difftest.chibarsq: Chi-Bar-Square Difference Test

Description

This function performs the chi-bar-square difference test to compare the random intercept cross-lagged panel model (RI-CLPM) and traditional cross-lagged panel model (CLPM) as discussed in Hamaker et al. (2015).

Usage

difftest.chibarsq(clpm, riclpm, alpha = 0.05, digits = 2, p.digits = 3,
                  write = NULL, append = TRUE, check = TRUE, output = TRUE)

Value

Returns an object of class misty.object, which is a list with following entries:

call: function call
type: type of analysis
model: data frame including all variables used in the analysis, i.e., indicators for the factor, grouping variable and cluster variable
args: specification of function arguments
model.fit: list of fitted lavaan objects specified in the argument riclpm and clpm
result: list with result tables, i.e., difftest for the chi-bar-square difference test and weights for the weights for the mixture of chi-square mixture distribution

Arguments

clpm: an object of class lavaan, i.e., a fitted random intercept cross-lagged panel model (RI-CLPM) with the variance and covariances of latent intercept factors fixed to zero. Note that a RI-CLPM with the variance of all random intercepts fixed to zero is statistically equivalent to the traditional cross-lagged panel model (CLPM).
riclpm: an object of class lavaan, i.e., a fitted random intercept cross-lagged panel model with variance and covariances of latent intercept factors freely estimated.
alpha: a numeric value indicating the type-I-risk, \(\alpha\).
digits: an integer value indicating the number of decimal places to be used for displaying results.
p.digits: an integer value indicating the number of decimal places to be used for displaying the p-values.
write: a character string naming a file for writing the output into either a text file with file extension ".txt" (e.g., "Output.txt") or Excel file with file extension ".xlsx" (e.g., "Output.xlsx"). If the file name does not contain any file extension, an Excel file will be written.
append: logical: if TRUE (default), output will be appended to an existing text file with extension .txt specified in write, if FALSE existing text file will be overwritten.
check: logical: if TRUE (default), argument specification is checked.
output: logical: if TRUE (default), output is shown on the console.

Author

Takuya Yanagida

Details

Random Intercept Cross-Lagged Panel Model (RI-CLPM): The RI-CLPM is an extension of the traditional cross-lagged panel model that disentangles the within-person process from stable between-person differences (Hamaker et al., 2015). In a bivariate RI-CLPM, each variable \(x\) and \(y\) is decomposed into a stable time-invariant trait-like component, captured with random intercept factors denoted by \(\kappa\) for variable \(x\) and \(\omega\) for variable \(y\) (see Figure 1 in Hamaker et al., 2015). Note that the CLPM is nested under the RI-CLPM, i.e., the RI-CLPM is statistically equivalent to the CLPM when fixing the variance of all random intercepts and their covariances to zero.
Chi-Bar-Square Difference Test: The \(\bar{\chi^2}\) difference test is used to compare the fit of the nested models CLPM and RI-CLPM based on a mixture of chi-square distributions to test the null hypothesis e.g., \(H_0: Var_{\kappa} = 0, Var_{\omega} = 0, Cov_{\kappa, \omega} = 0\). The chi-bar-square distribution is a weighted sum of different chi-square distributions with varying degrees of freedom resulting from parameters fixed at the boundaries of the parameter space as variances are non-negative values (Stoel et al., 2016).
Chi-Square Difference Test: The regular \(\chi^2\) difference test is conservative due to ignoring the mixture distribution, i.e., if it is statistically significant, we are certain that the chi-bar-square difference test will be significant too, while the reverse will not be the case. Accordingly, if researchers find it more important to detect a true CLPM than a true RI-CLPM, it is advised to use the regular chi-square difference test (Sukpan & Kuiper, 2026). It should also be mentioned that estimating a RI-CLPM when the CLPM is the true model may reduce statistical power due to estimating additional parameters, but it does not introduce bias (see Table 4 in Scott, 2021), while estimating a CLPM when the RI-CLPM is the true model introduces bias (see Table 3 in Scott, 2021).

References

Hamaker, E. L., Kuiper, R. M., & Grasman, R. P. (2015). A critique of the cross-lagged panel model. Psychological Methods, 20(1), 102-116. https://doi.org/10.1037/a0038889

Kuiper R (2026). ChiBarSq.DiffTest: Chi-bar-square difference test of the RI-CLPM versus the CLPM and more general. R package version 0.0.0.9000. https://github.com/rebeccakuiper/ChiBarSq.DiffTest

Mulder, J. D., & Hamaker, E. L. (2021). Three extensions of the random intercept cross-lagged panel model. Structural Equation Modeling: A Multidisciplinary Journal, 28(4), 638-648. https://doi.org/10.1080/10705511.2020.1784738

Scott, P. W. (2021). Accounting for time-varying inter-individual differences in trajectories when assessing cross-lagged models. Structural Equation Modeling, 28(3), 365-375. https://doi.org/10.1080/10705511.2020.1819815

Stoel, R. D., Garre, F. G., Dolan, C., & van den Wittenboer, G. (2006). On the likelihood ratio test in structural equation modeling when parameters are subject to boundary constraints. Psychological Methods, 11(4), 439-455. https://doi.org/10.1037/1082-989X.11.4.439

Sukpan, C., & Kuiper, R. M. (2026). Selecting the correct RI-CLPM using chi-square-type tests and AIC-type criteria. Structural Equation Modeling: A Multidisciplinary Journal, 1-14. https://doi.org/10.1080/10705511.2025.2592831

Examples

Run this code

if (FALSE) {
#————————————————————————————————————————————————————————————————————————————
# Step-wise Procedure (Sukpan & Kuiper, 2026)
#
# Note that only the first step is shown in this example:
# - CLPM versus RI-CLPM(Kappa)
# - CLPM versus RI-CLPM(Omega)
#
# Model specification based on code provided on the accompanying website of
# Mulder and Hamaker (2021)

#——————————————————————————————————————
## Model Specification: Cross-Lagged Panel Model (CLPM)
## i.e., Var(Kappa) = 0, Var(Omega) = 0, Cov(Kappa, Omega) = 0

mod.clpm <- '
  # Create between components (random intercepts)
  RIx =~ 1*x1 + 1*x2 + 1*x3
  RIy =~ 1*y1 + 1*y2 + 1*y3

  # Create within-person centered variables
  wx1 =~ 1*x1
  wx2 =~ 1*x2
  wx3 =~ 1*x3
  wy1 =~ 1*y1
  wy2 =~ 1*y2
  wy3 =~ 1*y3

  # Estimate lagged effects between within-person centered variables
  wx2 + wy2 ~ wx1 + wy1
  wx3 + wy3 ~ wx2 + wy2

  # Estimate covariance between within-person centered variables at first wave
  wx1 ~~ wy1 # Covariance

  # Estimate covariances between residuals of within-person centered variables
  wx2 ~~ wy2
  wx3 ~~ wy3

  # Fix variance and covariance of random intercepts to zero, i.e.,
  RIx ~~ 0*RIx
  RIy ~~ 0*RIy
  RIx ~~ 0*RIy

  # Estimate (residual) variance of within-person centered variables
  wx1 ~~ wx1
  wy1 ~~ wy1
  wx2 ~~ wx2
  wy2 ~~ wy2
  wx3 ~~ wx3
  wy3 ~~ wy3
'

#——————————————————————————————————————
## Model Specification: Random Intercept Cross-Lagged Panel Model RI-CLPM(Kappa)
## i.e., Var(Kappa) > 0, Var(Omega) = 0, Cov(Kappa, Omega) = 0

mod.ri.clpm.k <- '
  # Create between components (random intercepts)
  RIx =~ 1*x1 + 1*x2 + 1*x3
  RIy =~ 1*y1 + 1*y2 + 1*y3

  # Create within-person centered variables
  wx1 =~ 1*x1
  wx2 =~ 1*x2
  wx3 =~ 1*x3
  wy1 =~ 1*y1
  wy2 =~ 1*y2
  wy3 =~ 1*y3

  # Estimate lagged effects between within-person centered variables
  wx2 + wy2 ~ wx1 + wy1
  wx3 + wy3 ~ wx2 + wy2

  # Estimate covariance between within-person centered variables at first wave
  wx1 ~~ wy1

  # Estimate covariances between residuals of within-person centered variables
  wx2 ~~ wy2
  wx3 ~~ wy3

  # Fix variance of random intercept RIy and covariance with RIx to zero
  RIx ~~ RIx
  RIy ~~ 0*RIy
  RIx ~~ 0*RIy

  # Estimate (residual) variance of within-person centered variables
  wx1 ~~ wx1
  wy1 ~~ wy1
  wx2 ~~ wx2
  wy2 ~~ wy2
  wx3 ~~ wx3
  wy3 ~~ wy3
'

#——————————————————————————————————————
## Model Specification: Random Intercept Cross-Lagged Panel Model RI-CLPM(Omega)
## i.e., Var(Kappa) = 0, Var(Omega) > 0, Cov(Kappa, Omega) = 0

mod.ri.clpm.o <- '
  # Create between components (random intercepts)
  RIx =~ 1*x1 + 1*x2 + 1*x3
  RIy =~ 1*y1 + 1*y2 + 1*y3

  # Create within-person centered variables
  wx1 =~ 1*x1
  wx2 =~ 1*x2
  wx3 =~ 1*x3
  wy1 =~ 1*y1
  wy2 =~ 1*y2
  wy3 =~ 1*y3

  # Estimate lagged effects between within-person centered variables
  wx2 + wy2 ~ wx1 + wy1
  wx3 + wy3 ~ wx2 + wy2

  # Estimate covariance between within-person centered variables at first wave
  wx1 ~~ wy1 # Covariance

  # Estimate covariances between residuals of within-person centered variables
  wx2 ~~ wy2
  wx3 ~~ wy3

  # Fix variance of random intercept RIx and covariance with RIy to zero
  RIx ~~ 0*RIx
  RIy ~~ RIy
  RIx ~~ 0*RIy

  # Estimate (residual) variance of within-person centered variables
  wx1 ~~ wx1
  wy1 ~~ wy1
  wx2 ~~ wx2
  wy2 ~~ wy2
  wx3 ~~ wx3
  wy3 ~~ wy3
'

#——————————————————————————————————————
## Estimate Models
#
# Note that the example analysis cannot be conduct as the data set 'data'
# is not available.

# CLPM
fit.clpm <- lavaan(mod.clpm, data = data, estimator = "MLR")

# RI-CLPM(Kappa)
fit.ri.clpm.k <- lavaan(mod.ri.clpm.k, data = data, estimator = "MLR")

# RI-CLPM(Omega)
fit.ri.clpm.o <- lavaan(mod.ri.clpm.o, data = data, estimator = "MLR")

#···················
# Chi-Bar-Square Difference Test

# CLPM vs. RI-CLPM(Kappa)
difftest.chibarsq(fit.clpm, fit.ri.clpm.k)

# CLPM vs. RI-CLPM(Omega)
difftest.chibarsq(fit.clpm, fit.ri.clpm.o)
}

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