Details about the QR argument to rstanarm's modeling functions.

## Details

The QR argument is a logical scalar defaulting to FALSE, but if TRUE applies a scaled qr decomposition to the design matrix, $$X = Q^\ast R^\ast$$. If autoscale = TRUE (the default) in the call to the function passed to the prior argument, then $$Q^\ast = Q \sqrt{n-1}$$ and $$R^\ast = \frac{1}{\sqrt{n-1}} R$$. When autoscale = FALSE, $$R$$ is scaled such that the lower-right element of $$R^\ast$$ is $$1$$.

The coefficients relative to $$Q^\ast$$ are obtained and then premultiplied by the inverse of $$R^{\ast}$$ to obtain coefficients relative to the original predictors, $$X$$. Thus, when autoscale = FALSE, the coefficient on the last column of $$X$$ is the same as the coefficient on the last column of $$Q^\ast$$.

These transformations do not change the likelihood of the data but are recommended for computational reasons when there are multiple predictors. Importantly, while the columns of $$X$$ are almost generally correlated, the columns of $$Q^\ast$$ are uncorrelated by design, which often makes sampling from the posterior easier. However, because when QR is TRUE the prior argument applies to the coefficients relative to $$Q^\ast$$ (and those are not very interpretable), setting QR=TRUE is only recommended if you do not have an informative prior for the regression coefficients or if the only informative prior is on the last regression coefficient (in which case you should set autoscale = FALSE when specifying such priors).

For more details see the Stan case study The QR Decomposition For Regression Models at http://mc-stan.org/users/documentation/case-studies/qr_regression.html.

Stan Development Team. Stan Modeling Language Users Guide and Reference Manual. https://mc-stan.org/users/documentation/.