This vignette explains how to estimate ANalysis Of VAriance (ANOVA) models
using the stan_aov
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. We also demonstrate that Step 2 is not entirely automatic because it is sometimes necessary to specify some additional tuning parameters in order to obtain optimally efficient results.
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.
An ANOVA model can be considered a special case of the above linear regression model where each of the $K$ predictors in $\mathbf{x}$ is a dummy variable indicating membership in a group. An equivalent linear predictor can be written as $\mu_j = \alpha + \alpha_j$, which expresses the conditional expectation of the outcome in the $j$-th group as the sum of a common mean, $\alpha$, and a group-specific deviation from the common mean, $\alpha_j$.
If we view the ANOVA model as a special case of a linear regression model with
only dummy variables as predictors, then the model could be estimated using the
prior specification in the stan_lm
function. In fact, this is exactly how the
stan_aov
function is coded. These functions require the user to specify a
value for the prior location (by default the mode) of the $R^2$, the proportion
of variance in the outcome attributable to the predictors under a linear model.
This prior specification is appealing in an ANOVA context because of the
fundamental identity
$$SS_{\mbox{total}} = SS_{\mbox{model}} + SS_{\mbox{error}},$$
where $SS$ stands for sum-of-squares. If we normalize this identity, we obtain
the tautology $1 = R^2 + \left(1 - R^2\right)$ but it is reasonable to expect a
researcher to have a plausible guess for $R^2$ before conducting an ANOVA. See
the vignette for the stan_lm
function (regularized linear models) for more
information on this approach.
If we view the ANOVA model as a difference of means, then the model could be
estimated using the prior specification in the stan_lmer
function. In the
syntax popularized by the lme4 package, y ~ 1 + (1|group)
represents a
likelihood where $\mu_j = \alpha + \alpha_j$ and $\alpha_j$ is normally
distributed across the $J$ groups with mean zero and some unknown standard
deviation. The stan_lmer
function specifies that this standard deviation has
a Gamma prior with, by default, both its shape and scale parameters equal to
$1$, which is just an standard exponential distribution. However, the shape
and scale parameters can be specified as other positive values. This approach
also requires specifying a prior distribution on the standard deviation of the
errors that is independent of the prior distribution for each $\alpha_j$. See
the vignette for the stan_glmer
function (lme4-style models using
rstanarm) for more information on this approach.
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 4.3.1 analyzes an experiment where rats were subjected to different diets in order to see how much weight they gained. The experimental factors were whether their diet had low or high protein and whether the protein was derived from beef or cereal. Before seeing the data, one might expect that a moderate proportion of the variance in weight gain might be attributed to protein (source) in the diet. The frequentist ANOVA estimates can be obtained:
data("weightgain", package = "HSAUR3") coef(aov(weightgain ~ source * type, data = weightgain))
To obtain Bayesian estimates we can prepend stan_
to aov
and specify the
prior location of the $R^2$ as well as optionally the number of cores that the
computer is allowed to utilize:
library(rstanarm) post1 <- stan_aov(weightgain ~ source * type, data = weightgain, prior = R2(location = 0.5), adapt_delta = 0.999, seed = 12345) post1
print(post1)
Here we have specified adapt_delta = 0.999
to decrease the stepsize and
largely prevent divergent transitions. See the Troubleshooting section in the
main rstanarm vignette for more details about adapt_delta
. Also, our prior
guess that $R^2 = 0.5$ was overly optimistic. However, the frequentist estimates presumably overfit the data even more.
Alternatively, we could prepend stan_
to lmer
and specify the corresponding
priors
post2 <- stan_lmer(weightgain ~ 1 + (1|source) + (1|type) + (1|source:type), data = weightgain, prior_intercept = cauchy(), prior_covariance = decov(shape = 2, scale = 2), adapt_delta = 0.999, seed = 12345)
Comparing these two models using the loo
function in the loo package
reveals a negligible preference for the first approach that is almost entirely
due to its having a smaller number of effective parameters as a result of the
more regularizing priors. However, the difference is so small that it may seem
advantageous to present the second results which are more in line with a
mainstream Bayesian approach to an ANOVA model.
This vignette has compared and contrasted two approaches to estimating an ANOVA model with Bayesian techniques using the rstanarm package. They both have the same likelihood, so the (small in this case) differences in the results are attributable to differences in the priors.
The stan_aov
approach just calls stan_lm
and thus only requires a prior
location on the $R^2$ of the linear model. This seems rather easy to do in
the context of an ANOVA decomposition of the total sum-of-squares in the
outcome into model sum-of-squares and residual sum-of-squares.
The stan_lmer
approach just calls stan_glm
but specifies a normal prior
with mean zero for the deviations from $\alpha$ across groups. This is more
in line with what most Bayesians would do naturally --- particularly if the
factors were considered "random" --- but also requires a prior for $\alpha$,
$\sigma$, and the standard deviation of the normal prior on the group-level
intercepts. The stan_lmer
approach is very flexible and might be more
appropriate for more complicated experimental designs.
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