Aggregated predictor in a simple linear regression

Hi everybody!

In a simple linear regression, say I have a manifest outcome Y and a predictor X that is aggregated (i.e., the values of X are means of distributions of true parameters with known variances).

Would it be a sound modeling strategy to do the following:

data {
  vector[N] y;
  vector[N] X_means;
  vector<lower=0.0>[N] X_sds;

parameters {
  vector[N] X_i;
  real<lower=0.0> sd_residual;
  real beta_0;
  real beta_1;

model {
  X_i ~ normal(X_means, X_sds);
  y ~ normal(beta_0 + beta_1 * X_i, sd_residual);
  // Priors...

I couldn’t really figure out which part of the modeling world this belongs to, it is kind of a reverse-latent variable modeling, since we have the means and errors and are interested in the manifest values (that we don’t know). On the other hand, it’s not really mixed modeling either, at least I can’t manage to make it fit into the format.

I have to say I am a bit unsure if this is even permissable, because if you reparameterize X_i and plug it into the regression equation, it reads:

Y_i = \beta_0 + \beta_1(\bar{X}_i + \sigma_{X_i}X_i) + \varepsilon_i = \beta_0 + \beta_1\bar{X}_i + \beta_1\sigma_{X_i}X_i + \varepsilon_i

with X_i\sim N(0,1). Now how can we disentangle the terms \beta_1\sigma_{X_i}X_i and \varepsilon_i?

If it is actually a sensible model, I would be happy if you could tell me if this has a name and how I can find further information.

Thank you!

1 Like

This is valid (assuming the uncertainties in the X_i are independent and approximately Gaussian), and is often referred to as a “measurement error model”.

If \sigma_{X_i} is too large, then you won’t get identification. But it should be pretty straightforward to convince yourself that there won’t be identification problems if \sigma_{X_i} is sufficiently small. In the limit that \sigma_{X_i} approaches zero, this is just an ordinary linear regression.