tl;dr: I want to model a correlation between two latent parameters (estimated from observable data) and condition that correlation on another observable variable. I got stuck modeling the correlation, and extending the model to include conditioning of the correlation.

Imagine a situation in which two latent variables are correlated. E.g. Cognitive ability X in the child and in the parent. These cognitive abilities are not observed directly, but estimated from some observable performance in whatever test.

Generating the data:

```
library(MASS)
library(tidyverse)
## We have two variables (e.g. rates in a binomial or a poisson): rateC and rateP
## The two variables are generated according to a multivariate gaussian: they are correlated according to rho
## They are then used to generate performance in an aggregated binomial fashion (correct out of 30 trials)
n <- 200 # n of participants
trials = 30 # n of trials
muC <- 0.4 # pop level mean of child underlying performance rate
muP <- 0.8 # pop level mean of parent underlying performance rate
sigmaC <- 0.1 # pop level sd of child underlying performance rate
sigmaP <- 0.12 #pop level sd of parent underlying performance rate
rho <- 0.4 # correlation between parent and child performance rate
## Now we generate the covariance matrix
VarCovarM <- matrix(c(NA, NA, NA, NA), nrow = 2, ncol = 2)
VarCovarM[1,1] = sigmaC^2
VarCovarM[1,2] = rho * sigmaC * sigmaP
VarCovarM[2,1] = rho * sigmaC * sigmaP
VarCovarM[2,2] = sigmaP^2
vars <- mvrnorm(n = n, mu=c(muC, muP), Sigma = VarCovarM)
df <- tibble(
RateC = vars[,1],
RateP = vars[,2])
## Now we generate performance from rates
for (i in seq(n)){
temp <- tibble(
ID = i,
ScoreC = rbinom(1, size=trials, prob = df$RateC[i]),
ScoreP = rbinom(1, size=trials, prob = df$RateP[i]),
trials = trials
)
if (i==1){d <- temp} else{d<-rbind(d, temp)}
}
data <- list(
np = n,
nt = trials,
ScoreC = d$ScoreC,
ScoreP = d$ScoreP) # to be passed on to Stan
```

I then model this. Easy peasy in brms (just to make sure the setup works and I can recover the parameter values)

```
library(brms)
fC <- bf(ScoreC | trials(trials) ~ 1 + (1 | p | ID))
fP <- bf(ScoreP | trials(trials) ~ 1 + (1 | p | ID))
prior <- c(
prior(normal(0, 1), class = Intercept, resp=ScoreC),
prior(normal(0, 1), class = Intercept, resp=ScoreP),
prior(normal(0, .2), class = sd, resp=ScoreC),
prior(normal(0, .2), class = sd, resp=ScoreP)
)
m <- brm(
fC + fP + set_rescor(rescor=FALSE),
d,
family = binomial,
chains= 2,
cores= 2,
prior = prior,
sample_prior = T,
backend = "cmdstanr"
)
```

A bit less straightforward directly in Stan:

```
stan_file <- write_stan_file("
// The input data is a vector 'y' of length 'N'.
data {
int<lower=0> np; // n of participants
int<lower=0> nt; // n of trials
int ScoreC[np]; // vector of children scores
int ScoreP[np]; // vector of parents scores
}
// The parameters accepted by the model. Our model
// accepts two parameters 'mu' and 'sigma'.
parameters {
real mu[2];
real<lower=0> sigma[2];
real<lower=0,upper=1> Rate[2]; // rate of success for each person
real<lower=0,upper=1> p[np,2]; // rate of success for each person
real<lower=-1, upper=1> rho;
}
transformed parameters {
cov_matrix[2] T;
// Reparameterization
T[1,1] = square(sigma[1]);
T[1,2] = rho * sigma[1] * sigma[2];
T[2,1] = rho * sigma[1] * sigma[2];
T[2,2] = square(sigma[2]);
}
model {
// Priors
mu ~ normal(1, 1);
sigma ~ normal(0, 2);
rho ~ normal(0, 0.3);
// Data
p ~ multi_normal(mu, T);
for (i in 1:np)
ScoreC[i, 1] ~ binomial(nt, p[i,1]);
ScoreP[i, 2] ~ binomial(nt, p[i,2]);
}
")
```

Here I get the first issue:

p ~ multi_normal(mu, T);

is apparently misspecified:

Ill-typed arguments to ‘~’ statement. No distribution ‘multi_normal’ was found with the correct signature.

The second issue is how do I extend it to include a regression model generating rho. Let’s imagine that the correlation between parent and child cognitive function is (also) a function of how much time they spent together.

I’m thinking something like

```
for (n in 1:np)
rho[n] = ((inv_logit(InterceptRho + SlopeTimeRho * Time[n])*2)-1);
```

in the transformed parameters block. But any suggestion is welcome.

Generating data for this latter scenario:

```
library(MASS)
library(brms)
library(tidyverse)
n <- 200 # n of participants
trials = 30 # n of trials
muC <- 0.4 # pop level mean of child underlying performance rate
muP <- 0.8 # pop level mean of parent underlying performance rate
sigmaC <- 0.1 # pop level sd of child underlying performance rate
sigmaP <- 0.12 #pop level sd of parent underlying performance rate
## a 0-4 integer parameter representing some abstract way of measuring time spent together
Time <- round(runif(n, 0, 4), 0)
## Intercept and slope of the relation between time and rho
InterceptRho <- 0.2
SlopeRho <- 0.3
for (i in seq(n)){
rho <- inv_logit_scaled(InterceptRho + SlopeRho * Time[i], -1, 1)
VarCovarM <- matrix(c(NA, NA, NA, NA), nrow = 2, ncol = 2)
VarCovarM[1,1] = sigmaC^2
VarCovarM[1,2] = rho * sigmaC * sigmaP
VarCovarM[2,1] = rho * sigmaC * sigmaP
VarCovarM[2,2] = sigmaP^2
vars <- mvrnorm(n = 1, mu=c(muC, muP), Sigma = VarCovarM)
temp <- tibble(
RateC = vars[1],
RateP = vars[2],
rho,
Time = Time[i])
if (i==1){df <- temp} else{df <- rbind(df, temp)}
}
for (i in seq(n)){
temp <- tibble(
ID = i,
scoreC = rbinom(1, size=trials, prob = df$RateC[i]),
scoreP = rbinom(1, size=trials, prob = df$RateP[i]),
Time = Time[i]
)
if (i==1){d <- temp} else{d<-rbind(d, temp)}
}
```