David Kaplan & Heidi SweetmanUniversity of Delaware

Two Methodological Perspectives on the

Development of Mathematical Competencies in Young

Children: An Application of Continuous & Categorical Latent Variable Modeling

Topics To Be Covered…

• Growth mixture modeling (including conventional growth curve modeling)

• Latent transition analysis

• A Substantive Example: Math Achievement & ECLS-K

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

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

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

Two Perspectives on Conventional Growth Curve

ModelingThe 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

Two Perspectives on Conventional Growth Curve

ModelingThe 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

Measurement Portion of Growth Model

ii iKxy i = a p-

dimensional vector

measurement

intercepts

= a q-dimensional vector of

factors

Λ = a p x q matrix of

factor loadings

yi = p-dimensional

vector representing the empirical growth record for child i

K = p x k matrix of regression

coefficients relating the repeated

outcomes to a k – dimensional vector of

time-varying predictor variables xi

= p-dimensional vector of

measurement errors with a p x

p covariance matrix Θ

p = # of repeated measurements on the ECLS-K math proficiency testq = # of growth factorsk = # of time-varying predictorsS = # of time-invariant predictors

Structural Portion of Growth Model

= a q-dimensional vector that

contains the population

initial status & growth

parameters

B = a q x q matrix

containing coefficients

that relate the latent variables to each other

= a q-dimensional

vector of factors

Γ = q x s matrix of regression

coefficients relating the latent growth

factors to an s-dimensional vector of

time-invariant predictor variables z

= q-dimensional vector of

residuals with covariance matrix

Ψ

iiii ++= z

p = # of repeated measurements on the ECLS-K math proficiency testq = # of growth factorsk = # of time-varying predictorsS = # of time-invariant predictors

= random growth factor

allowing growth

factors to be related

to each and to time-

invariant predictors

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.

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

Growth Mixture Model

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

,ii ii Kxy

,iicicci ++= z

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 (Θ) and structural disturbance variances (Ψ) across classes

•Difference classes can show different relationship to a set of covariates z

,ii ii Kxy

,iicicci ++= z

Table 1.

Average Posterior Probabilities for the Three-Class Solution

Class 1 Class 2 Class 3

Class 1 0.882 0.027 0.090 Class 2 0.138 0.855 0.007 Class 3 0.138 0.001 0.861 Note: Class 1 = average developing; Class 2 = above average;

Class 3 = below average

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.

GMM Conclusions (cont’d)• 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.

GMM Conclusions (cont’d)• 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.

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

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

= 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

LTA Model

Proportion of individuals Y generating a

particular response y

S

p

S

qpqpkpjpipkpjpipprob

1 1|||||||)( yY

δ = proportion

of individuals

in latent status p at

time t

= the probability of response i to item 1 at time t

given membership in latent status p

pk '|pi | pj |

= the probability of

membership in latent status q at time t + 1

given membership in latent status p

at time t

pq|

pq|

t = 1st time of measurementt + 1 = 2nd time of measurementi’, i’’ = response categories 1, 2…I for 1st

indicatorj’, j’’ = response categories 1, 2…J for 2nd

indicatork’, k’’ = response categories 1, 2…K for

3rd indicatori’, j’, k’ = responses obtained at time 1i’’, j’’, k;’ = responses obtained at time t +

1p = latent status at time tq = latent status at time t + 1

Latent Class Model

ckcj

C

ccicprob '|'|

1'|)(

yY

= the proportion

of individuals

in latent class c.

c

= the probability of response i to item 1 at time t

given membership in latent status p

pi | = the probability of response i to item 2 at time t

given membership in latent status p

pj | = the probability of response i to item 3 at time t

given membership in latent status p

pk '|

Proportion of individuals Y generating a

particular response y

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

LTA Example (cont’d)

• 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.

Math Proficiency Levels in ECLS-K

Proficiency Level

Kindergarten/First GradeAssessment

Third GradeAssessment

1Number & Shape

Identifying some one-digit numeralsRecognizing geometric shapesReading all 1 & 2 digit numerals

Demonstrating understanding of place value in integers to hundreds place

2Relative Size

Recognizing geometric shapesUsing nonstandard units of length to compare the size of objects

Using knowledge of measurement and rate to solve word problems

3Ordinality &

Sequence

One-to-one counting up to 10 objectsRecognizing a sequence of patternsRecognizing the next number in a sequenceIdentifying ordinal position of an object

Recognizing more complex number patterns

4Add/Subtract

Solving simple addition and subtraction problems

Solving simple addition and subtraction problems

5Multiply/Divide

Solving simple multiplication and division problems

Solving simple multiplication and division problems

Table 2

Response probabilities for measuring latent status variable at each wave. Full Sample a

Math Proficiency Levelsb

Wave Latent Status OS AS MD Class Proportions Spring K Mod Skill 1.00 1.00 0.15 0.20 Low Skill 0.48 0.00 0.00 0.80 Fall 1st Mod Skill 1.00 1.00 0.19 0.35 Low Skill 0.62 0.00 0.00 0.65 Spring 1st 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 – prob(mastered). b OS = ordinality/sequence, AS = add/subtract, MD = multiply/divide.

Table 4. Logistic regression of dynamic latent status variable on kindergarten poverty status. Latent Status Estimate S.E. Est/S.E. Odds Ratio Wave Regression

Spring K Mod Skill on Belowpov -1.501 0.158 -9.492 0.223 Fall First Mod Skill on Belowpov -0.715 0.115 -6.235 0.489 Spring First Mod Skill on Belowpov -0.600 0.103 -5.833 0.549

Table 4. Logistic regression of dynamic latent status variable on kindergarten poverty status. Latent Status Estimate S.E. Est/S.E. Odds Ratio Wave Regression

Spring K Mod Skill on Belowpov -1.501 0.158 -9.492 0.223 Fall First Mod Skill on Belowpov -0.715 0.115 -6.235 0.489 Spring First Mod Skill on Belowpov -0.600 0.103 -5.833 0.549

LTA Conclusions

1. Two stable classes found across three waves.

2. Transition probabilities reflect some movement between classes over time.

3. Poverty status strongly relates to class membership but the strength of that relationship appears to change over time.

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.

General conclusion (cont’d)

• 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.

General conclusions (cont’d)• 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.