Team Activity 2 Final
Sarah Katie Hood and Hsin Wei Chen
2025-10-14
The purpose of this report is to examine the longitudinal relationships between kindergarten students’ internalizing and externalizing behavior problems and to explore how these behaviors are influenced by key demographic factors. Early behavioral patterns have been shown to predict later social and academic outcomes, making it important to identify early risk factors and developmental trajectories that contribute to persistent behavior problems across childhood. Grounded in ecological and developmental models of behavior, this study uses a cross-lagged panel model (CLPM) to assess both stability and reciprocal influences between internalizing and externalizing behaviors from Fall to Spring of the kindergarten year.
Specifically, the analysis addresses two research questions:
Do children’s behavior problems show stability during the kindergarten year, and are there cross-lagged relationships between internalizing and externalizing behaviors?
Do behavior problems differ across gender, race/ethnicity, or family socioeconomic status for children entering kindergarten?
The following sections present the specified structural model, model identification procedures, and results, including parameter estimates, model-implied matrices, and evaluation of model fit.
Cross-Lagged Panel Model Overview
The following model represents a cross-lagged panel model (CLPM) designed to examine the longitudinal relationships between kindergarten students’ internalizing and externalizing behavior problems across two time points—Fall and Spring. The model includes demographic covariates (Gender, Race/Ethnicity, and Socioeconomic Status) predicting the Fall behavior outcomes. Each behavior variable is modeled for stability over time (autoregressive paths), as well as cross-lagged effects capturing the influence of Fall internalizing on Spring externalizing behaviors and vice versa.
Correlated residuals were specified between internalizing and externalizing disturbances at each time point to account for shared unexplained variance.
Figure 1 below illustrates the hypothesized structural model, depicting both the autoregressive and cross-lagged pathways between children’s internalizing and externalizing behaviors from Fall to Spring. Solid directional arrows represent hypothesized predictive relationships, while curved bidirectional arrows indicate correlations between residuals and covariate. The two double arrows between represent of internalizing and externalizing behaviors correlate in Fall and Spring.
Figure 1
## lavaan 0.6-19 ended normally after 37 iterations
##
## Estimator ML
## Optimization method NLMINB
## Number of model parameters 34
##
## Number of observations 4999
## Number of missing patterns 1
##
## Model Test User Model:
## Standard Scaled
## Test Statistic 1372.907 1554.782
## Degrees of freedom 20 20
## P-value (Chi-square) 0.000 0.000
## Scaling correction factor 0.883
## Yuan-Bentler correction (Mplus variant)
##
## Model Test Baseline Model:
##
## Test statistic 8218.781 7892.795
## Degrees of freedom 36 36
## P-value 0.000 0.000
## Scaling correction factor 1.041
##
## User Model versus Baseline Model:
##
## Comparative Fit Index (CFI) 0.835 0.805
## Tucker-Lewis Index (TLI) 0.702 0.648
##
## Robust Comparative Fit Index (CFI) 0.834
## Robust Tucker-Lewis Index (TLI) 0.702
##
## Loglikelihood and Information Criteria:
##
## Loglikelihood user model (H0) -28109.079 -28109.079
## Scaling correction factor 1.294
## for the MLR correction
## Loglikelihood unrestricted model (H1) -27422.625 -27422.625
## Scaling correction factor 1.142
## for the MLR correction
##
## Akaike (AIC) 56286.157 56286.157
## Bayesian (BIC) 56507.735 56507.735
## Sample-size adjusted Bayesian (SABIC) 56399.695 56399.695
##
## Root Mean Square Error of Approximation:
##
## RMSEA 0.116 0.124
## 90 Percent confidence interval - lower 0.111 0.118
## 90 Percent confidence interval - upper 0.122 0.129
## P-value H_0: RMSEA <= 0.050 0.000 0.000
## P-value H_0: RMSEA >= 0.080 1.000 1.000
##
## Robust RMSEA 0.116
## 90 Percent confidence interval - lower 0.111
## 90 Percent confidence interval - upper 0.121
## P-value H_0: Robust RMSEA <= 0.050 0.000
## P-value H_0: Robust RMSEA >= 0.080 1.000
##
## Standardized Root Mean Square Residual:
##
## SRMR 0.066 0.066
##
## Parameter Estimates:
##
## Standard errors Sandwich
## Information bread Observed
## Observed information based on Hessian
##
## Regressions:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## X2TCHEXT ~
## X1TCHEXT (a1) 0.749 0.013 58.388 0.000 0.749 0.728
## X2TCHINT ~
## X1TCHINT (a2) 0.557 0.017 32.279 0.000 0.557 0.545
## X2TCHEXT ~
## X1TCHINT (b1) 0.004 0.015 0.260 0.794 0.004 0.003
## X2TCHINT ~
## X1TCHEXT (b2) 0.078 0.011 7.101 0.000 0.078 0.098
## X1TCHEXT ~
## sex_recod 0.269 0.017 16.074 0.000 0.269 0.219
## Black 0.042 0.028 1.494 0.135 0.042 0.023
## Hispanic -0.044 0.023 -1.865 0.062 -0.044 -0.029
## Other 0.006 0.027 0.241 0.809 0.006 0.003
## X12SESL -0.103 0.011 -9.175 0.000 -0.103 -0.134
## X1TCHINT ~
## sex_recod 0.045 0.013 3.370 0.001 0.045 0.047
## Black -0.058 0.022 -2.666 0.008 -0.058 -0.041
## Hispanic -0.020 0.019 -1.050 0.294 -0.020 -0.017
## Other -0.002 0.021 -0.099 0.921 -0.002 -0.001
## X12SESL -0.047 0.009 -5.334 0.000 -0.047 -0.079
##
## Covariances:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## .X1TCHEXT ~~
## .X1TCHINT 0.071 0.005 13.929 0.000 0.071 0.252
## .X2TCHEXT ~~
## .X2TCHINT 0.043 0.003 12.931 0.000 0.043 0.249
## sex_recode ~~
## Black 0.000 0.000 0.000
## Hispanic 0.000 0.000 0.000
## Other 0.000 0.000 0.000
## X12SESL 0.000 0.000 0.000
## Black ~~
## Hispanic 0.000 0.000 0.000
## Other 0.000 0.000 0.000
## X12SESL 0.000 0.000 0.000
## Hispanic ~~
## Other 0.000 0.000 0.000
## X12SESL 0.000 0.000 0.000
## Other ~~
## X12SESL 0.000 0.000 0.000
##
## Intercepts:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## .X2TCHEXT 0.438 0.025 17.586 0.000 0.438 0.692
## .X2TCHINT 0.572 0.025 23.107 0.000 0.572 1.175
## .X1TCHEXT 1.455 0.013 108.086 0.000 1.455 2.368
## .X1TCHINT 1.440 0.011 129.156 0.000 1.440 3.021
## sex_recode 0.510 0.007 72.118 0.000 0.510 1.020
## Black 0.130 0.005 27.334 0.000 0.130 0.387
## Hispanic 0.209 0.006 36.370 0.000 0.209 0.514
## Other 0.114 0.004 25.390 0.000 0.114 0.359
## X12SESL 0.009 0.011 0.764 0.445 0.009 0.011
##
## Variances:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## .X2TCHEXT 0.188 0.006 33.892 0.000 0.188 0.469
## .X2TCHINT 0.157 0.005 33.902 0.000 0.157 0.665
## .X1TCHEXT 0.352 0.009 37.876 0.000 0.352 0.933
## .X1TCHINT 0.225 0.007 31.951 0.000 0.225 0.990
## sex_recode 0.250 0.000 1784.739 0.000 0.250 1.000
## Black 0.113 0.004 32.137 0.000 0.113 1.000
## Hispanic 0.165 0.003 49.457 0.000 0.165 1.000
## Other 0.101 0.003 29.148 0.000 0.101 1.000
## X12SESL 0.637 0.011 57.020 0.000 0.637 1.000
##
## R-Square:
## Estimate
## X2TCHEXT 0.531
## X2TCHINT 0.335
## X1TCHEXT 0.067
## X1TCHINT 0.010Identification Assessment
Model context:
This cross-lagged panel model includes 9 observed variables:
5 exogenous covariates (Gender, Black, Hispanic, Other, SES)
and 4 endogenous behaviors (Fall Externalizing, Fall Internalizing,
Spring Externalizing, Spring Internalizing).
Residuals of internalizing and externalizing behaviors are correlated
within each time point.
1. t-rule (Order Condition)
- Number of observed variables (p): 9
- Number of distinct sample moments: t =54
- Number of estimated parameters (k):
- Covariates → Fall behaviors: 5 × 2 = 10
- Autoregressive paths: 2
- Cross-lagged paths: 2
- Residual variances (endogenous): 4
- Residual covariances (within wave): 2
- Variances (exogenous): 5
- Total k = 34 - Degrees of freedom: df = t - k = 54 - 34 =
20 - Conclusion:* The model satisfies the t-rule
and is overidentified (df = 20)** According to the T-Rule we
met this condition, which is necessary, but not sufficient. .
2. Null-B Rule
- The null-B rule applies only when there are no paths among
endogenous variables (B = 0).
- This model includes autoregressive and cross-lagged paths (Fall →
Spring), so B ≠ 0.
- Conclusion: The null-B rule does not apply
to this model.
3. Recursive Rule
- A recursive model has no feedback loops (acyclic) and
uncorrelated residuals.
- This model is acyclic (time-ordered), but residuals of Fall and Spring
behaviors are correlated.
- Conclusion: The recursive rule is not
satisfied, but this does not imply non-identification.
Overall Conclusion:
- t-rule: satisfied/met necessary-but not sufficient (overidentified, df
= 20) - Null-B: not applicable
- Recursive: not satisfied due to correlated residuals
- Therefore, the model is identified and suitable for
estimation under SEM.
Figure 2
Figure 2. Path diagram of the cross-lagged panel model illustrating associations between demographic covariates (Gender, Race/Ethnicity, SES) and kindergarteners’ internalizing and externalizing behavior problems across the school year. Solid lines represent statistically significant (p < .05) paths, while dashed lines represent non-significant effects; values on the arrows are standardized parameter estimates.
## <p style='font-size:90%'><em>Note.</em> Solid = p < .05; dashed = p ≥ .05. Labels show standardized estimates. Residual covariances are shown within Fall and within Spring.</p>Discussion
Overall, the model revealed several statistically significant relationships among demographic covariates and children’s behavior problems across the kindergarten year. Specifically, fall externalizing behaviors significantly predicted spring externalizing behaviors, indicating moderate stability in problem behaviors over time. Similarly, fall internalizing behaviors significantly predicted spring internalizing behaviors, supporting temporal continuity in internal emotional difficulties.
There were also significant cross-lagged effects, showing that higher fall externalizing behaviors predicted increases in spring internalizing behaviors, suggesting that outwardly directed behaviors early in the year may evolve into inward distress as children progress through school. However, the reverse pathway—from fall internalizing to spring externalizing—was not significant, indicating that internal distress may not directly manifest as externalized behaviors over this period.
Among the demographic covariates, gender (being a boy) and lower SES were significant predictors of higher fall externalizing scores, consistent with prior developmental and ecological research highlighting that boys and children from lower socioeconomic backgrounds often display more disruptive behaviors in early schooling. Other covariates (race/ethnicity indicators) did not show significant associations once SES and gender were included, suggesting these effects may operate indirectly through socioeconomic context.
Together, these findings underscore the stability and interrelation of early behavior problems, as well as the importance of addressing both externalizing and internalizing symptoms in kindergarten. They also highlight that early demographic risk factors—particularly gender and SES—are important contexts shaping children’s behavioral adjustment trajectories during the first school year.
Discussion of Model Fit from Normalized Residual Matrix
Large normalized residuals (|z| > 2) appeared mainly among demographic variables, especially between race/ethnicity and SES and between demographics and Fall externalizing behavior. This indicates the model fits well for the main behavioral paths but underfits the covariance structure among exogenous predictors, suggesting that relationships among race, SES, and gender are more complex than the model assumes.
To view the matrices see the Excel document.