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- Summary
- 1. Introduction
- 2. Methods
- 3. Simulation Study and numeric estimation procedures
- 4. Longitudinal Alzheimer’s Disease Study-Caregiver Stress Levels
- 5. Identifiability
- 6. Discussion
- REFERENCES

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Stat Med. Author manuscript; available in PMC 2017 April 30.

Published in final edited form as:

PMCID: PMC4821697

NIHMSID: NIHMS750375

Julia S. Benoit, PhD,^{1,}^{2} Wenyaw Chan, PhD,^{2} Sheng Luo, PhD,^{2} Hung-Wen Yeh, PhD,^{3} and Rachelle Doody, MD PhD^{4}

Understanding the dynamic disease process is vital in early detection, diagnosis, and measuring progression. Continuous-time Markov chain (CTMC) methods have been used to estimate state change intensities but challenges arise when stages are potentially misclassified. We present an analytical likelihood approach where the hidden state is modeled as a three-state CTMC model allowing for some observed states to be possibly misclassified. Covariate effects of the hidden process and misclassification probabilities of the hidden state are estimated without information from a ‘gold standard’ as comparison. Parameter estimates are obtained using a modified EM algorithm and identifiability of CTMC estimation is addressed. Simulation studies and an application studying Alzheimer Disease caregiver stress levels are presented. The method was highly sensitive to detecting true misclassification and did not falsely identify error in the absence of misclassification. In conclusion, we have developed a robust longitudinal method for analyzing categorical outcome data when classification of disease severity stage is uncertain and the purpose is to study the process’ transition behavior without a gold standard.

Early disease detection is fundamental in improving medical treatment design and intervention aimed at delaying disease progression, subsequently enhancing quality of life and in some cases reducing disease mortality. Unfortunately, disease staging may be subject to misclassification since true events are not directly observable. Proxy variables help explain unobservable phenomena in medical research but introduce biased estimates and can be especially concerning when the misclassified observable outcome is categorical. Solutions for inaccurate continuous outcomes usually include a random effect or the like in longitudinal settings. Remedies for categorical outcomes potentially observed with error continue to be studied. Further challenging is the lack of a “gold standard” for the targeted process (eg. Alzheimer’s disease staging). Multi-state transitional models are useful for quantifying disease staging and focus on the movement from one category where the interest lies in estimating the transition rates or intensities. Transition modeling approaches exist to examine misclassification in longitudinal studies where the outcome is categorical in both continuous and discrete time settings, specifically hidden Markov models. Discrete–time Hidden Markov Models (HMM) to examine misclassification have been considered by several authors ([1–5]). Focusing on continuous-time setting, two-state (binary outcome) Markov models accounting for misclassification have been studied using Bayesian [6–7] and classic EM approaches [8], among others [9]. Multi-state HMMs (more than two states) are more complicated and to date a reversible HMM with an analytical solution to simultaneously estimate transition rates and probability of misclassification (or sensitivity) has not been developed. Special cases of the multi-state CTMC approaches have been developed including semi-Markov [10] and irreversible hidden CTMC models [11].

Due to the complexity of the likelihood function of a general multi-state recurrent CTMC, methodology for exact solutions when the target state is observable is in its infancy. Li and Chan [12] developed a likelihood technique to estimate the transition rates of a three-state CTMC with a binary covariate, thus providing comparisons of transition probabilities between states for two groups. The aforementioned likelihood technique was later extended to include multiple covariates and also a practical interpretation of the process with covariates was provided [13, 14]. This research focuses on a possibly misclassified ternary recurrent outcome observed at irregular and varying time intervals among each individual. We propose methodology that accounts for possible misclassification of the outcomes modeled as a CTMC and further generalize the Baum-Welch algorithm to the three-state continuous-time Markov model with covariates.

In circumstances where data is not directly observable, (e.g., misclassified outcomes), unique parameter estimates may be difficult or impossible to identify (i.e. non-identifiable) in the case of ‘blocked’ or ‘hidden’ information. Parameter non-identifiability has been discussed in the literature with regards to HMMs [8, 15, 16], among others and is addressed in this research. The 3-state CTMC with one state subject to misclassification model and its likelihood are described in the following section (section 2). In section 3 we describe the estimation method implementing a modified EM algorithm for parameter estimation and conduct a simulation study to assess its performance. Applications of our method to analyze Alzheimer’s disease (AD) caregiver stress levels are described and results are presented in section 4. We make remarks on identifiability of CTMC estimation in section 5. This paper concludes with a discussion of our findings in section 6.

Consider a longitudinal study where *Z _{k}*(

$$\text{Pr}({Z}_{k}({t}_{k,s})|{Y}_{k}({t}_{k,1}),\u2025,{Y}_{k}({t}_{k,s}),{Z}_{k}({t}_{k,1}),\dots ,{Z}_{k}({t}_{k,s-1}))\phantom{\rule{0ex}{0ex}}=\text{Pr}({Z}_{k}({t}_{k,s})|{Y}_{k}({t}_{k,\mathrm{s}}))\phantom{\rule{0ex}{0ex}}={\epsilon}_{Y({t}_{k,s}),Z({t}_{k},s)},s=1,\dots {n}_{k}.$$

(1)

Furthermore, Equation 1 defines the probability that the observed state correctly classifies the hidden state of the process (or misclassification probability) {e.g., Pr(*Z _{k}*(

The three-state CTMC is fully described by the instantaneous transition rates, *q _{ij}*, the rate at which the process transitions from state ‘

$$R=\left[\begin{array}{ccc}\hfill -({q}_{12}+{q}_{13})\hfill & \hfill {q}_{12}\hfill & \hfill {q}_{13}\hfill \\ \hfill {q}_{21}\hfill & \hfill -({q}_{21}+{q}_{23})\hfill & \hfill {q}_{23}\hfill \\ \hfill {q}_{31}\hfill & \hfill {q}_{32}\hfill & \hfill -({q}_{31}+{q}_{32})\hfill \end{array}\right]$$

(2)

From the property of a continuous-time Markov chain, the sojourn time or amount of time a process stays in category ‘*i*’ before exiting follows an exponential distribution with mean ${(\sum _{l=1,l\ne i}^{3}{q}_{il})}^{-1}$ and is generally unobservable. At transition time, the probability of transitioning into state ‘*j*’ given that the process is currently in state ‘*i*’ is calculated as ${Q}_{ij}={q}_{ij}{(\sum _{l=1,l\ne i}^{3}{q}_{il})}^{-1}$. The transition rates make up the probability mechanism used to derive the probability of transition over a specific interval of time, *P _{ij}(t)*. Explicit algebraic formulas were derived for three scenarios [12] allowing us to write the likelihood function in terms of the transition probabilities, which are functions of the

The distribution of the initial observations, denoted as π_{i} = *P*{*Y*(0) = *i*}, where ‘*i*’ takes on the values 1, 2, or 3, under our model are also assumed indirectly observable as in Equation 1. We transformed the transition rates via a log link function to examine the linear combination of the covariates *x _{r}* by estimating coefficients β

$$\text{log}{q}_{ij}={\alpha}_{ij}+\sum _{r=1}^{p}{\beta}_{r}{x}_{r},\text{for}i\ne j,i,j=\mathit{1},\mathit{2},\mathit{3}\text{and}r=\mathit{1},\dots ,p.$$

(3)

By the Markov property of the hidden process and the basic probability, we can construct the likelihood function of *Z _{k}*(

$$P(Z({t}_{1})=z({t}_{1}),Z({t}_{2})=z({t}_{2}),\cdots ,Z({t}_{{n}_{k}})=z({t}_{{n}_{k}}))\phantom{\rule{0ex}{0ex}}=\sum _{\mathit{\text{all possible}}y\text{'}s}{\epsilon}_{y({t}_{1})z({t}_{1})}{\epsilon}_{y({t}_{2})z({t}_{2})}\cdots {\epsilon}_{y({t}_{{n}_{k}})z({t}_{{n}_{k}})}\times P(Y({t}_{1})=y({t}_{1}))\times {P}_{y({t}_{1})y({t}_{2})}({t}_{2}-{t}_{1})\cdots {P}_{y({t}_{{n}_{k}-1})y({t}_{{n}_{k}})}({t}_{{n}_{k}}-{t}_{{n}_{k}-1})$$

(4)

where ε_{y(t1)z(t1)} = 1 and *P*_{y(tm)y(tm+1)} (*t*_{m+1} − *t _{m}*) represents the likelihood that the hidden process is

$$L(\mathrm{\Theta})=\underset{k=1}{\overset{m}{\mathrm{\Pi}}}\left[\sum _{\mathit{\text{all}}y\text{'}s}P\right(Y({t}_{k,1})=y({t}_{k,1})\left)\right\{\underset{i=1}{\overset{{n}_{k}}{\mathrm{\Pi}}}\left({\epsilon}_{Y({t}_{k,i})Z({t}_{k,i})}\right)\{\underset{j=1}{\overset{{n}_{k}-1}{\mathrm{\Pi}}}({P}_{Y({t}_{k,j})Y({t}_{k,j+1})}({t}_{k,j+1}-{t}_{k,j}))\}\left\}\right]$$

(5)

where *t _{k,j}* = 0, for

The expectation of the complete log likelihood given the observed data and the parameter values of the ν ^{th} iteration can be expressed as

$$Q(\theta ,{\theta}^{(\nu )})=E[\mathrm{log}(L)|\mathit{\text{all}}z\text{'}s,{\theta}^{(\nu )}]\phantom{\rule{0ex}{0ex}}=\sum _{k=1}^{m}E[\text{log}\left\{\sum _{\mathit{\text{all}}y\text{'}s}P\right(Y({t}_{k,1})=y({t}_{k,1})\left)\right\{\underset{i=1}{\overset{{n}_{k}}{\mathrm{\Pi}}}\left({\epsilon}_{Y({t}_{k,i})Z({t}_{k,i})}\right)\{\underset{j=1}{\overset{{n}_{k}-1}{\mathrm{\Pi}}}({P}_{Y({t}_{k,j})Y({t}_{k,j+1})}({t}_{k,j+1}-{t}_{k,j}))\}\left\}\right\}|\mathit{\text{all}}z\text{'}s,{\theta}^{(\nu )}].$$

(6)

For calculation reduction during maximization a modified Baum-Welch [17] algorithm is proposed and derived beginning with the following forward-backward variables as

$${A}_{j}(i)=P(Z({t}_{1})=z({t}_{1}),Z({t}_{2})=z({t}_{2}),\cdots ,Z({t}_{j})=z({t}_{j}),Y({t}_{j})=i|\theta ),$$

(7a)

$${B}_{j}(i)=P(Z({t}_{j+1})=z({t}_{j+1}),Z({t}_{j+2})=z({t}_{j+2}),\cdots ,Z({t}_{{n}_{k}})=z({t}_{{n}_{k}})|\theta ,Y({t}_{j})=i)$$

(7b)

Equation 7a expresses the forward variable as the probability of the partially observed sequence until time *t _{j}* and the true state

$$P(Y({t}_{j})=w|\mathit{\text{all}}z\text{'}s,{\theta}^{(\nu )})=\frac{{A}_{j}(w){B}_{j}(w)}{\sum _{l=1}^{3}{A}_{{n}_{k}}(l)}$$

(8a)

$$P(Y({t}_{k,j-1})=w,Y({t}_{k,j})=l|\mathit{\text{all}}z\text{'}s,{\theta}^{(\nu )})=\frac{{A}_{k,j-1}(w){P}_{wl}({t}_{j}-{t}_{j-1}){\epsilon}_{lz\left({t}_{j}\right)}{B}_{k,j}(l)}{\sum _{i=1}^{3}{A}_{k,{n}_{k}}(i)}$$

(8b)

, where

$${A}_{1}\left(i\right)={\pi}_{i}{\epsilon}_{i,z({t}_{1})}\hspace{1em}{A}_{j+1}\left(l\right)=\left[\sum _{w=1}^{3}{A}_{j}\right(w){P}_{wl}({t}_{j+1}-{t}_{j})]{\epsilon}_{l,z({t}_{j+1})},\hspace{1em}j=1,\cdots ,{n}_{k}-1$$

(9a)

$${B}_{j}(w)=\sum _{l}{P}_{wl}({t}_{j+1}-{t}_{j}){\epsilon}_{l,z({t}_{j+1})}{B}_{j+1}(l),j=1,\cdots ,{n}_{k}-1,\text{when}j={n}_{k},{B}_{j+1}(l)=1\text{for all}l=1,2,3.$$

(9b)

Note that Equations 8 and 9 provides similar explanation as that of Baum-Welch algorithm. For example, *A*_{1}(*i*) represents the probability that the true initial state is *i* and the initial observed state is *z*(*t*_{1}) and *A*_{j+1}(*i*) represents the probability that the true state at *t*_{j+1} is *i* and all the observed states up to time *t*_{j+1}, and is expressed in *A _{j}*(). Similarly,

To evaluate the performance of the proposed method, a simulation study is conducted where 1,000 data sets were generated with N=1000 individuals. Assuming that transition rates, thus transition time depend on each individual’s covariates, one binary and one continuous, a three-state recurrent CTMC was simulated for each individual in each replicate. We assume the CTMC state is observable only at integer times 0,1,…,10 not at the actual transition times. Conditional on the hidden state of the process at time *t _{l}*, the observed outcomes follow a multinomial distribution with parameters ε

To improve the computational efficiency, we derived a data based procedure for calculating initial parameter values for the EM estimation. All possible combinations of individual covariates and their transition rate relationship from Equation 3 provided MLEs for *q _{ij}* and were then post-estimated to obtain regression coefficients for each parameter. By the same notion, starting values for misclassification parameters were obtained via post-estimation following our proposed polytomous logistic regression with covariates to predict the true state, and thus the proportion of misclassification from true state 2 was used as the starting parameter. Quasi-Newton optimization was used to find the MLE of the parameter set of the log-likelihood function and then compared to the true parameters using bias and coverage probability measures. For each parameter, empirical means, biases, standard errors, and coverage probabilities for estimation are presented in Table 1a–2b.

For 1,000 replicates, Tables 1a–2a show the bias of all estimated coefficients are negligible and the coverage probabilities are mostly above 90% for our proposed method under both misclassification scenarios (i.e. ε_{21} = ε_{23} and ε_{21} ≠ ε_{23}) and of varying rates (1%, 5%, and 10%). The biases of the misclassification parameter estimates are small and the coverage probabilities range from roughly 75% to 79%. Additionally, coverage probabilities of the estimated misclassification parameter increase with the rate of misclassification (Tables 1a–1b) suggesting that our method can detect sizeable amounts of misclassified outcomes; however, as the misclassification mechanism increases it may be difficult to capture the true parameters. The naïve analysis (Table 1a) with only a 1% imposed error rate yielded larger bias, poor coverage probability (3%–52%) and only 77% estimable data replicates, confirming that when data truly are misclassified using the naïve approach could lead to inaccurate conclusions. The misclassification modeling scenario where ε_{21} ≠ ε_{23} performs about the same as when ε_{21} = ε_{23} (Tables 1b & 2a) with respect to both estimated coefficients and misclassification rate estimates. Finally Table 2b demonstrates that when categorical outcomes are classified correctly our proposed method does not falsely provide a significant misclassification rate (aka specificity).

We have also conducted (results not presented) simulation studies to test the robustness of heterogeneities in progression rates. Specifically, we switched the direction of the process moving from state 3 to have a higher weight of moving to ‘2’ instead of ‘1’. Additionally, we have conducted a simulation study to assess the method with a small sample size. The results are similar. That means the proposed model is robust to at least some changes of parameters.

The proposed method’s ability to assess sensitivity of longitudinal outcome data without a “gold standard” is presented here. Data collected from January 1990 to September 2011 were extracted from the Baylor Alzheimer’s Disease and Memory Disorders Center. Patients referred and self-referred to the center with probable AD, defined using criteria from the National Institute of Neurological and Communicative Disorders and Stroke [18], were used in this study. Patients underwent comprehensive evaluation and socio-demographic information such as age, sex and years of education, medical history, and estimates of symptom duration [19] were collected at baseline. Further details regarding the baseline assessment, follow-up, and outcome diagnosis have been described elsewhere [20]. Intervals of time between visits varied among individuals as did the number of visits themselves. Patients were neuropsychologically evaluated at baseline and annually or on an as needed basis for medication management. The Mini-Mental State Examination (MMSE) [21] was among the tests implemented and aids in identifying dementia progression and severity, focusing on memory, attention and language. Scores range from 0 to 30 with lower scores indicating severe dementia. Additionally, the caregiver, a family member or friend spending the most time with the patient, provided information on their health and well-being. In this study, longitudinal self-rated stress levels of each caregiver were modeled as a continuous-time Markov chain with three categories (mild, moderate, and severe). Covariates baseline MMSE and caregiver relationship to the patient (spouse v. other) were examined to better understand the movement between stages of AD patient care giver stress levels. Self-reported stress levels were based on a construct of a caregiver questionnaire. At every time point past baseline, each prior measurement introduces bias to the subsequent measurements. Thus, baseline measures are treated as ‘true’ and all others are subject to misclassification. Patients with complete information on baseline MMSE, relationship to the caregiver, and whose caregivers provided at least two self-rated stress levels (i.e. at least one possible transition) were included in this analysis. Inter-observation time was calculated as the duration between two consecutive observations. This model’s ability to measure transitions over time while accounting for uneven intervals and number of observations allows the inclusion of patients with intermittent and monotone missing data under the assumption of missing completely at random.

A total of 952 patients had at least two caregiver self-rated stress levels and were included in the analysis. The average age of the patient was 74, ranging from 44–93, and the majority (68%) was female. The median number of visits was 3, ranging from 2–14, and baseline stress levels were distributed as 33% mild, 46% moderate, and 21% severe levels of stress. Three analyses were conducted to reflect the three models presented in Section 3. Note that for the proposed Model I we constrained *q*_{13}, *q*_{31} = 0 due to the relative few transitions taking place from 1 (‘mild’) to 3 (‘severe’) and from 3 to 1 between two visits.

Table 3 displays the parameter estimates obtained from the two proposed methods and from the naïve method that ignores the possibility of incorrectly observing a 1 or 3. The main finding in this analysis is the significance of the misclassification parameters and the variations of the intensity parameter estimates between the proposed and naïve methods. The overall estimate of correctly classified data differs between proposed methods M1 and M2 (M1: 1-.05-.10; M2: 1–2*.05), by about 5% (85% and 90%, respectively). When using all three approaches to estimate the movement between stages of AD patient care giver stress levels, parameters reflecting movement from state ‘1’ were not significant. After adjusting for patient relationship and baseline MMSE score, the proposed method suggests that at the time of change a caregiver who is moderately stressed is more likely to reach to increase to a ‘severe’ stress level than revert to mildly stressed.

Comparison of Proposed and Naïve approaches for estimating the parameters of movement between stages of AD patient care giver stress levels.

Note that the probability of this type of transition is .74 {exp(−2.43)/exp(−2.43)+exp(−3.48)} and .47 for the naïve approach. It is noteworthy to point out that the frequency of transitions from ‘mild’ to ‘severe’ and ‘severe’ to ‘mild’ is 1.8% and 2.4%%, respectively which is very few. The proposed method-M1 suggests that caregivers accurately report their stress levels 85% of the time and those patients moderately stressed are only slightly less likely to over-report their stress rather than under-report it.

Due to the nature of the process, a CTMC model may be non-identifiable when outcomes are recorded at pre-specified times with or without misclassification. If the sojourn time and state of change is recorded, the model will be fully identifiable. If the mean sojourn time is much longer than the inter-observation interval the non-identifiable problem will be almost negligible. A more likely scenario is that the mean sojourn time interval (*1/q _{ii}*) is shorter than the inter-observation time interval and stage changes could be missed between observational periods. In other words, an observed transition (outcome) could be reached by at least two different paths. The complexity of this problem increases when a misclassification parameter is added to the model. Under certain conditions the model specified in this paper can achieve a working level of identifiability. First when state 2 is observed it is not misclassified. So, an observed 2-to-1 transition can be used to compare with a 2-to-2 transition. If the true transited state is 2, the dynamic behavior of the observed 2-to-1 transition should be similar to that of the 2-to-2 transition. A similar comparison can also be applied to the observed 2-to-3 transition. Second, in our model, covariates X1 and X2 are not misclassified and are used to link the transition rates of the hidden states via a regression model. Thus, covariates can help amplify the inter-transition time and assist in identifying misclassification status. In other words, they can improve parameter identifiability. However, if all three states are possibly misclassified, this effect may be eliminated.

We have proposed a method to estimate parameters of a hidden three-state CTMC with covariates when some observed outcomes are potentially misclassified and to estimate the probabilities of misclassification (sensitivity rates) in the absence of ‘gold standard’ information. We allow movement between all possible states. This estimation method was able to recover the hidden parameters with varying levels of misclassification imposed (up to a total of 20%). Simulation studies revealed the disruption of naively analyzing even a small amount of misclassified data with uneven observational schedules among and within patients (Table 1a). Another attractive feature of the proposed method is that it detected almost negligible error rates when the data were correctly classified and does not falsely provide a significant misclassification rate (aka specificity). Finally, we have shown, without presenting the results, that this method is robust to heterogeneities in the progression rates, to smaller sample sizes (n=350), and shorter chains (reduced to 6 observation times).

The complexity of statistical methodology with potentially misclassified outcomes and the drawbacks of ignoring this scenario are well known. For irreversible multi-state processes, approaches exist to account for potential misclassification but to date no other methods take into account the more general case of recurrent multi-state processes subject to misclassification. Our analytical likelihood approach has been developed to confront this. The impact of this research contributes substantially to the situation where the outcome is truly hidden and the validity of the observed data is suspect yet also provides a flexible approach when the researcher is unsure of the nature of whether the true state is misclassified or not.

We have shown that our proposed method gives very different results from the naïve method in terms of significance and interpretation of covariate effects when using the naïve approach to estimate the movement between stages of AD patient care giver stress levels. One of the major difficulties in analyzing longitudinal categorical data using a Markov model is to mathematically prove the future distribution of the data depends on the present not the past at any time point. In our model, although the hidden process is assumed to be a CTMC, the observed data does not constitute a CTMC and hence, it is not possible to validate the Markov property of the underlying process.

Finally, this method can be applied to any longitudinal discrete data with or without possible misclassification if the purpose is to look at transition behavior. Diseases are often referred to in terms of progression (irreversible disease processes) and is a special case of our method. Thus, our method can handle both general and special cases of the 3-state CTMC with some observed states potentially misclassified.

Julia Benoit was supported by NIH grant 2T32GM074902-06.

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