Latent class analysis (LCA) is one type of finite mixture model that can be applied for a set of discrete response random variables. It summarizes the structure of population distribution by defining several partitions of population (latent classes) which cannot be observed directly. The latent class membership may be discovered by identifying a small number of representative response patterns of manifest variables. The LCA framework has been expanded to be utilized for more complicated data structures such as repeatedly measured data (Chung
A new type of LCA has been proposed in this article to deal with a latent group variable in a multivariate latent class profile analysis (MLCPA). Suppose that we have several sets of categorical item response variables that are repeatedly measured across the time stages. One set of item variables defines a discrete latent group variable via conventional LCA framework called a ‘latent group variable.’ The other sets of item response variables discover a sequence of multiple latent class variables to divide the population into homogeneous subgroups of those who have similar sequential patterns of latent class membership. We call this sequence of latent class variables a ‘latent profile variable’. Individuals with same latent profile membership will show similar sequential patterns for each latent class variable over time. Further, our proposed model reveals the association between the latent profile variable and the latent group variable using the conditional prevalence of latent profiles given in the latent group membership.
Parenting styles had been defined as four major categories: indulgent, permissive, authoritative, and authoritarian with respect to the characteristics of parent’s distance and responsibilities toward children (Maccoby and Martin, 1983). King
The rest of this article is as follows. The description of our proposed model and the estimation methods for the model parameters are presented in Sections 2 and 3. In Section 3.3, we examined the parameter estimation and inference procedure through empirical simulation; the simulation results are available in Appendix. In Section 4, we illustrate the practical usefulness of our new model by analyzing data from the NLSY97 using discrete item variables related to adolescents depression and drug-taking behavior such as alcohol drinking, cigarette smoking, and marijuana use. In Section 5, we summarize the paper and discuss future research areas.
A LCA is a classical methodology that divides the population into homogenous subgroups with respect to response patterns for manifest items. It postulates that a distribution of a set of categorical random variable is a mixture of finite classes with respective response patterns. Suppose there are
where
MLCPA has been introduced to explain the longitudinal patterns of latent class membership when there are multiple latent class variables (Lee
Let
where
Combining the LCA structure as group variable with MLCPA structure as an outcome, we propose MLCPA with latent group variable and refer it as GLCPA. The GLCPA postulates that the distribution of latent profile variable can be affected by another latent class variable which can be identified through the LCA structure. The proposed model is illustrated in Figure 1. A sequence of
Using the notation given in (2.2), the complete-data likelihood of the GLCPA for the
The complete likelihood in (2.3) consists of five parameters:
The primary measurement parameters (
The GLCPA assumes the following conditions: (1) the latent profile variable
The prevalence of the latent profile may also be affected by the individual’s covariates. Figure 1 shows that we can construct the multinomial logistic regression model by treating the latent profile variable as a response variable. While the conventional multinomial logistic regression utilizes observed values of the response variable and covariates, the regression on unobservable latent profile membership relates the covariates with posterior probabilities, which will be discussed later. Suppose we have a vector of covariates
where
In this section, we discuss the estimation and inference for the model parameters and the model selection procedure. We adopt the recursive expectation-maximization (EM) algorithm (Bartolucci
The parameter estimation in finite mixture model can be considered as a missing data problem, because the latent class membership cannot be observed directly. EM algorithm (Dempster
for
where marginal posterior probabilities can be computed as
The forward and backward probabilities allow us to obtain the posterior probabilities of latent class memberships at single time point (
The dimension of the overall posterior probability is reduced to
To deal with missing values in response variables, EM estimator for primary measurement parameters should be extended. Under the missing at random (MAR) assumption, the posterior probability
and
where
To include the covariate effects on the distribution of latent profiles,
Model selection in a complex latent structure is crucial because the interpretability of the model significantly differed by the number of latent classes for each latent variable. Likelihood-ratio test statistics (LRT) cannot be used for comparing different models since the models with different number of latent classes are not in a nested relationship (Collins and Lanza, 2010). Alternatively, we adopted AIC and BIC to assess a relative model fit among candidate models with a different number of classes; consequently, and the model with smaller AIC (or BIC) is preferred. For the absolute model fit, we used the parametric bootstrap
Models with all possible combinations can be candidates since the model selection procedure can entail tedious trials and errors because each latent variable may have a different number of classes. Jeon
The simulation study was designed to check if our estimation method provides consistent parameter estimates and asymptotic standard errors. We generated datasets and calculated the ML estimates using a recursive EM algorithm. We constructed 95% confidence interval for each parameters based on parameter estimates and standard errors; in addition, the empirical coverage of the confidence intervals were calculated from 100 iterations. Standard errors of estimates were obtained through an asymptotic variance-covariance matrix, by taking the negative inverse of the Hessian matrix. Data was simulated to have two time stages with two latent variables, a group latent variable with two classes, and a sample size of 500. Each latent variable was measured by four binary item-response variables with a latent profile variable designed to have a two-profile structure. The true
The NLSY97 is a longitudinal project that tracks the lives of a sample of American youth born between 1980–1984; 8,984 respondents ages from 14 to 17 were first interviewed in 1997. This ongoing project has been surveyed 10 times; respondents are now interviewed biennially. The response variables selected for the research are related with alcohol drinking, cigarette/marijuana smoking, and depression. Two sets of five items were used for measuring the two latent class variables, alcohol drinking and cigarette/marijuana smoking, respectively. An additional five items were used for measuring depression which serves as the latent group variable. Response variables related with depression were collected in 2000 when respondents were 17–20 years old, and the responses for alcohol drinking and cigarette/marijuana smoking were collected in 2000, 2002, and 2004.
To measure latent group variable (depression), we selected the following five survey questions: (a) How often you have been a nervous person in past month? (b) How often you felt calm and peaceful in past month? (c) How often you felt down and blue in past month? (d) How often you have been a happy person in past month? and (e) How often you depressed in last month? Response variables (b) and (d) were re-coded to be consistent in the manner that the higher response values implies more exposure to depression symptoms. In this way, we defined each binary manifest item indicating if the respondent had suffered that feeling at least one time or not, as Nervous, Not calm, Down and blue, Not happy, and Depressed, respectively.
For alcohol drinking, the following three survey items were selected and re-coded: (a) number of days drinking alcohol last 30 days (b) number of days having five or more drinks per day last 30 days (c) number of days drinking at schools or work per day last 30 days. The quantitative question (a) was used to create two binary manifest items on if one had drank alcohol in last 30 days (Current drinking), whether had ever drank for five or more days (Frequent drinking), and if one had drank for 20 or more days (Daily drinking). Questionnaire (b) and (c) were transformed into binary variables (Binge drinking and Drinking at school), having ‘yes’ if its value was higher than 0, ‘no’ otherwise.
The quantitative question for cigarette/marijuana smoking was transformed into two binary items of if one had ever smoked in the last 30 days (Current cigarette smoking), whether one had ever smoked in a daily manner for the last 30 days (Daily cigarette smoking), and if one had ever smoked 20 or more cigarettes per day in last 30 days (Heavy cigarette smoking). Finally, the variable Current marijuana smoking was ‘yes’ if one had ever smoked in last 30 days, and Frequent marijuana smoking was assigned to be ‘yes’ if one used marijuana more than 5 times in last 30 days. Table 1 shows the percentages of respondents who responded ‘yes’ to the 15 binary response variables, and the proportion of non-responses.
By introducing the GLCPA to drug-taking behavior and depression measurement items, we expect to study the following properties of the population: (a) What kinds of latent classes may be found for alcohol drinking, cigarette/marijuana smoking, and depression? (b) What kinds of common sequential patterns of alcohol drinking and cigarette/marijuana smoking can be identified? (c) How does the prevalence of latent profiles of alcohol drinking and cigarette/marijuana smoking change as the latent group membership of depression is varied?
We empirically fitted a series of conventional LCA models on each sets of response variable by increasing the number of latent classes from 2 to 5 and chose an appropriate model based on the AIC, BIC, and the model interpretability. Table 2 shows the goodness-of-fit statistics with the different number of classes for each latent variable. BIC selected a three-class model for depression and its bootstrap
The second step of model selection in GLCPA determined the number of latent profiles. As discussed in Section 3, we will treat the number of latent classes for each identified latent variable as given in the first step (three classes for depression and alcohol drinking and four classes for cigarette/marijuana smoking), and investigate a series of AIC and BIC values from various candidate models. We computed AIC and BIC values from different models whose number of latent profile are varied from 2 to 6. BIC showed the lowest value in the five-profile model. However, the class interpretations for the fourth and the fifth profile were obscure in a model with five profiles. Therefore, we adopted the four-profile structure as our final model. Given the selected latent structure, we tested if the primary measurement parameters can be equal across the time stages. This homogeneity assumption for
Finally, we included the covariates on the prevalence of the latent profile by incorporating the baseline multinomial logistic regression in GLCPA. We used gender (male/female) and race (white/black/others) as covariates, and obtained the estimated odds ratios to investigate their effect on identified latent profiles.
Table 3 shows the estimates of item-response probabilities for the three identified latent classes of the latent group variable, depression. The first latent class has probabilities that are lower than 0.5 for all binary responses, meaning that individuals in that class do not have any depression symptoms. Thus, this subgroup can be named as ‘not depressed.’ The second group has high probabilities for nervous, down and blue, and depressed items. Thus, the second latent groups can be considered as a ‘slightly depressed’ group. The third group has high probabilities for all items, except not happy item, so this group can be identified as ‘seriously depressed’ group.
Table 4 shows the primary measurement parameter estimates for the alcohol drinking variable. The
Table 5 shows the five classes of cigarette/marijuana smoking class variable and the estimated
Table 6 shows the estimated secondary measurement parameters (
We incorporated the multinomial logistic regression into our model in order to examine the effect of individual characteristics on latent profile membership. Table 7 shows the estimated odds ratios and the 95% confidence intervals. Profile 1 was set as the baseline category; therefore, the estimated parameters represent the odds ratios of belonging to a certain latent profile compared to Profile 1. We considered gender (female was set to be the baseline) and race (White was set to be the baseline) as covariates, and the estimated coefficients were transformed into odds ratios for interpretation. No covariate effect had a significant effect for ‘Not depressed’ group. For the ‘Slightly depressed’ group, male adolescents have 2.41 times higher odds for Profile 4 compared to the baseline (Profile 1) than females. Similarly, Blacks have 0.559 times lower odds for Profile 2 versus the baseline, 0.178 times lower odds for Profile 4 versus the baseline than White counterparts. For ‘Seriously depressed’ group, male adolescents have 4.21 times higher odds for Profile 4 compared to the baseline than females.
Finally, Table 8 shows the
Profile 1 was the most prevalent class (0.512) among the four profiles when the depression group was ‘Not depressed,’ but decreases to 0.325 for the ‘Seriously depressed’ group. Profile 2 showed relatively consistent proportions throughout all depression levels, ranging from 0.219 to 0.261. However, Profiles 3 and 4 showed an increasing trend as the level of depression becomes severe, from ‘Not Depressed’ to ‘Seriously Depressed.’ The estimated
We suggested a new type of latent variable model to examine the complex structure of categorical latent variables, especially in cases where we need to study the longitudinal trends of latent variables identified through a repeated measured manifest item. The GLCPA uncovers the three types of categorical latent variables: the first are the latent class variables explaining the associations among manifest items, the second is the latent profile variable that examines the longitudinal patterns of one or more latent class variables through repeatedly measured manifest items, and the third is a single latent variable that can be treated as group variable. The GLCPA may specify the conditional probability of individuals belonging to a certain latent profile given the identified latent group membership. Consequently, our proposed model can systemically specify the effect of a latent group membership on the probability of having a certain sequential patterns. We expect that this methodology can be widely applied in educational and psychological studies.
Through the analysis of NLSY97 data, we found four representative sequential patterns of young adolescents drug-taking behavior. These four common patterns identify the subgroups of a population not exposed to any type of substance use (Profile 1, ‘Not involved in any substance disorder’), who moved to severe alcohol drinker (Profile 2, ‘Heavy drinking advancer’), heavy cigarette smokers who were likely to transfer from non drinker to heavy drinker (Profile 3, ‘Heavy drinking advancer/Heavy cigarette smoker), and adolescents with serious drug-taking behavior (Profile 4, ‘Heavy substance user’). The proportions of the four latent profiles varied by the individual level of depression symptoms. Our proposed model discovered that the probability of not being exposed to the any type of drug-taking behavior decreases as the level of depression symptoms increase and that the prevalence of the adolescents with severe drug-taking behaviors also increases. This does not imply a causal relationship; however, such trend provides a quantitative indication of positive association between depression symptoms and drug-taking behavior. A rigid causal inference between the latent profile variable and the group latent variable represents a future research topic. For the causal inference approach in the conventional latent class structure (Lanza
The EM algorithm is widely adopted for the parameter estimation of the finite mixture model due to difficulties with unobservable structures. An EM algorithm provided a stable ML estimation; however, the computational cost was relatively huge compared to the other estimation strategies with the burden of computational complexity also becoming worse if the number of time stage increases. The recursive method discussed in Subsection 3.1 significantly reduced computational complexity by skipping the calculation of redundant posterior terms from (3.1). For the simulation result in the latent class profile analysis, see Chang and Chung (2013) which showed the superiority of recursive EM estimation to conventional EM in time efficiency. EM algorithm also required appropriate initial values to guarantee the converged solution to be a global maximum. To achieve global maximum, we used 100 different sets of starting values and chose the one with the highest likelihood as a final solution, which requires another huge calculation and time cost. A deterministic annealing EM algorithm ensures that a global maximum can be adopted to avoid the difficulty of choosing an appropriate initial value (Chang and Chung, 2013; Lee and Chung, 2017). We have now made a program for GLCPA written in R language (version 3.6.0) that is available on request.
Table A.1 and A.2 shows that the average of parameter estimates, mean square errors, and 95% coverage probabilities for strong and mixed primary measurement parameters (
Let
for
Here,
where
Here,
where
The Hessian matrix is the second derivatives of the log-likelihood with respect to all free parameters
where
The second derivatives of log-observed data likelihood with respect to
for
The second derivatives of log-observed data likelihood with respect to
for
The second derivatives of log-observed data likelihood with respect to
for
The second derivatives of log-observed data likelihood with respect to
for
The second derivatives of log-observed data likelihood with respect to
for
Since
where
This work was supported by Korea University Research Program (K1706321 to Chung) and Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2018R1D1A1B07045821 to Chung).
Percentage of ‘yes’ responses and non-responses to the items of the latent group (depression) and class variables (alcohol drinking and cigarette/marijuana smoking) over three waves
Latent variable | Manifest item | Year | |||||
---|---|---|---|---|---|---|---|
2000 | 2002 | 2004 | |||||
Yes | Missing | Yes | Missing | Yes | Missing | ||
Depression | Nervous | 62.86 | 9.69 | ||||
Not calm | 6.34 | 9.63 | |||||
Down and blue | 69.05 | 9.74 | |||||
Not happy | 2.64 | 9.69 | |||||
Depressed | 37.95 | 9.69 | |||||
Alcohol drinking | Current drinking | 42.51 | 9.48 | 51.04 | 11.11 | 52.21 | 17.11 |
Frequent drinking | 18.11 | 9.48 | 24.91 | 11.11 | 26.64 | 17.11 | |
Daily drinking | 4.41 | 9.69 | 4.72 | 11.11 | 2.54 | 17.93 | |
Binge drinking | 24.96 | 9.53 | 28.51 | 11.26 | 30.44 | 18.54 | |
Drinking at school | 8.77 | 9.43 | 6.85 | 11.06 | 5.94 | 16.81 | |
Cigarettes/marijuana smoking | Current cigarette smoking | 35.92 | 10.95 | 35.92 | 10.95 | 34.40 | 17.40 |
Daily cigarette smoking | 17.19 | 9.43 | 20.59 | 10.95 | 20.64 | 17.40 | |
Heavy cigarette smoking | 13.03 | 9.58 | 15.12 | 10.95 | 15.17 | 17.40 | |
Current marijuana smoking | 26.13 | 9.94 | 24.10 | 11.36 | 19.08 | 17.19 | |
Frequent marijuana smoking | 11.52 | 9.44 | 12.43 | 11.01 | 9.69 | 16.59 |
Goodness-of-fit measures for a series of LCA models with the different number of classes for each latent variable
Latent variable | Number of classes | AIC | BIC | Bootstrap |
---|---|---|---|---|
Depression | 2 | 7444.7 | 7506.1 | 0.00 |
3 | 7365.1 | 7460.1 | 0.08 | |
4 | 7364.9 | 7493.5 | 0.18 | |
5 | 7367.6 | 7529.6 | 0.77 | |
Alcohol drinking | 2 | 18246.5 | 18320.1 | 0.00 |
3 | 18092.4 | 18206.0 | 0.06 | |
4 | 18093.2 | 18247.0 | 0.42 | |
5 | 18105.2 | 18299.1 | 0.52 | |
Cigarette/marijuana smoking | 2 | 20802.1 | 20875.6 | 0.00 |
3 | 19669.7 | 19783.3 | 0.04 | |
4 | 18931.7 | 19085.4 | 0.52 | |
5 | 18942.4 | 19136.3 | 0.54 |
LCA = latent class analysis; AIC = Akaike information criterion; BIC = Bayesian information criterion.
The estimated item-response probabilities for the latent group variable, depression (
Manifest item | Latent group for depression | ||
---|---|---|---|
Not depressed | Slightly depressed | Seriously depressed | |
Nervous | 0.413 | 0.846 | 0.925 |
Not calm | 0.021 | 0.000† | 0.772 |
Down and blue | 0.423 | 0.961 | 0.949 |
Not happy | 0.000† | 0.012 | 0.280 |
Depressed | 0.062 | 0.603 | 0.764 |
^{†}The estimated probabilities are constrained to be zero or one.
The estimated item-response probabilities for the latent class variable, alcohol drinking (
Manifest item | Latent class for alcohol drinking | ||
---|---|---|---|
Non drinker | Current drinker | Heavy drinker | |
Current drinking | 0.064 | 1.000† | 1.000† |
Frequent drinking | 0.000† | 0.307 | 0.842 |
Daily drinking | 0.000† | 0.000† | 0.158 |
Binge drinking | 0.000† | 0.222 | 0.954 |
Drinking at school | 0.000† | 0.117 | 0.189 |
^{†}The estimated probabilities are constrained to be zero or one.
The estimated item-response probabilities for the latent class variable, cigarette/marijuana smoking (
Manifest item | Latent class for cigarette/marijuana smoking | |||
---|---|---|---|---|
Non smoker | Marijuana smoker | Heavy cigarette smoker | Heavy cigarette/marijuana smoker | |
Current cigarette smoking | 0.079 | 0.380 | 1.000† | 1.000† |
Daily cigarette smoking | 0.000† | 0.000† | 0.867 | 0.947 |
Heavy cigarette smoking | 0.000† | 0.000† | 0.524 | 0.581 |
Current marijuana smoking | 0.039 | 1.000† | 0.112 | 1.000† |
Frequent marijuana smoking | 0.000† | 0.443 | 0.000† | 0.737 |
The estimated probabilities are constrained to be zero or one.
The estimated conditional probabilities of the latent class membership for a given latent profile membership (
Profile | Year | Latent class for alcohol drinking | Latent class for cigarette/marijuana smoking | |||||
---|---|---|---|---|---|---|---|---|
Non drinker | Current drinker | Heavy drinker | Non smoker | Marijuana smoker | Heavy cigarette smoker | Heavy cigarette/marijuana smoker | ||
1 | 00 | 0.891 | 0.109 | 0.000† | 0.975 | 0.000† | 0.025 | 0.000† |
02 | 0.795 | 0.205 | 0.000† | 0.981 | 0.009 | 0.010 | 0.000† | |
04 | 0.690 | 0.272 | 0.038 | 0.950 | 0.011 | 0.039 | 0.000† | |
2 | 00 | 0.319 | 0.347 | 0.334 | 0.594 | 0.381 | 0.013 | 0.012 |
02 | 0.195 | 0.352 | 0.453 | 0.602 | 0.375 | 0.000† | 0.023 | |
04 | 0.167 | 0.352 | 0.481 | 0.636 | 0.302 | 0.023 | 0.039 | |
3 | 00 | 0.483 | 0.233 | 0.284 | 0.304 | 0.057 | 0.532 | 0.107 |
02 | 0.335 | 0.279 | 0.386 | 0.098 | 0.000† | 0.824 | 0.078 | |
04 | 0.339 | 0.223 | 0.438 | 0.146 | 0.000† | 0.818 | 0.036 | |
4 | 00 | 0.161 | 0.173 | 0.666 | 0.059 | 0.170 | 0.148 | 0.623 |
02 | 0.125 | 0.201 | 0.674 | 0.035 | 0.137 | 0.158 | 0.670 | |
04 | 0.053 | 0.182 | 0.765 | 0.030 | 0.084 | 0.270 | 0.616 |
^{†}The estimated probabilities are constrained to be zero or one.
The estimated odds ratio for the latent profile membership given a latent group variable (depression) and its 95% confidence interval (Profile 1 is the baseline)
Latent group for depression | Profile | Gender | Race | |
---|---|---|---|---|
Male | Black | Others | ||
Not depressed | 2 | 1.074 [0.615, 1.875] | 1.237 [0.673, 2.275] | 0.881[0.377, 2.059] |
3 | 1.279 [0.747, 2.189] | 1.197 [0.657, 2.179] | 0.682 [0.341, 1.363] | |
4 | 1.449 [0.609, 3.444] | 0.494 [0.177, 1.374] | 0.229 [0.040, 1.312] | |
Slightly depressed | 2 | 1.370 [0.865, 2.169] | 0.559 [0.328, 0.953] | 0.580 [0.312, 1.076] |
3 | 1.392 [0.902, 2.144] | 0.738 [0.447, 1.219] | 0.757 [0.452, 1.266] | |
4 | 2.410 [1.512, 3.838] | 0.178 [0.082, 0.384] | 0.409 [0.229, 0.729] | |
Seriously depressed | 2 | 0.983 [0.310, 3.111] | 1.288 [0.344, 4.815] | 0.834 [0.205, 3.392] |
3 | 0.950 [0.314, 2.880] | 1.045 [0.276, 3.955] | 0.902 [0.259, 3.146] | |
4 | 4.212 [1.003, 17.797] | 0.154 [0.014, 1.650] | 0.217 [0.036, 1.301] |
Estimated prevalence of latent profile membership for a given depression group
Latent profile | Latent group for depression | ||
---|---|---|---|
Not depressed | Slightly depressed | Seriously depressed | |
1 | 0.512 | 0.337 | 0.325 |
2 | 0.219 | 0.261 | 0.248 |
3 | 0.200 | 0.248 | 0.297 |
4 | 0.069 | 0.154 | 0.130 |
Average estimates (EST), mean square error (MSE), and coverage probability (CP) of 95% confidence intervals for parameter estimates (Strong)
Parameter | True | EST | MSE | CP | Parameter | True | EST | MSE | CP |
---|---|---|---|---|---|---|---|---|---|
0.90 | 0.901 | 0.0004 | 0.95 | 0.10 | 0.102 | 0.0003 | 0.97 | ||
0.90 | 0.902 | 0.0004 | 0.97 | 0.10 | 0.098 | 0.0004 | 0.97 | ||
0.90 | 0.897 | 0.0004 | 0.94 | 0.10 | 0.101 | 0.0004 | 0.97 | ||
0.90 | 0.904 | 0.0004 | 0.97 | 0.10 | 0.099 | 0.0004 | 0.96 | ||
0.10 | 0.097 | 0.0003 | 0.96 | 0.90 | 0.903 | 0.0005 | 0.95 | ||
0.10 | 0.101 | 0.0004 | 0.97 | 0.90 | 0.898 | 0.0004 | 0.92 | ||
0.10 | 0.101 | 0.0004 | 0.93 | 0.90 | 0.901 | 0.0004 | 0.94 | ||
0.10 | 0.102 | 0.0004 | 0.97 | 0.90 | 0.895 | 0.0004 | 0.94 | ||
0.10 | 0.099 | 0.0003 | 0.96 | 0.90 | 0.903 | 0.0004 | 0.96 | ||
0.10 | 0.104 | 0.0005 | 0.98 | 0.90 | 0.897 | 0.0005 | 0.97 | ||
0.10 | 0.097 | 0.0004 | 0.96 | 0.90 | 0.901 | 0.0003 | 0.94 | ||
0.10 | 0.097 | 0.0004 | 0.96 | 0.90 | 0.902 | 0.0008 | 0.96 | ||
0.90 | 0.896 | 0.0003 | 0.94 | 0.10 | 0.099 | 0.0004 | 0.97 | ||
0.90 | 0.900 | 0.0003 | 0.95 | 0.10 | 0.100 | 0.0004 | 0.95 | ||
0.90 | 0.896 | 0.0003 | 0.97 | 0.10 | 0.099 | 0.0003 | 0.95 | ||
0.90 | 0.899 | 0.0004 | 0.94 | 0.10 | 0.098 | 0.0003 | 0.92 | ||
0.80 | 0.799 | 0.0003 | 0.98 | 0.20 | 0.198 | 0.0004 | 0.98 | ||
0.20 | 0.201 | 0.0003 | 0.94 | 0.80 | 0.801 | 0.0004 | 0.96 | ||
0.20 | 0.202 | 0.0004 | 0.95 | 0.80 | 0.803 | 0.0003 | 0.99 | ||
0.80 | 0.802 | 0.0004 | 0.94 | 0.20 | 0.202 | 0.0004 | 0.97 | ||
−1.00 | −1.023 | 0.0713 | 0.97 | 1.00 | 1.042 | 0.0456 | 0.95 | ||
1.00 | 1.031 | 0.0899 | 0.98 | −1.00 | −1.007 | 0.0392 | 0.97 | ||
0.90 | 0.904 | 0.0007 | 0.99 | 0.10 | 0.101 | 0.0008 | 0.97 | ||
0.90 | 0.900 | 0.0005 | 0.97 | 0.10 | 0.103 | 0.0007 | 0.96 | ||
0.90 | 0.906 | 0.0009 | 0.97 | 0.10 | 0.099 | 0.0007 | 0.95 | ||
0.90 | 0.901 | 0.0006 | 0.96 | 0.10 | 0.102 | 0.0008 | 0.98 | ||
0.50 | 0.499 | 0.0005 | 0.95 |
Average estimates (EST), mean square error (MSE), and coverage probability (CP) of 95% confidence intervals for parameter estimates (Mixed)
Parameter | True | EST | MSE | CP | Parameter | True | EST | MSE | CP |
---|---|---|---|---|---|---|---|---|---|
0.90 | 0.898 | 0.0005 | 0.94 | 0.10 | 0.103 | 0.0003 | 0.98 | ||
0.90 | 0.900 | 0.0003 | 0.98 | 0.10 | 0.099 | 0.0004 | 0.97 | ||
0.70 | 0.703 | 0.0009 | 0.93 | 0.10 | 0.101 | 0.0004 | 0.96 | ||
0.70 | 0.700 | 0.0007 | 0.98 | 0.10 | 0.099 | 0.0004 | 0.96 | ||
0.10 | 0.096 | 0.0006 | 0.96 | 0.70 | 0.706 | 0.0005 | 0.94 | ||
0.10 | 0.101 | 0.0004 | 0.98 | 0.70 | 0.699 | 0.0004 | 0.92 | ||
0.30 | 0.301 | 0.0013 | 0.92 | 0.90 | 0.901 | 0.0014 | 0.94 | ||
0.30 | 0.300 | 0.0003 | 0.96 | 0.90 | 0.895 | 0.0009 | 0.94 | ||
0.10 | 0.099 | 0.0011 | 0.97 | 0.90 | 0.903 | 0.0006 | 0.95 | ||
0.10 | 0.104 | 0.0015 | 0.96 | 0.90 | 0.897 | 0.0006 | 0.94 | ||
0.30 | 0.299 | 0.0006 | 0.96 | 0.70 | 0.701 | 0.0010 | 0.94 | ||
0.30 | 0.300 | 0.0005 | 0.97 | 0.70 | 0.702 | 0.0008 | 0.97 | ||
0.90 | 0.896 | 0.0006 | 0.96 | 0.10 | 0.099 | 0.0004 | 0.94 | ||
0.90 | 0.900 | 0.0007 | 0.95 | 0.10 | 0.100 | 0.0005 | 0.95 | ||
0.70 | 0.697 | 0.0009 | 0.97 | 0.10 | 0.095 | 0.0008 | 0.95 | ||
0.70 | 0.697 | 0.0008 | 0.93 | 0.10 | 0.101 | 0.0010 | 0.93 | ||
0.80 | 0.799 | 0.0013 | 0.97 | 0.20 | 0.194 | 0.0012 | 0.98 | ||
0.20 | 0.196 | 0.0012 | 0.97 | 0.80 | 0.810 | 0.0018 | 0.96 | ||
0.20 | 0.190 | 0.0018 | 0.96 | 0.80 | 0.802 | 0.0014 | 0.97 | ||
0.80 | 0.796 | 0.0019 | 0.98 | 0.20 | 0.202 | 0.0012 | 0.97 | ||
−1.00 | −1.105 | 0.1525 | 0.97 | 1.00 | 1.011 | 0.1301 | 0.94 | ||
1.00 | 1.082 | 0.0694 | 0.98 | −1.00 | −1.025 | 0.0622 | 0.96 | ||
0.90 | 0.901 | 0.0008 | 0.98 | 0.10 | 0.101 | 0.0015 | 0.96 | ||
0.90 | 0.900 | 0.0013 | 0.97 | 0.10 | 0.101 | 0.0012 | 0.95 | ||
0.70 | 0.700 | 0.0009 | 0.96 | 0.30 | 0.298 | 0.0011 | 0.96 | ||
0.70 | 0.698 | 0.0015 | 0.94 | 0.30 | 0.297 | 0.0008 | 0.98 | ||
0.50 | 0.503 | 0.0007 | 0.94 |