Comparing Two-Profile LTA and CULTA Models
Ivan Jacob Agaloos Pesigan
Source:vignettes/sim-lta-2-profiles.Rmd
sim-lta-2-profiles.RmdWe 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 regular latent transition analysis (LTA) 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] 10000
m # measurement occasions
#> [1] 6
p # number of items
#> [1] 4
q # common trait dimension
#> [1] 1
# covariate parameters
mu_x
#> [1] 11.4009
sigma_x
#> [1] 24.67566
# profile membership and transition parameters
nu_0
#> [1] -3.563
kappa_0
#> [1] 0.122
alpha_0
#> [1] -3.586
beta_00
#> [1] 2.25
gamma_00
#> [1] 0.063
gamma_10
#> [1] 0.094
# trait parameters
psi_t
#> [,1]
#> [1,] 0.1
mu_t
#> [1] 0
psi_p
#> [,1] [,2] [,3] [,4]
#> [1,] 0.1 0.0 0.0 0.0
#> [2,] 0.0 0.1 0.0 0.0
#> [3,] 0.0 0.0 0.5 0.0
#> [4,] 0.0 0.0 0.0 0.5
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] 0.25
theta
#> [,1] [,2] [,3] [,4]
#> [1,] 0.15 0.00 0.00 0.00
#> [2,] 0.00 0.15 0.00 0.00
#> [3,] 0.00 0.00 0.15 0.00
#> [4,] 0.00 0.00 0.00 0.15
# 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 FitLTA2Profiles function fits the misspecified
two-profile LTA model using Mplus. Note:
This function requires that Mplus is already installed
on the system.
lta <- FitLTA2Profiles(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(lta, culta)
#> $fit
#> logLik df correction AIC BIC aBIC entropy
#> 2-profile LTA -330843.0 18 2.354802 661722.0 661851.8 661794.6 0.8513129
#> 2-profile CULTA -237508.2 33 1.000768 475082.4 475320.3 475215.5 0.9367669
#>
#> $test
#> chi_diff df_diff p_value
#> -299114.7 15.0 1.0Parameter Recovery
Parameter recovery was assessed by calculating the differences between the population values and the estimated profile-specific means and log-odds.
| Parameter | LTA Estimate | Difference | CULTA Estimate | Difference | |
|---|---|---|---|---|---|
| mu_10 | 2.253 | 1.3824160 | 0.8705840 | 2.2689973 | -0.0159973 |
| mu_20 | 1.493 | 1.1580352 | 0.3349648 | 1.5140045 | -0.0210045 |
| mu_30 | 1.574 | 1.2645449 | 0.3094551 | 1.6057853 | -0.0317853 |
| mu_40 | 1.117 | 1.0778100 | 0.0391900 | 1.1275251 | -0.0105251 |
| mu_11 | -0.278 | -0.4931759 | 0.2151759 | -0.2779016 | -0.0000984 |
| mu_21 | -0.165 | -0.3910813 | 0.2260813 | -0.1652498 | 0.0002498 |
| mu_31 | -0.199 | -0.4376782 | 0.2386782 | -0.1829942 | -0.0160058 |
| mu_41 | -0.148 | -0.4050322 | 0.2570322 | -0.1515997 | 0.0035997 |
| nu_0 | -3.563 | -1.1072981 | -2.4557019 | -3.6910557 | 0.1280557 |
| alpha_0 | -3.586 | -2.3408923 | -1.2451077 | -3.5634025 | -0.0225975 |
| kappa_0 | 0.122 | 0.0375012 | 0.0844988 | 0.1273813 | -0.0053813 |
| beta_00 | 2.250 | 2.2757322 | -0.0257322 | 2.1225611 | 0.1274389 |
| gamma_00 | 0.063 | 0.0198196 | 0.0431804 | 0.0676685 | -0.0046685 |
| gamma_10 | 0.094 | 0.0511943 | 0.0428057 | 0.0942463 | -0.0002463 |
Mplus Script Used
The LTA model was estimated using the following Mplus
script.
TITLE:
2-Profile LTA with Covariate;
DATA:
FILE = lta_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 = 20 4;
STSCALE = 5;
STITERATIONS = 10;
PROCESS = 1;
MODEL = NOCOV;
MODEL:
%OVERALL%
! 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 = lta_hqkdKzsBrLDoUq0A6l3I_estimates.dat;
RESULTS = lta_hqkdKzsBrLDoUq0A6l3I_results.dat;
TECH3 = lta_hqkdKzsBrLDoUq0A6l3I_tech3.dat;
TECH4 = lta_hqkdKzsBrLDoUq0A6l3I_tech4.dat;
FILE = lta_hqkdKzsBrLDoUq0A6l3I_cprobs.dat;
SAVE = CPROBABILITIES;