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Two Methodological Perspectives on the Development of Mathematical Competencies in Young Children: An Application of Continuous & Categorical Latent Variable Modeling. David Kaplan & Heidi Sweetman University of Delaware. Topics To Be Covered….
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Two Methodological Perspectives on the Development of Mathematical Competencies in Young Children: An Application of Continuous & Categorical Latent Variable Modeling
David Kaplan & Heidi Sweetman
University of Delaware
The Multilevel Modeling Perspective
The Structural Equation Modeling Perspective
yi = pdimensional vector representing the empirical growth record for child i
Λ= a p x q matrix of factor loadings
= a qdimensional vector of factors
= pdimensional vector of measurement errors with a p x p covariance matrixΘ
n= a pdimensional vector measurement intercepts
K =p x k matrix of regression coefficients relating the repeated outcomes to a k – dimensional vector of timevarying predictor variables xi
p = # of repeated measurements on the ECLSK math proficiency test
q = # of growth factors
k = # of timevarying predictors
S = # of timeinvariant predictors
B=a q x q matrix containing coefficients that relate the latent variables to each other
= random growth factor allowing growth factors to be related to each and to timeinvariant predictors
= qdimensional vector of residuals with covariance matrix Ψ
= a qdimensional vector of factors
= a qdimensional vector that contains the population initial status & growth parameters
Γ= q x s matrix of regression coefficients relating the latent growth factors to an sdimensional vector of timeinvariant predictor variables z
p = # of repeated measurements on the ECLSK math proficiency test
q = # of growth factors
k = # of timevarying predictors
S = # of timeinvariant predictors
t + 1= 2nd time of measurement
i’, i’’ = response categories 1, 2…I for 1st indicator
j’, j’’ = response categories 1, 2…J for 2nd indicator
k’, k’’ = response categories 1, 2…K for 3rd indicator
i’, j’, k’ = responses obtained at time 1
i’’, j’’, k;’ = responses obtained at time t + 1
p = latent status at time t
q = latent status at time t + 1
LTA Model= the probability of membership in latent status q at time t + 1 given membership in latent status p at time t
δ= proportion of individuals in latent status p at time t
= the probability of response i to item 1at time t given membership in latent status p
= the probability of response i to item 2 at time t given membership in latent status p
= the probability of response i to item 3 at time t given membership in latent status p
Proportion of individuals Y generating a particular response y
Proportion of individuals Y generating a particular response y
= the proportion of individuals in latent class c.
= the probability of response i to item 1at time t given membership in latent status p
= the probability of response i to item 2 at time t given membership in latent status p
= the probability of response i to item 3 at time t given membership in latent status p
Steps in LTA
1. Separate LCAs for each wave
2. LTA for all waves – calculation of transition probabilities.
3. Addition of poverty variable
a
Response
probabilities for measuring latent
status
variable at each wave.
Full Sample
b
Math Proficiency Levels
Wave
Latent Status
OS
AS
MD
Class Proportions
Spring K
Mod Skill
1.00
1.00
0.15
0.20
Low Skil
l
0.48
0.00
0.00
0.80
st
Fall
1
Mod Skill
1.00
1.00
0.
19
0.35
Low Skill
0.
62
0.
00
0.
0
0
0.65
st
Spring 1
Mod Skill
1.00
1.00
0.34
0.74
Low Skill
0.
78
0.00
0.00
0.26
a
Response probabilities are for mastered items
. Response probabilities for non

mastered items can be
computed from
1
–
mastered).
prob(
b
OS = ordinality/sequence, AS = add/subtract, MD = multiply/divide.