What Multi-Level Modeling Can Teach Us About Single-Level ... · Mplus Web Talks: No. 1 Recorded June 2020 ... Review, critique and recommendations. ... Bengt Muthen´ Mplus Web Talks:

What Multi-Level Modeling Can Teach UsAbout Single-Level Modeling & Vice Versa:

The Case of Latent Transition Analysis

Bengt [email protected]

Mplus: www.statmodel.com

Mplus Web Talks: No. 1Recorded June 2020

This talk builds on the keynote address planned for the June 2020 ModernModeling Methods Conference at UCONN that was cancelled due to the coronavirus. I thank Tihomir Asparouhov and Ellen Hamaker for comments and Noah

Hastings for assistance.Bengt Muthen Mplus Web Talks: No. 1 1/ 80

www.statmodel.com

Outline

Modeling:StatisticsPsychologyPsychometrics

Applications of LTA:Mood data (Eid & Langeheine, 2003)Dating and sexual risk behavior data (Lanza & Collins, 2008)Reading proficiency (Kaplan, 2008)Is LTA the right type of model?

Multi-level time series analysis (DSEM)

Appendix A: Sample size requirements

Appendix B: Computing time

Appendix C: New features in Mplus

Bengt Muthen Mplus Web Talks: No. 1 2/ 80

Hidden Markov - Latent Transition Analysis

u: observed categorical variable (latent class indicator)c: latent categorical variable (latent class variable)

u1 u3u2

c1 c2 c3

u4

c4

u5

c5

u11

c1 c2 c3

u12 u31 u32u21 u22


LTA References

u11

c1 c2 c3

u12 u31 u32u21 u22

Langeheine (1988). New developments in latent class theory. InLangeheine & Rost (eds), Latent Trait and Latent Class Models

Graham et al. (1991). Modelling transitions in latent stage-sequentialprocesses: A substance use prevention example. Journal of Consultingand Clinical Psychology

Collins & Wugalter (1992). Latent class models for stage-sequentialdynamic latent variables. Multivariate Behavioral Research

Langeheine & van de Pol (2002). Latent Markov chains. In Hagenaars& McCutcheon (eds), Applied Latent Class Analysis

Collins & Lanza (2010). Latent Class and Latent Transition Analysis.


An LTA Example: Dating Data

Lanza & Collins (2008). A new SAS procedure for latent transitionanalysis: Transitions in dating and sexual risk behavior.Developmental Psychology, 44, 446-456.

Data:

National Longitudinal Survey of Youth (NLSY97), N=29373 time points one year apart starting at ages 17-18

Items:

Past-year number of dating partners (0, 1, 2 or more)Past-year number of sexual partners (0, 1, 2 or more)Exposed to STD in past year (no, yes)


Dating Data: Measurement Probabilities for 5 Classes

Past year Nondaters Daters Monogamous Multi-safe Multi-exposed

# dating partners0 0.789 0.008 0.096 0.031 0.0251 0.166 0.214 0.641 0.026 0.044

>=2 0.045 0.778 0.262 0.943 0.932# sex partners

0 0.975 0.948 0.000 0.104 0.0001 0.014 0.048 0.961 0.258 0.118

>=2 0.011 0.004 0.039 0.638 0.882Exposed to STD

No 1.000 1.000 0.385 1.000 0.187Yes 0.000 0.000 0.615 0.000 0.813


Dating Data: Transition Probabilities

Past year Nondaters Daters Monogamous Multi-safe Multi-exposed

Nondaters 0.627 0.197 0.096 0.038 0.042Daters 0.023 0.626 0.173 0.086 0.091

Monogamous 0.034 0.032 0.679 0.056 0.199Multi-safe 0.036 0.000 0.177 0.584 0.203

Multi-exposed 0.021 0.033 0.201 0.055 0.690


LTA Features

u11

c1 c2 c3

u12 u31 u32u21 u22

1 Measurement probabilities: P(Ut|Ct) - LCA for each time point,measurement invariance across time

2 Initial status probabilities: P(C1)

3 Transition probabilities: P(C2|C1), P(C3|C2)

Extensions:

Mover-Stayer modeling

Multiple-group analysis: Measurement invariance

Covariates: Influencing latent class probabilities and transitionprobabilities


What’s Missing in These Models?

u1 u3u2

c1 c2 c3

u4

c4

u5

c5

u11

c1 c2 c3

u12 u31 u32u21 u22

Single indicator per time pointA statistical perspective

Multilevel modeling: Level 1 = time, level 2 = subjectRandom effects, especially random intercepts

A substantive perspectiveTrait theory in psychologyBetween-subject differences that are stable over time

Multiple indicators per time pointA psychometric perspective

Measurement non-invarianceMultilevel factor analysisMultilevel latent class analysis


A Quick Reminder of Logistic Regression

1

0.5

0

P(Ui = 1|X1i,X2i) =1

1+ e−(β0+β1 X1i+β2 X2i). (1)

Logistic regression with a binary covariate X1 and a continuouscovariate X2. In factor analysis and IRT, X2 can be a factor and inLTA, X1 can be a latent class variable.


Logistic Regression Continued

P(Ui = 1|X1i,X2i) =1

1+ e−(β0+β1 X1i+β2 X2i). (2)

Recalling that the odds is P(U = 1)/[P(U = 0)], this implies a linearequation for the log odds, also referred to as the logit:

logit[P(U = 1|X1,X2)] = β0 +β1 X1 +β2 X2. (3)

Interpretation of the influence of X on U in terms of odds ratios. Forinstance, the influence of X1 on U is eβ1 .

See also chapter 5 of the book:

Muthen, Muthen & Asparouhov (2016). Regression and MediationAnalysis Using Mplus.http://www.statmodel.com/Mplus_Book.shtml


http://www.statmodel.com/Mplus_Book.shtml

Measurement Probabilities as a Function of Latent Classes

Binary latent class indicator U, latent class variable C

P(U = 1|C = c) = 1/(1+ e−Lc), where L is the logit for class c

Lc = logit[P(U = 1|C = c)] = β0c =−τc (threshold)

High logit gives high probability, high τ gives low probability

L = 0 and τ = 0 give P = 0.5

L =−1 and τ = 1 give P = 0.27

L = 1 and τ =−1 give P = 0.73

How do you get L from P?

L = log odds = log[P(U = 1)/P(U = 0)]


Longitudinal Data: Random Intercept Logistic Regression

Two representations of regression of a binary U on a covariate X attime t for subject j using a random intercept uj:

Logit [P(Utj = 1|Xtj)] = uj +β Xtj, (4)

uj = β0 + εj. V(εj) = σ2u (5)

Two-level, long format: Single-level, wide format:

x1 x2 x3

u

1 1 1

u1 u2 u3

β β β

u

Between (subject):Within (time):

xt

ut

β


An Aside: Panel Data, Time-Invariant Omitted Variables,Fixed vs Random Effects

Logit [P(Utj = 1|Xtj)] = uj +β Xtj = β0 + εj (+γ Zj)+β Xtj.

x1 x2 x3

u

1 1 1

u1 u2 u3

β β β

Random effects modeling is criticizedfor not including the likely correlationbetween uj and Xtj. But this is resolvedby standard multilevel modeling:

Logit [P(Utj = 1|Xtj)] = uj +βw (Xtj−X.j),

uj = β0 +βb X.j +δj

Raudenbush & Bryk (2002). Hierarchical Linear Models. Sage. See,e.g., Table 5.11

Asparouhov & Muthen (2019). Latent variable centering of predictorsand mediators in multilevel and time-series models. StructuralEquation Modeling, 26, 119-142


Panel Data Fixed vs Random Effects: References

Wooldridge (2002). Econometric Analysis of Cross Sectional andPanel Data. The MIT Press.

Allison (2009). Fixed effects regression models. Sage

Bollen & Brandt (2010). A general panel model with random and fixedeffects: A structural equations approach. Social Forces, 89, 1-34

Bell & Jones (2015). Explaining fixed effects: Random effectmodeling of time-series cross-sectional and panel data. PoliticalScience Research and Methods, 3 , 133-153

Bou & Satorra (2017). Univariate versus multivariate modeling ofpanel data: Model specification and goodness-of fit testing.Organizational Research Methods

Antonakis et al. (2019). On ignoring the random effects assumption inmultilevel models: Review, critique and recommendations.Organizational Research Methods

Hamaker & Muthen (2020). The fixed versus random effects debateand how it relates to centering in multilevel modeling. Psych Methods


Single-Indicator LTA (Hidden Markov)

u

1 1 1

u1 u2 u3 u1 u2 u3

c1 c2 c3 c1 c2 c3

The Hidden Markov model on the left can use the fastbackward-forward Baum-Welch algorithm for ML estimation

The random intercept model on the right loses this simplicity - the U’sare no longer independent conditional on the Cs

ML estimation needs numerical integration with much heaviercomputationsBartolucci et al. (2012). Latent Markov Models for LongitudinalData


Hidden Markov Modeling with a Random Intercept

u

1 1 1

u1 u2 u3

c1 c2 c3

Altman (2007). Mixed hidden Markov models. Journal of the AmericanStatistical Association, 102, 201-210.

Relapsing-remitting multiple sclerosis patients, N=39

Symptoms worsen and then improve in alternating periods ofrelapse and remission

Outcome: Number of lesions in the brain (count variable)

T=24 (monthly scans for two years)

2 latent classes (hidden states): Relapse versus remission


What’s Missing?

Single indicator per time point

A statistical perspectiveMultilevel modeling: Level 1 = time, level 2 = subjectRandom effects, especially random intercepts


Multiple indicators per time point

A psychometric perspectiveMeasurement non-invarianceMultilevel factor analysisMultilevel latent class analysis


Latent Trait-State Analysis

Kenny & Zautra (1995). The trait-state-error model for multiwavedata. Journal of Consulting and Clinical Psychology.

y1 y3y2

s1 s2 s3

y4

s4

y5

s5

t

t represents a continuous latent trait variable

y1-y5 represent continuous observed variables

s1-s5 represent continuous latent state variables


Latent Trait-State Analysis Continued

Cole et al. (2005). Empirical and conceptual problems withlongitudinal trait-state models: Introducing a trait-state-occasionmodel. Psychological Methods

Kenny & Zautra (2001). Trait-state models for longitudinal data. In A.Sayer & L. M. Collins (Eds.), New methods for the analysis of change(pp. 243-263). Washington, DC: American Psychological Association.- STARTS model

Wagner at al. (2015). Self-esteem is mostly stable across youngadulthood: Evidence from latent STARTS models. Journal ofPersonality

Ludtke et al. (2018). More stable estimation of the STARTS model: ABayesian approach using Markov Chain Monte Carlo techniques.Psychological Methods


Cross-Lagged Panel Modeling (CLPM)

z1

y1

z3

y3

z2

y2

Hamaker, Kuiper, Grasman (2015). A critique of the cross-laggedpanel model. Psychological Methods, 1, 102-116.


Hamaker et al. (2015) Critique of CLPM

z1

y1

z3

y3

z2

y2

”if stability of constructs is to some extent of a trait-like, timeinvariantnature, the autoregressive relationships of the CLPM fail to adequatelyaccount for this”

”As a result, the lagged parameters that are obtained with the CLPM donot represent the actual within-person relationships over time”

”this may lead to erroneous conclusions regarding the presence,predominance, and sign of causal influences

an alternative model ... separates the within-person process fromstable between-person differences through the inclusion of randomintercepts”


Random Intercept Cross-Lagged Panel Model (RI-CLPM)

z1

y1

wy1

wz1

y

z

z3

y3

wy3

wz3

z2

y2

wy2

wz2

Hamaker et al. (2015)Mplus scripts:http://www.statmodel.com/RI-CLPM.shtml


http://www.statmodel.com/RI-CLPM.shtml

What’s Missing? Multiple-Indicator Case

Single indicator per time point

A statistical perspectiveMultilevel modeling: Level 1 = time, level 2 = subjectRandom effects, especially random intercepts


Multiple indicators per time pointA psychometric perspective

Measurement non-invarianceMultilevel factor analysisMultilevel latent class analysis


Multiple Indicators in Cross-Sectional Data:Capturing Response Bias in Self-Report Questionnaires

Individual differences in response styles e.g. due to acquiescence orsocial desirability:

Wolf (1940). Factor analysis to 1940. Psychometric Monograph No. 3

Individual differences in intercepts and loadings

Maydeu-Olivares & Coffman (2006). Random intercept factoranalysis. Psychological Methods

Individual differences captured by an added response style factorwith factor loadings fixed at 1, that is, a random intercept:Different individuals are allowed to have different intercept values

Schmalbach et al. (2020). From bi-dimensionality touni-dimensionality in self-report questionnaires. European Journal ofPsychological Assessment.

Applications to established questionnaires for depression,procrastination, life orientation, self-esteem, etc


Random Intercept Factor Analysis

1 1 11 1

u

u uu3 4 5u2u1

f

λ λ λ λ λ54321

Random intercept factor analysis for a binary factor indicator Ur(r = 1, . . . ,p), individual i, and factor f :

logit[P(Uri = 1|fi)] =νr +ui +λr fi, (6)

ui ∼ N(0,σ2), (7)

where ui is a random intercept factor with loadings fixed at 1 for all thefactor indicators. A single added parameter: σ2.


Random Intercept Factor Analysis Conclusions

Measurement non-invariance captured by a random intercept:Allows better estimation of the model of interestSuggests that measurement non-invariance is ok if modeled

But is the model realistic?Different indicators may have different random intercept variance

We need a design where each indicator is multiply measuredMultilevel data such as different students nested in differentschools so that an intercept is allowed to vary across schoolsLongitudinal data where time is nested within individual so thatan intercept for each indicator is allowed to vary acrossindividuals


Multiple Indicators in Cross-Sectional Data:Multilevel Factor Analysis

Within

Between

fb

u1 u2 u3

fw

Students within schools:

Typically more within-level factors than between-level factorsfb interpretation different from fw interpretation


Multilevel Factor Analysis Origins

Yij = ν +YBj +YWij (8)

= ν +ΛB fBj + εBj︸︷︷︸+ΛW fWij + εWij︸︷︷︸, (9)

with covariance structure V(Yij) = ΣB +ΣW where each covariancematrix has a factor model structure, Λ Ψ Λ′+Θ.

Harnqvist (1978). Primary mental abilities of collective and individuallevels. Journal of Educational Psychology

Goldstein & McDonald (1988). A general model for the analysis ofmultilevel data. Psychometrika

McDonald & Goldstein (1989). Balanced versus unbalanced designsfor linear structural relations in two-level data. British Journal ofMathematical and Statistical Psychology

Muthen (1994). Multilevel covariance structure analysis. SociologicalMethods & Research


Multilevel Factor Analysis: The Random Intercepts Aspect

For each binary Uij factor indicator, the model

logit[P(Uij = 1|.)] = ν +λB fBj + εBj +λW fWij (10)

can be expressed by the two equations (level-1 and level-2):

logit[P(Uij = 1|.)] = uj +λW fWij , (11)

uj = ν +λB fBj + εBj, (12)

This is in line with two-level regression where the intercept uj israndom, varying across Level-2 units (schools)


Random Intercept as Measurement Non-Invariance



Can be interpreted as measurement non-invariance with respect to theintercept - expressed as a random effect:

Continuous outcomes: Jak et al. (2013; 2014) in SEM, Muthen &Asparouhov in SM&RCategorical outcomes: Fox (2005) BJMSP, Fox & Glas (2001)Psychometrika

Measurement non-invariance ok because you model it


Random Intercept for the Factor: Multilevel IRT

Fox (2010). Bayesian Item Response Theory

Fox & Verhagen (2011). Random item effects modeling forcross-national survey data



Multilevel IRT assumes λB = λW , εB = 0:

logit[P(Uij = 1|.)] = uj +λ fWij , (17)

uj = ν +λ fBj +0, (18)

I.e. a random intercept for the factor f , not a separate one for each U:

logit[P(Uij = 1|.)] = ν +λ fij, (19)

fij = fBj + fWij . (20)


Two Types of Multilevel Factor Models

fw

u1 u2 u3 u4 u5

(a) (b)

Between

u2 u3u1 u4

fb

u5

f

u1 u2 u3 u4 u5

Within

Between

Within

u2 u3u1 u4

fb

u5


Multilevel Factor Analysis versus Latent Class Analysis

c

u1 u2 u3 u4 u5

u2 u3u1 u4

f

u5

fw

u1 u2 u3 u4 u5

u2 u3u1 u4

fb

u5

(a)

Within

Between

c#1 c#2

u1

c

u2 u3 u4 u5

(b)

(c)

c#1 c#2

u1

c

u2 u3 u4 u5

(b)

Within

Between

Within

Between

Within

Between


Two Approaches to Multilevel Latent Class Analysis

Vermunt (2003, 2008), Asparouhov-Muthen (2008), Henry-Muthen(2010)

c

u1 u2 u3 u4 u5

u2 u3u1 u4

f

u5

fw

u1 u2 u3 u4 u5

u2 u3u1 u4

fb

u5

(a)

Within

Between

c#1 c#2

u1

c

u2 u3 u4 u5

(b)

(c)

c#1 c#2

u1

c

u2 u3 u4 u5

(b)

Within

Between

Within

Between

Within

Between


Multilevel Latent Class Analysis: An Application

Henry & Muthen (2010). Multilevel latent class analysis: Anapplication of adolescent smoking typologies with individual andcontextual predictors. Structural Equation Modeling

Smoking behavior for 10,772 9th grade females in 206 ruralcommunities

Six categorical indicators of three latent classes of student smokingbehavior

Random intercepts for the indicators

Across-community variation in the random intercepts found for severalindicators. Variation related to proportion of youth living in poverty

E.g. “Most friends are smokers” had a much larger probability of beingendorsed in communities with a large poverty proportion


Going Deeper into Multilevel Factor Analysis

Within

Between

fb

u1 u2 u3

fw

Cross-sectional: Students within schools

But the case of time points within individuals is also multilevel - sowhy not apply this model to longitudinal data?


What Multilevel Factor Modeling Can Teach Us:Longitudinal Factor Model for T=2

==

=u11 u12 u13 u u u21 22 23

fw1 fw2

fb

u1 u2 u3

Within

Between

fb

u1 u2 u3

fw


Longitudinal Factor AnalysisBased on Multilevel Thinking

==

=u11 u12 u13 u u u21 22 23

fw1 fw2

fb

u1 u2 u3

Model extended to include auto-regression for the fw factors

Corresponds to a cross-sectional model where students within schoolsinfluence each other

This multilevel-based longitudinal model is quite different from the”naive” longitudinal model which uses only the bottom part


Longitudinal Factor Analysis

Hamaker et al. (2016). Using a few snapshots to distinguish mountainsfrom waves: Weak factorial invariance in the context of trait-stateresearch. Multivariate Behavioral Research. - Based on multilevel FA

Argues that factors obtained from cross-sectional factor analysis arepartly determined by between- and partly by within-person covariancestructure (Cattell, 1978): “An uninterpretable blend” (R & B, 2002)

Multilevel-based longitudinal factor analysis allows differencesbetween between-person, trait-like factor structure and within-person,state-like factor structure

T=2 is sufficient to determine the within-person factor structure. Noneed for intensive longitudinal data to estimate a factor model for eachperson (N=1 analysis; Cattell’s P-technique)

Relates to work in Joreskog (1970; 1979), Raffalovich-Bohrnstedt(1987) in SM&R, Marsh-Grayson (1994) in SEM, Eid (1996) inMPRO, Dumenci-Windle (1996) in MBR, Geiser-Lockhart (2012) inPsychological Methods, Geiser et al. (2015) in Behav Res


A Longitudinal Factor Analysis Application

Depression in adolescents (Dumenci & Windle, 1996); used inHamaker et al. (2016)

Four waves of data for 372 males and 433 females

Four subscales of the CES-D: Depressed affect, positive affect, somaticcomplaints, interpersonal. One factor on both “levels”

BIC for the naive model model versus the multilevel-based model, bothwith AR(1):

Males: 26123 versus 25465Females: 32216 versus 31423

The naive model gives inflated reliabilities (standardized loadings)relative to fw and inflated AR

Are we interested in the reliability wrt to fw or fb or both or thecombination?


What Does Longitudinal Factor Analysis Imply forCross-Sectional Factor Analysis (and LCA)?

Generate data from a longitudinal factor analysis model

Analyze one time point using regular factor analysis

For example:Using data generated by 1 between factor with different loadingsacross indicators and 1 within factor, regular factor analysis maymistakenly settle on 2 factorsBut the model that the data were generated from doesn’t havevariation across individuals in 2 factors - but variation in 1 factoracross individuals and variation in 1 factor across time

Non-aggregatable model

Similar situation in cross-sectional multilevel settings:Muthen (1989). Latent variable modeling in heterogeneouspopulations. Psychometrika, 54:4, 557-585Muthen & Satorra (1995). Complex sample data in structuralequation modeling. Sociological Methodology, 25, 267-316


Conclusions from the Modeling Part of the Talk

Modeling Lessons Learned

Statistics:Random intercept

Psychology:Latent trait

Psychometrics:Measurement Non-Invariance


Back to Multiple-Indicator Latent Transition Analysis

u11

c1 c2 c3

u12 u31 u32u21 u22

1 Initial status probabilities: P(C1)

2 Transition probabilities: P(C2|C1), P(C3|C2)

3 Measurement probabilities: P(Ut|Ct) - LCA for each time point,measurement invariance across time

- Now it is clear what is missing


Random Intercept LTA (RI-LTA)

u11

c1 c2 c3

u12 u31 u32u21 u22

f

λ1 λ1 λ1 λ2λ2λ2

Muthen & Asparouhov (2020). Random intercept latent transitionanalysis (RI-LTA). Under review in Psychological Methods.

Papers and Mplus scripts:http://www.statmodel.com/RI-LTA.shtml


http://www.statmodel.com/RI-LTA.shtml

Features of RI-LTA

u11

c1 c2 c3

u12 u31 u32u21 u22

f


Fits the data much better than regular LTA which is unnecessarilyrestrictive and gives distorted results

Does not confound between- and within-subject sources of variation:Ct−1 −> Ct represents a within-subject process free of time-invariantbetween-subject differences (trait differences; meas. non-invariance)

Latent class indicators correlate over time beyond what is captured bythe latent class correlations over time:

Tends to reduce the probability of subjects staying in the sameclass as compared to regular LTAReduces the need for Mover-Stayer modeling because Moversand Stayers can be captured by different random intercept values


Software Improvements for LTA and RI-LTA

RI-LTA can be time consuming due to numerical integration andneeding many random starts to find the global maximum.Mplus Version 8.4 released last November:

Significant speed improvements for computationally demandingmixture models such as with LTA and RI-LTA using a new three-stagerandom starts search and using specialized algorithms drawing onBaum-Welch ideas

Asparouhov & Muthen (2019). Random Starting Values andMultistage Optimization (Technical Report: http://www.statmodel.com/download/StartsUpdate.pdf)

”A 20 hours computation in Mplus 8.3 can be done in Mplus 8.4in less than 15 minutes, by utilizing the advantages of thethree-stage estimation, the Baum-Welch algorithm, as well asupdated hardware (i9-9900k Intel CPU)”

Substantially simplified output for mixture models with multiple latentclass variables


http://www.statmodel.com/download/StartsUpdate.pdf

http://www.statmodel.com/download/StartsUpdate.pdf

An Aside: Different Types of Random Effects

Continuous - normal. Main model used in Muthen-Asparouhov (2020)

Discrete - non-parametric (one extra latent class variable):Latent trait related:

Eid & Langeheine (1999). The measurement of consistency andoccasion specificity with latent class models: A new model and itsapplicaton to the measurement of affect. Psychological MethodsEid & Langeheine (2003). Separating stable from variableindividuals in longitudinal studies by mixture distribution models.Measurement: Interdisciplinary Research and Perspectives

Random effects related for hidden Markov:Maruotti (2011). Mixed hidden Markov models for longitudinaldata: An overview. International Statistical ReviewBartolucci et al (2012). Latent Markov Models for LongitudinalData

The discrete alternative using a binary random effect variable ispresented in Muthen-Asparouhov (2020) as based on theEid-Langeheine parameterization. It is somewhat less time consumingbut gave a worse BIC than the continuous-normal model in allapplications. More than 2 categories was not needed


Outline



Multi-level time series analysis (DSEM)

Appendix A: Sample size requirements




Regular LTA Fits Worse than RI-LTA Most of the Time

Analyses of 4 data sets from the LTA literature (data athttp://www.statmodel.com/RI-LTA.shtml):Table 1: Model fitting results

Life Satisfaction (non-stationary) Mood (stationary)N=5147, T=5, R=1, J=5 N=494, T=4, R=2, J=2

Model # par’s logL BIC Model # par’s logL BIC

Regular LTA 11 -15326 30745 Regular LTA 7 -2053 4150

RI-LTA 12 -15267 30637 RI-LTA 9 -2018 4093

Reading proficiency (non-stationary) Dating and sexual risk behavior (stationary)N=3574, T=4, R=5, J=3 N=2933, T=3, R=3, J=5

Regular LTA 35 -21793 43873 Regular LTA 49 -16202 32796

RI-LTA 40 -20329 40985 RI-LTA 52 -16043 32502

1


http://www.statmodel.com/RI-LTA.shtml

Mood Data. Eid & Langeheine (2003)

German Multidimensional Mood Questionnaire data, N=494, ages17-77 (Steyer et al., 1997)

4 time points 3 weeks apart

2 binary items based on ratings of momentary sadness and unhappiness

Stationary model (equal transition probabilities over time)

Eid & Langeheine (2003). Separating stable from variable individualsin longitudinal studies by mixture distribution models. Measurement:Interdisciplinary Research and Perspectives


Model Fit for Mood Data

Model # parameters log likelihood BIC

Standard

1 Regular LTA 7 -2053 41502 RI-LTA 9 -2018 4093


Mood Data Measurement Probability Estimates

Classes

Regular LTA

High mood Low moodIndicator

Sad 0.089 0.902Unhappy 0.038 0.857

RI-LTA

High mood Low moodIndicator

Sad 0.286 0.748Unhappy 0.164 0.804


Mood Data Transition Probability Estimates

2 latent classes: High mood and Low mood

Classes

Regular LTA

Classes High mood Low moodHigh mood 0.803 0.197Low mood 0.248 0.752

RI-LTA

Classes High mood Low moodHigh mood 0.691 0.309Low mood 0.486 0.514


Model Fit for Mood Data: Mover-Stayer Modeling


Standard


Mover-Stayer


Unlike regular LTA, RI-LTA does not need a Mover-Stayercomponent

Large negative and positive intercept factor scores captureStayers


Transitions in Dating and Sexual Risk BehaviorLanza and Collins (2008) Developmental Psychology

Data:

National Longitudinal Survey of Youth (NLSY97), N=29373 time points one year apart starting at ages 17-18

Items:

Past-year number of dating partners (0, 1, 2 or more)Past-year number of sexual partners (0, 1, 2 or more)Exposed to STD in past year (no, yes)


Model Results for Dating Data

Regular LTA (# pars = 49, logL = -16202, BIC = 32796):5 classes: Nondaters, Daters, Monogamous, Multipartner-Safe,Multipartner-Exposed

RI-LTA (# pars = 52, logL = -16043, BIC = 32502):5 classes but only Nondaters, Daters, and Multipartner-Exposedcorrespond to the solution obtained with regular LTATwo different classes: (1) Monogamous wrt sex partner but withmultiple dating partners, (2) Monogamous-ExposedTransition probabilities:

Past year Nondaters Daters Monogamous? Mono-exposed Multi-exposed

Nondaters 0.583 0.264 0.029 0.058 0.067Daters 0.014 0.660 0.183 0.013 0.130

Monogamous? 0.008 0.067 0.405 0.200 0.320Mono-exposed 0.071 0.007 0.040 0.639 0.243Multi-exposed 0.031 0.050 0.172 0.127 0.619


Model Fit for Dating DataAdding a Mover-Stayer Component

Table 1: Model fitting results for Dating data

Model # par’s logL BIC

Standard

Regular LTA 49 -16202 32796RI-LTA 52 -16043 32502

Mover-Stayer


1


Dating Data Estimates

RI-LTA allows a continuum of Staying instead of a Stayer class:

RI-LTA model captures staying also via different intercept factor values

RI-LTA does not need a Stayer class or the Stayer class is smaller

Regular LTA Mover-Stayer model estimates 20% StayersRI-LTA Mover-Stayer model estimates 9% Stayers

RI-LTA makes it possible to learn more about the indicators:

Estimated RI-LTA intercept factor loadings (all significant):

# dating partners = 1.55# sex partners = 4.22 (strongest indicator of the random interceptfactor, that is, the time-invariant trait)exposed to STD = 1.51


Reading Proficiency. Kaplan (2008)

Early Childhood Longitudinal Study (ECLS), N = 3574

4 time points: Fall and Spring of Kindergarten and Fall and Spring ofGrade 1

5 binary items representing a stage-sequential process:

Basic reading skills of letter recognitionBeginning soundsEnding letter soundsSight wordsWords in context

3 latent classes corresponding to 3 stages of learning

Kaplan (2008). An overview of Markov chain methods for the study ofstage-sequential developmental processes. Developmental Psychology


Model Fit for Reading Data


Standard



Reading Data Measurement Probability Estimates

Class 1 = low alphabet knowledge, Class 2 = early word reading,Class 3 = early reading comprehension

Regular LTA RI-LTA

Classes Classes1 2 3 1 2 3

Letrec 0.505 0.994 1.000 0.627 0.939 0.979Begin 0.066 0.917 0.984 0.303 0.806 0.941Ending 0.013 0.661 0.972 0.167 0.630 0.904Sight 0.000 0.051 0.985 0.020 0.208 0.808WIC 0.000 0.001 0.509 0.005 0.058 0.460


Reading Data Latent Class Probabilities

Regular LTA

Fall K Spring K Fall 1st Spring 1st

ClassProbabilities

1 0.694 0.235 0.142 0.0412 0.284 0.635 0.627 0.1543 0.023 0.130 0.232 0.805

RI-LTA

Fall K Spring K Fall 1st Spring 1st

ClassProbabilities

1 0.948 0.161 0.040 0.0102 0.049 0.818 0.880 0.0173 0.003 0.022 0.080 0.973


Reading Data Transition Probabilities

Regular LTA

1 2 3 1 2 3

TransitionProbabilities Fall K -– Spring K Spring K -– Fall 1st

1 0.338 0.649 0.012 0.596 0.401 0.0022 0.001 0.652 0.348 0.002 0.837 0.1613 0.000 0.000 1.000 0.002 0.003 0.994

RI-LTA

1 2 3 1 2 3

TransitionProbabilities Fall K -– Spring K Spring K -– Fall 1st

1 0.170 0.820 0.010 0.240 0.742 0.0182 0.000 0.819 0.181 0.001 0.931 0.0683 0.000 0.000 1.000 0.022 0.000 0.978


The Meaning of the Random Intercept Factor:The Measurement Non-Invariance Perspective

u11

c1 c2 c3

u12 u31 u32u21 u22

f


For binary latent class indicator Ur, individual i, at time t, and class c,

logit[P(Urit = 1|Cit = c, fi) =−τrc +λr fi, (21)

for threshold τ and random intercept factor loading λ . A high logitvalue implies a high probability of U = 1, e.g. λ > 0, fi > 0.

In the presentation of measurement probabilities, the random interceptfactor f was integrated out, that is, considering the overall conditionalprobability P(Urit = 1|Cit = c)

Individuals with different fi values have different conditionalprobabilities P(Urit = 1|Cit = c, fi) as shown in (21)

This means that the random intercept factor f can be seen as capturingmeasurement non-invariance


Measurement Probabilities for the Reading Datafrom a Measurement Non-Invariance Perspective

Regular LTA RI-LTA

Classes Classes1 2 3 1 2 3


RI-LTA

f =−0.5 f =+0.5



Exploring Measurement Non-Invariance in the Reading Data

RI-LTA

f =−0.5 f =+0.5


RI-LTA, allowing measurement non-invariance across poverty groups

Poverty (19%) Non-poverty (81%)


The random intercept factor f can perhaps be viewed as a readingpreparedness dimension. It is strongly related to poverty


RI-LTA Model Variations

Lag-2, lag-3 etc for the latent class variables

Distal outcomes

Time trend

Multiple processes (cross-lagged RI-LTA)

Multilevel RI-LTA


Is LTA the Right Type of Model? Classes vs Factors

Latent classes or latent dimensions?

Latent transition analysis versus Longitudinal factor analysis (LFA)

LFA has many factors - high-dimensional integration for ML due tocategorical outcomes but easy for Bayes

LFA number of dimensions = T*(number of within factors) +number of of indicators/time point (one between factor)

Mood: 4*1 + 2 = 6, Dating: 3*1 + 3 = 6, Reading: 4*1 + 5 = 9

==

=u11 u12 u13 u u u21 22 23

fw1 fw2

fb

u1 u2 u3


Is LTA the Right Type of Model: RI-LTA vs LFA Using ML

Mood data:

RI-LTA: 9 par’s, logL =-2018, BIC = 4093LFA: 16 par’s, logL = -2007, BIC = 4113

Dating data:

RI-LTA: 52 par’s, logL = -16043, BIC = 32502LFA: 20 par’s, logL = -17278, BIC = 34715

Reading data:

RI-LTA: 40 par’s, logL = -20329, BIC = 40984LFA: 29 par’s, logL = -19939, BIC = 40115LFA indicator variances (random intercepts; measurementnon-invariance): Letrec = 14.8, Begin = 6.9, Ending = 6.0, Sight= 6.0, WIC = 16.1LFA factor means increase significantly, except between Spring Kand Fall 1st

Can we use a model that fits much worse than another model?


Outline



Multi-level time series analysis (DSEM)Appendix A: Sample size requirements




The Vice Versa: What Single-Level Modeling Can Teach UsAbout Multi-Level Modeling

u11

c1 c2 c3

u12 u31 u32u21 u22

f


How do you do analyze relations between variables at different timepoints using two-level modeling?

Multilevel time series analysis (time and individual)Other random effects can be added: Random slopes, randomvariances, random AR, random transition probabilitiesMany time points required - intensive longitudinal data (EMA,ESM, daily diary)


Dynamic Structural Equation Modeling (DSEM)

Single-level, wide Two-level, long, time series (DSEM)

u11

c1 c2 c3

u12 u31 u32u21 u22

f


Within (time)

u1 u2

f

λ λ

ct-1 ct

u1t u2t

Between (subject)

1 2

DSEM data are in long format:2 outcomes per time point results in 2 columns of data (not 2*T)

Across-time effects specified by using lags: C ON C&1 (lag 1)Bengt Muthen Mplus Web Talks: No. 1 73/ 80

Dynamic Factor Analysis

fb φ

Between

φfwtfwt-1

Within


Example: Affective InstabilityIn Ecological Momentary Assessment

Jahng S., Wood, P. K.,& Trull, T. J., (2008). Analysis of AffectiveInstability in Ecological Momentary Assessment: Indices UsingSuccessive Difference and Group Comparison via MultilevelModeling. Psychological Methods, 13, 354-375

An example of the growing amount of EMA data

84 outpatient subjects: 46 meeting borderline personality disorder(BPD) and 38 meeting MDD or DYS

Each individual is measured several times a day for 4 weeks for total ofabout 100 assessments

A mood factor for each individual is measured with 21 self-ratedcontinuous items

The research question is if the BPD group demonstrates more temporalnegative mood instability than the MDD/DYS group


Dynamic Structural Equation Modeling

Two-level analysis: randomeffects varying across subjectsCross-classified analysis (oftime and subject): randomeffects varying across subjectsand time

fb φ

Between

φfwtfwt-1

Within

Dynamic structural equation modeling (DSEM):

Asparouhov, Hamaker & Muthen (2017). Dynamic Latent ClassAnalysis. Structural Equation Modeling: A MultidisciplinaryJournal, 24:2, 257-269Asparouhov, Hamaker & Muthen (2018). Dynamic structuralequation models. Structural Equation Modeling: AMultidisciplinary Journal, 25:3, 359-388

Papers at http://www.statmodel.com/TimeSeries.shtml


http://www.statmodel.com/TimeSeries.shtml

Appendix A: Sample Size Requirements

LTA typically needs large samples (> 1,000)

RI-LTA (limited Monte Carlo simulation study in Muthen-Asparouhov,2020; binary latent class indicators):

T ≥ 3: Good performance at N ≥ 1,000T = 2: N > 4,000 needed

RI-LTA needs larger samples than regular LTA (assuming eachmodel is correct)But if data have been generated by RI-LTA, RI-LTA estimates arebetter than regular LTA at sample size as low as N=500

DSEM (continuous outcomes):

Schultzberg-Muthen (2018). Number of subjects and time pointsneeded for multilevel time series analysis: A simulation study ofdynamic structural equation modeling. SEM journalT ≥ 10 for minimal models, otherwise T ≥ 20-50


Appendix B: RI-LTA Computing time

Timings using different Intel CPUs (hours:minutes:seconds)

i7-7700k i7-7700k i9-9900k i9-9900kReg LTA RI-LTA RI-LTA RI-LTA

Proc=8 Proc=12

Mood (N=494, 00:00:04 00:00:12 - -T=4, 2 classes)

Dating (N=2937, 00:00:41 00:06:00 00:03:31 00:03:25T=3, 5 classes)

Dating (N=2937, 00:05:00 00:40:00 00:21:00 00:19:00T=3, 5 classes,adding 4 covariates)


Appendix C: New Mplus Featuresin Version 8.3 and Version 8.4

http://www.statmodel.com/verhistory.shtml

Mixture modeling:Significant speed improvements for computationally demandingmixture models, particularly with multiple latent class variablessuch as with Latent Transition AnalysisSubstantially simplified output for mixture models with multiplelatent class variables

Bayesian analysis:Significant speed improvements for Bayesian computations usinga new parallelized computing approach (use PROCESSORS = 8)Bayesian estimation of two-level models with latent variableinteractions using the XWITH option. This is especially helpfulfor models with moderation where ML has problemsExpanded Bayesian fit statistics in 3 areas: improved posteriorpredictive p-values (PPP) when there are missing data, BayesianCFI/TLI/RMSEA including confidence intervals, and a Bayesianversion of the Wald test of parameter restrictions using theMODEL TEST command


http://www.statmodel.com/verhistory.shtml

Some Recent Mplus-Related Papers

Asparouhov & Muthen (2019). Random Starting Values and MultistageOptimization (Technical Report)

Asparouhov & Muthen (2019). Bayes Parallel Computation: Choosing thenumber of processors (Technical Report)

Asparouhov & Muthen (2019). Bayesian estimation of single and multilevelmodels with latent variable interactions. Forthcoming in Structural EquationModeling

Asparouhov & Muthen (2019). Advances in Bayesian Model Fit Evaluation forStructural Equation Models. Forthcoming in Structural Equation Modeling

Asparouhov & Muthen (2019). Latent variable centering of predictors andmediators in multilevel and time-series models. Structural Equation Modeling:A Multidisciplinary Journal, 26, 119-142.

Asparouhov & Muthen (2020). Comparison of models for the analysis ofintensive longitudinal data. Structural Equation Modeling: A MultidisciplinaryJournal, 27(2) 275-297

Muthen & Asparouhov (2020). Random intercept latent transition analysis(RI-LTA). Under review in Psychological Methods


Documents

What Multi-Level Modeling Can Teach Us About Single-Level ... · Mplus Web Talks: No. 1 Recorded June 2020 ... Review, critique and recommendations. ... Bengt Muthen´ Mplus Web Talks: