Comparing Two-Profile RILTA and CULTA Models

library(manCULTA)

We generate data using the CULTA model with two latent profiles, where profile membership depends on a covariate and profile transitions follow a multinomial structure. However, for model fitting, we impose a simpler structure by fitting a random intercept latent transition analysis (RILTA) model. We then compare this misspecified model to the correctly specified two-profile CULTA model.

Data Generation

# 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 FitRILTA2Profiles function fits the misspecified two-profile RILTA model using Mplus. Note: This function requires that Mplus is already installed on the system.

rilta <- FitRILTA2Profiles(data = data)

The FitCULTA2Profiles function fits the correct two-profile 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.

culta <- FitCULTA2Profiles(data = data)

Model Comparison

The anova function can be used to compare the two fitted models.

anova(rilta, culta)
#> $fit
#>                    logLik df correction      AIC     BIC     aBIC   entropy
#> 2-profile RILTA -6981.619 22   1.615733 14007.24 14079.8 14010.10 0.8836678
#> 2-profile CULTA -5850.928 33   1.023671 11767.86 11876.7 11772.15 0.7169988
#> 
#> $test
#>  chi_diff   df_diff   p_value 
#> -14093.62     11.00      1.00

Parameter Recovery

Parameter recovery was assessed by calculating the relative bias of the estimated profile-specific means, log-odds, and residual variances.

Parameter Recovery
	Parameter	RILTA Estimate	Relative Bias	CULTA Estimate	Relative Bias
mu_10	2.253	2.2035760	-0.0219370	2.1694682	-0.0370758
mu_20	1.493	1.6653136	0.1154143	1.5565449	0.0425619
mu_30	1.574	1.6534203	0.0504576	1.5795206	0.0035074
mu_40	1.117	1.3752747	0.2312218	1.1605009	0.0389444
mu_11	-0.278	-0.4207472	0.5134792	-0.2859422	0.0285691
mu_21	-0.165	-0.2640176	0.6001068	-0.0880921	-0.4661086
mu_31	-0.199	-0.2999945	0.5075099	-0.1544862	-0.2236876
mu_41	-0.148	-0.3141873	1.1228870	-0.0510851	-0.6548303
nu_0	-0.405	-0.0013525	-0.9966605	-0.2154072	-0.4681304
alpha_0	-0.500	-0.4055947	-0.1888107	-0.3451484	-0.3097032
kappa_0	0.100	0.4158691	3.1586911	0.3926138	2.9261377
beta_00	0.850	0.8192341	-0.0361952	0.6689811	-0.2129634
gamma_00	0.200	0.0853273	-0.5733633	0.2315390	0.1576949
gamma_10	0.200	0.1172341	-0.4138297	0.2815591	0.4077955
theta_11	0.200	0.8114558	3.0572789	0.2216670	0.1083351
theta_22	0.200	0.6660180	2.3300901	0.2296140	0.1480698
theta_33	0.200	0.6553469	2.2767343	0.2072222	0.0361108
theta_44	0.200	0.6757793	2.3788964	0.1752072	-0.1239641

Mplus Script Used

The RILTA model was estimated using the following Mplus script.

TITLE:
  2-Profile RILTA with Covariate;

DATA:
  FILE = rilta_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%
    ! random intercept --------------------------------------------------

    !! loadings
    !!! t = 0
    f BY y1t0* (lambdat1);
    f BY y2t0 (lambdat2);
    f BY y3t0 (lambdat3);
    f BY y4t0 (lambdat4);
    !!! t = 1
    f BY y1t1* (lambdat1);
    f BY y2t1 (lambdat2);
    f BY y3t1 (lambdat3);
    f BY y4t1 (lambdat4);
    !!! t = 2
    f BY y1t2* (lambdat1);
    f BY y2t2 (lambdat2);
    f BY y3t2 (lambdat3);
    f BY y4t2 (lambdat4);
    !!! t = 3
    f BY y1t3* (lambdat1);
    f BY y2t3 (lambdat2);
    f BY y3t3 (lambdat3);
    f BY y4t3 (lambdat4);
    !!! t = 4
    f BY y1t4* (lambdat1);
    f BY y2t4 (lambdat2);
    f BY y3t4 (lambdat3);
    f BY y4t4 (lambdat4);
    !!! t = 5
    f BY y1t5* (lambdat1);
    f BY y2t5 (lambdat2);
    f BY y3t5 (lambdat3);
    f BY y4t5 (lambdat4);

    !! latent mean
    [ f@0 ];

    !! latent variance
    f@1;

    ! 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);

    ! 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);

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

    ! covariate
    c2 ON x (gamma10);

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

    ! covariate
    c3 ON x (gamma00);

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

    ! covariate
    c3 ON x (gamma10);

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

    ! covariate
    c4 ON x (gamma00);

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

    ! covariate
    c4 ON x (gamma10);

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

    ! covariate
    c5 ON x (gamma00);

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

    ! covariate
    c5 ON x (gamma10);

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

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


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
  theta11 > 0;
  theta22 > 0;
  theta33 > 0;
  theta44 > 0;

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

SAVEDATA:
  ESTIMATES = rilta_lrIq31gaoeSlXSleIeIz_estimates.dat;
  RESULTS = rilta_lrIq31gaoeSlXSleIeIz_results.dat;
  TECH3 = rilta_lrIq31gaoeSlXSleIeIz_tech3.dat;
  TECH4 = rilta_lrIq31gaoeSlXSleIeIz_tech4.dat;
  FILE = rilta_lrIq31gaoeSlXSleIeIz_cprobs.dat;
  SAVE = CPROBABILITIES;

Ivan Jacob Agaloos Pesigan

Data Generation

Model Fitting

Model Comparison

Parameter Recovery

Mplus Script Used