Right-Sizing Growth Mixture Models for Longitudinal Data - PowerPoint Presentation

Slide1

Right-Sizing Growth Mixture Models for Longitudinal Data

Phillip Wood, Wolfgang Wiedermann, Douglas SteinleyUniversity of MissouriSlide2

Some Questions We Wish We Could Answer with Longitudinal Data

Are there Different Types of Learners? Slow Versus QuickSmooth trajectories versus Bumpy (Stage-like transitions)Late-Onset Versus Early OnsetAcross the Lifespan or During the Course of MeasurementAre there Subsets of People who Onset, Persist, or Recover with Respect to Some Behavior Over Time? Who Are They?

Do Some Groups of People Experience Developmentally Limited Periods of Problem Behaviors, Moratoria of No Change Relative to their Peers or Delayed Onset of Normative Change?

Are there Qualitative Differences in Rate of Change?

Longitudinal (or Age-Invariant) Diagnostic Types

Do Some Groups of Individuals Exhibit Distinct Ages of Onset?

Is Age of Onset Associated with Different Patterns of Persistence?

Are Patterns of

Cormorbidity

Linked Over Time for Some Groups of Individuals?Slide3

Growth/Decline

= Joint change in Variability and LevelChange = Change in Level Alone

Longitudinal data frequently show a pattern of increasing variability related to level.

Conceptually,

Growth

reflects learning or maturation. Similar linkage associated with decreases over time may reflect

decline

.By Change, mean differences are observed over time with no change to observed variability. (i.e., no inter-individual differences in mean differences over time)Historically, Growth/Decline have been studied using Hierarchical Linear ModelsTo Answer Questions about Initial Level and Subsequent CourseTo Explain how Individuals Arrived at Final Level (Less Frequently) Success of HLM led to extensions of Growth Mixture Models Using Linear or Polynomial Measurement Model

Time

Score

Score

TimeSlide4

Growth Mixture Modeling – A Way to Identify Different Types of Learners or Changers?

GMM Uses a Growth Curve Model as:A way to identify multiple, unobserved sub-populations which are similar in terms ofTheir longitudinal change or overall level

A

post-hoc

method for identification

and description of group differences in change

.“HLM-like” models commonly used: Basis functions of Linear Slope Intercept, Quadratic or other Polynomial Curves.Slide5

Linear Quadratic Growth Mixture Model

For Example,

q

I

,

q

L

, and qQ Vary Across Four Groups

“Cat’s Cradle” cf Sher, Steinley

, Jackson, 2011Slide6

But What If Change Is Not Linear or Polynomial?

Conceptually, Growth May Not Follow a Parametric FormDevelopmentally limited periods of elevationMoratoriaDelayed OnsetAlthough some mathematical Form (such as a quadratic or other polynomial curve) may approximate this

With large numbers of measurement occasions prohibitively high numbers of polynomial factors are needed. (i.e., Y=bI+bX+bX

2

+bX

3

…)

Functional form of growth by be obscured if insufficient powers are considered.No conceptual reason for such a high dimensional form.Do Other More Complex Models Exist?

Time

Time

Score

ScoreSlide7

What About the Assumption of a Random Intercept?

What if there is no Random Intercept Factor?

(i.e., our Constructs “behave” like the physical properties of height or weight)

If so, the HLM model contains a superfluous variance component

What if only a Random Intercept Factor is really present?

(i.e., our variables represent only

interindividual

differences in elevation with no differences in mean over time?)

If so, a model of “no growth” may be sufficient for identifying groups.

Alternatively, subgroups may be based on change patterns (only elevation).

Would a Simpler Model Be Better?Slide8

Existing Alternatives

Other Finite Mixture Classification Approaches Have Been UsedLatent Class Analysis with Polynomial Contrasts (i.e., GMM with Null y matrix)Latent Profile Analysis with Common Unstructured

y

matrix

K-Means Cluster Approaches (i.e., Diagonal

y

matrix)None of These Approaches, However, Capture the Notion of “Growth” as Patterned Covariation Over Time Thought to Underlie “Growth/Decline.” Groups are defined based on Change (i.e. common “level” only).May be “close enough for government work” but do other growth models exist?Slide9

Other Growth Models Besides Linear or Polynomial Curves:

I: Single Factor Models

Factor Mean (FM)

(McArdle & Epstein, 1987)

Factor Mean Shift

(FM)-Shift

McDonald (1967)Slide10

Other Growth Models Besides Linear or Polynomial Curves:

II Two Factor Models

Free Curve (FCSI)

Meredith &

Tisak

(1984)

Two Factor Growth

(Additional factors

are also possible)Slide11

For Example, One Can Extract Two Latent Groups With FM-Shift Model

!Factor Mean Shift Code Snippet for five variables;%c#1% [y1@0] [y2@0]

[y3@0]

[y4@0]

[y5@0];

[

i

*0] (g1imean); [s*0.5] (g1smean); s by y1*1 (G1L1) y2*1 (G1L2) y3*0.5 (G1L3) y4*0.5 (G1L4)

y5*0.5 (G1L5); i@0;

s@1; i with s@0 (g1iscov); y1*1 (g1e1); y2*1 (g1e2); y3*1 (g1e3);

y4*1 (g1e4); y5*1 (g1e5); %c#2% [y1@0] [y2@0] [y3@0]

[y4@0] [y5@0];[i*-0.1] (g2imean); [s*1] (g2smean); s by y1*3 (G2L1) y2*2 (G2L2) y3*1 (G2L3) y4*1 (G2L4) y5*1 (G2L5); i@0;

s@1;

i

with s@0;

y1*2 (g2e1);

y2*2 (g2e2);

y3*2 (g2e3);

y4*2 (g2e4);

y5*2 (g2e5);Slide12

Free Curve (Latent Basis) Growth Curve to Identify Two Subgroups

%c#1% !Code Snippet

[y1@0]

[y2@0]

[y3@0]

[y4@0]

[y5@0];

[i*0] (g1imean); [s*0.5] (g1smean); s by y1*1 (G1L1) y2*1 (G1L2)

y3*0.5 (G1L3) y4*0.5 (G1L4)

y5*0.5 (G1L5); i*1 (g1ivar); s@1; i with s@0; y1*1 (g1e1); y2*1 (g1e2);

y3*1 (g1e3); y4*1 (g1e4); y5*1 (g1e5);

%c#2%

[y1@0]

[y2@0]

[y3@0]

[y4@0]

[y5@0];

[i*-0.1] (g2imean);

[s*1] (g2smean);

s

by

y1*3 (G2L1)

y2*2 (G2L2)

y3*1 (G2L3)

y4*1 (G2L4)

y5*1 (G2L5);

i*1 (g2ivar);

s@1;

i

with

s@0;

y1*2 (g2e1);

y2*2 (g2e2);

y3*2 (g2e3);

y4*2 (g2e4);

y5*2 (g2e5); Slide13

“Right-Sizing” Single Group Growth Curve Models

Three Step ProcedureDetermine DimensionalityExplore Parsimonious Confirmatory Factor Models (i.e., Manifest Variables Freely Estimated)

Tau-Equivalent Factor Model (i.e., Random Intercept)

Congeneric Factor Model (i.e., Factor Model with Freely Estimated Loadings)

Random Intercept Factor Model (i.e., Simultaneous Tau and Congeneric Factors)

Multi-Factor Model

Triangular Decomposition

Simple StructureBifactor ModelsMulti-Factor Models with Additional Tau-Equivalent ModelCan Models in II Parsimoniously Explain Mean Effects?Wood, P. K., Steinley, D., Jackson, K.M. (2015).

Right-Sizing Statistical Models for Longitudinal Data. Psychological Methods, 20(4), 470-488.Slide14

There is No Reason, Conceptually, That These Models Could Not Also Be Described as “Growth” or “Change” Within a Finite Mixture Model

Such models do, however, vary in terms of their complexity in terms of:DimensionalityParsimony (Discussed in a Moment)Additionally, multi-group models can provide an additional framework for mixture models. This amounts of a type of multi-group invariance exercise, except that group membership is a latent, as opposed to manifest, variable.

This leads to a variety of possible models depending on whether factor loadings, variances/

covariances

, intercepts and error variances are thought to be invariant across groups.Slide15

Multi-Group Alternative: Reference Centered Groups (Factorial Invariance) Two Group Example for FM Model

Reference Group

Additional Group

As proposed by, e.g.,

Dolan, C. V.,

Molenaar

, P. C. M., &

Boomsma

, D. I. (1989). LISREL analysis of twin data with structured means. 

Behavior Genetics, 19(1), 51-62.Slide16

Candidate Factor Models

Factor Models, then, can be arranged from simpler, more parsimonious models to more complex, less parsimonious. For Groups=2:Slide17

“Right-Sizing” Growth Mixture Models

(Multiple Models Exist for

Modeling Mean Level)

Alternatively, consider all possibilities in Steps 3 & 4Slide18

Could It Work in Practice?

Numerous Models are Being Compared Which Vary in Terms ofNumber of Latent VariablesPatterning of Latent VariablesWays in Which Mean Levels Are ModeledNumber of latent groupsEven if this approach did work for large sample sizes, would it work for the smaller sample sizes often considered?Slide19

Conditions for Simulation

Amount of change over time for the two “change groups” set at one standard deviation between highest and lowest scoresroughly similar to the improvement during the undergraduate years in critical thinking and reflective judgment (Pascarella

&

Terenzini

, 2005

),

1/4

that found in studies in elementary students in such variables as vocabulary (e.g., Osborne and Suddick 1967; 1972). Error variances selected to produce measures with internal consistency estimates of .55 to .71 for the consistently low and high groups0.50 to 0.83, for the two change groups with an average of 0.62

Sample Size Conditions:

Equally Sized Groups, Ntotal=2000, 3000, 4000, 8000Slide20

A Monte Carlo Exploration Parameter Matrices

Latent Groups

Parameters

1

2

3

4

Factor Loadings (

l

)

Y11130

Y21212

Y3

.5

1

0

2

Y4

.5

1

1

2

y5

.5

1

2

2

Latent Groups

Parameters

1

2

3

4

Factor Variances (

y

)

Intercept

1

1

1

1

Slope

1

1

1

1

Latent Groups

Parameters

1

2

3

4

Error Variances (

q

e

)

Y1

1

2

1

2

Y2

1

2

1

2

Y3

1

2

1

2

Y4

1

2

1

2

y51212

Latent GroupsParameters1234Factor Means (q)Intercept1111Slope1111

In the true model, groups vary

as a function of

loadings (

l

),

intercept variance, (

y

i

)

factor means (

q

)

and

error variances (

e

)Slide21

Time for the “Big Reveal”

The Shape (and Interpretation) of 4 Group Models Depends, of Course, on Measurement Model. The Following 4 Group Solutions are Based on the Same Data

Looks like:

Mason

, S. T., Corry, N., Gould, N. F.,

Amoyal

, N., Gabriel, V.,

Wiechman-Askay, S., ... & Fauerbach, J. A. (2010). Growth curve trajectories of distress in burn patients. Journal of Burn Care & Research

, 31(1), 64-72.

Looks like:Sher, K. J., Jackson, K. M., & Steinley, D. (2011). Alcohol use trajectories and the ubiquitous cat's cradle: cause for concern?. Journal of Abnormal Psychology, 120

(2), 322.Linear Basis ModelQuadratic Basis ModelSlide22

True Recovered Four Group FCSI ModelSlide23

Candidate Models Fit to the Data

Eleven Types of GMM’s Considered: FCSI FCSI-Reference FM

FM-Reference

FM-Shift

Linear

Linear-Reference Quadratic Profile Repeated Measures H0 Repeated Measures HaltLatent Classes: 3, 4, 5, 644 Models in AllSlide24

Correct Model Chosen Across Simulations

Model

N=2000

N=3000

N=4000

N=8000

3gFM

1.5

0

0

0

3gFCSI53.5

23.5

3.5

0

4gFCSI

2.5

51.5

88.5

95

4gFMShift

1

5

4

3

4gRepMeas

27

3.5

0

0

4gFM

0.5

1

0

0

5gFCSI

0

0

1.5

5gRepMeas

12

3.5

0

0

6gRepMeas

2

12

4

0.5

(and 35 Other

Models Never Chosen)

Model

N=2000

N=3000

N=4000

N=8000

3gFM

3

0

0

0

3gFCSI

107

47

7

0

4gFCSI

5

103

177

190

4gFMShift

2

10

8

6

4gRepMeas

54

7

0

0

4gFM

1

2

0

0

5gFCSI

0

0

3

5gRepMeas

24

7

0

0

6gRepMeas42481(and 35 Other Models Never Chosen)

Percent

Number Out of 200Slide25

So, to Conclude, Conceptually

If a researcher has access to a large data set, and a relatively reliably identified phenomenon: The researcher may useful employ Growth Mixture Modeling, but exploration of different base measurement models may identify differential patterns of growth different than those suggested by polynomial models.If, however, the researcher has access to an only moderately-sized data set:

Use of a simpler base model, such as the repeated measures model, which assumes a tau-equivalent measurement model of the construct may be preferable.

If the researcher has access to only a small sized data set or is researching an unreliable construct

A simpler base model, such as an independence model may be all that one can state regarding time-bound relationships.

Slide26

Limitations and extensions

Of course, we have a few- (as well as a few regrets…)Slide27

Limitations

Mixed Structures (latent groups with different factor models) not consideredSome PossibilitiesModel trimming of nonsignificant variance components for some but not all groupsBayesian EstimationDifferent Proportions of Latent ClassesMissing Data PatternsDifferent factor patterns by subgroup is an intermediary model between the ergodic criticisms of aggregation and single subject research aggregation (DFA, State-Space & DSEM)Slide28

Extensions

Inspection of recovered solutions suggests additional parsimonious models in addition to trimming nonsignificant latent componentsEstimated factor loadings may suggest a parametric form (e.g., von Bertalanffy, Gompertz, Schnute)

Factor

loadings

which are proportional across groups may

suggest

subgroups with different variances but common factor structure.

Researchers interested in growth curves which explain the data in terms of level and correlated change relative a given time point can respecify the models in oblique terms. Doing this from the outset, though, not only results in more non-nested relationships between basis models, but encounters more minimization problems.Differential Change Models May Be QuickerE.g., DeSolve, or Other Differential Equation ModelsSlide29

Thanks for your time!

Fini