Please help with Heywood cases and poor fit for latent growth curve model

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Jennifer S

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Sep 26, 2025, 3:58:29 PM (14 days ago) Sep 26

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Hi all,

A team member and I could use some additional help as we are trying to run analyses with a second-order latent growth curve model. After trying all we can come up with on our end and speaking with our department statistician, we still have significant negative variances and fit indices that do not meet pre-specified criteria and are at a dead end.

Here is the code and fit indices for our preregistered model specification:

LGC_cog <-'
Cog_T1=~1*immc_m0_z3+p1*delmc_m0_z3+p2*ccc_m0_z3
Cog_T2=~1*immc_m6_z3+p1*delmc_m6_z3+p2*ccc_m6_z3
Cog_T3=~1*immc_m18_z3+p1*delmc_m18_z3+p2*ccc_m18_z3

# Estimates
intercept=~1*Cog_T1+1*Cog_T2+1*Cog_T3
slope=~-3*Cog_T1+-2*Cog_T2+0*Cog_T3

# Intercepts
intercept ~1
slope ~1
Cog_T1 ~ 0*1
Cog_T2 ~ 0*1
Cog_T3 ~ 0*1
immc_m0_z3 ~ 0*1
immc_m6_z3 ~ 0*1
immc_m18_z3 ~ 0*1
delmc_m0_z3 ~ 0*1
delmc_m6_z3 ~ 0*1
delmc_m18_z3 ~ 0*1
ccc_m0_z3 ~ 0*1
ccc_m6_z3 ~ 0*1
ccc_m18_z3 ~ 0*1

# Variances
immc_m0_z3~~v1*immc_m0_z3
delmc_m0_z3~~v2*delmc_m0_z3
ccc_m0_z3~~v3*ccc_m0_z3

immc_m6_z3~~v1*immc_m6_z3
delmc_m6_z3~~v2*delmc_m6_z3
ccc_m6_z3~~v3*ccc_m6_z3

immc_m18_z3~~v1*immc_m18_z3
delmc_m18_z3~~v2*delmc_m18_z3
ccc_m18_z3~~v3*ccc_m18_z3

intercept ~~ intercept
slope ~~ slope
intercept ~~ slope
Cog_T1 ~~ Cog_T1
Cog_T2 ~~ Cog_T2
Cog_T3 ~~ Cog_T3
'

Fit indices for original model: CFI 0.708, RMSEA 0.229, SRMR 0.169; significant negative variances for Cog_T1 (-0.174) and Cog_T3 (-0.632)

Since then, we have removed measurement invariance, added covariance terms between immc&delmc due to high correlation, and added covariances within indicators to improve model fit, which you will see reflected in the code for the most up-to-date model below. We have also tried changing the estimator to MLR, relaxed the constraint on equal residual covariance (cv), relaxed the constraint on equal residual variances (v1, v2, v3), and relaxed cv and v1/v2/v3 in the same model. These did not improve fit nor fix negative variances and thus are not included in the most up-to-date model specification. Below is the current model specification and output - does anyone have any suggestions for additional things we could try regarding model fit and/or negative variances? Thank you very much!

LGC_cog <-'
Cog_T1=~1*immc_m0_z3+delmc_m0_z3+ccc_m0_z3
Cog_T2=~1*immc_m6_z3+delmc_m6_z3+ccc_m6_z3
Cog_T3=~1*immc_m18_z3+delmc_m18_z3+ccc_m18_z3

# Estimates
intercept=~1*Cog_T1+1*Cog_T2+1*Cog_T3
slope=~-3*Cog_T1+-2*Cog_T2+0*Cog_T3

# Intercepts
intercept ~1
slope ~1
Cog_T1 ~ 0*1
Cog_T2 ~ 0*1
Cog_T3 ~ 0*1
immc_m0_z3 ~ 0*1
immc_m6_z3 ~ 0*1
immc_m18_z3 ~ 0*1
delmc_m0_z3 ~ 0*1
delmc_m6_z3 ~ 0*1
delmc_m18_z3 ~ 0*1
ccc_m0_z3 ~ 0*1
ccc_m6_z3 ~ 0*1
ccc_m18_z3 ~ 0*1

# Covariances (immc and delmc are highly correlated r = ~0.8)
immc_m0_z3~~cv*delmc_m0_z3
immc_m6_z3~~cv*delmc_m6_z3
immc_m18_z3~~cv*delmc_m18_z3

# Covariances Within Indicators
ccc_m0_z3~~ccc_m6_z3
ccc_m0_z3~~ccc_m18_z3
ccc_m6_z3~~ccc_m18_z3

delmc_m0_z3~~delmc_m6_z3
delmc_m0_z3~~delmc_m18_z3
delmc_m6_z3~~delmc_m18_z3

immc_m0_z3~~immc_m6_z3
immc_m0_z3~~immc_m18_z3
immc_m6_z3~~immc_m18_z3

# Variances

immc_m0_z3~~v1*immc_m0_z3
delmc_m0_z3~~v2*delmc_m0_z3
ccc_m0_z3~~v3*ccc_m0_z3

immc_m6_z3~~v1*immc_m6_z3
delmc_m6_z3~~v2*delmc_m6_z3
ccc_m6_z3~~v3*ccc_m6_z3

immc_m18_z3~~v1*immc_m18_z3
delmc_m18_z3~~v2*delmc_m18_z3
ccc_m18_z3~~v3*ccc_m18_z3

intercept ~~ intercept
slope ~~ slope
intercept ~~ slope
Cog_T1 ~~ Cog_T1
Cog_T2 ~~ Cog_T2
Cog_T3 ~~ Cog_T3
'

fit_LGC_cog <- sem(LGC_cog, data=df, estimator='ML',missing='fiml')

OUTPUT

summary(fit_LGC_cog, fit.measures=TRUE, standardized=TRUE, rsquare=TRUE)
lavaan 0.6-19 ended normally after 59 iterations

Estimator ML
Optimization method NLMINB
Number of model parameters 35
Number of equality constraints 8

Number of observations 290
Number of missing patterns 5

Model Test User Model:

Test statistic 119.495
Degrees of freedom 27
P-value (Chi-square) 0.000

Model Test Baseline Model:

Test statistic 2167.413
Degrees of freedom 36
P-value 0.000

User Model versus Baseline Model:

Comparative Fit Index (CFI) 0.957
Tucker-Lewis Index (TLI) 0.942

Robust Comparative Fit Index (CFI) 0.959
Robust Tucker-Lewis Index (TLI) 0.945

Loglikelihood and Information Criteria:

Loglikelihood user model (H0) -2288.298
Loglikelihood unrestricted model (H1) -2228.550

Akaike (AIC) 4630.596
Bayesian (BIC) 4729.682
Sample-size adjusted Bayesian (SABIC) 4644.060

Root Mean Square Error of Approximation:

RMSEA 0.109
90 Percent confidence interval - lower 0.089
90 Percent confidence interval - upper 0.129
P-value H_0: RMSEA <= 0.050 0.000
P-value H_0: RMSEA >= 0.080 0.992

Robust RMSEA 0.112
90 Percent confidence interval - lower 0.091
90 Percent confidence interval - upper 0.134
P-value H_0: Robust RMSEA <= 0.050 0.000
P-value H_0: Robust RMSEA >= 0.080 0.994

Standardized Root Mean Square Residual:

SRMR 0.104

Parameter Estimates:

Standard errors Standard
Information Observed
Observed information based on Hessian

Latent Variables:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
Cog_T1 =~
immc_m0_z3 1.000 0.610 0.826
delmc_m0_z3 1.043 0.064 16.389 0.000 0.636 0.727
ccc_m0_z3 0.842 0.089 9.507 0.000 0.513 0.511
Cog_T2 =~
immc_m6_z3 1.000 1.115 0.937
delmc_m6_z3 0.790 0.042 19.001 0.000 0.882 0.826
ccc_m6_z3 0.475 0.048 9.922 0.000 0.530 0.523
Cog_T3 =~
immc_m18_z3 1.000 0.974 0.920
delmc_m18_z3 0.887 0.047 18.933 0.000 0.864 0.821
ccc_m18_z3 0.481 0.055 8.723 0.000 0.469 0.477
intercept =~
Cog_T1 1.000 2.325 2.325
Cog_T2 1.000 1.271 1.271
Cog_T3 1.000 1.455 1.455
slope =~
Cog_T1 -3.000 -2.041 -2.041
Cog_T2 -2.000 -0.744 -0.744
Cog_T3 0.000 0.000 0.000

Covariances:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
.immc_m0_z3 ~~
.dlmc_m0_3 (cv) 0.095 0.017 5.497 0.000 0.095 0.379
.immc_m6_z3 ~~
.dlmc_m6_3 (cv) 0.095 0.017 5.497 0.000 0.095 0.379
.immc_m18_z3 ~~
.dlmc_18_3 (cv) 0.095 0.017 5.497 0.000 0.095 0.379
.ccc_m0_z3 ~~
.ccc_m6_z3 0.537 0.054 9.931 0.000 0.537 0.720
.ccc_m18_3 0.494 0.055 9.021 0.000 0.494 0.663
.ccc_m6_z3 ~~
.ccc_m18_3 0.545 0.056 9.815 0.000 0.545 0.731
.delmc_m0_z3 ~~
.dlmc_m6_3 0.161 0.028 5.736 0.000 0.161 0.445
.dlmc_18_3 0.140 0.030 4.700 0.000 0.140 0.386
.delmc_m6_z3 ~~
.dlmc_18_3 0.164 0.029 5.595 0.000 0.164 0.453
.immc_m0_z3 ~~
.immc_m6_3 -0.038 0.027 -1.415 0.157 -0.038 -0.220
.immc_18_3 -0.005 0.028 -0.198 0.843 -0.005 -0.032
.immc_m6_z3 ~~
.immc_18_3 -0.029 0.033 -0.869 0.385 -0.029 -0.166
intercept ~~
slope 0.477 0.065 7.354 0.000 0.811 0.811

Intercepts:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
intercept -0.021 0.065 -0.324 0.746 -0.015 -0.015
slope 0.130 0.016 8.400 0.000 0.314 0.314
.Cog_T1 0.000 0.000 0.000
.Cog_T2 0.000 0.000 0.000
.Cog_T3 0.000 0.000 0.000
.immc_m0_z3 0.000 0.000 0.000
.immc_m6_z3 0.000 0.000 0.000
.immc_m18_z3 0.000 0.000 0.000
.delmc_m0_z3 0.000 0.000 0.000
.delmc_m6_z3 0.000 0.000 0.000
.delmc_m18_z3 0.000 0.000 0.000
.ccc_m0_z3 0.000 0.000 0.000
.ccc_m6_z3 0.000 0.000 0.000
.ccc_m18_z3 0.000 0.000 0.000

Variances:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
.immc_m0_3 (v1) 0.173 0.031 5.575 0.000 0.173 0.317
.dlmc_m0_3 (v2) 0.362 0.034 10.795 0.000 0.362 0.472
.ccc_m0_z3 (v3) 0.746 0.053 13.957 0.000 0.746 0.739
.immc_m6_3 (v1) 0.173 0.031 5.575 0.000 0.173 0.122
.dlmc_m6_3 (v2) 0.362 0.034 10.795 0.000 0.362 0.318
.ccc_m6_z3 (v3) 0.746 0.053 13.957 0.000 0.746 0.727
.immc_18_3 (v1) 0.173 0.031 5.575 0.000 0.173 0.154
.dlmc_18_3 (v2) 0.362 0.034 10.795 0.000 0.362 0.327
.ccc_m18_3 (v3) 0.746 0.053 13.957 0.000 0.746 0.772
intercept 2.010 0.229 8.769 0.000 1.000 1.000
slope 0.172 0.024 7.215 0.000 1.000 1.000
.Cog_T1 -0.326 0.049 -6.631 0.000 -0.877 -0.877
.Cog_T2 0.452 0.064 7.036 0.000 0.364 0.364
.Cog_T3 -1.061 0.157 -6.752 0.000 -1.118 -1.118

R-Square:
Estimate
immc_m0_z3 0.683
delmc_m0_z3 0.528
ccc_m0_z3 0.261
immc_m6_z3 0.878
delmc_m6_z3 0.682
ccc_m6_z3 0.273
immc_m18_z3 0.846
delmc_m18_z3 0.673
ccc_m18_z3 0.228
Cog_T1 NA
Cog_T2 0.636
Cog_T3 NA

Edward Rigdon

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Sep 26, 2025, 4:25:36 PM (14 days ago) Sep 26

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Some thoughts--maybe not useful, but at least you know people are paying attention...

Your time model is linear but the systematic rise in residual variances / decline in R2 across time makes me wonder if there is nonlinearity. Have you plotted these data and is the linear model plausible? Is there a time-varying covariate that might partially account for changing behavior across time?

You need more time points to actually model, say, a quadratic time term, but as an approximation, you could play with different time intervals. Right now your data says month 0, month 6 and month 18 so you label time points -3, -2 and 0. Suppose instead of 0 you use something like +3 (that's a W.A.G.), but maybe try +1, +3, +5 and see if things get better.

Do your constituent factor models behave well? You only have three indicators each, and a factor model of just 3 indicators is saturated, but you still would have estimates of loadings and residual variances--you could see if those make sense at that very basic level.

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Stas Kolenikov

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Sep 27, 2025, 6:14:24 AM (13 days ago) Sep 27

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If you drop the model over time and just consider this as a CFA model with three factors, how does that fit?

What is the nature of the indicators that you have? Are those test scores that are kinda sorta continuous? Are those Likert scales with five values?

-- Stas Kolenikov, PhD, PStat (ASA, SSC)

-- Principal Statistician, NORC @NORCnews

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Jennifer S

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Sep 30, 2025, 1:07:10 PM (10 days ago) Sep 30

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Thank you both for your input so far!

- We have inspected and plotted the data, and cognition over time seems linear

- all cognitive variables are continuous, originally on a 0-100 scale

- We did conduct CFAs, where everything appeared normal. Below is the code and output for that cognition CFA. Any additional inputs or advice would be greatly appreciated!

model_cog_all <- '

Cog_T1 =~ immc_m0_z3 + delmc_m0_z3 + ccc_m0_z3

immc_m0_z3 ~~ immc_m0_z3

delmc_m0_z3 ~~ delmc_m0_z3

ccc_m0_z3 ~~ ccc_m0_z3

Cog_T2 =~ immc_m6_z3 + delmc_m6_z3 + ccc_m6_z3

immc_m6_z3 ~~ immc_m6_z3

delmc_m6_z3 ~~ delmc_m6_z3

ccc_m6_z3 ~~ ccc_m6_z3

Cog_T3 =~ immc_m18_z3 + delmc_m18_z3 + ccc_m18_z3

immc_m18_z3 ~~ immc_m18_z3

delmc_m18_z3 ~~ delmc_m18_z3

ccc_m18_z3 ~~ ccc_m18_z3

fit_cog_all <- cfa(model_cog_all, data = df)

summary(fit_cog_all, fit.measures=TRUE)

lavaan 0.6-19 ended normally after 36 iterations

Estimator ML

Optimization method NLMINB

Number of model parameters 21

Used Total

Number of observations 229 290

Model Test User Model:

Test statistic 507.854

Degrees of freedom 24

P-value (Chi-square) 0.000

Model Test Baseline Model:

Test statistic 1913.559

Degrees of freedom 36

P-value 0.000

User Model versus Baseline Model:

Comparative Fit Index (CFI) 0.742

Tucker-Lewis Index (TLI) 0.613

Loglikelihood and Information Criteria:

Loglikelihood user model (H0) -2131.044

Loglikelihood unrestricted model (H1) -1877.117

Akaike (AIC) 4304.088

Bayesian (BIC) 4376.197

Sample-size adjusted Bayesian (SABIC) 4309.640

Root Mean Square Error of Approximation:

RMSEA 0.297

90 Percent confidence interval - lower 0.275

90 Percent confidence interval - upper 0.319

P-value H_0: RMSEA <= 0.050 0.000

P-value H_0: RMSEA >= 0.080 1.000

Standardized Root Mean Square Residual:

SRMR 0.155

Parameter Estimates:

Standard errors Standard

Information Expected

Information saturated (h1) model Structured

Latent Variables:

Estimate Std.Err z-value P(>|z|)

Cog_T1 =~

immc_m0_z3 1.000

delmc_m0_z3 1.149 0.061 18.870 0.000

ccc_m0_z3 0.608 0.101 6.027 0.000

Cog_T2 =~

immc_m6_z3 1.000

delmc_m6_z3 0.927 0.049 18.943 0.000

ccc_m6_z3 0.489 0.061 8.055 0.000

Cog_T3 =~

immc_m18_z3 1.000

delmc_m18_z3 0.993 0.047 20.915 0.000

ccc_m18_z3 0.449 0.062 7.213 0.000

Covariances:

Estimate Std.Err z-value P(>|z|)

Cog_T1 ~~

Cog_T2 0.578 0.065 8.949 0.000

Cog_T3 0.553 0.062 8.931 0.000

Cog_T2 ~~

Cog_T3 0.866 0.093 9.327 0.000

Variances:

Estimate Std.Err z-value P(>|z|)

.immc_m0_z3 0.121 0.017 7.120 0.000

.delmc_m0_z3 0.132 0.021 6.367 0.000

.ccc_m0_z3 0.877 0.083 10.534 0.000

.immc_m6_z3 0.215 0.032 6.801 0.000

.delmc_m6_z3 0.265 0.033 8.111 0.000

.ccc_m6_z3 0.687 0.066 10.444 0.000

.immc_m18_z3 0.191 0.027 7.014 0.000

.delmc_m18_z3 0.198 0.027 7.189 0.000

.ccc_m18_z3 0.686 0.065 10.511 0.000

Cog_T1 0.432 0.052 8.314 0.000

Cog_T2 0.947 0.109 8.651 0.000

Cog_T3 0.880 0.101 8.743 0.000

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;]]

Jennifer S

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Sep 30, 2025, 1:07:18 PM (10 days ago) Sep 30

to lavaan

Thanks to you both!

- We did plot the data, and the relationship between cognition and time seems linearly related.

- We will post the CFA for cognition, with its 3 indicators at all 3 timepoints, below. We think the loadings and residual variances seem to be fine.

- Each indicator is a continuous score, originally on a scale of 0-100.

Please let us know if this helps come up with any additional things for us to try or consider, and thank you again!

model_cog_all <- '

Cog_T1 =~ immc_m0_z3 + delmc_m0_z3 + ccc_m0_z3

immc_m0_z3 ~~ immc_m0_z3

delmc_m0_z3 ~~ delmc_m0_z3

ccc_m0_z3 ~~ ccc_m0_z3

Cog_T2 =~ immc_m6_z3 + delmc_m6_z3 + ccc_m6_z3

immc_m6_z3 ~~ immc_m6_z3

delmc_m6_z3 ~~ delmc_m6_z3

ccc_m6_z3 ~~ ccc_m6_z3

Cog_T3 =~ immc_m18_z3 + delmc_m18_z3 + ccc_m18_z3

immc_m18_z3 ~~ immc_m18_z3

delmc_m18_z3 ~~ delmc_m18_z3

ccc_m18_z3 ~~ ccc_m18_z3

fit_cog_all <- cfa(model_cog_all, data = df)

summary(fit_cog_all, fit.measures=TRUE)

lavaan 0.6-19 ended normally after 36 iterations

Estimator ML

Optimization method NLMINB

Number of model parameters 21

Used Total

Number of observations 229 290

Model Test User Model:

Test statistic 507.854

Degrees of freedom 24

P-value (Chi-square) 0.000

Model Test Baseline Model:

Test statistic 1913.559

Degrees of freedom 36

P-value 0.000

User Model versus Baseline Model:

Comparative Fit Index (CFI) 0.742

Tucker-Lewis Index (TLI) 0.613

Loglikelihood and Information Criteria:

Loglikelihood user model (H0) -2131.044

Loglikelihood unrestricted model (H1) -1877.117

Akaike (AIC) 4304.088

Bayesian (BIC) 4376.197

Sample-size adjusted Bayesian (SABIC) 4309.640

Root Mean Square Error of Approximation:

RMSEA 0.297

90 Percent confidence interval - lower 0.275

90 Percent confidence interval - upper 0.319

P-value H_0: RMSEA <= 0.050 0.000

P-value H_0: RMSEA >= 0.080 1.000

Standardized Root Mean Square Residual:

SRMR 0.155

Parameter Estimates:

Standard errors Standard

Information Expected

Information saturated (h1) model Structured

Latent Variables:

Estimate Std.Err z-value P(>|z|)

Cog_T1 =~

immc_m0_z3 1.000

delmc_m0_z3 1.149 0.061 18.870 0.000

ccc_m0_z3 0.608 0.101 6.027 0.000

Cog_T2 =~

immc_m6_z3 1.000

delmc_m6_z3 0.927 0.049 18.943 0.000

ccc_m6_z3 0.489 0.061 8.055 0.000

Cog_T3 =~

immc_m18_z3 1.000

delmc_m18_z3 0.993 0.047 20.915 0.000

ccc_m18_z3 0.449 0.062 7.213 0.000

Covariances:

Estimate Std.Err z-value P(>|z|)

Cog_T1 ~~

Cog_T2 0.578 0.065 8.949 0.000

Cog_T3 0.553 0.062 8.931 0.000

Cog_T2 ~~

Cog_T3 0.866 0.093 9.327 0.000

Variances:

Estimate Std.Err z-value P(>|z|)

.immc_m0_z3 0.121 0.017 7.120 0.000

.delmc_m0_z3 0.132 0.021 6.367 0.000

.ccc_m0_z3 0.877 0.083 10.534 0.000

.immc_m6_z3 0.215 0.032 6.801 0.000

.delmc_m6_z3 0.265 0.033 8.111 0.000

.ccc_m6_z3 0.687 0.066 10.444 0.000

.immc_m18_z3 0.191 0.027 7.014 0.000

.delmc_m18_z3 0.198 0.027 7.189 0.000

.ccc_m18_z3 0.686 0.065 10.511 0.000

Cog_T1 0.432 0.052 8.314 0.000

Cog_T2 0.947 0.109 8.651 0.000

Cog_T3 0.880 0.101 8.743 0.000

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