# "Directionality" of relations between two variables in model w measurement error

I have a simple univariate linear model with uncertainty in both variables. The variables are roughly on the same scale (centered at 0 and within a -3 to +3 range), with roughly equivalent levels of uncertainty.

I then build 4 linear models (with brms): 2 null ones and two symmetric ones:

x | se(x_se) ~ 1
y | se(y_se) ~ 1

x | se(x_se) ~ 1 + me(y, sdx=y_se)
y | se(y_se) ~ 1 + me(x, sdx=x_se)

Priors are put only on the intercept and beta: in both cases normal(0,0.3).

Now y as a function of x is credibly better than the null model (using LOOIC and stacking weights), while x as a function of y cannot really be preferred to the null model, with very different effect size.

However, if I remove the measurement errors, suddenly I get very symmetric models (same beta estimate for y ~ x and x ~ y) and neither of them is credibly better than the null model.

So what I am wondering is: where does the asymmetry come from when including measurement errors? And, is the directionality of the results at all interpretable? I am not talking about causality, obviously, but I’m still curious about whether the asymmetry has any interpretable meaning.

se() does not the same thing as me() as the former still predicts the manifest variable. If you want to predict latent responses with measurement error, go for `mi(y_se)`.

of course. Thanks, now all my simulations make sense :-)