This vignette explains how to estimate linear models using the `stan_lm`

function in the **rstanarm** package.

Steps 3 and 4 are covered in more depth by the vignette entitled "How to Use the
**rstanarm** Package". This vignette focuses on Step 1 when the likelihood is
the product of independent normal distributions.

The goal of the **rstanarm** package is to make Bayesian estimation of common
regression models routine. That goal can be partially accomplished by providing
interfaces that are similar to the popular formula-based interfaces to
frequentist estimators of those regression models. But fully accomplishing that
goal sometimes entails utilizing priors that applied researchers are unaware
that they prefer. These priors are intended to work well for any data that a
user might pass to the interface that was generated according to the assumptions
of the likelihood function.

It is important to distinguish between priors that are easy for applied
researchers to *specify* and priors that are easy for applied researchers to
*conceptualize*. The prior described below emphasizes the former but we outline
its derivation so that applied researchers may feel more comfortable utilizing
it.

The likelihood for one observation under a linear model can be written as a conditionally normal PDF $$\frac{1}{\sigma_{\epsilon} \sqrt{2 \pi}} e^{-\frac{1}{2} \left(\frac{y - \mu}{\sigma_{\epsilon}}\right)^2},$$ where $\mu = \alpha + \mathbf{x}^\top \boldsymbol{\beta}$ is a linear predictor and $\sigma_{\epsilon}$ is the standard deviation of the error in predicting the outcome, $y$. The likelihood of the entire sample is the product of $N$ individual likelihood contributions.

It is well-known that the likelihood of the sample is maximized when the
sum-of-squared residuals is minimized, which occurs when
$$ \widehat{\boldsymbol{\beta}} = \left(\mathbf{X}^\top \mathbf{X}\right)^{-1}
\mathbf{X}^\top \mathbf{y}, $$
$$ \widehat{\alpha} = \overline{y} - \overline{\mathbf{x}}^\top
\widehat{\boldsymbol{\beta}}, $$
$$ \widehat{\sigma}*{\epsilon}^2 =
\frac{\left(\mathbf{y} - \widehat{\alpha} - \mathbf{X} \widehat{
\boldsymbol{\beta}}\right)^\top
\left(\mathbf{y} - \widehat{\alpha} - \mathbf{X} \widehat{
\boldsymbol{\beta}}\right)}{N},$$
where $\overline{\mathbf{x}}$ is a vector that contains the sample means of the
$K$ predictors, $\mathbf{X}$ is a $N \times K$ matrix of _centered* predictors,
$\mathbf{y}$ is a $N$-vector of outcomes and $\overline{y}$ is the sample mean
of the outcome.

The `lm`

function in R actually performs a QR decomposition of the design
matrix, $\mathbf{X} = \mathbf{Q}\mathbf{R}$, where
$\mathbf{Q}^\top \mathbf{Q} = \mathbf{I}$ and $\mathbf{R}$ is upper triangular.
Thus, the OLS solution for the coefficients can be written as
$\left(\mathbf{X}^\top \mathbf{X}\right)^{-1} \mathbf{X}^\top \mathbf{y} =
\mathbf{R}^{-1} \mathbf{Q}^\top \mathbf{y}$.
The `lm`

function utilizes the QR decomposition for numeric stability reasons,
but the QR decomposition is also useful for thinking about priors in a Bayesian
version of the linear model. In addition, writing the likelihood in terms of
$\mathbf{Q}$ allows it to be evaluated in a very efficient manner in Stan.

The key innovation in the `stan_lm`

function in the **rstanarm** package is the
prior for the parameters in the QR-reparameterized model. To understand
this prior, think about the equations that characterize the maximum likelihood
solutions before observing the data on $\mathbf{X}$ and especially $\mathbf{y}$.

What would the prior distribution of $\boldsymbol{\theta} = \mathbf{Q}^\top \mathbf{y}$ be? We can write its $k$-th element as $\theta_k = \rho_k \sigma_Y \sqrt{N - 1}$ where $\rho_k$ is the correlation between the $k$th column of $\mathbf{Q}$ and the outcome, $\sigma_Y$ is the standard deviation of the outcome, and $\frac{1}{\sqrt{N-1}}$ is the standard deviation of the $k$ column of $\mathbf{Q}$. Then let $\boldsymbol{\rho} = \sqrt{R^2}\mathbf{u}$ where $\mathbf{u}$ is a unit vector that is uniformly distributed on the surface of a hypersphere. Consequently, $R^2 = \boldsymbol{\rho}^\top \boldsymbol{\rho}$ is the familiar coefficient of determination for the linear model.

An uninformative prior on $R^2$ would be standard uniform, which is a special case of a Beta distribution with both shape parameters equal to $1$. A non-uniform prior on $R^2$ is somewhat analogous to ridge regression, which is popular in data mining and produces better out-of-sample predictions than least squares because it penalizes $\boldsymbol{\beta}^\top \boldsymbol{\beta}$, usually after standardizing the predictors. An informative prior on $R^2$ effectively penalizes $\boldsymbol{\rho}\top \boldsymbol{\rho}$, which encourages $\boldsymbol{\beta} = \mathbf{R}^{-1} \boldsymbol{\theta}$ to be closer to the origin.

Lewandowski, Kurowicka, and Joe (2009) derives a distribution for a correlation
matrix that depends on a single shape parameter $\eta > 0$, which implies the
variance of one variable given the remaining $K$ variables is
$\mathrm{Beta}\left(\eta,\frac{K}{2}\right)$. Thus, the $R^2$ is distributed $\mathrm{Beta}\left(\frac{K}{2},\eta\right)$ and any prior information about
the location of $R^2$ can be used to choose a value of the hyperparameter
$\eta$. The `R2(location, what)`

function in the **rstanarm** package supports
four ways of choosing $\eta$:

`what = "mode"`

and`location`

is some prior mode on the $\left(0,1\right)$ interval. This is the default but since the mode of a $\mathrm{Beta}\left(\frac{K}{2},\eta\right)$ distribution is $\frac{\frac{K}{2} - 1}{\frac{K}{2} + \eta - 2}$ the mode only exists if $K > 2$. If $K \leq 2$, then the user must specify something else for`what`

.`what = "mean"`

and`location`

is some prior mean on the $\left(0,1\right)$ interval, where the mean of a $\mathrm{Beta}\left(\frac{K}{2},\eta\right)$ distribution is $\frac{\frac{K}{2}}{\frac{K}{2} + \eta}$.`what = "median"`

and`location`

is some prior median on the $\left(0,1\right)$ interval. The median of a $\mathrm{Beta}\left(\frac{K}{2},\eta\right)$ distribution is not available in closed form but if $K > 2$ it is approximately equal to $\frac{\frac{K}{2} - \frac{1}{3}}{\frac{K}{2} + \eta - \frac{2}{3}}$. Regardless of whether $K > 2$, the`R2`

function can numerically solve for the value of $\eta$ that is consistent with a given prior median utilizing the quantile function.`what = "log"`

and`location`

is some (negative) prior value for $\mathbb{E} \ln R^2 = \psi\left(\frac{K}{2}\right)- \psi\left(\frac{K}{2}+\eta\right)$, where $\psi\left(\cdot\right)$ is the`digamma`

function. Again, given a prior value for the left-hand side it is easy to numerically solve for the corresponding value of $\eta$.

There is no default value for the `location`

argument of the `R2`

function.
This is an *informative* prior on $R^2$, which must be chosen by the user
in light of the research project. However, specifying `location = 0.5`

is
often safe, in which case $\eta = \frac{K}{2}$ regardless of whether `what`

is `"mode"`

, `"mean"`

, or `"median"`

. In addition, it is possible to specify
`NULL`

, in which case a standard uniform on $R^2$ is utilized.

We set $\sigma_y = \omega s_y$ where $s_y$ is the sample standard deviation of the outcome and $\omega > 0$ is an unknown scale parameter to be estimated. The only prior for $\omega$ that does not contravene Bayes' theorem in this situation is Jeffreys prior, $f\left(\omega\right) \propto \frac{1}{\omega}$, which is proportional to a Jeffreys prior on the unknown $\sigma_y$, $f\left(\sigma_y\right) \propto \frac{1}{\sigma_y} = \frac{1}{\omega \widehat{\sigma}_y} \propto \frac{1}{\omega}$. This parameterization and prior makes it easy for Stan to work with any continuous outcome variable, no matter what its units of measurement are.

It would seem that we need a prior for $\sigma_{\epsilon}$, but our prior beliefs about $\sigma_{\epsilon} = \omega s_y \sqrt{1 - R^2}$ are already implied by our prior beliefs about $\omega$ and $R^2$. That only leaves a prior for $\alpha = \overline{y} - \overline{\mathbf{x}}^\top \mathbf{R}^{-1} \boldsymbol{\theta}$. The default choice is an improper uniform prior, but a normal prior can also be specified such as one with mean zero and standard deviation $\frac{\sigma_y}{\sqrt{N}}$.

The previous sections imply a posterior distribution for $\omega$, $\alpha$, $\mathbf{u}$, and $R^2$. The parameters of interest can then be recovered as generated quantities:

- $\sigma_y = \omega s_y$
- $\sigma_{\epsilon} = \sigma_y \sqrt{1 - R^2}$
- $\boldsymbol{\beta} = \mathbf{R}^{-1} \mathbf{u} \sigma_y \sqrt{R^2 \left(N-1\right)}$

The implementation actually utilizes an improper uniform prior on
$\ln \omega$. Consequently, if $\ln \omega = 0$, then the marginal standard
deviation of the outcome *implied by the model* is the same as the sample
standard deviation of the outcome. If $\ln \omega > 0$, then the marginal
standard deviation of the outcome implied by the model exceeds the sample
standard deviation, so the model overfits the data. If $\ln \omega < 0$,
then the marginal standard deviation of the outcome implied by the model is
less than the sample standard deviation, so the model *underfits* the data or
that the data-generating process is nonlinear. Given the regularizing nature
of the prior on $R^2$, a minor underfit would be considered ideal if the goal
is to obtain good out-of-sample predictions. If the model badly underfits or
overfits the data, then you may want to reconsider the model.

We will utilize an example from the **HSAUR3** package by Brian S. Everitt and
Torsten Hothorn, which is used in their 2014 book
*A Handbook of Statistical Analyses Using R (3rd Edition)* (Chapman & Hall /
CRC). This book is frequentist in nature and we will show how to obtain the
corresponding Bayesian results.

The model in section 5.3.1 analyzes an experiment where clouds were seeded
with different amounts of silver iodide to see if there was increased rainfall.
This effect could vary according to covariates, which (except for `time`

) are
interacted with the treatment variable. Most people would probably be skeptical
that cloud hacking could explain very much of the variation in rainfall and
thus the prior mode of the $R^2$ would probably be fairly small.

The frequentist estimator of this model can be replicated by executing

data("clouds", package = "HSAUR3") ols <- lm(rainfall ~ seeding * (sne + cloudcover + prewetness + echomotion) + time, data = clouds) round(coef(ols), 3)

Note that we have *not* looked at the estimated $R^2$ or $\sigma$ for the
ordinary least squares model. We can estimate a Bayesian version of this model
by prepending `stan_`

to the `lm`

call, specifying a prior mode for $R^2$, and
optionally specifying how many cores the computer may utilize:

library(rstanarm) post <- stan_lm( rainfall ~ seeding * (sne + cloudcover + prewetness + echomotion) + time, data = clouds, prior = R2(location = 0.2), seed = 12345 ) post

```
print(post)
```

In this case, the "Bayesian point estimates", which are represented by the
posterior medians, appear quite different from the ordinary least squares
estimates. However, the `log-fit_ratio`

(i.e. $\ln \omega$) is quite small,
indicating that the model only slightly overfits the data when the prior
derived above is utilized. Thus, it would be safe to conclude that the ordinary
least squares estimator considerably overfits the data since there are only
$24$ observations to estimate $12$ parameters with and no prior information
on the parameters.

Also, it is not obvious what the estimated average treatment effect is since the
treatment variable, `seeding`

, is interacted with four other correlated
predictors. However, it is easy to estimate or visualize the average treatment
effect (ATE) using **rstanarm**'s `posterior_predict`

function.

clouds_cf <- clouds clouds_cf$seeding[] <- "yes" y1_rep <- posterior_predict(post, newdata = clouds_cf) clouds_cf$seeding[] <- "no" y0_rep <- posterior_predict(post, newdata = clouds_cf) qplot(x = c(y1_rep - y0_rep), geom = "histogram", xlab = "Estimated ATE")

As can be seen, the treatment effect is not estimated precisely and is as almost as likely to be negative as it is to be positive.

The prior derived above works well in many situations and is quite simple
to *use* since it only requires the user to specify the prior location of
the $R^2$. Nevertheless, the implications of the prior are somewhat difficult
to *conceptualize*. Thus, it is perhaps worthwhile to compare to another
estimator of a linear model that simply puts independent Cauchy priors on the
regression coefficients. This simpler approach can be executed by calling the
`stan_glm`

function with `family = gaussian()`

and specifying the priors:

simple <- stan_glm( rainfall ~ seeding * (sne + cloudcover + prewetness + echomotion) + time, data = clouds, family = gaussian(), prior = cauchy(), prior_intercept = cauchy(), seed = 12345 )

We can compare the two approaches using an approximation to Leave-One-Out
(LOO) cross-validation, which is implemented by the `loo`

function in the
**loo** package.

(loo_post <- loo(post)) loo_compare(loo_post, loo(simple))

The results indicate that the first approach is expected to produce better out-of-sample predictions but the Warning messages are at least as important. Many of the estimated shape parameters for the Generalized Pareto distribution are above $0.5$ in the model with Cauchy priors, which indicates that these estimates are only going to converge slowly to the true out-of-sample deviance measures. Thus, with only $24$ observations, they should not be considered reliable. The more complicated prior derived above is stronger --- as evidenced by the fact that the effective number of parameters is about half of that in the simpler approach and $12$ for the maximum likelihood estimator --- and only has a few of the $24$ Pareto shape estimates in the "danger zone". We might want to reexamine these observations

plot(loo_post, label_points = TRUE)

because the posterior is sensitive to them but, overall, the results seem tolerable.

In general, we would expect the joint prior derived here to work better when there are many predictors relative to the number of observations. Placing independent, heavy-tailed priors on the coefficients neither reflects the beliefs of the researcher nor conveys enough information to stabilize all the computations.

This vignette has discussed the prior distribution utilized in the `stan_lm`

function, which has the same likelihood and a similar syntax as the `lm`

function in R but adds the ability to expression prior beliefs about the
location of the $R^2$, which is the familiar proportion of variance in the
outcome variable that is attributable to the predictors under a linear model.
Since the $R^2$ is a well-understood bounded scalar, it is easy to specify
prior information about it, whereas other Bayesian approaches require the
researcher to specify a joint prior distribution for the regression
coefficients (and the intercept and error variance).

However, most researchers have little inclination to specify all these prior
distributions thoughtfully and take a short-cut by specifying one prior
distribution that is taken to apply to all the regression coefficients as if
they were independent of each other (and the intercept and error variance).
This short-cut is available in the `stan_glm`

function and is described in more
detail in other **rstanarm** vignettes for Generalized Linear Models (GLMs),
which can be found by navigating up one level.

We are optimistic that this prior on the $R^2$ will greatly help in accomplishing
our goal for **rstanarm** of making Bayesian estimation of regression models
routine. The same approach is used to specify a prior in ANOVA models (see
`stan_aov`

) and proportional-odds models for ordinal outcomes (see `stan_polr`

).

Finally, the `stan_biglm`

function can be used when the design matrix is too
large for the `qr`

function to process. The `stan_biglm`

function inputs the
output of the `biglm`

function in the **biglm** package, which utilizes an
incremental QR decomposition that does not require the entire dataset to be
loaded into memory simultaneously. However, the `biglm`

function needs to be
called in a particular way in order to work with `stan_biglm`

. In particular,
The means of the columns of the design matrix, the sample mean of the outcome, and
the sample standard deviation of the outcome all need to be passed to the `stan_biglm`

function, as well as a flag indicating whether the model really does include an
intercept. Also, the number of columns of the design matrix currently cannot
exceed the number of rows. Although `stan_biglm`

should run fairly quickly and
without much memory, the resulting object is a fairly plain `stanfit`

object
rather than an enhanced `stanreg`

object like that produced by `stan_lm`

. Many
of the enhanced capabilities of a `stanreg`

object depend on being able to
access the full design matrix, so doing posterior predictions, posterior checks,
etc. with the output of `stan_biglm`

would require some custom R code.

Lewandowski, D., Kurowicka D., and Joe, H. (2009).
Generating random correlation matrices based on vines and extended onion method.
*Journal of Multivariate Analysis*. **100**(9), 1989--2001.

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