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