vars n mean sd median min max range se
id 1 6165 1391.95 811.09 1393.00 1.00 2812.00 2811.00 10.33
sad 2 6119 1.47 0.75 1.00 1.00 4.00 3.00 0.01
blues 3 6109 1.35 0.74 1.00 1.00 4.00 3.00 0.01
depress 4 6130 1.52 0.82 1.00 1.00 4.00 3.00 0.01
happy 5 6071 1.70 1.00 1.00 1.00 4.00 3.00 0.01
enjoy 6 6063 1.54 0.94 1.00 1.00 4.00 3.00 0.01
hopeful 7 5693 1.94 1.19 1.00 1.00 4.00 3.00 0.02
t 8 6165 2.36 2.38 3.00 0.00 6.00 6.00 0.03
td 9 6165 0.24 0.24 0.30 0.00 0.60 0.60 0.00
sex 10 6165 1.60 0.49 2.00 1.00 2.00 1.00 0.01
age 11 6165 2.20 1.24 2.00 1.00 5.00 4.00 0.02
ymean 12 6165 1.59 0.64 1.50 1.00 4.00 3.00 0.01
yz 13 6165 0.03 1.01 -0.11 -0.91 3.87 4.77 0.01
female 14 6165 0.60 0.49 1.00 0.00 1.00 1.00 0.01
agec70 15 6165 0.30 0.62 0.20 -0.30 1.70 2.00 0.01Multilevel Approaches to Modeling Growth and Change in Latent Variables
Rich Jones
2024-01-27
Multilevel Curve of Factors
Sometimes there is too much longitudinal data to keep the data “wide”. Growth can be modeled more easily if we use a “long” data framework and a multilevel approach to growth.
But how to specify the model?
Note: This presentation treats factor indicators as continuous indicators.
I will work through an example using CESD responses collected from the New Haven Site of the EPESE (Established Populations for the Epidemiologic Study of the Elderly) study. The four-category responses of six CESD questions will be treated as continuous indicators of a common underlying trait.
These data are public and can be obtained from ICPSR.
Descriptives
Latent Growth Curve Model (LGCM)
Modeling considerations: LGCM
This is a typical LGCM for a univariate outcome.
For my first model, I’m going to use an outcome yz which is a baseline normalized z-score based on the observed items. It’s not a latent variable. But by normalizing at the baseline, it will give an approximation of the results we should expect when we move to the latent variable modeling framework.
yz0, yz3, and yz6 are the normalized scores for the mean of 6 CESD questions observed at the baseline and 3 and 6 year follow-up of the EPESE.
Libraries
Wide data
Descriptives of wide data
vars n mean sd median min max range se
id 1 2762 1401.29 810.50 1400.50 1.00 2812.00 2811.00 15.42
female 2 2762 0.58 0.49 1.00 0.00 1.00 1.00 0.01
agec70 3 2762 0.36 0.65 0.20 -0.30 1.70 2.00 0.01
yz0 4 2728 0.00 1.00 -0.38 -0.91 3.87 4.77 0.02
yz3 5 2015 0.03 1.01 -0.27 -0.91 3.87 4.77 0.02
yz6 6 1422 0.08 1.03 -0.11 -0.91 3.87 4.77 0.03Use MplusAutomation to prepare data set for Mplus
Mplus ordinary LGC model
TITLE: LGCM CESD score
DATA: FILE = cesd.dat ;
VARIABLE: NAMES = id female agec70 yz0 yz3 yz6;
IDVARIABLE = id ;
MISSING = . ;
MODEL: i s | yz0@0 yz3@0.3 yz6@0.6 ; ! 10 year units
yz0-yz6 (e) ;
i s on agec70 female ;Mplus VERSION 8.11 (Mac)
MUTHEN & MUTHEN
04/10/2025 12:32 PM
INPUT INSTRUCTIONS
TITLE: LGCM CESD score
DATA: FILE = cesd.dat ;
VARIABLE: NAMES = id female agec70 yz0 yz3 yz6;
IDVARIABLE = id ;
MISSING = . ;
MODEL: i s | yz0@0 yz3@0.3 yz6@0.6 ; ! 10 year units
yz0-yz6 (e) ;
i s on agec70 female ;
INPUT READING TERMINATED NORMALLY
LGCM CESD score
SUMMARY OF ANALYSIS
Number of groups 1
Number of observations 2762
Number of dependent variables 3
Number of independent variables 2
Number of continuous latent variables 2
Observed dependent variables
Continuous
YZ0 YZ3 YZ6
Observed independent variables
FEMALE AGEC70
Continuous latent variables
I S
Variables with special functions
ID variable ID
Estimator ML
Information matrix OBSERVED
Maximum number of iterations 1000
Convergence criterion 0.500D-04
Maximum number of steepest descent iterations 20
Maximum number of iterations for H1 2000
Convergence criterion for H1 0.100D-03
Input data file(s)
cesd.dat
Input data format FREE
SUMMARY OF DATA
Number of missing data patterns 7
COVARIANCE COVERAGE OF DATA
Minimum covariance coverage value 0.100
PROPORTION OF DATA PRESENT
Covariance Coverage
YZ0 YZ3 YZ6 FEMALE AGEC70
________ ________ ________ ________ ________
YZ0 0.988
YZ3 0.719 0.730
YZ6 0.507 0.491 0.515
FEMALE 0.988 0.730 0.515 1.000
AGEC70 0.988 0.730 0.515 1.000 1.000
UNIVARIATE SAMPLE STATISTICS
UNIVARIATE HIGHER-ORDER MOMENT DESCRIPTIVE STATISTICS
Variable/ Mean/ Skewness/ Minimum/ % with Percentiles
Sample Size Variance Kurtosis Maximum Min/Max 20%/60% 40%/80% Median
YZ0 0.000 1.378 -0.908 29.62% -0.908 -0.589 -0.377
2728.000 1.000 1.699 3.867 0.51% -0.112 0.684
YZ3 0.027 1.409 -0.908 27.00% -0.908 -0.377 -0.271
2015.000 1.029 1.768 3.867 0.50% -0.112 0.684
YZ6 0.081 1.172 -0.908 27.78% -0.908 -0.377 -0.112
1422.000 1.053 0.910 3.867 0.21% 0.047 0.949
FEMALE 0.584 -0.342 0.000 41.56% 0.000 0.000 1.000
2762.000 0.243 -1.883 1.000 58.44% 1.000 1.000
AGEC70 0.364 0.673 -0.300 34.65% -0.300 0.200 0.200
2762.000 0.419 -0.681 1.700 8.76% 0.200 1.200
THE MODEL ESTIMATION TERMINATED NORMALLY
MODEL FIT INFORMATION
Number of Free Parameters 10
Loglikelihood
H0 Value -8447.120
H1 Value -8443.931
Information Criteria
Akaike (AIC) 16914.240
Bayesian (BIC) 16973.477
Sample-Size Adjusted BIC 16941.704
(n* = (n + 2) / 24)
Chi-Square Test of Model Fit
Value 6.378
Degrees of Freedom 5
P-Value 0.2711
RMSEA (Root Mean Square Error Of Approximation)
Estimate 0.010
90 Percent C.I. 0.000 0.030
Probability RMSEA <= .05 1.000
CFI/TLI
CFI 0.998
TLI 0.997
Chi-Square Test of Model Fit for the Baseline Model
Value 736.587
Degrees of Freedom 9
P-Value 0.0000
SRMR (Standardized Root Mean Square Residual)
Value 0.013
MODEL RESULTS
Two-Tailed
Estimate S.E. Est./S.E. P-Value
I |
YZ0 1.000 0.000 999.000 999.000
YZ3 1.000 0.000 999.000 999.000
YZ6 1.000 0.000 999.000 999.000
S |
YZ0 0.000 0.000 999.000 999.000
YZ3 0.300 0.000 999.000 999.000
YZ6 0.600 0.000 999.000 999.000
I ON
AGEC70 0.113 0.029 3.961 0.000
FEMALE 0.217 0.037 5.801 0.000
S ON
AGEC70 0.158 0.078 2.027 0.043
FEMALE -0.013 0.094 -0.135 0.892
S WITH
I -0.045 0.061 -0.741 0.458
Intercepts
YZ0 0.000 0.000 999.000 999.000
YZ3 0.000 0.000 999.000 999.000
YZ6 0.000 0.000 999.000 999.000
I -0.177 0.030 -5.871 0.000
S 0.196 0.075 2.607 0.009
Residual Variances
YZ0 0.595 0.021 28.943 0.000
YZ3 0.595 0.021 28.943 0.000
YZ6 0.595 0.021 28.943 0.000
I 0.400 0.031 12.916 0.000
S 0.327 0.183 1.788 0.074
QUALITY OF NUMERICAL RESULTS
Condition Number for the Information Matrix 0.416E-02
(ratio of smallest to largest eigenvalue)
Beginning Time: 12:32:28
Ending Time: 12:32:28
Elapsed Time: 00:00:00
MUTHEN & MUTHEN
3463 Stoner Ave.
Los Angeles, CA 90066
Tel: (310) 391-9971
Fax: (310) 391-8971
Web: www.StatModel.com
Support: Support@StatModel.com
Copyright (c) 1998-2024 Muthen & MuthenSave output from Mplus Engine
Multilevel model for change
LGCM vs Multilevel model
Next I’ll look at a multilevel modeling approach to measuring change in CESD symptoms. This means using the data in LONG format (rather than WIDE in the LGCM). Still looking at a standardized z-score for CESD symptoms: no latents yet for CESD.
Data setup
df.cesd <- df %>%
select(which(names(df) %in% c("id", "td","yz","agec70","female")))
psych::describe(df.cesd, type = 3, skew = FALSE, ranges = TRUE, na.rm = TRUE) vars n mean sd median min max range se
id 1 6165 1391.95 811.09 1393.00 1.00 2812.00 2811.00 10.33
td 2 6165 0.24 0.24 0.30 0.00 0.60 0.60 0.00
yz 3 6165 0.03 1.01 -0.11 -0.91 3.87 4.77 0.01
female 4 6165 0.60 0.49 1.00 0.00 1.00 1.00 0.01
agec70 5 6165 0.30 0.62 0.20 -0.30 1.70 2.00 0.01Use MplusAutomation to prepare data set for Mplus
Mplus Multilevel model
TITLE: MLM CESD score
DATA: FILE = cesd.dat ;
VARIABLE: NAMES = id td yz female agec70;
MISSING = . ;
WITHIN = td ;
BETWEEN = agec70 female ;
CLUSTER = id ;
ANALYSIS: TYPE = TWOLEVEL RANDOM ;
OUTPUT: TECH1;
MODEL: %WITHIN%
s | yz on td ;
%BETWEEN%
yz on agec70 female ;
s on agec70 female ;
yz with s ;Mplus VERSION 8.11 (Mac)
MUTHEN & MUTHEN
04/10/2025 12:32 PM
INPUT INSTRUCTIONS
TITLE: MLM CESD score
DATA: FILE = cesd.dat ;
VARIABLE: NAMES = id td yz female agec70;
MISSING = . ;
WITHIN = td ;
BETWEEN = agec70 female ;
CLUSTER = id ;
ANALYSIS: TYPE = TWOLEVEL RANDOM ;
OUTPUT: TECH1;
MODEL: %WITHIN%
s | yz on td ;
%BETWEEN%
yz on agec70 female ;
s on agec70 female ;
yz with s ;
*** WARNING
One or more individual-level variables have no variation within a
cluster for the following clusters.
Variable Cluster IDs with no within-cluster variation
YZ 762 765 769 50 776 1497 1510 792 1538 1559 425 96 1590 836 1665 1666 1682 853
860 1716 457 889 899 1773 1777 1778 1783 1800 930 1843 1846 255 1872 1874 257
1883 259 1899 1923 501 969 1939 1969 990 1991 1992 1997 1999 1004 513 2024 1015
524 2070 2090 2092 2109 2159 2162 2165 564 2236 2242 25 2318 2330 2338 329 601
2371 2374 1179 603 2382 2393 611 2398 2402 2411 331 2452 8 159 2471 1241 653
163 2542 1270 1283 1293 1295 1301 2598 668 2617 2620 1316 2623 2625 1322 1323
2632 1328 2648 2650 2679 2684 173 2686 2690 361 2707 1356 363 709 2750 180 2783
1399 1404 195 740 396 1419 1948 2634 346 347 1477 1484 747 1561 781 384 1636
1641 812 1651 1663 1684 1692 1694 1696 399 843 1724 1737 1744 1745 857 1760
865 414 418 895 924 1869 23 1903 218 108 111 997 498 2023 2034 2045 2061 2067
2082 2086 1051 1062 526 2173 2179 260 122 2199 2216 1120 2252 1149 2290 2291
1156 1157 276 2321 1169 2325 278 279 2353 2357 284 588 2392 132 598 2432 2444
2469 615 2483 298 1258 1265 1278 2525 2526 630 2539 2547 2554 2577 639 2586
648 1325 1342 34 2680 675 2694 1383 1393 1396 703 2793 704 2798 73 1445
1 WARNING(S) FOUND IN THE INPUT INSTRUCTIONS
MLM CESD score
SUMMARY OF ANALYSIS
Number of groups 1
Number of observations 6165
Number of dependent variables 1
Number of independent variables 3
Number of continuous latent variables 1
Observed dependent variables
Continuous
YZ
Observed independent variables
TD FEMALE AGEC70
Continuous latent variables
S
Variables with special functions
Cluster variable ID
Within variables
TD
Between variables
FEMALE AGEC70
Estimator MLR
Information matrix OBSERVED
Maximum number of iterations 100
Convergence criterion 0.100D-05
Maximum number of EM iterations 500
Convergence criteria for the EM algorithm
Loglikelihood change 0.100D-02
Relative loglikelihood change 0.100D-05
Derivative 0.100D-03
Minimum variance 0.100D-03
Maximum number of steepest descent iterations 20
Maximum number of iterations for H1 2000
Convergence criterion for H1 0.100D-03
Optimization algorithm EMA
Input data file(s)
cesd.dat
Input data format FREE
SUMMARY OF DATA
Number of missing data patterns 1
Number of clusters 2762
COVARIANCE COVERAGE OF DATA
Minimum covariance coverage value 0.100
PROPORTION OF DATA PRESENT
Covariance Coverage
YZ TD FEMALE AGEC70
________ ________ ________ ________
YZ 1.000
TD 1.000 1.000
FEMALE 1.000 1.000 1.000
AGEC70 1.000 1.000 1.000 1.000
UNIVARIATE SAMPLE STATISTICS
UNIVARIATE HIGHER-ORDER MOMENT DESCRIPTIVE STATISTICS
Variable/ Mean/ Skewness/ Minimum/ % with Percentiles
Sample Size Variance Kurtosis Maximum Min/Max 20%/60% 40%/80% Median
YZ 0.028 1.339 -0.908 28.34% -0.908 -0.377 -0.112
6165.000 1.022 1.522 3.867 0.44% -0.112 0.684
TD 0.236 0.395 0.000 44.25% 0.000 0.000 0.300
6165.000 0.057 -1.306 0.600 23.07% 0.300 0.600
FEMALE 0.584 -0.342 0.000 41.56% 0.000 0.000 1.000
2762.000 0.243 -1.883 1.000 58.44% 1.000 1.000
AGEC70 0.364 0.673 -0.300 34.65% -0.300 0.200 0.200
2762.000 0.419 -0.681 1.700 8.76% 0.200 1.200
THE MODEL ESTIMATION TERMINATED NORMALLY
MODEL FIT INFORMATION
Number of Free Parameters 10
Loglikelihood
H0 Value -8447.120
H0 Scaling Correction Factor 1.2863
for MLR
Information Criteria
Akaike (AIC) 16914.241
Bayesian (BIC) 16981.507
Sample-Size Adjusted BIC 16949.730
(n* = (n + 2) / 24)
MODEL RESULTS
Two-Tailed
Estimate S.E. Est./S.E. P-Value
Within Level
Residual Variances
YZ 0.595 0.028 21.362 0.000
Between Level
S ON
AGEC70 0.158 0.083 1.891 0.059
FEMALE -0.013 0.091 -0.142 0.887
YZ ON
AGEC70 0.113 0.030 3.813 0.000
FEMALE 0.217 0.036 5.951 0.000
YZ WITH
S -0.046 0.075 -0.609 0.543
Intercepts
YZ -0.177 0.027 -6.598 0.000
S 0.196 0.069 2.847 0.004
Residual Variances
YZ 0.400 0.040 9.943 0.000
S 0.328 0.220 1.493 0.135
QUALITY OF NUMERICAL RESULTS
Condition Number for the Information Matrix 0.230E-02
(ratio of smallest to largest eigenvalue)
TECHNICAL 1 OUTPUT
PARAMETER SPECIFICATION FOR WITHIN
NU
YZ TD
________ ________
0 0
LAMBDA
YZ TD
________ ________
YZ 0 0
TD 0 0
THETA
YZ TD
________ ________
YZ 0
TD 0 0
ALPHA
YZ TD
________ ________
0 0
BETA
YZ TD
________ ________
YZ 0 0
TD 0 0
PSI
YZ TD
________ ________
YZ 1
TD 0 0
PARAMETER SPECIFICATION FOR BETWEEN
NU
YZ FEMALE AGEC70
________ ________ ________
0 0 0
LAMBDA
S YZ FEMALE AGEC70
________ ________ ________ ________
YZ 0 0 0 0
FEMALE 0 0 0 0
AGEC70 0 0 0 0
THETA
YZ FEMALE AGEC70
________ ________ ________
YZ 0
FEMALE 0 0
AGEC70 0 0 0
ALPHA
S YZ FEMALE AGEC70
________ ________ ________ ________
2 3 0 0
BETA
S YZ FEMALE AGEC70
________ ________ ________ ________
S 0 0 4 5
YZ 0 0 6 7
FEMALE 0 0 0 0
AGEC70 0 0 0 0
PSI
S YZ FEMALE AGEC70
________ ________ ________ ________
S 8
YZ 9 10
FEMALE 0 0 0
AGEC70 0 0 0 0
STARTING VALUES FOR WITHIN
NU
YZ TD
________ ________
0.000 0.000
LAMBDA
YZ TD
________ ________
YZ 1.000 0.000
TD 0.000 1.000
THETA
YZ TD
________ ________
YZ 0.000
TD 0.000 0.000
ALPHA
YZ TD
________ ________
0.000 0.000
BETA
YZ TD
________ ________
YZ 0.000 0.000
TD 0.000 0.000
PSI
YZ TD
________ ________
YZ 0.511
TD 0.000 0.028
STARTING VALUES FOR BETWEEN
NU
YZ FEMALE AGEC70
________ ________ ________
0.000 0.000 0.000
LAMBDA
S YZ FEMALE AGEC70
________ ________ ________ ________
YZ 0.000 1.000 0.000 0.000
FEMALE 0.000 0.000 1.000 0.000
AGEC70 0.000 0.000 0.000 1.000
THETA
YZ FEMALE AGEC70
________ ________ ________
YZ 0.000
FEMALE 0.000 0.000
AGEC70 0.000 0.000 0.000
ALPHA
S YZ FEMALE AGEC70
________ ________ ________ ________
0.000 0.028 0.000 0.000
BETA
S YZ FEMALE AGEC70
________ ________ ________ ________
S 0.000 0.000 0.000 0.000
YZ 0.000 0.000 0.000 0.000
FEMALE 0.000 0.000 0.000 0.000
AGEC70 0.000 0.000 0.000 0.000
PSI
S YZ FEMALE AGEC70
________ ________ ________ ________
S 1.000
YZ 0.000 0.511
FEMALE 0.000 0.000 0.120
AGEC70 0.000 0.000 0.000 0.192
Beginning Time: 12:32:28
Ending Time: 12:32:29
Elapsed Time: 00:00:01
MUTHEN & MUTHEN
3463 Stoner Ave.
Los Angeles, CA 90066
Tel: (310) 391-9971
Fax: (310) 391-8971
Web: www.StatModel.com
Support: Support@StatModel.com
Copyright (c) 1998-2024 Muthen & MuthenCollect fits
model1Results <- MplusAutomation::readModels(target="model1.out")
model2Results <- MplusAutomation::readModels(target="model2.out")
Fits <- MplusAutomation::SummaryTable(list(model1Results, model2Results),
keepCols = c("Title","Parameters","NDependentVars",
"NContinuousLatentVars","Observations","LL", "AIC", "BIC", "ChiSqM_Value",
"ChiSqM_DF","RMSEA_Estimate","SRMR","CFI"))
FitsT <- as.data.frame(t(Fits))
# Add the variable names as the first column
FitsT$Feature <- rownames(FitsT)
# Adjust the column names
colnames(FitsT) <- c("Model 1", "Model 2", "Feature")
# Reset the rownames
rownames(FitsT) <- NULL
# Reorder
FitsT <- FitsT[c("Feature", "Model 1", "Model 2")]Compare fits
| Feature | Model 1 | Model 2 |
|---|---|---|
| Title | LGCM CESD score | MLM CESD score |
| Parameters | 10 | 10 |
| NDependentVars | 3 | 1 |
| NContinuousLatentVars | 2 | 1 |
| Observations | 2762 | 6165 |
| LL | -8447.12 | -8447.12 |
| AIC | 16914.24 | 16914.24 |
| BIC | 16973.48 | 16981.51 |
| ChiSqM_Value | 6.378 | NA |
| ChiSqM_DF | 5 | NA |
| RMSEA_Estimate | 0.01 | NA |
| SRMR | 0.013 | NA |
| CFI | 0.998 | NA |
These models are equivalent by LL. The BICs are discussed on the next slide.
About fits
If you just looked at the BIC, you might think that there was a small advantage for LGCM over MLM. But the loglikelihood (\(LL\)) parameters reveal these models are equivalent. The BIC advantage comes from how the sample size is calculated.
The AIC is \(-2LL + 2r\) where \(r\) is the number of free parameters. The BIC is \(-2LL + r + ln(N)\) where \(N\) is the sample size. The BIC and the AIC will only be equal when \(r = ln(N)\). Moreover, because of how the data is structured for the MLM analysis, Mplus considers the sample size to be 6165 (person observations \(\times\) occasions) for the MLM and only 2762 (person observations) for the LGCM. Therefore, the BICs are not comparable between the LGCM and MLM models.
Let’s compare parameter estimates and see what’s the same and what’s different.
LGCM_parameters <- model1Results$parameters$unstandardized %>% mutate(PARAMETER = str_c(paramHeader,param, sep = " "))
MLM_parameters <- model2Results$parameters$unstandardized %>% mutate(PARAMETER = str_c(paramHeader,param, sep = " "))
combinedParameters <- bind_rows(LGCM_parameters, MLM_parameters)BY parameters
| paramHeader | param | est | se | est_se | pval | PARAMETER | BetweenWithin |
|---|---|---|---|---|---|---|---|
| I.| | YZ0 | 1.0 | 0 | 999 | 999 | I.| YZ0 | NA |
| I.| | YZ3 | 1.0 | 0 | 999 | 999 | I.| YZ3 | NA |
| I.| | YZ6 | 1.0 | 0 | 999 | 999 | I.| YZ6 | NA |
| S.| | YZ0 | 0.0 | 0 | 999 | 999 | S.| YZ0 | NA |
| S.| | YZ3 | 0.3 | 0 | 999 | 999 | S.| YZ3 | NA |
| S.| | YZ6 | 0.6 | 0 | 999 | 999 | S.| YZ6 | NA |
If the column BetweenWithin is NA, that means the result is from the LGCM.
Means and Intercepts parameters
| paramHeader | param | est | se | est_se | pval | PARAMETER | BetweenWithin |
|---|---|---|---|---|---|---|---|
| Intercepts | YZ0 | 0.000 | 0.000 | 999.000 | 999.000 | Intercepts YZ0 | NA |
| Intercepts | YZ3 | 0.000 | 0.000 | 999.000 | 999.000 | Intercepts YZ3 | NA |
| Intercepts | YZ6 | 0.000 | 0.000 | 999.000 | 999.000 | Intercepts YZ6 | NA |
| Intercepts | I | -0.177 | 0.030 | -5.871 | 0.000 | Intercepts I | NA |
| Intercepts | S | 0.196 | 0.075 | 2.607 | 0.009 | Intercepts S | NA |
| Intercepts | YZ | -0.177 | 0.027 | -6.598 | 0.000 | Intercepts YZ | Between |
| Intercepts | S | 0.196 | 0.069 | 2.847 | 0.004 | Intercepts S | Between |
The intercepts for the LEVEL (Intercept I for LGCM model and Intercept YZ for the MLM model) are the same, but the standard errors are smaller for the MLM model. Sme for the intercepts for the SLOPES.
Variances parameters
| paramHeader | param | est | se | est_se | pval | PARAMETER | BetweenWithin |
|---|---|---|---|---|---|---|---|
| Residual.Variances | YZ0 | 0.595 | 0.021 | 28.943 | 0.000 | Residual.Variances YZ0 | NA |
| Residual.Variances | YZ3 | 0.595 | 0.021 | 28.943 | 0.000 | Residual.Variances YZ3 | NA |
| Residual.Variances | YZ6 | 0.595 | 0.021 | 28.943 | 0.000 | Residual.Variances YZ6 | NA |
| Residual.Variances | I | 0.400 | 0.031 | 12.916 | 0.000 | Residual.Variances I | NA |
| Residual.Variances | S | 0.327 | 0.183 | 1.788 | 0.074 | Residual.Variances S | NA |
| Residual.Variances | YZ | 0.595 | 0.028 | 21.362 | 0.000 | Residual.Variances YZ | Within |
| Residual.Variances | YZ | 0.400 | 0.040 | 9.943 | 0.000 | Residual.Variances YZ | Between |
| Residual.Variances | S | 0.328 | 0.220 | 1.493 | 0.135 | Residual.Variances S | Between |
By constraining the residual variances to be equal in the LGCM, we obtain the same results set in the LGCM as the MLM model. For these parameters the standard errors are smaller for the LGCM model.
Multilevel factor model
Now I’ll turn to latent variable modeling for depression.
First I’ll use a multilevel confirmatory factor analysis model (MLCFA). Many examples of this kind of a model can be found in the literature. However, this model comes with some difficulties. A factor measurement model must be specified at the WITHIN and BETWEEN level. These need not be the same, but I am ignorant regarding guidance on whether, how, and why these measurement models should be different. I keep them the same.
Additionally, the MLCFA model allows for residual variances at the factor indicator level to be distributed at the WITHIN and BETWEEN levels. I am just as confused about what to do about that. As I will describe, I don’t allow for factor indicator residual variances at the BETWEEN level in my parameterization.
Modeling considerations: MLCFA
This is the model set-up for the multilevel factor analysis with regression on time in study as a random effect. Observed indicators of depression (y1-y6) are both within and between level variables (and not specified as WITHIN or BETWEEN). A within-level factor is specified (fw) that is identified by fixing the first factor loading (I run a series of preliminary models to find the factor loading that returns a total, single-level, baseline-only common latent variable variance of 1.0, and fix to that value). The fixed parameters are shown in purlple with “@” label. fw is regressed on time and this is declared a random effect. If we had within-level (i.e., time-varying) covariates, we would include them as illustrated with the dashed box and regressions “xw”.
At the between level, I have a between level common latent variable (fb). We assume the factor loadings at the between level and within level are equal. (This assumption is not necessary but I don’t have a reason to do anything else. I am not sure of the implications of having the measurement slopes be equal versus different at the BETWEEN and WITHIN levels). The item intercepts are modeled at the BETWEEN level. All but one are freely estimated. I fix one so that a mean for fb will be identified. The value to which the first item intercept is fixed is derived from the same preliminary models described previously for setting the metric of the factor loadings. The first item intercept is fixed to a value that returns a latent variable mean of 0 at baseline in a single level baseline only model.
The indicators have residual variances at the WITHIN and BETWEEN levels. I will fix these to 0 at the BETWEEN level because I am concerned that not doing so will rob the latent slope S of variance. But, we may have to play around with that.
Data setup
items <- c("sad", "blues", "depress", "happy", "enjoy", "hopeful")
df.cesd <- df %>%
select(which(names(df) %in% c("id", "td",items,"agec70","female")))
# Create new variables y1-y6 that correspond to the variables in items
for (i in seq_along(items)) {
df.cesd[[paste0("y", i)]] <- df.cesd[[items[i]]]
}
df.cesd <- df.cesd %>% select(-all_of(items))Descriptives for MLFA data
vars n mean sd median min max range se
id 1 6165 1391.95 811.09 1393.0 1.0 2812.0 2811.0 10.33
td 2 6165 0.24 0.24 0.3 0.0 0.6 0.6 0.00
female 3 6165 0.60 0.49 1.0 0.0 1.0 1.0 0.01
agec70 4 6165 0.30 0.62 0.2 -0.3 1.7 2.0 0.01
y1 5 6119 1.47 0.75 1.0 1.0 4.0 3.0 0.01
y2 6 6109 1.35 0.74 1.0 1.0 4.0 3.0 0.01
y3 7 6130 1.52 0.82 1.0 1.0 4.0 3.0 0.01
y4 8 6071 1.70 1.00 1.0 1.0 4.0 3.0 0.01
y5 9 6063 1.54 0.94 1.0 1.0 4.0 3.0 0.01
y6 10 5693 1.94 1.19 1.0 1.0 4.0 3.0 0.02Use MplusAutomation to prepare data set for Mplus
Preliminary model 1
Find the measurement slope at baseline for y1 such that the factor has a variance of 1.
First step: find R-squared for fw
Mplus VERSION 8.11 (Mac)
MUTHEN & MUTHEN
04/10/2025 12:32 PM
INPUT INSTRUCTIONS
TITLE: MLFA Preliminary Model 1
DATA: FILE = cesd.dat ;
VARIABLE: NAMES = id t female agec70 y1 y2 y3 y4 y5 y6 ;
USEVARIABLES = female agec70 y1-y6 ;
USEOBSERVATIONS = (t==0) ;
MISSING = . ;
OUTPUT: STDYX ; TECH4 ; SVALUES;
MODEL: fw by y1-y6* (l1-l6);
fw@1;
fw on female agec70 ;
INPUT READING TERMINATED NORMALLY
MLFA Preliminary Model 1
SUMMARY OF ANALYSIS
Number of groups 1
Number of observations 2728
Number of dependent variables 6
Number of independent variables 2
Number of continuous latent variables 1
Observed dependent variables
Continuous
Y1 Y2 Y3 Y4 Y5 Y6
Observed independent variables
FEMALE AGEC70
Continuous latent variables
FW
Estimator ML
Information matrix OBSERVED
Maximum number of iterations 1000
Convergence criterion 0.500D-04
Maximum number of steepest descent iterations 20
Maximum number of iterations for H1 2000
Convergence criterion for H1 0.100D-03
Input data file(s)
cesd.dat
Input data format FREE
SUMMARY OF DATA
Number of missing data patterns 22
COVARIANCE COVERAGE OF DATA
Minimum covariance coverage value 0.100
PROPORTION OF DATA PRESENT
Covariance Coverage
Y1 Y2 Y3 Y4 Y5
________ ________ ________ ________ ________
Y1 0.993
Y2 0.987 0.993
Y3 0.991 0.989 0.996
Y4 0.987 0.982 0.986 0.989
Y5 0.986 0.981 0.986 0.981 0.988
Y6 0.927 0.925 0.929 0.924 0.923
FEMALE 0.993 0.993 0.996 0.989 0.988
AGEC70 0.993 0.993 0.996 0.989 0.988
Covariance Coverage
Y6 FEMALE AGEC70
________ ________ ________
Y6 0.931
FEMALE 0.931 1.000
AGEC70 0.931 1.000 1.000
UNIVARIATE SAMPLE STATISTICS
UNIVARIATE HIGHER-ORDER MOMENT DESCRIPTIVE STATISTICS
Variable/ Mean/ Skewness/ Minimum/ % with Percentiles
Sample Size Variance Kurtosis Maximum Min/Max 20%/60% 40%/80% Median
Y1 1.458 1.812 1.000 65.26% 1.000 1.000 1.000
2709.000 0.545 3.142 4.000 3.77% 1.000 2.000
Y2 1.345 2.371 1.000 77.44% 1.000 1.000 1.000
2708.000 0.552 5.035 4.000 4.36% 1.000 2.000
Y3 1.522 1.718 1.000 63.55% 1.000 1.000 1.000
2716.000 0.686 2.339 4.000 6.04% 1.000 2.000
Y4 1.650 1.189 1.000 66.22% 1.000 1.000 1.000
2697.000 1.001 -0.071 4.000 7.45% 1.000 3.000
Y5 1.465 1.735 1.000 75.91% 1.000 1.000 1.000
2694.000 0.815 1.599 4.000 5.86% 1.000 2.000
Y6 1.998 0.643 1.000 56.87% 1.000 1.000 1.000
2539.000 1.543 -1.319 4.000 20.40% 2.000 4.000
FEMALE 0.585 -0.345 0.000 41.50% 0.000 0.000 1.000
2728.000 0.243 -1.881 1.000 58.50% 1.000 1.000
AGEC70 0.363 0.675 -0.300 34.60% -0.300 0.200 0.200
2728.000 0.418 -0.674 1.700 8.72% 0.200 1.200
THE MODEL ESTIMATION TERMINATED NORMALLY
MODEL FIT INFORMATION
Number of Free Parameters 20
Loglikelihood
H0 Value -18938.943
H1 Value -18682.363
Information Criteria
Akaike (AIC) 37917.886
Bayesian (BIC) 38036.113
Sample-Size Adjusted BIC 37972.566
(n* = (n + 2) / 24)
Chi-Square Test of Model Fit
Value 513.160
Degrees of Freedom 19
P-Value 0.0000
RMSEA (Root Mean Square Error Of Approximation)
Estimate 0.098
90 Percent C.I. 0.090 0.105
Probability RMSEA <= .05 0.000
CFI/TLI
CFI 0.889
TLI 0.843
Chi-Square Test of Model Fit for the Baseline Model
Value 4495.941
Degrees of Freedom 27
P-Value 0.0000
SRMR (Standardized Root Mean Square Residual)
Value 0.049
MODEL RESULTS
Two-Tailed
Estimate S.E. Est./S.E. P-Value
FW BY
Y1 0.505 0.014 37.026 0.000
Y2 0.503 0.014 36.267 0.000
Y3 0.619 0.015 41.107 0.000
Y4 0.604 0.020 30.811 0.000
Y5 0.521 0.018 29.135 0.000
Y6 0.446 0.026 17.047 0.000
FW ON
FEMALE 0.276 0.043 6.341 0.000
AGEC70 0.113 0.033 3.412 0.001
Intercepts
Y1 1.358 0.020 68.105 0.000
Y2 1.245 0.020 62.374 0.000
Y3 1.398 0.023 59.995 0.000
Y4 1.530 0.026 59.738 0.000
Y5 1.363 0.023 59.997 0.000
Y6 1.909 0.028 68.479 0.000
Residual Variances
Y1 0.285 0.010 28.819 0.000
Y2 0.293 0.010 28.719 0.000
Y3 0.294 0.012 24.630 0.000
Y4 0.630 0.021 30.503 0.000
Y5 0.538 0.017 31.176 0.000
Y6 1.338 0.039 34.266 0.000
FW 1.000 0.000 999.000 999.000
QUALITY OF NUMERICAL RESULTS
Condition Number for the Information Matrix 0.215E-01
(ratio of smallest to largest eigenvalue)
STANDARDIZED MODEL RESULTS
STDYX Standardization
Two-Tailed
Estimate S.E. Est./S.E. P-Value
FW BY
Y1 0.692 0.013 53.584 0.000
Y2 0.685 0.013 51.685 0.000
Y3 0.756 0.012 63.787 0.000
Y4 0.610 0.015 39.784 0.000
Y5 0.584 0.016 36.638 0.000
Y6 0.363 0.020 18.541 0.000
FW ON
FEMALE 0.134 0.021 6.433 0.000
AGEC70 0.072 0.021 3.426 0.001
Intercepts
Y1 1.837 0.039 46.880 0.000
Y2 1.674 0.037 44.665 0.000
Y3 1.688 0.039 43.574 0.000
Y4 1.528 0.035 43.688 0.000
Y5 1.508 0.034 43.868 0.000
Y6 1.537 0.032 47.359 0.000
Residual Variances
Y1 0.521 0.018 29.148 0.000
Y2 0.531 0.018 29.222 0.000
Y3 0.428 0.018 23.883 0.000
Y4 0.628 0.019 33.534 0.000
Y5 0.659 0.019 35.462 0.000
Y6 0.868 0.014 60.901 0.000
FW 0.976 0.006 151.557 0.000
R-SQUARE
Observed Two-Tailed
Variable Estimate S.E. Est./S.E. P-Value
Y1 0.479 0.018 26.792 0.000
Y2 0.469 0.018 25.842 0.000
Y3 0.572 0.018 31.893 0.000
Y4 0.372 0.019 19.892 0.000
Y5 0.341 0.019 18.319 0.000
Y6 0.132 0.014 9.271 0.000
Latent Two-Tailed
Variable Estimate S.E. Est./S.E. P-Value
FW 0.024 0.006 3.724 0.000
MODEL COMMAND WITH FINAL ESTIMATES USED AS STARTING VALUES
fw BY y1*0.50532 (l1);
fw BY y2*0.50318 (l2);
fw BY y3*0.61882 (l3);
fw BY y4*0.60374 (l4);
fw BY y5*0.52103 (l5);
fw BY y6*0.44589 (l6);
fw ON female*0.27581;
fw ON agec70*0.11268;
[ y1*1.35775 ];
[ y2*1.24453 ];
[ y3*1.39801 ];
[ y4*1.53031 ];
[ y5*1.36285 ];
[ y6*1.90864 ];
y1*0.28463;
y2*0.29333;
y3*0.29381;
y4*0.62958;
y5*0.53844;
y6*1.33815;
fw@1;
TECHNICAL 4 OUTPUT
ESTIMATES DERIVED FROM THE MODEL
ESTIMATED MEANS FOR THE LATENT VARIABLES
FW FEMALE AGEC70
________ ________ ________
0.202 0.585 0.363
S.E. FOR ESTIMATED MEANS FOR THE LATENT VARIABLES
FW FEMALE AGEC70
________ ________ ________
0.028 0.009 0.012
EST./S.E. FOR ESTIMATED MEANS FOR THE LATENT VARIABLES
FW FEMALE AGEC70
________ ________ ________
7.242 62.018 29.327
TWO-TAILED P-VALUE FOR ESTIMATED MEANS FOR THE LATENT VARIABLES
FW FEMALE AGEC70
________ ________ ________
0.000 0.000 0.000
ESTIMATED COVARIANCE MATRIX FOR THE LATENT VARIABLES
FW FEMALE AGEC70
________ ________ ________
FW 1.025
FEMALE 0.068 0.243
AGEC70 0.051 0.013 0.418
S.E. FOR ESTIMATED COVARIANCE MATRIX FOR THE LATENT VARIABLES
FW FEMALE AGEC70
________ ________ ________
FW 0.007
FEMALE 0.011 0.007
AGEC70 0.014 0.006 0.011
EST./S.E. FOR ESTIMATED COVARIANCE MATRIX FOR THE LATENT VARIABLES
FW FEMALE AGEC70
________ ________ ________
FW 151.557
FEMALE 6.370 36.932
AGEC70 3.625 2.099 36.932
TWO-TAILED P-VALUE FOR ESTIMATED COVARIANCE MATRIX FOR THE LATENT VARIABLES
FW FEMALE AGEC70
________ ________ ________
FW 0.000
FEMALE 0.000 0.000
AGEC70 0.000 0.036 0.000
ESTIMATED CORRELATION MATRIX FOR THE LATENT VARIABLES
FW FEMALE AGEC70
________ ________ ________
FW 1.000
FEMALE 0.137 1.000
AGEC70 0.077 0.040 1.000
S.E. FOR ESTIMATED CORRELATION MATRIX FOR THE LATENT VARIABLES
FW FEMALE AGEC70
________ ________ ________
FW 0.000
FEMALE 0.021 0.000
AGEC70 0.021 0.019 0.000
EST./S.E. FOR ESTIMATED CORRELATION MATRIX FOR THE LATENT VARIABLES
FW FEMALE AGEC70
________ ________ ________
FW 999.000
FEMALE 6.566 999.000
AGEC70 3.661 2.104 999.000
TWO-TAILED P-VALUE FOR ESTIMATED CORRELATION MATRIX FOR THE LATENT VARIABLES
FW FEMALE AGEC70
________ ________ ________
FW 0.000
FEMALE 0.000 0.000
AGEC70 0.000 0.035 0.000
Beginning Time: 12:32:29
Ending Time: 12:32:29
Elapsed Time: 00:00:00
MUTHEN & MUTHEN
3463 Stoner Ave.
Los Angeles, CA 90066
Tel: (310) 391-9971
Fax: (310) 391-8971
Web: www.StatModel.com
Support: Support@StatModel.com
Copyright (c) 1998-2024 Muthen & MuthenSave and read results from first preliminary model
Find the R-squared
Second step: constrain residual variance of fw to 1-Rsquared, and check TECH4
Mplus VERSION 8.11 (Mac)
MUTHEN & MUTHEN
04/10/2025 12:32 PM
INPUT INSTRUCTIONS
TITLE: MLFA Preliminary Model 2
DATA: FILE = cesd.dat ;
VARIABLE: NAMES = id t female agec70 y1 y2 y3 y4 y5 y6 ;
USEVARIABLES = female agec70 y1-y6 ;
USEOBSERVATIONS = (t==0) ;
MISSING = . ;
OUTPUT: STDYX ; TECH4 ; SVALUES;
MODEL: fw by y1-y6* (l1-l6);
fw@0.976; !1-R-squared
fw on female agec70 ;
INPUT READING TERMINATED NORMALLY
MLFA Preliminary Model 2
SUMMARY OF ANALYSIS
Number of groups 1
Number of observations 2728
Number of dependent variables 6
Number of independent variables 2
Number of continuous latent variables 1
Observed dependent variables
Continuous
Y1 Y2 Y3 Y4 Y5 Y6
Observed independent variables
FEMALE AGEC70
Continuous latent variables
FW
Estimator ML
Information matrix OBSERVED
Maximum number of iterations 1000
Convergence criterion 0.500D-04
Maximum number of steepest descent iterations 20
Maximum number of iterations for H1 2000
Convergence criterion for H1 0.100D-03
Input data file(s)
cesd.dat
Input data format FREE
SUMMARY OF DATA
Number of missing data patterns 22
COVARIANCE COVERAGE OF DATA
Minimum covariance coverage value 0.100
PROPORTION OF DATA PRESENT
Covariance Coverage
Y1 Y2 Y3 Y4 Y5
________ ________ ________ ________ ________
Y1 0.993
Y2 0.987 0.993
Y3 0.991 0.989 0.996
Y4 0.987 0.982 0.986 0.989
Y5 0.986 0.981 0.986 0.981 0.988
Y6 0.927 0.925 0.929 0.924 0.923
FEMALE 0.993 0.993 0.996 0.989 0.988
AGEC70 0.993 0.993 0.996 0.989 0.988
Covariance Coverage
Y6 FEMALE AGEC70
________ ________ ________
Y6 0.931
FEMALE 0.931 1.000
AGEC70 0.931 1.000 1.000
UNIVARIATE SAMPLE STATISTICS
UNIVARIATE HIGHER-ORDER MOMENT DESCRIPTIVE STATISTICS
Variable/ Mean/ Skewness/ Minimum/ % with Percentiles
Sample Size Variance Kurtosis Maximum Min/Max 20%/60% 40%/80% Median
Y1 1.458 1.812 1.000 65.26% 1.000 1.000 1.000
2709.000 0.545 3.142 4.000 3.77% 1.000 2.000
Y2 1.345 2.371 1.000 77.44% 1.000 1.000 1.000
2708.000 0.552 5.035 4.000 4.36% 1.000 2.000
Y3 1.522 1.718 1.000 63.55% 1.000 1.000 1.000
2716.000 0.686 2.339 4.000 6.04% 1.000 2.000
Y4 1.650 1.189 1.000 66.22% 1.000 1.000 1.000
2697.000 1.001 -0.071 4.000 7.45% 1.000 3.000
Y5 1.465 1.735 1.000 75.91% 1.000 1.000 1.000
2694.000 0.815 1.599 4.000 5.86% 1.000 2.000
Y6 1.998 0.643 1.000 56.87% 1.000 1.000 1.000
2539.000 1.543 -1.319 4.000 20.40% 2.000 4.000
FEMALE 0.585 -0.345 0.000 41.50% 0.000 0.000 1.000
2728.000 0.243 -1.881 1.000 58.50% 1.000 1.000
AGEC70 0.363 0.675 -0.300 34.60% -0.300 0.200 0.200
2728.000 0.418 -0.674 1.700 8.72% 0.200 1.200
THE MODEL ESTIMATION TERMINATED NORMALLY
MODEL FIT INFORMATION
Number of Free Parameters 20
Loglikelihood
H0 Value -18938.943
H1 Value -18682.363
Information Criteria
Akaike (AIC) 37917.886
Bayesian (BIC) 38036.113
Sample-Size Adjusted BIC 37972.566
(n* = (n + 2) / 24)
Chi-Square Test of Model Fit
Value 513.160
Degrees of Freedom 19
P-Value 0.0000
RMSEA (Root Mean Square Error Of Approximation)
Estimate 0.098
90 Percent C.I. 0.090 0.105
Probability RMSEA <= .05 0.000
CFI/TLI
CFI 0.889
TLI 0.843
Chi-Square Test of Model Fit for the Baseline Model
Value 4495.941
Degrees of Freedom 27
P-Value 0.0000
SRMR (Standardized Root Mean Square Residual)
Value 0.049
MODEL RESULTS
Two-Tailed
Estimate S.E. Est./S.E. P-Value
FW BY
Y1 0.511 0.014 37.026 0.000
Y2 0.509 0.014 36.267 0.000
Y3 0.626 0.015 41.107 0.000
Y4 0.611 0.020 30.811 0.000
Y5 0.527 0.018 29.135 0.000
Y6 0.451 0.026 17.047 0.000
FW ON
FEMALE 0.272 0.043 6.341 0.000
AGEC70 0.111 0.033 3.412 0.001
Intercepts
Y1 1.358 0.020 68.105 0.000
Y2 1.245 0.020 62.374 0.000
Y3 1.398 0.023 59.995 0.000
Y4 1.530 0.026 59.738 0.000
Y5 1.363 0.023 59.997 0.000
Y6 1.909 0.028 68.479 0.000
Residual Variances
Y1 0.285 0.010 28.819 0.000
Y2 0.293 0.010 28.719 0.000
Y3 0.294 0.012 24.630 0.000
Y4 0.630 0.021 30.503 0.000
Y5 0.538 0.017 31.176 0.000
Y6 1.338 0.039 34.266 0.000
FW 0.976 0.000 999.000 999.000
QUALITY OF NUMERICAL RESULTS
Condition Number for the Information Matrix 0.219E-01
(ratio of smallest to largest eigenvalue)
STANDARDIZED MODEL RESULTS
STDYX Standardization
Two-Tailed
Estimate S.E. Est./S.E. P-Value
FW BY
Y1 0.692 0.013 53.584 0.000
Y2 0.685 0.013 51.685 0.000
Y3 0.756 0.012 63.787 0.000
Y4 0.610 0.015 39.784 0.000
Y5 0.584 0.016 36.638 0.000
Y6 0.363 0.020 18.541 0.000
FW ON
FEMALE 0.134 0.021 6.433 0.000
AGEC70 0.072 0.021 3.426 0.001
Intercepts
Y1 1.837 0.039 46.880 0.000
Y2 1.674 0.037 44.665 0.000
Y3 1.688 0.039 43.574 0.000
Y4 1.528 0.035 43.688 0.000
Y5 1.508 0.034 43.868 0.000
Y6 1.537 0.032 47.359 0.000
Residual Variances
Y1 0.521 0.018 29.148 0.000
Y2 0.531 0.018 29.222 0.000
Y3 0.428 0.018 23.883 0.000
Y4 0.628 0.019 33.534 0.000
Y5 0.659 0.019 35.462 0.000
Y6 0.868 0.014 60.901 0.000
FW 0.976 0.006 151.557 0.000
R-SQUARE
Observed Two-Tailed
Variable Estimate S.E. Est./S.E. P-Value
Y1 0.479 0.018 26.792 0.000
Y2 0.469 0.018 25.842 0.000
Y3 0.572 0.018 31.893 0.000
Y4 0.372 0.019 19.892 0.000
Y5 0.341 0.019 18.319 0.000
Y6 0.132 0.014 9.271 0.000
Latent Two-Tailed
Variable Estimate S.E. Est./S.E. P-Value
FW 0.024 0.006 3.724 0.000
MODEL COMMAND WITH FINAL ESTIMATES USED AS STARTING VALUES
fw BY y1*0.51150 (l1);
fw BY y2*0.50933 (l2);
fw BY y3*0.62638 (l3);
fw BY y4*0.61112 (l4);
fw BY y5*0.52740 (l5);
fw BY y6*0.45134 (l6);
fw ON female*0.27248;
fw ON agec70*0.11132;
[ y1*1.35775 ];
[ y2*1.24453 ];
[ y3*1.39801 ];
[ y4*1.53031 ];
[ y5*1.36285 ];
[ y6*1.90864 ];
y1*0.28463;
y2*0.29333;
y3*0.29381;
y4*0.62958;
y5*0.53844;
y6*1.33814;
fw@0.97600;
TECHNICAL 4 OUTPUT
ESTIMATES DERIVED FROM THE MODEL
ESTIMATED MEANS FOR THE LATENT VARIABLES
FW FEMALE AGEC70
________ ________ ________
0.200 0.585 0.363
S.E. FOR ESTIMATED MEANS FOR THE LATENT VARIABLES
FW FEMALE AGEC70
________ ________ ________
0.028 0.009 0.012
EST./S.E. FOR ESTIMATED MEANS FOR THE LATENT VARIABLES
FW FEMALE AGEC70
________ ________ ________
7.242 62.018 29.327
TWO-TAILED P-VALUE FOR ESTIMATED MEANS FOR THE LATENT VARIABLES
FW FEMALE AGEC70
________ ________ ________
0.000 0.000 0.000
ESTIMATED COVARIANCE MATRIX FOR THE LATENT VARIABLES
FW FEMALE AGEC70
________ ________ ________
FW 1.000
FEMALE 0.068 0.243
AGEC70 0.050 0.013 0.418
S.E. FOR ESTIMATED COVARIANCE MATRIX FOR THE LATENT VARIABLES
FW FEMALE AGEC70
________ ________ ________
FW 0.007
FEMALE 0.011 0.007
AGEC70 0.014 0.006 0.011
EST./S.E. FOR ESTIMATED COVARIANCE MATRIX FOR THE LATENT VARIABLES
FW FEMALE AGEC70
________ ________ ________
FW 151.557
FEMALE 6.370 36.932
AGEC70 3.625 2.099 36.932
TWO-TAILED P-VALUE FOR ESTIMATED COVARIANCE MATRIX FOR THE LATENT VARIABLES
FW FEMALE AGEC70
________ ________ ________
FW 0.000
FEMALE 0.000 0.000
AGEC70 0.000 0.036 0.000
ESTIMATED CORRELATION MATRIX FOR THE LATENT VARIABLES
FW FEMALE AGEC70
________ ________ ________
FW 1.000
FEMALE 0.137 1.000
AGEC70 0.077 0.040 1.000
S.E. FOR ESTIMATED CORRELATION MATRIX FOR THE LATENT VARIABLES
FW FEMALE AGEC70
________ ________ ________
FW 0.000
FEMALE 0.021 0.000
AGEC70 0.021 0.019 0.000
EST./S.E. FOR ESTIMATED CORRELATION MATRIX FOR THE LATENT VARIABLES
FW FEMALE AGEC70
________ ________ ________
FW 999.000
FEMALE 6.566 999.000
AGEC70 3.661 2.104 999.000
TWO-TAILED P-VALUE FOR ESTIMATED CORRELATION MATRIX FOR THE LATENT VARIABLES
FW FEMALE AGEC70
________ ________ ________
FW 0.000
FEMALE 0.000 0.000
AGEC70 0.000 0.035 0.000
Beginning Time: 12:32:29
Ending Time: 12:32:29
Elapsed Time: 00:00:00
MUTHEN & MUTHEN
3463 Stoner Ave.
Los Angeles, CA 90066
Tel: (310) 391-9971
Fax: (310) 391-8971
Web: www.StatModel.com
Support: Support@StatModel.com
Copyright (c) 1998-2024 Muthen & MuthenCheck TECH4 to make sure variance of FW is 1
Loading and Intercept for item 1
Here I use a custom made function readSvalues chatGPT made for me. It’s in the guts of this QMD document. It’s better to use SVALUES because estimates are reported to six decimal places precision.
Third step: constrain factor loading in y1, free residual variance of fw, and check TECH4
Mplus VERSION 8.11 (Mac)
MUTHEN & MUTHEN
04/10/2025 12:32 PM
INPUT INSTRUCTIONS
TITLE: MLFA Preliminary Model 3
DATA: FILE = cesd.dat ;
VARIABLE: NAMES = id t female agec70 y1 y2 y3 y4 y5 y6 ;
USEVARIABLES = female agec70 y1-y6 ;
USEOBSERVATIONS = (t==0) ;
MISSING = . ;
OUTPUT: STDYX ; TECH4 ;
MODEL: fw BY y1@0.51150 (l1);
fw by y2-y6* (l2-l6);
fw*;
fw on female agec70 ;
[y1@1.35775];
[fw*];
INPUT READING TERMINATED NORMALLY
MLFA Preliminary Model 3
SUMMARY OF ANALYSIS
Number of groups 1
Number of observations 2728
Number of dependent variables 6
Number of independent variables 2
Number of continuous latent variables 1
Observed dependent variables
Continuous
Y1 Y2 Y3 Y4 Y5 Y6
Observed independent variables
FEMALE AGEC70
Continuous latent variables
FW
Estimator ML
Information matrix OBSERVED
Maximum number of iterations 1000
Convergence criterion 0.500D-04
Maximum number of steepest descent iterations 20
Maximum number of iterations for H1 2000
Convergence criterion for H1 0.100D-03
Input data file(s)
cesd.dat
Input data format FREE
SUMMARY OF DATA
Number of missing data patterns 22
COVARIANCE COVERAGE OF DATA
Minimum covariance coverage value 0.100
PROPORTION OF DATA PRESENT
Covariance Coverage
Y1 Y2 Y3 Y4 Y5
________ ________ ________ ________ ________
Y1 0.993
Y2 0.987 0.993
Y3 0.991 0.989 0.996
Y4 0.987 0.982 0.986 0.989
Y5 0.986 0.981 0.986 0.981 0.988
Y6 0.927 0.925 0.929 0.924 0.923
FEMALE 0.993 0.993 0.996 0.989 0.988
AGEC70 0.993 0.993 0.996 0.989 0.988
Covariance Coverage
Y6 FEMALE AGEC70
________ ________ ________
Y6 0.931
FEMALE 0.931 1.000
AGEC70 0.931 1.000 1.000
UNIVARIATE SAMPLE STATISTICS
UNIVARIATE HIGHER-ORDER MOMENT DESCRIPTIVE STATISTICS
Variable/ Mean/ Skewness/ Minimum/ % with Percentiles
Sample Size Variance Kurtosis Maximum Min/Max 20%/60% 40%/80% Median
Y1 1.458 1.812 1.000 65.26% 1.000 1.000 1.000
2709.000 0.545 3.142 4.000 3.77% 1.000 2.000
Y2 1.345 2.371 1.000 77.44% 1.000 1.000 1.000
2708.000 0.552 5.035 4.000 4.36% 1.000 2.000
Y3 1.522 1.718 1.000 63.55% 1.000 1.000 1.000
2716.000 0.686 2.339 4.000 6.04% 1.000 2.000
Y4 1.650 1.189 1.000 66.22% 1.000 1.000 1.000
2697.000 1.001 -0.071 4.000 7.45% 1.000 3.000
Y5 1.465 1.735 1.000 75.91% 1.000 1.000 1.000
2694.000 0.815 1.599 4.000 5.86% 1.000 2.000
Y6 1.998 0.643 1.000 56.87% 1.000 1.000 1.000
2539.000 1.543 -1.319 4.000 20.40% 2.000 4.000
FEMALE 0.585 -0.345 0.000 41.50% 0.000 0.000 1.000
2728.000 0.243 -1.881 1.000 58.50% 1.000 1.000
AGEC70 0.363 0.675 -0.300 34.60% -0.300 0.200 0.200
2728.000 0.418 -0.674 1.700 8.72% 0.200 1.200
THE MODEL ESTIMATION TERMINATED NORMALLY
MODEL FIT INFORMATION
Number of Free Parameters 20
Loglikelihood
H0 Value -18938.943
H1 Value -18682.363
Information Criteria
Akaike (AIC) 37917.886
Bayesian (BIC) 38036.113
Sample-Size Adjusted BIC 37972.566
(n* = (n + 2) / 24)
Chi-Square Test of Model Fit
Value 513.160
Degrees of Freedom 19
P-Value 0.0000
RMSEA (Root Mean Square Error Of Approximation)
Estimate 0.098
90 Percent C.I. 0.090 0.105
Probability RMSEA <= .05 0.000
CFI/TLI
CFI 0.889
TLI 0.843
Chi-Square Test of Model Fit for the Baseline Model
Value 4495.941
Degrees of Freedom 27
P-Value 0.0000
SRMR (Standardized Root Mean Square Residual)
Value 0.049
MODEL RESULTS
Two-Tailed
Estimate S.E. Est./S.E. P-Value
FW BY
Y1 0.512 0.000 999.000 999.000
Y2 0.509 0.017 29.924 0.000
Y3 0.626 0.019 32.169 0.000
Y4 0.611 0.024 25.839 0.000
Y5 0.527 0.021 24.824 0.000
Y6 0.451 0.028 15.973 0.000
FW ON
FEMALE 0.273 0.043 6.314 0.000
AGEC70 0.111 0.033 3.410 0.001
Intercepts
Y1 1.358 0.000 999.000 999.000
Y2 1.245 0.015 83.167 0.000
Y3 1.398 0.017 83.430 0.000
Y4 1.530 0.020 76.044 0.000
Y5 1.363 0.018 75.157 0.000
Y6 1.909 0.025 75.391 0.000
FW 0.000 0.039 0.000 1.000
Residual Variances
Y1 0.285 0.010 28.819 0.000
Y2 0.293 0.010 28.720 0.000
Y3 0.294 0.012 24.630 0.000
Y4 0.630 0.021 30.503 0.000
Y5 0.538 0.017 31.176 0.000
Y6 1.338 0.039 34.266 0.000
FW 0.976 0.053 18.513 0.000
QUALITY OF NUMERICAL RESULTS
Condition Number for the Information Matrix 0.205E-01
(ratio of smallest to largest eigenvalue)
STANDARDIZED MODEL RESULTS
STDYX Standardization
Two-Tailed
Estimate S.E. Est./S.E. P-Value
FW BY
Y1 0.692 0.013 53.586 0.000
Y2 0.685 0.013 51.689 0.000
Y3 0.756 0.012 63.785 0.000
Y4 0.610 0.015 39.782 0.000
Y5 0.584 0.016 36.637 0.000
Y6 0.363 0.020 18.541 0.000
FW ON
FEMALE 0.134 0.021 6.434 0.000
AGEC70 0.072 0.021 3.426 0.001
Intercepts
Y1 1.837 0.025 73.524 0.000
Y2 1.674 0.032 52.613 0.000
Y3 1.688 0.032 53.023 0.000
Y4 1.528 0.030 50.287 0.000
Y5 1.508 0.030 49.994 0.000
Y6 1.537 0.031 49.746 0.000
FW 0.000 0.039 0.000 1.000
Residual Variances
Y1 0.521 0.018 29.148 0.000
Y2 0.531 0.018 29.221 0.000
Y3 0.428 0.018 23.884 0.000
Y4 0.628 0.019 33.535 0.000
Y5 0.659 0.019 35.463 0.000
Y6 0.868 0.014 60.901 0.000
FW 0.976 0.006 151.549 0.000
R-SQUARE
Observed Two-Tailed
Variable Estimate S.E. Est./S.E. P-Value
Y1 0.479 0.018 26.793 0.000
Y2 0.469 0.018 25.845 0.000
Y3 0.572 0.018 31.893 0.000
Y4 0.372 0.019 19.891 0.000
Y5 0.341 0.019 18.319 0.000
Y6 0.132 0.014 9.271 0.000
Latent Two-Tailed
Variable Estimate S.E. Est./S.E. P-Value
FW 0.024 0.006 3.724 0.000
TECHNICAL 4 OUTPUT
ESTIMATES DERIVED FROM THE MODEL
ESTIMATED MEANS FOR THE LATENT VARIABLES
FW FEMALE AGEC70
________ ________ ________
0.200 0.585 0.363
S.E. FOR ESTIMATED MEANS FOR THE LATENT VARIABLES
FW FEMALE AGEC70
________ ________ ________
0.028 0.009 0.012
EST./S.E. FOR ESTIMATED MEANS FOR THE LATENT VARIABLES
FW FEMALE AGEC70
________ ________ ________
7.206 62.018 29.327
TWO-TAILED P-VALUE FOR ESTIMATED MEANS FOR THE LATENT VARIABLES
FW FEMALE AGEC70
________ ________ ________
0.000 0.000 0.000
ESTIMATED COVARIANCE MATRIX FOR THE LATENT VARIABLES
FW FEMALE AGEC70
________ ________ ________
FW 1.000
FEMALE 0.068 0.243
AGEC70 0.050 0.013 0.418
S.E. FOR ESTIMATED COVARIANCE MATRIX FOR THE LATENT VARIABLES
FW FEMALE AGEC70
________ ________ ________
FW 0.054
FEMALE 0.011 0.007
AGEC70 0.014 0.006 0.011
EST./S.E. FOR ESTIMATED COVARIANCE MATRIX FOR THE LATENT VARIABLES
FW FEMALE AGEC70
________ ________ ________
FW 18.542
FEMALE 6.344 36.932
AGEC70 3.623 2.099 36.932
TWO-TAILED P-VALUE FOR ESTIMATED COVARIANCE MATRIX FOR THE LATENT VARIABLES
FW FEMALE AGEC70
________ ________ ________
FW 0.000
FEMALE 0.000 0.000
AGEC70 0.000 0.036 0.000
ESTIMATED CORRELATION MATRIX FOR THE LATENT VARIABLES
FW FEMALE AGEC70
________ ________ ________
FW 1.000
FEMALE 0.137 1.000
AGEC70 0.077 0.040 1.000
S.E. FOR ESTIMATED CORRELATION MATRIX FOR THE LATENT VARIABLES
FW FEMALE AGEC70
________ ________ ________
FW 0.000
FEMALE 0.021 0.000
AGEC70 0.021 0.019 0.000
EST./S.E. FOR ESTIMATED CORRELATION MATRIX FOR THE LATENT VARIABLES
FW FEMALE AGEC70
________ ________ ________
FW 999.000
FEMALE 6.567 999.000
AGEC70 3.660 2.104 999.000
TWO-TAILED P-VALUE FOR ESTIMATED CORRELATION MATRIX FOR THE LATENT VARIABLES
FW FEMALE AGEC70
________ ________ ________
FW 0.000
FEMALE 0.000 0.000
AGEC70 0.000 0.035 0.000
Beginning Time: 12:32:29
Ending Time: 12:32:29
Elapsed Time: 00:00:00
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Copyright (c) 1998-2024 Muthen & MuthenCheck TECH4 to make sure variance of FW is 1
MLFA
TITLE: MLFA CESD EPESE
DATA: FILE = cesd.dat ;
VARIABLE: NAMES = id t female agec70 y1 y2 y3 y4 y5 y6 ;
MISSING = . ;
WITHIN = t ;
BETWEEN = agec70 female ;
CLUSTER = id ;
ANALYSIS: TYPE = TWOLEVEL RANDOM ;
MODEL: %WITHIN%
fw BY y1@0.51150 (l1);
fw by y2-y6* (l2-l6);
s | fw on t ;
%BETWEEN%
fb BY y1@0.51150 (l1);
fb by y2-y6* (l2-l6);
y1-y6@0;
fb on agec70 female ;
s on agec70 female ;
fb with s ;
[y1@1.35775];
[fb*];Mplus VERSION 8.11 (Mac)
MUTHEN & MUTHEN
04/10/2025 12:32 PM
INPUT INSTRUCTIONS
TITLE: MLFA CESD EPESE
DATA: FILE = cesd.dat ;
VARIABLE: NAMES = id t female agec70 y1 y2 y3 y4 y5 y6 ;
MISSING = . ;
WITHIN = t ;
BETWEEN = agec70 female ;
CLUSTER = id ;
ANALYSIS: TYPE = TWOLEVEL RANDOM ;
MODEL: %WITHIN%
fw BY y1@0.51150 (l1);
fw by y2-y6* (l2-l6);
s | fw on t ;
%BETWEEN%
fb BY y1@0.51150 (l1);
fb by y2-y6* (l2-l6);
y1-y6@0;
fb on agec70 female ;
s on agec70 female ;
fb with s ;
[y1@1.35775];
[fb*];
*** WARNING
One or more individual-level variables have no variation within a
cluster for the following clusters.
Variable Cluster IDs with no within-cluster variation
Y1 154 319 484 815 1046 1528 1791 2164 2451 557 762 765 1458 767 1460 1466 769
770 1476 50 1482 1483 776 404 1496 1497 1498 1503 1507 210 786 787 1526 790
1534 792 1538 413 420 1546 1551 797 800 1558 1563 425 1575 96 1578 814 1581
429 1585 1586 1590 1596 1598 816 1607 1611 1620 1628 831 1646 213 1656 840 1665
1666 846 1671 848 851 853 854 1699 860 1708 444 1711 1713 1716 217 1722 98 869
221 872 875 454 1747 882 457 224 889 1766 103 899 1770 1774 1777 1778 107 1783
471 1789 1797 35 1800 476 1807 1808 920 1818 248 930 1839 1843 114 1846 1863
252 254 941 255 1872 943 1876 1880 257 1883 259 950 952 1894 1899 1901 1917
966 1923 501 969 502 1938 1943 975 1945 1962 1968 1973 986 1975 1976 1977 1980
1991 1992 1994 1997 2001 2007 2010 1004 1005 1007 2024 514 2028 1016 2055 1030
2064 524 2070 2074 1041 125 2080 2081 287 2087 2088 2092 2093 2095 58 3 2100
539 2109 2111 2121 2123 134 1072 1073 2137 543 304 2140 546 549 61 2156 2159
2162 64 556 2167 1092 2175 2184 1096 564 2187 2196 2203 1104 1105 2207 147 149
2225 2229 2232 2236 2242 2245 2247 2250 1122 2259 2260 2265 1123 2267 2281 578
2284 2285 2286 314 2304 1140 25 2311 2312 2318 2330 68 1150 1154 2338 1163 329
599 601 2367 2371 2374 1179 2378 2379 603 2382 2385 2386 2390 2391 2393 157
2398 2400 2402 612 2411 331 2425 2433 2449 2456 8 159 622 10 2470 632 1241 2488
1247 635 1250 1251 2515 2519 2522 1262 2524 1266 2531 2533 2536 2537 2542 1270
1272 1273 340 665 1283 2572 2573 2579 1292 1293 2588 1300 1301 2598 668 164
2610 671 2617 1315 2620 1316 2623 348 2625 1322 44 2632 1328 682 2646 686 2648
2650 1340 2665 2668 2673 2676 2679 1346 1347 2684 173 2686 1351 2702 2707 1355
1356 699 362 2713 363 2717 2719 2731 1365 1368 2750 1371 180 2757 2766 1389
2773 194 2782 2783 375 1399 1402 1404 385 195 740 100 196 87 396 1419 1423 1162
1224 1414 92 1948 758 2224 2497 2634 2799 346 347 1472 349 1477 732 736 1484
1487 1491 739 354 743 356 746 747 748 1508 1509 750 1512 357 753 1520 1522 757
1530 1531 1533 763 764 364 1544 1548 1549 1556 774 1561 778 1569 779 1572 781
174 784 1584 374 176 789 376 1591 378 379 1599 381 382 801 1616 1617 384 1627
1632 808 1635 1636 1638 1639 1641 178 811 1644 812 1648 1651 1652 179 1659 1663
20 819 820 821 182 823 825 393 1684 1686 1690 1691 1692 395 1694 1696 1697 1701
183 1706 1707 399 1714 841 185 843 1721 845 1724 88 1727 1729 1732 1733 1734
89 1736 1737 405 408 1744 1745 856 857 190 411 1755 1760 864 415 418 1772 873
874 1779 1781 193 1788 21 1794 93 432 1803 891 198 895 1813 200 1827 903 904
1833 906 46 443 910 47 1842 445 446 915 1848 916 1852 1854 917 1857 447 924
451 452 928 1869 208 455 209 23 459 939 1888 212 101 946 1896 467 102 1903 1906
1909 953 1915 954 1920 1925 474 1929 961 1937 962 1940 1942 965 222 1947 479
108 1954 970 11 972 1964 1965 974 1972 110 980 983 1982 984 987 111 991 1990
229 230 994 231 1996 997 493 1000 2005 2006 2009 495 112 498 2023 1011 2025
2026 2029 503 238 2034 2035 1018 2045 53 240 1024 2051 2054 1025 1026 2061 507
1029 508 2067 2073 1038 2082 2085 2086 1048 516 517 1051 2094 1053 12 1057 1058
1062 2108 1065 1067 2120 2122 1069 28 2125 2128 526 117 1075 253 1078 2141 2142
2147 2150 256 2158 2161 2171 1094 1097 2179 2183 2189 2190 544 122 2197 2198
2199 2201 1110 1113 60 2214 2216 2219 1116 2226 62 1120 1124 2238 268 2244 1130
2252 1132 2254 2255 561 562 270 2270 566 1143 2275 2276 568 1147 1149 2287 2288
2291 571 2295 1156 2301 2302 1157 2305 2307 2315 2316 276 1168 2321 1169 2323
1170 2325 2327 1171 1173 1174 278 1177 279 2340 280 131 1182 2352 2353 2355
2357 1186 2362 584 2366 1189 283 2370 2372 285 588 1196 1197 1204 591 2392 132
594 595 596 2403 1216 598 288 1219 1220 2414 1221 2417 1222 2422 1223 290 2432
2434 1232 2439 2442 2444 1239 2453 608 2463 609 2466 610 2468 2469 294 295 1252
615 2480 2483 2484 298 1258 619 299 2498 1265 624 2511 2516 2520 1278 2525 2526
1281 630 1286 2539 2540 1288 135 1290 2547 633 136 2550 305 2554 2558 1297 2563
2565 1299 137 2571 2577 639 2586 308 2595 2600 1319 2603 2605 647 2612 1325
1326 2629 2631 1342 2640 2642 142 2649 663 664 2661 2662 34 4 2671 673 2678
323 2680 1363 675 324 2689 2694 2695 2697 1372 2701 2703 1375 148 1382 1383
2721 2724 2725 2726 2727 2729 2730 2734 69 2739 71 152 1396 2751 1398 1401 2762
696 2767 2771 2775 2776 2778 700 2784 2786 703 2792 2793 704 2796 705 2798 2800
1420 1421 2806 1424 156 73 1432 711 75 76 715 716 717 1443 1445 718 1449 1457
345 1463
Y2 687 793 1071 1618 1669 2096 2104 2369 557 2205 762 765 766 1458 767 1460 1462
1466 1467 769 770 1475 1476 50 1480 1481 1482 1483 776 404 1497 782 1507 409
786 1523 787 790 792 1538 413 420 1551 797 800 1563 425 1575 96 1581 429 1585
1586 1590 1596 1607 817 826 1615 434 1620 1628 831 1640 835 840 1665 1666 214
1673 848 1682 851 215 853 1689 1699 858 860 1708 444 1711 1716 1717 1722 869
872 875 1747 884 457 30 224 889 463 892 1766 103 899 1773 1774 1777 1783 1789
472 908 1797 35 1800 912 476 1808 1816 1818 113 478 248 929 930 1835 931 1839
481 1843 114 1846 485 1862 1863 936 941 255 1872 943 1874 1876 1877 257 1883
259 1889 950 1899 1901 499 1917 1923 501 968 969 1938 1939 1943 116 37 1962
1969 1973 986 1975 1976 1981 1984 990 1989 1991 1992 1997 1998 1999 2001 2007
2010 999 1004 1005 1007 512 513 2024 514 518 1015 1016 1017 2036 1023 2049 1030
2064 2065 524 2070 123 2074 125 2081 287 2084 2088 535 2091 2092 2093 1050 2097
58 3 2100 1068 2113 1070 2121 2126 1072 2131 2133 1073 2137 543 2140 549 61
2159 2162 558 2167 2175 2184 1096 564 2187 565 1103 2195 2203 1105 2207 2210
149 2225 572 2229 313 2232 2236 2242 2245 2247 2250 2251 1122 2259 2265 2267
1131 2281 578 2285 2286 314 580 2300 1137 2304 25 2318 2330 68 1150 2334 2338
1163 329 1175 601 2364 2367 602 2371 2374 1179 1180 603 2382 2383 2385 2386
604 2390 2391 2393 157 611 1192 2398 2399 2402 2409 2410 2411 1199 331 2424
614 2433 2441 1211 2449 1215 2452 2456 8 159 10 2470 632 1234 634 1241 1242
1246 2488 2489 635 337 2509 650 2517 2519 2522 2524 1266 2531 2533 2534 2536
2548 1270 1272 1273 340 2561 665 1283 2572 2573 666 2581 1300 1301 2598 668
2608 2610 2617 2620 1316 2623 2625 1322 1323 44 2632 1328 167 45 682 686 2648
2650 1338 2652 1340 2657 2663 2665 2668 2673 2679 1346 1347 2684 173 2686 2690
83 2698 1351 2702 2707 1356 2713 363 2717 2719 709 1365 2741 1368 1370 2750
1371 180 2760 2766 1389 2773 194 2782 2783 2787 375 727 1399 1402 2801 1404
733 195 740 100 389 550 196 87 1419 1423 1162 1224 1304 1354 1902 1948 1428
758 2224 2497 2634 2670 1438 1465 346 347 728 1472 1473 349 1477 732 162 1484
1488 1490 16 739 354 39 743 356 1502 746 747 1508 1512 1513 357 1516 753 754
1520 1521 1522 755 359 1530 1533 1535 761 169 170 364 1544 1548 1549 1550 771
772 1556 1557 775 1561 171 1564 5 1569 779 781 174 784 1584 374 788 789 376
377 1593 1595 378 1597 379 1603 1605 1609 798 799 382 801 383 1616 1617 384
1621 1622 1624 1625 1627 1632 1633 808 1635 1636 1637 1638 1639 809 1641 178
811 1644 1645 812 1647 1648 1650 1651 1652 179 1657 388 1659 1662 1663 20 181
818 819 820 1670 821 1672 182 1677 823 824 393 394 1684 828 830 1691 1692 395
1694 1696 1697 1698 1701 834 183 1706 1707 837 399 1714 841 185 1718 843 1721
1724 88 1727 402 403 1732 1733 1734 1736 1737 405 407 408 1743 1744 1745 855
857 411 1753 1754 1755 1758 1760 864 866 414 1765 415 871 418 1772 873 874 877
1779 1781 192 1788 21 428 93 430 432 1802 1803 891 433 1806 198 895 1810 1811
896 1813 1814 22 1820 1822 437 439 903 904 201 1833 906 1836 442 910 446 1847
916 1851 1853 1854 917 1856 447 205 924 452 928 1869 455 209 1875 23 459 1879
1881 461 1887 1888 212 945 946 1896 466 467 102 1903 1904 24 1906 1909 1913
1915 104 218 473 474 106 1933 1937 962 1940 1942 965 2 1947 479 1950 108 1952
1953 1954 970 1957 1958 11 1960 972 1964 974 482 1971 977 225 980 1978 1979
1982 987 989 111 991 1990 229 231 1996 997 493 1000 2004 2005 2006 495 232 112
498 2022 2023 1011 1013 1014 237 238 2034 2035 1018 1019 2039 2045 53 240 1024
2051 2052 2053 2054 1026 2057 2061 2062 507 1029 508 2067 2071 242 2073 1036
1037 1038 511 244 2082 245 1045 2085 2086 54 515 1048 517 1051 2094 1053 1057
1058 2102 1061 1062 2108 2110 1064 2119 2120 2122 1069 28 2125 2127 2128 2129
526 117 1075 253 1078 2142 2147 533 256 2161 120 1090 538 2168 2170 2171 2173
1094 2179 2182 2183 1099 1100 2189 2190 2191 2192 261 122 2197 2199 2201 2202
547 548 1110 1112 2216 2217 2218 2219 2222 2223 551 2226 62 1120 2233 2234 267
1124 1125 2238 2240 2241 2243 2244 1127 2248 1129 1130 2252 1132 2254 2255 562
563 1138 270 2269 1143 128 2276 2278 2280 1147 1149 2287 2290 2291 2296 2298
1155 1156 2301 2302 1157 2307 2308 1160 1161 2314 2315 2316 276 2319 1168 2321
1169 2323 1170 2325 2327 2328 1173 130 278 1177 279 2339 2340 280 2343 2345
131 1182 1183 2352 2353 2355 2356 2357 1186 584 1188 2368 2370 2372 285 588
1195 1196 1198 1200 2384 286 2387 2388 1203 1204 591 2392 592 132 594 595 1213
1214 2403 2405 2408 598 1220 2413 2414 2415 1221 2417 1222 2420 2422 2427 290
2432 1231 2434 2435 1232 2438 2439 291 2442 2443 2444 2447 2448 606 1239 2453
607 608 2459 1245 2463 609 610 2469 294 1252 615 2480 616 2483 2484 2485 298
1258 2493 619 2495 299 2498 2500 1263 1265 301 623 626 2511 628 2516 1276 2520
1278 302 2525 2526 1281 2530 630 631 1286 2539 2541 1288 135 2547 136 2550 1294
2553 2554 2555 2558 2562 2564 2565 2566 1299 137 638 2574 2577 1303 639 138
1309 642 2586 2587 1311 645 2595 2603 2605 2606 647 648 2609 139 2612 1324 1325
1326 310 1329 311 1335 2629 2630 1337 659 1342 2640 1343 2645 2649 663 664 66
2656 2661 2662 34 146 4 2669 2671 1360 2677 2678 2680 1363 2682 675 324 1367
2687 2689 2694 2695 679 2697 2701 1373 2704 1375 1376 2710 148 683 1380 2714
2716 1382 2718 1383 2720 2721 2722 1384 2725 2726 2727 2729 2730 1386 1387 2734
69 2739 70 152 1393 2748 1396 333 2752 691 1400 1401 2758 2762 1405 695 696
2770 2771 153 2775 2776 2778 700 1413 2784 2785 2786 702 703 2789 2792 2793
704 1418 2796 705 2798 1420 1421 1422 2806 1424 156 73 1432 711 76 342 715 716
717 1441 1445 718 1449 1451 1453 1457 1463
Y3 154 2369 762 765 1458 767 1460 1467 769 770 1475 1476 50 1480 1482 1483 776
1493 1496 1497 786 1526 1534 792 1538 1541 1551 800 1558 1559 421 1563 425 1568
1575 96 1581 429 1585 1590 1596 826 1615 434 1620 1628 831 1640 835 836 840
1665 1666 436 848 860 444 1711 1712 1713 1716 1717 98 872 1738 1739 456 1751
884 457 889 1766 899 1773 1774 1777 1778 107 1783 1789 908 1797 35 1800 912
476 1807 1808 920 1816 1818 478 248 930 1835 1839 1841 1843 114 1846 485 1863
254 1867 941 255 1872 1874 1876 947 1880 257 1883 259 1889 950 952 1899 1901
499 1917 1923 501 968 969 502 1943 975 1945 1951 37 1962 1968 1969 1973 1975
1977 1981 990 1989 1991 1992 1994 1997 1999 2001 2007 2010 999 1004 1007 2024
518 1015 1016 1017 2036 519 2042 2047 1023 2064 524 2069 2070 2074 2077 2081
287 2084 2091 2092 2093 2097 58 2100 539 1068 2109 2113 2123 1072 2131 1073
543 304 2140 549 1083 61 2162 2165 558 2167 2175 2184 1096 564 2195 1105 2208
2210 2212 149 2225 572 1117 2229 2232 2236 2242 2247 2249 2265 1123 2267 2271
1126 2273 2281 578 2285 2286 314 2300 2304 25 2311 1144 2318 2320 2330 68 1150
320 322 1163 329 599 2354 601 2367 602 2371 2374 1179 2378 603 2382 2383 2385
2386 604 2391 2393 611 2398 2399 2402 2409 2410 2411 2424 2425 2429 2441 1211
2449 1215 2452 2456 8 159 622 2470 1234 1246 2488 1247 635 1250 2507 650 2517
2522 1262 1266 2531 163 2542 2548 1270 2561 665 1283 2572 2579 666 2581 1301
2592 1302 2598 667 668 2610 671 2617 2620 2623 348 2625 1322 44 2632 1328 676
167 682 2646 2648 2650 1340 2657 49 2668 2673 2676 2679 1346 1347 2684 173 2686
2690 83 2698 2702 2707 1356 699 363 2717 2719 1364 1365 1368 2750 180 2760 187
2766 1389 194 724 2782 2783 727 1402 380 1404 195 740 100 389 87 1419 1423 1162
1224 1304 1354 1414 92 1948 758 2224 2634 2670 1435 1439 346 347 1472 1473 36
349 1477 732 1484 1490 16 739 1494 38 354 743 747 1509 1512 1513 357 1516 753
1520 1522 359 1531 1532 1533 364 1544 1547 1548 1549 1556 1561 171 1564 1566
1569 779 85 781 174 784 1584 176 376 1594 1595 378 379 1599 795 1603 381 799
382 801 1616 384 1621 1622 1624 1627 1631 1632 1633 808 1636 1639 1641 178 811
1644 1645 812 1651 179 1657 388 1663 181 819 820 821 182 1677 823 824 393 1684
828 1688 830 1691 1692 395 1694 1695 1696 1697 833 1701 183 1706 1707 399 185
1718 843 844 1724 88 1726 1727 1732 1733 1734 89 1737 405 407 1743 1744 1745
1746 856 857 189 411 1755 1758 191 1760 864 1765 416 871 418 1772 1779 1780
1781 192 1788 21 432 1801 1802 1803 891 433 895 1813 1821 1822 1825 200 439
1833 1836 442 47 445 1848 916 1852 1853 1854 917 1857 205 924 452 1869 209 934
23 1879 1881 939 1888 946 1896 467 1900 1903 24 1906 1907 1908 218 474 1929
1937 962 1940 2 108 1952 1953 970 1957 1958 11 972 1967 483 1971 977 225 980
1978 1979 486 984 111 1990 229 994 231 1996 996 997 493 494 1000 1001 2005 2006
2008 495 498 1011 2025 2026 1013 1014 237 238 2034 2035 1018 2045 53 240 2050
2051 2053 2054 1026 2061 507 1029 508 2067 2071 2073 1036 1037 1038 244 2082
245 1045 2085 2086 54 517 1051 1052 2094 1053 1057 1058 1061 1062 1064 2118
2120 250 2122 1069 28 2125 2127 2129 526 1075 253 1078 2142 531 2147 2150 14
256 2161 258 1090 538 2173 1094 1097 2180 2183 1098 1099 2189 2192 122 2197
2198 2201 1110 2216 2218 2219 2223 62 1120 2230 554 2234 1124 2238 2240 2244
1127 1129 1130 2252 1132 2255 2261 2269 566 1143 2277 568 2280 1146 1147 1149
273 2287 2288 2290 2291 2295 2296 129 2302 1157 2307 1160 1161 575 1165 2315
2316 276 1168 2321 1169 2323 1170 2325 2327 2328 1174 130 278 1177 279 280 2345
131 1182 2352 2353 282 2355 2357 2358 1186 2361 584 2368 2372 588 1196 589 1200
2384 1201 286 2387 1203 591 2392 132 594 595 1214 2403 2407 598 288 1220 2413
2414 2415 2417 2422 600 290 2432 2434 1232 2438 2439 291 292 2442 2443 2444
2448 2453 2455 607 2459 2463 610 2468 2469 1252 1254 615 2480 2483 2484 2485
2486 298 1258 618 2493 619 2495 299 2498 1265 2511 1274 2516 2520 1278 2525
2526 630 631 1284 2539 2547 1294 2553 2554 2555 2558 2565 1299 137 2574 2576
2577 2578 639 138 1309 2586 2587 2603 2605 2606 648 2612 2613 1325 1326 310
1337 1342 2640 1343 660 2649 663 2656 2660 2661 2662 34 4 2669 2671 2674 2680
675 1367 2688 2689 2691 2694 2695 679 2697 2701 1373 2704 1375 1376 681 2710
148 2714 2716 2718 1383 2720 2721 2724 2725 2726 2727 2730 1386 1387 2734 152
1393 2746 2748 1396 1398 1400 1401 2758 2762 696 2771 2775 2776 700 2781 1413
2784 2785 2786 703 2792 2793 704 1418 2796 705 2798 1422 706 2806 156 73 1432
711 75 1437 715 716 1445 718 1447 1449 1453 1457 723 345 1463
Y4 154 319 484 605 815 1012 1046 1528 1786 1849 2549 557 762 765 766 1456 1458
767 1460 769 770 50 1480 1483 776 404 1497 1501 1505 1510 1523 787 790 1536
792 1538 413 1541 1546 1551 1558 1563 425 1575 96 1578 814 1585 1590 1592 1600
816 1607 817 826 1615 434 1628 831 1640 836 213 1656 840 1665 1666 436 846 848
1681 851 860 1708 1711 1712 1713 1716 1717 98 875 1738 880 456 1747 1748 883
884 457 224 889 892 893 899 1773 1777 1778 107 1783 1789 905 1797 1800 476 929
930 1835 481 1841 1843 114 1845 1846 485 1860 1862 936 252 255 1872 943 1874
1880 1883 1889 956 1899 1901 499 1917 1923 501 1939 1943 1951 1962 1969 1973
986 1976 1983 992 993 1989 1991 1992 57 1997 1998 1999 2010 1004 1007 512 513
2024 2028 518 1015 1016 1017 1023 272 2065 524 2070 2074 2077 287 2084 2087
2093 2095 2097 58 3 539 1068 2109 2111 2113 134 1072 2131 543 304 2140 546 1083
61 2162 2165 2167 1092 2184 564 2203 1105 2208 2210 2225 1117 2229 2242 2247
2249 1123 2284 314 580 2300 1137 25 582 2312 2313 1144 2318 2320 151 2330 68
1150 1163 328 329 601 2364 2371 2374 1179 1180 2378 2382 2383 2386 2390 2391
2393 2395 1192 2398 2402 612 2409 2410 2411 331 2425 2429 2433 1209 2456 8 159
622 2470 2471 1250 2503 2507 2509 650 653 2517 2519 2522 2524 163 2533 2534
2536 2542 1270 1272 340 2561 1283 2572 2573 1292 1293 1301 2598 667 668 2608
2610 671 2617 1315 2620 1316 2623 2625 1322 1323 44 2632 1328 682 2648 2650
2652 1340 2665 2668 2676 2679 1347 2684 173 2686 698 2707 1355 1356 362 2719
709 2735 1365 2747 180 186 2760 84 194 2782 2783 727 1399 1404 195 740 100 389
196 396 1419 1423 1162 1224 1304 1948 1428 758 2634 2799 1435 1438 1465 346
347 728 1472 36 1477 732 1484 1490 1494 354 356 1502 746 747 1512 1513 753 1520
1522 755 1531 1533 17 1544 1549 1555 1561 171 1564 1569 1572 1573 1577 784 1582
374 176 788 789 376 378 1599 1601 795 1603 1606 799 384 1627 1629 1633 1635
1636 178 1644 1645 812 1649 1651 387 1663 181 819 182 823 824 394 1684 830 1692
1694 1696 1701 834 183 1706 184 399 1714 1715 841 185 1718 843 844 1724 88 402
1733 1734 1737 407 1744 1745 1746 857 190 411 1753 861 1758 191 1760 864 865
414 418 1772 874 1779 1785 21 431 432 1802 895 200 1827 903 1833 906 46 1836
47 445 446 1847 1852 1854 1857 205 924 928 1869 455 48 23 1887 212 945 1896
466 467 1900 1903 216 953 218 1924 473 1929 1933 1937 962 1942 965 2 1947 108
1952 1954 970 1957 11 1965 1967 974 482 977 225 1978 1979 984 987 111 230 231
997 494 1002 2008 232 1006 112 498 2023 2025 2026 236 238 2034 2035 1020 2045
240 2051 2054 1026 2061 507 508 2067 1033 242 1036 511 2082 245 1045 2086 517
1051 12 1062 2110 1064 1067 2117 2119 1069 28 2125 2127 526 1075 253 2142 13
533 2150 14 120 1090 2171 2173 2179 1098 1099 2188 2192 544 122 263 2198 2200
2202 1110 60 2215 2216 2218 2221 2223 2226 62 1120 2233 2234 2238 2243 2244
1127 2252 2254 2255 559 560 2261 1139 2269 128 568 1149 2288 2290 2291 2295
2296 1155 2301 1157 2305 1165 2314 276 2321 1169 1170 2325 2327 1173 1174 130
278 279 2340 280 2345 131 2352 2355 2356 2357 2358 2362 1188 2372 588 1195 1196
1197 1200 1201 286 1203 1204 591 2392 132 1213 1214 1216 2407 598 1219 2413
2415 1221 2417 2420 2422 1228 2432 2435 1232 2437 2438 2439 291 2442 2444 2447
2448 606 1238 2453 2455 2459 1245 2463 2468 2469 294 615 2481 616 2483 2484
2487 298 1258 2491 619 2495 2498 2500 1265 1274 2516 2520 2521 1278 2525 2526
1281 2530 630 2539 2541 1288 2547 633 136 2554 2555 2558 1297 2564 2566 638
2574 2576 2577 639 1309 2586 2587 645 2593 2595 1320 2612 1325 2616 1326 310
311 655 2631 1339 1342 2640 660 142 2655 2656 317 34 4 2671 2672 2678 2680 2682
675 324 2689 2691 2694 2697 2701 2704 1376 684 2716 1382 2718 1383 2720 2724
2725 2726 2730 1387 2739 2740 2742 1393 1396 1401 2762 1405 696 2775 2781 701
2785 703 2789 2790 2793 704 2798 1420 1422 73 1432 75 76 716 717 1441 1444 1445
718 1449 1453 1457 345
Y5 154 223 319 484 1046 1528 1741 2164 2451 2528 2557 557 886 762 765 766 1456
767 1467 769 770 1476 50 1480 1481 1482 1483 776 404 1497 1505 1510 787 790
1536 792 1538 413 1541 1551 1552 797 1558 1559 1563 425 1568 1575 96 1578 814
1581 1585 1588 1590 1596 1607 817 822 826 1615 434 1620 1628 831 1640 835 1656
840 1665 1666 846 1671 1681 1682 851 215 1689 854 1699 860 1708 1711 1713 1716
1717 98 872 1738 1739 454 1749 883 884 457 224 889 463 1766 893 899 1770 1773
1777 107 1783 471 1789 905 472 1797 909 35 1800 476 1818 921 1826 929 930 1839
481 1841 1843 1846 1855 1860 1862 936 252 254 1867 1868 941 255 1872 943 1876
1877 947 1880 1883 259 1889 950 1899 1901 499 1923 501 968 969 502 1939 1943
1945 116 1951 37 1959 1962 1969 506 1973 1976 1977 1980 990 992 993 1991 1992
57 1997 1999 121 2001 2010 1004 2013 1005 1007 513 2024 518 1015 1017 1023 2055
1030 522 2064 524 2070 123 2074 2080 287 2084 2088 2091 2093 2097 58 2114 1070
134 2126 1072 2131 2133 543 546 549 61 2156 2162 64 2165 558 1092 1093 564 2203
1105 2207 2208 2210 147 149 2229 2232 2235 2236 2242 2245 2247 2249 2250 1122
2259 2265 1123 1126 578 2284 2286 1137 1140 25 2311 2312 1144 2318 2320 151
2330 1150 2338 329 2351 2354 601 2367 602 2371 2374 1179 2379 603 2382 2383
2385 2386 604 2391 2393 2395 2398 2399 2400 2402 612 2409 2411 2433 617 1211
1215 2452 2456 8 159 2471 634 1242 1246 2489 1247 635 2503 643 338 2507 2509
2522 1262 163 2533 2534 2536 2542 1270 1272 1273 340 2561 1283 2573 1293 2588
1300 1301 2592 1302 2598 667 668 164 2610 343 671 2617 2620 2623 348 2625 1322
1323 44 2632 1328 676 167 2639 682 2646 686 2648 2650 2652 1340 2665 2668 2676
2679 2684 173 2686 2690 83 361 2698 2702 698 2707 1356 363 2719 709 2731 1365
2747 2750 180 186 2761 187 84 1389 2773 194 2783 727 1399 380 1404 385 2807
733 195 740 100 389 87 396 1419 1423 1162 1224 1425 1304 1414 92 1948 758 2497
2634 2670 400 1436 1438 346 347 1469 1472 36 1477 732 1484 1488 1490 1491 39
743 356 1502 747 1508 1512 1513 1516 753 1520 1521 1522 1524 359 1529 1530 1531
1533 1535 18 763 364 1544 1545 1549 1557 367 775 1561 171 5 1566 1567 1569 779
85 1577 371 174 784 176 788 376 377 1603 1606 381 1608 798 799 801 804 805 384
1621 1624 1629 1632 1633 1635 1636 1637 1639 1641 178 811 1645 812 1651 179
388 1659 1662 1663 181 182 1676 823 393 394 1684 828 830 1691 1692 395 1694
1696 1697 1698 1701 834 1706 1707 837 399 1714 1715 841 185 1719 843 844 1724
88 1727 402 1732 1733 89 1736 1737 850 1744 1745 857 190 411 1753 1755 862 1758
191 1760 864 414 415 416 418 1772 873 874 1779 1781 192 1788 21 432 1802 1803
198 895 1810 22 1821 439 903 904 201 1833 906 442 443 910 47 446 1847 1848 1851
1852 1857 447 924 451 452 928 1869 208 455 209 48 23 460 1887 1888 212 945 946
1896 465 467 1900 1903 1909 1910 1913 1915 954 1924 473 1929 1933 1936 962 1940
1942 964 965 2 1947 479 1950 108 1952 1954 1957 1958 1960 972 1964 1967 482
977 225 980 981 1978 1979 1982 984 111 991 1990 230 994 997 494 1000 2003 2005
1002 1006 2014 112 498 2021 2025 2029 1014 238 2034 1018 2037 2039 1020 1021
2045 240 1024 2051 2054 1026 2058 2061 507 508 2067 1033 2071 242 2082 1045
2086 1048 516 1051 12 1061 1062 2107 2108 1063 2110 1064 1065 1066 1067 2118
2119 250 2125 2127 2128 526 117 1075 253 1078 2141 13 2147 2148 2150 14 256
2158 2161 120 258 1089 1090 2168 2170 1094 2179 2181 2182 1098 2188 2189 2192
122 263 2197 2198 2201 548 60 2214 2216 2217 1116 2221 2222 2223 2226 62 1120
2233 2234 1124 1125 2238 2241 268 1127 2248 1129 2252 1132 2255 559 560 561
2261 562 563 1139 2269 1143 128 2276 2277 568 1147 1149 2287 2288 2290 2291
571 1152 2295 129 1155 1156 2301 1157 1161 275 1165 2315 276 2319 1168 2321
1169 1170 2325 2326 2328 1173 1174 130 278 1177 279 2340 280 2343 2345 2346
131 2353 282 2355 2356 2357 2359 584 1188 2368 2372 588 1195 1196 589 1200 2384
1201 286 1204 591 2392 132 594 596 1213 1214 2403 1216 2405 2407 598 1219 2413
2415 1221 2417 1222 2422 2432 2434 2435 1232 2437 2438 2442 2443 2444 2447 2448
606 1238 1239 2453 2455 1244 1245 2463 610 2468 2469 294 615 2480 616 2483 2484
298 1258 2491 2498 2500 1264 1265 301 623 625 626 2511 2516 1276 1277 1278 2525
2526 1281 2530 630 631 1284 2539 2540 2541 1288 2543 2547 633 136 305 2553 2554
2555 1297 2562 2564 2565 1299 2569 638 2576 2577 1303 639 1307 1309 2586 2587
1311 2593 2595 1320 2605 648 139 1325 1326 310 651 311 1335 2633 1339 1342 2640
1343 2644 142 662 2656 2661 34 4 2669 2671 2672 673 2677 2678 2680 2682 675
324 2691 2694 2697 2699 1372 681 2710 148 683 684 1382 1383 2725 2726 1386 1387
685 2737 69 2740 2746 1396 2751 1405 695 2770 2771 697 153 2775 2781 1413 2784
2785 702 703 2789 2793 704 2798 2800 1420 1421 1422 706 1424 156 1427 73 1429
710 75 76 715 716 1441 1443 1444 1445 1449 1450 345
Y6 29 67 143 154 239 341 360 373 448 510 553 605 687 707 933 938 979 1046 1115
1191 1282 1381 1406 1417 1479 1667 1687 1742 1756 1762 1764 1786 1809 1878 1895
1956 1987 2012 2096 2266 2292 2369 2375 2404 2428 2496 2523 2528 2756 557 802
2205 762 765 1448 1458 1460 1462 1466 769 1475 1476 50 776 1493 404 1497 409
1523 1538 1552 421 425 96 814 1581 1590 1596 826 1615 434 835 1665 1666 1681
854 1699 860 1708 444 1712 1717 98 221 872 875 1738 1749 882 883 884 457 889
899 1777 1778 1783 905 472 908 35 1800 912 921 113 1826 248 929 930 1839 1843
1846 1855 252 941 255 1872 943 1874 1876 947 1883 259 1889 950 952 264 501 968
975 976 116 986 1976 1984 990 1991 1992 1994 1997 121 1004 2024 514 1016 1017
1030 2066 2070 125 2080 2081 2088 535 2090 2092 3 2100 2113 2114 2131 2133 2137
304 2140 2159 2162 2165 1095 564 2187 2195 2203 2208 2210 147 149 1117 2237
2242 2249 1123 2273 2281 2284 2297 580 25 2318 2320 151 2330 1154 328 329 2351
1175 601 2364 2367 2371 2374 1179 603 2383 2393 611 2398 2399 2400 2402 2411
331 2429 2441 1215 2452 8 159 10 632 1234 2503 2509 2517 2519 2531 163 2534
2536 2545 1270 1273 340 1283 2573 666 2588 1300 1301 2592 1302 2598 667 668
2608 343 2617 1315 2620 1316 2623 2625 1322 1328 2639 2646 2650 1338 2652 49
2665 2676 2679 173 2686 2698 2702 698 2707 1356 363 2717 2731 1366 2747 180
187 84 2782 2783 727 380 1404 2807 735 740 389 196 396 1419 1425 1414 1948 1428
2497 2634 2799 1436 1438 346 347 1469 1470 36 1477 351 736 1484 1488 354 356
1502 747 748 750 1513 1516 1521 1524 757 1529 1531 1532 17 364 1545 1548 366
367 1572 1573 85 788 377 1593 1595 1599 86 1603 796 1606 799 383 805 384 1624
1627 807 1635 1636 1641 811 812 1651 1652 388 1663 20 1670 824 393 394 1684
1688 1692 1694 1696 1697 832 399 1714 1719 843 1721 1724 1726 1727 1729 1734
89 1737 405 90 1743 1744 1745 855 857 1754 191 1760 414 415 416 417 871 418
1775 424 427 1794 431 1802 1803 891 197 198 895 1811 1825 439 1829 201 46 446
1854 917 1856 1857 919 924 452 927 1869 208 209 48 1875 23 1879 937 939 1887
944 946 466 1900 1903 24 1907 1908 1909 104 1922 218 106 1933 961 1937 1940
964 965 1947 479 1950 108 1957 1958 11 1960 973 1965 977 225 1978 227 989 111
991 230 231 997 2004 2005 495 1006 112 498 2020 2023 2029 1014 2034 1018 2045
1024 2058 2061 2062 2067 1032 2071 2073 1042 2082 1045 2086 1048 517 1051 1053
12 1058 1062 1063 1065 1066 2118 250 28 2129 526 1074 2135 529 530 2142 13 2147
2153 256 2158 1089 1090 538 31 2173 1094 2181 1098 2192 122 263 2197 2198 2201
548 2209 1112 2216 1116 551 1120 2234 2240 2243 2244 1127 1129 2252 561 1139
1143 1149 2289 2291 571 2302 1157 2314 276 2319 2321 1169 2325 2327 2328 130
278 279 2343 2345 2346 1183 2357 2361 1189 284 285 588 1195 1203 1204 2392 1208
132 594 1214 2403 598 1219 2415 2420 600 1229 2431 2432 2434 2444 2447 2448
2455 2459 609 2469 615 616 2483 298 1258 2491 2493 619 299 2500 1265 626 627
1276 1278 2525 2526 2530 630 631 1285 2539 2540 1288 135 2547 633 305 2554 2558
2563 2564 2576 2577 2578 639 138 1309 2586 2587 1317 2600 2612 1325 655 1334
141 658 1342 142 662 2649 66 317 34 2669 673 2680 1363 675 324 2687 2694 679
1372 1373 2704 1376 2710 684 2716 1383 2724 2727 2730 1386 1387 2733 69 152
1396 1401 2770 2777 700 1413 2784 2785 702 703 2789 2793 704 1418 705 2798 1421
1422 706 156 73 1432 711 76 1443 1444 1445 1447 160 721 1457
1 WARNING(S) FOUND IN THE INPUT INSTRUCTIONS
MLFA CESD EPESE
SUMMARY OF ANALYSIS
Number of groups 1
Number of observations 6165
Number of dependent variables 6
Number of independent variables 3
Number of continuous latent variables 3
Observed dependent variables
Continuous
Y1 Y2 Y3 Y4 Y5 Y6
Observed independent variables
T FEMALE AGEC70
Continuous latent variables
FW S FB
Variables with special functions
Cluster variable ID
Within variables
T
Between variables
FEMALE AGEC70
Estimator MLR
Information matrix OBSERVED
Maximum number of iterations 100
Convergence criterion 0.100D-05
Maximum number of EM iterations 500
Convergence criteria for the EM algorithm
Loglikelihood change 0.100D-02
Relative loglikelihood change 0.100D-05
Derivative 0.100D-03
Minimum variance 0.100D-03
Maximum number of steepest descent iterations 20
Maximum number of iterations for H1 2000
Convergence criterion for H1 0.100D-03
Optimization algorithm EMA
Input data file(s)
cesd.dat
Input data format FREE
SUMMARY OF DATA
Number of missing data patterns 32
Number of clusters 2762
COVARIANCE COVERAGE OF DATA
Minimum covariance coverage value 0.100
PROPORTION OF DATA PRESENT
Covariance Coverage
Y1 Y2 Y3 Y4 Y5
________ ________ ________ ________ ________
Y1 0.993
Y2 0.984 0.991
Y3 0.989 0.986 0.994
Y4 0.982 0.977 0.981 0.985
Y5 0.981 0.976 0.981 0.975 0.983
Y6 0.920 0.918 0.921 0.915 0.913
T 0.993 0.991 0.994 0.985 0.983
FEMALE 0.993 0.991 0.994 0.985 0.983
AGEC70 0.993 0.991 0.994 0.985 0.983
Covariance Coverage
Y6 T FEMALE AGEC70
________ ________ ________ ________
Y6 0.923
T 0.923 1.000
FEMALE 0.923 1.000 1.000
AGEC70 0.923 1.000 1.000 1.000
UNIVARIATE SAMPLE STATISTICS
UNIVARIATE HIGHER-ORDER MOMENT DESCRIPTIVE STATISTICS
Variable/ Mean/ Skewness/ Minimum/ % with Percentiles
Sample Size Variance Kurtosis Maximum Min/Max 20%/60% 40%/80% Median
Y1 1.472 1.794 1.000 64.36% 1.000 1.000 1.000
6119.000 0.564 3.045 4.000 4.13% 1.000 2.000
Y2 1.353 2.330 1.000 76.49% 1.000 1.000 1.000
6109.000 0.552 4.888 4.000 4.35% 1.000 2.000
Y3 1.522 1.704 1.000 63.41% 1.000 1.000 1.000
6130.000 0.676 2.312 4.000 5.76% 1.000 2.000
Y4 1.697 1.058 1.000 63.05% 1.000 1.000 1.000
6071.000 1.009 -0.352 4.000 7.15% 1.000 3.000
Y5 1.543 1.474 1.000 71.24% 1.000 1.000 1.000
6063.000 0.885 0.751 4.000 6.28% 1.000 2.000
Y6 1.944 0.721 1.000 56.44% 1.000 1.000 1.000
5693.000 1.412 -1.136 4.000 17.13% 2.000 3.000
T 0.236 0.395 0.000 44.25% 0.000 0.000 0.300
6165.000 0.057 -1.306 0.600 23.07% 0.300 0.600
FEMALE 0.584 -0.342 0.000 41.56% 0.000 0.000 1.000
2762.000 0.243 -1.883 1.000 58.44% 1.000 1.000
AGEC70 0.364 0.673 -0.300 34.65% -0.300 0.200 0.200
2762.000 0.419 -0.681 1.700 8.76% 0.200 1.200
THE MODEL ESTIMATION TERMINATED NORMALLY
MODEL FIT INFORMATION
Number of Free Parameters 26
Loglikelihood
H0 Value -42123.918
H0 Scaling Correction Factor 1.6585
for MLR
Information Criteria
Akaike (AIC) 84299.835
Bayesian (BIC) 84474.728
Sample-Size Adjusted BIC 84392.107
(n* = (n + 2) / 24)
MODEL RESULTS
Two-Tailed
Estimate S.E. Est./S.E. P-Value
Within Level
FW BY
Y1 0.512 0.000 999.000 999.000
Y2 0.504 0.015 33.522 0.000
Y3 0.608 0.016 38.119 0.000
Y4 0.606 0.023 26.600 0.000
Y5 0.554 0.022 24.955 0.000
Y6 0.413 0.023 17.916 0.000
Residual Variances
Y1 0.287 0.011 25.740 0.000
Y2 0.283 0.012 23.965 0.000
Y3 0.283 0.013 21.315 0.000
Y4 0.619 0.019 31.867 0.000
Y5 0.562 0.019 29.718 0.000
Y6 1.232 0.022 55.136 0.000
FW 0.527 0.046 11.350 0.000
Between Level
FB BY
Y1 0.512 0.000 999.000 999.000
Y2 0.504 0.015 33.522 0.000
Y3 0.608 0.016 38.119 0.000
Y4 0.606 0.023 26.600 0.000
Y5 0.554 0.022 24.955 0.000
Y6 0.413 0.023 17.916 0.000
FB ON
AGEC70 0.103 0.033 3.101 0.002
FEMALE 0.263 0.041 6.398 0.000
S ON
AGEC70 0.206 0.095 2.165 0.030
FEMALE 0.039 0.100 0.390 0.697
FB WITH
S 0.000 0.101 0.000 1.000
Intercepts
Y1 1.358 0.000 999.000 999.000
Y2 1.240 0.008 150.954 0.000
Y3 1.384 0.010 145.024 0.000
Y4 1.561 0.014 114.171 0.000
Y5 1.421 0.012 117.583 0.000
Y6 1.854 0.017 107.373 0.000
S 0.184 0.074 2.481 0.013
FB -0.005 0.031 -0.147 0.883
Residual Variances
Y1 0.000 0.000 999.000 999.000
Y2 0.000 0.000 999.000 999.000
Y3 0.000 0.000 999.000 999.000
Y4 0.000 0.000 999.000 999.000
Y5 0.000 0.000 999.000 999.000
Y6 0.000 0.000 999.000 999.000
S 0.177 0.296 0.598 0.550
FB 0.500 0.058 8.670 0.000
QUALITY OF NUMERICAL RESULTS
Condition Number for the Information Matrix 0.369E-05
(ratio of smallest to largest eigenvalue)
Beginning Time: 12:32:29
Ending Time: 12:32:31
Elapsed Time: 00:00:02
MUTHEN & MUTHEN
3463 Stoner Ave.
Los Angeles, CA 90066
Tel: (310) 391-9971
Fax: (310) 391-8971
Web: www.StatModel.com
Support: Support@StatModel.com
Copyright (c) 1998-2024 Muthen & MuthenSave output
[1] TRUE[1] TRUEIt only takes 3 seconds
Compare MLM and MLFA
BY parameters
| paramHeader | param | est | se | est_se | pval | BetweenWithin | PARAMETER | model |
|---|---|---|---|---|---|---|---|---|
| FW.BY | Y1 | 0.512 | 0.000 | 999.000 | 999 | Within | FW.BY Y1 | MLFA |
| FW.BY | Y2 | 0.504 | 0.015 | 33.522 | 0 | Within | FW.BY Y2 | MLFA |
| FW.BY | Y3 | 0.608 | 0.016 | 38.119 | 0 | Within | FW.BY Y3 | MLFA |
| FW.BY | Y4 | 0.606 | 0.023 | 26.600 | 0 | Within | FW.BY Y4 | MLFA |
| FW.BY | Y5 | 0.554 | 0.022 | 24.955 | 0 | Within | FW.BY Y5 | MLFA |
| FW.BY | Y6 | 0.413 | 0.023 | 17.916 | 0 | Within | FW.BY Y6 | MLFA |
| FB.BY | Y1 | 0.512 | 0.000 | 999.000 | 999 | Between | FB.BY Y1 | MLFA |
| FB.BY | Y2 | 0.504 | 0.015 | 33.522 | 0 | Between | FB.BY Y2 | MLFA |
| FB.BY | Y3 | 0.608 | 0.016 | 38.119 | 0 | Between | FB.BY Y3 | MLFA |
| FB.BY | Y4 | 0.606 | 0.023 | 26.600 | 0 | Between | FB.BY Y4 | MLFA |
| FB.BY | Y5 | 0.554 | 0.022 | 24.955 | 0 | Between | FB.BY Y5 | MLFA |
| FB.BY | Y6 | 0.413 | 0.023 | 17.916 | 0 | Between | FB.BY Y6 | MLFA |
Means and Intercepts parameters
| paramHeader | param | est | se | est_se | pval | BetweenWithin | PARAMETER | model |
|---|---|---|---|---|---|---|---|---|
| Intercepts | YZ | -0.177 | 0.027 | -6.598 | 0.000 | Between | Intercepts YZ | MLM |
| Intercepts | S | 0.196 | 0.069 | 2.847 | 0.004 | Between | Intercepts S | MLM |
| Intercepts | Y1 | 1.358 | 0.000 | 999.000 | 999.000 | Between | Intercepts Y1 | MLFA |
| Intercepts | Y2 | 1.240 | 0.008 | 150.954 | 0.000 | Between | Intercepts Y2 | MLFA |
| Intercepts | Y3 | 1.384 | 0.010 | 145.024 | 0.000 | Between | Intercepts Y3 | MLFA |
| Intercepts | Y4 | 1.561 | 0.014 | 114.171 | 0.000 | Between | Intercepts Y4 | MLFA |
| Intercepts | Y5 | 1.421 | 0.012 | 117.583 | 0.000 | Between | Intercepts Y5 | MLFA |
| Intercepts | Y6 | 1.854 | 0.017 | 107.373 | 0.000 | Between | Intercepts Y6 | MLFA |
| Intercepts | S | 0.184 | 0.074 | 2.481 | 0.013 | Between | Intercepts S | MLFA |
| Intercepts | FB | -0.005 | 0.031 | -0.147 | 0.883 | Between | Intercepts FB | MLFA |
Variances parameters
| paramHeader | param | est | se | est_se | pval | BetweenWithin | PARAMETER | model |
|---|---|---|---|---|---|---|---|---|
| Residual.Variances | YZ | 0.595 | 0.028 | 21.362 | 0.000 | Within | Residual.Variances YZ | MLM |
| Residual.Variances | YZ | 0.400 | 0.040 | 9.943 | 0.000 | Between | Residual.Variances YZ | MLM |
| Residual.Variances | S | 0.328 | 0.220 | 1.493 | 0.135 | Between | Residual.Variances S | MLM |
| Residual.Variances | Y1 | 0.287 | 0.011 | 25.740 | 0.000 | Within | Residual.Variances Y1 | MLFA |
| Residual.Variances | Y2 | 0.283 | 0.012 | 23.965 | 0.000 | Within | Residual.Variances Y2 | MLFA |
| Residual.Variances | Y3 | 0.283 | 0.013 | 21.315 | 0.000 | Within | Residual.Variances Y3 | MLFA |
| Residual.Variances | Y4 | 0.619 | 0.019 | 31.867 | 0.000 | Within | Residual.Variances Y4 | MLFA |
| Residual.Variances | Y5 | 0.562 | 0.019 | 29.718 | 0.000 | Within | Residual.Variances Y5 | MLFA |
| Residual.Variances | Y6 | 1.232 | 0.022 | 55.136 | 0.000 | Within | Residual.Variances Y6 | MLFA |
| Residual.Variances | FW | 0.527 | 0.046 | 11.350 | 0.000 | Within | Residual.Variances FW | MLFA |
| Residual.Variances | Y1 | 0.000 | 0.000 | 999.000 | 999.000 | Between | Residual.Variances Y1 | MLFA |
| Residual.Variances | Y2 | 0.000 | 0.000 | 999.000 | 999.000 | Between | Residual.Variances Y2 | MLFA |
| Residual.Variances | Y3 | 0.000 | 0.000 | 999.000 | 999.000 | Between | Residual.Variances Y3 | MLFA |
| Residual.Variances | Y4 | 0.000 | 0.000 | 999.000 | 999.000 | Between | Residual.Variances Y4 | MLFA |
| Residual.Variances | Y5 | 0.000 | 0.000 | 999.000 | 999.000 | Between | Residual.Variances Y5 | MLFA |
| Residual.Variances | Y6 | 0.000 | 0.000 | 999.000 | 999.000 | Between | Residual.Variances Y6 | MLFA |
| Residual.Variances | S | 0.177 | 0.296 | 0.598 | 0.550 | Between | Residual.Variances S | MLFA |
| Residual.Variances | FB | 0.500 | 0.058 | 8.670 | 0.000 | Between | Residual.Variances FB | MLFA |
Curve of Factors via Multilevel Modeling
The multilevel curve of factors (MLCOF) model might be totally new. But, as I show in the very last slide (spoiler alert) is equivalent to the MLCFA model as I have parameterized these models.
The MLGOF model works by using a “trick” to get the common factor intercept at the between level without having to specify a between level measurement model for the factor indicators. It’s an old trick that was used back in the day to access the meanstructure of structural equation models before these were more readily available to the programmer: regress on a constant.
Modeling considerations: MLC OF
This is the model set-up for the multilevel growth-of-factors model. This model is conceptually similar to the MLFA model, although we remove the BETWEEN level factor model for the items, and we “trick” Mplus into estimating a random intercept for the latent factor at the between level by regressing it on a constant (k) and declaring that regression to be random.
As with the MLFA model, observed indicators of depression (y1-y6) are both within and between level variables (and not specified as WITHIN or BETWEEN in Mplus input). A within-level factor is specified (fw) that is identified by fixing the first factor loading (as described in the MLFA model; fixed parameters are shown in purlple). fw is regressed on time and this is declared a random effect and assigned the label “s”. fw is regressed on a constant variable “k”, and this is also declared random and assigned the label “i”.
At the between level, we model the item intercepts (fixing the first item’s intercept as described in the MLFA model to identify the mean of i) and the between-level residual variances of y1-y6 are fixed to 0, for reasons described in MLFA model setup.
Data setup
Use MplusAutomation to prepare data set for Mplus. You actually have to compute the constant and add it to the data set output to Mplus.
Mplus MLGOF model
Multilevel Growth of Factors is what I’m calling this model
TITLE: Growth of Factors MLM
DATA: FILE = cesdlongk.dat ;
VARIANCES = NOCHECK ; ! this is critical b/c of k
VARIABLE: NAMES = id td female agec70 y1 y2 y3 y4 y5 y6 k ;
MISSING = . ;
WITHIN = td k ;
BETWEEN = agec70 female ;
CLUSTER = id ;
ANALYSIS: TYPE = TWOLEVEL RANDOM ;
MODEL: %WITHIN%
fw BY y1@0.51150 (l1);
fw by y2-y6* (l2-l6);
s | fw on td ; ! random slope with respect to td
i | fw on k ; ! random intercept (constant)
%BETWEEN%
i s on agec70 female ;
i with s ;
y1-y6@0 ; ! force all residual variance to within or i, s
[y1@1.35775]; ! needed for identification of intercept(i)
[y2-y6*];Mplus VERSION 8.11 (Mac)
MUTHEN & MUTHEN
04/10/2025 12:32 PM
INPUT INSTRUCTIONS
TITLE: Growth of Factors MLM
DATA: FILE = cesdlongk.dat ;
VARIANCES = NOCHECK ; ! this is critical b/c of k
VARIABLE: NAMES = id td female agec70 y1 y2 y3 y4 y5 y6 k ;
MISSING = . ;
WITHIN = td k ;
BETWEEN = agec70 female ;
CLUSTER = id ;
ANALYSIS: TYPE = TWOLEVEL RANDOM ;
MODEL: %WITHIN%
fw BY y1@0.51150 (l1);
fw by y2-y6* (l2-l6);
s | fw on td ; ! random slope with respect to td
i | fw on k ; ! random intercept (constant)
%BETWEEN%
i s on agec70 female ;
i with s ;
y1-y6@0 ; ! force all residual variance to within or i, s
[y1@1.35775]; ! needed for identification of intercept(i)
[y2-y6*];
*** WARNING
One or more individual-level variables have no variation within a
cluster for the following clusters.
Variable Cluster IDs with no within-cluster variation
Y1 154 319 484 815 1046 1528 1791 2164 2451 557 98
Y2 687 793 1071 1618 1669 2096 2104 2369 557 2205
Y3 154 2369
Y4 154 319 484 605 815 1012 1046 1528 1786 1849 2549 557 98
Y5 154 223 319 484 1046 1528 1741 2164 2451 2528 2557 557 886 98
Y6 29 67 143 154 239 341 360 373 448 510 553 605 687 707 933 938 979 1046 1115
1191 1282 1381 1406 1417 1479 1667 1687 1742 1756 1762 1764 1786 1809 1878 1895
1956 1987 2012 2096 2266 2292 2369 2375 2404 2428 2496 2523 2528 2756 557 802
2205 50 444 1783 905 952 1994 2452 180 2576 1325 1342
1 WARNING(S) FOUND IN THE INPUT INSTRUCTIONS
Growth of Factors MLM
SUMMARY OF ANALYSIS
Number of groups 1
Number of observations 6165
Number of dependent variables 6
Number of independent variables 4
Number of continuous latent variables 3
Observed dependent variables
Continuous
Y1 Y2 Y3 Y4 Y5 Y6
Observed independent variables
TD FEMALE AGEC70 K
Continuous latent variables
FW S I
Variables with special functions
Cluster variable ID
Within variables
TD K
Between variables
FEMALE AGEC70
Estimator MLR
Information matrix OBSERVED
Maximum number of iterations 100
Convergence criterion 0.100D-05
Maximum number of EM iterations 500
Convergence criteria for the EM algorithm
Loglikelihood change 0.100D-02
Relative loglikelihood change 0.100D-05
Derivative 0.100D-03
Minimum variance 0.100D-03
Maximum number of steepest descent iterations 20
Maximum number of iterations for H1 2000
Convergence criterion for H1 0.100D-03
Optimization algorithm EMA
Input data file(s)
cesdlongk.dat
Input data format FREE
SUMMARY OF DATA
Number of missing data patterns 32
Number of clusters 2762
COVARIANCE COVERAGE OF DATA
Minimum covariance coverage value 0.100
PROPORTION OF DATA PRESENT
Covariance Coverage
Y1 Y2 Y3 Y4 Y5
________ ________ ________ ________ ________
Y1 0.993
Y2 0.984 0.991
Y3 0.989 0.986 0.994
Y4 0.982 0.977 0.981 0.985
Y5 0.981 0.976 0.981 0.975 0.983
Y6 0.920 0.918 0.921 0.915 0.913
TD 0.993 0.991 0.994 0.985 0.983
K 0.993 0.991 0.994 0.985 0.983
FEMALE 0.993 0.991 0.994 0.985 0.983
AGEC70 0.993 0.991 0.994 0.985 0.983
Covariance Coverage
Y6 TD K FEMALE AGEC70
________ ________ ________ ________ ________
Y6 0.923
TD 0.923 1.000
K 0.923 1.000 1.000
FEMALE 0.923 1.000 1.000 1.000
AGEC70 0.923 1.000 1.000 1.000 1.000
UNIVARIATE SAMPLE STATISTICS
UNIVARIATE HIGHER-ORDER MOMENT DESCRIPTIVE STATISTICS
Variable/ Mean/ Skewness/ Minimum/ % with Percentiles
Sample Size Variance Kurtosis Maximum Min/Max 20%/60% 40%/80% Median
Y1 1.472 1.794 1.000 64.36% 1.000 1.000 1.000
6119.000 0.564 3.045 4.000 4.13% 1.000 2.000
Y2 1.353 2.330 1.000 76.49% 1.000 1.000 1.000
6109.000 0.552 4.888 4.000 4.35% 1.000 2.000
Y3 1.522 1.704 1.000 63.41% 1.000 1.000 1.000
6130.000 0.676 2.312 4.000 5.76% 1.000 2.000
Y4 1.697 1.058 1.000 63.05% 1.000 1.000 1.000
6071.000 1.009 -0.352 4.000 7.15% 1.000 3.000
Y5 1.543 1.474 1.000 71.24% 1.000 1.000 1.000
6063.000 0.885 0.751 4.000 6.28% 1.000 2.000
Y6 1.944 0.721 1.000 56.44% 1.000 1.000 1.000
5693.000 1.412 -1.136 4.000 17.13% 2.000 3.000
TD 0.236 0.395 0.000 44.25% 0.000 0.000 0.300
6165.000 0.057 -1.306 0.600 23.07% 0.300 0.600
K 1.000 0.000 1.000 100.00% 1.000 1.000 1.000
6165.000 0.000 0.000 1.000 100.00% 1.000 1.000
FEMALE 0.584 -0.342 0.000 41.56% 0.000 0.000 1.000
2762.000 0.243 -1.883 1.000 58.44% 1.000 1.000
AGEC70 0.364 0.673 -0.300 34.65% -0.300 0.200 0.200
2762.000 0.419 -0.681 1.700 8.76% 0.200 1.200
WARNING: THE SAMPLE VARIANCE OF K IS 0.000.
THE MODEL ESTIMATION TERMINATED NORMALLY
MODEL FIT INFORMATION
Number of Free Parameters 26
Loglikelihood
H0 Value -42123.918
H0 Scaling Correction Factor 1.6585
for MLR
Information Criteria
Akaike (AIC) 84299.835
Bayesian (BIC) 84474.728
Sample-Size Adjusted BIC 84392.107
(n* = (n + 2) / 24)
MODEL RESULTS
Two-Tailed
Estimate S.E. Est./S.E. P-Value
Within Level
FW BY
Y1 0.512 0.000 999.000 999.000
Y2 0.504 0.015 33.521 0.000
Y3 0.608 0.016 38.118 0.000
Y4 0.606 0.023 26.600 0.000
Y5 0.554 0.022 24.956 0.000
Y6 0.413 0.023 17.916 0.000
Residual Variances
Y1 0.287 0.011 25.740 0.000
Y2 0.283 0.012 23.964 0.000
Y3 0.283 0.013 21.314 0.000
Y4 0.619 0.019 31.867 0.000
Y5 0.562 0.019 29.718 0.000
Y6 1.232 0.022 55.136 0.000
FW 0.527 0.046 11.351 0.000
Between Level
I ON
AGEC70 0.103 0.033 3.100 0.002
FEMALE 0.263 0.041 6.396 0.000
S ON
AGEC70 0.206 0.095 2.166 0.030
FEMALE 0.039 0.100 0.393 0.695
I WITH
S 0.000 0.101 -0.002 0.999
Means
Y1 1.358 0.000 999.000 999.000
Y2 1.240 0.008 150.954 0.000
Y3 1.384 0.010 145.026 0.000
Y4 1.561 0.014 114.172 0.000
Y5 1.421 0.012 117.583 0.000
Y6 1.854 0.017 107.374 0.000
Intercepts
S 0.184 0.074 2.478 0.013
I -0.005 0.031 -0.145 0.885
Variances
Y1 0.000 0.000 999.000 999.000
Y2 0.000 0.000 999.000 999.000
Y3 0.000 0.000 999.000 999.000
Y4 0.000 0.000 999.000 999.000
Y5 0.000 0.000 999.000 999.000
Y6 0.000 0.000 999.000 999.000
Residual Variances
S 0.178 0.296 0.602 0.547
I 0.500 0.058 8.671 0.000
QUALITY OF NUMERICAL RESULTS
Condition Number for the Information Matrix 0.369E-05
(ratio of smallest to largest eigenvalue)
Beginning Time: 12:32:31
Ending Time: 12:32:33
Elapsed Time: 00:00:02
MUTHEN & MUTHEN
3463 Stoner Ave.
Los Angeles, CA 90066
Tel: (310) 391-9971
Fax: (310) 391-8971
Web: www.StatModel.com
Support: Support@StatModel.com
Copyright (c) 1998-2024 Muthen & MuthenSave output
[1] TRUE[1] TRUEMultiple Indicator LGCM
Modeling considerations: MILGCM
This is the model set-up for the multiple indicators LGCM. It’s too much of a model for the Mplus DEMO version, as in our case it will involve 18 dependent variables (y1-y6 at each of 3 observations). These models are challenging to specify properly, and can take a long time to estimate with maximum likelihood methods because of the relatively large number of latent variables over which to integrate.
But, with only 3 observation time points, it is not too onerous to specify this model, and provides a useful comparison to our multilevel approaches.
I will use constraints on y1 (intercepts and measurement slopes) as in previous multilevel models to place the results on a comparable scale.
Data setup
items <- c("sad", "blues", "depress", "happy", "enjoy", "hopeful")
df.cesd <- df %>%
select(which(names(df) %in% c("id", "t",items,"agec70","female")))
# Create new variables y1-y6 that correspond to the variables in items
for (i in seq_along(items)) {
df.cesd[[paste0("y", i)]] <- df.cesd[[items[i]]]
}
df.cesd <- df.cesd %>% select(-all_of(items))
# Convert 't' into a factor to prevent it from being treated as a function
df.cesd$t <- as.factor(df.cesd$t)
# Reshape the dataframe to long format
long_df <- df.cesd %>%
pivot_longer(
cols = starts_with("y"),
names_to = "y",
values_to = "y_values"
) %>%
# Ensure each combination of id and t has a unique row for each y
group_by(id, t) %>%
mutate(y_number = row_number()) %>%
ungroup()
# Create a time variable that interacts with 'y_number' and 't'
long_df <- long_df %>%
mutate(y_time = paste0("y", t, y_number))
# Then, reshape back to wide format with new names for 'y' variables
wide_df <- long_df %>%
pivot_wider(
id_cols = c(id, female, agec70),
names_from = y_time,
values_from = y_values,
names_prefix = ""
)View the resulting dataframe
# A tibble: 6 × 21
id female agec70 y01 y02 y03 y04 y05 y06 y31 y32 y33
<int> <int> <dbl> <int> <int> <int> <int> <int> <int> <int> <int> <int>
1 1 0 1.7 2 2 2 3 3 3 NA NA NA
2 2 1 1.7 1 1 1 1 1 1 2 1 1
3 3 1 1.2 2 1 1 3 3 4 2 1 2
4 4 0 1.2 1 1 1 1 1 1 1 1 1
5 5 1 1.2 NA 1 2 NA NA NA 1 1 1
6 6 0 1.2 1 1 1 1 1 4 NA NA NA
# ℹ 9 more variables: y34 <int>, y35 <int>, y36 <int>, y61 <int>, y62 <int>,
# y63 <int>, y64 <int>, y65 <int>, y66 <int>Use MplusAutomation to generate data for Mplus
Mplus MILGCM
TITLE: LGCM CESD score
DATA: FILE = "cesdwide.dat";
VARIABLE: NAMES = id female agec70
y01 y02 y03 y04 y05 y06
y31 y32 y33 y34 y35 y36
y61 y62 y63 y64 y65 y66;
IDVARIABLE = id ;
MISSING = . ;
MODEL: fw0 BY y01@0.51150 (l1);
fw3 BY y31@0.51150 (l1);
fw6 BY y61@0.51150 (l1);
fw0 by y02-y06* (l2-l6);
fw3 by y32-y36* (l2-l6);
fw6 by y62-y66* (l2-l6);
[y01@1.35775];
[y31@1.35775];
[y61@1.35775];
[y02-y06*] (nu2-nu6) ;
[y32-y36*] (nu2-nu6) ;
[y62-y66*] (nu2-nu6) ;
y01 y31 y61 (te1) ;
y02 y32 y62 (te2) ;
y03 y33 y63 (te3) ;
y04 y34 y64 (te4) ;
y05 y35 y65 (te5) ;
y06 y36 y66 (te6) ;
y01 with y31 (te11) ;
y31 with y61 (te11) ;
y02 with y32 (te22) ;
y32 with y62 (te22) ;
y03 with y33 (te33) ;
y33 with y63 (te33) ;
y04 with y34 (te44) ;
y34 with y64 (te44) ;
y05 with y35 (te55) ;
y35 with y65 (te55) ;
y06 with y36 (te66) ;
y36 with y66 (te66) ;
y01 with y61 ;
y02 with y62 ;
y03 with y63 ;
y04 with y64 ;
y05 with y65 ;
y06 with y66 ;
[fw0@0] ;
[fw3@0] ;
[fw6@0] ;
fw0* (psi1) ;
fw3* (psi1) ;
fw6* (psi1) ;
i by fw0-fw6@1 ;
s by fw0@0 fw3@0.3 fw6@0.6 ; ! per decade
i with s ;
i s on agec70 female ;
[i*] ;
[s*] ;Mplus VERSION 8.11 (Mac)
MUTHEN & MUTHEN
04/10/2025 12:32 PM
INPUT INSTRUCTIONS
TITLE: LGCM CESD score
DATA: FILE = "cesdwide.dat";
VARIABLE: NAMES = id female agec70
y01 y02 y03 y04 y05 y06
y31 y32 y33 y34 y35 y36
y61 y62 y63 y64 y65 y66;
IDVARIABLE = id ;
MISSING = . ;
MODEL: fw0 BY y01@0.51150 (l1);
fw3 BY y31@0.51150 (l1);
fw6 BY y61@0.51150 (l1);
fw0 by y02-y06* (l2-l6);
fw3 by y32-y36* (l2-l6);
fw6 by y62-y66* (l2-l6);
[y01@1.35775];
[y31@1.35775];
[y61@1.35775];
[y02-y06*] (nu2-nu6) ;
[y32-y36*] (nu2-nu6) ;
[y62-y66*] (nu2-nu6) ;
y01 y31 y61 (te1) ;
y02 y32 y62 (te2) ;
y03 y33 y63 (te3) ;
y04 y34 y64 (te4) ;
y05 y35 y65 (te5) ;
y06 y36 y66 (te6) ;
y01 with y31 (te11) ;
y31 with y61 (te11) ;
y02 with y32 (te22) ;
y32 with y62 (te22) ;
y03 with y33 (te33) ;
y33 with y63 (te33) ;
y04 with y34 (te44) ;
y34 with y64 (te44) ;
y05 with y35 (te55) ;
y35 with y65 (te55) ;
y06 with y36 (te66) ;
y36 with y66 (te66) ;
y01 with y61 ;
y02 with y62 ;
y03 with y63 ;
y04 with y64 ;
y05 with y65 ;
y06 with y66 ;
[fw0@0] ;
[fw3@0] ;
[fw6@0] ;
fw0* (psi1) ;
fw3* (psi1) ;
fw6* (psi1) ;
i by fw0-fw6@1 ;
s by fw0@0 fw3@0.3 fw6@0.6 ; ! per decade
i with s ;
i s on agec70 female ;
[i*] ;
[s*] ;
INPUT READING TERMINATED NORMALLY
LGCM CESD score
SUMMARY OF ANALYSIS
Number of groups 1
Number of observations 2762
Number of dependent variables 18
Number of independent variables 2
Number of continuous latent variables 5
Observed dependent variables
Continuous
Y01 Y02 Y03 Y04 Y05 Y06
Y31 Y32 Y33 Y34 Y35 Y36
Y61 Y62 Y63 Y64 Y65 Y66
Observed independent variables
FEMALE AGEC70
Continuous latent variables
FW0 FW3 FW6 I S
Variables with special functions
ID variable ID
Estimator ML
Information matrix OBSERVED
Maximum number of iterations 1000
Convergence criterion 0.500D-04
Maximum number of steepest descent iterations 20
Maximum number of iterations for H1 2000
Convergence criterion for H1 0.100D-03
Input data file(s)
cesdwide.dat
Input data format FREE
SUMMARY OF DATA
Number of missing data patterns 140
COVARIANCE COVERAGE OF DATA
Minimum covariance coverage value 0.100
PROPORTION OF DATA PRESENT
Covariance Coverage
Y01 Y02 Y03 Y04 Y05
________ ________ ________ ________ ________
Y01 0.981
Y02 0.975 0.980
Y03 0.979 0.977 0.983
Y04 0.975 0.970 0.974 0.976
Y05 0.974 0.969 0.974 0.969 0.975
Y06 0.915 0.914 0.917 0.913 0.911
Y31 0.711 0.711 0.711 0.708 0.707
Y32 0.707 0.706 0.706 0.703 0.702
Y33 0.711 0.711 0.711 0.708 0.707
Y34 0.701 0.700 0.701 0.698 0.696
Y35 0.699 0.699 0.699 0.696 0.694
Y36 0.642 0.642 0.642 0.639 0.638
Y61 0.501 0.500 0.501 0.499 0.498
Y62 0.502 0.501 0.502 0.500 0.499
Y63 0.501 0.501 0.501 0.499 0.499
Y64 0.498 0.498 0.499 0.496 0.496
Y65 0.498 0.498 0.499 0.496 0.496
Y66 0.479 0.478 0.479 0.477 0.476
FEMALE 0.981 0.980 0.983 0.976 0.975
AGEC70 0.981 0.980 0.983 0.976 0.975
Covariance Coverage
Y06 Y31 Y32 Y33 Y34
________ ________ ________ ________ ________
Y06 0.919
Y31 0.665 0.724
Y32 0.662 0.715 0.720
Y33 0.665 0.722 0.716 0.725
Y34 0.657 0.712 0.706 0.711 0.714
Y35 0.655 0.710 0.703 0.710 0.702
Y36 0.603 0.652 0.648 0.652 0.646
Y61 0.467 0.485 0.484 0.485 0.481
Y62 0.468 0.487 0.485 0.486 0.482
Y63 0.467 0.486 0.485 0.486 0.482
Y64 0.465 0.483 0.482 0.483 0.479
Y65 0.465 0.484 0.482 0.483 0.480
Y66 0.447 0.464 0.462 0.463 0.459
FEMALE 0.919 0.724 0.720 0.725 0.714
AGEC70 0.919 0.724 0.720 0.725 0.714
Covariance Coverage
Y35 Y36 Y61 Y62 Y63
________ ________ ________ ________ ________
Y35 0.712
Y36 0.644 0.654
Y61 0.479 0.446 0.510
Y62 0.480 0.447 0.507 0.512
Y63 0.479 0.446 0.508 0.508 0.511
Y64 0.476 0.444 0.504 0.505 0.505
Y65 0.478 0.445 0.506 0.505 0.505
Y66 0.457 0.429 0.485 0.486 0.487
FEMALE 0.712 0.654 0.510 0.512 0.511
AGEC70 0.712 0.654 0.510 0.512 0.511
Covariance Coverage
Y64 Y65 Y66 FEMALE AGEC70
________ ________ ________ ________ ________
Y64 0.508
Y65 0.504 0.508
Y66 0.483 0.483 0.488
FEMALE 0.508 0.508 0.488 1.000
AGEC70 0.508 0.508 0.488 1.000 1.000
UNIVARIATE SAMPLE STATISTICS
UNIVARIATE HIGHER-ORDER MOMENT DESCRIPTIVE STATISTICS
Variable/ Mean/ Skewness/ Minimum/ % with Percentiles
Sample Size Variance Kurtosis Maximum Min/Max 20%/60% 40%/80% Median
Y01 1.458 1.812 1.000 65.26% 1.000 1.000 1.000
2709.000 0.545 3.142 4.000 3.77% 1.000 2.000
Y02 1.345 2.371 1.000 77.44% 1.000 1.000 1.000
2708.000 0.552 5.035 4.000 4.36% 1.000 2.000
Y03 1.522 1.718 1.000 63.55% 1.000 1.000 1.000
2716.000 0.686 2.339 4.000 6.04% 1.000 2.000
Y04 1.650 1.189 1.000 66.22% 1.000 1.000 1.000
2697.000 1.001 -0.071 4.000 7.45% 1.000 3.000
Y05 1.465 1.735 1.000 75.91% 1.000 1.000 1.000
2694.000 0.815 1.599 4.000 5.86% 1.000 2.000
Y06 1.998 0.643 1.000 56.87% 1.000 1.000 1.000
2539.000 1.543 -1.319 4.000 20.40% 2.000 4.000
Y31 1.469 1.826 1.000 65.07% 1.000 1.000 1.000
2001.000 0.575 3.102 4.000 4.35% 1.000 2.000
Y32 1.368 2.260 1.000 74.90% 1.000 1.000 1.000
1988.000 0.552 4.646 4.000 4.33% 1.000 2.000
Y33 1.502 1.764 1.000 65.05% 1.000 1.000 1.000
2003.000 0.667 2.508 4.000 5.64% 1.000 2.000
Y34 1.723 0.959 1.000 59.79% 1.000 1.000 1.000
1972.000 0.963 -0.499 4.000 6.03% 2.000 3.000
Y35 1.589 1.301 1.000 67.55% 1.000 1.000 1.000
1966.000 0.880 0.298 4.000 5.49% 1.000 3.000
Y36 1.870 0.837 1.000 56.81% 1.000 1.000 1.000
1806.000 1.266 -0.861 4.000 13.79% 2.000 3.000
Y61 1.505 1.716 1.000 61.60% 1.000 1.000 1.000
1409.000 0.584 2.786 4.000 4.54% 1.000 2.000
Y62 1.350 2.352 1.000 76.93% 1.000 1.000 1.000
1413.000 0.553 4.966 4.000 4.39% 1.000 2.000
Y63 1.548 1.594 1.000 60.81% 1.000 1.000 1.000
1411.000 0.667 2.006 4.000 5.39% 1.000 2.000
Y64 1.749 0.950 1.000 61.55% 1.000 1.000 1.000
1402.000 1.080 -0.618 4.000 8.13% 1.000 3.000
Y65 1.630 1.278 1.000 67.43% 1.000 1.000 1.000
1403.000 1.006 0.152 4.000 8.20% 1.000 3.000
Y66 1.941 0.707 1.000 55.12% 1.000 1.000 1.000
1348.000 1.348 -1.109 4.000 15.43% 2.000 3.000
FEMALE 0.584 -0.342 0.000 41.56% 0.000 0.000 1.000
2762.000 0.243 -1.883 1.000 58.44% 1.000 1.000
AGEC70 0.364 0.673 -0.300 34.65% -0.300 0.200 0.200
2762.000 0.419 -0.681 1.700 8.76% 0.200 1.200
THE MODEL ESTIMATION TERMINATED NORMALLY
MODEL FIT INFORMATION
Number of Free Parameters 38
Loglikelihood
H0 Value -42032.198
H1 Value -41231.306
Information Criteria
Akaike (AIC) 84140.396
Bayesian (BIC) 84365.497
Sample-Size Adjusted BIC 84244.759
(n* = (n + 2) / 24)
Chi-Square Test of Model Fit
Value 1601.785
Degrees of Freedom 187
P-Value 0.0000
RMSEA (Root Mean Square Error Of Approximation)
Estimate 0.052
90 Percent C.I. 0.050 0.055
Probability RMSEA <= .05 0.051
CFI/TLI
CFI 0.879
TLI 0.877
Chi-Square Test of Model Fit for the Baseline Model
Value 11854.776
Degrees of Freedom 189
P-Value 0.0000
SRMR (Standardized Root Mean Square Residual)
Value 0.049
MODEL RESULTS
Two-Tailed
Estimate S.E. Est./S.E. P-Value
FW0 BY
Y01 0.512 0.000 999.000 999.000
Y02 0.504 0.011 46.171 0.000
Y03 0.605 0.012 49.330 0.000
Y04 0.602 0.015 39.212 0.000
Y05 0.549 0.014 38.065 0.000
Y06 0.405 0.018 22.643 0.000
FW3 BY
Y31 0.512 0.000 999.000 999.000
Y32 0.504 0.011 46.171 0.000
Y33 0.605 0.012 49.330 0.000
Y34 0.602 0.015 39.212 0.000
Y35 0.549 0.014 38.065 0.000
Y36 0.405 0.018 22.643 0.000
FW6 BY
Y61 0.512 0.000 999.000 999.000
Y62 0.504 0.011 46.171 0.000
Y63 0.605 0.012 49.330 0.000
Y64 0.602 0.015 39.212 0.000
Y65 0.549 0.014 38.065 0.000
Y66 0.405 0.018 22.643 0.000
I BY
FW0 1.000 0.000 999.000 999.000
FW3 1.000 0.000 999.000 999.000
FW6 1.000 0.000 999.000 999.000
S BY
FW0 0.000 0.000 999.000 999.000
FW3 0.300 0.000 999.000 999.000
FW6 0.600 0.000 999.000 999.000
I ON
AGEC70 0.104 0.032 3.212 0.001
FEMALE 0.265 0.042 6.254 0.000
S ON
AGEC70 0.205 0.087 2.365 0.018
FEMALE 0.038 0.104 0.363 0.716
I WITH
S 0.009 0.078 0.110 0.912
Y01 WITH
Y31 0.028 0.006 4.476 0.000
Y61 0.013 0.009 1.417 0.157
Y31 WITH
Y61 0.028 0.006 4.476 0.000
Y02 WITH
Y32 0.008 0.007 1.264 0.206
Y62 0.004 0.009 0.412 0.680
Y32 WITH
Y62 0.008 0.007 1.264 0.206
Y03 WITH
Y33 0.025 0.007 3.342 0.001
Y63 0.014 0.010 1.325 0.185
Y33 WITH
Y63 0.025 0.007 3.342 0.001
Y04 WITH
Y34 0.052 0.014 3.803 0.000
Y64 0.092 0.019 4.945 0.000
Y34 WITH
Y64 0.052 0.014 3.803 0.000
Y05 WITH
Y35 0.073 0.012 6.049 0.000
Y65 0.047 0.017 2.850 0.004
Y35 WITH
Y65 0.073 0.012 6.049 0.000
Y06 WITH
Y36 0.168 0.026 6.557 0.000
Y66 0.099 0.036 2.729 0.006
Y36 WITH
Y66 0.168 0.026 6.557 0.000
Intercepts
Y01 1.358 0.000 999.000 999.000
Y02 1.241 0.010 121.140 0.000
Y03 1.386 0.012 120.192 0.000
Y04 1.562 0.014 109.603 0.000
Y05 1.422 0.013 105.906 0.000
Y06 1.861 0.017 106.982 0.000
Y31 1.358 0.000 999.000 999.000
Y32 1.241 0.010 121.140 0.000
Y33 1.386 0.012 120.192 0.000
Y34 1.562 0.014 109.603 0.000
Y35 1.422 0.013 105.906 0.000
Y36 1.861 0.017 106.982 0.000
Y61 1.358 0.000 999.000 999.000
Y62 1.241 0.010 121.140 0.000
Y63 1.386 0.012 120.192 0.000
Y64 1.562 0.014 109.603 0.000
Y65 1.422 0.013 105.906 0.000
Y66 1.861 0.017 106.982 0.000
FW0 0.000 0.000 999.000 999.000
FW3 0.000 0.000 999.000 999.000
FW6 0.000 0.000 999.000 999.000
I -0.008 0.036 -0.212 0.832
S 0.187 0.083 2.245 0.025
Residual Variances
Y01 0.286 0.007 43.204 0.000
Y02 0.281 0.007 42.980 0.000
Y03 0.283 0.008 37.245 0.000
Y04 0.622 0.014 45.493 0.000
Y05 0.565 0.012 45.807 0.000
Y06 1.236 0.024 50.962 0.000
Y31 0.286 0.007 43.204 0.000
Y32 0.281 0.007 42.980 0.000
Y33 0.283 0.008 37.245 0.000
Y34 0.622 0.014 45.493 0.000
Y35 0.565 0.012 45.807 0.000
Y36 1.236 0.024 50.962 0.000
Y61 0.286 0.007 43.204 0.000
Y62 0.281 0.007 42.980 0.000
Y63 0.283 0.008 37.245 0.000
Y64 0.622 0.014 45.493 0.000
Y65 0.565 0.012 45.807 0.000
Y66 1.236 0.024 50.962 0.000
FW0 0.552 0.031 17.887 0.000
FW3 0.552 0.031 17.887 0.000
FW6 0.552 0.031 17.887 0.000
I 0.482 0.042 11.558 0.000
S 0.146 0.231 0.632 0.527
QUALITY OF NUMERICAL RESULTS
Condition Number for the Information Matrix 0.513E-03
(ratio of smallest to largest eigenvalue)
Beginning Time: 12:32:33
Ending Time: 12:32:33
Elapsed Time: 00:00:00
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Copyright (c) 1998-2024 Muthen & MuthenSave output from Mplus Engine
file.copy("formplus.out", "milgcm.out", overwrite = TRUE)
file.copy("formplus.inp", "milgcm.inp", overwrite = TRUE)
milgcmResults <- MplusAutomation::readModels(target="milgcm.out")
milgcm_parameters <- milgcmResults$parameters$unstandardized %>%
mutate(PARAMETER = str_c(paramHeader,param, sep = " "))Mplus VERSION 8.11 (Mac)
MUTHEN & MUTHEN
04/10/2025 12:32 PM
INPUT INSTRUCTIONS
TITLE: LGCM CESD score
DATA: FILE = "cesdwide.dat";
VARIABLE: NAMES = id female agec70
y01 y02 y03 y04 y05 y06
y31 y32 y33 y34 y35 y36
y61 y62 y63 y64 y65 y66;
IDVARIABLE = id ;
MISSING = . ;
MODEL: fw0 BY y01@0.51150 (l1);
fw3 BY y31@0.51150 (l1);
fw6 BY y61@0.51150 (l1);
fw0 by y02-y06* (l2-l6);
fw3 by y32-y36* (l2-l6);
fw6 by y62-y66* (l2-l6);
[y01@1.35775];
[y31@1.35775];
[y61@1.35775];
[y02-y06*] (nu2-nu6) ;
[y32-y36*] (nu2-nu6) ;
[y62-y66*] (nu2-nu6) ;
y01 y31 y61 (te1) ;
y02 y32 y62 (te2) ;
y03 y33 y63 (te3) ;
y04 y34 y64 (te4) ;
y05 y35 y65 (te5) ;
y06 y36 y66 (te6) ;
! y01 with y31 (te11) ;
! y31 with y61 (te11) ;
! y02 with y32 (te22) ;
! y32 with y62 (te22) ;
! y03 with y33 (te33) ;
! y33 with y63 (te33) ;
! y04 with y34 (te44) ;
! y34 with y64 (te44) ;
! y05 with y35 (te55) ;
! y35 with y65 (te55) ;
! y06 with y36 (te66) ;
! y36 with y66 (te66) ;
! y01 with y61 ;
! y02 with y62 ;
! y03 with y63 ;
! y04 with y64 ;
! y05 with y65 ;
! y06 with y66 ;
[fw0@0] ;
[fw3@0] ;
[fw6@0] ;
fw0* (psi1) ;
fw3* (psi1) ;
fw6* (psi1) ;
i by fw0-fw6@1 ;
s by fw0@0 fw3@0.3 fw6@0.6 ; ! per decade
i with s ;
i s on agec70 female ;
[i*] ;
[s*] ;
INPUT READING TERMINATED NORMALLY
LGCM CESD score
SUMMARY OF ANALYSIS
Number of groups 1
Number of observations 2762
Number of dependent variables 18
Number of independent variables 2
Number of continuous latent variables 5
Observed dependent variables
Continuous
Y01 Y02 Y03 Y04 Y05 Y06
Y31 Y32 Y33 Y34 Y35 Y36
Y61 Y62 Y63 Y64 Y65 Y66
Observed independent variables
FEMALE AGEC70
Continuous latent variables
FW0 FW3 FW6 I S
Variables with special functions
ID variable ID
Estimator ML
Information matrix OBSERVED
Maximum number of iterations 1000
Convergence criterion 0.500D-04
Maximum number of steepest descent iterations 20
Maximum number of iterations for H1 2000
Convergence criterion for H1 0.100D-03
Input data file(s)
cesdwide.dat
Input data format FREE
SUMMARY OF DATA
Number of missing data patterns 140
COVARIANCE COVERAGE OF DATA
Minimum covariance coverage value 0.100
PROPORTION OF DATA PRESENT
Covariance Coverage
Y01 Y02 Y03 Y04 Y05
________ ________ ________ ________ ________
Y01 0.981
Y02 0.975 0.980
Y03 0.979 0.977 0.983
Y04 0.975 0.970 0.974 0.976
Y05 0.974 0.969 0.974 0.969 0.975
Y06 0.915 0.914 0.917 0.913 0.911
Y31 0.711 0.711 0.711 0.708 0.707
Y32 0.707 0.706 0.706 0.703 0.702
Y33 0.711 0.711 0.711 0.708 0.707
Y34 0.701 0.700 0.701 0.698 0.696
Y35 0.699 0.699 0.699 0.696 0.694
Y36 0.642 0.642 0.642 0.639 0.638
Y61 0.501 0.500 0.501 0.499 0.498
Y62 0.502 0.501 0.502 0.500 0.499
Y63 0.501 0.501 0.501 0.499 0.499
Y64 0.498 0.498 0.499 0.496 0.496
Y65 0.498 0.498 0.499 0.496 0.496
Y66 0.479 0.478 0.479 0.477 0.476
FEMALE 0.981 0.980 0.983 0.976 0.975
AGEC70 0.981 0.980 0.983 0.976 0.975
Covariance Coverage
Y06 Y31 Y32 Y33 Y34
________ ________ ________ ________ ________
Y06 0.919
Y31 0.665 0.724
Y32 0.662 0.715 0.720
Y33 0.665 0.722 0.716 0.725
Y34 0.657 0.712 0.706 0.711 0.714
Y35 0.655 0.710 0.703 0.710 0.702
Y36 0.603 0.652 0.648 0.652 0.646
Y61 0.467 0.485 0.484 0.485 0.481
Y62 0.468 0.487 0.485 0.486 0.482
Y63 0.467 0.486 0.485 0.486 0.482
Y64 0.465 0.483 0.482 0.483 0.479
Y65 0.465 0.484 0.482 0.483 0.480
Y66 0.447 0.464 0.462 0.463 0.459
FEMALE 0.919 0.724 0.720 0.725 0.714
AGEC70 0.919 0.724 0.720 0.725 0.714
Covariance Coverage
Y35 Y36 Y61 Y62 Y63
________ ________ ________ ________ ________
Y35 0.712
Y36 0.644 0.654
Y61 0.479 0.446 0.510
Y62 0.480 0.447 0.507 0.512
Y63 0.479 0.446 0.508 0.508 0.511
Y64 0.476 0.444 0.504 0.505 0.505
Y65 0.478 0.445 0.506 0.505 0.505
Y66 0.457 0.429 0.485 0.486 0.487
FEMALE 0.712 0.654 0.510 0.512 0.511
AGEC70 0.712 0.654 0.510 0.512 0.511
Covariance Coverage
Y64 Y65 Y66 FEMALE AGEC70
________ ________ ________ ________ ________
Y64 0.508
Y65 0.504 0.508
Y66 0.483 0.483 0.488
FEMALE 0.508 0.508 0.488 1.000
AGEC70 0.508 0.508 0.488 1.000 1.000
UNIVARIATE SAMPLE STATISTICS
UNIVARIATE HIGHER-ORDER MOMENT DESCRIPTIVE STATISTICS
Variable/ Mean/ Skewness/ Minimum/ % with Percentiles
Sample Size Variance Kurtosis Maximum Min/Max 20%/60% 40%/80% Median
Y01 1.458 1.812 1.000 65.26% 1.000 1.000 1.000
2709.000 0.545 3.142 4.000 3.77% 1.000 2.000
Y02 1.345 2.371 1.000 77.44% 1.000 1.000 1.000
2708.000 0.552 5.035 4.000 4.36% 1.000 2.000
Y03 1.522 1.718 1.000 63.55% 1.000 1.000 1.000
2716.000 0.686 2.339 4.000 6.04% 1.000 2.000
Y04 1.650 1.189 1.000 66.22% 1.000 1.000 1.000
2697.000 1.001 -0.071 4.000 7.45% 1.000 3.000
Y05 1.465 1.735 1.000 75.91% 1.000 1.000 1.000
2694.000 0.815 1.599 4.000 5.86% 1.000 2.000
Y06 1.998 0.643 1.000 56.87% 1.000 1.000 1.000
2539.000 1.543 -1.319 4.000 20.40% 2.000 4.000
Y31 1.469 1.826 1.000 65.07% 1.000 1.000 1.000
2001.000 0.575 3.102 4.000 4.35% 1.000 2.000
Y32 1.368 2.260 1.000 74.90% 1.000 1.000 1.000
1988.000 0.552 4.646 4.000 4.33% 1.000 2.000
Y33 1.502 1.764 1.000 65.05% 1.000 1.000 1.000
2003.000 0.667 2.508 4.000 5.64% 1.000 2.000
Y34 1.723 0.959 1.000 59.79% 1.000 1.000 1.000
1972.000 0.963 -0.499 4.000 6.03% 2.000 3.000
Y35 1.589 1.301 1.000 67.55% 1.000 1.000 1.000
1966.000 0.880 0.298 4.000 5.49% 1.000 3.000
Y36 1.870 0.837 1.000 56.81% 1.000 1.000 1.000
1806.000 1.266 -0.861 4.000 13.79% 2.000 3.000
Y61 1.505 1.716 1.000 61.60% 1.000 1.000 1.000
1409.000 0.584 2.786 4.000 4.54% 1.000 2.000
Y62 1.350 2.352 1.000 76.93% 1.000 1.000 1.000
1413.000 0.553 4.966 4.000 4.39% 1.000 2.000
Y63 1.548 1.594 1.000 60.81% 1.000 1.000 1.000
1411.000 0.667 2.006 4.000 5.39% 1.000 2.000
Y64 1.749 0.950 1.000 61.55% 1.000 1.000 1.000
1402.000 1.080 -0.618 4.000 8.13% 1.000 3.000
Y65 1.630 1.278 1.000 67.43% 1.000 1.000 1.000
1403.000 1.006 0.152 4.000 8.20% 1.000 3.000
Y66 1.941 0.707 1.000 55.12% 1.000 1.000 1.000
1348.000 1.348 -1.109 4.000 15.43% 2.000 3.000
FEMALE 0.584 -0.342 0.000 41.56% 0.000 0.000 1.000
2762.000 0.243 -1.883 1.000 58.44% 1.000 1.000
AGEC70 0.364 0.673 -0.300 34.65% -0.300 0.200 0.200
2762.000 0.419 -0.681 1.700 8.76% 0.200 1.200
THE MODEL ESTIMATION TERMINATED NORMALLY
MODEL FIT INFORMATION
Number of Free Parameters 26
Loglikelihood
H0 Value -42124.312
H1 Value -41231.306
Information Criteria
Akaike (AIC) 84300.624
Bayesian (BIC) 84454.640
Sample-Size Adjusted BIC 84372.030
(n* = (n + 2) / 24)
Chi-Square Test of Model Fit
Value 1786.012
Degrees of Freedom 199
P-Value 0.0000
RMSEA (Root Mean Square Error Of Approximation)
Estimate 0.054
90 Percent C.I. 0.051 0.056
Probability RMSEA <= .05 0.004
CFI/TLI
CFI 0.864
TLI 0.871
Chi-Square Test of Model Fit for the Baseline Model
Value 11854.776
Degrees of Freedom 189
P-Value 0.0000
SRMR (Standardized Root Mean Square Residual)
Value 0.052
MODEL RESULTS
Two-Tailed
Estimate S.E. Est./S.E. P-Value
FW0 BY
Y01 0.512 0.000 999.000 999.000
Y02 0.504 0.011 47.045 0.000
Y03 0.608 0.012 50.818 0.000
Y04 0.606 0.015 40.521 0.000
Y05 0.554 0.014 39.517 0.000
Y06 0.413 0.017 23.804 0.000
FW3 BY
Y31 0.512 0.000 999.000 999.000
Y32 0.504 0.011 47.045 0.000
Y33 0.608 0.012 50.818 0.000
Y34 0.606 0.015 40.521 0.000
Y35 0.554 0.014 39.517 0.000
Y36 0.413 0.017 23.804 0.000
FW6 BY
Y61 0.512 0.000 999.000 999.000
Y62 0.504 0.011 47.045 0.000
Y63 0.608 0.012 50.818 0.000
Y64 0.606 0.015 40.521 0.000
Y65 0.554 0.014 39.517 0.000
Y66 0.413 0.017 23.804 0.000
I BY
FW0 1.000 0.000 999.000 999.000
FW3 1.000 0.000 999.000 999.000
FW6 1.000 0.000 999.000 999.000
S BY
FW0 0.000 0.000 999.000 999.000
FW3 0.300 0.000 999.000 999.000
FW6 0.600 0.000 999.000 999.000
I ON
AGEC70 0.103 0.032 3.208 0.001
FEMALE 0.263 0.042 6.237 0.000
S ON
AGEC70 0.206 0.086 2.387 0.017
FEMALE 0.039 0.104 0.376 0.707
I WITH
S 0.000 0.077 -0.002 0.998
Intercepts
Y01 1.358 0.000 999.000 999.000
Y02 1.240 0.010 125.416 0.000
Y03 1.384 0.011 126.372 0.000
Y04 1.561 0.013 116.563 0.000
Y05 1.421 0.013 113.257 0.000
Y06 1.854 0.016 114.537 0.000
Y31 1.358 0.000 999.000 999.000
Y32 1.240 0.010 125.416 0.000
Y33 1.384 0.011 126.372 0.000
Y34 1.561 0.013 116.563 0.000
Y35 1.421 0.013 113.257 0.000
Y36 1.854 0.016 114.537 0.000
Y61 1.358 0.000 999.000 999.000
Y62 1.240 0.010 125.416 0.000
Y63 1.384 0.011 126.372 0.000
Y64 1.561 0.013 116.563 0.000
Y65 1.421 0.013 113.257 0.000
Y66 1.854 0.016 114.537 0.000
FW0 0.000 0.000 999.000 999.000
FW3 0.000 0.000 999.000 999.000
FW6 0.000 0.000 999.000 999.000
I -0.005 0.036 -0.127 0.899
S 0.184 0.083 2.212 0.027
Residual Variances
Y01 0.287 0.007 43.859 0.000
Y02 0.283 0.006 43.585 0.000
Y03 0.283 0.007 37.986 0.000
Y04 0.620 0.013 46.176 0.000
Y05 0.562 0.012 46.570 0.000
Y06 1.232 0.024 51.571 0.000
Y31 0.287 0.007 43.859 0.000
Y32 0.283 0.006 43.585 0.000
Y33 0.283 0.007 37.986 0.000
Y34 0.620 0.013 46.176 0.000
Y35 0.562 0.012 46.570 0.000
Y36 1.232 0.024 51.571 0.000
Y61 0.287 0.007 43.859 0.000
Y62 0.283 0.006 43.585 0.000
Y63 0.283 0.007 37.986 0.000
Y64 0.620 0.013 46.176 0.000
Y65 0.562 0.012 46.570 0.000
Y66 1.232 0.024 51.571 0.000
FW0 0.527 0.030 17.527 0.000
FW3 0.527 0.030 17.527 0.000
FW6 0.527 0.030 17.527 0.000
I 0.500 0.041 12.103 0.000
S 0.178 0.227 0.782 0.434
QUALITY OF NUMERICAL RESULTS
Condition Number for the Information Matrix 0.567E-03
(ratio of smallest to largest eigenvalue)
Beginning Time: 12:32:33
Ending Time: 12:32:33
Elapsed Time: 00:00:00
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In the remainder of this presentation I will tabulate comparable parameter estimates from the various models we have considered
Table 1. Growth model levels and slopes: intercept estimates
| Model | Level | se | Slope | se |
|---|---|---|---|---|
| MLM(Yz) | -0.177 | 0.027 | 0.196 | 0.069 |
| MLCFA | -0.005 | 0.031 | 0.184 | 0.074 |
| MLGOF | -0.005 | 0.031 | 0.184 | 0.074 |
| MILGCM | -0.008 | 0.036 | 0.187 | 0.083 |
Table 2. Growth model levels and slopes: residual variances
| Model | Level | se | Slope | se |
|---|---|---|---|---|
| MLM(Yz) | 0.4 | 0.04 | 0.328 | 0.22 |
| MLCFA | 0.5 | 0.058 | 0.177 | 0.296 |
| MLGOF | 0.5 | 0.058 | 0.178 | 0.296 |
| MILGCM | 0.482 | 0.042 | 0.146 | 0.231 |
MLM(Yz) residual variances are from the BETWEEN model part.
Table 3. Growth model LEVEL parameter: effect of agec70 and female sex
| Model | agec70 |
se | female |
se |
|---|---|---|---|---|
| MLM(Yz) | 0.113 | 0.03 | 0.217 | 0.036 |
| MLCFA | 0.103 | 0.033 | 0.263 | 0.041 |
| MLGOF | 0.103 | 0.033 | 0.263 | 0.041 |
| MILGCM | 0.104 | 0.032 | 0.265 | 0.042 |
Table 4. Growth model SLOPE parameter: effect of agec70 and female sex
| Model | agec70 |
se | female |
se |
|---|---|---|---|---|
| MLM(Yz) | 0.158 | 0.083 | -0.013 | 0.091 |
| MLCFA | 0.206 | 0.095 | 0.039 | 0.1 |
| MLGOF | 0.206 | 0.095 | 0.039 | 0.1 |
| MILGCM | 0.205 | 0.087 | 0.038 | 0.104 |
Table 5. Residual covariance of LEVEL and SLOPE
| Model | est | se |
|---|---|---|
| MLM(Yz) | -0.046 | 0.075 |
| MLCFA | 0 | 0.101 |
| MLGOF | 0 | 0.101 |
| MILGCM | 0.009 | 0.078 |
Table 6. Fit information
| MLFA | MLGOF | MILGCM no rescov | MILGCM | |
|---|---|---|---|---|
| Observations | 6165 | 6165 | 2762 | 2762 |
| Parameters | 26 | 26 | 26 | 38 |
| LL | -42123.918 | -42123.918 | -42124.312 | -42032.198 |
| LLCorrectionFactor | 1.6585 | 1.6585 | NA | NA |
| AIC | 84299.835 | 84299.835 | 84300.624 | 84140.396 |
| BIC | 84474.728 | 84474.728 | 84454.64 | 84365.497 |
| elapsed_time | 00:00:02 | 00:00:02 | 00:00:00 | 00:00:00 |
The multilevel confirmatory factor analysis (MLCFA) and multilevel growth of factors (MLGOF) models, as I have parameterized them, are equivalent. The MILGCM (no rescov) [the MILGCM model without the item-level residual covariances over time] has a very similar LL to these two models as well. The MILGCM provides the best fit according to loglikelihood (LL). The \(\chi^2\) difference test on LL for the MILGCM (no rescov) relative to MILGCM model indicates the residual covariances provide significant improvement in model fit (\(\chi^2 = 184, \text{df} = 12, P < .001\)).
Notes:
(1) As discussed previously, BIC from “wide” and “long” data layouts are not comparable.
(2) The multilevel model for the observed composite CESD score, MLM(Yz), is not comparable to the other models in terms of fit, and therefore is not tabulated.
(fin)
