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.