---
title: "Estimating Generalized (Non-)Linear Models with Group-Specific Terms with rstanarm"
author: "Jonah Gabry and Ben Goodrich"
date: "`r Sys.Date()`"
output: 
  html_vignette:
    toc: yes
---
<!--
%\VignetteEngine{knitr::rmarkdown}
%\VignetteIndexEntry{stan_glmer: GLMs with Group-Specific Terms}
-->
```{r, child="children/SETTINGS-knitr.txt"}
```
```{r, child="children/SETTINGS-gg.txt"}
```

# Introduction

This vignette explains how to use the `stan_lmer`, `stan_glmer`, `stan_nlmer`, 
and `stan_gamm4` functions in the __rstanarm__ package to estimate linear and
generalized (non-)linear models with parameters that may vary across groups.
Before continuing, we recommend reading the vignettes (navigate up one level) for 
the various ways to use the `stan_glm` function. The _Hierarchical Partial Pooling_ 
vignette also has examples of both `stan_glm` and `stan_glmer`.

# GLMs with group-specific terms

Models with this structure are refered to by many names: multilevel models,
(generalized) linear mixed (effects) models (GLMM), hierarchical (generalized)
linear models, etc. In the simplest case, the model for an outcome can be written as
$$\mathbf{y} = \alpha + \mathbf{X} \boldsymbol{\beta} + \mathbf{Z} \mathbf{b} + \boldsymbol{\epsilon},$$
where $\mathbf{X}$ is a matrix predictors that is analogous to that in Generalized
Linear Models and $\mathbf{Z}$ is a matrix that encodes deviations in the 
predictors across specified groups.

The terminology for the unknowns in the model is diverse. To frequentists, the
error term consists of $\mathbf{Z}\mathbf{b} + \boldsymbol{\epsilon}$ and the
observations within each group are _not_ independent conditional on $\mathbf{X}$
alone. Since, $\mathbf{b}$ is considered part of the random error
term, frequentists allow themselves to make distributional assumptions about
$\mathbf{b}$, invariably that it is distributed multivariate normal with mean
vector zero and structured covariance matrix $\boldsymbol{\Sigma}$. If $\epsilon_i$
is also distributed (univariate) normal with mean zero and standard deviation
$\sigma$, then $\mathbf{b}$ can be integrated out, which implies
$$\mathbf{y} \thicksim \mathcal{N}\left(\alpha + \mathbf{X}\boldsymbol{\beta}, \sigma^2 \mathbf{I}+\mathbf{Z}^\top \boldsymbol{\Sigma} \mathbf{Z} \right),$$
and it is possible to maximize this likelihood function by choosing proposals 
for the parameters $\alpha$, $\boldsymbol{\beta}$, and (the free elements of) $\boldsymbol{\Sigma}$.

Consequently, frequentists refer to $\mathbf{b}$ as the _random effects_ because
they capture the random deviation in the effects of predictors from one group to
the next. In contradistinction, $\alpha$ and $\boldsymbol{\beta}$ are referred to
as _fixed effects_ because they are the same for all groups. Moreover, $\alpha$ 
and $\boldsymbol{\beta}$ persist in the model in hypothetical replications of the
analysis that draw the members of the groups afresh every time, whereas $\mathbf{b}$
would differ from one replication to the next. Consequently, $\mathbf{b}$ is not
a "parameter" to be estimated because parameters are unknown constants that are
fixed in repeated sampling.

Bayesians condition on the data in-hand without reference to repeated sampling
and describe their _beliefs_ about the unknowns with prior distributions before
observing the data. Thus, the likelihood in a simple hierarchical model in
__rstarnarm__ is
$$\mathbf{y} \thicksim \mathcal{N}\left(\alpha + \mathbf{X}\boldsymbol{\beta} + \mathbf{Z}\mathbf{b}, \sigma^2 \mathbf{I}\right)$$
and the observations are independent conditional on $\mathbf{X}$ and $\mathbf{Z}$.
In this formulation, there are

* intercept(s) and coefficients that are _common across groups_
* deviations in the intercept(s) and / or coefficients that _vary across groups_

Bayesians are compelled to state their prior beliefs about all unknowns and the
usual assumption (which is maintained in __rstanarm__) is that 
$\mathbf{b} \thicksim \mathcal{N}\left(\mathbf{0},\boldsymbol{\Sigma}\right),$
but it is then necessary to state prior beliefs about $\boldsymbol{\Sigma}$, in
addition to $\alpha$, $\boldsymbol{\beta}$, and $\sigma$.

One of the many challenges of fitting models to data comprising multiple 
groupings is confronting the tradeoff between validity and precision. An analysis 
that disregards between-group heterogeneity can yield parameter estimates that 
are wrong if there is between-group heterogeneity but would be relatively precise 
if there actually were no between-group heterogeneity. Group-by-group analyses, on 
the other hand, are valid but produces estimates that are relatively imprecise. 
While complete pooling or no pooling of data across groups is sometimes called for, 
models that ignore the grouping structures in the data tend to underfit or overfit 
(Gelman et al.,2013). Hierarchical modeling provides a compromise by allowing parameters
to vary by group at lower levels of the hierarchy while estimating common
parameters at higher levels. Inference for each group-level parameter is
informed not only by the group-specific information contained in the data but
also by the data for other groups as well. This is commonly referred to as
_borrowing strength_ or _shrinkage_.

In __rstanarm__, these models can be estimated using the `stan_lmer` and 
`stan_glmer` functions, which are similar in syntax to the `lmer` and `glmer` 
functions in the __lme4__ package. However, rather than performing (restricted) 
maximum likelihood (RE)ML estimation, Bayesian estimation is performed via MCMC.
The Bayesian model adds independent prior distributions on the regression 
coefficients (in the same way as `stan_glm`) as well as priors on the terms of a
decomposition of the covariance matrices of the group-specific parameters. These
priors are discussed in greater detail below.

# Priors on covariance matrices

In this section we discuss a flexible family of prior distributions for the
unknown covariance matrices of the group-specific coefficients.


### Overview

For each group, we assume the vector of varying slopes and intercepts is a 
zero-mean random vector following a multivariate Gaussian distribution with an 
unknown covariance matrix to be estimated. Unfortunately, 
expressing prior information about a covariance matrix is not intuitive and can 
also be computationally challenging. When the covariance matrix is not $1\times 1$, 
it is often both much more intuitive and 
efficient to work instead with the __correlation__ matrix and variances. When the
covariance matrix is $1\times 1$, we still denote it as $\boldsymbol{\Sigma}$ but
most of the details in this section do not apply.

The variances are in turn decomposed into the product of
a simplex vector (probability vector) and the trace of the implied covariance matrix,
which is defined as the sum of its diagonal elements. Finally, this trace is set 
equal to the product of the order of the matrix and the
square of a scale parameter. This implied prior on a covariance matrix is represented 
by the `decov` (short for decomposition of covariance) function in __rstanarm__.
  
### Details

Using the decomposition described above, the prior used for a correlation
matrix $\Omega$ is called the LKJ distribution and has a probability density function
proportional to the determinant of the correlation matrix raised to a power of
$\zeta$ minus one:

$$ f(\Omega | \zeta) \propto \text{det}(\Omega)^{\zeta - 1}, \quad \zeta > 0. $$

The shape of this prior depends on the value of the regularization parameter, $\zeta$ 
in the following ways:

* If $\zeta = 1$ (the default), then the LKJ prior is jointly uniform over all 
correlation matrices of the same dimension as $\Omega$. 
* If $\zeta > 1$, then the mode of the distribution is the identity matrix. The
larger the value of $\zeta$ the more sharply peaked the density is at the
identity matrix.
* If $0 < \zeta < 1$, then the density has a trough at the identity matrix.

The $J \times J$ covariance matrix $\Sigma$ of a random vector $\boldsymbol{\theta} =
(\theta_1, \dots, \theta_J)$ has diagonal entries ${\Sigma}_{jj} = \sigma^2_j =
\text{var}(\theta_j)$. Therefore, the trace of the covariance matrix is equal to the 
sum of the variances. We set the trace equal to the product of the order of the 
covariance matrix and the square of a positive scale parameter $\tau$:

$$\text{tr}(\Sigma) = \sum_{j=1}^{J} \Sigma_{jj} = J\tau^2.$$

The vector of variances is set equal to the product of a simplex vector 
$\boldsymbol{\pi}$ --- which is non-negative and sums to 1 --- and the scalar trace: 
$J \tau^2 \boldsymbol{\pi}$. Each element $\pi_j$ of $\boldsymbol{\pi}$ then 
represents the proportion of the trace (total variance) attributable to the 
corresponding variable $\theta_j$.

For the simplex vector $\boldsymbol{\pi}$ we use a symmetric Dirichlet prior,
which has a single _concentration_ parameter $\gamma > 0$:

* If $\gamma = 1$ (the default), then the prior is jointly uniform over the space
of simplex vectors with $J$ elements. 
* If $\gamma > 1$, then the prior mode corresponds to all variables having the 
same (proportion of total) variance, which can be used to ensure that the
posterior variances are not zero. As the concentration parameter approaches
infinity, this mode becomes more pronounced.
* If $0 < \gamma < 1$, then the variances are more polarized.

If all the elements of $\boldsymbol{\theta}$ were multiplied by the same number
$k$, the trace of their covariance matrix would increase by a factor of $k^2$. 
For this reason, it is sensible to use a scale-invariant prior for $\tau$. 
We choose a Gamma distribution, with shape and scale parameters both set to $1$
by default, implying a unit-exponential distribution. Users can set the shape 
hyperparameter to some value greater than one to ensure that the posterior trace
is not zero. In the case where $\boldsymbol{\Sigma}$ is $1\times 1$, this shape
parameter is the cross-group standard deviation in the parameters and its square
is the variance.

# Comparison with __lme4__

There are several advantages to estimating these models using __rstanarm__ 
rather than the __lme4__ package. There are also a few drawbacks. In
this section we briefly discuss what we find to be the two most important 
advantages as well as an important disadvantage.

### Advantage: better uncertainty estimates

While __lme4__ uses (restricted) maximum likelihood (RE)ML estimation, 
__rstanarm__ enables full Bayesian inference via MCMC to be performed. It is
well known that (RE)ML tends to underestimate uncertainties because it relies on
point estimates of hyperparameters. Full Bayes, on the other hand, propagates
the uncertainty in the hyperparameters throughout all levels of the model and 
provides more appropriate estimates of uncertainty for models that consist of a 
mix of common and group-specific parameters.

### Advantage: incorporate prior information

The `stan_glmer` and `stan_lmer` functions allow the user to specify 
prior distributions over the regression coefficients as well as any unknown 
covariance matrices. There are various reasons to specify priors, from helping 
to stabilize computation to incorporating important information into an analysis
that does not enter through the data.

### Disadvantage: speed

The benefits of full Bayesian inference (via MCMC) come with a cost. Fitting 
models with (RE)ML will tend to be much faster than fitting a similar model using 
MCMC. Speed comparable to __lme4__ can be obtained with __rstanarm__ using 
approximate Bayesian inference via the mean-field and full-rank variational algorithms 
(see `help("rstanarm-package", "rstanarm")` for details). These 
algorithms can be useful to narrow the set of candidate models in large problems, 
but MCMC should always be used for final statistical inference.

# Relationship to `glmer`

In the __lme4__ package, there is a fundamental distinction between the way that
Linear Mixed Models and Generalized Linear Mixed Models are estimated. In Linear
Mixed Models, $\mathbf{b}$ can be integrated out analytically, leaving a likelihood
function that can be maximized over proposals for the parameters. To estimate a
Linear Mixed Model, one can call the `lmer` function.

Generalized Linear Mixed Models are appropriate when the conditional mean of the 
outcome is determined by an inverse link function, 
$\boldsymbol{\mu} = g\left(\alpha + \mathbf{X} \boldsymbol{\beta} + \mathbf{Z}\mathbf{b}\right)$.
If $g\left(\cdot\right)$ is not the identity function, then it is not possible
to integrate out $\mathbf{b}$ analytically and numerical integration must be used.
To estimate a Generalized Linear Mixed Model, one can call the `glmer` function
and specify the `family` argument.

In the __rstanarm__ package, there is no such fundamental distinction; in fact
`stan_lmer` simply calls `stan_glmer` with `family = gaussian(link = "identity")`.
Bayesians do not (have to) integrate $\mathbf{b}$ out of the likelihood and if
$\mathbf{b}$ is not of interest, then the margins of its posterior distribution
can simply be ignored.

# Relationship to `gamm4`

The __rstanarm__ package includes a `stan_gamm4` function that is similar to the
`gamm4` function in the __gamm4__ package, which is in turn similar to the `gamm`
function in the __mgcv__ package. The substring `gamm` stands for Generalized
Additive Mixed Models, which differ from Generalized Additive Models (GAMs) due
to the presence of group-specific terms that can be specified with the syntax
of __lme4__. Both GAMs and GAMMs include nonlinear functions of (non-categorical)
predictors called "smooths". In the example below, so-called "thin-plate splines"
are used to model counts of roaches where we might fear that the number of 
roaches in the current period is an exponentially increasing function of the 
number of roaches in the previous period. Unlike `stan_glmer`, in `stan_gamm4`
it is necessary to specify group-specific terms as a one-sided formula that
is passed to the `random` argument as in the `lme` function in the __nlme__
package.

```{r, results = "hide"}
library(rstanarm)
data(roaches)
roaches$roach1 <- roaches$roach1 / 100
roaches$log_exposure2 <- log(roaches$exposure2)
post <- stan_gamm4(
  y ~ s(roach1) + treatment + log_exposure2,
  random = ~(1 | senior),
  data = roaches, 
  family = neg_binomial_2, 
  QR = TRUE,
  cores = 2,
  chains = 2, 
  adapt_delta = 0.99,
  seed = 12345
)
```
```{r}
plot_nonlinear(post)
```

Here we see that the relationship between past and present roaches is estimated to 
be nonlinear. For a small number of past roaches, the function is steep and then
it appears to flatten out, although we become highly uncertain about the function
in the rare cases where the number of past roaches is large.

# Relationship to `nlmer`

The `stan_gamm4` function allows designated predictors to have a nonlinear effect
on what would otherwise be called the "linear" predictor in Generalized Linear Models.
The `stan_nlmer` function is similar to the `nlmer` function in the __lme4__ package,
and essentially allows a wider range of nonlinear functions that relate the linear
predictor to the conditional expectation of a Gaussian outcome.

To estimate an example model with the `nlmer` function in the __lme4__ package, we
start by rescaling the outcome and main predictor(s) by a constant
```{r}
data("Orange", package = "datasets")
Orange$age <- Orange$age / 100
Orange$circumference <- Orange$circumference / 100
```
Although doing so has no substantive effect on the inferences obtained, it is
numerically much easier for Stan and for __lme4__ to work with variables whose
units are such that the estimated parameters tend to be single-digit numbers that
are not too close to zero. The `nlmer` function requires that the user pass 
starting values to the ironically-named self-starting non-linear function:
```{r, warning=TRUE}
startvec <- c(Asym = 2, xmid = 7.25, scal = 3.5)
library(lme4)
nm1 <- nlmer(circumference ~ SSlogis(age, Asym, xmid, scal) ~ Asym|Tree,
             data = Orange, start = startvec)
summary(nm1)
```
Note the warning messages indicating difficulty estimating the variance-covariance
matrix. Although __lme4__ has a fallback mechanism, the need to utilize it suggests
that the sample is too small to sustain the asymptotic assumptions underlying the
maximum likelihood estimator.

In the above example, we use the `SSlogis` function, which is a lot like the logistic
CDF, but with an additional `Asym` argument that need not be one and indicates what
value the function approaches for large values of the first argument. In this case,
we can interpret the asymptote as the maximum possible circumference for an orange.
However, this asymptote is allowed to vary from tree to tree using the `Asym | Tree`
syntax, which reflects an assumption that the asymptote for a randomly-selected
tree deviates from the asymptote for the population of orange trees in a Gaussian
fashion with mean zero and an unknown standard deviation.

The `nlmer` function supports user-defined non-linear functions, whereas the 
`stan_nlmer` function only supports the pre-defined non-linear functions starting with
`SS` in the __stats__ package, which are
```{r, echo = FALSE}
grep("^SS[[:lower:]]+", ls("package:stats"), value = TRUE)
```
To fit essentially the same model using Stan's implementation of MCMC, we add a
`stan_` prefix
```{r, results = "hide"}
post1 <- stan_nlmer(circumference ~ SSlogis(age, Asym, xmid, scal) ~ Asym|Tree,
                    data = Orange, cores = 2, seed = 12345, init_r = 0.5)
```
```{r}
post1
```

In `stan_nlmer`, it is not necessary to supply starting values; however, in this
case it was necessary to specify the `init_r` argument so that the randomly-chosen
starting values were not more than $0.5$ away from zero (in the unconstrained
parameter space). The default value of $2.0$ produced suboptimal results.

As can be seen, the posterior medians and estimated standard deviations in the
MCMC case are quite similar to the maximum likelihood estimates and estimated
standard errors. However, `stan_nlmer` produces uncertainty estimates
for the tree-specific deviations in the asymptote, which are considerable.
```{r}
plot(post1, regex_pars = "^[b]")
```


As can be seen, the age of the tree has a non-linear effect on the predicted
circumference of the tree (here for a out-of-sample tree):
```{r}
nd <- data.frame(age = 1:20, Tree = factor("6", levels = 1:6))
PPD <- posterior_predict(post1, newdata = nd)
PPD_df <- data.frame(age = as.factor(rep(1:20, each = nrow(PPD))),
                     circumference = c(PPD))
ggplot(PPD_df, aes(age, circumference)) + geom_boxplot()
```

If we were pharmacological, we could evaluate drug concentration using a
first-order compartment model, such as
```{r, eval = FALSE}
post3 <- stan_nlmer(conc ~ SSfol(Dose, Time, lKe, lKa, lCl) ~ 
                    (0 + lKe + lKa + lCl | Subject), data = Theoph,
                    cores = 2, seed = 12345, 
                    QR = TRUE, init_r = 0.25, adapt_delta = 0.999)
pairs(post3, regex_pars = "^l")
pairs(post3, regex_pars = "igma")
```
However, in this case the posterior distribution is bimodal Thus, you 
should always be running many chains when using Stan, especially `stan_nlmer`.

# Conclusion

There are model fitting functions in the **rstanarm** package that can do
essentially all of what can be done in the **lme4** and **gamm4** packages
--- in the sense that they can fit models with multilevel structure and / or
nonlinear relationships --- and propagate the uncertainty in the parameter
estimates to the predictions and other functions of interest. The documentation
of **lme4** and **gamm4** has various warnings that acknowledge that the 
estimated standard errors, confidence intervals, etc. are not entirely correct,
even from a frequentist perspective. 

A frequentist point estimate would also completely miss the second mode in the
last example with `stan_nlmer`. Thus, there is considerable reason to prefer
the **rstanarm** variants of these functions for regression modeling. The only
disadvantage is the execution time required to produce an answer that properly
captures the uncertainty in the estimates of complicated models such as these.

