Tables and SEPath and LISREL Programs to accompany

Empirical and Conceptual Problems with Longitudinal Trait-State Models:

Introducing a Trait-State-Occasion Model

David A. Cole

Nina C. Martin

James H. Steiger

Psychological Methods, March 2005


Table 1

Likelihood of Improper solutions in TSE Models as a Function of Sample Size, Number of Waves, and Ratio of Occasion/Trait Variance: Occasion Factor Stability = .20

Design / / NCa / Mean
c2 / Mean RMSEA / OR
only / SH
only / Both
OR & SH
N = 200
4 waves / 20/60 / 0.07 / 5.06 / 0.02 / 0.62 / 0.03 / 0.02
40/40 / 0.02 / 5.52 / 0.024 / 0.68 / 0.03 / 0.01
60/20 / 0.01 / 5.71 / 0.025 / 0.72 / 0.01 / 0.00
8 waves / 20/60 / 0.00 / 31.86 / 0.015 / 0.48 / 0.01 / 0.01
40/40 / 0.00 / 32.53 / 0.016 / 0.44 / 0.01 / 0.00
60/20 / 0.00 / 31.72 / 0.014 / 0.45 / 0.00 / 0.00
12 waves / 20/60 / 0.00 / 73.71 / 0.011 / 0.39 / 0.01 / 0.00
40/40 / 0.00 / 75.27 / 0.012 / 0.40 / 0.00 / 0.00
60/20 / 0.00 / 74.57 / 0.012 / 0.43 / 0.00 / 0.00
N = 500
4 waves / 20/60 / 0.08 / 5.08 / 0.013 / 0.51 / 0.02 / 0.03
40/40 / 0.03 / 5.08 / 0.013 / 0.57 / 0.01 / 0.00
60/20 / 0.00 / 5.41 / 0.014 / 0.66 / 0.01 / 0.00
8 waves / 20/60 / 0.00 / 31.28 / 0.009 / 0.39 / 0.03 / 0.00
40/40 / 0.00 / 31.81 / 0.01 / 0.38 / 0.01 / 0.00
60/20 / 0.00 / 31.09 / 0.008 / 0.40 / 0.02 / 0.00
12 waves / 20/60 / 0.00 / 74.23 / 0.008 / 0.26 / 0.03 / 0.00
40/40 / 0.00 / 74.40 / 0.008 / 0.27 / 0.01 / 0.00
60/20 / 0.00 / 73.83 / 0.007 / 0.31 / 0.00 / 0.00
N = 1000
4 waves / 20/60 / 0.08 / 5.23 / 0.010 / 0.46 / 0.04 / 0.03
40/40 / 0.02 / 5.27 / 0.010 / 0.49 / 0.02 / 0.00
60/20 / 0.00 / 5.32 / 0.010 / 0.52 / 0.02 / 0.00
8 waves / 20/60 / 0.00 / 31.36 / 0.006 / 0.28 / 0.03 / 0.00
40/40 / 0.00 / 31.28 / 0.006 / 0.33 / 0.01 / 0.00
60/20 / 0.00 / 31.11 / 0.006 / 0.34 / 0.01 / 0.00
12 waves / 20/60 / 0.00 / 73.15 / 0.005 / 0.19 / 0.03 / 0.00
40/40 / 0.00 / 74.29 / 0.006 / 0.24 / 0.02 / 0.00
60/20 / 0.00 / 74.09 / 0.006 / 0.26 / 0.01 / 0.00

Note. NC = nonconvergence; OR = out-of-range estimates; SH = singular Hessian matrix. Degrees of freedom are 5 for 4-waves, 31 for 8 waves, and 73 for 12 waves.

a Percent of nonconvergence within first 500 cases. These cases were dropped, and more cases (without convergence problems) were run to replace them, bringing every cell size to 500.


Table 2

Likelihood of Improper solutions in TSE Models as a Function of Sample Size, Number of Waves, and Ratio of Occasion/Trait Variance: Occasion Factor Stability = .50

Design / / NC a / Mean
c2 / Mean RMSEA / OR
only / SH
only / Both
OR & SH
N = 200
4 waves / 20/60 / 0.03 / 5.08 / 0.020 / 0.54 / 0.00 / 0.00
40/40 / 0.00 / 5.08 / 0.020 / 0.56 / 0.00 / 0.00
60/20 / 0.00 / 5.30 / 0.022 / 0.67 / 0.00 / 0.00
8 waves / 20/60 / 0.00 / 31.61 / 0.014 / 0.10 / 0.00 / 0.00
40/40 / 0.00 / 31.26 / 0.014 / 0.07 / 0.00 / 0.00
60/20 / 0.00 / 31.34 / 0.014 / 0.09 / 0.00 / 0.00
12 waves / 20/60 / 0.00 / 74.71 / 0.011 / 0.01 / 0.00 / 0.00
40/40 / 0.00 / 74.46 / 0.012 / 0.01 / 0.00 / 0.00
60/20 / 0.00 / 75.81 / 0.013 / 0.03 / 0.00 / 0.00
N = 500
4 waves / 20/60 / 0.03 / 4.97 / 0.013 / 0.39 / 0.01 / 0.00
40/40 / 0.00 / 5.01 / 0.013 / 0.39 / 0.00 / 0.00
60/20 / 0.00 / 5.19 / 0.013 / 0.43 / 0.00 / 0.00
8 waves / 20/60 / 0.00 / 30.85 / 0.008 / 0.01 / 0.00 / 0.00
40/40 / 0.00 / 31.18 / 0.009 / 0.01 / 0.00 / 0.00
60/20 / 0.00 / 31.58 / 0.009 / 0.01 / 0.00 / 0.00
12 waves / 20/60 / 0.00 / 74.54 / 0.008 / 0.00 / 0.00 / 0.00
40/40 / 0.00 / 74.45 / 0.008 / 0.00 / 0.00 / 0.00
60/20 / 0.00 / 73.60 / 0.007 / 0.00 / 0.00 / 0.00
N = 1000
4 waves / 20/60 / 0.01 / 4.87 / 0.008 / 0.24 / 0.01 / 0.00
40/40 / 0.00 / 4.92 / 0.009 / 0.20 / 0.00 / 0.00
60/20 / 0.00 / 4.97 / 0.009 / 0.23 / 0.00 / 0.00
8 waves / 20/60 / 0.00 / 31.37 / 0.006 / 0.00 / 0.00 / 0.00
40/40 / 0.00 / 31.93 / 0.007 / 0.00 / 0.00 / 0.00
60/20 / 0.00 / 30.57 / 0.006 / 0.00 / 0.00 / 0.00
12 waves / 20/60 / 0.00 / 72.75 / 0.005 / 0.00 / 0.00 / 0.00
40/40 / 0.00 / 73.60 / 0.005 / 0.00 / 0.00 / 0.00
60/20 / 0.00 / 72.77 / 0.005 / 0.00 / 0.00 / 0.00

Note. NC = nonconvergence; OR = out-of-range estimates; SH = singular Hessian matrix. Degrees of freedom are 5 for 4-waves, 31 for 8 waves, and 73 for 12 waves.

a Percent of nonconvergence within first 500 cases. These cases were dropped, and more cases (without convergence problems) were run to replace them, bringing every cell size to 500.
Table 3

Likelihood of Improper solutions in TSE Models as a Function of Sample Size, Number of Waves, and Ratio of Occasion/Trait Variance: Occasion Factor Stability = .80

Design / / NC a / Mean
c2 / Mean RMSEA / OR
only / SH
only / Both
OR & SH
N = 200
4 waves / 20/60 / 0.07 / 5.05 / 0.020 / 0.60 / 0.01 / 0.02
40/40 / 0.01 / 5.23 / 0.021 / 0.56 / 0.00 / 0.00
60/20 / 0.00 / 5.21 / 0.021 / 0.59 / 0.00 / 0.00
8 waves / 20/60 / 0.00 / 31.23 / 0.013 / 0.13 / 0.00 / 0.00
40/40 / 0.00 / 30.79 / 0.013 / 0.10 / 0.00 / 0.00
60/20 / 0.00 / 31.49 / 0.014 / 0.24 / 0.00 / 0.00
12 waves / 20/60 / 0.00 / 74.64 / 0.012 / 0.00 / 0.00 / 0.00
40/40 / 0.00 / 74.09 / 0.011 / 0.00 / 0.00 / 0.00
60/20 / 0.00 / 74.93 / 0.012 / 0.04 / 0.00 / 0.00
N = 500
4 waves / 20/60 / 0.04 / 4.99 / 0.012 / 0.47 / 0.01 / 0.02
40/40 / 0.01 / 5.17 / 0.013 / 0.43 / 0.00 / 0.00
60/20 / 0.00 / 5.17 / 0.014 / 0.46 / 0.00 / 0.00
0.00 / 0.00 / 0.00
8 waves / 20/60 / 0.00 / 31.09 / 0.009 / 0.03 / 0.00 / 0.00
40/40 / 0.00 / 31.21 / 0.009 / 0.02 / 0.00 / 0.00
60/20 / 0.00 / 30.62 / 0.008 / 0.06 / 0.00 / 0.00
0.00 / 0.00 / 0.00
12 waves / 20/60 / 0.00 / 73.21 / 0.007 / 0.00 / 0.00 / 0.00
40/40 / 0.00 / 73.75 / 0.007 / 0.00 / 0.00 / 0.00
60/20 / 0.00 / 73.60 / 0.007 / 0.00 / 0.00 / 0.00
N = 1000
4 waves / 20/60 / 0.03 / 5.04 / 0.009 / 0.35 / 0.01 / 0.02
40/40 / 0.00 / 5.29 / 0.010 / 0.38 / 0.00 / 0.00
60/20 / 0.00 / 5.20 / 0.010 / 0.38 / 0.00 / 0.00
0.00 / 0.00 / 0.00
8 waves / 20/60 / 0.00 / 31.54 / 0.007 / 0.00 / 0.00 / 0.00
40/40 / 0.00 / 31.09 / 0.006 / 0.00 / 0.00 / 0.00
60/20 / 0.00 / 31.27 / 0.006 / 0.02 / 0.00 / 0.00
0.00 / 0.00 / 0.00
12 waves / 20/60 / 0.00 / 71.92 / 0.005 / 0.00 / 0.00 / 0.00
40/40 / 0.00 / 74.04 / 0.006 / 0.00 / 0.00 / 0.00
60/20 / 0.00 / 72.86 / 0.005 / 0.00 / 0.00 / 0.00

Note. NC = nonconvergence; OR = out-of-range estimates; SH = singular Hessian matrix. Degrees of freedom are 5 for 4-waves, 31 for 8 waves, and 73 for 12 waves.

a Percent of nonconvergence within first 500 cases. These cases were dropped, and more cases (without convergence problems) were run to replace them, bringing every cell size to 500.


Table 4

Likelihood of Out-of-range Estimates in TSE Models for Each Parameter Type (among solutions with at least one out-of-range estimate): Occasion Factor Stability = .20

Design / / # OR
cases / Likelihood of out-of-range variance estimate for
Trait / Occasion / Error / Residual
N = 200
4 waves / 20/60 / 311 / 0.17 / 0.10 / 0.65 / 0.08
40/40 / 341 / 0.28 / 0.03 / 0.66 / 0.02
60/20 / 360 / 0.29 / 0.01 / 0.68 / 0.01
8 waves / 20/60 / 239 / 0.04 / 0.15 / 0.80 / 0.02
40/40 / 218 / 0.03 / 0.04 / 0.94 / 0.00
60/20 / 223 / 0.04 / 0.00 / 0.96 / 0.00
12 waves / 20/60 / 196 / 0.00 / 0.06 / 0.94 / 0.00
40/40 / 202 / 0.00 / 0.01 / 0.99 / 0.00
60/20 / 215 / 0.00 / 0.00 / 1.00 / 0.00
N = 500
4 waves / 20/60 / 255 / 0.21 / 0.04 / 0.73 / 0.02
40/40 / 286 / 0.24 / 0.00 / 0.75 / 0.00
60/20 / 332 / 0.30 / 0.00 / 0.70 / 0.00
8 waves / 20/60 / 194 / 0.02 / 0.08 / 0.90 / 0.00
40/40 / 189 / 0.01 / 0.00 / 0.99 / 0.00
60/20 / 199 / 0.00 / 0.00 / 1.00 / 0.00
12 waves / 20/60 / 130 / 0.00 / 0.02 / 0.98 / 0.00
40/40 / 135 / 0.00 / 0.00 / 1.00 / 0.00
60/20 / 157 / 0.00 / 0.00 / 1.00 / 0.00
N =1000
4 waves / 20/60 / 230 / 0.15 / 0.01 / 0.84 / 0.00
40/40 / 247 / 0.14 / 0.00 / 0.86 / 0.00
60/20 / 260 / 0.18 / 0.00 / 0.82 / 0.00
8 waves / 20/60 / 139 / 0.00 / 0.01 / 0.99 / 0.00
40/40 / 164 / 0.00 / 0.00 / 1.00 / 0.00
60/20 / 172 / 0.00 / 0.00 / 1.00 / 0.00
12 waves / 20/60 / 96 / 0.00 / 0.00 / 1.00 / 0.00
40/40 / 119 / 0.00 / 0.00 / 1.00 / 0.00
60/20 / 129 / 0.00 / 0.00 / 1.00 / 0.00

Note. Percentages for parameter 4 are not shown since zero or negative values are legitimate for this parameter.


Table 5

Likelihood of Out-of-range Estimates in TSE Models for Each Parameter Type (among solutions with at least one out-of-range estimate): Occasion Factor Stability = .50

Design / / # OR
cases / Likelihood of out-of-range variance estimate for
Trait / Occasion / Error / Residual
N = 200
4 waves / 20/60 / 271 / 0.46 / 0.03 / 0.49 / 0.01
40/40 / 282 / 0.47 / 0.01 / 0.51 / 0.00
60/20 / 336 / 0.54 / 0.00 / 0.46 / 0.00
8 waves / 20/60 / 51 / 0.10 / 0.02 / 0.88 / 0.00
40/40 / 36 / 0.00 / 0.00 / 1.00 / 0.00
60/20 / 44 / 0.07 / 0.00 / 0.93 / 0.00
12 waves / 20/60 / 3 / 0.00 / 0.33 / 0.67 / 0.00
40/40 / 4 / 0.00 / 0.00 / 1.00 / 0.00
60/20 / 14 / 0.00 / 0.00 / 1.00 / 0.00
N = 500
4 waves / 20/60 / 197 / 0.48 / 0.00 / 0.52 / 0.00
40/40 / 197 / 0.47 / 0.00 / 0.53 / 0.00
60/20 / 215 / 0.54 / 0.00 / 0.46 / 0.00
8 waves / 20/60 / 4 / 0.00 / 0.00 / 1.00 / 0.00
40/40 / 6 / 0.00 / 0.00 / 1.00 / 0.00
60/20 / 6 / 0.00 / 0.00 / 1.00 / 0.00
12 waves / 20/60 / 0 / -- / -- / -- / --
40/40 / 0 / -- / -- / -- / --
60/20 / 0 / -- / -- / -- / --
N = 1000
4 waves / 20/60 / 119 / 0.46 / 0.00 / 0.54 / 0.00
40/40 / 100 / 0.40 / 0.00 / 0.60 / 0.00
60/20 / 115 / 0.53 / 0.00 / 0.47 / 0.00
8 waves / 20/60 / 0 / -- / -- / -- / --
40/40 / 0 / -- / -- / -- / --
60/20 / 0 / -- / -- / -- / --
12 waves / 20/60 / 0 / -- / -- / -- / --
40/40 / 0 / -- / -- / -- / --
60/20 / 0 / -- / -- / -- / --

Note. Percentages for parameter 4 are not shown since zero or negative values are legitimate for this parameter.
Table 6

Likelihood of Out-of-range Estimates in TSE Models for Each Parameter Type (among solutions with at least one out-of-range estimate): Occasion Factor Stability = .80

Design / / # OR
cases / Likelihood of out-of-range variance estimate for
Trait / Occasion / Error / Residual
N = 200
4 waves / 20/60 / 298 / 0.61 / 0.05 / 0.30 / 0.03
40/40 / 280 / 0.75 / 0.00 / 0.25 / 0.00
60/20 / 296 / 0.77 / 0.00 / 0.23 / 0.00
8 waves / 20/60 / 66 / 0.98 / 0.00 / 0.02 / 0.00
40/40 / 49 / 1.00 / 0.00 / 0.00 / 0.00
60/20 / 122 / 1.00 / 0.00 / 0.00 / 0.00
12 waves / 20/60 / 1 / 1.00 / 0.00 / 0.00 / 0.00
40/40 / 1 / 1.00 / 0.00 / 0.00 / 0.00
60/20 / 18 / 1.00 / 0.00 / 0.00 / 0.00
N = 500
4 waves / 20/60 / 237 / 0.75 / 0.00 / 0.24 / 0.00
40/40 / 214 / 0.88 / 0.00 / 0.12 / 0.00
60/20 / 228 / 0.95 / 0.00 / 0.05 / 0.00
8 waves / 20/60 / 16 / 1.00 / 0.00 / 0.00 / 0.00
40/40 / 9 / 1.00 / 0.00 / 0.00 / 0.00
60/20 / 28 / 1.00 / 0.00 / 0.00 / 0.00
12 waves / 20/60 / 0 / -- / -- / -- / --
40/40 / 0 / -- / -- / -- / --
60/20 / 2 / 1.00 / 0.00 / 0.00 / 0.00
N = 1000
4 waves / 20/60 / 176 / 0.89 / 0.00 / 0.11 / 0.00
40/40 / 192 / 0.99 / 0.00 / 0.01 / 0.00
60/20 / 188 / 1.00 / 0.00 / 0.00 / 0.00
8 waves / 20/60 / 2 / 1.00 / 0.00 / 0.00 / 0.00
40/40 / 0 / -- / -- / -- / --
60/20 / 9 / 1.00 / 0.00 / 0.00 / 0.00
12 waves / 20/60 / 0 / -- / -- / -- / --
40/40 / 0 / -- / -- / -- / --
60/20 / 0 / -- / -- / -- / --

Note. Percentages for parameter 4 are not shown since zero or negative values are legitimate for this parameter.

Table 7

TSE Model Population Parameters and Two Estimates: One Based on Replications with No Out-of-Range Values (Est. 1) and One Based on Replications with Out-of-Range Values Set to Zero (Est. 2). Stability = .20.

Design / Trait variance / Occasion variance / Error variance / Stability beta / Residual variance
Actual / Est. 1 / Est. 2 / Actual / Est. 1 / Est. 2 / Actual / Est. 1 / Est. 2 / Actual / Est. 1 / Est. 2 / Actual / Est. 1 / Est. 2
N = 200
4-wave / .60 / 0.54 / 0.60 / .20 / 0.18 / 0.43 / .20 / 0.27 / 0.33 / .20 / 0.46 / 0.29 / .192 / 0.127 / 0.305
.40 / 0.34 / 0.41 / .40 / 0.30 / 0.58 / .20 / 0.35 / 0.45 / .20 / 0.49 / 0.36 / .384 / 0.231 / 0.438
.20 / 0.18 / 0.21 / .60 / 0.41 / 0.69 / .20 / 0.40 / 0.55 / .20 / 0.39 / 0.31 / .576 / 0.365 / 0.601
8-wave / .60 / 0.58 / 0.60 / .20 / 0.16 / 0.41 / .20 / 0.25 / 0.34 / .20 / 0.41 / 0.21 / .192 / 0.131 / 0.329
.40 / 0.38 / 0.40 / .40 / 0.28 / 0.59 / .20 / 0.33 / -- / .20 / 0.40 / 0.18 / .384 / 0.240 / 0.559
.20 / 0.18 / 0.20 / .60 / 0.41 / 0.78 / .20 / 0.40 / -- / .20 / 0.38 / 0.18 / .576 / 0.358 / 0.747
12-wave / .60 / 0.58 / 0.61 / .20 / 0.16 / 0.40 / .20 / 0.24 / -- / .20 / 0.33 / 0.13 / .192 / 0.145 / 0.375
.40 / 0.40 / 0.40 / .40 / 0.30 / 0.59 / .20 / 0.31 / -- / .20 / 0.32 / 0.14 / .384 / 0.269 / 0.583
.20 / 0.20 / 0.20 / .60 / 0.45 / 0.80 / .20 / 0.35 / -- / .20 / 0.31 / 0.15 / .576 / 0.417 / 0.782
N = 500
4-wave / .60 / 0.56 / 0.60 / .20 / 0.17 / 0.45 / .20 / 0.26 / 0.33 / .20 / 0.48 / 0.31 / .192 / 0.130 / 0.309
.40 / 0.35 / 0.41 / .40 / 0.30 / 0.59 / .20 / 0.34 / 0.46 / .20 / 0.45 / 0.32 / .384 / 0.237 / 0.469
.20 / 0.16 / 0.20 / .60 / 0.44 / 0.69 / .20 / 0.39 / 0.57 / .20 / 0.40 / 0.33 / .576 / 0.389 / 0.597
8-wave / .60 / 0.60 / 0.60 / .20 / 0.16 / 0.40 / .20 / 0.24 / -- / .20 / 0.34 / 0.16 / .192 / 0.145 / 0.360
.40 / 0.39 / 0.41 / .40 / 0.31 / 0.60 / .20 / 0.29 / -- / .20 / 0.32 / 0.13 / .384 / 0.286 / 0.586
.20 / 0.19 / 0.20 / .60 / 0.47 / 0.81 / .20 / 0.34 / -- / .20 / 0.31 / 0.14 / .576 / 0.429 / 0.778
12-wave / .60 / 0.60 / 0.60 / .20 / 0.17 / 0.40 / .20 / 0.23 / -- / .20 / 0.29 / 0.11 / .192 / 0.158 / 0.388
.40 / 0.40 / 0.40 / .40 / 0.34 / 0.60 / .20 / 0.25 / -- / .20 / 0.26 / 0.13 / .384 / 0.326 / 0.589
.20 / 0.20 / 0.20 / .60 / 0.51 / 0.81 / .20 / 0.28 / -- / .20 / 0.26 / 0.15 / .576 / 0.487 / 0.780
N = 1000
4-wave / .60 / 0.57 / 0.60 / .20 / 0.16 / 0.44 / .20 / 0.27 / 0.33 / .20 / 0.47 / 0.23 / .192 / 0.127 / 0.341
.40 / 0.35 / 0.41 / .40 / 0.30 / 0.58 / .20 / 0.35 / 0.47 / .20 / 0.45 / 0.23 / .384 / 0.237 / 0.515
.20 / 0.16 / 0.21 / .60 / 0.46 / 0.73 / .20 / 0.37 / 0.58 / .20 / 0.37 / 0.25 / .576 / 0.398 / 0.670
8-wave / .60 / 0.60 / 0.60 / .20 / 0.18 / 0.40 / .20 / 0.22 / -- / .20 / 0.29 / 0.10 / .192 / 0.167 / 0.392
.40 / 0.40 / 0.41 / .40 / 0.33 / 0.60 / .20 / 0.27 / -- / .20 / 0.28 / 0.13 / .384 / 0.309 / 0.587
.20 / 0.20 / 0.20 / .60 / 0.51 / 0.80 / .20 / 0.29 / -- / .20 / 0.26 / 0.15 / .576 / 0.481 / 0.780
12-wave / .60 / 0.60 / 0.60 / .20 / 0.19 / 0.40 / .20 / 0.21 / -- / .20 / 0.24 / 0.10 / .192 / 0.182 / 0.395
.40 / 0.40 / 0.40 / .40 / 0.37 / 0.60 / .20 / 0.23 / -- / .20 / 0.24 / 0.13 / .384 / 0.347 / 0.588
.20 / 0.20 / 0.20 / .60 / 0.54 / 0.80 / .20 / 0.26 / -- / .20 / 0.23 / 0.15 / .576 / 0.516 / 0.781

Note. Estimates based on fewer than 30 cases are not reported.