We are interested in understanding how children’s interpersonal skills, executive function, and demographics upon entering kindergarten are associated with their externalizing problem behaviors (e.g., disruptive, aggressive, non-compliant, hyperactivity) at the end of kindergarten. Externalizing problem behaviors can significantly impact a child’s functioning and development. These behaviors may disrupt family life, strain relationships with peers and adults, and hinder academic progress. It is important for parents, caregivers, and educators to provide support and seek appropriate interventions when necessary to help children with externalizing behaviors develop more adaptive and prosocial skills. Early intervention and a supportive environment can be crucial in addressing and managing externalizing behaviors in children.
The data utilized in this study were obtained from the Early Childhood Longitudinal Study – 2011. Children, their families, teachers, schools, and care providers provided information on children’s cognitive, social, emotional, and physical development. ECLS-K:2011 is designed to provide comprehensive and reliable data that can be used to describe and to better understand children’s development and experiences in the elementary grades and how children’s early experiences relate to their later development, learning, and experiences in school.
First, we ran descriptives on gender, race, and early childhood education center attendance. Table 1.1 below shows the results. From the table you can see the overall sample was n=4999. Of the 4999, 2450 (49%) were female and 2549 (51%) were male.
## sex n percent
## Female 2450 49.0%
## Male 2549 51.0%
## Total 4999 100.0%Table 1.2 shows the breakdown of students by race and percentages. The majority of students in the sample were white, 54.7% with Hispanic making up 20.7% of the sample followed by Black/African American making up 13%. A full breakdown of the sample by race is shown below.
## race n percent
## White, non-Hispanic 2732 54.7%
## Black/African American, non-Hispanic 650 13.0%
## Hispanic, race specified 1033 20.7%
## Hispanic, no race specified 13 0.3%
## Asian, non-Hispanic 281 5.6%
## Native Hawaiian/Pacific Islander 13 0.3%
## American Indian/Alaska Native 42 0.8%
## Two or more races, non-Hispanic 235 4.7%
## Total 4999 100.0%When looking descriptively, we also looked at students attendance in an early learning center or not. Table 1.3 shows that 43.7 percent of the students did not attend an early learning center and 56.3% of students did attend and early learning center.
## ece n percent
## No ECE center 2187 43.7%
## ECE center 2812 56.3%
## Total 4999 100.0%After looking at descriptives we created a SEM model by regressing spring externalizing behaviors on children’s sex.
a. What does \(\alpha\)
(alpha) represent in this model?
\(\alpha\) represents the intercept, or
the expected score on externalizing behaviors (\(X2TCHEXT\)) when \(sex = 0\) (the reference group). The
average score for females (reference group) is
1.503, so the average score for males
is 1.759.
b. What does \(\gamma\)
(gamma) represent in this model?
\(\gamma\) represents the slope, or the
expected change in externalizing behaviors (\(X2TCHEXT\)) for a one-unit change in \(sex\_recode\). Because \(sex\_recode\) is coded 0 = Female and 1 =
Male, the coefficient \(\gamma =
0.256\) means that males score on average 0.256 points
higher on externalizing behaviors than females.
c. What do we assume about the distribution of the residuals?
How would you check these assumptions?
We assume the residuals (\(\zeta\)) are
normally distributed with a mean of 0 and a variance of \(\psi\).
d. Write the estimated model for the unstandardized
solution.
\[Y_{i} = 1.503 + 0.256 \cdot sex_{i} +
\zeta_{i}\]
e. Write the estimated model for the partially standardized
solution.
\[Y_{i} = 2.377 + 0.404 \cdot sex_{i} +
\zeta_{i}\]
f. Report \(R^2\).
The model explained approximately \(R^2 =
0.041\), meaning that 4.1% of the variance in externalizing
behaviors (\(X2TCHEXT\)) is explained
by sex.
g. Interpret the slope for both estimated
models.
For the unstandardized model, the average spring externalizing score for
males is 1.759, which is higher than the average for females (reference
group).
For the partially standardized model, the average spring externalizing score for males is 2.781, indicating a higher standardized score compared to females.
## lavaan 0.6-19 ended normally after 1 iteration
##
## Estimator ML
## Optimization method NLMINB
## Number of model parameters 3
##
## Number of observations 4999
##
## Model Test User Model:
##
## Test statistic 0.000
## Degrees of freedom 0
##
## Parameter Estimates:
##
## Standard errors Standard
## Information Expected
## Information saturated (h1) model Structured
##
## Regressions:
## Estimate Std.Err z-value P(>|z|) Std.nox
## X2TCHEXT ~
## sex_recode 0.256 0.018 14.595 0.000 0.404
##
## Intercepts:
## Estimate Std.Err z-value P(>|z|) Std.nox
## .X2TCHEXT 1.503 0.013 120.143 0.000 2.377
##
## Variances:
## Estimate Std.Err z-value P(>|z|) Std.nox
## .X2TCHEXT 0.383 0.008 49.995 0.000 0.959## X2TCHEXT
## 0.041After analyzing the first model, we created a second model to add more demographic variables, including age at kindergarten entry (X1AGEENT), socioeconomic status (X12SESL), and whether the child attended an early childhood education center (Center indicator variable).
Write the estimated model for the unstandardized solution.
\[Y_{i} = 1.738 + 0.257 \cdot sex_{i} - 0.004
\cdot age_{i} - 0.112 \cdot SES_{i} + 0.068 \cdot Center_{i} +
\zeta_{i}\]
Write the estimated model for the fully standardized
solution.
\[Z_{Y_{i}} = 2.748 + 0.203 \cdot Z_{sex_{i}}
- 0.031 \cdot Z_{age_{i}} - 0.142 \cdot Z_{SES_{i}} + 0.053 \cdot
Z_{Center_{i}} + \zeta_{i}\]
Report \(R^2\). The model explained approximately \(R^2 = 0.062\), meaning that 6.2% of the variance in externalizing behaviors (\(X2TCHEXT\)) is explained by the predictors (sex, age, SES, and center).
What predictors (if any) are non-significant? All predictors are significant.
Interpret the unstandardized and fully standardized partial regression coefficient for socioeconomic status (SES). For the unstandardized model, for every one-unit increase in SES, externalizing behaviors decreased by 0.112 points. For the fully standardized model, for every one-unit increase in SES, externalizing behaviors decreased by 0.142 standard deviations.
Discuss the stability (or instability) of the slope estimate for sex. The slope estimate for sex appears stable. The standard error (SE) for sex is 0.017, which is relatively small, indicating a precise estimate. Additionally, the z-value is 14.805, which is extremely high (greater than 10), further indicating that the estimate is highly stable and unlikely to fluctuate across samples.
## lavaan 0.6-19 ended normally after 1 iteration
##
## Estimator ML
## Optimization method NLMINB
## Number of model parameters 6
##
## Number of observations 4999
##
## Model Test User Model:
##
## Test statistic 0.000
## Degrees of freedom 0
##
## Parameter Estimates:
##
## Standard errors Standard
## Information Expected
## Information saturated (h1) model Structured
##
## Regressions:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## X2TCHEXT ~
## sex_recode 0.257 0.017 14.805 0.000 0.257 0.203
## X1AGEENT -0.004 0.002 -2.264 0.024 -0.004 -0.031
## X12SESL -0.112 0.011 -10.183 0.000 -0.112 -0.142
## Center 0.068 0.018 3.825 0.000 0.068 0.053
##
## Intercepts:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## .X2TCHEXT 1.738 0.121 14.379 0.000 1.738 2.748
##
## Variances:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## .X2TCHEXT 0.375 0.008 49.995 0.000 0.375 0.938## X2TCHEXT
## 0.062We created a third model to add Fall interpersonal skills (X1TCHPER) and executive function (attentional focus X1ATTNFS and inhibitory control X1INBCNT).
Write the estimated model for the unstandardized solution.
\[ Y_{i} = 3.149 + 0.075\,sex_{i} + 0.001\,age_{i} - 0.031\,SES_{i} + 0.062\,Center_{i} - 0.208\,interpersonal_{i} + 0.030\,attention_{i} - 0.235\,inhibitory_{i} + \zeta_{i} \]
Write the estimated model for the fully standardized solution.
\[ Z_{Y_{i}} = 4.979 + 0.059 \cdot Z_{sex_{i}} + 0.008 \cdot Z_{age_{i}} - 0.039 \cdot Z_{SES_{i}} + 0.048 \cdot Z_{Center_{i}} - 0.207 \cdot Z_{interpersonal_{i}} + 0.062 \cdot Z_{attention_{i}} - 0.475 \cdot Z_{inhibitory_{i}} + \zeta_{i} \]
Report \(R^2\).
The model explained approximately \(R^2 =
0.367\), meaning that 36.7% of the variance in externalizing
behaviors (\(X2TCHEXT\)) is explained
by the predictors.
What predictors (if any) are non-significant? Age was the only non-significant predictor in the model (\(p = 0.475\)).
What can you say about the effect of sex on externalizing
behaviors after controlling for interpersonal skills and executive
function?
Once interpersonal skills and executive function are included in the
model, the effect of sex explains much less of the unique variance in
externalizing behaviors. The effect size dropped from 0.256 to 0.075.
While this effect remains statistically significant, it represents a
small effect size when controlling for these additional
predictors.
Examining the fully standardized estimates, what are the two
strongest predictors of externalizing behaviors at the end of
kindergarten?
The two strongest predictors are interpersonal skills \((\beta = -0.207)\) and inhibitory control
\((\beta = -0.475)\).
Discuss any issues in considering only the fully standardized
estimates.
There is no such thing as a standard deviation (SD) increase in a binary
variable like Center or Sex, so the fully standardized estimates for
these predictors are not directly interpretable. Standardized
coefficients assume a continuous variable where a 1 SD is meaningful,
which doesn’t apply to binary predictors.
## lavaan 0.6-19 ended normally after 1 iteration
##
## Estimator ML
## Optimization method NLMINB
## Number of model parameters 9
##
## Number of observations 4999
##
## Model Test User Model:
##
## Test statistic 0.000
## Degrees of freedom 0
##
## Parameter Estimates:
##
## Standard errors Standard
## Information Expected
## Information saturated (h1) model Structured
##
## Regressions:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## X2TCHEXT ~
## sex_recode 0.075 0.015 5.043 0.000 0.075 0.059
## X1AGEENT 0.001 0.002 0.714 0.475 0.001 0.008
## X12SESL -0.031 0.009 -3.352 0.001 -0.031 -0.039
## Center 0.062 0.015 4.216 0.000 0.062 0.048
## X1TCHPER -0.208 0.015 -14.033 0.000 -0.208 -0.207
## X1ATTNFS 0.030 0.009 3.271 0.001 0.030 0.062
## X1INBCNT -0.235 0.010 -23.598 0.000 -0.235 -0.475
##
## Intercepts:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## .X2TCHEXT 3.149 0.104 30.227 0.000 3.149 4.979
##
## Variances:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## .X2TCHEXT 0.253 0.005 49.995 0.000 0.253 0.633## X2TCHEXT
## 0.367(OpenAI, 2025) utilized to help with coding for this markdown.