Finite Markov chains with absorbing states and mis-specified random effects: application to cognitive data

Abstract

Finite Markov chains with absorbing states are valuable tools for analyzing longitudinal data with categorical responses. However, defining the one-step transition probabilities in terms of fixed and random effects presents challenges due to the large number of unknown parameters involved. To address this, we employ a marginal model to estimate the fixed effects across various choices of the distribution governing the random effects. Subsequently, we utilize an h-likelihood method to estimate the random effects based on these fixed effect estimates. The estimation approach is applied to analyze longitudinal cognitive data from the Nun Study. Our findings highlight that the fixed effects remain relatively robust across a wide range of assumptions. However, the analysis of random effects utilizing tools such as AIC, Q-Q plots, and gradient plots appears to be sensitive to mis-specifications in the distribution of the random effects. Our proposed approach allows researchers to verify the assumptions of random effects and provides more accurate estimation of these effects. Additionally, the precisely estimated random effects enable researchers to identify individuals at high risk for absorbing states (e.g., incurable diseases) and to determine the progression rates for certain diseases.

1. Introduction

When investigating the progression of participants toward a specific chronic disease in a longitudinal dataset, multi-state models serve as a valuable tool. Assuming that participants are observed at equally spaced time intervals, Markov chains can be employed for data analysis. The modeling of finite Markov chains may be complex and include a combination of transient and absorbing states; the use of a one-step transition matrix (P matrix) can be essential [1–4]. The probabilities in this transition matrix are linked to a series of covariates by applying multinomial logistic regressions to each row. To further address clustering effects at the subject level or between-subject heterogeneity, unobserved random effects should be included in addition to the fixed effects when modeling transition probabilities (refer to, for example, [1,2,4,5]). In typical practice, the research focus is on estimating, inferring and interpreting the fixed effects by maximizing the marginal likelihood function, with the random effects being integrated out under a strong parametric assumption. It is also well known that the estimation of fixed effects is robust to the choice of the distributions of the random effects (see, for example, [5–7]), and thus typically the normal distribution is assumed for the random effects to simplify the mathematical formulation and save the computational cost. Additionally, Huang [8] noted that the consistency of MLEs (maximum likelihood estimates) for the fixed effect and variance components in homoscedastic non-linear mixture models (NLMMs) can be proved when the response is linear in the random effects.

Instead of focusing on investigating the fixed effects, many researchers have directed their attention towards the analysis of random effects. Several studies have delved into inferring the parametric form or shape of random effects within the context of linear mixed models. Discussions on this topic can be found in Ghidey et al. [9], Shen and Louis [10], Zhang and Davidian [11], Ghidey et al. [12], and Verbeke and Lesaffre [13]. In a linear mixed model, Zhang et al. [14] estimated subject-specific rates for disease progression to predict the realized values of random effects. Additionally, Selby et al. [15] examined the variance of random effects while adjusting for subject characteristics in a linear mixed model. Furthermore, McCulloch and Neuhaus [7] investigated the impact of shape mis-specification of the distributions of the random effects in different inference targets under the framework of generalized linear mixed models (GLMMs). They concluded that mis-specification has no effect on testing within-cluster covariates, and remains robust in estimating between-cluster covariates and random effect variance, but results in moderate to large bias in the estimation of the intercept.

One question involved in analyzing random effects is the diagnosis of the random effect assumptions, which may be difficult since random effects usually are unobservable. There are several proposed approaches in the literature. For example, Lange and Ryan [16] used the empirical Bayes estimates to check the normality of the distribution of random effects; Huang [8] proposed test statistics by comparing the MLEs from modeling the original and coarsened data; Tchetgen and Coull [17] used the testing methods by comparing the MLEs from the marginal and conditional likelihood functions; and some researchers used more flexible distributions of random effects to mitigate the negative outcomes from mis-specified assumptions. However, these methods are either restricted in certain models or require complex computational techniques. Verbeke and Molenberghs [18] proposed a simple graphical diagnostic tool based on the gradient function to evaluate the adequacy of the fit with an assumed distribution of random effects. The evaluation is based on the marginal likelihood and it thus does not require any complex calculation and is capable of dealing with a wide range of mixed models.

This study focuses on a scenario involving an extensive number of random effects, emphasizing the interpretation of these effects alongside their estimation with mis-specified distributions of random effects. To estimate these random effects, we extend the three-step estimation approach introduced by Wang et al. [5], which is built upon the h-likelihood framework proposed by Lee and Nelder [19] and Lee and Nelder [20]. The new extension allows us to estimate the random effects with a wide range of assumptions. Traditionally, within the h-likelihood framework, the inference regarding the fixed effects relies on the marginal distribution of the response by integrating out the random effects, while the inference about random effects is based on the conditional likelihood given the response, which in turn often depends on the fixed effects. In the adopted three-step estimation approach, the estimated fixed effects obtained from the marginal likelihood replace the fixed effects in the conditional likelihood of random effects given the response.

The proposed concept is assessed via a cognitive dataset. In this real-world application, we assumed various parametric distributions for the random effects and compared their performances. In the estimation process, most software designed for nonlinear mixed effects models necessitates the assumption of normality f or the random effects. To handle non-normally distributed random effects within the Generalized Linear Mixed Model (GLMM) framework, integrating the probability integral transformation with standard software becomes beneficial for obtaining MLEs [21]. For this study, the NLMIXED package within SAS was employed for conducting the first step of estimation, while R was utilized for the second and third steps of estimation.

The rest of this study is structured as follows: the estimation approach and diagnostic tool are discussed in Section 2; the application to the cognitive data is given in Section 3; and the study is summarized with a discussion and conclusion in Section 4.

2. The Proposed Model

Let $\mathbf{\text{Y}}=\left({\mathbf{\text{Y}}}_1,\cdots ,{\mathbf{\text{Y}}}_N\right)$, where ${\mathbf{\text{Y}}}_i\text{=}\left(Y_{i,1},\cdots ,Y_{i,T_i}\right)$, represents the observed response variable. Here, $T_i$ denotes the number of observations for the $i\text{th}$ participant. Let $\mathbf{\text{X}}=\left({\mathbf{\text{X}}}_1,\cdots ,{\mathbf{\text{X}}}_N\right)$ be the covariate matrix, where each component ${\mathbf{\text{X}}}_i=\left(X_{i,j},\cdots ,X_{i,T_i}\right)$ denotes the covariate matrix for $i\text{th}$ participant and ${\mathbf{\text{X}}}_{i,j}=\left(X_{ij1},\cdots ,X_{ijp}\right)$. $\text{β}$ represents the parameter vector for the fixed effects, and $\mathbf{\text{V}}=\left({\mathbf{\text{V}}}_1,\cdots ,{\mathbf{\text{V}}}_N\right)$ represents the unobservable random effects, with $N$ denoting the total number of participants.

In the realm of a finite Markov chain, a state is defined as absorbing when there is no possibility of moving out of it. Conversely, a state that permits transitions is termed as transient. In our modeling approach, we are examining a total of $K$ states, where the first $k_1>0$ states are transient and the subsequent $k_2=K-k_1>0$ states are absorbing. The response $\mathbf{\text{Y}}$ takes values from the state space. If we further assume the first-order Markov property holds, we can define a one-step transition matrix (see Table 1), where $p_{r,c}=p\left(Y_{i,j}=c|Y_{i,j-1}=r\right)$ denotes the transition probability from a prior state $Y_{i,j-1}=r$ to a current state $Y_{i,j}=c$ and ${\sum }_{c=1}^K{p}_{r,c}=1$ for $r=1,\cdots ,k_1$, $c=1,\cdots ,K$.

Table 1.

One-step transition matrix

	$\text{Current}\text{States}\left(c\right)$
$\text{Prior States}\left(r\right)$	$1$	$\cdots$	$k_1$	$k_1+1$	$\cdots$	$K$
$1$	$p_{1,1}$	$\cdots$	$p_{1,k_1}$	$p_{1,k_1+1}$	$\cdots$	$p_{1,K}$
$⋮$	$⋮$	$⋮$	$⋮$	$⋮$	$⋮$	$⋮$
$k_1$	$p_{k_1,1}$	$\cdots$	$p_{k_1,k_1}$	$p_{k_1,k_1+1}$	$\cdots$	$p_{k_1,K}$

The distribution function for the $i\text{th}$ participant then can be defined as the product of the $T_i$ transition probabilities. That is,

$$f\left({\mathbf{\text{Y}}}_i\right)=\prod _{j=1}^{T_i}p\left(Y_{i,j}|Y_{i,j-1}\right),$$ (1)

where $Y_{i,0}$ denotes the initial state for the transition.

Given a prior state, the current state (response $Y_{i,j}$) takes discrete nominal values from the state space and it is natural to use a multinomial logistic regression to link these values with the covariates. To be more specific, given a prior state $Y_{i,j-1}=r$ and letting the first state be the reference category, we have

$$\log \frac{p\left(Y_{i,j}=c|Y_{i,j-1}=r,\mathit{\text{β}},{\mathbf{\text{X}}}_{i,j}\right)}{p\left(Y_{i,j}=1|Y_{i,j-1}=r,\mathit{\text{β}},{\mathbf{\text{X}}}_{i,j}\right)}={\alpha }_c+{\mathit{\text{β}}}_{r,c}^T{\mathbf{\text{X}}}_{i,j},$$ (2)

where $\mathit{\text{β}}$ is defined by combining all fixed effect parameters, $\mathit{\text{β}}={\left({\alpha }_2,\cdots ,{\alpha }_K,{\mathit{\text{β}}}_{1,2}^T,\cdots ,{\mathit{\text{β}}}_{1,K}^T,\cdots ,{\mathit{\text{β}}}_{k_1,K}^T\right)}^T$ and ${\mathbf{\text{X}}}_{i,j}$ is the $p$-dimensional covariates for the $i\text{th}$ participant at the $j\text{th}$ observation.

If we further consider the cluster effect from the participant, Equation 2 is modified as

$$\log \frac{p\left(Y_{i,j}=c|Y_{i,j-1}=r,\mathit{\text{β}},{\mathbf{\text{V}}}_i,{\mathbf{\text{X}}}_{i,j},{\mathbf{\text{Z}}}_i\right)}{p\left(Y_{i,j}=1|Y_{i,j-1}=r,\mathit{\text{β}},{\mathbf{\text{V}}}_i,{\mathbf{\text{X}}}_{i,j},{\mathbf{\text{Z}}}_i\right)}={\alpha }_c+{\mathit{\text{β}}}_{r,c}^T{\mathbf{\text{X}}}_{i,j}+{\mathbf{\text{V}}}_i^T{\mathbf{\text{Z}}}_i,$$ (3)

where ${\mathbf{\text{Z}}}_i$ is the random effect design vector and ${\mathbf{\text{V}}}_i$ is a vector of unknown random effects. Then, the transition probabilities can be solved by Equation 3, derived from the multinomial logistic regression.

$$p\left(Y_{i,j}=c|Y_{i,j-1}=r,\mathit{\text{β}},{\mathbf{\text{V}}}_i,{\mathbf{\text{X}}}_{i,j},{\mathbf{\text{Z}}}_i\right)=\left\{\begin{array}{cc}\frac{1}{1+{\sum }_{h=2}^K\exp \left({\alpha }_h+{\mathit{\text{β}}}_{rh}^T{\mathbf{\text{X}}}_{i,j}+{\mathbf{\text{V}}}_i^T{\mathbf{\text{Z}}}_i\right)}& \text{if}c=1\\ \frac{\exp \left({\alpha }_c+{\mathit{\text{β}}}_{rc}^T{\mathbf{\text{X}}}_{i,j}+{\mathbf{\text{V}}}_i^T{\mathbf{\text{Z}}}_i\right)}{1+{\sum }_{h=2}^K\exp \left({\alpha }_h+{\mathit{\text{β}}}_{rh}^T{\mathbf{\text{X}}}_{i,j}+{\mathbf{\text{V}}}_i^T{\mathbf{\text{Z}}}_i\right)}& \text{if}c>1\end{array}\right.$$

Plugging the transition probabilities into Equation 1 defines the distribution functions which depend on the covariates. That is,

$$f\left({\mathbf{\text{Y}}}_i|\mathit{\text{β}},{\mathbf{\text{V}}}_i,{\mathbf{\text{X}}}_{i,j},{\mathbf{\text{Z}}}_i\right)=\prod _{j=1}^{T_i}p\left(Y_{i,j}=c|Y_{i,j-1}=r,\mathit{\text{β}},{\mathbf{\text{V}}}_i,{\mathbf{\text{X}}}_{i,j},{\mathbf{\text{Z}}}_i\right).$$ (4)

The participants are commonly assumed to be independent of each other and the overall conditional distribution function of $\mathbf{\text{Y}}$ given $\mathbf{\text{V}}$, $\mathit{\text{β}}$, $\text{σ}$, $\mathbf{\text{X}}$ and $\mathbf{\text{Z}}$ from $N$ participants is defined as

$$f\left(\mathbf{\text{Y}}|\mathit{\text{β}},\mathbf{\text{V}},\mathbf{\text{X}},\mathbf{\text{Z}}\right)=\prod _{i=1}^N\prod _{j=1}^{T_i}p\left(Y_{i,j}=c|Y_{i,j-1}=r,\mathit{\text{β}},{\mathbf{\text{V}}}_i,{\mathbf{\text{X}}}_{i,j},{\mathbf{\text{Z}}}_i\right).$$ (5)

Multiplying Equation 5 by the distribution function of $\mathbf{\text{V}}$ yields the joint distribution function of $\mathbf{\text{Y}}$ and $\mathbf{\text{V}}$ given $\mathit{\text{β}}$, $\text{σ}$, $\mathbf{\text{X}}$ and $\mathbf{\text{Z}}$. This expression defines the joint likeli-hood function of parameters $\mathit{\text{β}}$ and $\text{σ}$ and unobservable random effects $\mathbf{\text{V}}$, denoted as $L\left(\mathit{\text{β}},\text{σ},\mathbf{\text{V}}|\mathbf{\text{Y}},\mathbf{\text{X}},\mathbf{\text{Z}}\right)$, where $\text{σ}$ represents the parameter vector determining the distri-bution of the random effects, $f\left(\mathbf{\text{V}}|\text{σ},\mathbf{\text{Z}}\right)$. That is,

$$L\left(\mathit{\text{β}},\text{σ},\mathbf{\text{V}}|\mathbf{\text{Y}},\mathbf{\text{X}},\mathbf{\text{Z}}\right)=f\left(\mathbf{\text{Y}},\mathbf{\text{V}}|\mathit{\text{β}},\text{σ},\mathbf{\text{X}},\mathbf{\text{Z}}\right)=f\left(\mathbf{\text{Y}}|\mathit{\text{β}},\mathbf{\text{V}},\mathbf{\text{X}},\mathbf{\text{Z}}\right)\times f\left(\mathbf{\text{V}}|\text{σ},\mathbf{\text{Z}}\right).$$

2.1. h-Likelihood

The proposed model of using multinomial logistic regression for each row in the one-step transition matrix falls into the category of hierarchical GLMM. Following the proposed work in Lee et al. [22], we factor the joint likelihood function of $\mathit{\text{β}}$, $\text{σ}$ and $\mathbf{\text{V}}$ to the conditional likelihood function of $\mathit{\text{β}}$ given the random effects and the marginal likelihood function of $\text{σ}$ and $\mathbf{\text{V}}$ derived from the random effect distribution function solely. That is, $L\left(\mathit{\text{β}},\text{σ},\mathbf{\text{V}}|\mathbf{\text{Y}},\mathbf{\text{X}},\mathbf{\text{Z}}\right)=L_1\left(\mathit{\text{β}}|\mathbf{\text{Y}},\mathbf{\text{V}},\mathbf{\text{X}},\mathbf{\text{Z}}\right)L_2\left(\text{σ},\mathbf{\text{V}}|\mathbf{\text{Z}}\right)$. We assume $\mathbf{\text{V}}$ is a vector of independently and identically distributed (i.i.d.) random variables with the distribution function $f\left({\mathbf{\text{V}}}_i|\text{σ}\right)$. The estimates from optimizing the likelihood function and its logarithmic transformation are equivalent; as the latter version is preferred in practice, we work with the log-likelihood function, which also leads to the h-likelihood $h\left(\mathit{\text{β}},\text{σ},\mathbf{\text{V}}\right)=\log \left[L_1\left(\mathit{\text{β}}|\mathbf{\text{Y}},\mathbf{\text{V}},\mathbf{\text{X}},\mathbf{\text{Z}}\right)\right]+\log \left[L_2\left(\text{σ},\mathbf{\text{V}}|\mathbf{\text{Z}}\right)\right]=l_1\left(\mathit{\text{β}}|\mathbf{\text{Y}},\mathbf{\text{V}},\mathbf{\text{X}},\mathbf{\text{Z}}\right)+l_2\left(\text{σ},\mathbf{\text{V}}|\mathbf{\text{Z}}\right)$.

In principle the h-likelihood can be evaluated for each choice of $\left(\mathit{\text{β}},\mathbf{\text{V}}\right)$, but the optimization involves too many unknowns including parameters and unobservable random effects. Instead, we adopt the following 3-step strategy proposed by Wang et al. [5] and summarize it here:

• Step 1: Integrate out $\mathbf{\text{V}}$ from the joint likelihood function of $\mathit{\text{β}}$, $\text{σ}$ and $\mathbf{\text{V}}$, $L\left(\mathit{\text{β}},\text{σ},\mathbf{\text{V}}|\mathbf{\text{Y}},\mathbf{\text{X}},\mathbf{\text{Z}}\right)$, to get the likelihood function $L\left(\mathit{\text{β}},\text{σ}|\mathbf{\text{Y}},\mathbf{\text{X}}\right)$. Optimize $L\left(\mathit{\text{β}},\text{σ}|\mathbf{\text{Y}},\mathbf{\text{X}}\right)$ to obtain the MLE of $\mathit{\text{β}}$ and $\text{σ}$, $\left(\widehat{\mathit{\text{β}}},\widehat{\text{σ}}\right)$.

  Several approximations to this integration are discussed in the literature; we selected the Gaussian approximation defined as follows (see, e.g. [1]):

  $$\begin{array}{l}l\left(\mathit{\text{β}},\text{σ}|\mathbf{\text{Y}},\mathbf{\text{X}}\right)=\log \left(L\left(\mathit{\text{β}},\text{σ}|\mathbf{\text{Y}},\mathbf{\text{X}}\right)\right)\\ \approx \sum _{i=1}^N\log \left\{\sum _{m=1}^M{w}_m\prod _{c=1}^{T_i}p\left(Y_{i,c}|Y_{i,c-1},\mathit{\text{β}},{\mathbf{\text{v}}}_m,{\mathbf{\text{X}}}_{i,j},{\mathbf{\text{Z}}}_i\right)f\left({\mathbf{\text{v}}}_m|\text{σ},{\mathbf{\text{Z}}}_i\right)\right\},\end{array}$$

  where $w_m$ and ${\mathbf{\text{v}}}_m$ are the weights and abscissas for the approximation respect-tively.

• Step 2: Substitute the MLE of $\left(\mathit{\text{β}},\text{σ}\right)$ from Step 1 into the h-likelihood yields $h\left(\widehat{\mathit{\text{β}}},\widehat{\text{σ}},\mathbf{\text{V}}\right)$ and solve $\frac{\partial h\left(\widehat{\mathit{\text{β}}},\widehat{\text{σ}},\mathbf{\text{V}}\right)}{\partial \mathbf{\text{V}}}=0$ to obtain the MLE of $\mathbf{\text{V}}$. This usually involves an iterative procedure such as Newton-Raphson.

• Step 3: The covariance matrix for the MLE of $\mathbf{\text{V}}$ is $\mathbf{\text{A}}={\left[{\mathbf{\text{H}}}_{11}-{\mathbf{\text{H}}}_{12}{\mathbf{\text{H}}}_{11}^{-1}{\mathbf{\text{H}}}_{12}\right]}^{-1}$, where the Hessian matrix

  $$\mathbf{\text{H}}=\left(\begin{array}{cc}{H}_{11}& H_{12}\\ H_{21}& H_{22}\end{array}\right)=\left(\begin{array}{cc}-\frac{{\partial }^{2}h}{\partial {\mathit{\text{β}}}^2}& -\frac{{\partial }^{2}h}{\partial \mathit{\text{β}}\partial V}\\ -\frac{{\partial }^{2}h}{\partial V\partial \mathit{\text{β}}}& -\frac{{\partial }^{2}h}{\partial V^2}\end{array}\right).$$

2.2. Probability Integral Transformation

In this study, we propose to investigate a wide range of assumptions on random effects, and select a small set of assumptions based on the model selection criteria, such as AIC and BIC. In fitting the GLMM model, the commonly used procedure PROC NLMIXED in SAS requires the normal assumptions for the random effects. To fit the model without normal assumptions on random effects, Nelson et al. [21] proposed using the probability integral transformation along with the standard PROC NLMIXED procedure to get the MLEs for the parameter vector. In common practice, the MLEs for the fixed effects are obtained by optimizing the marginal likelihood functions with the random effects being integrated out. That is,

$$\begin{array}{l}\left(\widehat{\mathit{\text{β}}},\widehat{\text{σ}}\right)={\mathrm{argmax}}_{\mathit{\text{β}},\text{σ}}L\left(\mathit{\text{β}},\text{σ},\mathbf{\text{V}}|\mathbf{\text{Y}},\mathbf{\text{X}},\mathbf{\text{Z}}\right)\\ ={\mathrm{argmax}}_{\mathit{\text{β}},\text{σ}}\prod _{i=1}^N{\int }_{{\mathbf{\text{V}}}_i}\left[\prod _{j=1}^{T_i}p\left({\mathbf{\text{Y}}}_{i,j}|{\mathbf{\text{Y}}}_{i,j-1},{\mathbf{\text{X}}}_{i,j},{\mathbf{\text{V}}}_i,\mathit{\text{β}},{\mathbf{\text{Z}}}_i\right)f\left({\mathbf{\text{V}}}_i|\text{σ},{\mathbf{\text{Z}}}_i\right)\right]d{\mathbf{\text{V}}}_i.\end{array}$$ (6)

The optimization (6) does not have a closed form and approximation is required. In most of the available tools, ${\mathbf{\text{V}}}_i$ is assumed to be normally distributed and numerical integration is used to obtain the estimates. To accommodate this frequently used procedure for fitting the GLMM with normal assumption, probability integral transformation could be used for the nonnormal case. According to Nelson et al. [21], if ${\mathbf{\text{R}}}_i$ follows a standard normal distribution (i.e., ${\mathbf{\text{R}}}_i\sim N\left(0,1\right)$), then ${\mathbf{\text{U}}}_i=\Phi \left({\mathbf{\text{R}}}_i\right)\sim \text{Unif}\left(0,1\right)$, where $\Phi$ denotes the cumulative distribution function (CDF) of $N\left(0,1\right)$. With the probability integral transformation, the CDF of ${\mathbf{\text{V}}}_i$, $F_{\text{σ}}\left({\mathbf{\text{V}}}_i\right)\sim \text{Unif}\left(0,1\right)$. Then, ${\mathbf{\text{V}}}_i=F_{\text{σ}}^{-1}\left({\mathbf{\text{U}}}_i\right)=F_{\text{σ}}^{-1}\left(\Phi \left({\mathbf{\text{R}}}_i\right)\right)$ has the non-normal random effect distribution, where $F_{\text{σ}}^{-1}$ is the inverse CDF of ${\mathbf{\text{V}}}_i$. Thus, the complex integral in terms of the non-normal random effect ${\mathbf{\text{V}}}_i$ can be simplified to the integral with respect to the normal random effect ${\mathbf{\text{R}}}_i$. More specifically, optimization (6) can be rewritten as

$$\left(\widehat{\mathit{\text{β}}},\widehat{\text{σ}}\right)={\mathrm{argmax}}_{\mathit{\text{β}},\text{σ}}\prod _{i=1}^N{\int }_{{\mathbf{\text{R}}}_i}\left[\prod _{j=1}^{T_i}p\left({\mathbf{\text{Y}}}_{i,j}|{\mathbf{\text{Y}}}_{i,j-1},{\mathbf{\text{X}}}_{i,j},F_{\text{σ}}^{-1}\left(\Phi \left({\mathbf{\text{R}}}_i\right)\right),\mathit{\text{β}},{\mathbf{\text{Z}}}_i\right)f\left({\mathbf{\text{R}}}_i\right)\right]d{\mathbf{\text{R}}}_i,$$ (7)

where $f\left({\mathbf{\text{R}}}_i\right)$ is the standard normal probability density function.

2.3. Diagnostic Tool

The key idea in the gradient function diagnostic tool proposed by Verbeke and Molenberghs [18] lies in checking whether an alternative random effect distribution $\mathbf{\text{G}}$ other than the assumed one $\mathbf{\text{V}}$ can achieve a larger log marginal likelihood. Motivated by it, Verbeke and Molenberghs [18] used the gradient function, developed from the directional derivative of the log marginal likelihood function $l_1$ at $\mathbf{\text{V}}$ to the direction of $\mathbf{\text{G}}$, to check the adequacy of the model fit. The directional derivative and gradient function are defined as

$$\Upsilon \left(\mathbf{\text{V}},\mathbf{\text{G}}\right)={\left.\underset{\alpha \to 0}{\lim }\frac{\partial l_1\left(\mathbf{\text{Y}},\mathit{\text{β}}|\left(1-\alpha \right)\mathbf{\text{V}}+\alpha \mathbf{\text{G}},\mathbf{\text{X}},\mathbf{\text{Z}}\right)}{\partial \alpha }\right|}_{\alpha =0}$$ (8)

and

$$\Delta \left(\mathbf{\text{V}},\mathbf{\text{v}}\right)=\frac{1}{N}\sum _i\frac{p_i\left({\mathbf{\text{Y}}}_{i,j}|{\mathbf{\text{Y}}}_{i,j-1},{\mathbf{\text{X}}}_{i,j},\mathbf{\text{v}},\mathit{\text{β}},{\mathbf{\text{Z}}}_i\right)}{{\int }_{{\mathbf{\text{V}}}_i}\left[{\prod }_{j=1}^{T_i}{p}_i\left({\mathbf{\text{Y}}}_{i,j}|{\mathbf{\text{Y}}}_{i,j-1},{\mathbf{\text{X}}}_{i,j},{\mathbf{\text{V}}}_i,\mathit{\text{β}},{\mathbf{\text{Z}}}_i\right)f\left({\mathbf{\text{V}}}_i|\text{σ},{\mathbf{\text{Z}}}_i\right)\right]d{\mathbf{\text{V}}}_i},$$ (9)

where in the numerator, the random effects are assumed to be $\mathbf{\text{v}}$ rather than being sampled from the distribution ${\mathbf{\text{V}}}_i$, and $\mathbf{\text{v}}$ is a supporting point for ${\mathbf{\text{V}}}_i$. If the assumed random effect distribution fits the model adequately without competing alternatives, the gradient function should be equal to 1 in the supporting region of ${\mathbf{\text{V}}}_i$. Thus, in visualizing the gradient function with all possible supporting points, 1 serves as the benchmark for comparison. To be more specific, if the distribution of random effects are mis-specified, the gradient function is expected to be significantly different from 1.

3. APPLICATION TO COGNITIVE DATA

In this section, we applied the proposed methods to investigate the data from the Nun Study. The original cohort consisted of 672 religious sisters who were members of the School Sisters of Notre Dame. They were all born before 1917 (i.e., 75+ years at the start of the study in 1991) and at the study baseline were living in the midwestern, eastern and southern United States. Participants received cognitive assessments approximately annually. The sample was reduced to 461 participants after excluding participates who had only one assessment, were missing APOE -ϵ4 data, or who had dementia at baseline. The number of cognitive assessments for each participant ranged from 2 to 12, with an average of 6.4 and a standard deviation of 3.1. At each visit, the participants were classified into one of the following five states:

• 1 = Intact Cognition (IC)

• 2 = Mild Impairment (MI)

• 3 = Global Impairment (GI)

• 4 = Dementia

• 5 = Death without Dementia.

The first three are transient states and the last two are absorbing states. We refer to the detailed definitions and descriptions of the states and participant classifications in Tyas et al. [23] and Iraniparast et al. [24]. The covariates we consider here include age, years of education ( ≤ 12, = 16, and > 16) and APOE -ϵ4 carrier status (yes or no). The one-step transition frequency matrix is given in Table 2.

Table 2.

One-step transition frequency table

Prior State	Current State					Total
	Intact Cognition	Mild Impairment	Global Impairment	Dementia	Death without Dementia
Intact Cognition	593	197	54	5	48	897
Mild Impairment	177	697	136	82	83	1175
Global Impairment	16	39	184	75	94	408
Total	786	933	374	162	225	2480

Assuming the normal distribution for the random effect with mean 0 and variance ${\sigma }^2$, and that the proportionality holds in the GLMM, the coefficients an d standard errors (in parentheses) for the fixed effects are given in Table 3.

Table 3.

Fixed effect coefficient estimates (standard errors; p-value) for the three prior transient states using intact cognition as the reference state, proportionality in the GLMM, and a normal distribution with mean 0 random effect.

Risk Factor	Mild Impairment	Global Impairment	Dementia	Death
Intercept	0.727 (0.362; 0.046)	2.029 (0.340; 0.000)	1.199 (0.362; 0.001)	1.557 (0.351; 0.000)
Education ≤ 12 years	2.028 (0.504; 0.000)	2.125 (0.531; 0.000)	1.952 (0.565; 0.001)	1.540 (0.562; 0.006)
Education = 16 years	0.675 (0.242; 0.006)	0.714 (0.264; 0.007)	0.580 (0.295; 0.050)	0.425 (0.277; 0.125)
Age in years	0.133 (0.021; 0.000)	0.218 (0.024; 0.000)	0.214 (0.026; 0.000)	0.243 (0.025; 0.000)
APOE-ϵ4	1.173 (0.336; 0.001)	1.632 (0.356; 0.000)	1.745 (0.381; 0.000)	1.420 (0.374; 0.000)
Prior State: Intact Cognition	−0.790 (0.362; 0.030)	−3.369 (0.354; 0.000)	−4.882 (0.563; 0.000)	−2.857 (0.366; 0.000)
Prior State: Mild Impairment	0.390 (0.350; 0.266)	−2.668 (0.332; 0.000)	−2.288 (0.351; 0.000)	−2.492 (0.346; 0.000)

To investigate the effect from mis-specification and to screen the most plausible random effect assumption, the AIC criterion is used. The data are refitted using a wide range of competing distributions for the random effect, including a normal distribution with two parameters (mean and variance), Weibull distribution with two parameters (shape and scale), Gamma distribution with two parameters (shape and scale), log-normal distribution with two parameters ( logarithm of location and logarithm of scale), exponential distribution with one parameter (scale), and ${\chi }^2$ distribution with one parameter (degrees of freedom), among others. For each non-normal distribution, the model is fitted using the probability integral transformation along with the standard procedure in PROC NLMIXED in SAS. The model selection criterion AIC is calculated for each model to check adequacy of the fit. Among all the choices, the assumptions of a normal distribution with mean 0 and a Weibull distribution yielded the smallest AIC values (5520.5 and 5521, respectively). The difference between these choices is not obvious, especially since the Weibull distribution has an additional parameter compared to the normal distribution with mean 0. In terms of the fixed effects, the two assumptions generated consistent results (Table 3).

Since it is hard to separate Weibull from the normal distribution with mean 0, we will investigate the random effects estimated from these two assumptions through the Q-Q plot and the diagnostics tool by gradient functions. Figure 1 shows the Q-Q plot made from the random effect estimates based o n t he normal distribution with mean 0 (Figure 1a) and Weibull distribution (Figure 1b) assumptions respectively. The Q-Q plot of the standardized random effect estimates assuming a normal distribution with mean 0 for random effects shows a lack of fit (Figure 1a). This plot deviates from a straight line with a curvature around 0 due to a wider range for the negative standardized random effect estimates compared to the positive standardized random effect estimates. To mitigate the curvature, a mixture of two normal distributions is assumed for the random effects and the curvature is smoothed out as suggested in Verbeke and Molenberghs [18]. However, the improvement is modest. A greater improvement is seen in Figure 1b where a Weibull distribution is assumed for the random effects. In this plot, the standardized random effect estimates are closely aligned around the fitted line, which indicates a good fit. In addition, unlike the normal case where all the standarized random effect estimates are concentrated around 0, the standardized random effect estimates from the Weibull assumption are more scattered, which indicates better order preservation.

Figure 1.

Q-Q plot for random effects estimated from normal vs. Weibull distribution assumptions

Figure 2 plots the estimated random effects from the GLMM with normal (Figure 2a) and Weibull distribution assumptions (Figure 2b). The two subplots look very similar with the center blue line being close to 1 and intervals tightly encompassing the blue line especially when the ${\mathbf{\text{V}}}_i$ takes larger values, which indicates a good model fit. However, if we compare the plot from the normal distribution assumption to that from the Weibull assumption, the latter is slightly better since it converges to 1 faster than the former and the interval towards the left end is more reasonable. The findings align with expectations because gradient plots reflect information from both random and fixed effects through the gradients of the log-likelihood function. In our analysis, the fixed effects are consistent under both assumptions, resulting in overall similarity in the shapes of the gradient plots. However, the convergence to the benchmark value of 1 differs, driven by the distinct contributions of the random effects under each assumption. Specifically, at lower values of ${\mathbf{\text{V}}}_i$, the gradient plot under the normal distribution assumption deviates significantly from the benchmark value, indicating a poorer fit. In contrast, the gradient plot for the Weibull distribution assumption remains close to the benchmark value even at smaller values of ${\mathbf{\text{V}}}_i$, supporting the preference for the Weibull distribution assumption.

Figure 2.

Gradient plot for random effects estimated from normal vs. Weibull distribution assumptions

Although the normal distribution with mean 0 assumption for random effects shows the best fit according to the model evaluation criteria AIC, the Weibull distribution assumption shows better model fit according to both the standardized random effect estimates in the Q-Q plot and both the random and fixed effect estimates in the gradient function diagnostic plot, albeit having a slightly larger AIC. One noteworthy observation from the Nun Study data analysis is the importance of verifying the usual assumption of normally distributed random effects in practice; it is essential not to rely solely on model fitting adequacy criteria as the verification tool.

We then graphed the residuals estimated from both the normal distribution with mean 0 and Weibull distribution assumptions for the random effects, revealing a clear linear relationship (see Figure 3). This linearity can be attributed to the consistency of the fixed effects for both assumptions. Additionally, it explains why the AIC values from the models with these two assumptions are closely aligned.

Figure 3.

Scatterplot of residuals of normal vs Weibull distribution assumptions

Because the Weibull assumption is preferable based on the visualizations for both fixed and random effect estimates, we further analyze the random effects from the Weibull assumption by splitting the participants into three groups based on the standardized random effect estimate for each participant: the top 5%, the middle 90%, and the bottom 5%. Within each subgroup, the risk factors and mean number of transi-tions are reported in Table 4. The top 5%, as identified by their random effects values, are considered at high risk for developing dementia or facing mortality. This classifi-cation is based on their random effects having the largest positive contributions to the associated probabilities. However, as detailed in Table 4, the top 5% group is distinguished by higher levels of education, the absence of the APOE-ϵ4 gene (associated with dementia), and a younger average age, characteristics that suggest a compara-tively lower risk profile with the terminal events expected to occur later. Contrary to this expectation, the observed outcomes indicate that terminal events occurred ear-lier, underscoring a poor model fit when relying solely on fixed effects. This limitation arises because fixed effects alone fail to account for participant heterogeneity. In con-trast, incorporating random effects effectively captures this heterogeneity, providing a more accurate representation of the data. These findings align with the concept of cognitive reserve, where those with higher levels of education are hypothesized to have a shorter time to death once diagnosed (although not to dementia). In contrast, the bottom 5% were more likely to carry the APOE -ϵ4 gene, were older at baseline and had fewer years of education; almost all the members of this subgroup experienced a terminal event after one visit. To explain the observations here, we further investigate the baseline cognitive states and last known states across the three subgroups (see Tables 5 and 6).

Table 4.

Risk factors and transitions across subgroups defined by the random effect estimates

Category	Number of Participants	Years of education (%)			APOE-ϵ4 carrier (%)	Baseline age Years (mean ± SD)	Number of Transitions (mean ± SD)
		≤ 12	16	> 16
Bottom 5%	23	17.4	56.5	26.1	30.4	87.6 ± 5.7	1.0 ± 0.2
Middle 90%	415	10.6	48.2	41.2	19.3	82.2 ± 4.8	5.4 ± 2.9
Top 5%	23	0.0	13.0	87.0	0.0	78.2 ± 1.8	9.5 ± 0.8
P–value	-	= 0.001			0.025	< 0.0001	< 0.0001

Table 5.

Baseline cognitive state by subgroup split by the random effect estimates(%)

Category	Baseline Cognitive State
	Intact Cognition	Mild Impairment	Global Impairment
Bottom 5%	0.0	30.4	69.6
Middle 90%	31.9	59.7	8.4
Top 5%	32.6	67.4	0.0

P-value < 0.028 by Fisher’s exact test

Table 6.

Last known state by subgroup split by the random effect estimates (%)

Category	Final State
	Intact Cognition	Mild Impairment	Global Impairment	Dementia	Death without dementia
Bottom 5%	0.0	0.0	0.0	95.7	4.3
Middle 90%	7.4	4.6	1.8	33.2	53.1
Top 5%	0.0	26.1	17.4	21.7	34.8

Table 5 shows the baseline cognitive state distribution across the three subgroups. The top 5% were more likely to start in the prior states 1 (intact cognitive) or 2 (mild impairment) than in state 3 (global impairment). Hence, the top 5% were less likely to have a serious health issue at baseline since none entered the cohort with an ADL (activity of daily living) deficit. The latter statement is in sharp contrast to the bottom 5%.

Table 6 gives the last known state by subgroup. We observed that the terminating events (states 4 [dementia] and 5 [death without dementia]) varied substantially by category: 100% in the bottom 5% subgroup, 86.3% in the middle 90% subgroup, and 56.5% in the top 5% subgroup. The bottom 5% were highly likely to terminate with a dementia diagnosis due to their high risks as documented in Table 4. If a terminal event occurred among the top 5%, they were more likely to die without a dementia diagnosis than progress to a dementia. The latter may be attributed at least partially to the high years of education among this subgroup; prior work in the Nun Study shows that education is protective against a dementia diagnosis, see Iraniparast et al. [24].

4. Discussion

In fitting the mixture models, the usual assumption of normal distribution for random effects is not always correct and thus needs to be verified. In this study, we proposed a method to estimate the random effects with a wide range of assumptions and, for the competitive models, further analyses were conducted to verify the assumptions through Q-Q plots for random effect estimates and gradient plots for both the random and fixed effect estimates. Based on both Q-Q plots and gradient plots, we concluded that the Weibull distribution provides a better fit than the normal distribution with mean 0 assumption for the random effects in the application to Nun Study data, even though the normal distribution assumption gives a slightly smaller AIC value.

Through investigation of the Nun Study data with different random effect distribution assumptions, we found the estimated fixed effects to be consistent since these estimates are based on the marginal model (Step 1). However, the random effect estimates are sensitive to the choice of the probability distributions as shown by the Nun Study data, which is best fitted in Steps 2 and 3 of the algorithm by a Weibull random effect.

The gradient plot is based on the fixed effects, ma king the estimation of these effects crucial. For example, the plots in [18] show clear discrepancies between the two proposed assumptions, mainly due to differences in the fixed effects. In their example, the advocated assumption, a mixture of three normal distributions, was able to address the issues raised by the single normal distribution and generated different fixed effects, leading to a more meaningful diagnostic plot. However, the gradient function plot may not distinguish between two assumptions with consistent fixed effects, as seen in GLMM, which is a limitation of this method. In the application to the Nun Study data, although the fixed effects are similar across both normal and Weibull distribution assumptions and result in similar gradient function diagnostic plots, we can see that the Weibull distribution is slightly better.

Research related to random effects is important because it helps us identify subsets of individuals in a cohort who either take longer or move more quickly to reach competing absorbing states. This identification enhances our understanding of the variability in individual outcomes within a population. In specific medical research, analyzing random effects enables researchers to discover patterns of different progression rates, which can inform treatment and intervention strategies. Understanding why certain individuals progress more rapidly can provide insights into the underlying biological, genetic, or environmental factors contributing to their faster transition. For example, in the analysis of the Nun Study data, the top 5% of participants categorized by estimated random effects are expected to transition to terminal states faster than the remaining groups. By investigating the baseline cognitive and last known states, we understand that the top 5% subgroup is more likely to transition to death without dementia, while the bottom 5% subgroup is more inclined to develop dementia. Additionally, in a more general setting, this knowledge can aid in the development of predictive models to identify high-risk individuals in other cohorts, leading to better resource allocation and more effective prevention strategies.

Appendix

Appendix A. MLE of ${\mathbf{\text{V}}}_i$ and its estimated standard error

The estimated random effects for the 461 participants, along with their associated standard errors under the Weibull distribution assumption, are presented in Tables A1 and A2.

Table A1.

MLE of ${\mathbf{\text{V}}}_i$ (sd) for participants 1–270

1–45	46–90	91–135	136–180	181–225	226–270
4.306(1.341)	3.932(0.569)	4.440(0.887)	4.633(0.583)	4.745(0.466)	4.756(0.851)
4.288(0.959)	4.402(0.668)	3.818(0.427)	3.629(0.802)	4.940(0.661)	4.624(0.931)
4.327(1.092)	4.556(1.166)	4.333(0.758)	4.556(0.955)	3.316(0.557)	2.533(0.42)
4.315(0.919)	3.542(0.586)	4.311(1.079)	4.567(0.822)	5.106(0.589)	3.215(0.447)
4.310(0.840)	4.390(1.854)	4.713(0.591)	3.304(0.457)	4.349(1.327)	4.289(0.850)
3.142(1.013)	4.784(0.642)	4.467(1.215)	3.835(0.583)	3.888(0.505)	3.118(0.448)
4.419(1.832)	4.522(0.554)	3.446(0.663)	4.923(1.034)	4.377(0.561)	3.400(0.592)
3.119(0.759)	4.026(0.443)	4.698(0.670)	4.691(0.875)	4.274(1.377)	4.627(0.728)
4.275(0.973)	3.678(0.569)	4.519(0.629)	4.710(0.475)	4.430(1.824)	4.722(1.662)
4.428(0.662)	4.398(0.460)	4.605(0.815)	3.847(0.495)	4.289(0.963)	3.188(0.646)
4.761(1.645)	3.145(0.616)	4.491(0.507)	3.432(0.527)	4.452(0.726)	3.255(0.462)
4.705(1.670)	4.509(1.010)	4.437(0.536)	4.687(0.909)	4.311(1.332)	4.592(0.942)
4.437(1.280)	4.451(1.274)	4.187(0.822)	3.998(0.629)	4.408(1.841)	4.233(0.627)
4.643(1.700)	4.687(0.950)	3.383(0.513)	3.926(0.478)	4.305(1.921)	4.781(1.056)
4.391(1.009)	3.553(0.796)	4.633(1.038)	4.664(0.503)	4.296(1.929)	3.236(0.466)
3.036(0.450)	3.305(0.701)	4.006(0.626)	3.701(0.773)	4.488(0.985)	3.639(0.457)
4.153(0.470)	4.589(0.619)	3.842(0.621)	5.011(1.572)	4.335(0.641)	3.127(0.454)
4.310(1.080)	4.712(0.918)	4.843(0.575)	4.332(0.659)	3.751(0.452)	4.291(1.933)
3.820(0.866)	3.949(0.654)	4.515(1.188)	4.903(0.860)	4.437(1.215)	4.394(0.611)
3.779(0.702)	4.830(0.822)	3.541(0.540)	3.039(0.783)	3.417(0.653)	4.826(1.619)
4.053(0.627)	3.494(0.443)	4.730(0.697)	4.171(0.456)	4.834(1.616)	4.413(1.024)
4.940(1.578)	4.781(1.637)	3.810(0.748)	4.632(0.706)	4.495(0.852)	3.417(0.525)
4.110(0.475)	3.251(0.478)	4.879(0.681)	4.568(1.738)	4.291(1.120)	4.376(0.689)
4.314(1.111)	3.403(0.560)	4.711(0.859)	4.622(0.948)	4.415(1.284)	4.484(0.450)
4.307(1.082)	4.573(1.221)	3.707(0.704)	3.899(0.500)	4.353(1.325)	4.864(0.834)
3.539(0.619)	3.919(0.649)	4.590(0.659)	4.708(0.804)	4.673(0.581)	4.712(0.507)
4.521(0.565)	4.792(1.237)	3.747(0.819)	5.135(1.522)	4.708(0.903)	4.486(1.790)
4.553(0.680)	3.225(0.486)	4.441(0.505)	4.779(0.475)	4.316(1.089)	4.677(0.916)
4.444(0.969)	3.785(0.531)	3.083(0.424)	5.329(1.036)	4.365(1.055)	4.024(0.479)
4.457(0.575)	3.922(0.894)	4.70(0.470)	4.404(0.919)	4.186(0.585)	4.657(1.693)
4.316(1.911)	4.320(0.622)	4.575(0.608)	4.628(0.646)	4.368(1.316)	3.151(0.477)
4.433(0.706)	4.563(1.224)	4.398(0.702)	4.629(1.196)	3.483(0.802)	4.496(0.844)
3.417(1.010)	4.208(0.482)	3.567(0.635)	3.406(0.516)	4.624(0.561)	3.464(0.619)
3.715(0.641)	4.310(1.078)	3.902(1.034)	4.373(0.503)	4.673(0.993)	4.346(1.328)
4.503(0.694)	4.494(1.782)	4.107(0.737)	5.007(1.573)	4.348(0.890)	3.538(0.634)
3.792(0.566)	4.276(1.123)	4.893(0.861)	4.801(0.917)	2.807(0.536)	4.294(1.931)
4.383(0.678)	4.461(1.218)	4.410(0.531)	4.494(0.942)	4.562(1.742)	3.036(0.459)
3.832(0.466)	4.568(1.135)	3.889(0.893)	4.447(0.595)	4.525(0.574)	4.488(1.786)
4.787(0.586)	3.634(0.843)	4.808(0.914)	4.832(0.537)	3.332(0.530)	3.058(0.474)
4.338(0.559)	4.327(1.317)	3.476(0.565)	3.255(0.441)	3.472(0.516)	4.907(0.457)
4.596(0.977)	4.659(0.951)	3.83(0.443)	4.565(1.745)	4.412(1.032)	5.226(1.049)
4.143(0.568)	4.646(0.888)	5.112(0.862)	4.553(0.506)	3.720(0.461)	4.564(0.500)
4.714(0.638)	4.699(0.785)	3.775(0.585)	3.843(0.447)	4.519(0.979)	4.450(0.473)
4.779(0.646)	4.872(1.033)	4.781(0.532)	4.269(0.808)	3.010(0.460)	4.979(0.510)
4.746(0.538)	4.411(0.942)	3.522(0.532)	3.656(0.509)	4.150(0.734)	4.412(0.596)

Table A2.

MLE of ${\mathbf{\text{V}}}_i$ (sd) continue for participants 271–461

271–302	303–334	335–366	367–398	399–430	431–461
4.469(0.468)	4.308(1.336)	3.916(0.640)	5.140(0.813)	3.821(0.452)	4.372(0.813)
4.956(1.101)	4.297(1.350)	3.626(0.463)	3.387(0.543)	4.494(0.671)	3.635(0.464)
4.810(0.914)	4.358(1.027)	3.760(0.623)	4.399(0.499)	4.320(1.071)	3.747(0.524)
4.600(0.632)	4.293(1.108)	4.441(1.016)	4.566(1.222)	4.580(0.591)	4.480(0.859)
4.546(0.969)	4.408(1.840)	4.510(1.007)	3.393(0.497)	3.120(0.448)	3.410(0.806)
4.062(0.473)	4.298(1.927)	4.336(0.831)	3.296(0.435)	3.338(0.442)	4.081(0.580)
3.202(0.499)	4.562(1.742)	4.588(0.743)	4.486(1.016)	4.302(0.855)	4.597(0.689)
3.398(0.455)	4.327(1.314)	3.844(0.576)	4.782(0.778)	4.773(0.477)	4.695(0.501)
4.960(1.016)	4.504(0.684)	4.478(1.084)	5.063(0.662)	4.556(0.849)	4.487(0.693)
4.396(0.552)	4.415(0.989)	4.345(1.073)	3.335(0.484)	3.417(0.482)	4.348(0.468)
4.489(0.854)	4.441(0.964)	3.366(0.644)	4.662(1.101)	3.363(0.511)	3.517(0.459)
4.286(0.455)	4.301(0.752)	4.389(1.041)	4.013(0.469)	3.213(0.605)	4.393(1.851)
4.325(0.823)	4.363(0.737)	3.825(0.608)	4.349(0.568)	3.622(0.603)	4.793(0.508)
4.739(0.706)	4.402(0.582)	4.311(0.951)	4.844(0.522)	4.992(0.458)	4.838(0.580)
4.611(0.713)	3.795(0.529)	4.441(1.011)	4.008(0.570)	4.415(0.707)	4.559(0.470)
4.519(0.459)	4.306(0.668)	3.737(0.520)	3.152(0.436)	4.323(0.939)	4.275(0.820)
5.278(1.037)	4.425(1.230)	4.707(0.480)	4.336(1.308)	3.215(0.494)	3.908(0.916)
4.763(0.792)	4.342(1.890)	4.634(1.705)	4.455(0.533)	4.620(0.602)	4.874(0.775)
4.791(1.059)	4.576(0.674)	4.769(1.642)	4.341(1.073)	4.030(0.448)	4.826(0.665)
3.283(0.437)	4.531(0.997)	4.475(0.862)	3.823(0.501)	4.290(0.649)	3.295(0.422)
4.792(0.814)	4.408(1.841)	3.190(0.782)	5.004(0.812)	3.254(0.560)	4.673(0.709)
3.634(0.499)	4.376(1.864)	4.745(1.067)	3.786(0.532)	3.806(0.538)	3.914(0.509)
4.841(0.546)	4.568(0.827)	3.813(0.579)	2.873(0.445)	4.360(1.072)	5.013(1.091)
4.650(0.450)	4.534(0.710)	4.241(0.618)	3.436(0.497)	3.795(0.449)	3.957(0.617)
2.949(0.463)	4.428(0.786)	4.524(0.727)	4.569(0.449)	3.589(0.790)	5.347(1.475)
4.869(0.460)	4.315(0.743)	4.397(0.763)	4.417(0.630)	4.582(1.136)	4.426(1.827)
4.668(0.951)	4.606(1.144)	4.745(0.496)	3.446(0.522)	4.736(0.899)	3.729(0.469)
4.285(1.364)	4.435(0.853)	4.588(0.497)	4.288(0.475)	3.694(0.511)	4.137(0.452)
2.701(1.127)	4.560(1.146)	3.677(0.572)	4.853(0.836)	4.644(0.557)	4.302(1.358)
4.356(1.060)	4.357(1.353)	2.992(0.422)	5.087(0.550)	5.051(1.005)	4.352(0.921)
4.280(1.942)	4.550(0.673)	2.801(0.436)	4.934(1.023)	4.179(0.676)	3.123(0.478)
3.852(0.464)	4.348(0.924)	4.996(1.095)	3.881(0.859)	4.885(0.773)	/

Appendix B. Fixed effects from Weibull distribution assumption

This section presents the fixed effect estimates for the model under the Weibull distribution assumption with two parameters for the random effects. As shown in Table B1, the results closely resemble those obtained under the Normal distribution assumption (Table 3).

Table B1.

Fixed effect coefficient estimates (standard errors; p-value) for the three prior transient states using intact cognition as the reference state, proportionality in the GLMM, and a Weibull distribution

Risk Factor	Mild Impairment	Global Impairment	Dementia	Death
Intercept	−3.710 (1.220; 0.003)	−2.405 (1.214; 0.048)	−3.236 (1.220; 0.008)	−2.876 (1.219; 0.019)
Education ≤ 12 years	2.145 (0.493; 0.000)	2.243 (0.520; 0.000)	2.073 (0.555; 0.000)	1.657 (0.551; 0.003)
Education = 16 years	0.706 (0.246; 0.004)	0.746 (0.267; 0.005)	0.613 (0.298; 0.040)	0.457 (0.279; 0.103)
Age in years	0.134 (0.021; 0.000)	0.219 (0.024; 0.000)	0.216 (0.027; 0.000)	0.244 0.025; 0.000)
APOE-ϵ4	1.152 (0.330; 0.001)	1.609 (0.350; 0.000)	1.724 (0.376; 0.000)	1.395 (0.368; 0.000)
Prior State: Intact Cognition	−0.741 (0.363; 0.042)	−3.325 (0.355; 0.000)	−4.838 (0.564; 0.000)	−2.813 (0.367; 0.000)
Prior State: Mild Impairment	0.395 (0.353; 0.263)	−2.664 (0.333; 0.000)	−2.284 (0.352; 0.000)	−2.487 (0.348; 0.000)

DATA AVAILABILITY STATEMENT

The data that support the findings of this study are available from the corresponding author upon reasonable request.