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 \(1\times 1\), \(\tau\) is the cross-group standard deviation in the parameters and its square is the variance (so the Gamma prior with its shape and scale directly applies to the cross-group standard deviation in the parameters).
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 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
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:
Warning in vcov.merMod(object, use.hessian = use.hessian): variance-covariance matrix computed from finite-difference Hessian is
not positive definite or contains NA values: falling back to var-cov estimated from RX
Warning in vcov.merMod(object, correlation = correlation, sigm = sig): variance-covariance matrix computed from finite-difference Hessian is
not positive definite or contains NA values: falling back to var-cov estimated from RX
Nonlinear mixed model fit by maximum likelihood ['nlmerMod']
Formula: circumference ~ SSlogis(age, Asym, xmid, scal) ~ Asym | Tree
Data: Orange
AIC BIC logLik deviance df.resid
-49.2 -41.4 29.6 -59.2 30
Scaled residuals:
Min 1Q Median 3Q Max
-1.9170 -0.5421 0.1754 0.7116 1.6820
Random effects:
Groups Name Variance Std.Dev.
Tree Asym 0.100149 0.31646
Residual 0.006151 0.07843
Number of obs: 35, groups: Tree, 5
Fixed effects:
Estimate Std. Error t value
Asym 1.9205 0.1558 12.32
xmid 7.2791 0.3444 21.14
scal 3.4807 0.2631 13.23
Correlation of Fixed Effects:
Asym xmid
xmid 0.384
scal 0.362 0.762
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
[1] "SSasymp" "SSasympOff" "SSasympOrig" "SSbiexp" "SSfol"
[6] "SSfpl" "SSgompertz" "SSlogis" "SSmicmen" "SSweibull"
To fit essentially the same model using Stan’s implementation of MCMC, we add a stan_ prefix
stan_nlmer
family: gaussian [inv_SSlogis]
formula: circumference ~ SSlogis(age, Asym, xmid, scal) ~ Asym | Tree
observations: 35
------
Median MAD_SD
Asym 1.9 0.1
xmid 7.2 0.4
scal 3.4 0.3
Auxiliary parameter(s):
Median MAD_SD
sigma 0.1 0.0
Error terms:
Groups Name Std.Dev.
Tree Asym 0.311
Residual 0.089
Num. levels: Tree 5
------
* For help interpreting the printed output see ?print.stanreg
* For info on the priors used see ?prior_summary.stanreg
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.
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):
If we were pharmacological, we could evaluate drug concentration using a first-order compartment model, such as
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.
