1. An Introduction to Latent Curve Models
Instructor: Shelley Blozis, UC Davis

2. Outline Longitudinal panel study design Latent Curve Models
Missing Data
Some examples of fitting latent curve models using Mplus and SAS PROC MIXED

3. Longitudinal Panel Design
A sample of subjects is observed at multiple points in time
The timing of the measurements can be the same for all individuals
Fixed Occasions Design
Or the timing can vary between individuals
Varying Occasions Design

4. Latent Growth Curve Models
One of several different statistical methods for the analysis of longitudinal panel data
Assume that all individuals in a population have the same functional form
But the parameters of the function vary between individuals This allows for the individual curves to vary

5. Brief Detour: common factor analysis
The latent curve model is based on a latent variable model

6. Common factor model X1 = λ1 + δ1 X2 = λ2 + δ2 X3 = λ3 + δ3 X = Λξ + δ
X : set of manifest variables
Λ : factor loading matrix
ξ : factors (latent variables)
δ : set of uniquenesses

7. Common factor model with a mean structure
X = τ + Λξ + δ
X is the set of manifest variables
τ is the set of intercepts
Λ is the factor loading matrix
ξ is the set of factors (latent variables)
δ is the set of uniquenesses

8. Latent Variables
Multiple manifest variables serve as indicators of an underlying, unobserved variable
Indicators are reflective of the construct
Example: intelligence, self-esteem, life satisfaction

9. Similar to the common factor analysis model with the mean structure, the latent curve model is used to account for the means, variances and covariances of the observed scores measured over time

10. Making the distinction
Factor analysis
The factors of a factor analysis model represent unobservable variables, such as intelligence or self-esteem
In a latent curve model, the factors represent characteristics of change
Not latent variables in the standard sense
Represent unobservable characteristics of change

11. Let yti be an observed test score
Longitudinal panel study: Intellectual test score for n = 100 children assessed up to four times at different ages
Let yti be an observed test score
t is an occasion, t = 1,…,4
i is the individual

12. Using this framework, specify a form of change for the response
A latent curve model
yi = Ληi + εi
Based on a common factor model with a mean structure
Using this framework, specify a form of change for the response

13. If a measured response is modeled by a linear function of time, then a latent curve model could be specified as
where

14. Model intellectual test scores using a linear function of children’s age
Individual i’s expected rate of change
The expected value of yi at Agei = 0
The expected value of yi at (Agei – 5)

15. Linear Growth Model
But often the intent is to describe variation between individuals in the coefficients and possibly account for this variation
From this model we can estimate each individual’s set of coefficients
We can also estimate the population trajectory

16. Use the individual-specific coefficients to estimate each person’s trajectory
We can then work with the model to describe variation between individuals in their response level and rate of change

17. Linear growth model The coefficients are each a sum of
a fixed effect (a constant for the population) and
a random effect (unique to the individual)

18. Model Assumptions The residual
Assume that the responses of each individual follow an underlying trajectory; here we assume linear growth
The observations are assumed to be due to this trajectory plus other factors that are captured in the residual
Other factors include possible measurement error, as well as other factors apart from time

19. An illustration Observed scores that includes measurement error
“inherent” trajectory for an individual
Scores free of measurement error

20. Model Assumptions The residual
Assume that the responses of each individual follow an underlying trajectory; here we assume linear growth
The observations are assumed to be due to this trajectory plus other factors that are captured in the residual
Other factors include possible measurement error, as well as other factors apart from time
Possible assumptions about the residuals
Independent between individuals
Independent within individuals
Constant variance across time

21. Model Assumptions The person-specific coefficients
Assume the coefficients vary between individuals according to the random effects, b0i and b1i

22. An illustration The ‘typical’ trajectory
“inherent” trajectory for an individual
Observed scores
Source: Harring & Blozis (2014), Behav Res, 46,

23. Model Assumptions The person-specific coefficients
Assume the coefficients vary between individuals according to the random effects, b0i and b1i
Each individual can have a curve that is unique from those of others
Under the model, b0i and b1i are often assumed to be normally distributed, means equal to 0
Each has a variance to describe individual differences
The two random effects can covary

24. Model Assumptions The population model Describes the typical response
Figure adapted from Blozis & Harring (2016), Structural Equation Modeling, 23(6),

25. Latent Curve Models versus Multilevel Models
We have options in how we approach the analysis
Latent curve models
As a structural equation model (SEM)
Multilevel models
It’s possible to specify equivalent models under the two approaches, and by applying the same estimation methods, obtain identical results

26. Estimating the model: A latent curve model vs. a multilevel model
Using Mplus to take the latent curve model approach and SAS PROC MIXED (or R) to take the multilevel model approach
Fit a linear growth model to repeated measures of intellectual ability

27. Latent Curve Model Approach
The intercept and slope are the latent variables
Time (Age) is incorporated into the model as specific and constrained values of the factor loadings, all possibly unique to each person

28. Factor loadings, specified as fixed, constrained to specific values
The means of ‘Int’ and ‘Slope’ relate to the population model
The variances of ‘Int’ and ‘Slope’ relate to variation in the level and rate of change across individuals
‘Int’ and ‘Slope’ free to covary

29. Data model: yi = Ληi + εi
The factor loading matrix, Λ, reflects assumptions about the pattern of change in the response variable
Each column is known as a ‘basis function’
For linear change, Λ has two columns
The first is a column of ones to represent the intercept
The second is a column with values equal to the times of measurement
In our example, time is represented by the child’s ages at each assessment
Λ = 1
5.58
7.83
10.58
17.17

30. Data model: yi = Ληi + εi
The factor, ηi, is unknown and varies across individuals
For an individual, the elements that make up ηi represent different aspects of change in y
For linear change, ηi contains two factors
ηi = (η0i,η1i )'
η0i is individual i’s intercept
η1i is individual i’s slope
The factors are weights, each linked to a basis function defined in the factor matrix

31. Data model: yi = Ληi + εi According to the model
An observed score is modeled as a weighted linear combination of the basis functions plus residual
Underlying trajectory for an individual is given by Ληi
The residual is the difference between the observed scores and the individual’s underlying trajectory

32. Estimation using Mplus
Obtain maximum likelihood estimates of model parameters
Estimate
Factors means
Factor variances and their covariance
A common variance of the time-specific residuals

33. Setting up the data file
Using the latent variable model approach, set up the data file in wide format

34. Save as ascii file; no header
Indicate missing data

35. Mplus syntax for fitting a linear growth model
random effects
The | symbol is used in conjunction with TYPE=RANDOM to name and define the random effects

36. Mplus syntax for fitting a linear growth model
Specifying ‘maximum likelihood’ estimation – we have other options for estimation
The model assumes that the variances of the residuals are constant across time
“(1)” will constrain the residual variances to be equal

37. Results from Mplus

38. Fitting the model using a multilevel model approach
Using SAS PROC MIXED
Using the same estimator as was used in Mplus (maximum likelihood)
PROC MIXED requires data in long-format

39. Data file in wide format
Data for the first child (famid = 1)
Data for the 2nd child (famid = 2)

40. Bring data into SAS

41. PROC MIXED syntax for fitting a linear growth model with random coefficients

42. Results from SAS PROC MIXED
Fixed Intercept and Slope
Variance of the random intercept
Covariance between the random intercept and slope
Variance of the random slope

43. Mplus and PROC MIXED comparison

44. Missing Data
Missing data are often encountered in longitudinal panel studies
Participants may miss one or more of the planned assessments, including those who drop from a study and do not return

45. Intelligence test scores Only 16% of cases have complete data for all 4 waves

46. How latent curve models and multilevel models handle missing data
Even though many participants have incomplete data for the 4 waves, all participants are included in the analysis
Due to the method of estimation
Maximum likelihood
Does not require complete data for the response variable

47. In the intelligence study, scores are studied according to Age that differs between children
For children who have complete data for all 4 waves, Λ is composed of 4 rows, 2 columns, ages are unique to the child
For children who have data for waves 1-3 but not 4, Λ is composed of 3 rows, 2 columns, ages are unique to the child 1
5.58
7.83
10.58
17.17 1
2.33
5.17
11.42

48. Due to the way in which the parameters of the models are estimated, missing response data (e.g., intelligence test scores) are handled
This is different from other statistical methods, such as ANOVA, that require complete data for all cases for estimation

49. Assumptions about missing data
For complete-case methods (e.g., ANOVA), data are assumed to be missing completely at random (MCAR)
MCAR = whether data are missing or not is independent of the missing data as well as any observed data
For available-case methods (e.g., latent curve models) data are assumed to be missing at random (MAR)
MAR = whether data are missing or not is independent of the missing values; possibly related to any observed data

50. Back to our example
Linear growth is one possible model to describe test scores
Can we improve on model fit by considering a nonlinear form of change, such as by applying a quadratic growth model?

51. Interpretation Age is centered at 5 years Intercept Linear slope
Quadratic slope

52. Slight modification of the Mplus syntax

53. Quadratic Growth Model - Results

54. Comparison of Model Fit
Linear Growth
Quadratic Growth

55. For nested models, we can use the likelihood ratio test to compare fit
Calculate the deviance for each model
Deviance = -2*loglikelihood
Statistic: chi-square = difference in deviances
Linear growth: deviance = -2*( ) =
Quadratic growth: deviance = -2*( ) =
Chi-square = =
df = difference in degrees of freedom between models
df = 4
Chi-square of with 4 df, p < .0001

56. Accounting for between-person differences in the random effects
Mother’s IQ test score
Repeated measures of child test score

57. Mplus syntax for a conditional growth model

58. Conditional Growth Model - Results

59. Conditional Growth Model - Results

60. Other structures are possible
One more model: Evaluate our assumptions about the within-subject residuals
Typical assumption
Residuals are independent with constant variance across time
Other structures are possible
Autocorrelation
Independent with heterogeneous variances across time

61. Allow for autocorrelation between adjacent residuals

62. Results

63. Just an introduction to latent curve models
As a framework for the analysis of longitudinal panel data, these models offer many possibilities
Are there any benefits to a latent curve model approach versus multilevel?
Yes!

64. Resources
Duncan, T. E., Duncan, S. C., & Strycker, L. A. (2006). An introduction to latent variable growth curve modeling: Concepts, issues, and application (2nd ed.). Mahwah, NJ: Lawrence Erlbaum Associates, Inc.
Bollen, K. A., & Curran, P.J. (2006). Latent curve models: A structural equation perspective. Hoboken, NJ: Wiley.
Singer, J. D. (1998). Using SAS PROC MIXED to fit multilevel models, hierarchical models, and individual growth models. Journal of Educational and Behavioral Statistics, 23,
Singer, J. D., & Willett, J. B. (2003). Applied longitudinal data analysis: Modeling change and event occurrence. New York: Oxford University Press.