Introduction

This vignette explains how to estimate ANalysis Of VAriance (ANOVA) models using the stan_aov function in the rstanarm package

The four steps of a Bayesian analysis are

1. Specify a joint distribution for the outcome(s) and all the unknowns, which typically takes the form of a marginal prior distribution for the unknowns multiplied by a likelihood for the outcome(s) conditional on the unknowns. This joint distribution is proportional to a posterior distribution of the unknowns conditional on the observed data
2. Draw from posterior distribution using Markov Chain Monte Carlo (MCMC).
3. Evaluate how well the model fits the data and possibly revise the model.
4. Draw from the posterior predictive distribution of the outcome(s) given interesting values of the predictors in order to visualize how a manipulation of a predictor affects (a function of) the outcome(s).

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.

Likelihood

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, \(\alpha_j\).

Priors

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.

Example

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))

         (Intercept)         sourceCereal              typeLow 
               100.0                -14.1                -20.8 
sourceCereal:typeLow 
                18.8 

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

stan_aov
 family:       gaussian [identity]
 formula:      weightgain ~ source * type
 observations: 40
 predictors:   4
------
                     Median MAD_SD
(Intercept)           98.9    4.6 
sourceCereal         -12.9    6.2 
typeLow              -18.8    6.4 
sourceCereal:typeLow  16.9    8.4 

Auxiliary parameter(s):
              Median MAD_SD
R2             0.2    0.1  
log-fit_ratio  0.0    0.1  
sigma         14.7    1.7  

ANOVA-like table:
                    Median MAD_SD
Mean Sq source      550.6  416.5 
Mean Sq type        982.0  601.2 
Mean Sq source:type 714.9  673.5 

------
* For help interpreting the printed output see ?print.stanreg
* For info on the priors used see ?prior_summary.stanreg

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.

Conclusion

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.