Two-Profile Common and Unique Latent Transition Analysis

library(manCULTA)

This vignette describes the data generation process for a two-profile CULTA model with a covariate. The CULTA model incorporates covariates, latent categorical variables, trait components, state components, and profile-specific means to simulate longitudinal data with latent profile transitions. Data generation is demonstrated at the end of the vignette using the GenCULTA2Profiles function, which integrates all components into a single function call.

Let $i \in \left\{ 1, \ldots, n \right\}$ denote the index for individuals, let $t \in \left\{ 0, \ldots, m - 1 \right\}$ denote the index measurement occasions, let $k \in \left\{ 1, \ldots, p \right\}$ denote the index items, and let $c \in \left\{ 0, 1 \right\}$ be the index of the two latent profiles (profile 0 and profile 1). Let $q$ be the trait dimension, $q = 1$ in this context.

# dimensions
n # number of individuals
#> [1] 200
m # measurement occasions
#> [1] 6
p # number of items
#> [1] 4
q # common trait dimension
#> [1] 1

Covariate

The covariate is generated from a normal distribution with mean $\mu_X$ and $\sigma_X$ variance.

# covariate parameters
mu_x 
#> [1] 0
sigma_x
#> [1] 1

Latent Categorical Variables

Latent categorical variables represent profile membership for each individual at each measurement occasion. In a two-profile model, profile membership is influenced by covariates and previous profile status, following a logistic formulation. We distinguish between:

Initial profile membership (baseline time point)
Profile transitions across subsequent time points

We describe both components below.

Initial Profile Membership

For the first measurement occasion ( $t = 0$ ), profile membership is determined by the following log-odds for belonging to profile 0 (with profile 1 as the reference category): $\begin{equation} \left( \begin{array}{cc} \nu_{0} + \kappa_{0} \times \mathrm{Covariate} & 0 \\ \end{array} \right) . \end{equation}$ The corresponding probability of belonging to each profile is given by: $\begin{equation} \left( \begin{array}{cc} \frac{ \exp \left( \nu_{0} + \kappa_{0} \times \mathrm{Covariate} \right) }{ \exp \left( \nu_{0} + \kappa_{0} \times \mathrm{Covariate} \right) + 1 } & \frac{1}{ \exp \left( \nu_{0} + \kappa_{0} \times \mathrm{Covariate} \right) + 1 } \\ \end{array} \right) . \end{equation}$ Profile membership at the first occasion is sampled based on these probabilities.

Profile Transitions

For subsequent occasions ( $t = 1, \ldots, m - 1$ ), profile transitions depend on the profile at the previous occasion and the covariate. The log-odds for transitioning to profile 0 at time $t$ are given by: $\begin{equation} \left( \begin{array}{cc} \alpha_{0} + \beta_{00} + \gamma_{00} \times \mathrm{Covariate} & 0 \\ \alpha_{0} + \gamma_{01} \times \mathrm{Covariate} & 0 \\ \end{array} \right) . \end{equation}$ The probability of transitioning to each profile is computed as: $\begin{equation} \left( \begin{array}{cc} \frac{ \exp \left( \alpha_{0} + \beta_{00} + \gamma_{00} \times \mathrm{Covariate} \right) }{ \exp \left( \alpha_{0} + \beta_{00} + \gamma_{00} \times \mathrm{Covariate} \right) + 1 } & \frac{1}{ \exp \left( \alpha_{0} + \beta_{00} + \gamma_{00} \times \mathrm{Covariate} \right) + 1 } \\ \frac{ \exp \left( \alpha_{0} + \gamma_{01} \times \mathrm{Covariate} \right) }{ \exp \left( \alpha_{0} + \gamma_{01} \times \mathrm{Covariate} \right) + 1 } & \frac{1}{ \exp \left( \alpha_{0} + \gamma_{01} \times \mathrm{Covariate} \right) + 1 } \\ \end{array} \right) . \end{equation}$ Profile membership for each subsequent time point is sampled using these transition probabilities, based on the individual’s covariate value and previous profile.

# profile membership and transition parameters
nu_0
#> [1] -0.405
kappa_0
#> [1] 0.1
alpha_0
#> [1] -0.5
beta_00
#> [1] 0.85
gamma_00
#> [1] 0.2
gamma_10
#> [1] 0.2

Trait Components

The trait variate captures between-person differences and is composed of a shared (common) component and item-specific (unique) components. The full decomposition is given by: $\begin{equation} \mathrm{Trait}_{i} = \mathrm{Common\ Trait\ Loading} \times \mathrm{Common\ Trait}_{i} + \mathrm{Unique\ Trait}_{i} . \end{equation}$ We describe each component below.

Common Trait

The common trait $\mathrm{Common\ Trait}_{i}$ represents shared individual differences that influence all items uniformly. It is drawn from a normal distribution with mean $\mu_T$ and variance $\psi_T$ : $\begin{equation} \mathrm{Common\ Trait}_{i} \sim \mathcal{N} \left( \mu_T, \psi_T \right) \end{equation}$

The influence of the common trait on each item is determined by the $p \times q$ common trait loading,

Unique Traits

The unique trait component $\mathrm{Unique\ Trait}_{k, i}$ captures item-specific stable differences and is drawn from a multivariate normal distribution: $\begin{equation} \mathrm{Unique\ Trait}_{i} \sim \mathcal{N} \left( \boldsymbol{\mu}_p, \boldsymbol{\Psi}_{p \times p} \right) \end{equation}$

Combined Trait Variate

The trait variate for item $k$ and individual $i$ is obtained by combining the common and unique trait components: $\begin{equation} \mathrm{Trait}_{k, i} = \mathrm{Common\ Trait\ Loading}_{k} \times \mathrm{Common\ Trait}_{i} + \mathrm{Unique\ Trait}_{k, i} . \end{equation}$ The common trait component introduces shared variance across items, while the unique trait component allows for item-specific differences not explained by the common trait.

# trait parameters
psi_t
#>      [,1]
#> [1,]  0.3
mu_t
#> [1] 0
psi_p
#>      [,1] [,2] [,3] [,4]
#> [1,]  0.3  0.0  0.0  0.0
#> [2,]  0.0  0.3  0.0  0.0
#> [3,]  0.0  0.0  0.3  0.0
#> [4,]  0.0  0.0  0.0  0.3
mu_p
#> [1] 0 0 0 0
common_trait_loading
#>      [,1]
#> [1,]    1
#> [2,]    1
#> [3,]    1
#> [4,]    1

State Components

The state variate is composed of two parts: a common state shared across items, and unique states specific to each item. The full decomposition is given by: $\begin{equation} \mathrm{State}_{k, i, t} = \mathrm{Common\ State\ Loading}_{k} \times \mathrm{Common\ State}_{i, t} + \mathrm{Unique\ State}_{k, i, t} . \end{equation}$ We describe each component below.

Common State

The common state $\mathrm{Common\ State}_{i, t}$ evolves over time following a first-order autoregressive process: $\begin{equation} \mathrm{Common\ State}_{i, t} = \phi_{c} \times \mathrm{Common\ State}_{i, t - 1} + \zeta_{i, t} . \end{equation}$ The initial common state is drawn from a normal distribution: $\begin{equation} \mathrm{Common\ State}_{i, 0} \sim \mathcal{N} \left( 0, \psi_{s_{0}} \right) . \end{equation}$ The innovation term $\zeta_{i, t}$ is normally distributed: $\begin{equation} \zeta_{i, t} \sim \mathcal{N} \left( 0, \psi_{s} \right) . \end{equation}$ The autoregressive parameter $\phi_{c}$ depends on latent profile membership $c$ : $\begin{equation} \phi_{c} = \phi_{0} + \left( \phi_{1} - \phi_{0} \right) c . \end{equation}$ Here, $\phi_{0}$ and $\phi_{1}$ represent the autoregressive coefficients for profiles coded as 0 and 1, respectively.

Unique State

The unique state $\mathrm{Unique\ State}_{k, i, t}$ captures item-specific deviations and is drawn from a multivariate normal distribution: $\begin{equation} \mathrm{Unique\ State}_{i, t} \sim \mathcal{N} \left( 0, \boldsymbol{\theta} \right) \end{equation}$ where $\boldsymbol{\theta}$ is the item-level covariance matrix for the unique state component.

Combined State Variate

The state variate for item $k$ , individual $i$ , and time $t$ combines the common and unique state components: $\begin{equation} \mathrm{State}_{k, i, t} = \mathrm{Common\ State\ Loading}_{k} \times \mathrm{Common\ State}_{i, t} + \mathrm{Unique\ State}_{k, i, t} \end{equation}$ The common state loading parameter $\mathrm{Common\ State\ Loading}_{k}$ controls the influence of the shared state on each item.

# state parameters
common_state_loading
#>      [,1]
#> [1,]    1
#> [2,]    1
#> [3,]    1
#> [4,]    1
phi_0
#> [1] 0
phi_1
#> [1] 0.311
psi_s0
#> [1] 1
psi_s
#> [1] 0.5
theta
#>      [,1] [,2] [,3] [,4]
#> [1,]  0.2  0.0  0.0  0.0
#> [2,]  0.0  0.2  0.0  0.0
#> [3,]  0.0  0.0  0.2  0.0
#> [4,]  0.0  0.0  0.0  0.2

Trait-State Plus Profile Specific Means

The observed variable is given by $\begin{equation} Y_{k, i, t} = \mu_{k, c} + \mathrm{Trait}_{k, i} + \mathrm{State}_{k, i, t} \end{equation}$ where $\mu_{k, c}$ is the profile specific mean, while $\mathrm{Trait}_{k, i}$ and $\mathrm{State}_{k, i, t}$ correspond to the trait and state components of the model.

# profile-specific means
mu_profile
#>       [,1]   [,2]
#> [1,] 2.253 -0.278
#> [2,] 1.493 -0.165
#> [3,] 1.574 -0.199
#> [4,] 1.117 -0.148

Data Generation

The GenCULTA2Profiles function puts together all the components described above to generate data using a single function call.

# complete list of R function arguments

# random seed for reproducibility
set.seed(42)

# dimensions
n # number of individuals
#> [1] 200
m # measurement occasions
#> [1] 6
p # number of items
#> [1] 4
q # common trait dimension
#> [1] 1

# covariate parameters
mu_x 
#> [1] 0
sigma_x
#> [1] 1

# profile membership and transition parameters
nu_0
#> [1] -0.405
kappa_0
#> [1] 0.1
alpha_0
#> [1] -0.5
beta_00
#> [1] 0.85
gamma_00
#> [1] 0.2
gamma_10
#> [1] 0.2

# trait parameters
psi_t
#>      [,1]
#> [1,]  0.3
mu_t
#> [1] 0
psi_p
#>      [,1] [,2] [,3] [,4]
#> [1,]  0.3  0.0  0.0  0.0
#> [2,]  0.0  0.3  0.0  0.0
#> [3,]  0.0  0.0  0.3  0.0
#> [4,]  0.0  0.0  0.0  0.3
mu_p
#> [1] 0 0 0 0
common_trait_loading
#>      [,1]
#> [1,]    1
#> [2,]    1
#> [3,]    1
#> [4,]    1

# state parameters
common_state_loading
#>      [,1]
#> [1,]    1
#> [2,]    1
#> [3,]    1
#> [4,]    1
phi_0
#> [1] 0
phi_1
#> [1] 0.311
psi_s0
#> [1] 1
psi_s
#> [1] 0.5
theta
#>      [,1] [,2] [,3] [,4]
#> [1,]  0.2  0.0  0.0  0.0
#> [2,]  0.0  0.2  0.0  0.0
#> [3,]  0.0  0.0  0.2  0.0
#> [4,]  0.0  0.0  0.0  0.2

# profile-specific means
mu_profile
#>       [,1]   [,2]
#> [1,] 2.253 -0.278
#> [2,] 1.493 -0.165
#> [3,] 1.574 -0.199
#> [4,] 1.117 -0.148

data <- GenCULTA2Profiles(
  n = n,
  m = m,
  mu_x = mu_x,
  sigma_x = sigma_x,
  nu_0 = nu_0,
  kappa_0 = kappa_0,
  alpha_0 = alpha_0,
  beta_00 = beta_00,
  gamma_00 = gamma_00,
  gamma_10 = gamma_10,
  mu_t = mu_t,
  psi_t = psi_t,
  mu_p = mu_p,
  psi_p = psi_p,
  common_trait_loading = common_trait_loading,
  common_state_loading = common_state_loading,
  phi_0 = phi_0,
  phi_1 = phi_1,
  psi_s0 = psi_s0,
  psi_s = psi_s,
  theta = theta, 
  mu_profile = mu_profile
)

Model Fitting

The FitCULTA2Profiles function fits the model using Mplus. Note: This function requires that Mplus is already installed on the system. To speed up model fitting, consider using the ncores argument to leverage multiple cores.

fit <- FitCULTA2Profiles(data = data)

summary(fit)
#>               est     se       z      p    2.5%   97.5%
#> mu_10      2.1695 0.0815 26.6118 0.0000  2.0097  2.3293
#> mu_20      1.5565 0.0779 19.9824 0.0000  1.4039  1.7092
#> mu_30      1.5795 0.0750 21.0706 0.0000  1.4326  1.7264
#> mu_40      1.1605 0.0698 16.6377 0.0000  1.0238  1.2972
#> lambda_t2  0.9352 0.0958  9.7663 0.0000  0.7475  1.1229
#> lambda_s2  0.9350 0.0408 22.9064 0.0000  0.8550  1.0150
#> lambda_t3  0.9094 0.1310  6.9413 0.0000  0.6526  1.1662
#> lambda_s3  0.9566 0.0459 20.8264 0.0000  0.8666  1.0466
#> lambda_t4  0.6841 0.1047  6.5324 0.0000  0.4788  0.8893
#> lambda_s4  1.0039 0.0452 22.1997 0.0000  0.9152  1.0925
#> theta_11   0.2217 0.0173 12.8371 0.0000  0.1878  0.2555
#> theta_22   0.2296 0.0122 18.8169 0.0000  0.2057  0.2535
#> theta_33   0.2072 0.0124 16.7196 0.0000  0.1829  0.2315
#> theta_44   0.1752 0.0178  9.8294 0.0000  0.1403  0.2101
#> phi_0     -0.0100 0.0669 -0.1489 0.8817 -0.1411  0.1212
#> psi_t      0.4327 0.1040  4.1612 0.0000  0.2289  0.6365
#> psi_p_11   0.2222 0.0485  4.5845 0.0000  0.1272  0.3172
#> psi_p_22   0.2469 0.0382  6.4665 0.0000  0.1721  0.3218
#> psi_p_33   0.2741 0.0500  5.4859 0.0000  0.1762  0.3720
#> psi_p_44   0.3041 0.0450  6.7518 0.0000  0.2158  0.3924
#> psi_s0     1.1566 0.2344  4.9337 0.0000  0.6971  1.6160
#> psi_s      0.4781 0.0639  7.4873 0.0000  0.3530  0.6033
#> mu_11     -0.2859 0.0898 -3.1853 0.0014 -0.4619 -0.1100
#> mu_21     -0.0881 0.0757 -1.1634 0.2447 -0.2365  0.0603
#> mu_31     -0.1545 0.0771 -2.0043 0.0450 -0.3056 -0.0034
#> mu_41     -0.0511 0.0758 -0.6742 0.5002 -0.1996  0.0974
#> phi_1      0.3327 0.0676  4.9230 0.0000  0.2002  0.4651
#> nu_0      -0.2154 0.1909 -1.1283 0.2592 -0.5896  0.1588
#> alpha_0   -0.3451 0.1259 -2.7405 0.0061 -0.5920 -0.0983
#> kappa_0    0.3926 0.2298  1.7083 0.0876 -0.0578  0.8431
#> beta_00    0.6690 0.1680  3.9811 0.0001  0.3396  0.9983
#> gamma_00   0.2315 0.1087  2.1308 0.0331  0.0186  0.4445
#> gamma_10   0.2816 0.1137  2.4768 0.0133  0.0588  0.5044

Parameter Recovery

Parameter recovery was assessed by calculating the relative bias of the estimated parameters.

Parameter Recovery
	Parameter	Estimate	Relative Bias
mu_10	2.253	2.1694682	-0.0370758
mu_20	1.493	1.5565449	0.0425619
mu_30	1.574	1.5795206	0.0035074
mu_40	1.117	1.1605009	0.0389444
lambda_t2	1.000	0.9352381	-0.0647619
lambda_s2	1.000	0.9349552	-0.0650448
lambda_t3	1.000	0.9094139	-0.0905861
lambda_s3	1.000	0.9565730	-0.0434270
lambda_t4	1.000	0.6840685	-0.3159315
lambda_s4	1.000	1.0038704	0.0038704
theta_11	0.200	0.2216670	0.1083351
theta_22	0.200	0.2296140	0.1480698
theta_33	0.200	0.2072222	0.0361108
theta_44	0.200	0.1752072	-0.1239641
phi_0	0.000	-0.0099583	-Inf
psi_t	0.300	0.4327078	0.4423593
psi_p_11	0.300	0.2221968	-0.2593441
psi_p_22	0.300	0.2469447	-0.1768509
psi_p_33	0.300	0.2741133	-0.0862891
psi_p_44	0.300	0.3040932	0.0136440
psi_s0	1.000	1.1565711	0.1565711
psi_s	0.500	0.4781312	-0.0437376
mu_11	-0.278	-0.2859422	0.0285691
mu_21	-0.165	-0.0880921	-0.4661086
mu_31	-0.199	-0.1544862	-0.2236876
mu_41	-0.148	-0.0510851	-0.6548303
phi_1	0.311	0.3326714	0.0696828
nu_0	-0.405	-0.2154072	-0.4681304
alpha_0	-0.500	-0.3451484	-0.3097032
kappa_0	0.100	0.3926138	2.9261377
beta_00	0.850	0.6689811	-0.2129634
gamma_00	0.200	0.2315390	0.1576949
gamma_10	0.200	0.2815591	0.4077955

Methods

The FitCULTA2Profiles function has a number of methods including the following:

print
summary
converged
confint
coef
vcov
logLik
AIC
BIC
entropy
anova

Mplus Script Used

TITLE:
  2-Profile CULTA with Covariate;

DATA:
  FILE = culta_data.dat;

VARIABLE:
  NAMES =
    id
    x
    y1t0
    y2t0
    y3t0
    y4t0
    y1t1
    y2t1
    y3t1
    y4t1
    y1t2
    y2t2
    y3t2
    y4t2
    y1t3
    y2t3
    y3t3
    y4t3
    y1t4
    y2t4
    y3t4
    y4t4
    y1t5
    y2t5
    y3t5
    y4t5
  ;
  USEVARIABLES =
    x
    y1t0
    y2t0
    y3t0
    y4t0
    y1t1
    y2t1
    y3t1
    y4t1
    y1t2
    y2t2
    y3t2
    y4t2
    y1t3
    y2t3
    y3t3
    y4t3
    y1t4
    y2t4
    y3t4
    y4t4
    y1t5
    y2t5
    y3t5
    y4t5
  ;
  IDVARIABLE = id;
  CLASSES =
    c0(2)
    c1(2)
    c2(2)
    c3(2)
    c4(2)
    c5(2)
  ;

ANALYSIS:
  TYPE = MIXTURE;
  STARTS = 500 100;
  STSCALE = 2;
  STITERATIONS = 200;
  PROCESS = 24;
  MODEL = NOCOV;

MODEL:
  %OVERALL%
    ! common trait ------------------------------------------------------

    !! factor loadings
    !!! t = 0
    t BY y1t0@1;
    t BY y2t0 (lambdat2);
    t BY y3t0 (lambdat3);
    t BY y4t0 (lambdat4);
    !!! t = 1
    t BY y1t1@1;
    t BY y2t1 (lambdat2);
    t BY y3t1 (lambdat3);
    t BY y4t1 (lambdat4);
    !!! t = 2
    t BY y1t2@1;
    t BY y2t2 (lambdat2);
    t BY y3t2 (lambdat3);
    t BY y4t2 (lambdat4);
    !!! t = 3
    t BY y1t3@1;
    t BY y2t3 (lambdat2);
    t BY y3t3 (lambdat3);
    t BY y4t3 (lambdat4);
    !!! t = 4
    t BY y1t4@1;
    t BY y2t4 (lambdat2);
    t BY y3t4 (lambdat3);
    t BY y4t4 (lambdat4);
    !!! t = 5
    t BY y1t5@1;
    t BY y2t5 (lambdat2);
    t BY y3t5 (lambdat3);
    t BY y4t5 (lambdat4);

    !! latent mean
    [ t@0 ];

    !! latent variance
    t (psit);

    ! unique traits -----------------------------------------------------

    !! factor loadings
    !!! k = 1
    u1 BY y1t0@1;
    u1 BY y1t1@1;
    u1 BY y1t2@1;
    u1 BY y1t3@1;
    u1 BY y1t4@1;
    u1 BY y1t5@1;
    !!! k = 2
    u2 BY y2t0@1;
    u2 BY y2t1@1;
    u2 BY y2t2@1;
    u2 BY y2t3@1;
    u2 BY y2t4@1;
    u2 BY y2t5@1;
    !!! k = 3
    u3 BY y3t0@1;
    u3 BY y3t1@1;
    u3 BY y3t2@1;
    u3 BY y3t3@1;
    u3 BY y3t4@1;
    u3 BY y3t5@1;
    !!! k = 4
    u4 BY y4t0@1;
    u4 BY y4t1@1;
    u4 BY y4t2@1;
    u4 BY y4t3@1;
    u4 BY y4t4@1;
    u4 BY y4t5@1;

    !! latent means
    [ u1@0 ];
    [ u2@0 ];
    [ u3@0 ];
    [ u4@0 ];

    !! latent variances
    u1 (psip1);
    u2 (psip2);
    u3 (psip3);
    u4 (psip4);

    ! common states -----------------------------------------------------

    !! factor loadings
    !!! t = 0
    s0 BY y1t0@1;
    s0 BY y2t0 (lambdas2);
    s0 BY y3t0 (lambdas3);
    s0 BY y4t0 (lambdas4);
    !!! t = 1
    s1 BY y1t1@1;
    s1 BY y2t1 (lambdas2);
    s1 BY y3t1 (lambdas3);
    s1 BY y4t1 (lambdas4);
    !!! t = 2
    s2 BY y1t2@1;
    s2 BY y2t2 (lambdas2);
    s2 BY y3t2 (lambdas3);
    s2 BY y4t2 (lambdas4);
    !!! t = 3
    s3 BY y1t3@1;
    s3 BY y2t3 (lambdas2);
    s3 BY y3t3 (lambdas3);
    s3 BY y4t3 (lambdas4);
    !!! t = 4
    s4 BY y1t4@1;
    s4 BY y2t4 (lambdas2);
    s4 BY y3t4 (lambdas3);
    s4 BY y4t4 (lambdas4);
    !!! t = 5
    s5 BY y1t5@1;
    s5 BY y2t5 (lambdas2);
    s5 BY y3t5 (lambdas3);
    s5 BY y4t5 (lambdas4);

    !! latent means
    [ s0@0 ];
    [ s1@0 ];
    [ s2@0 ];
    [ s3@0 ];
    [ s4@0 ];
    [ s5@0 ];

    !! latent variance of s0
    s0 (psis0);

    !! variance of the process noise
    s1 (psis);
    s2 (psis);
    s3 (psis);
    s4 (psis);
    s5 (psis);

    ! unique states -----------------------------------------------------

    !! variances
    !!! t = 0
    y1t0 (theta11);
    y2t0 (theta22);
    y3t0 (theta33);
    y4t0 (theta44);
    !!! t = 1
    y1t1 (theta11);
    y2t1 (theta22);
    y3t1 (theta33);
    y4t1 (theta44);
    !!! t = 2
    y1t2 (theta11);
    y2t2 (theta22);
    y3t2 (theta33);
    y4t2 (theta44);
    !!! t = 3
    y1t3 (theta11);
    y2t3 (theta22);
    y3t3 (theta33);
    y4t3 (theta44);
    !!! t = 4
    y1t4 (theta11);
    y2t4 (theta22);
    y3t4 (theta33);
    y4t4 (theta44);
    !!! t = 5
    y1t5 (theta11);
    y2t5 (theta22);
    y3t5 (theta33);
    y4t5 (theta44);

    ! constrained intercepts --------------------------------------------

    !! t = 0
    [ y1t0@0 ];
    [ y2t0@0 ];
    [ y3t0@0 ];
    [ y4t0@0 ];
    !! t = 1
    [ y1t1@0 ];
    [ y2t1@0 ];
    [ y3t1@0 ];
    [ y4t1@0 ];
    !! t = 2
    [ y1t2@0 ];
    [ y2t2@0 ];
    [ y3t2@0 ];
    [ y4t2@0 ];
    !! t = 3
    [ y1t3@0 ];
    [ y2t3@0 ];
    [ y3t3@0 ];
    [ y4t3@0 ];
    !! t = 4
    [ y1t4@0 ];
    [ y2t4@0 ];
    [ y3t4@0 ];
    [ y4t4@0 ];
    !! t = 5
    [ y1t5@0 ];
    [ y2t5@0 ];
    [ y3t5@0 ];
    [ y4t5@0 ];

    ! lta ---------------------------------------------------------------

    !! initial profile membership
    [ c0#1 ] (nu0);
    c0#1 ON x (kappa0);

    !! profile transitions
    [ c1#1 ] (alpha0);
    [ c2#1 ] (alpha0);
    [ c3#1 ] (alpha0);
    [ c4#1 ] (alpha0);
    [ c5#1 ] (alpha0);
    c1#1 ON c0#1 (beta00);
    c2#1 ON c1#1 (beta00);
    c3#1 ON c2#1 (beta00);
    c4#1 ON c3#1 (beta00);
    c5#1 ON c4#1 (beta00);

MODEL c0:
  %c0#1%
    ! profile specific means
    [ y1t0 ] (mu10);
    [ y2t0 ] (mu20);
    [ y3t0 ] (mu30);
    [ y4t0 ] (mu40);

    ! covariate
    c1 ON x (gamma00);

  %c0#2%
    ! profile specific means
    [ y1t0 ] (mu11);
    [ y2t0 ] (mu21);
    [ y3t0 ] (mu31);
    [ y4t0 ] (mu41);

    ! covariate
    c1 ON x (gamma10);

MODEL c1:
  %c1#1%
    ! profile specific means
    [ y1t1 ] (mu10);
    [ y2t1 ] (mu20);
    [ y3t1 ] (mu30);
    [ y4t1 ] (mu40);

    ! covariate
    c2 ON x (gamma00);

    ! inertia
    s1 ON s0 (phi0);

  %c1#2%
    ! profile specific means
    [ y1t1 ] (mu11);
    [ y2t1 ] (mu21);
    [ y3t1 ] (mu31);
    [ y4t1 ] (mu41);

    ! covariate
    c2 ON x (gamma10);

    ! inertia
    s1 ON s0 (phi1);

MODEL c2:
  %c2#1%
    ! profile specific means
    [ y1t2 ] (mu10);
    [ y2t2 ] (mu20);
    [ y3t2 ] (mu30);
    [ y4t2 ] (mu40);

    ! covariate
    c3 ON x (gamma00);

    ! inertia
    s2 ON s1 (phi0);

  %c2#2%
    ! profile specific means
    [ y1t2 ] (mu11);
    [ y2t2 ] (mu21);
    [ y3t2 ] (mu31);
    [ y4t2 ] (mu41);

    ! covariate
    c3 ON x (gamma10);

    ! inertia
    s2 ON s1 (phi1);

MODEL c3:
  %c3#1%
    ! profile specific means
    [ y1t3 ] (mu10);
    [ y2t3 ] (mu20);
    [ y3t3 ] (mu30);
    [ y4t3 ] (mu40);

    ! covariate
    c4 ON x (gamma00);

    ! inertia
    s3 ON s2 (phi0);

  %c3#2%
    ! profile specific means
    [ y1t3 ] (mu11);
    [ y2t3 ] (mu21);
    [ y3t3 ] (mu31);
    [ y4t3 ] (mu41);

    ! covariate
    c4 ON x (gamma10);

    ! inertia
    s3 ON s2 (phi1);

MODEL c4:
  %c4#1%
    ! profile specific means
    [ y1t4 ] (mu10);
    [ y2t4 ] (mu20);
    [ y3t4 ] (mu30);
    [ y4t4 ] (mu40);

    ! covariate
    c5 ON x (gamma00);

    ! inertia
    s4 ON s3 (phi0);

  %c4#2%
    ! profile specific means
    [ y1t4 ] (mu11);
    [ y2t4 ] (mu21);
    [ y3t4 ] (mu31);
    [ y4t4 ] (mu41);

    ! covariate
    c5 ON x (gamma10);

    ! inertia
    s4 ON s3 (phi1);

MODEL c5:
  %c5#1%
    ! profile specific means
    [ y1t5 ] (mu10);
    [ y2t5 ] (mu20);
    [ y3t5 ] (mu30);
    [ y4t5 ] (mu40);

    ! inertia
    s5 ON s4 (phi0);

  %c5#2%
    ! profile specific means
    [ y1t5 ] (mu11);
    [ y2t5 ] (mu21);
    [ y3t5 ] (mu31);
    [ y4t5 ] (mu41);

    ! inertia
    s5 ON s4 (phi1);


MODEL CONSTRAINT:
  ! means for the first profile are higher than the second
  mu10 > mu11;
  mu20 > mu21;
  mu30 > mu31;
  mu40 > mu41;

  ! make sure variances are greater than zero
  psit > 0;
  psip1 > 0;
  psip2 > 0;
  psip3 > 0;
  psip4 > 0;
  psis0 > 0;
  psis > 0;
  theta11 > 0;
  theta22 > 0;
  theta33 > 0;
  theta44 > 0;

OUTPUT:
  TECH1
  TECH3
  TECH4
  TECH7
  TECH8
  TECH12
  TECH13
  TECH15
  ENTROPY;

SAVEDATA:
  ESTIMATES = culta_W7WVUNrII0x9ieu95cVv_estimates.dat;
  RESULTS = culta_W7WVUNrII0x9ieu95cVv_results.dat;
  TECH3 = culta_W7WVUNrII0x9ieu95cVv_tech3.dat;
  TECH4 = culta_W7WVUNrII0x9ieu95cVv_tech4.dat;
  FILE = culta_W7WVUNrII0x9ieu95cVv_cprobs.dat;
  SAVE = CPROBABILITIES;

Ivan Jacob Agaloos Pesigan