LGCM Quick Reference

Fast lookup for syntax, fit indices, and troubleshooting. For step-by-step learning, see the Walkthrough. For conceptual background, see the Overview.

Jump to: Syntax · Estimation · Extract Output · Model Comparison · Fit Indices · Errors & Fixes · Extensions · Resources

lavaan Syntax

Basic Linear Growth

model <- '
  intercept =~ 1*y1 + 1*y2 + 1*y3 + 1*y4 + 1*y5
  slope     =~ 0*y1 + 1*y2 + 2*y3 + 3*y4 + 4*y5
'
fit <- growth(model, data = data_wide)
summary(fit, fit.measures = TRUE, standardized = TRUE)

Equal Residual Variances

y1 ~~ rv*y1
y2 ~~ rv*y2
y3 ~~ rv*y3
y4 ~~ rv*y4
y5 ~~ rv*y5

Fixed Slope (No Variance)

slope ~~ 0*slope
intercept ~~ 0*slope

Predictor of Growth

intercept ~ predictor
slope ~ predictor

Quadratic Growth

intercept =~ 1*y1 + 1*y2 + 1*y3 + 1*y4 + 1*y5
slope     =~ 0*y1 + 1*y2 + 2*y3 + 3*y4 + 4*y5
quad      =~ 0*y1 + 1*y2 + 4*y3 + 9*y4 + 16*y5

Piecewise Growth

intercept =~ 1*y1 + 1*y2 + 1*y3 + 1*y4 + 1*y5
slope1    =~ 0*y1 + 1*y2 + 2*y3 + 2*y4 + 2*y5
slope2    =~ 0*y1 + 0*y2 + 0*y3 + 1*y4 + 2*y5

Latent Basis (Freed Loadings)

intercept =~ 1*y1 + 1*y2 + 1*y3 + 1*y4 + 1*y5
slope     =~ 0*y1 + y2 + y3 + y4 + 1*y5

Correlated Adjacent Residuals

y1 ~~ y2
y2 ~~ y3
y3 ~~ y4
y4 ~~ y5

Multigroup Growth

fit <- growth(model, data = data_wide, group = "group_var")

Estimation & Missing Data

Scenario	Estimator	Code
Complete data	ML (default)	`growth(model, data)`
Missing data (MAR)	FIML	`missing = "fiml"`
Non-normal data	MLR	`estimator = "MLR"`
Ordinal outcomes	WLSMV	`estimator = "WLSMV"`

FIML for missing data:

fit <- growth(model, data = data_wide, missing = "fiml")

Robust SEs (non-normality):

fit <- growth(model, data = data_wide, estimator = "MLR")

Ordinal outcomes:

fit <- growth(model, data = data_wide, estimator = "WLSMV")

Extract Output

Fit Indices

fitmeasures(fit, c("chisq", "df", "pvalue", "cfi", "rmsea", "srmr"))

Growth Factor Means

parameterEstimates(fit) %>%
  filter(op == "~1", lhs %in% c("intercept", "slope"))

Growth Factor Variances/Covariances

parameterEstimates(fit) %>%
  filter(op == "~~", lhs %in% c("intercept", "slope"),
         rhs %in% c("intercept", "slope"))

Standardized Estimates

parameterEstimates(fit, standardized = TRUE) %>%
  select(lhs, op, rhs, est, se, std.all)

R² (Variance Explained)

inspect(fit, "r2")

Residual Correlations

resid(fit, type = "cor")$cov %>% round(3)

Modification Indices

modindices(fit, sort = TRUE, minimum.value = 10)

Model Comparison

Nested Models (LRT)

anova(fit_constrained, fit_full)

Robust LRT (for MLR fits)

anova(fit1, fit2, method = "satorra.bentler.2001")

Information Criteria

AIC(fit1); AIC(fit2)  # Lower = better
BIC(fit1); BIC(fit2)  # Lower = better

Fit Indices

Index	Good	Acceptable	Interpretation
χ² p-value	> .05	> .01	Non-significant = adequate fit
CFI	≥ .95	≥ .90	Comparative fit (vs. null model)
RMSEA	≤ .06	≤ .08	Population misfit estimate
SRMR	≤ .08	≤ .10	Avg. residual correlation

Parameters

Parameter	Symbol	Interpretation
Intercept mean	μᵢ	Average score at Time = 0
Slope mean	μₛ	Average change per time unit
Intercept variance	ψᵢᵢ	Individual differences in starting level
Slope variance	ψₛₛ	Individual differences in change rate
I-S covariance	ψᵢₛ	Start-change relationship
Residual variance	θ	Unexplained variance at each wave

Time Coding

Centering	Loadings	Intercept =
Wave 1 (default)	0, 1, 2, 3, 4	Score at Wave 1
Midpoint	-2, -1, 0, 1, 2	Score at Wave 3
Final wave	-4, -3, -2, -1, 0	Score at Wave 5
Actual months	0, 3, 6, 12, 24	Score at baseline

Slope interpretation unchanged by centering; intercept moves to the zero-point.

Common Errors & Fixes

Issue	Symptom	Fix
Negative variance	`slope variance = -2.4`	Constrain to 0 or simplify model
Non-convergence	"Model did not converge"	Check N, outliers; provide start values
Huge SEs	SE > estimate	Parameter poorly identified; simplify
Poor fit	CFI < .90, RMSEA > .10	Check linearity; consider quadratic
Variable name mismatch	lavaan error about variables	Use `names(data)` to verify spelling

Troubleshooting

Non-Positive Definite Matrix

Very high correlations between time points
Variance estimates near zero
Try constraining residual variances equal

Negative Variance (Heywood Case)

slope ~~ 0*slope           # Constrain to zero
intercept ~~ 0*slope       # Also fix covariance

Non-Convergence

Increase iterations: control = list(iter.max = 10000)
Check sample size relative to model complexity
Simplify model (fewer free parameters)

Flag Problematic Estimates

parameterEstimates(fit) %>%
  filter(op == "~~", est < 0)

Interpretation Pitfalls

Pitfall

These mistakes are common but avoidable:

Mistake	Reality
"Everyone improved" (slope mean = 2)	Check slope variance—many may have slope ≤ 0
"High starters declined" (negative I-S corr)	Negative correlation = slower growth, not decline
"Good fit = correct model"	Good fit ≠ true model; consider alternatives
Comparing intercepts across studies	Intercept meaning depends on time coding

For detailed examples and fixes, see LGCM Overview.

Quick Formulas

Intercept-slope correlation:

r_is <- cov_is / sqrt(var_i * var_s)

Proportion with positive slopes:

pnorm(0, mean = slope_mean, sd = sqrt(slope_var), lower.tail = FALSE)

Degrees of freedom:

fitmeasures(fit, "df")

Advanced Extensions

Time-Invariant Predictors (TIC)

Baseline characteristics predicting growth:

model <- '
  intercept =~ 1*y1 + 1*y2 + 1*y3 + 1*y4 + 1*y5
  slope     =~ 0*y1 + 1*y2 + 2*y3 + 3*y4 + 4*y5

  intercept ~ treatment + age
  slope     ~ treatment + age
'

Coefficient	Interpretation
`intercept ~ treatment`	Group difference in starting level
`slope ~ treatment`	Group difference in rate of change

Time-Varying Covariates (TVC)

Concurrent predictors at each wave:

model <- '
  intercept =~ 1*y1 + 1*y2 + 1*y3 + 1*y4 + 1*y5
  slope     =~ 0*y1 + 1*y2 + 2*y3 + 3*y4 + 4*y5

  y1 ~ x1
  y2 ~ x2
  y3 ~ x3
  y4 ~ x4
  y5 ~ x5
'

Other Extensions

Extension	Description	Notes
Growth Mixture Models	Latent subgroups with different trajectories	Complex; requires specialized packages
Parallel Process	Two outcomes growing together	Relate intercepts/slopes across processes
Latent Change Score	Change between adjacent waves	Tests dynamic coupling

Path Diagram

              ┌─────────────┐          ┌─────────────┐
              │  Intercept  │          │    Slope    │
              │     (I)     │◄────────►│     (S)     │
              └──────┬──────┘          └──────┬──────┘
                     │                        │
     ┌───────┬───────┼───────┬───────┐       │
     │1      │1      │1      │1      │1      │
     ▼       ▼       ▼       ▼       ▼       │
   ┌───┐   ┌───┐   ┌───┐   ┌───┐   ┌───┐    │
   │y1 │   │y2 │   │y3 │   │y4 │   │y5 │    │
   └─┬─┘   └─┬─┘   └─┬─┘   └─┬─┘   └─┬─┘    │
     ▲       ▲       ▲       ▲       ▲       │
     │0      │1      │2      │3      │4      │
     └───────┴───────┴───────┴───────┴───────┘

Intercept loadings: all = 1
Slope loadings: 0, 1, 2, 3, 4 (encode time)

Resources

Open-access resources for learning and applying LGCM.

Articles

Article	Focus
Duncan & Duncan (2004). The ABC's of LGM: An Introductory Guide to Latent Variable Growth Curve Modeling.	Conceptual introduction to LGCM — intercept/slope factors, path diagrams, growth parameters
Biesanz, Deeb-Sossa, Papadakis, Bollen, & Curran (2004). The Role of Coding Time in Estimating and Interpreting Growth Curve Models.	How time coding choices affect intercept meaning and growth factor (co)variances

Online

lavaan tutorial: Growth models
QuantDev tutorials
CenterStat (YouTube — Curran & Bauer lectures on longitudinal methods)

Software

Package	Use Case
lavaan	Primary choice; free, flexible, active development
OpenMx	Maximum flexibility; steeper learning curve
lme4 / nlme	MLM approach; equivalent for basic models
semTools	lavaan extensions (measurement invariance)
lcmm	Latent class / mixture growth models
Mplus	Gold standard for complex/mixture models; licensed
Stata (sem)	Good documentation; integrates with Stata workflow

Tutorial	Focus
LGCM Basic	Linear growth with lavaan
LGCM Basic (OpenMx)	Linear growth with OpenMx
LGCM Multiple Groups	Group comparisons
LGCM Time-Invariant Covariates	Predictors of growth
LGCM Time-Invariant Covariates (OpenMx)	Predictors of growth with OpenMx
LGCM Nesting	Nested/clustered data
MLGCM	Multivariate growth
MLGCM (OpenMx)	Multivariate growth with OpenMx

LGCM Overview — When to use, key concepts, mathematical notation
LGCM Walkthrough — Step-by-step worked example
LMM Overview — Linear mixed models for continuous outcomes
GLMM Overview — Generalized mixed models for non-continuous outcomes
GEE Overview — Population-averaged alternative to mixed models

LGCM Quick Reference

lavaan Syntax

Basic Linear Growth

Equal Residual Variances

Fixed Slope (No Variance)

Predictor of Growth

Quadratic Growth

Piecewise Growth

Latent Basis (Freed Loadings)

Correlated Adjacent Residuals

Multigroup Growth

Estimation & Missing Data

Extract Output

Fit Indices

Growth Factor Means

Growth Factor Variances/Covariances

Standardized Estimates

R² (Variance Explained)

Residual Correlations

Modification Indices

Model Comparison

Nested Models (LRT)

Robust LRT (for MLR fits)

Information Criteria

Fit Indices

Parameters

Time Coding

Common Errors & Fixes

Troubleshooting

Non-Positive Definite Matrix

Negative Variance (Heywood Case)

Non-Convergence

Flag Problematic Estimates

Interpretation Pitfalls

Quick Formulas

Advanced Extensions

Time-Invariant Predictors (TIC)

Time-Varying Covariates (TVC)

Other Extensions

Path Diagram

Resources

Articles

Online

Software

Related Tutorials

Related Guides