David Kaplan & Heidi Sweetman University of Delaware

1. 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

2. Topics To Be Covered… • Growth mixture modeling (including conventional growth curve modeling) • Latent transition analysis • A Substantive Example: Math Achievement & ECLS-K

3. 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

4. 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

5. 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

6. 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

7. 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

8. Measurement Portion of Growth Model yi = p-dimensional vector representing the empirical growth record for child i Λ= a p x q matrix of factor loadings  = a q-dimensional vector of factors  = p-dimensional vector of measurement errors with a p x p covariance matrixΘ 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

9. 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 Ψ = a q-dimensional vector of factors = a q-dimensional vector that contains the population initial status & growth parameters Γ= 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

10. 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.

11. 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

12. Growth Mixture Model • The conventional growth curve model can be rewritten with the subscript c to reflect the presence of trajectory classes

13. 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

19. 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.

20. 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.

21. 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.

22. 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

23. 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

24. t = 1st time of measurement 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

25. 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 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

26. 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

27. 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.

28. Math Proficiency Levels in ECLS-K

29. Table 2 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.

32. 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.

33. 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.

34. 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.

35. 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.