David Kaplan

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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

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Topics To Be Covered

• Growth mixture modeling (including conventional
  growth curve modeling)
• Latent transition analysis
• A Substantive Example Math Achievement ECLS-K

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Math Achievement in the U.S.

• Third International Mathematics Science Study
  (TIMMS) has led to increased interest in
  understanding how students develop mathematical
  competencies
• Advances in statistical methodologies such as
  structural equation modeling (SEM) and multilevel
  modeling now allow for more sophisticated
  analysis of math competency growth trajectories.
• Work by Jordan, Hanich Kaplan (2002) has begun
  to investigate the shape of early math
  achievement growth trajectories using these more
  advanced methodologies

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Early Childhood Longitudinal Study-Kindergarten
(ECLS-K)

• Longitudinal study of children who began
  kindergarten in the fall of 1998
• Study employed three stage probability sampling
  to obtain nationally representative sample
• Sample was freshened in first grade so it is
  nationally representative of the population of
  students who began first grade in fall 1999

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Data Gathering for ECLS-K

• Data gathered on the entire sample
• Fall kindergarten (fall 1998)
• Spring kindergarten (spring 1999)
• Spring first grade (spring 2000)
• Spring third grade (spring 2002)
• Additionally, 27 of cohort sub-sampled in fall
  of first grade (fall 1999)
• Initial sample included 22,666 students.
• Due to attrition, there are 13,698 with data
  across the four main time points

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Two Perspectives on Conventional Growth Curve
Modeling

• The Multilevel Modeling Perspective
• Level 1 represents intra-individual differences
  in growth over time
• Time-varying predictors can be included at level
  1
• Level 1 parameters include individual intercepts
  and slopes that are modeled at level 2
• Level 2 represents variation in the intercept and
  slopes modeled as functions of time-invariant
  individual characteristics
• Level 3 represents the parameters of level 2
  modeled as a function of a level 3 unit of
  analysis such as the school or classroom

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Two Perspectives on Conventional Growth Curve
Modeling

• The Structural Equation Modeling Perspective
• Measurement portion links repeated measures of an
  outcome to latent growth factors via a factor
  analytic specification.
• Structural Portion links latent growth factors to
  each other and to individual level predictors
• Advantages
• Flexibility in treating measurement error in the
  outcomes and predictors
• Ability to be extended to latent class models

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Measurement Portion of Growth Model
? a q-dimensional vector of factors
? a p x q matrix of factor loadings
yi p-dimensional vector representing the
empirical growth record for child i
? p-dimensional vector of measurement errors
with a p x p covariance matrix T
n a p-dimensional vector measurement intercepts
K p x k matrix of regression coefficients
relating the repeated outcomes to a k
dimensional vector of time-varying predictor
variables xi
p of repeated measurements on the ECLS-K
math proficiency test q of growth factors k
of time-varying predictors S of
time-invariant predictors
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Structural Portion of Growth Model
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 time-invariant
predictors
? q-dimensional vector of residuals with
covariance matrix ?
? a q-dimensional vector of factors
? a q-dimensional vector that contains the
population initial status growth parameters
G q x s matrix of regression coefficients
relating the latent growth factors to an
s-dimensional vector of time-invariant predictor
variables z
p of repeated measurements on the ECLS-K
math proficiency test q of growth factors k
of time-varying predictors S of
time-invariant predictors
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Limitation of Conventional Growth Curve Modeling

• Conventional growth curve modeling assumes that
  the manifest growth trajectories are a sample
  from a single finite population of individuals
  characterized by a single average status
  parameter a single average growth rate.

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Growth Mixture Modeling (GMM)

• Allows for individual heterogeneity or individual
  differences in rates of growth
• Joins conventional growth curve modeling with
  latent class analysis
• under the assumption that there exists a mixture
  of populations defined by unique trajectory
  classes
• Identification of trajectory class membership
  occurs through latent class analysis
• Uncover clusters of individuals who are alike
  with respect to a set of characteristics measured
  by a set of categorical outcomes

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Growth Mixture Model

• The conventional growth curve model can be
  rewritten with the subscript c to reflect the
  presence of trajectory classes

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The Power of GMM (Assuming the time scores are
constant across the cases)

• ?c captures different growth trajectory shapes
• Relationships between growth parameters in Bc are
  allowed to be class-specific
• Model allows for differences in measurement error
  variances (T) and structural disturbance
  variances (?) across classes
• Difference classes can show different
  relationship to a set of covariates z

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GMM Conclusions

• Three growth mixture classes were obtained.
• Adding the poverty indicator yields interesting
  distinctions among the trajectory classes and
  could require that the classes be renamed.

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GMM Conclusions (contd)

• We find a distinct class of high performing
  children who are above poverty. They come in
  performing well.
• Most come in performing similarly, but
  distinctions emerge over time.

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GMM Conclusions (contd)

• We might wish to investigate further the middle
  group of kids those who are below poverty but
  performing more like their above poverty
  counterparts.
• Who are these kids?
• Such distinctions are lost in conventional growth
  curve modeling.

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Latent Transition Analysis(LTA)

• LTA examines growth from the perspective of
  change in qualitative status over time
• Latent classes are categorical factors arising
  from the pattern of response frequencies to
  categorical items
• Unlike continuous latent variables (factors),
  categorical latent variables (latent classes)
  divide individuals into mutually exclusive groups

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Development of LTA

• Originally, Latent Class Analysis relied on one
  single manifest indicator of the latent variable
• Advances in Latent Class Analysis allowed for
  multiple manifest categorical indictors of the
  categorical latent variable
• This allowed for the development of LTA
• In LTA the arrangement of latent class
  memberships defines an individual's latent status
• This makes the calculation of the probability of
  moving between or across latent classes over time
  possible

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LTA Model
t 1st time of measurement t 1 2nd time of
measurement i, i response categories 1, 2I
for 1st indicator j, j response categories
1, 2J for 2nd indicator k, k response
categories 1, 2K 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
the probability of membership in latent
status q at time t 1 given membership in latent
status p at time t
d proportion of individuals in latent status p
at time t
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
the probability of response i to item 1
at time t given membership in latent status p
Proportion of individuals Y generating a
particular response y
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Latent Class Model
Proportion of individuals Y generating a
particular response y
the proportion of individuals in latent class
c.
the probability of response i to item 1
at 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
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LTA Example

• Steps in LTA
• 1. Separate LCAs for each wave
• 2. LTA for all waves calculation of
  transition probabilities.
• 3. Addition of poverty variable

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LTA Example (contd)

• For this analysis, we use data from (1) end of
  kindergarten, (2) beginning of first, and (3) end
  of first.
• We use proficiency levels 3-5.
• Some estimation problems due to missing data in
  some cells. Results should be treated with
  caution.

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Math Proficiency Levels in ECLS-K
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LTA Conclusions

• Two stable classes found across three waves.
• Transition probabilities reflect some movement
  between classes over time.
• Poverty status strongly relates to class
  membership but the strength of that relationship
  appears to change over time.

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General Conclusions

• We presented two perspectives on the nature of
  change over time in math achievement
• Growth mixture modeling
• Latent transition analysis
• While both results present a consistent picture
  of the role of poverty on math achievement, the
  perspectives are different.

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General conclusion (contd)

• GMM is concerned with continuous growth and the
  role of covariates in differentiating growth
  trajectories.
• LTA focuses on stage-sequential development over
  time and focuses on transition probabilities.

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General conclusions (contd)

• Assuming we can conceive of growth in mathematics
  (or other academic competencies) as continuous or
  stage-sequential, value is added by employing
  both sets of methodologies.