According to the 2017 National Survey on Drug Use and Health (NSDUH), about 13.3% of adolescents aged 12 to 17 in the United States have experienced a depressive episode (Center for Behavioral Health Statistics and Quality, 2018). Adolescent depression is a major concern for youth development that relates to physical development and cognitive development as well as success in school, work, and society. Several researchers have suggested that depression in adolescence is strongly linked with anxiety, but knowledge of the mechanisms underlying this link remains inconclusive (Garber and Weersing, 2010; Horn and Wuyek, 2010; Schleider
In many studies, adolescent mental health have been inferred from responses to several related questionnaires rather than directly measured as a single scale, but it has been rarely considered as a latent variable. Some research has written recently used latent class analysis (LCA) to categorically classify subpopulations for depressive and anxiety symptoms (van Lang
In this regard, this paper develops generalized latent class models that reflect cross-sectional dependence as well as longitudinal dependence between latent variables. In order to explain the dynamic changes or developments of multiple latent variables over time, we propose two types of new LC models of JLCA combined with the LTA concept and JLCA combined with the LCPA concept. For the parameter estimation, maximum likelihood estimation using an EM algorithm are used. As an empirical analysis, we apply our models to examine stage-sequential patterns of adolescent emotional well-being considering parent-adolescent relationships, using the data from Angel
The rest of the paper is organized as follows. In Section 2, the model concepts and descriptions of the proposed JLCA with latent transition (JLTA) and JLCA with latent profile (JLCPA) are presented. Section 3 describes estimation methods for the parameters and model diagnosis. In Section 4, empirical analysis results are represented, and Section 5 concludes the paper.
JLCA explores the joint action or association of multiple latent variables only for a static phenomena; therefore, we develop two JLC variation models by incorporating the JLCA with two typical latent class (LC) models for longitudinal data of LCPA and LTA. These two models allow the investigation of changes in multiple latent class variables over time that reflect longitudinal dependence.
Joint LCPA (JLCPA) is an extended LCPA where multiple latent variables are allowed to be associated and their latent paths can be tracked using sets of repeatedly measured manifest items. JLCPA also postulates a joint latent variable composed of several latent variables at each measurement time point. Figure 1 illustrats JLCA at each time
Suppose that there are
The likelihood of JLCPA is based on the following assumptions: (a) both the joint class membership and the joint class profile membership are related to the manifest items only through the class membership; (b) the manifest items are conditionally independent given a class membership; (c) given a joint class membership, each class membership is unrelated; and (d) the joint class membership is unrelated within a joint class profile. Assumptions (b), (c), and (d) indicate local independence which allows the inference of the unobserved classes, joint class, and joint class profile.
The observed-data likelihood, the likelihood of manifest items, can be derived by marginalizing the complete-data likelihood over all the considered latent variables
The second model is a joint LTA (JLTA) that identifies the marginal joint class prevalence at the initial time and latent transitions over two consecutive time points, instead of a latent profile or common pathways of joint class. We assume that the sequence of
JLTA also requires the local independence assumptions for meaningful inferences about all latent variables. The observed-data likelihood can be obtained as
Expectation-Maximization (EM) algorithm is the most widely used estimation method for LC-type models (Mooijaart and Van der Heijden, 1992; Jeon
E-step calculates the expectation of the log-complete data likelihood under the current estimates of the parameters. The expectation is expressed as a function of posterior probabilities that individual
For the invariant
E-step computes the conditional probability that the
Using the likelihood given in ( 3.5), the expectation of the log-complete data likelihood can be obtained in terms of
The M-step focuses on the maximization of the expected complete-data likelihood of JLCA model with latent transition with respect to the model parameters. The resulting parameter estimators are
The parameters of
It is important to assess the model with objective fit measures to understand whether the fitted model is appropriate to the data. LC model which properly reflects the underlying latent structure of the data enables us draw valid statistical inferences. Pearson
The study of Angel
For the study, we used 6 manifest items on the adolescent depression, 6 items measuring adolescent anxiety. Questionnaires on depression behaviors are “do you...?” (a) often feel no interest in things; (b) often feel lonely; (c) often feel blue; (d) often feel worthless; (e) often feel hopeless about future; (f) sometimes think about suicide. Items on anxiety behaviors are whether do you (a) often feel nervous; (b) often feel tense; (c) often feel scared suddenly; (d) often feel panic; (e) often feel restless; (f) often feel fearful. For the interpretational simplicity, all the polychotomous items were changed into dichotomous indicating true or not.
JLCPA and JLTA have several latent variables and complicated structure over time which can cause identifiability problem. Prior to the construction of JLCPA (or JLTA) model, it is required to specify the appropriate latent structure of each latent variable. The joint latent structure of the multiple latent variables can be determined, once the latent structure of the each latent variable is specified (Jeon
Table 1 presents AIC, BIC and bootstrap
Figure 2 represents the
Based on the pre-specified latent structures, the optimal number of joint latent class was selected. We called the identified joint latent class variable as
Figure 3 represents the
JLCPA has an extra number of latent component (latent profile) to be determined based on the pre-specified latent class and joint class structures. Table 3 presents the goodness-of-fit for JLTA and JLCPA with different number of profile, and the JLCPA model with 2-profile showed the smallest BIC and the model with 4-profile showed the smallest AIC. Considering model interpretability and law of parsimony, we decided to choose 2-profile latent model with 3-joint class, 3-class for
We estimated the parameters of our interest under the selected model structure. Table 4 shows the estimated
Table 5 represents the probability of belonging to the class. From JLCPA results, the first joint class can be labeled as the
Based on these latent structure, the conditional probabilities of joint class membership for a particular class profile (the tertiary measurement parameters of JLCPA) identified the representative sequential patterns of
Table 7 presents the estimates of marginal probabilities at wave 1 and transition probabilities from JLTA. At wave 1, the class membership of
We now expand the discussion by considering the relationship with parents that affect the emotional well-being of adolescents, and perform our JLCPA according to the parent-adolescent-relationship. For the parent-adolescent relationship, we divided sample youths into subgroups using 7 items at wave 1: whether do you (a) seek your caregiver’s point of view; (b) tell your caregiver about problem; (c) trust your caregiver; (d) feel satisfied discussing problems with your caregiver; (e) rarely get upset about your caregiver; (f) never hesitate to bother your caregiver with problem; (g) rarely feel angry with your caregiver. Here, all the polychotomous items were also changed into dichotomous.
The optimal segmentation was determined as three groups by LCA in terms of model interpretation and simplicity (AIC = 5848.8, BIC = 5957.6). The subgroups were labeled as ‘
This paper proposes joint latent class analysis for longitudinal data in two different approaches of the as joint latent class profile analysis (JLCPA) and joint latent transition analysis (JLTA). By reflecting the dependencies among latent variables in cross-sectional and longitudinal view, our models allows an examination of the associations among latent variables of different attributes as well as an investigation of the joint stage-sequential progression for the mixture of multiple latent class variables. These models can be applied to other research subjects such as disease progression considering gene-gene interactions, or poly-substance use behavior patterns consisted of alcohol and cigarettes. We applied our models to investigate the sequential patterns of adolescents’ emotional well-being, depression and anxiety, which are repeatedly measured as several survey items. Empirical results revealed three joint latent classes of normal emotions, mild symptoms, and severe symptoms, under both JLCPA and JLTA. JLCPA seems more useful to interpret than JLTA as time points to consider increase. Our JLCPA results presented two representative profiles for the adolescent emotional well-being over time, and their tendencies showed different according to the parent-adolescent-relationship subgroups. The proposed models can be generalized to incorporate covariates of individual characteristics, such as gender, race, and age. However, such inclusion of covariates may cause computational complexity requiring alternative estimation methods, which are left to future studies.
This work was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) by the Ministry of Education (2017R1C1B5077065 to Saebom Jeon and 2018R1D1A1B07045821 to Hwan Chung).
Model concept of joint latent class profile analysis.
The estimated
The estimated
Model structure of the selected joint latent class profile analysis.
Selection of the number of latent class for each latent component at wave 1
Latent variable | Number of classes | AIC | BIC | Bootstrap |
---|---|---|---|---|
Depression | 2 | 4822.1 | 4883.6 | 0.00 |
3 | 4807.0 | 4901.6 | 0.05 | |
4 | 4799.6 | 4927.4 | 0.39 | |
5 | 4801.6 | 4962.5 | 0.54 | |
Anxiety | 2 | 4585.0 | 4646.5 | 0.00 |
3 | 4524.9 | 4619.5 | 0.71 | |
4 | 4527.2 | 4654.9 | 0.65 | |
5 | 4536.3 | 4697.2 | 0.48 |
AIC = Akaike information criterion; BIC = Bayesian information criterion.
Selection of the number of joint latent class with different number of classes at wave 1
Joint latent variable | Number of classes | AIC | BIC | |
---|---|---|---|---|
Emotional well-being | 2 | 25963.75 | 26602.59 | 0.00 |
3 | 25640.17 | 26350.00 | 1.00 | |
4 | 25678.25 | 26459.07 | 1.00 |
AIC = Akaike information criterion; BIC = Bayesian information criterion.
Diagnostic statistics for JLTA model and JLCPA models with the different number of profiles
Model | Number of profiles | AIC | BIC |
---|---|---|---|
JLCPA | 2 | 31898.03 | 32409.08 |
3 | 31893.10 | 32494.99 | |
4 | 31805.02 | 32495.93 | |
JLTA | N/A | 31879.14 | 32413.87 |
JLTA = joint latent transition analysis; JLCPA = joint latent class profile analysis; AIC = Akaike information criterion; BIC = Bayesian information criterion.
The estimated
Manifest item | Latent class for Depression | |||||
---|---|---|---|---|---|---|
JLCPA | JLTA | |||||
Normal | Mild-depression | Severe-depression | Normal | Mild-depression | Severe-depression | |
Feeling no interest in things | 0.134 | 0.590 | 0.922 | 0.132 | 0.585 | 0.922 |
Feeling lonely | 0.061 | 0.508 | 0.953 | 0.059 | 0.501 | 0.947 |
Feeling blue | 0.045 | 0.412 | 0.861 | 0.043 | 0.410 | 0.859 |
Feeling worthless | 0.006 | 0.154 | 0.791 | 0.006 | 0.150 | 0.791 |
Feeling hopeless about future | 0.052 | 0.315 | 0.763 | 0.050 | 0.313 | 0.762 |
Sometimes think about suicide | 0.011 | 0.119 | 0.482 | 0.012 | 0.117 | 0.480 |
Manifest item | Latent class for Anxiety | |||||
---|---|---|---|---|---|---|
JLCPA | JLTA | |||||
Normal | Mild-anxiety | Severe-anxiety | Normal | Mild-anxiety | Severe-anxiety | |
Feeling nervous | 0.079 | 0.455 | 0.893 | 0.077 | 0.448 | 0.890 |
Feeling tense | 0.038 | 0.401 | 0.775 | 0.035 | 0.396 | 0.771 |
Feeling scared suddenly | 0.032 | 0.256 | 0.858 | 0.031 | 0.252 | 0.846 |
Feeling panic | 0.004 | 0.106 | 0.662 | 0.004 | 0.103 | 0.649 |
Feeling restless | 0.063 | 0.411 | 0.827 | 0.060 | 0.405 | 0.826 |
Feeling fearful | 0.010 | 0.147 | 0.812 | 0.010 | 0.140 | 0.807 |
JLCPA = joint latent class profile analysis; JLTA = joint latent transition analysis.
The estimated
Latent variable | Class | Joint class for Emotional well-being | |||||
---|---|---|---|---|---|---|---|
JLCPA | JLTA | ||||||
Normal emotions | Mild symptoms | Severe symptoms | Normal emotions | Mild symptoms | Severe symptoms | ||
Depression | Normal | 0.937 | 0.014 | 0.000 | 0.923 | 0.001 | 0.000 |
Mild-depression | 0.063 | 0.931 | 0.023 | 0.076 | 0.931 | 0.005 | |
Severe-depression | 0.000 | 0.055 | 0.977 | 0.001 | 0.068 | 0.995 | |
Anxiety | Normal | 0.993 | 0.017 | 0.000 | 0.977 | 0.000 | 0.000 |
Mild-anxiety | 0.006 | 0.967 | 0.139 | 0.023 | 0.975 | 0.093 | |
Severe-anxiety | 0.001 | 0.015 | 0.861 | 0.001 | 0.025 | 0.907 |
JLCPA = joint latent class profile analysis; JLTA = joint latent transition analysis.
The tertiary measurements estimates of
Profile | Joint class | Wave 1 | Wave 2 | Wave 3 |
---|---|---|---|---|
Deterioration (71.6%) | Normal emotions | 0.621 | 0.541 | 0.477 |
Mild symptoms | 0.259 | 0.370 | 0.388 | |
Severe symptoms | 0.120 | 0.089 | 0.135 | |
Polarization (28.4%) | Normal emotions | 0.238 | 0.267 | 0.405 |
Mild symptoms | 0.518 | 0.502 | 0.291 | |
Severe symptoms | 0.234 | 0.231 | 0.304 |
JLCPA = joint latent class profile analysis.
The initial joint class prevalence estimates of
Joint Class | Joint class for Emotional well-being | |||||||
---|---|---|---|---|---|---|---|---|
Wave 2 | Wave3 | |||||||
Normal emotions | Mild symptoms | Severe symptoms | Normal emotions | Mild symptoms | Severe symptoms | |||
Normal emotions (0.503) | 0.736 | 0.228 | 0.036 | 0.698 | 0.271 | 0.032 | ||
Mild symptoms (0.390) | Wave 1 | 0.334 | 0.559 | 0.107 | Wave 2 | 0.332 | 0.502 | 0.166 |
Severe symptoms (0.107) | 0.176 | 0.275 | 0.549 | 0.227 | 0.394 | 0.380 |
JLTA = joint latent transition analysis.
The contribution of manifest items for subgrouping parent-adolescent relationship
Manifest item | Class for Parent-adolescent relationship | ||
---|---|---|---|
Trust & no anger | Trust & squabbling | Apathy | |
Seek parent’s attention | 0.937 | 0.958 | 0.425 |
Communication | 0.937 | 0.863 | 0.139 |
Trust their parents | 0.985 | 0.975 | 0.624 |
Not discussing problems w/them | 0.782 | 0.348 | 0.530 |
Rarely getting upset | 0.754 | 0.118 | 0.391 |
Rarely avoiding parents | 0.697 | 0.362 | 0.439 |
Rarely feeling angry | 0.812 | 0.462 | 0.337 |
The estimated tertiary measurements according to the relationship with parent
Relationship group | Profile | Joint class | Wave 1 | Wave 2 | Wave 3 |
---|---|---|---|---|---|
Trust & no anger | Deterioration | Normal emotions | 0.870 | 0.879 | 0.759 |
Mild symptoms | 0.108 | 0.121 | 0.241 | ||
Severe symptoms | 0.022 | 0.000 | 0.000 | ||
Polarization | Normal emotions | 0.382 | 0.073 | 0.123 | |
Mild symptoms | 0.617 | 0.812 | 0.708 | ||
Severe symptoms | 0.001 | 0.115 | 0.169 | ||
Trust & squabbling | Deterioration | Normal emotions | 0.450 | 0.571 | 0.584 |
Mild symptoms | 0.526 | 0.385 | 0.325 | ||
Severe symptoms | 0.024 | 0.044 | 0.091 | ||
Polarization | Normal emotions | 0.000 | 0.017 | 0.117 | |
Mild symptoms | 0.510 | 0.458 | 0.483 | ||
Severe symptoms | 0.490 | 0.525 | 0.400 | ||
Apathy | Deterioration | Normal emotions | 0.440 | 0.467 | 0.382 |
Mild symptoms | 0.510 | 0.533 | 0.498 | ||
Severe symptoms | 0.050 | 0.000 | 0.120 | ||
Polarization | Normal emotions | 0.000 | 0.139 | 0.466 | |
Mild symptoms | 0.092 | 0.115 | 0.000 | ||
Severe symptoms | 0.908 | 0.746 | 0.534 |