---
title: "The R Package clustGLMM - Clustering Longitudinal Data of Mixed Type"
author: "Jan Vávra, Bettina Grün, Gertraud Malsiner-Walli, Arnošt Komárek"
date: "`r Sys.Date()`"
output: 
  bookdown::html_document2:
    base_format: rmarkdown::html_vignette
bibliography: "clustGLMM.bib"
vignette: >
  %\VignetteIndexEntry{The R Package clustGLMM - Clustering Longitudinal Data of Mixed Type}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

```{r setup, include=FALSE}
knitr::opts_chunk$set(echo = FALSE, warning = FALSE, message = FALSE, fig.path = file.path(tempdir(), "figure-latex/"))
oldopt <- options(tinytex.verbose = TRUE, width = 90, digits = 3)
# options(width = 110) # does not change the template
library("kableExtra")
library("clustGLMM")

opar <- par(no.readonly = TRUE)

combine <- function(x, sep, width) {
  cs <- cumsum(nchar(x))
  remaining <- if (any(cs[-1] > width)) combine(x[c(FALSE, cs[-1] > width)], sep, width)
  c(paste(x[c(TRUE, cs[-1] <= width)], collapse= sep), remaining)
}
prettyPrint <- function(x, sep = " ", linebreak = "\n\t", width = getOption("width")) {
    x <- strsplit(x, sep)[[1]]
    paste(combine(x, sep, width), collapse = paste(sep, linebreak, collapse = ""))
}  
```

**Abstract** 

Package **clustGLMM** provides tools for Bayesian estimation of finite mixtures of multivariate generalized linear mixed models to cluster longitudinal data. A multivariate regression setup with several dependent variables of mixed type (continuous, count, binary, ordinal and nominal) is covered where random effects have a joint distribution across all dependent variables. For each cluster, different regression coefficients and random effect distributions may be assumed to characterize and differentiate them. The package provides functionality to suitably specify the model using a formula interface, perform fully Bayesian estimation based on Markov chain Monte Carlo sampling with data augmentation and assess the fitted model using post-processing tools to investigate convergence, identify cluster distributions and classify (new) observations.

# Introduction {#sec:introduction}

Clustering longitudinal multivariate data is a key task in many fields, including medicine, psychology, and the social sciences, where repeated measurements of heterogeneous types are commonly collected for individuals. These data usually consist of a combination of continuous, count, binary, ordinal, and nominal outcome variables which are predicted through explanatory variables. Modeling such data structures in a clustering context using a multivariate regression setup requires the flexibility to jointly account for individual random effects and group-specific heterogeneity.

This indicates the need for a unified, flexible, and extensible implementation for clustering mixed-type longitudinal data within a fully probabilistic framework based on a mixture model of multivariate generalized linear mixed models. Several R packages address only parts of this problem as they only allow to fit simplified versions of this model. Package **lcmm** [@Proust-Lima-etal2017] provides tools to estimate latent class mixed models for continuous and categorical outcomes, but does not cover the full range of outcome types from the generalized linear modeling framework. Package **flexmix** [@Leisch04flexmix] supports finite mixture modeling with customizable components, with the currently available model drivers not covering the non-Gaussian case of a mixed model and a multivariate approach requiring the conditional independence assumption. Packages such as **MCMCglmm** [@Hadfield10MCMCglmm] and **mixAK** [@KomarekKomarkovaJSS14mixAK] offer Bayesian inference via Markov chain Monte Carlo (MCMC) sampling for generalized linear mixed models and mixture models, but require specialized formulations and offer limited clustering functionalities for mixed outcomes.

To address these limitations, we introduce the R package **clustGLMM**, which implements a Bayesian model-based clustering framework for longitudinal data with mixed outcome types as described by @Vavra-etal2024. The package models the data using a mixture of generalized linear mixed models (GLMMs), allowing for a flexible specification of outcome distributions and random effects. Inference is performed via MCMC sampling, enabling a full posterior characterization of both, model parameters and latent cluster assignments.

Package **clustGLMM** draws on principles from general-purpose Bayesian modeling tools such as **brms** [@Burkner17brms], which provides a high-level interface to Bayesian GLMMs via Stan [@RStan] for MCMC sampling, and **coda** [@Plummer06coda] for MCMC output analysis. By focusing specifically on clustering, **clustGLMM** fills a gap between general-purpose Bayesian modeling tools and more specialized latent class or mixture modeling packages, offering a practical and extensible solution for researchers working with complex longitudinal data. In particular, a number of post-processing procedures are provided to  inspect and assess the posterior MCMC draws, but also to identify cluster distributions and classify observations to allow for a convenient use in clustering applications. 

<!-- All computations and graphics in this paper have been done with **clustGLMM** version `r packageVersion("clustGLMM")` and R version `r getRversion()` using R Markdown. The newest release version of **clustGLMM** is always available from the Comprehensive R Archive Network (CRAN) at [https://CRAN.R-project.org/package=clustGLMM](https://CRAN.R-project.org/package=clustGLMM "clustGLMM"), an up-to-date version of this paper is contained as a package vignette. -->

# Methodology {#sec:methodology}

In this section, we briefly outline the model for clustering multivariate longitudinal data of mixed-type implemented in package **clustGLMM**. In addition to the model specification, we also discuss the prior structure and the hyperparameters required for the Bayesian approach. More details can be found in @Vavra-etal2024.

## Modeling longitudinal mixed-type outcomes {#subsec:glmms}

We start with explaining the necessary notation. We consider a multivariate regression setting where the indices for the outcomes are contained in the set $\mathcal{R}$. The index $r\in\mathcal{R}$ denotes one specific outcome variable and $\mathcal{R}$ is the union of five disjoint sets of indices: $\mathcal{R}^{\mathsf{Num}}$, $\mathcal{R}^{\mathsf{Poi}}$, $\mathcal{R}^{\mathsf{Bin}}$, $\mathcal{R}^{\mathsf{Ord}}$, $\mathcal{R}^{\mathsf{Cat}}$. Each set contains the indices for one of the five different outcome types considered: numeric, count, binary, ordinal and nominal. 

We assume that the data provided is collected for $n$ individuals with observations possibly made for each individual at different observation points resulting in a potentially different number of observations for each individual. We denote outcome $r\in\mathcal{R}$ of individual $i=1, \ldots, n$ from the $j$-th time point, $j = 1, \ldots, n_i$, by $Y_{ij}^r$, which together form a data set of $N = \sum_{i=1}^n n_i$ rows. These observations are accompanied by explanatory variables which are split into two regression vectors $\mathbf{X}_{ijr}$ and $\mathbf{Z}_{ijr}$ to distinguish the model matrix for the fixed and random effects in the mixed-effects model, respectively. Note that each outcome $r$ is allowed to have a different set of regressors. Thus, the linear predictor 
$$
\eta_{ijr} = \mathbf{X}_{ijr}^\top \boldsymbol{\beta}_r + \mathbf{Z}_{ijr}^\top \mathbf{b}_{i}^r,
$$
with fixed effects $\boldsymbol{\beta}_r$ and individual-specific random effects $\mathbf{b}_{i}^r$ represents the main part for modeling outcome $r$. 

For each outcome we consider one of five different types of GLMMs, i.e., different specifications of the response distribution and the link function which relates the mean to the linear predictor. In particular, we distinguish the following regression models: 

* We consider normal linear mixed-effects models for a quantitative (numeric) outcome $r\in\mathcal{R}^{\mathsf{Num}}$. This requires an additional dispersion parameter which captures the variability of the regression errors. As usual in Bayesian statistics, we parameterize the normal distribution via the precision parameter $\tau_r>0$. 
* For count outcomes, $r\in\mathcal{R}^{\mathsf{Poi}}$, we consider the Poisson distribution with the log link. This specification does not require any additional model parameter. 
* Binary outcomes, $r\in\mathcal{R}^{\mathsf{Bin}}$, are modeled by a logistic regression with the logit link function $\mathsf{logit}(x) = \log (x/(1-x))$. 

If the number of categories, $K^r$, is higher than 2, we consider the following two specifications:

* If an ordering of the categories (labeled $0 < \ldots < K^r -1$) is possible, we apply the proportional odds assumption, i.e., the log-odds differ only in intercepts $c_{r,k}$. In this case the intercept is excluded and this effect is modeled through a separate parameter because the following ordering constraint needs to be satisfied: $-\infty = c_{r,-1} < c_{r,0} < \cdots < c_{r,K^r-1} = \infty$, which requires a special prior and sampling approach. 
* If categories (labeled again as $0, \ldots, K^r -1$) cannot be ordered or the proportional odds assumption does not apply, then we create a linear predictor specific for each category, $\eta_{ijrk}$, with category-specific fixed effects $\boldsymbol{\beta}_{r,k}$. The random effects $\mathbf{b}_{ik}^r$ might also either be category-specific or alternatively common to all categories except the first. For identifiability, we fix the first category to zero: $\boldsymbol{\beta}_{r,0} = \mathbf{0}$ and $\mathbf{b}_{i0}^r$ leading to $\eta_{ijr0} = 0$.

The probability distributions of the observed outcomes given the linear predictors and auxiliary model parameters can be summarized in the following way:
\begin{align*}
  \left. Y_{ij}^r \,\right\vert\, \boldsymbol{\beta}_r, \mathbf{b}_i^r, \mathbf{X}_{ijr}, \mathbf{Z}_{ijr} 
  &\sim 
  \mathsf{N}\left(\eta_{ijr}, \; \tau_r^{-1}\right), 
  &\qquad r \in\mathcal{R}^{\mathsf{Num}}, 
  \\
  \left. Y_{ij}^r \,\right\vert\, \boldsymbol{\beta}_r, \mathbf{b}_i^r, \mathbf{X}_{ijr}, \mathbf{Z}_{ijr} 
  &\sim 
  \mathsf{Pois}\left(\exp\left\{\eta_{ijr}\right\}\right), 
  &\qquad r \in\mathcal{R}^{\mathsf{Poi}},
  \\
  \mathsf{P} \left[\left. Y_{ij}^r = 1 \,\right\vert\, \boldsymbol{\beta}_r, \mathbf{b}_i^r, \mathbf{X}_{ijr}, \mathbf{Z}_{ijr} \right] 
  &= 
  \mathsf{logit}^{-1} \left(\eta_{ijr}\right), 
  &\qquad r \in\mathcal{R}^{\mathsf{Bin}},
  \\
  \mathsf{P} \left[\left. Y_{ij}^r > k \,\right\vert\, c_{r,k}, \boldsymbol{\beta}_r, \mathbf{b}_i^r, \mathbf{X}_{ijr}, \mathbf{Z}_{ijr} \right] 
  &= 
  \mathsf{logit}^{-1} \left(\eta_{ijr} - c_{r,k}\right), 
  &\qquad k = 0, 1, \ldots, K^r-1, \;r \in\mathcal{R}^{\mathsf{Ord}},
  \\
  \mathsf{P} \left[\left. Y_{ij}^r = k \,\right\vert\, \boldsymbol{\beta}_r, \mathbf{b}_i^r, \mathbf{X}_{ijr}, \mathbf{Z}_{ijr} \right] 
  &= 
  \frac{\exp (\eta_{ijkr})}{1 + \sum\limits_{\kappa=1}^{K^r-1} \exp (\eta_{ij\kappa r})}, 
  &\qquad k = 0, 1, \ldots, K^r-1, \;r \in\mathcal{R}^{\mathsf{Cat}}.
\end{align*}

Assuming independence among the set of outcomes for the same individual is often unrealistic. Hence, we opt for a joint modeling approach which captures the dependency between the observations of the same individual. In particular, we stack the random effects for each individual $i$ into one vector $\mathbf{b}_{i} = \{\mathbf{b}_{i}^r, r\in\mathcal{R}\}$ with length denoted by $d^R$ and assume that they follow independently across individuals a multivariate normal distribution, $\mathbf{b}_{i} \sim \mathsf{N}_{d^R}(\mathbf{0}, \boldsymbol{\Sigma})$ where $\boldsymbol{\Sigma}$ is a positive definite matrix. This implies that the outcomes for the same individual are marginally dependent. However, they are still independent conditional on the random effects. 

## Model-based clustering of GLMMs {#subsec:mbc-glmms}

The model described in the previous section assumes that all individuals within the random sample follow the same, homogeneous model conditional on the explanatory variables. In case unobserved heterogeneity is present in the sample, this approach will fail. We thus extend our model and consider a model-based clustering approach where we assume the presence of $G$ latent groups. Each group follows the model outlined in the previous section but possibly differs in the values of the parameters. E.g., the groups may differ in the fixed-effects regression parameters, in the error precision of the numeric outcome and / or in the dependence structure of the random effects. Our implementation allows to flexibly choose which model parameters are group-specific and which ones are fixed to be the same across all latent groups. 

The specification which parameters are group-specific and which are the same across all latent groups will usually depend on what is assumed to characterize the latent groups. In addition imposing the restriction that some of the effects are the same across groups simplifies the model and facilitates estimation and interpretation. In the following, we will denote group-specific parameters with a superscript $(g)$, e.g., $\tau_r^{(g)}$. Parameters without this superscript are parameters which are the same across groups or represent the set of the corresponding parameters across all groups depending on context. Turning group-specificity on or off will be made en bloc for auxiliary model parameters like $\tau_r$, $\mathbf{c}_r$ or $\boldsymbol{\Sigma}$. For the fixed effects, a more selective approach is usually pursued with some effects being assumed to be the same across all groups, while others differ across groups. I.e., we split the fixed-effects contribution to the linear predictor into two parts such that the linear predictor for group $g$ takes the form of 
$$
\eta_{ijr}^{(g)} = \mathbf{F}_{ijr}^\top \boldsymbol{\beta}_r^F + \mathbf{X}_{ijr}^\top \boldsymbol{\beta}_r^{(g)} + \mathbf{Z}_{ijr}^\top \mathbf{b}_{ir},
$$
where $\boldsymbol{\beta}_r^F$ are the fixed effects shared among all groups  and $\boldsymbol{\beta}_r^{(g)}$ are the fixed effects specific to group $g$. Naturally, the sets of regressors $\mathbf{F}_{ijr}$ and $\mathbf{X}_{ijr}$ need to be mutually exclusive to avoid identifiability issues. 

Each individual $i$ belongs to one of the groups. Group membership is assumed to be unobserved and given by the latent group allocation indicator $U_i \in \{1, \ldots, G\}$. The group size corresponds to the marginal group allocation probabilities $w_g = \mathsf{P}(U_i = g)$ that sum up to one over $g = 1, \ldots, G$. Marginalizing over the latent group allocation indicators, we obtain a mixture distribution where the components consist of multivariate generalized linear mixed models and the probability density function (pdf) is given by:
\begin{equation}
  p(\mathbb{Y}_i \,\vert\, \mathcal{C}_i)
  = \sum\limits_{g=1}^{G} 
  w_g \int 
  p\left(\mathbb{Y}_i \,\left\vert\, U_i = g, \mathbf{b}_i;\, \boldsymbol{\beta}^F, \boldsymbol{\beta}^{(g)}, \boldsymbol{\tau}^{(g)}, \mathbf{c}^{(g)}, \mathcal{C}_i \right.\right)
  p\left(\mathbf{b}_i \,\left\vert\, U_i = g;\, \boldsymbol{\Sigma}^{(g)}\right.\right)
  \mathrm{d} \mathbf{b}_i,
  (\#eq:pdf-int-bi-Ui)
\end{equation}
where $p(\,\vert\,)$ denotes the conditional pdf, $\mathbb{Y}_i = \{Y_{ij}^r, \, j = 1, \ldots, n_i, \, r \in\mathcal{R}\}$ gathers all outcome values observed for individual $i$ and $\mathcal{C}_i = \{\mathbb{F}_{i}, \mathbb{X}_{i}, \mathbb{Z}_{i}\}$ gathers the covariates for individual $i$. Due to the general model specification with different distributions for the dependent variables, the integral over the random effects $\mathbf{b}_i$ cannot be analytically evaluated. In Bayesian estimation using MCMC sampling, we avoid these integrals by not marginalizing over the random effects, but performing Bayesian data augmentation and working with the conditional distributions. 

## Back transformation after standardization {#standardize-back}

In Bayesian mixture modeling, the priors on the group-specific parameters in the components need to be proper to obtain a proper posterior, and in general, hyperparameters inducing a certain regularization are preferable. We standardize numeric outcome variables and non-intercept regressors by default to ensure that the priors with the default hyperparameters impose a comparable shrinkage on the regression coefficients, error precision and random effect covariance regardless of the specific scaling of the variables. This standardization does not only change the fixed and random effects (especially intercepts), but also the precision $\tau_r$, the ordered intercepts $\mathbf{c}_{r}$ as well as the covariance matrix $\boldsymbol{\Sigma}$. In order to be able to interpret the parameter estimates with respect to the original scaling, the back-transformed parameters are reported in the output.

Below, we detail how we transform
the parameter estimates back to be able to interpret them
with respect to the original scaling of the variables.

First, consider the current parameterization of a numeric outcome $Y_{ijr}$:
$$
Y_{ijr} = \beta_{r,0}^{(g)} + \mathbf{F}_{ij}^\top \boldsymbol{\beta}_r^F + \mathbf{X}_{ijr}^\top \boldsymbol{\beta}_r^{(g)} + b_{i,0}^r+ \mathbf{Z}_{ijr}^\top \mathbf{b}_{i}^r + \varepsilon_{ijr}.
$$
where we explicitly separated the group-specific intercept parameter
$\beta_{r,0}^{(g)}$ as well as the random intercepts $b_{ir,0}$, and
added the noise $\varepsilon_{ijr}$. 

We switch to the equivalent version of this model with standardized regressors
(and numeric outcome) using
\begin{multline*} 
\dfrac{Y_{ijr} - m_r^Y}{S_r^Y} =
\widetilde{\beta}_{r,0}^{(g)}
+ 
\sum\limits_{l\neq 0}\dfrac{F_{ijr,l} - m_{r,l}^F}{S_{r,l}^{F}}\widetilde{\beta}_{r,l}^F 
+ 
\sum\limits_{l\neq 0}\dfrac{X_{ijr,l} - m_{r,l}^X}{S_{r,l}^{X}}\widetilde{\beta}_{r,l}^{(g)} 
+ % \\ +
\widetilde{b}_{i,0}^r 
+ 
\sum\limits_{l\neq 0}\dfrac{Z_{ijr,l} - m_{r,l}^Z}{S_{r,l}^{Z}}\widetilde{b}_{i,l}^r 
+ 
\widetilde{\varepsilon}_{ijr},
\end{multline*}
where $m^\bullet_\bullet$ and
 $S^\bullet_\bullet$ denote the corresponding means and the standard deviations.

This transformation does not only affects the regression effects but
also the precision $\tau_r$ as well as the covariance matrix
$\boldsymbol{\Sigma}$ of the random effects.

A similar decomposition of the linear predictor is performed for the
other types of outcomes as well. However, naturally in this case
$m_r^Y = 0$ and $S_r^Y = 1$. 

In the following we outline the necessary transformations to obtain
the original interpretation of model parameters (without $\sim$) from
the model fitted with standardized regressors (with $\sim$). In
particular, the following transformations (performed in this order)
provide the original model parameters:
\begin{align}
\widetilde{\beta}_{r,l}^F
&\;\rightsquigarrow\; 
\beta_{r,l}^F = \dfrac{S_r^Y}{S_{r,l}^F} \widetilde{\beta}_{r,l},
\\
\widetilde{\beta}_{r,l}^{(g)} 
&\;\rightsquigarrow\;
\beta_{r,l}^{(g)} = \dfrac{S_r^Y}{S_{r,l}^X} \widetilde{\beta}_{r,l}^{(g)},
\\
\widetilde{\beta}_{r,0}^{(g)} 
&\;\rightsquigarrow\; 
\beta_{r,0}^{(g)} = m_r^Y + S_r^Y \widetilde{\beta}_{r,0}^{(g)} 
- \sum\limits_{l\neq 0} m_{r,l}^F \beta_{r,l}^F 
- \sum\limits_{l\neq 0} m_{r,l}^X \beta_{r,l}^{(g)},
\\
\widetilde{b}_{i,l}^r 
&\;\rightsquigarrow\; 
b_{i,l}^r = \dfrac{S_r^Y}{S_{r,l}^Z} \widetilde{b}_{i,l}^r,
\\
\widetilde{b}_{i,0}^r 
&\;\rightsquigarrow\; 
b_{i,0}^r = S_r^Y \widetilde{b}_{i,0}^r 
- \sum\limits_{l\neq 0} m_{r,l}^Z b_{i,k}^r,
\\
\widetilde{\tau}_r^{(g)}
&\;\rightsquigarrow\; 
\tau_r^{(g)} = \dfrac{\widetilde{\tau}_r^{(g)}}{\left(S_r^Y\right)^2},
\\
\widetilde{\mathbf{c}}_r^{(g)}
&\;\rightsquigarrow\; 
c_{r,k}^{(g)} = \widetilde{c}_{r,k}^{(g)} 
+ \sum\limits_{l} m_{r,l}^F \beta_{r,l}^F 
+ \sum\limits_{l} m_{r,l}^X \beta_{r,l}^{(g)},
\\
L_r &\;=\;
\begin{pmatrix}
  1 & -\dfrac{m_{r,1}^Z}{S_{r,l}^Z} & -\dfrac{m^Z_{\ldots}}{S^Z_{\ldots}} \\
  0 & \dfrac{1}{S_{r,1}^Z} & 0 \\
  0 & 0 & \dfrac{1}{S^Z_{\ldots}}
\end{pmatrix},
\\
L &\;=\; \mathrm{diag}
\left\{S_r^Y
  L_{\mathsf{Num}}, L_{\mathsf{Poi}}, L_{\mathsf{Bin}}, L_{\mathsf{Ord}}, L_{\mathsf{Cat}} 
\right\},
\\
\widetilde{\boldsymbol{\Sigma}}^{(g)}
&\;\rightsquigarrow\; 
\boldsymbol{\Sigma}^{(g)} = 
L \widetilde{\boldsymbol{\Sigma}}^{(g)} L^\top.
\end{align}

In the derivations above, we considered only exactly one outcome for
each available type to simplify notation when indicating how the
$\boldsymbol{\Sigma}$ matrix needs to be transformed using a block
diagonal matrix $L$.

All formulae above are under the full generality of group-specificity
of all model parameters. The intercept and ordered intercepts have to
be group-specific in order to make the transformation possible if
there is at least one group-specific regression effect. If there is no
group-specific regression effect, then the intercept is allowed to
also be common to all groups. Regression effects for the categorical
outcome are transformed in the same manner, but require an additional
index for each non-baseline category level.

## Priors {#subsec:priors}

Within the Bayesian context, we need to specify prior distributions for all model parameters. We choose conditionally conjugate priors whenever possible to obtain well known full-conditional distributions. We distinguish between selecting priors for the component-specific parameters and the component sizes. 

All the fixed effects are given an independent centered normal prior, where the scale is adjusted depending on the outcome type. For numeric outcomes, the precision parameter $\tau_r$ is included within the normal prior for fixed effects $\boldsymbol{\beta}_r^F$ and $\boldsymbol{\beta}_r^{(g)}$. The precision $\tau_r$ is given a gamma prior to complete the normal-gamma prior family for linear regression models. For the ordered intercepts of ordinal outcomes, we use an implicit prior by imposing a symmetric Dirichlet prior on the induced category probabilities given the ordered intercepts and a zero effect of the linear predictor:
\begin{align*}
\pi_{r,k}^{(g)} 
&= \mathsf{P}\left[Y_{ijr} = k \,\vert\, \eta_{ijr} = 0, U_i = g\right]
= \mathsf{logit}^{-1}(c_{r,k-1}^{(g)}) - \mathsf{logit}^{-1}(c_{r,k}^{(g)}).
\end{align*}

Pursuing a conditional conjugate specification, we use the Inverse Wishart distribution as prior for the positive definite covariance matrix $\boldsymbol{\Sigma}$ (or group-specific $\boldsymbol{\Sigma}^{(g)}$). This is equivalent to assuming that $\boldsymbol{\Sigma}^{-1}$ follows a Wishart distribution with some scale matrix $\mathbb{Q}$. Since, this prior specification is very sensitive to the choice of $\mathbb{Q}$, we decided to use a hierarchical prior structure. I.e., $\mathbb{Q}$ also is a random parameter with an analogous inverse Wishart prior structure.

We select the prior on the component sizes assuming that the number of clusters is not known and that a *sparse finite mixture* approach [@Malsiner-Walli-etal2016] is pursued to determine a suitable number of clusters. In particular, we specify an overfitting mixture, where the number of components is clearly larger than the foreseen number of clusters in the data, and assign a symmetric Dirichlet distribution to the component sizes with a small value for the parameter $e_0$. The lower the value of $e_0$, the higher the probability that only a few components get substantial weight while the remaining components have negligible size. We use this prior setting to determine the suitable number of clusters by focusing on the number of non-empty components. To reduce the impact of a specific choice of $e_0$, we  use a hierarchical prior specification imposing a gamma prior on $e_0$, still in line with our sparse approach, i.e., putting a lot of mass at small values of $e_0$. 

We summarize our choices for the prior distributions in Table\ 
\@ref(tab:prior-table). This table also includes the default choices
for the hyperparameters and indicates the names used in the
**clustGLMM** package.

```{r prior-table, tab.cap = "Overview of prior distributions and their hyperparameters including the default values."}
ptab <- data.frame(Notation = c("$\\beta_{lr}^F$, $\\beta_{lr}^{(g)}$",
                                "$\\tau_r$",
                                "$\\beta_{lr}^F$, $\\beta_{lr}^{(g)}$",
                                "$\\mathbf{c}_r$",
                                "$\\mathbf{b}_i$",
                                "$\\boldsymbol{\\Sigma}^{-1}$",
                                "$\\mathbb{Q}^{-1}$",
                                "$w_g$", 
                                "$e_0$"), 
                   Distribution = c("$\\mathsf{N}(0,\\tau_r^{-1} s_{\\mathsf{Num}}^2)$", 
                                    "$\\Gamma(a_\\tau,b_\\tau)$",
                                    "$\\mathsf{N}(0,s_r^2)$", 
                                    "transformed $\\mathsf{Dir}(\\alpha)$",
                                    "$\\mathsf{N}_{d^R}(0,\\boldsymbol{\\Sigma})$",
                                    "$\\mathsf{W}_{d^R}(\\nu_0+d^R, \\mathbb{Q})$",
                                    "$\\mathsf{W}_{d^R}(\\nu_1+d^R, v\\mathbb{I})$",
                                    "$\\mathsf{Dir}(e_0)$",
                                    "$\\Gamma(a_e,b_e)$"), 
                   Hyperparameter = c("$s_{\\mathsf{Num}} = 1$",
                                      "$a_\\tau = 1$, $b_\\tau = 10$",
                                      "$s_{r} = 1$",
                                      "$\\alpha = 1$",
                                      "",
                                      "$\\nu_0=1$",
                                      "$\\nu_1=1$, $v=100$",
                                      "",
                                      "$a_e = 1, b_e = 100$"),
                   `**clustGLMM** name` = c("$\\mathtt{beta\\_num\\_fix\\_sd}$, $\\mathtt{beta\\_num\\_sd}$",
                                 "$\\mathtt{prec\\_num\\_shp}$, $\\mathtt{prec\\_num\\_rte}$",
                                 "$\\mathtt{beta\\_type\\_fix\\_sd}$, $\\mathtt{beta\\_type\\_sd}$",
                                 "$\\mathtt{api\\_prior}$",
                                 "",
                                 "$\\mathtt{InvSigma\\_df}$",
                                 "$\\mathtt{InvQ\\_df}$, $\\mathtt{InvV}$",
                                 "",
                                 "$\\mathtt{e0\\_shp}$, $\\mathtt{e0\\_rte}$"),
                   Note = c("or with $\\tau_r^{(g)}$, for conjugacy",
                            "or for $\\tau_r^{(g)}$",
                            "$s_r$ specific to type not each outcome",
                            "for $\\pi_{r,k}^{(g)} = \\mathsf{logit}^{-1}(c_{r,k-1}^{(g)}) - \\mathsf{logit}^{-1}(c_{r,k}^{(g)})$",
                            "or with $\\boldsymbol{\\Sigma}^{(g)}$",
                            "or for $\\boldsymbol{\\Sigma}^{-(g)} := \\left(\\boldsymbol{\\Sigma}^{(g)}\\right)^{-1}$",
                            "or for $\\mathbb{Q}^{-(g)} := \\left(\\mathbb{Q}^{(g)}\\right)^{-1}$",
                            "symmetric, $e_0$ randomized",
                            "sparse finite mixture with $a_e/b_e$ low"),
                   row.names = c("Fixed effects ($\\mathsf{Num}$)", 
                                 "Error precision",
                                 "Fixed effects (other)", 
                                 "Ordered intercepts",
                                 "Random effects", 
                                 "Precision matrix",
                                 "Inverse scale matrix",
                                 "Component sizes",
                                 "Sparsity"),
                   check.names = FALSE)
if (requireNamespace("kableExtra", quietly = TRUE) & requireNamespace("dplyr", quietly = TRUE)) {
  kable(ptab, booktabs = TRUE, escape = FALSE) |> kableExtra::landscape()
}else{
  kable(ptab, booktabs = TRUE, escape = FALSE)
}
```

# Inference {#sec:inference}

In the following, we summarize our approach to approximate the posterior distribution of the model parameters. This posterior is too complex to be evaluated directly and thus we approximate it using MCMC sampling. The presence of latent variables such as the component allocation indicators $U_i$, random effects $\mathbf{b}_i$ or possibly even missing outcome observations can easily be accounted for during MCMC sampling of the posterior. Based on the data augmentation principle, we combine all primary model parameters, random prior hyperparameters and latent variables into one large set of parameters and latent variables $\boldsymbol{\theta}$:
$$
\boldsymbol{\theta} = \left\{ \boldsymbol{\beta}^F, \boldsymbol{\beta}^{(g)}, \boldsymbol{\tau}^{(g)}, \mathbf{c}^{(g)}, \boldsymbol{\Sigma}^{(g)}; \mathbf{w}, e_0, \mathbb{Q}; \mathbf{b}_i, U_i, 
\;g=1,\ldots, G, i=1,\ldots,n   \right\}.
$$
The posterior distribution of $\boldsymbol{\theta}$ given by a usual Bayes formula
$$
p(\boldsymbol{\theta} \, | \, \mathbb{Y}, \mathcal{C}) 
\propto 
p(\mathbb{Y} \, | \, \boldsymbol{\theta}, \mathcal{C}) 
\, 
p(\boldsymbol{\theta} \, | \, \mathcal{C})
$$
is the target distribution, where $\mathbb{Y} = \{\mathbb{Y}_i, i = 1, \ldots, n\}$ and $\mathcal{C} = \{\mathcal{C}_i, i = 1, \ldots, n\}$ gather outcomes and covariates for all individuals.
Pursuing an MCMC approach, we create a Markov chain consisting of samples of $\boldsymbol{\theta}$ that could eventually be considered as a dependent random sample from the target distribution. We use these samples for posterior inference about model parameters and transformations thereof. 

## MCMC sampling {#subsec:mcmc}

To create such a Markov chain, we pursue a Gibbs sampling approach where possible. We divide $\boldsymbol{\theta}$ into suitable smaller blocks $\boldsymbol{\theta}_l$ and iteratively sample every block from its full-conditional distribution:
\begin{equation}
  p(\boldsymbol{\theta}_l \, | \cdots) :=  
  p(\boldsymbol{\theta}_l \, | \, \mathbb{Y}, \boldsymbol{\theta}_{-l}, \mathcal{C}),
%  \propto 
%  p(\mathbb{Y} \, | \, \boldsymbol{\theta}, \mathcal{C}) 
%  \, 
%  p(\boldsymbol{\theta} \, | \, \mathcal{C}), 
  (\#eq:pdf-full-cond)
\end{equation}
where the full-conditional distribution is viewed as a function of $\boldsymbol{\theta}_l$ and other parameter values $\boldsymbol{\theta}_{-l}  = \boldsymbol{\theta} \setminus \boldsymbol{\theta}_l$ are kept fixed. 
We carefully set up the prior distributions for the model parameters to ensure conditional conjugacy where possible. Therefore, many parameters can be directly sampled from well known distribution families and the only challenge is the efficient computation of the parameters of the full-conditional distributions. 

However, not every full-conditional distribution falls into a known family of distributions. In this case, we use a Metropolis-Hastings (M-H) approach. In each M-H step, a move from the current value to a new value of the parameter is proposed and the new value is accepted with prescribed probability or alternatively the current value is retained. The challenge when implementing an M-H step is to find a suitable proposal distribution that sufficiently well resembles the full-conditional distribution in order to be able to obtain a suitable acceptance rate. Here, we only briefly outline the M-H approach pursued; more details on the implementation can be found in the supplementary material to @Vavra-etal2024. 

The first step consists of transforming the block of interest $\boldsymbol{\theta}_l$ so that it has unbounded support, $\boldsymbol{\omega} = t(\boldsymbol{\theta_l}) \in \mathbb{R}^d$, e.g., we log-transform $e_0$ or transform the ordered intercepts to probabilities $\pi$ using the softmax function. Then, the log-pdf of the full-conditional distribution for the transformed parameter $\boldsymbol{\omega}$ is approximated by the Taylor expansion of the second order:
$$
\ell (\boldsymbol{\omega}) := \log p (\boldsymbol{\omega} | \cdots) 
\approx 
\mathrm{const.} - 
\frac{1}{2} \left(\boldsymbol{\omega}-\widehat{\boldsymbol{\omega}}\right)^\top
\left[- \left.\frac{\partial^2 \ell(\boldsymbol{\omega})}{\partial \boldsymbol{\omega} \partial \boldsymbol{\omega}^\top }\right|_{\boldsymbol{\omega}=\widehat{\boldsymbol{\omega}}} \right]
\left(\boldsymbol{\omega}-\widehat{\boldsymbol{\omega}}\right).
$$
This Taylor expansion reduces to a quadratic form with negative Hessian matrix in the middle if evaluated at the maximum point $\widehat{\boldsymbol{\omega}}$ found by the Newton-Raphson (N-R) method, i.e., where $\frac{\partial}{\partial \boldsymbol{\omega}} \ell(\widehat{\boldsymbol{\omega}}) = \mathbf{0}$. Locally, the log-pdf resembles the log-pdf of a centered multivariate normal distribution. Since the M-H step works best if the proposal distribution is close to the target distribution, we use the negative Hessian matrix (obtained as a by-product of the N-R method) as the precision matrix for the proposal distribution of direction in which to move from the current value of the transformed block $\boldsymbol{\theta}_l$. This distribution can be adjusted by a multiplicative constant $c_{\boldsymbol{\theta}_l}$ to increase or decrease the potential step size. Hence,
$$
\boldsymbol{\omega}^{\mathrm{proposal}} = \boldsymbol{\omega}^{\mathrm{current}} + \mathbf{s}, 
\qquad \text{where} \quad 
\mathbf{s} \sim 
\mathsf{N}_d\left(\mathbf{0}, c_{\boldsymbol{\theta}_l} \left[- \left.\frac{\partial^2 \ell(\boldsymbol{\omega})}{\partial \boldsymbol{\omega} \partial \boldsymbol{\omega}^\top }\right|_{\boldsymbol{\omega}=\widehat{\boldsymbol{\omega}}} \right]^{-1}\right).
$$
The new value is accepted with probability 
$$
\min\left\{1, \, \exp(\ell(\boldsymbol{\omega}^{\mathrm{proposal}}) - \ell(\boldsymbol{\omega}^{\mathrm{current}}))\right\},
$$
otherwise the old value is retained. Finally, the new value of $\boldsymbol{\omega}^{\mathrm{new}}$ is transformed back to  $\boldsymbol{\theta}_l^{\mathrm{new}} = t^{-1}(\boldsymbol{\omega}^{\mathrm{new}})$.

## Clustering probabilities and deviance {#subsec:clust-prob-deviance}

Using an MCMC sampling approach with data augmentation, we can avoid approximating the integral \@ref(eq:pdf-int-bi-Ui) to estimate the model parameters. However, to determine some quantities of interest such as the clustering probabilities or evaluating the deviance, this integration over the random effects needs to be performed. In particular, the clustering probabilities $u_{i,g}$ of an individual $i$ to belong to component $g$ can be viewed as parametric functions of the primary model parameters and the data of the individual $i$ and are proportional to
\begin{align*}
u_{i,g}(\boldsymbol{\beta}, &\boldsymbol{\tau}, \mathbf{c}, \boldsymbol{\Sigma}, \mathbf{w} \,|\, \mathbb{Y}_i, \mathcal{C}_i) 
\propto\\
&w_g 
\int 
p\left(\mathbb{Y}_i \,\left\vert\, U_i = g, \mathbf{b}_i;\, \boldsymbol{\beta}^F, \boldsymbol{\beta}^{(g)}, \boldsymbol{\tau}^{(g)}, \mathbf{c}^{(g)}, \mathcal{C}_i \right.\right)
p\left(\mathbf{b}_i \,\left\vert\, U_i = g;\, \boldsymbol{\Sigma}^{(g)}\right.\right)
\mathrm{d} \mathbf{b}_i.
\end{align*}
We reformulate the integrand as a function $\exp\{h^{(g)}\}$ in variable $\mathbf{b}_i$, i.e., as $\int \exp\{h^{(g)}(\mathbf{b}_i)\} \mathrm{d} \mathbf{b}_i$, to apply an approximation method. The Laplace approximation relies on the Taylor expansion of the second order of the function $h^{(g)}$, assuming that it is differentiable and concave. Using the N-R method the maximum $\widehat{\mathbf{b}}_i^{(g)}$ and the negative Hessian matrix $H^{(g)}$ at the maximum are found. These are then used in the Laplace approximation:
\begin{equation}
  \int \exp\left\{h^{(g)}(\mathbf{b}_i)\right\} \mathrm{d} \mathbf{b}_i
  \approx 
  |H^{(g)}|^{-\frac{1}{2}} \exp\left\{h\left(\widehat{\mathbf{b}}_i^{(g)}\right)\right\}. 
  (\#eq:integral-bi-approx)
\end{equation}
We use these approximations to obtain estimates for the clustering probabilities $u_{i,g}$ defined above. The normalizing denominator for the proportionality is an approximation to $p(\mathbb{Y}_i \,|\, \mathcal{C}_i)$ given in \@ref(eq:pdf-int-bi-Ui), which can be used to determine the contribution of a single individual to the deviance defined as $-2\log p(\mathbb{Y}_i \,|\, \mathcal{C}_i)$.

In general, however, the Laplace approximation of the integral is rather crude. To reduce the approximation error, we also implemented Adaptive Gaussian Quadrature, a natural generalization of the Laplace approximation [@PinheiroChao06]. The idea is to evaluate the function $h^{(g)}$ on a suitable grid of $\mathbf{b}_i$ values determined by the maximal point $\widehat{\mathbf{b}}_i^{(g)}$ and $H^{(g)}$-scaled roots of the Hermite polynomial of prescribed order. The first order polynomial has only one root (0), thus yielding the Laplace approximation. The number of points where function $h^{(g)}$ needs to be evaluated increases exponentially with the order of the polynomial. Further details can be found in the supplementary material to @Vavra-etal2024.

These approximations have to be done for all considered cluster labels $g$ and for every sampled set of model parameters within the MCMC chain to explore the posterior distribution of $u_{i,g}$ properly. Moreover, the full deviance requires the separate evaluation for each individual within the dataset. This is usually computationally very expensive and, hence, one might either need to rely on crude approximations or make use of parallel computing for better approximations.

# Implementation {#implementation}

In the following, we discuss the implementation of the methodology
in package **clustGLMM** in more detail. An overview of the typical workflow and
the key functions is given in Figure\ \@ref(fig:workflow-diagram). In
the following subsections we go through each of the steps and discuss
it in detail.

```{r workflow-diagram, fig.cap="Typical workflow with **clustGLMM** and key functions.", out.width="100%", dev="png"}
if (requireNamespace("DiagrammeR", quietly = TRUE) & 
    requireNamespace("DiagrammeRsvg", quietly = TRUE) & 
    requireNamespace("rsvg", quietly = TRUE)) {
  graph <- DiagrammeR::grViz("digraph {
    graph[layout = dot, rankdir = TB]
    
    node[shape = rectangle, style = rounded, fontname = Helvetica, penwidth = 2, fontsize = 15, fillcolor = gold, style = filled]  
    rec1[label = '@@1']
    rec2[label = '@@2']
    rec3[label = 'Visual diagnostics']
    rec3a[label = 'Satisfied', shape = diamond, fillcolor = LemonChiffon, style = filled]
    rec4[label = 'Within chain post-processing']
    rec5[label = 'Across chain post-processing', fillcolor = LemonChiffon]
    
    node[shape = rectangle, style = rounded, fontname = courier, penwidth = 2, fillcolor = grey70, style = filled] 
    code1[label = 'data, family, formula, varying, ...']
    code2[label = 'clustGLMM()']
    code3[label = 'as_coda(), plot()']
    code3a[label = 'If not, return to previous steps (adjust settings or continue in sampling)', fillcolor = white, fontname = Helvetica]
    code4[label = 'post_processing()']
    code5[label = 'plot(, which=clusters_kerneldensity), permute_cluster_labels()']
    
    node[shape = rectangle, style = rounded, fontname = courier, penwidth = 2, fillcolor = grey70, style = filled]
    code6a[label = 'summary()']
    code6b[label = 'predict()']
    code6c[label = 'clustering_probabilities_and_deviance()']
    
    subgraph cluster6 {
      graph[layout = fdp, shape = rectangle, style = rounded, fillcolor = gold, style = filled]
      
      label = ''
      node[shape = rectangle, style = rounded, fontname = Helvetica, penwidth = 2, fillcolor = LemonChiffon, style = filled]
      rec6[label = 'Inference', fillcolor = gold, shape = none]
      rec6a[label = 'Posterior description with the use of coda']
      rec6b[label = 'Reconstruct linear predictor']
      rec6c[label = 'Approximate posterior of clustering probabilities and deviance']
    }
    
    # edge definitions with the node IDs
    edge[arrowhead = normal, arrowtail = none]
    rec1 -> rec2 -> rec3 -> rec3a -> rec4 -> rec5 -> rec6
    
    edge[arrowhead = none, arrowtail = none]
    {{rank=same rec1 -> code1}}
    {{rank=same rec2 -> code2}}
    {{rank=same rec3 -> code3}}
    {{rank=same rec3a -> code3a}}
    {{rank=same rec4 -> code4}}
    {{rank=same rec5 -> code5}}
    {{rank=same rec6a -> code6a}}
    {{rank=same rec6b -> code6b}}
    {{rank=same rec6c -> code6c}}
    
    edge[arrowhead = normal, arrowtail = none, color = white, maxlen=1]
    rec6 -> rec6a -> rec6b -> rec6c 
    
    }
    
    # Substitutions with R code are possible:
    [1]: paste0('Prepare inputs')
    [2]: paste0('Sample chains')
    ",
    height = 500, width = 600)
  svg_text <- DiagrammeRsvg::export_svg(graph)
  rsvg::rsvg_png(charToRaw(svg_text), file = file.path(tempdir(), "workflow-diagram.png"))
  knitr::include_graphics(file.path(tempdir(), "workflow-diagram.png"))
}else{
  knitr::include_graphics(file.path(tempdir(), "workflow-diagram.png"))
}
```

## Prepare inputs {#impl-prepare-inputs}

First, we discuss the arguments which need to be specified for the main fitting function `clustGLMM()` and describe the expected format of the required inputs. The overview of the arguments of function `clustGLMM()` reads:

```{r}
cat(prettyPrint(prompt(clustGLMM::clustGLMM, 
                       filename = NA)$usage[[2]], sep = ", ", 
                linebreak = paste("\n", paste(rep(" ", 2), collapse = ""), sep= ""),
                width = 60))
```

The outcomes with their types, i.e., their distributions, are specified in the `family` argument. This argument is provided as a named vector indicating the outcome types named by the outcome names matching the column name in `data`. The following types are available:

* `"num"`, `"gaussian"` for numeric outcomes, $r\in\mathcal{R}^{\mathsf{Num}}$,
* `"poi"`, `"poisson"` for count outcomes, $r\in\mathcal{R}^{\mathsf{Poi}}$,
* `"bin"`, `"bernoulli"` for binary outcomes, $r\in\mathcal{R}^{\mathsf{Bin}}$,
* `"ord"`, `"cumulative"` for ordinal outcomes, $r\in\mathcal{R}^{\mathsf{Ord}}$,
* `"cat"`, `"categorical"` for categorical outcomes, $r\in\mathcal{R}^{\mathsf{Cat}}$, where the first category is considered the baseline.

Internally, outcomes are ordered to ensure that the types are in the same order as listed above. <!-- Ensuring this ordering eases the estimation of the covariance matrix $\boldsymbol{\Sigma}$. --> The `family` returned by `clustGLMM()` then follows this ordering.

Our implementation allows a different model formula for each outcome. All formulas are passed in the argument `formula` as a named list with names indicating the outcomes. For each outcome, three formulas (of class `"formula"`) are required:

* `fixed` for fixed effects common to all clusters,
* `group` for cluster-specific fixed effects,
* `random` for `id`-specific random effects.

In addition there is also the option to include an element named `offset` which contains the column name of the offset variable, i.e., the values that shift the linear predictor without any unknown coefficient. By default, no `offset` is considered. Sharing the same terms in both, the `fixed` and `group`, formulas is allowed. By default, this is resolved by removing duplicated terms from `fixed` and having them only included in the model matrix corresponding to the `group` formula. For example, if both formulas imply that an intercept is included in the model matrix, a cluster-specific intercept is estimated. Note that the intercept term is removed from the formula for ordinal outcomes, since this is captured by the model parameter `c_ord` (or its transformations `a_ord` or `pi_ord`). The `"clustglmm"` object returned by `clustGLMM()` contains lists (`lfixnames`, `lgrpnames`, `lrannames`) of covariate names which can be matched to the corresponding regression coefficients `beta...[]` or random effects `b[,]`.

Data are passed in the `data` argument as a `data.frame` which contains all outcomes, regressors and a column with name specified in the argument `id` that indicates which rows correspond to the same subject. The values in `id` are internally mapped to `1:n` where `n` is the number of unique values in `id`. This mapping is returned in the `"clustglmm"` object as element `numbered_unique_ids`. The maximum number of components to include in the mixture model is specified as an integer value with the argument `G` (allowing also the value 1 to induce no clustering).

The next four arguments provide more details regarding the model specification, the model estimation via MCMC sampling as well as how sampling results are returned. These arguments are expected to have the following structure: 

* `varying` - a named vector of Booleans controlling cluster-specificity of other model parameters, 
* `save` - a named vector of Booleans controlling which sampled model parameters should be returned, 
* `param` - a named list of prior hyperparameters, 
* `tuning` - a named list with named lists for integer and double values that adjust the sampler, especially the M-H proposal steps constructed by the use of the N-R method. 

For a full description of each argument see the documentation. If an argument is missing, then the default choices constructed by `default_varying()`, `default_save()`, `default_param()`, and `default_tuning()` are used. We recommend to call these functions, change the default object returned as required and use the updated object in the call to `clustGLMM()`.

The argument `inits` expects a list of initial values used to start each of the chains. The structure is the same as the output `last` of `clustGLMM()`. The output `last` can be used to easily continue sampling. It suffices to call `clustGLMM()` with the same settings and specify `inits = ...$last`. Note that if standardization is in effect, these initial values relate to the transformed scale and not the original one, which is used for the returned samples. If the argument `inits` is missing, then initial values are determined in an automatic way based on a random partition of the observations for each chain. 

It remains to choose the number of chains using `nchains` (chains will be labeled by `1:nchains`, with `G` allowed to vary across chains through specification via an integer vector), the length of the chains to be sampled with `iter` and the standardization indicator `standardize`. The last argument specifies the format in which the samples should be returned via `howsave`. We recommend to use the option `"data.fame"` which implies that for each chain, all model parameters are returned in one huge data frame with carefully named columns based on the following pseudocode:

```{r matrix-column-name, eval = FALSE, include = TRUE, echo = TRUE}
paste0(parameter, # parameter ~ character(1)
       if (is.outcome.specific) {"_", outcome}, # outcome ~ character(1)
       if (is.group.specific) {"(", g, ")")}, # g ~ integer(1)
       if (is.not.0D) {"[", d[1], if(is.2D) {",", d[2]}, "]"} # d ~ integer(2)
) # e.g., "beta_num_income(1)[1]"
```

This means that the naming convention indicates that first the
parameter name and outcome type are given (e.g., `"beta"` and `"num"`)
separated by `"_"` and followed by the outcome name (e.g., `"income"`)
also separated by `"_"`. Then the component number in round
brackets is included (e.g., `"(1)"`) and the dimension by square
brackets (e.g., `"[1]"`).

Alternatively, we also provide a second option corresponding to
`"list"`. Using this option, all sampled values are saved in a
carefully structured list with the following layers: chain, parameter
name, outcome name (for outcome-specific parameters), cluster (for
cluster-specific parameters). The leaves of this structure are arrays
(2D or 3D) depending on the subset of parameters.

## Sample chains {#impl-sample-chains}

After careful preparation of all inputs, `clustGLMM()` can safely be
called to samples the MCMC chains. In this section we describe each of
the steps this function performs.

First, missing arguments are replaced with their default values. A warning is issued in case argument `save` specified that crucial parameters are not to be returned. Subject indices `id` are relabeled and outcomes are suitably ordered. Then, each outcome is checked for compatibility with its given type:

* `num` - values have to be numeric, infinite values are not allowed, otherwise an error occurs. If `standardize = TRUE`, `scale()` is applied with `center` and `scale` determined by `mean` and `sd` while ignoring missing values.
* `poi` - values are checked to be finite non-negative integers, otherwise an error occurs.
* `bin` - `factor()` is applied and an error occurs if the number of unique values is not 2. Internally, the first level is considered to be 0, the other 1.
* `ord` - if not yet of class `"ordered"`, `factor(., ordered = TRUE)` is called. Internally, the ordered categories are replaced with 0, 1, ..., `Kord`.
* `cat` - if not yet of class `"factor"`, `factor()` is called. Internally, the ordered categories are replaced with 0 (baseline), 1, ..., `Kcat`.

Ordinal and categorical outcomes are not allowed to have less than 3 levels (as for 2 levels one may use the binary type), but there is no upper limit for the number of categories. As the number of parameters grows with the levels, the user needs to take care not to specify a variable with many unique values to be a categorical outcome. Note that `Kord` and `Kcat` correspond to the actual number of categories minus 1 (as the numbering starts at 0).

Next, the `formula` is processed and model matrices are created. All model matrices are combined into one large model matrix with uniquely named columns. This is computationally more efficient because we expect many columns to be simultaneously  used for several outcomes. For each outcome, a list of the column names that result from the `fixed`, `group` and `random` formulas is created to select the suitable columns from the full model matrix (`lfixnames`, `lgrpnames`, `lrannames`). These lists also determine the order in which the effects are named and saved. For example, `"[1]"` in `"beta_num_income(1)[1]"` indicates that this cluster-specific regression coefficient is tied to the first element of the corresponding `lgrpnames[["income"]]` value. During these preparations, we ensure that not the columns are included in `fixed` and `group` and eliminate the column names common to `fixed` and `group` from `fixed` and keep them only in `group`.

If `standardize = TRUE`, the ability to transform parameters back to the original scale is checked. An error is triggered if the intercept term (column `"(Intercept)"`) does not appear among the `fixed` and `group` effects. In case of group-specific effects, the `group` formula is ensured to include an intercept. An ordinal outcome has no explicit intercept within the formula, but the parameter of the ordered intercepts needs to be group-specific (`varying["c_ord"] = TRUE`) if there are some group-specific effects. In case of standardization, random effects have to include an intercept and, otherwise, an error is triggered. The scaling of columns other than `"id"` or `"(Intercept)"` is performed with `scale(, center = TRUE, scale = TRUE)`. The centers and scales used are saved in named vectors `centers` and `scales`.
Finally, note that `NA` values for regressors are not allowed.

Next, initial values are determined if the argument `inits` is missing. First, a random partition of observations into the given number of components `G` is created. Then, for each outcome, the corresponding model (`lm`, `glm(family = poisson)`, `glm(family = binomial)`, `MASS::polr`, `nnet::multinom`) is fitted with common effects for the fixed and group-specific effects and all available data. The estimates corresponding to fixed effects are used as their initial values. Then, the same model is fitted on each subset of data from the random partition to obtain the initial values for the group-specific regression coefficients. If an error occurs, e.g., due to small group size in the partition, then the estimates from the fit with the complete data are used instead. The random effects are sampled randomly from a centered normal distribution with standard deviation given by `param$init_b_sd`. The precision matrix for the random effects is initialized randomly as a diagonal matrix with diagonal elements sampled from a gamma distribution with shape `1/param$init_b_sd^2` and unit rate.

The model contains a lot of parameters of very different nature (outcome-specific, group-specific, varying dimension). In order to cope with these differences, a matrix `settings` is created for each chain where each row contains all necessary information to properly handle one of the model parameters. Since the number of components could be different for different chains, a list of such matrices for each chain is returned. The most important columns of this matrix provide information on the dimension size (with or without group-specificity). This information is crucial to store the sampled values of the parameters. For outcome-specific parameters (e.g., regression coefficients), the lists `yspecd1` and `yspecd2` contain the total dimension sizes per each outcome.

The MCMC sampling is by far the most time-consuming part of this function. The pure R implementation was painfully slow and, therefore, C routines were implemented to do the heavy lifting. This requires additional preparation steps, since only vectors of integers or doubles can be passed down via `.C()`. During this preparation, for example, IDs and cluster allocation indicators are shifted to start from 0 and not 1, the labels for the categorical outcomes of the first and last category are switched (as the C functions assume the last category to be the baseline), `NA` values are replaced with zeros. Moreover, the initial values are concatenated in the order expected by the C function. These values are duplicated into a vector of the same length representing the current parameter values. Both vectors are accompanied by a structure of pointers pointing to the starts of each of the parameters. When a new value of the parameter is sampled (and accepted), the corresponding cells are overwritten. Simultaneously, these new values are saved into storage vectors which were prepared separately for each model parameter including also the transformed ones (standard deviations, correlations, ...). The C code regularly prints the percentage of sampled iterations so that the user may estimate the remaining time to finish the sampling step. After the sampling, the vector containing the last values is transformed back into the R format. In this way, the last sampled values `$last` can be used again as initial values. Depending on the save format indicated in `howsave`, the stored values are processed into the final `$draws` output with the help of `from_C_to_...` functions. If standardization was applied, then the `$draws` (but not `$last`) are transformed back to enable the interpretation on the original scale. The details about this back-transformation are given in `r if (knitr::is_html_output()) "[the Back transformation after standardization section](#standardize-back)" else "Section \\ref{standardize-back}"`.

This sampling is performed sequentially for all chains. In case this is very time-consuming, parallelization across chains might be attempted. This can be achieved by writing a function that calls `clustGLMM()` with the same settings for `nchains = 1` in parallel and then combines the results into one object. In addition, high memory demands might be caused, not only by the vast number of parameters of large dimension sizes but also due to auxiliary variables. For example, the C functions use a 4-component predictor dedicated to the fixed, group, random and offset part, and the final predictor is the sum of these four components. When some of the effects are updated, only the corresponding part of the predictor gets updated and the rest remains untouched. This approach is even more expensive for categorical outcomes where a separate predictor is kept for every non-baseline category. To reduce the memory demands, all operations with symmetric matrices are handled using only the upper triangle.

Another potential source for long sampling times could be due to difficulty when determining a suitable proposal distribution via the N-R method initiated with the last known value. In case the N-R method does not converge or does not meet the required tolerance `tuning$double$tolerance` after the maximal number of steps `tuning$integer$maxiter`, the N-R procedure is started again from a different point (corresponding to the current best approximation scaled by a value uniformly sampled from $(0, 1)$). This is repeated until the required tolerance is met or the maximal number of N-R reiterations `tuning$integer$maxnrep` is reached. If the N-R method is not successfully terminated, the current best solution is used for the proposal distribution. Since this search for an optimal proposal distribution can be very time-consuming, the proposal distributions are updated only every `tuning$integer$freq_proposal_update`-th iteration. Since the acceptance rate could be very low, there is also an option to repeat the proposal step for the same parameters several times with `tuning$integer$times_proposal > 1`. Faster computation times could be achieved by adjusting these tuning parameters. However, a faster evaluation could also come at the cost of lower acceptance rates. We recommend to update the proposal distribution frequently when the stationary distribution has not been reached yet. The proposal distribution is scaled by `tuning$double$const_proposal_...` where `...` stands for any parameter updated via the M-H proposal. Similar to the prior hyperparameters, their value is fixed for every underlying parameter, e.g., one cannot set different values for two outcomes of the same type.

Latent variables and model parameters are sampled in the following order (using either a Gibbs or M-H sampling step):

* `naY` - missing outcome values are generated from the underlying model given the current allocation of the corresponding individual. These replace the `NA` values in the vector used for storing the outcome values. It is also possible to sample the potential outcome values if the individual had been in other clusters by setting `varying["naY"]` to `TRUE`.
* `beta_num`, `beta_num_fix` - regression coefficients for numeric outcomes (Gibbs).
* `prec_num` - precision error term for numeric outcomes and its transformation to standard deviation `sd_num` and variance `var_num` (Gibbs).
* `beta_poi`, `beta_poi_fix` - regression coefficients for count outcomes (M-H).
* `beta_bin`, `beta_bin_fix` - regression coefficients for binary outcomes (M-H).
* `beta_ord`, `beta_ord_fix` - regression coefficients for ordinal outcomes (M-H).
* `a_ord` - ordered intercepts transformed into the unrestricted space (M-H), then transformed back to the original and prior interpretation `c_ord`, `pi_ord`.
* `beta_cat`, `beta_cat_fix` - regression coefficients for categorical outcomes (M-H).
* `InvSigma` - precision matrix for random effects (Gibbs) and its transformations: covariance matrix `Sigma`, standard deviations from diagonal `sdSigma`, correlation matrix `corSigma` and determinant `detInvSigma`.
* `InvQ` - inverse of the Wishart scaling matrix for the prior for `InvSigma` (Gibbs) and its transformations `Q` and determinant `detInvQ`.
* `b` - random effects (M-H). 
* `w` - component weights (Gibbs).
* `pUig` - full-conditional clustering probabilities and then the new allocation indicators `U`. As by-product we obtain the cluster occupancy numbers `ng` and the number of non-empty clusters `Gplus`. In this step, also `loglik` is determined, which corresponds to the "in-sample log-likelihood". In particular, to obtain the log-likelihood value, integration over latent variables (i.e., the allocation indicators and random effects) is not performed and the last sampled values are inserted. 
* `e0` - Dirichlet parameter sampled on the log-scale and then transformed back (M-H). 

Note that the steps for `InvSigma`, `InvQ` and random effects are dropped if there are no random effects. In such a case the outcomes as well as the observations (rows) for the same ID are assumed to be independent. This allows to also fit regular cluster-wise regressions without including random effects. 

Before all the outputs of `clustGLMM()` are returned in a list, a crude clustering is performed for each chain separately which does not account for potential label switching. First, the sampled allocation indicators `U` (with values in 1, ..., G) are summarized for each individual. The matrix `clustering` then contains the most frequent cluster label for each individual in the rows and each chain in the columns. The matrix `certainty` then contains the proportion of these most frequent labels for each individual and chain. Then, the most frequent number of non-empty components `modeGplus` is determined for each chain. A list `clusters` then contains for each chain the `modeGplus` labels included in `clustering`. These labels will be used for plotting. The other components are considered empty and ignored for plotting, unless specified otherwise.

Finally, `call` summarizes the sample sizes as well as the formulas specified for all outcomes: first, the fixed effects, then the random effects (in the order assumed by `InvSigma`) and finally the group-specific effects. Printing objects of class `"clustglmm"` prints the content of `call` followed by the summary of the crude clustering for each chain separately.

## Visual diagnostics {#impl-visual-diagnostics}

Given that some users may already be familiar with already established diagnostic tools for assessing convergence of a MCMC chain, we provide function `as_coda()` to convert an object of class `"clustglmm"` to a `"mcmc.list"` object as defined in package **coda** [@Plummer06coda]. This enables easy use of all diagnostic tools available in that package.

The use of these tools, however, might be complicated by the large dimension of the parameter space of our model, which can easily contain hundreds of parameters and even thousands of values in case latent id-specific variables are also considered. We thus provide a set of plotting functions which provide diagnostic plots for selected specific individual parameters related to a given parameter name:

* `plot_ACF_param` - plots the auto-correlation function (ACF), use `plot_ACF` for a single scalar parameter choice.
* `plot_ECDF_param` - plots the empirical cumulative distribution function (ECDF), use `plot_ECDF` for a single scalar parameter choice.
* `plot_kerneldensity_param` - plots the kernel density estimate, use `plot_kerneldensity` for a single scalar parameter choice.
* `plot_traceplots_param` - provides trace plots (value vs. iteration), use `plot_traceplots` for a single scalar parameter choice.
* `plot_clusters_ECDF_param` - compares cluster-specific parameters across (non-empty) clusters via ECDF, use `plot_clusters` for a single scalar parameter choice. 
* `plot_clusters_kerneldensity_param` - compares cluster-specific parameters across (non-empty) clusters via kernel density estimates, use `plot_clusters` for a single scalar parameter choice. 

These functions are wrapped by the `plot` method for `"clustglmm"` objects that covers all of these options. The trace plots of total cluster occupation numbers `ng` are shown by default. Based on these plots, the burn-in period for dropping of samples, a suitable thinning of the samples as well as even a set of suitable iterations may be selected. The plots are automatically organized into a matrix of convenient sizes by setting `par(mfrow = ...)`. This can be prevented by setting `setparmfrow = FALSE`. 

By default, `plot(mcmc)` produces trace plots of the cluster occupancy numbers with the use of `plot_ng_trace_chain_split()` where separate plots are created for each chain and each plot contains the trace plots for all clusters within one chain. This default plot allows to easily identify which clusters are empty (or in the process of emptying). 
For all other specifications, the  `plot` method for `"clustglmm"` objects plots all chains together into one plot but using only a single model parameter. If this is also desired for `what = "ng"`, set `ng_trace_chain_split = FALSE`. <!-- In addition, the `plot` method is also able to arrange plots for parameters of the same purpose, e.g., all regression coefficients for certain type of outcome. -->

In case a univariate parameter is of interest, `plot_diagnostics()` displays the four plots consisting of ACF, ECDF, kernel density and trace plot on a $2 \times 2$ grid. To select the univariate parameter properly, one needs to provide (if applicable) `what` (the name of the parameter), `yspec` (to which outcome it belongs), `gspec` (to which group it corresponds) and `dimspec` (the specific dimension). The `settings` matrix provides the information on what has to be supplied for which parameter. There is also the possibility to include a line in the plot indicating the true value of the parameter, if known. 
Table \@ref(tab:settings-table) summarizes the key parameter attributes.

```{r get-some-mcmc, echo = FALSE, results = FALSE, eval = TRUE, message = FALSE}
data("longitudinal_mixed_type_data")
id <- "i"

family <- c("num", "poi", "bin", "ord", "cat")
names(family) <- Ys <- c("ynum", "ypoi", "ybin", "yord", "ycat")

## Create list of formulae for each outcome
formula <- list()
for(y in Ys){
  formula[[y]] <- list()
  formula[[y]]$fixed <- ~ f
  formula[[y]]$group <- ~ x
  formula[[y]]$random <- ~ 1
  formula[[y]]$offset <- ""
}

Gmax <- 5     # maximal number of mixture components to be considered
nchains <- 2  # number of chains to be sampled 

## No inits specified
set.seed(123456789)
mcmc <- clustGLMM(formula = formula, id = id, family = family, 
                  data = longitudinal_mixed_type_data, 
                  G = Gmax, iter = 10, nchains = nchains)
```


```{r settings-table, echo = FALSE, tab.cap = "Information on the model parameters stored in \\texttt{settings}: D = 0 indicates scalar, D = 1 vector, D = 2 matrix. g indicates group-specificity (? depends on choice). y indicates if specific to an outcome. yd1, yd2 indicates if specific to an outcome in the first or second dimension. N, P, B, O, C indicate the outcome type. sym indicates whether the matrix is symmetric and diag whether it contains the diagonal. "}
m <-mcmc$settings[[1]][c("D", "gspec", "isy", "ydepd1", "ydepd2", "ynums", "ypois", "ybins", "yords", "ycats", "sym", "diag")]
m[c("prec_num", "var_num", "sd_num", "c_ord", "a_ord", "pi_ord", "InvSigma", "Sigma", "sdSigma", "corSigma", "detInvSigma", "InvQ", "Q", "detInvQ", "naY"), "gspec"] <- "?"
for(col in c("gspec", "isy", "ydepd1", "ydepd2", "ynums", "ypois", "ybins", "yords", "ycats", "sym", "diag")){
  m[m[,col] == TRUE, col] <- "T"
  m[m[,col] == FALSE, col] <- "F"
}
colnames(m) <- c("D", "g", "y", "yd1", "yd2", "N", "P", "B", "O", "C", "sym", "diag")
kable(m, booktabs = TRUE)
```

If the visual diagnostic tools suggest that the MCMC sampling has not converged to the stationary distribution yet, sampling can be continued by calling `clustGLMM()` with the same input arguments and `inits = ...$last`. If the results seem not suitable, for example, from a clustering perspective, it could be helpful to adjust the input arguments or start sampling with a different seed. 

## Within-chain post-processing {#impl-post-processing}

In case of a clear clustering structure in the data and suitable model specification, one potentially may  skip this step. Otherwise, this step is necessary to resolve issues due to iterations recorded during sampling where the numbers of non-empty components differed within one chain or due to the meaning of the labels being ambiguous because of label switching between iterations. Function `post_processing()` allows to address these issues by implementing suggestions by @FruhwirthSchnatter11 and @Malsiner-Walli-etal2026clips. The function retains only iterations where the number of non-empty components corresponds to the most common number of non-empty components `Gplus` saved as `modeGplus` in the `"clustglmm"` object and resolves label-switching to enable inference about cluster-specific parameters to be based on an identified model. 

The function implements the following steps for each chain separately:

0. If sampling results are saved in format `"list"`, convert them to format `"data.frame"` and then back after the last step. This function requires format `"data.frame"`.
1. Take the most frequent `Gplus` value - `modeGplus` - for the number of non-empty components across the sampled chain and retain only iterations where `Gplus = modeGplus`. <!-- The indices are stored in `inds` and their length is `iters`. -->
2. Select all cluster-specific parameters with the exception of `w`, `U`, `pUig`, unless specific columns to be used with `kmeans()` are indicated.
3. Gather the cluster-specific parameters from non-empty components for each retained iteration. Each parameter will have `modeGplus` values.
4. Stack the values for the same parameter into one long column for every considered cluster-specific parameter, resulting in a tall matrix where the number of rows correspond to the number of iterations retained times `modeGplus`. 
5. Apply $K$-means clustering using `kmeans()` to the matrix from 4. with `centers = modeGplus`, and `iter.max = 200` and `nstart = 30` as default options. Ideally, rows corresponding to the same iteration are assigned to different centroids in the $K$-means clustering solution. In this case, values are permuted accordingly. If all rows for one iteration are not assigned to `modeGplus` different clusters, then this iteration is eliminated. In particular, values for cluster-specific parameters are replaced with `NA`. The summary of `post_processing()` informs about the percentage of iterations removed.
6. Apply the same permutations to the rest of the group-specific parameters and transform the parameters back to the wide format for the iterations retained.
7. Create the data frame of posterior draws by joining cluster-invariant samples, permuted (and aligned) group-specific and clustering-related parameters.
8. Update all other items of the `"clustglmm"` object, accordingly, including in particular `clustering`, `certainty`, `G` (now `modeGplus`, possibly different among chains), `last`, `call` (summary of the post-processing) and many others.

The return value of this post-processing step is again of class `"clustglmm"` and all plotting functions should still work with the reduced set of iterations. 

## Across-chain post-processing {#impl-permute-cluster-labels}

Even though label switching has been resolved in the previous step for each chain, the interpretation of the labels may still not match across multiple chains. This is one of the drawbacks of random initialization and not fixing the meaning of the clusters a-priori. <!-- One way to deal with it would be to take favorite chain and use its `last` state as initial value for every other chain and continue in sampling. -->  To resolve the label switching across chains, we provide function `permute_cluster_labels()`. The user can supply a list of permutations `perm` (one for each chain) and all quantities with a clustering index in the given `"clustglmm"` object are permuted. Suitable permutations could for example be determined by inspecting the plots for cluster-specific parameters across chains. If, however, the `perm` argument is not specified, the Hungarian algorithm is used to align cluster labels of all chains with the first chain. The cost matrix for the Hungarian algorithm is determined based on the root mean square error between cluster-specific parameters.

Once the cluster labels are aligned across chains, inference may be based on the sampled values from all chains similarly to the usual behavior in the **coda** package. In some cases, such an alignment may not be possible, e.g.,  when the number of non-empty components does not match across chains. We recommend to then proceed performing inference for each chain separately, potentially focusing on the chain providing the most reasonable and the best interpretable results. 

## Inference {#impl-inference}

### Posterior description {#impl-summary}

A first basic posterior exploration may be performed using the `summary` method for objects of class `"clustglmm"`. The outputs are obtained using the `summary` method for `"mcmc"` object with `na.rm = TRUE` after transforming the `"clustglmm"` object with `as_coda` for each chain separately. The `print` method automatically prints a `quantiles` table and the posterior mean from `statistics` for all chains contained in `$chains`, unless specified differently. Since the list of model parameters can be long, we divided them into several groups: cluster-invariant parameters (`"inv_param"`), clustering-related parameters (`"clust_param"`, i.e., `Gplus`, `e0` and the component sizes), latent variables (`"latent_param"`) and group-specific parameters (`"group_param"`, i.e., the cluster-specific regression coefficients). Users can specify the groups of parameters which should be printed via argument `which`. Argument `which` may specify several groups and also include `"mcmc"` (to obtain a chain overview) and `"clustering"` (to obtain a clustering overview). The summary of the clustering is based on `clustering` and `certainty` in the `"clustglmm"` object where all subjects are classified into the cluster with the highest certainty. In addition, the posterior distribution of the model parameters can also be explored visually using the `plot` method.  <!-- If some subjects should remain unclassified (label 0) due to insufficient certainty level, this has to be done manually by the user. -->

We also provide functions for plotting the outcomes against one of the continuous covariates: `plot_num_vs_x_grouped()` and `plot_cat_vs_x_grouped()`. The argument `group` specifies the name of a column in `data` which partitions the subjects into groups, e.g., the resulting clustering. Every cluster is plotted with one of the colors obtained with `colorspace::rainbow_hcl()` with the color slate-grey being reserved for label 0 (unclassified). 

With `plot_num_vs_x_grouped` the relationship of outcome and continuous covariate `x` is depicted by spaghetti plots. Observations corresponding to the same subject (i.e., having the same ID) are joined with thin segments (ordered by the $x$-axis covariate `x`). A thick line of darker color shade then corresponds to the `loess.smooth` curve. Since the clusters can overlap, we recommend to plot subsets of the data separately for better visualization. If `x` is a factor, then overlapping boxplots with transparent colors are plotted for each level and group. 

<!-- TODO - think about implementing of plots only for subset of groups instead of all of them (default).-->

With `plot_cat_vs_x_grouped` the relationship of a factor outcome and a continuous covariate `x` is depicted by spineplots. For each cluster separately, stacked proportions of outcome labels are plotted for categories of `x` determined by `xbreaks`. By default, `x` is categorized using 11 equidistant points from minimum to maximum as breaks. If `x` is already a factor, then the spineplot is created using this categorization. 


### Linear predictor {#impl-predict}

Function `plot_num_vs_x_grouped()` plots loess curves which are quick and easy to plot, yet do not represent the posterior estimates. Therefore, we also provide a `predict` method for `"clustglmm"` objects that constructs the linear predictor for given `newdata`. For example, the majority of covariates could be fixed at specific values and the remaining covariates may be systematically varied on a grid and then the `predict` method used to capture their effect. 

Two methods are implemented to obtain a posterior linear predictor point estimate. The plug-in method takes the posterior point estimates of the regression coefficients and computes the corresponding linear combinations yielding a point estimate of the same type. The other method takes the posterior samples of the regression coefficients and computes linear combinations for all iterations. In this way, samples to approximate the full posterior are obtained, and, for example, credible intervals (for individual points, not the whole curve) could be computed.

### Clustering probabilities and deviance {#impl-clust-prob-deviance}

After MCMC sampling, estimates of the posterior probabilities for an observation to be assigned to each of the clusters might be obtained based on the sampled allocation indicators. In some cases, these estimates could be rather crude. An alternative approach, which potentially results in better estimates, builds on the fact that the posterior probability of a subject to belong to one of the clusters could be determined directly for specific parameter values, in case one integrates over the random effects. Function `clustering_probabilities_and_deviance()` provides this by resolving the integral using numerical approximation using Laplace approximation or Adaptive Gaussian Quadrature with Hermite polynomials. The `data` argument does not necessarily need to specify the same dataset, which was used to fit the model. It could potentially also contain newly observed data (for which we do not have estimates of cluster allocation indicators `U` or random effects `b`). To speed up the computation, parts of this functions are written in C, which requires similar internal preparation of inputs as for function `clustGLMM()`. 

<!-- From our experience, these computations can take a long time. We thus implemented the function in a way that allows for easy parallelization. The function can focus not only on a given chain, but also on specific iterations specified by arguments `start`, `end` and `thin`. For example, using `thin=10` and `start=1`, .... `start=10`, we divided the computation of one chain into 10 jobs that were computed in parallel on computational cluster. Then, after proper aggregation of these outputs one obtains a `"clustglmm"` object and can use the already established plotting functions.-->

The returned object is of class `"clustglmm"` and contains the following additional variables:

* `pUig_int` - group-specific vector of clustering probabilities (random effects integrated out) for given individuals (unique id values aligned with `1:n`), e.g., `pUig_int(2)[1]` is the probability that the first individual is assigned to cluster 2. 
* `dev_i` - vector of individual contributions to the deviance.
* `deviance` - a scalar value corresponding to the sum of all individual contributions. To correspond to the deviance of the model, the data specified in the `data` has to include all observations, on which the model has been fitted.

Moreover, `clustering` and `certainty` of the returned `"clustglmm"` object are now based on the posterior mean of the clustering probabilities. Also, the `call` argument has been changed to inform about the use of `clustering_probabilities_and_deviance()`. 



# Illustrative use of the package {#sec:psurvey}

We created an artificial longitudinal dataset `psurvey` and included it in package **clustGLMM**. We use this dataset to demonstrate the use of the package and illustrate a broad range of the functionality covered. The dataset emulates respondents observed repeatedly during their age span of 18--40 years. The total number of observations $n_i$ per person $i$ differs and the  observation times are uniformly distributed across the range of age years 18--40. Variable `age` is the predictor variable in the regression models, i.e., we are interested to assess how the outcome variables change with age for different groups of respondents.

We observe three outcomes of different type:

* `income` (numeric): the total income in the last year (in artificial units),
* `ntrips` (count): the number of vacation trips undertaken in the last year,
* `satisfaction` (ordinal): the level of life satisfaction (0=dissatisfied < 1=neither-nor < 2=satisfied).

We generated the data assuming three latent groups which differ in the regression coefficient parameters, i.e., how the mean of the outcome variables depends on age. In particular, the numeric outcome variable alone does not allow to differentiate between the three groups. The three outcome variables in combination, however, allow us to distinguish between the three groups.

We load the dataset and inspect the first few entries:

```{r psurvey-load, echo = TRUE}
data("psurvey", package = "clustGLMM")
head(psurvey)
```

```{r psurvey-n}
n <- length(unique(psurvey$i))
```

The dataset contains in total observations for $n = `r n`$
respondents.  Column `i` represents the id variable for the
respondents and `j` gives the observation number (time point) for that
respondent.  The three outcome variables are accompanied by the
`age` of a person when interviewed, which is the only explanatory
variable.

The dataset `psurvey` does not contain any missing values. In order to
be able to show that package **clustGLMM** can also be used when
there are missing values in the outcome variables, we insert missing
values into each outcome and use this dataset in the following. 

```{r psurvey-n-nmis, warning=FALSE, echo=TRUE}
set.seed(31415)
nmis <- 3
na_indices <- na_true_vals <- list()
for (y in c("income", "ntrips", "satisfaction")) {
  na_indices[[y]] <- sort(sample(seq_len(nrow(psurvey)), nmis))
  na_true_vals[[y]] <- psurvey[na_indices[[y]], y]
  is.na(psurvey[na_indices[[y]], y]) <- TRUE
}
```

We use functions `plot_num_vs_x_grouped()` and
`plot_cat_vs_x_grouped()` from the package to inspect how the outcomes
depend on age. The argument `add_lowess` adds a smoothing line to
indicate the trend. Note that `_grouped` in the name of the plot
function indicates that potentially an argument `group` could be
specified which relates to a known classification of observations. We
assume that no classification is available and leave that argument
unspecified which means that the default of no group assignment is
used.

```{r psurvey-outcomes-vs-age, echo=TRUE, fig.align="center", fig.width=10, fig.height=4, fig.cap="The evolution of income, number of trips and satisfaction level by age.", out.width="100%"}
par(mar = c(4, 4, 1, 0.5), mfrow = c(1, 3))
plot_num_vs_x_grouped(psurvey, "income", x = "age", id = "i", add_lowess = TRUE)
plot_num_vs_x_grouped(psurvey, "ntrips", x = "age", id = "i", add_lowess = TRUE)
plot_cat_vs_x_grouped(psurvey, "satisfaction", x = "age") 
```

Figure \@ref(fig:psurvey-outcomes-vs-age) shows that on aggregate
there is an increasing trend in the numerical outcome `income`, that
also the average number of trips changes over time, and that the
satisfaction distribution is rather stable over time. However, the
true data generating mechanism consists of three different groups with
different evolution profiles by age. This heterogeneity is not easily
discernible from these plots. We will demonstrate that the proposed
model is able to infer the group structure and the different evolution
profiles of these underlying groups.

## Preparing inputs {#subsec:psurvey-prepare-inputs}

We start by preparing all necessary input to call the main function
`clustGLMM()`. In particular, we create a named vector `family` of
length equal to the number of outcomes which specifies the asumed family type
of each outcome. The family type names are not unique and one can for
example either use `"num"` or `"gaussian"` to indicate that the
outcome is numeric and a normal distribution should be used, i.e., the
following two specifications of `family` are equivalent for our example:

```{r psurvey-family, echo=TRUE}
family <- c(income = "num", ntrips = "poi", satisfaction = "ord") 
family <- c(income = "gaussian", ntrips = "poisson", satisfaction = "cumulative")
```

Then, for each outcome a regression model is specified.  In
particular, we assume no fixed effects and group-specific effects of
age. We assume that `ntrips` and `satisfaction` depend linearly on
`age`, while for `income` also a squared effect of `age` is
included. We also include random intercepts using `random = ~ 1`.

```{r psurvey-splines, echo=TRUE}
formula <- list()
formula[["income"]] <- list(fixed = ~ 0, group = ~ age + I(age^2), random = ~ 1)
for (y in c("ntrips", "satisfaction")) {
  formula[[y]] <- list(fixed = ~ 0, group = ~ age, random = ~ 1)
}
```

Function `default_varying()` returns a named logical vector indicting
for other parameters which could potentially be group-specified, whether they are to be
fitted group-specific. The default values are:

```{r psurvey-defaults, echo = TRUE}
varying <- default_varying()
varying
```

The standard deviation of the error of a numeric outcome thus is by
default estimated group-specific. In the following we change that and
impose the restriction that the standard deviation is the same across
groups.

```{r psurvey-defaults-1, echo = TRUE}
varying["prec_num"] <- FALSE
```

Function `default_save()` returns a named logical vector which can be
used to indicate for different parameters if their generated chains should be saved and
returned. The default setting implies that both the chains and the posterior estimates of
the missing values are not returned and this can be changed by setting
the argument `naY` to `TRUE`.

```{r psurvey-defaults-2, echo = TRUE}
save <- default_save(naY = TRUE)
```

The default setting of the priors on the model parameters are obtained
using `default_param()`. We change the rate of the Gamma prior on the
precision of the normal prior on the numeric outcome to `15`.

```{r psurvey-defaults-3, echo=TRUE}
param <- default_param(prec_num_rte = 15)  
```

In the following, we use the default options for the argument `tuning`
when calling `clustGLMM()` which are obtained with `default_tuning()`.

## MCMC sampling {#subsec:psurvey-sample-chains}

We sample just one randomly initialized chain of short length for $G =
10$, i.e., specifying the number of components considerably higher
than the true number of clusters. We call `clustGLMM()` and specify
the regression model using `formula`. Argument `id` specifies the
grouping variable for the random effects and `family` the distribution
of the outcome variables and the data frame to evaluate the formula is
in `data`. 

```{r psurvey-mcmc-initial, echo=TRUE, eval=TRUE, results=FALSE}
set.seed(12345678)
mcmc_burnin <- clustGLMM(formula, id = "i", family, data = psurvey, G = 10, 
                         varying = varying, save = save, param = param, nchains = 1, 
                         iter = 1000, standardize = TRUE, howsave = "data.frame")
```

The setting for the MCMC sampling is specified by the number of
components `G`, the number of parallel chains `nchains` and the number
of iterations `iter`. The further specifications of the model include
the restrictions on the parameters given in `varying` and the prior
settings in `param`. The parameters to save and return are indicated
in `save`.  The argument `standardize` defines if the numeric outcomes
and covariates are standardized before analysis. Argument `howsave`
specifies how the MCMC draws are returned. We set `howsave = "data.frame"`
to return them as a data frame which allows to easily access the draws of
the different parameters.

If `inits` is not provided, function `clustGLMM()` initializes the
MCMC sampling by randomly assigning observations to equally sized
components and, based on the obtained partition of the data,
estimating the model parameters by separate regression models (without
random effects). The sampling is then implemented in C to speed up
computations. Finally, an object of class `"clustglmm"` is returned
which contains among others information on the fitted model, the
posterior draws, a list with the last sampled values. The MCMC draws
returned consist of all draws without omitting any iterations as
burn-in. 

Trace plots of the MCMC draws of the parameters or inferred quantities
allow to assess convergence. In particular, inspecting the evolution
of the component occupancy numbers `ng` is of interest in this
context. We can use the `plot` method for this. We can easily assess
how `ng` changes separately for each component by setting
`ng_trace_chain_split = FALSE`.

```{r psurvey-mcmc-burnin-plot-ng, echo=TRUE, results=FALSE, fig.align="center", fig.width=10, fig.height=4, fig.cap="Traceplots - component occupancy numbers during MCMC sampling for the 10 components.", fig.pos="H", out.width="100%"}
plot(mcmc_burnin, ng_trace_chain_split = FALSE)
```

Figure \@ref(fig:psurvey-mcmc-burnin-plot-ng) clearly shows that some
of the `r mcmc_burnin$G[1]` components are immediately emptied and that
`r length(unique(mcmc_burnin$last[[1]]$U))` non-empty components remain
in the end. One can continue sampling from the last iteration using:

```{r psurvey-mcmc, echo=TRUE, results=FALSE}
set.seed(1234)
mcmc <- clustGLMM(formula, id = "i", family, data = psurvey, G = 10, 
                  varying = varying, save = save, param = param, iter = 1000, 
                  inits = mcmc_burnin$last, nchains = 1, standardize = TRUE)
```
	
Note that sampling MCMC chains can take several seconds or minutes,
depending on the size of the dataset, the complexity of the model, the
number of chains and the number of iterations. The `print` method
without specifying any arguments provides an overview on the dataset,
the defined regression models for the different outcomes and the
simulation specification used for MCMC sampling.

```{r psurvey-mcmc-print, echo = TRUE, results=TRUE}
print(mcmc)
```

## Visual diagnostics  {#subsec:psurvey-visual-diagnostics}

The default plot for objects returned by `clustGLMM()` combines the
trace plots of component occupancy numbers `ng` for each chain into
one plot (with argument `ng_train_chain_split = TRUE` by default). In
addition argument `what` can be used to specify another parameter to
be plotted, e.g., the precision of the numeric outcome, and `yspec` to
indicate the outcome variable. To combine the two plots into one plot
we specify `par()` accordingly and set argument `setparmfrow` to
`FALSE`.

```{r psurvey-mcmc-ng-prec-num, echo=TRUE, results = FALSE, fig.align="center", fig.width=10, fig.height=4, fig.cap="Trace plots - cluster occupancy numbers (left) and precision of the errors for income (right).", fig.pos="H", out.width="80%"}
par(mfrow = c(1, 2))
plot(mcmc, ng_trace_chain_split = TRUE, setparmfrow = FALSE)
plot(mcmc, what = "prec_num", yspec = "income", setparmfrow = FALSE)
```

Figure \@ref(fig:psurvey-mcmc-ng-prec-num) contains the default output
of the `plot` method on the left-hand side. The plot indicates that
the three filled components slightly change in size over the
iterations, but in general their size is rather stable. This indicates
that the sampler has already converged to a specific clustering
solution and this chain is suitable for cluster-specific
inference. If, by contrast, still strong changes in size were
observable or, in particular, if the emptying of some components was
still discernible, one should consider continuing sampling and
dropping iterations where superfluous components are not emptied yet.

Even if the trace plots of the component sizes indicate that the
stationary distribution is obtained, this might not be the case for
other model parameters. One can check the trace plot of each model
parameter by specifying the `what` argument in a suitable way. Figure
\@ref(fig:psurvey-mcmc-ng-prec-num) on the right showcases this output
for the precision parameter of the error for the numeric outcome
`income`.  The possible parameter names are explained on the man page
obtained using `help("default_save")`. See also Table 2 in the
supplementary material.

One might also be interested in focusing on a single specific scalar
model parameter and inspect all four implemented plot types: trace
plot, kernel density estimate, empirical cumulative distribution
function (ECDF) and auto-correlation function (ACF). These plots can be
obtained with function `plot_diagnostics()`. 
<!-- This function also has -->
<!-- the option to add a line for the true value of the parameters, if -->
<!-- known.  -->
Note that we specify the outcome to which the parameter belongs
(`yspec`), but neither group (`gspec` empty, precision is not
considered group-specific) nor dimension (`dimspec` empty, scalar
parameter). When in doubt about the specification, check
`mcmc$settings[[1]]`. Moreover, every plotting function is equipped
with an option to trim iterations from a burn-in phase and thin the
remaining samples to reduce auto-correlation.

```{r psurvey-mcmc-beta-3-num, echo=TRUE, results=FALSE, fig.align="center", fig.width=8, fig.height=5.6, fig.cap="The four implemented plots - regression coefficient for squared age for income.", fig.pos="t", out.width="80%"}
plot_diagnostics(mcmc, what = "beta_num", yspec = "income", dimspec = 3, gspec = 3,
                 burnin = 0, thin = 2)
```

The plots in Figure \@ref(fig:psurvey-mcmc-beta-3-num) show that the
effect of the squared age on income in component 3 
differs from zero, indicating that the effect of age on income is
nonlinear in this component.

## Within-chain post-processing {#subsec:psurvey-post-processing}

We take inspiration from the CliPS approach
[@Malsiner-Walli-etal2026clips] to obtain an identified model and
solve the label switching issue. In particular, we cluster the MCMC
draws of the parameters of the filled components based on the
label-invariant point process representation using $k$-means. This
post-processing step is implemented in the function
`post_processing()` which can be applied to objects returned by
`clustGLMM()`. This function focuses only on the draws where the
number of non-empty components `Gplus` is equal to the most frequent
number of non-empty components. In our example the most frequent
number of non-empty components is `r mcmc$modeGplus[1]`.  Only draws
of iterations where the number of non-empty components corresponds
exactly to `Gplus` are kept for further analysis. Additionally, draws
of parameters sampled from empty components are dropped and the
remaining draws are relabeled to 
`r paste(1:mcmc$modeGplus[1], collapse = ", ")`.
In addition, the labels of each draw are - if possible - permuted
according to the $k$-means clustering solution. Parameter draws where the
$k$-means clustering solution does not induce a permutation are
replaced with `NA`.

```{r psurvey-post-processing, echo = TRUE, results = TRUE}
set.seed(123456)
(mcmc_pp <- post_processing(mcmc))
```

The `print` method of the post-processed MCMC draws summarizes how
successful this procedure was. In this example, some iterations were
dropped because their draws did not contain exactly 
`r mcmc$modeGplus[1]` filled components. However, for all remaining draws
the label switching issue could be resolved. The returned object thus
only contains the draws of these iterations for the non-empty components with
suitable labels from 1 to `r mcmc$modeGplus[1]`.

Note that the success of this post-processing step may vary depending on the
parameters used for clustering, the random initialization used for
`kmeans()` as well as its argument specification, which can be changed
by setting the arguments `cols`, `iter.max` and `nstart`.

## Inference {#subsec:psurvey-inference}

### Describing the posteriors {#subsubsec:psurvey-summary}

After successfully obtaining an identified model, we inspect the
posterior distribution of the model parameters to perform inference
for cluster-invariant as well as group-specific parameters. In
particular the group-specific parameters are of interest to
characterize the clusters. An overview can be obtained via the
`summary()` method which applies the `summary()` method for `"mcmc"`
objects as defined by the **coda** package for each chain separately
and prints only posterior mean and quantiles. In addition, there is
also the option to convert the draws into a `"mcmc.list"` object with
function `as_coda()` and utilize the framework for this kind of
objects.

Printing the posterior inference results for all model parameters
would produce a very long output. Hence, we decided to split the
summary into several parts.  We start with the subset of
cluster-invariant model parameters where the name contains the pattern
`"income"`:

```{r psurvey-mcmc-summary-inv-param, echo = TRUE, results = TRUE, R.options = list(digits = 2)}
sum_mcmc <- summary(mcmc_pp, burnin = 0, thin = 1)
print(sum_mcmc, which = "inv_param", pattern = "income")
```

The only cluster-invariant parameter for the outcome `income` is the
precision of the error.  We thus obtain the summary of the posterior
of the precision parameter of the error for outcome `income` as well
as the derived quantities, standard deviation and variance.

The summary for group-specific parameters reports the statistics for
every group-specific parameter. In the following, we only summarize
the group-specific parameters for cluster 1 where the parameter names
contain the pattern `"ntrips"`.

```{r psurvey-summary-group-param, echo = TRUE, results = TRUE, R.options = list(digits = 2)}
print(sum_mcmc, which = "group_param", clusters = 1, pattern = "ntrips")
```

The differences between posterior distributions of parameters across
different clusters can also easily be visualized using the
specification `"clusters_..."` for `which` argument in the `plot` method.

```{r psurvey-mcmc-pp-beta-num-1, echo = TRUE, results=TRUE, fig.align="center", fig.width=10, fig.height=4, fig.cap="The regression coefficients of income - posterior kernel density estimates, clusters distinguished by color.", fig.pos="t", out.width="100%"}
plot(mcmc_pp, what = "beta_num", yspec = "income", which = "clusters_kerneldensity")
```

Figure \@ref(fig:psurvey-mcmc-pp-beta-num-1) clearly demonstrates that
one cluster differs from the other two that seem to be rather similar,
in regard to `income`.

Next we provide the summary of the model parameters related to the
component weights:

```{r psurvey-mcmc-summary-clust-param, echo = TRUE, results = TRUE}
print(sum_mcmc, which = "clust_param")
```

Finally, we might be interested in inspecting the predictive
distributions for the `r nmis` missing values for `income`:

```{r psurvey-mcmc-summary-latent-param, echo = TRUE, results = TRUE}
print(sum_mcmc, which = "latent_param", pattern = "^naY_income")
```

We also plot the kernel density estimates of the posterior
distributions of the missing values. We add a dashed seegrean line
indicating the true value to the plot as well as dotted black lines to
indicate the 2.5% and the 97.5% quantiles of the posterior
distributions. Figure \@ref(fig:psurvey-mcmc-naY-income) shows that the true values
are always covered by these 95% credible intervals.

```{r psurvey-mcmc-naY-income, echo=TRUE, results=TRUE, fig.align="center", fig.width=10, fig.height=4, fig.cap="Predictive distribution for missing outcome values of `income`.", fig.pos="H", out.width="100%"}
par(mar = c(4, 4, 0.5, 0.5), mfrow = c(1, 3))
y <- "income"
for (j in seq_len(nmis)) {
  plot_kerneldensity(mcmc_pp, what = "naY", yspec = "income", dimspec = j, labcex = 0.8)
  abline(v = as.numeric(na_true_vals[["income"]][j]), col = "seagreen", lty = 2)
  abline(v = sum_mcmc$coda_summary[[1]]$quantiles[paste0("naY_income[", j, "]"),
         c("2.5%", "97.5%")], col = "black", lty = 3)
}
```

## Extracting specific MCMC draws {#subsec:psurvey-return-object}

It is also possible to extract specific draws from the rich MCMC output
included in the object returned by `clustGLMM()`. To understand how to
select specific parameters, an overview over the sampled parameters and
how to address them is helpful. The names can be inspected using
`str(mcmc$param_names)`:

```{r psurvey-mcmc-names, echo=TRUE}
str(mcmc$param_names, vec.len = 1)
```

E.g.,
`beta_num_income(4)[3]` contains the draws of coefficient three in the
regression formula for `income` in component 4, whereas
`prec_num_income` contains the draws of the precision for the numeric
outcome income. The occupation numbers of component 5 are stored in 
`mcmc$draws[[1]][["ng(5)"]]`, while the allocations of the third
observation are stored in `mcmc$draws[[1]][["U[3]"]]`.  The names can be
used to access the draws directly and, e.g., plot a histogram.

Draws of specific parameters such as the latent cluster assignments
$U$ of each respondent or the components sizes $w$ can easily be
extracted from the `draws` object in the return object by selecting
the list element of the relevant chain, in this case chain 1, and the
columns which start with `U` or `w`:

```{r, echo=TRUE}
U <- mcmc$draws[[1]][, grep("^U", colnames(mcmc$draws[[1]]))]
w <- mcmc$draws[[1]][, grep("^w", colnames(mcmc$draws[[1]]))]
# or using dplyr
library("dplyr")
U <- mcmc$draws[[1]] |> select(starts_with("U"))
w <- mcmc$draws[[1]] |> select(starts_with("w"))
```

<!-- ```{r psurvey-mcmc-histograms, echo=TRUE, results = FALSE, fig.align="center", fig.width=10, fig.height=4, fig.cap="Histogram of the sampled coefficient of squared age of income for component 4 (left) and sampled precision of income (right)", fig.pos="H", out.width="100%"} -->
<!-- par(mar = c(4, 4, 1, 0.5), mfrow = c(1, 2)) -->
<!-- hist(mcmc$draws[[1]]$`beta_num_income(4)[3]`, -->
<!--      main = "coefficient of squared age of income for g=4", xlab = "draws") -->
<!-- hist(mcmc$draws[[1]]$`prec_num_income`,  main = "precision of income", -->
<!--      xlab = "draws") -->
<!-- ``` -->

### Inspecting the posterior predictive distributions {#subsubsec:psurvey-predict}

Inspecting the summaries of the posterior distributions provides some
insights into how the clusters differ. However, the easiest way to
interpret the clusters is to inspect and compare cluster-specific
predictions. In our example, we are interested in how the predicted
cluster-specific outcomes vary by age. The `predict` function is
designed to follow the usual use of `predict` in R.  The functions
expects as input `newdata` and a single outcome variable name `y`.  By
default (`type = "link"`) the posterior distribution of the linear
predictor consisting of both, fixed and group-specific, parameters is
returned.

```{r psurvey-predict-basic-use, echo = TRUE, eval = TRUE}
newdata <- data.frame(age = seq(18, 40, by = 0.2))
pred1 <- predict(mcmc_pp, newdata, y = "ntrips", type = "link")
```

The function returns a list of `fit`, `lwr` and `upr` containing the
posterior mean and equal-tailed credible intervals of prescribed level
for each cluster separately. In addition, predictions of the mean on
the response scale (`type = "response"`) as well as of the response
(`type = "posterior_predictive_response"`) might also be obtained.

```{r psurvey-predict-basic-use-2, echo = TRUE, eval = TRUE}
pred2 <- predict(mcmc_pp, newdata, y = "ntrips", type = "response") 
pred3 <- predict(mcmc_pp, newdata, y = "ntrips",                    
                 type = "posterior_predictive_response")            
```

In the following we plot the cluster-specific predicted values for `ntrips`.

```{r  psurvey-prediction, echo=FALSE, results=FALSE, fig.align="center", fig.width=10, fig.height=4, fig.cap="Cluster-specific predictions of `ntrips`. Linear predictor (left), predicted mean response with 95% credible intervals (middle) and samples from the predictive distribution (right).", fig.pos="H", out.width="100%", echo = TRUE, eval = TRUE}
par(mar = c(4, 4, 1, 0.5), mfrow = c(1, 3))
COL <- colorspace::rainbow_hcl(mcmc_pp$modeGplus[1], c = 80, l = 50)

plot(c(18, 40), c(0, 3), type = "n", xlab = "age", ylab = expression(x * beta))
for (g in 1:mcmc_pp$modeGplus[1]) {
  lines(newdata$age, pred1$fit[, g], col = COL[g], lwd = 2)
  lines(newdata$age, pred1$lwr[, g], col = COL[g], lty = 2, lwd = 1)
  lines(newdata$age, pred1$upr[, g], col = COL[g], lty = 2, lwd = 1)
}  
legend("topleft", legend = 1:mcmc_pp$modeGplus[1], title = "Cluster", 
       col = COL, lty = 1, lwd = 2, bty = "n")
plot(c(18, 40), c(0, 8), type = "n", xlab = "age", ylab = expression(exp(x * beta)))
for (g in 1:mcmc_pp$modeGplus[1]) {
  lines(newdata$age, pred2$fit[, g], col = COL[g], lwd = 2)
  lines(newdata$age, pred2$lwr[, g], col = COL[g], lty = 2, lwd = 1)
  lines(newdata$age, pred2$upr[, g], col = COL[g], lty = 2, lwd = 1)
}
plot(c(18, 40), c(0, 20), type = "n", xlab = "age",
     ylab = expression(Y ~ "~" ~ Pois(exp(x * beta))))
for (g in 1:mcmc_pp$modeGplus[1]) {
  lines(newdata$age, pred3$fit[, g], col = COL[g], lwd = 2)
  lines(newdata$age, pred3$lwr[, g], col = COL[g], lty = 2, lwd = 1)
  lines(newdata$age, pred3$upr[, g], col = COL[g], lty = 2, lwd = 1)
}    
```

```{r psurvey-predict-income-and-ntrips, echo=FALSE, results=FALSE, fig.align="center", fig.width=10, fig.height=4, fig.cap="The estimated spline curves for income (left) and number of trips (right).", fig.pos="H", out.width="100%"}
# psurvey[, "Uclustering"] <- mcmc_pp$clustering[psurvey$i, 1]
# 
# age_grid <- seq(18, 40, by = 0.1)
# newdata <- data.frame(age = age_grid)
# 
# burnin <- 0
# thin <- 1
# COL <- colorspace::rainbow_hcl(mcmc_pp$modeGplus[1], c = 80, l = 50)
# 
# par(mfrow = c(1, 2), mar = c(4, 4, 1, 0.5))
# for (y in c(mcmc_pp$Nums, mcmc_pp$Pois)) {
#   plot_num_vs_x_grouped(psurvey, y = y, x = "age", g = "Uclustering",
#                         id = "i", legend_placement = "topleft", add_lowess = FALSE)
# 
#   pred <- predict(mcmc_pp, newdata, y = y, type = "response", method = "samples",
#                   burnin = burnin, thin = thin, chain = 1)
# 
#   for (g in mcmc_pp$clusters[[1]]) {
#     col <- paste0("(", g, ")")
#     lines(x = age_grid, y = pred$fit[, col], col = COL[g], lwd = 3)
#     lines(x = age_grid, y = pred$lwr[, col], col = COL[g], lty = 2)
#     lines(x = age_grid, y = pred$upr[, col], col = COL[g], lty = 2)
#   }
# 
#   ## Posterior predictive distribution for each cluster separately
#   pred <- predict(mcmc_pp, newdata, y = y,
#                   type = "posterior_predictive_response",
#                   posterior_predictive_by_cluster = TRUE,
#                   method = "samples",
#                   burnin = burnin, thin = thin, chain = 1)
#   for (g in mcmc_pp$clusters[[1]]) {
#     lines(x = age_grid, y = pred$lwr[, g], col = COL[g], lty = 1)
#     lines(x = age_grid, y = pred$upr[, g], col = COL[g], lty = 1)
#   }
# }
```

<!-- For the ordinal response, we plot the posterior estimates of the -->
<!-- specific linear predictors based on the mean as well as the lower and -->
<!-- upper bound for the 95% credible interval. We complement this plot -->
<!-- with a visualization of the induced cluster-specific posterior -->
<!-- probabilities for each category of the ordinal response. Finally, we -->
<!-- also visualize the observed data using spineplots by splitting them -->
<!-- into groups given the cluster classifications obtained with the fitted -->
<!-- model. -->

<!-- ```{r psurvey-predict-satisfaction, echo=FALSE, results=FALSE, fig.align="center", fig.width=10, fig.height=5, fig.cap="Estimated spline curves for satisfaction with shifting by intercepts (left), probability of attaining each ordinal level (middle) and spineplot for each group separately in time (right).", fig.pos="H", out.width="100%"} -->
<!-- psurvey[, "Uclustering"] <- mcmc_pp$clustering[psurvey$i, 1] -->

<!-- burnin <- 0 -->
<!-- thin <- 1 -->
<!-- age_grid <- seq(18, 40, by = 0.1) -->
<!-- COL <- colorspace::rainbow_hcl(mcmc_pp$modeGplus[1], c = 80, l = 50) -->

<!-- par(mfrow = c(1, 2)) -->
<!-- for (y in mcmc_pp$Ords) { -->
<!--   maxK <- mcmc_pp$Kord[y] -->
<!--   pred <- predict(mcmc_pp, newdata, y = y, type = "link", -->
<!--                   what = "fg", method = "samples", -->
<!--                   burnin = burnin, thin = thin, chain = 1) -->

<!--   par(mar = c(4, 4, 2, 0.5)) -->
<!--   plot(x = 0, y = 0, type = "n", -->
<!--        xlim = range(age_grid), ylim = range(unlist(pred)), -->
<!--        xlab = "age", ylab = "Linear predictor - ordered intercept", -->
<!--        main = y) -->
<!--   for (k in 1:maxK) { -->
<!--     for (g in mcmc_pp$clusters[[1]]) { -->
<!--       lines(x = age_grid, y = pred$fit[, paste0("(", g, ")"), paste0("k=", k)], -->
<!--             col = COL[g], lwd = 3) -->
<!--       lines(x = age_grid, y = pred$lwr[, paste0("(", g, ")"), paste0("k=", k)], -->
<!--             col = COL[g], lty = 2) -->
<!--       lines(x = age_grid, y = pred$upr[, paste0("(", g, ")"), paste0("k=", k)], -->
<!--             col = COL[g], lty = 2) -->
<!--     } -->
<!--   } -->

<!--   # Predicted probabilities -->
<!--   pred <- predict(mcmc_pp, newdata, y = y, type = "response", -->
<!--                   what = "fg", method = "samples", -->
<!--                   burnin = burnin, thin = thin, chain = 1) -->

<!--   par(mar = c(4, 4, 2, 2.5)) -->
<!--   plot(x = 0, y = 0, type = "n", -->
<!--        xlim = range(age_grid), ylim = range(1, maxK+2), -->
<!--        xlab = "age", ylab = "Predicted probability for level k", -->
<!--        yaxt = "n", -->
<!--        main = y) -->
<!--   axis(2, at = 0.5+1:(maxK+1), labels = paste0("k=", 0:maxK)) -->
<!--   for (p in c(0.2, 0.5, 0.8)) { -->
<!--     axis(4, at = p+1:(maxK+1), labels = rep(p, maxK+1), las = 1, cex = 0.8) -->
<!--   } -->
<!--   for (k in 1:(maxK+1)) { -->
<!--     for (g in mcmc_pp$clusters[[1]]) { -->
<!--       lines(x = age_grid, y = k+pred$fit[, paste0("(", g, ")"), paste0("k=", k)], -->
<!--             col = COL[g], lwd = 3) -->
<!--       lines(x = age_grid, y = k+pred$lwr[, paste0("(", g, ")"), paste0("k=", k)], -->
<!--             col = COL[g], lty = 2) -->
<!--       lines(x = age_grid, y = k+pred$upr[, paste0("(", g, ")"), paste0("k=", k)], -->
<!--             col = COL[g], lty = 2) -->
<!--     } -->
<!--   } -->
<!-- } -->
<!-- ``` -->

### Obtaining clustering probabilities and deviance {#subsubsec:psurvey-clust-prob-deviance}


The clustering of the observations is based on the sampled allocation
indicators.  A quick summary of the clustering can be displayed by
adjusting the `print` statement:

```{r psurvey-mcmc-pp-clustering, echo = TRUE, results = TRUE}
print(mcmc_pp, which = "clustering")
```

The most commonly sampled label for each respondent can be obtained
using `mcmc_pp$clustering` while the relative frequency of this
cluster assignment is in `mcmc_pp$certainty` (with both returning a
matrix with dimension equal to the number of respondents times number
of chains).

This clustering can be compared to the true cluster labels which are
available in the data object `psurvey_latent` using:

```{r psurvey-misclassification-U-samples, echo=TRUE, results=TRUE}
data("psurvey_latent", package = "clustGLMM")
psurvey1 <- merge(psurvey, psurvey_latent, by = c("i", "j"))
psurvey1 <- psurvey1[psurvey1$j == 1, ]
psurvey1[, "Uclustering"] <- mcmc_pp$clustering[psurvey1$i, 1]
with(psurvey1, table(g, Uclustering))
with(psurvey1, mclust::classError(g, Uclustering)$errorRate)
```

After suitable relabeling, the misclassification error is indeed very
low (`r with(psurvey1, mclust::classError(g, Uclustering)$errorRate)`).
To permute the labels and align them with the true labels function
`permute_cluster_labels()` can be used.  This function can also be
used to align the cluster labels across multiple chains. Using `perm`, we
specify how to permute the labels.

```{r psurvey-permuting-labels, echo = TRUE, eval = TRUE}
perm <- list(c(3, 1, 2))             
mcmc_perm <- permute_cluster_labels(mcmc_pp, perm = perm)
```

However, for consistency in the analysis and interpretation of the
clusters we continue working with the original post-processed MCMC
sample `mcmc_pp` where the labels remain unpermuted.

Finally, we plot the observed data again, but also include the
information to which cluster each respondent is assigned using
argument `group`.

```{r psurvey-final-clustering, echo=TRUE, results=FALSE, fig.align="center", fig.width=10, fig.height=4, fig.cap="Estimated clustering of the data. Colors refer to clusters, smoothed mean function is added.", fig.pos="H", out.width="100%"}
psurvey$Uclustering <- mcmc_pp$clustering[psurvey$i, 1]
par(mar = c(4, 4, 1, 0.5), mfrow = c(1, 3))
plot_num_vs_x_grouped(psurvey, "income", x = "age",  id = "i", group = "Uclustering",
                      add_lowess = TRUE, legend_placement = "topleft")
plot_num_vs_x_grouped(psurvey, "ntrips", x = "age",  id = "i", group = "Uclustering",
                      add_lowess = TRUE, legend_placement = "topleft")
plot_cat_vs_x_grouped(psurvey, "satisfaction", x = "age", group = "Uclustering")
```

Figure \@ref(fig:psurvey-final-clustering) clearly shows the differences
between the clusters which were not visible in Figure
\@ref(fig:psurvey-outcomes-vs-age).  Note that using the true cluster labels
available in `data("psurvey_latent", package = "clustGLMM")`, we would
essentially obtain the same differences up to label permutation.

<!-- Overall, the results may be interpreted in the following way: Group 3 -->
<!-- (blue) represents respondents who started to work earlier compared to -->
<!-- members of the other groups, presumably because they did not pursue a -->
<!-- university degree. That might explain their higher income at the age -->
<!-- 18-23 years and the slower growth of their income later in their -->
<!-- life. They start going on trips at an earlier age than other groups, -->
<!-- but due to their lower income they cannot go as often at older age. -->

<!-- Groups 1 (red) and also 2 (green) represent respondents who have a low -->
<!-- income at young age but then have a steep increase in income after an -->
<!-- age of 25 years. This might be due to them staying longer in education -->
<!-- and earning a university degree. Comparing these two groups further -->
<!-- indicates that respondents in group 1 travel a lot, in particular -->
<!-- between 25 and 30 years and again after an age of 35 years while also -->
<!-- showing an increase in satisfaction levels. By contrast, respondents -->
<!-- in group 2 rarely go on trips after the age of 20 years and show a -->
<!-- reduction in level of satisfaction when they grow older. -->

So far we considered the classification of respondents based on the
sampled indicators. Alternatively, the posterior distribution of the
clustering probabilities $u_{i,g}$ could be determined and used to
obtain clustering probabilities for already observed or new
respondents. $u_{i,g}$ is the probability induced by the model that observation $i$ belongs
to component $g$ where the random effects are integrated out. <!-- E.g., -->
<!-- the probabilities for observation $i=1$ to belong to component -->
<!-- $g=1,2,3$ is stored in `pmcmc$draws[[1]][["pUig_int(1)[1]"]]`, -->
<!-- `pmcmc$draws[[1]][["pUig_int(2)[1]"]]` and -->
<!-- `pmcmc$draws[[1]][["pUig_int(3)[1]"]]`. -->

Function `clustering_probabilities_and_deviance()` evaluates these
clustering probabilities at carefully chosen iterations (which could
be parallelized), see \@ref(eq:integral-bi-approx).

```{r psurvey-clustering-probabilities-and-deviance, echo=TRUE, results = FALSE}
pmcmc <- clustering_probabilities_and_deviance(
    mcmc_pp, data = psurvey, id = "i",
    start = 1, end = mcmc_pp$iter, thin = 10, chains = 1, NGQ = 1)
```

Using `data = psurvey`, these clustering probabilities are determined
for all original respondents.  We reduce the samples by using only
everth 10-th sample (using thinning with `thin = 10`) and the rather
crude Laplace approximation (`NGQ = 1`).

<!-- ```{r psurvey-clustering-probabilities-and-deviance-print, echo=TRUE, results = TRUE} -->
<!-- pmcmc -->
<!-- ``` -->

We compare the resulting clustering based on the clustering
probabilities to the clustering obtained based on the sampled
indicators to see that they align rather well:

```{r psurvey-clustering-probabilities-confusion, echo=TRUE, eval = TRUE}
psurvey1[, "clustering"] <- pmcmc$clustering[psurvey1$i, 1]
with(psurvey1, table(Uclustering, clustering))
```

In the following, we visualize the posterior distributions of the
clustering probabilities. We focus on subjects where the uncertainty
is high. We plot the probabilities for respondents with the four
lowest certainties to be able to inspect the competing clusters for
these observations. We use the ECDF rather than the kernel density
estimate here, since the density reaches extremely high values for one
cluster, which makes the other two clusters almost flat despite being
the two competing ones.

```{r psurvey-clust-probs-posterior, echo=TRUE, eval=TRUE, results=TRUE, fig.align="center", fig.width=10, fig.height=4, fig.cap="Posterior distribution of clustering probabilities for subjects with the lowest certainty level.", fig.pos="H", out.width="100%"}
par(mar = c(4, 4, 2.5, 0.5), mfrow = c(1, 4))
for (i in order(pmcmc$certainty[, 1])[1:4]) {
  plot_clusters(pmcmc, what = "pUig_int", dimspec = i, 
                chains = 1, setparmfrow = FALSE, doKern = FALSE, thin = 10)
  mtext(paste0("true g=", perm[[1]][psurvey1$g[psurvey1$i == i]]), 
        side = 3, line = 0.5, font = 2, cex = 0.8)
  legend("bottomright", legend = 1:mcmc_pp$modeGplus[1], title = "Cluster", 
       col = COL, lty = 1, bty = "n")
}
```

The `clustering_probabilities_and_deviance()` function may also be
used for out-of-sample predictions. It does not use any samples of the
latent data specific for each respondent. Hence, it can be used to
evaluate the clustering probabilities $u_{\mathsf{new},g}$ for a newly
observed respondent with outcomes $\mathbb{Y}_{\mathsf{new}}$ and
covariates $\mathcal{C}_{\mathsf{new}}$. Moreover, the probabilities
could be estimated sequentially as the number of observations per
respondent increases using `dynamic_prob = TRUE`.
In this case we use all samples and a better approximation with `NGQ = 3`:

```{r psurvey-probability-new-data, echo = TRUE, eval = TRUE, message = FALSE, warning = FALSE, results = FALSE}
newdata <- data.frame(i = 501, 
                      age = seq(19, 37, length.out = 10),
                      income = seq(110, 180, length.out = 10),
                      ntrips = 0:9,
                      satisfaction = c(0, 0, 0, 1, 1, 1, 1, 2, 2, 2))
mcmc_new <- clustering_probabilities_and_deviance(
    mcmc_pp, data = newdata, id = "i", dynamic_prob = TRUE,
    start = 1, end = mcmc_pp$iter, thin = 1, NGQ = 3)                                              
```

We visualize the sequentially estimated clustering probabilities for
this respondent to indicate how cluster assignment varies when more
information is available for this respondent.

```{r psurvey-probability-new-data-plot, echo = TRUE, eval = TRUE, fig.width=10, fig.height=5, fig.align="center", fig.cap="Posterior quantiles (2.5%, 50%, 97.5%) of clustering probabilities (dynamically in time) for newly observed respondent.", fig.pos="t", out.width="80%"}
sum_new <- summary(as_coda(mcmc_new))$quantiles
fit <- matrix(sum_new[-c(1:2), "50%"], ncol = 3, byrow = FALSE)
lwr <- matrix(sum_new[-c(1:2), "2.5%"], ncol = 3, byrow = FALSE)
upr <- matrix(sum_new[-c(1:2), "97.5%"], ncol = 3, byrow = FALSE)

par(mfrow = c(1, 1), mar = c(4, 4, 0.5, 0.5))
matplot(x = newdata$age, y = fit, type = "l", col = COL, lwd = 3, lty = 1, 
        xlab = "Age", ylab = "Clustering probability")
matplot(x = newdata$age, y = lwr, type = "l", col = COL, lwd = 2, lty = 2, add = TRUE)
matplot(x = newdata$age, y = upr, type = "l", col = COL, lwd = 2, lty = 2, add = TRUE)
legend("topleft", legend = 1:3, title = "Cluster", col = COL, lwd = 3, lty = 1, bty = "n")
```

As more and more observations for this new respondent are available
the clustering probabilities evolve in time as depicted in Figure
\@ref(fig:psurvey-probability-new-data-plot). The high income at young
age lead to rather high clustering probabilities for one custer, which
shifts to another cluster due to the steady income growth and the high
number of trips and level of satisfaction. These results also indicate
how the clustering uncertainty decreases with more information being
available for the individual.

# Summary {#sec:summary}

We introduced the package **clustGLMM**, which provides an MCMC
implementation for clustering longitudinal data of mixed type. The
package combines the framework of GLMMs with finite mixtures to allow
for flexible regression modeling of different outcome types and link
observations from the same individual through correlated random
effects as well as cluster memberships.

In particular, the package allows to include five different types of
outcomes in the regression models. Each outcome type may possibly be
modeled with a different formula, and the user can specify which
effects should differentiate the clusters. The functions provided do
not only enable efficient MCMC sampling based on data augmentation,
but also suitable post-processing and inspection of the fitted model
to ease the use of this model class in data analysis. Specific
features covered by the package are that it allows to deal with
missing observations, label switching, an unknown number of clusters,
insufficient burn-in iterations, prediction and the clustering of new
observations. We demonstrated the ease of usage as well as the broad
range of functionality covered on an artificial dataset.

<!-- Although we consider our implementation to be very flexible, there -->
<!-- some potential limitations for the user. For example, the prior -->
<!-- distributions (i.e., their hyperparameters) are restricted to be the -->
<!-- same for all components and within each type of outcome. Currently, -->
<!-- differences are only possible for different outcome types, which may -->
<!-- be insufficient for some applications. A similar problem holds for the -->
<!-- model parameters $\tau$ (the error precision of numeric outcomes) and -->
<!-- $c$ (the ordered intercepts for ordered outcomes), their -->
<!-- group-specificity is applied uniformly to all outcomes. Cluster -->
<!-- identification could be also resolved with a semi-supervised approach -->
<!-- where labels of some individuals are assumed to be known in -->
<!-- advance. Finally, there is also a limitation in the parameterization -->
<!-- of each outcome type: we do not allow for different link functions or -->
<!-- families. However, we opted for these simplifications to obtain a -->
<!-- suitable trade-off between flexibility and ease of use. -->

# References

```{r final, include=FALSE}
par(opar)
```
