1, 1];
}
}
```

I run the Stan code above with the following R code.

```
library("rstan")
### Generate fake data
set.seed(100)
# sample size
N <- 100
# True parameter values
alpha <- -0.2
beta <- 0.5
sigma <- 0.1
# predictor values (x) and Delta P values
while (TRUE) {
x <- runif(N, -1, 1)
deltaP <- alpha + beta * x + rnorm(N, sd = sigma)
if (all(deltaP <= 1) & all(deltaP >= -1)) break
}
# theta values
theta1 <- theta2 <- numeric(N)
for (i in 1:N) {
if (deltaP[i] > 0) {
theta1[i] <- runif(1, deltaP[i], 1)
theta2[i] <- theta1[i] - deltaP[i]
} else {
theta2[i] <- runif(1, abs(deltaP[i]), 1)
theta1[i] <- theta2[i] + deltaP[i]
}
}
# denoms and noms
denom1 <- sample(N, replace = TRUE)
denom2 <- sample(N, replace = TRUE)
nom1 <- rbinom(N, denom1, theta1)
nom2 <- rbinom(N, denom2, theta2)
### fit the model
fit <- stan(file = 'xxx.stan',
data = list(
N = N,
denom1 = denom1,
denom2 = denom2,
nom1 = nom1,
nom2 = nom2,
x = x
))
```

This runs, but I also get the following message:

```
DIAGNOSTIC(S) FROM PARSER:
Info:
Left-hand side of sampling statement (~) may contain a non-linear transform of a parameter or local variable.
If it does, you need to include a target += statement with the log absolute determinant of the Jacobian of the transform.
Left-hand-side of sampling statement:
deltaP[i] ~ normal(...)
```

I only have a vague understanding of Jacobian, but I believe it is necessary when parameters are transformed nonlinearly as it alters the shape of variable distribution. What I am not sure of is whether the case above (`deltaP = theta1 - theta2`

) equates with nonlinear transformation, and if it does, what kind of Jacobian adjustments are necessary (or if there are any other ways to circumvent the issue).

When I repeated the above code 1,000 times with different seeds and examined the distribution of the mean of the posterior distributions in the three focal parameters (i.e., `alpha`

, `beta`

, `sigma`

), 70.5% of `alpha`

, 20.1% of `beta`

, and 37.4% of `sigma`

were above the true value (see the figure below), which makes me suspect they may be biased and the bias could be due to the failure to include Jacobian adjustments.