LATENT TRANSITION
ANALYSIS
A Gentle Introduction…. Hopefully
Angela B. Bradford, PhD, LMFT
School of Family Life
Brigham Young University
Background
• Mixture Models (aka “finite mixture models”)- Models
based on the idea that there are multiple
characteristically different sub-populations within the
population, and that those subpopulations are not
directly observable. Mixture models characterize and
estimate parameters for those sub-populations
• Different Genders are directly observable (well…. they used to
be), so that’s not an example; Different kinds of “Awesome
Assistant Professors” may not be directly observable, so a mixture
model could help characterize them
• Latent Transition Analysis is a type of Mixture Model and
an extension of the Latent Class Analysis
“But what is a Latent Class Analysis?” you
ask.
• Latent Class Analysis identifies unobservable groups (or
categories) within a population, using observed variables/
indicators.
• Similar to a factor analysis, but the latent variable is
categorical, rather than continuous.
• LCA usually refers to models in which the indicators are
categorical; Latent Profile Analysis (LPA) usually refers to
the models in which the indicators are continuous.
• Aside from why you’ve theoretically chosen to model continuous
data by “chunking it up” into categories, there are very few- if anypractical differences in modeling and interpretation of results.
Latent Class Analysis/ Latent Profile
Analysis
e1
e2
e3
X1
X2
X3
C
These are your conditional
item probabilities or
loadings…. Will represent
means of each X in an
LPA.
Latent Transition Analysis
e11
e21
e31
e12
e22
e32
X11
X21
X31
X12
X22
X32
C1
C2
This is the transition in
class memberships from
Time 1 to Time 2
So why conduct an LTA?
• Ideal for modeling categorical, multivariate constructs
developmentally (where growth modeling is not as wellsuited)
• Useful for identifying profiles of individuals that move into
categories that are more/less functional, risky, or whatever
• Can thereby inform prevention and intervention efforts
• Others?
How to conduct an LTA
(from Nylund, 2007)
• Step 0: Study descriptive statistics
• Step 1: Study measurement model alternatives for each
•
•
•
•
time point (using LCA / LPA)
Step 2: Explore transitions based on cross-sectional
results; Test measurement invariance if the same number
and types of classes emerge in Step 1
Step 3: Explore specification of the latent transition model
without covariates
Step 4: Include covariates in the LTA model
Step 5: Include distal outcomes and advanced modeling
extensions (e.g., multi-level LTA)
An Example (for Example’s sake only)
• Adolescents from families with conflict/ interpersonal stress
often have negative outcomes, particularly when there are
parent-child problems, such as psychological control (Barber,
1996; Patterson, DeBaryshe, & Ramsey, 1989; Steeger &
Gondoli, 2013; Vinita & Saroj, 2012)
• Many adolescents, however, show resilience to those stressors
• The literature has focused singularly on resilience factors,
rather than profiles of resilience that could inform intervention
• Such factors are internal locus of control, self-esteem, and emotional
regulation (Alvord & Grados, 2005; Masten & Coatsworth, 1998).
Cont.
• Because adolescent romantic relationships are linked to
many facets of well-being (Adler-Baeder, Kerpelman,
Schramm, & Paulk, 2007), teaching them healthy
relationship skills may improve their resilience.
• There has been some work indicating that adolescents
improve in these individual domains after participation in
Relationship Education classes (e.g., Adler-Baeder et al.,
2007). However, no work on whether youth in difficult
family situations develop more functional “profiles” of
resilience after RE participation has been done.
Sample
• Participants are drawn from a sample of adolescents who
received an average of 12 hours of RE as part of the
Alabama Health Marriage and Relationship Education
Initiative (www.alabamamarriage.org).
• Analytic sample (N=125) is comprised of youth who
reported scores ≥1 standard deviation removed from the
mean on measures of parental psychological control,
parent support, and family harmony.
• I can send you demographic information if you really want
it.
Measures
• Internal locus of control
• Self-esteem
• Reflective Coping (similar to emotional regulation)
• Covariates:
• Conflict Management difference score (improvement in conflict
management)
Step 2a: Explore transitions based on cross-sectional results
• Each LCA gives you class counts and proportions; use
these to create a crosstabulation table
Time 1
Time 2
Class 1
Low resilience
.213
.264
Class 2
Moderate
resilience w/ low
ILOC
.787
.736
Step 2b: Test measurement invariance if the same number and
types of classes emerge in Step 1
• You can use the usual Likelihood Ratio Difference Test
(Likelihood Ratio distributed as chi-square, so a delta chisquare test) and the AIC and BIC
• You can have:
• Full Noninvariance- Conditional Item Probabilities (w/ categorical
indicators) or Loadings are freely estimated for each timepoint
• Partial Invariance- Some loadings are constrained others are freely
estimated for each timepoint
• Full Invariance- All loadings are constrained to be equal across
time
• In my case, tests indicated the model should be specified as
fully noninvariant
• This means my classes are different at Time 2 than they were at
Time 1
Step 3: Explore specification of the latent
transition model without covariates
These are latent transition probabilities from Mplus output.
Full Invariance
Full NonInvariance
T2 Class 1
(resilient)
T2 Class 2
(non)
T2 Class 1 T2 Class 2
(non)
(resilient)
T1 Class 1
(resilient)
.805
.195
T1 Class 1
(non)
.292
.708
T1 Class 2
(non)
.126
.874
T1 Class 2
(resilient)
.178
.822
Step 4: Include covariates in the LTA
model
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X11
X21
X31
X12
X22
X32
C1
C2
x
Covariate effects
(multinomial logistic regression)
Latent Class Pattern 1 1
Estimate
C2#1
ON
CONMGT
0.704
S.E.
Estimate/ S.E.
p-value
0.405
1.740
0.082
0.363
1.085
0.278
Latent Class Pattern 2 1
C2#1
ON
CONMGT
0.394
If these were significant, you could look at the odds ratio to help you interpret what
they mean. (E.g., The odds ratio of C2#1 ON conmgt is e(.704) = 2.02, so for every
unit increase in conmgt, people are 102% more likely to stay in class 1 vs. moving
to class 2 at Time 2.)
Relevant Mplus code (LCA or LPA)
IDVARIABLE=ID; !Tells you which of your variables is the identifier for each case
CLASSES = c (2); !Specifies that there should be two classes in the model. When model testing, this is the
value you change as you examine how many classes are the most appropriate
USEVARIABLES T1iloc T1esteem T1refcop; !Which variables are indicators of your classes
MISSING are all(999); !What missing values are in my dataset
ANALYSIS: TYPE = MIXTURE; !It's a mixture model
LRTSTARTS= 50 10 75 20; !If you have difficulty with convergence, you can increased starts and stops
Plot:
type is plot3; !Gives you a plot of the indicator means for each class
series is T1iloc (1) T1esteem (2) T1refcop (3); !Defines which variable is 1st, 2nd, and so on
SAVEDATA: !Saves a new SPSS file that gives you the probability of each case belonging to each of the
classes and the most likely class membership
file is T1classes.sav;
save is cprobabilities;
OUTPUT: SAMPSTAT STANDARDIZED TECH11 TECH14; !The usual; TECH14 gives you the LMR and
BLRT tests of model fit, testing the hypothesis that k-1 classes is better fitting than k classes
LTA, no covariates
IDVARIABLE=id; !Which variable is the identifier
CLASSES = c1 (2) c2(2); !Specifies how many time points (c1 is one and c2 is the second) and how many classes
each timepoint has (2 in each)
USEVARIABLES T1iloc T1esteem T1refcop T2iloc T2esteem T2refcop; !Which variables are indicators of the latent
classes at each timepoint
MISSING are all(999); !Missing values
ANALYSIS: TYPE = MIXTURE; !Still a mixture model
MODEL: %OVERALL% !The overall model specification
c2 ON c1; !regresses c2 on c1 (for the latent transition probabilities)
MODEL c1: !Specification/ measurement model for each class at Time 1
%c1#1% !Time 1, class 1
[T1iloc]; !These loadings/means can be constrained when testing measurement invariance across time
[T1esteem];
[T1refcop];
%c1#2% !Time 1, class 2
[T1iloc];
[T1esteem];
[T1refcop];
LTA specification (cont.)
MODEL c2: !Specification/ measurement model for each class at Time 2
%c2#1% !Time 2, class 1
[T2iloc];
[T2esteem];
[T2refcop];
%c2#2% !Time 2, class 2
[T2iloc];
[T2esteem];
[T2refcop];
OUTPUT: SAMPSTAT STANDARDIZED TECH1 TECH8 TECH15;
LTA with covariates
• Everything up to the model specification is the same (w/ the addition of the covariate to
USEVARIABLES).
MODEL: %OVERALL%
c2 ON c1; !Latent transition specification
c1 ON covariate; !Tests the covariate's influence on c1 (on which the transition probability is dependent)
MODEL c1: %c1#1%
[T1iloc];
[T1esteem];
[T1refcop];
c2 ON covariate; !Tests the interaction of the transition and the covariate for those starting in class 1
%c1#2%
[T1iloc];
[T1esteem];
[T1refcop];
c2 ON covariate; !Tests the interaction of the transition and the covariate for those starting in class 2
MODEL c2:
%c2#1%
[T2iloc];
[T2esteem];
[T2refcop];
%c2#2%
[T2iloc];
[T2esteem];
[T2refcop];
OUTPUT: SAMPSTAT STANDARDIZED TECH1 TECH8 TECH15;
Any other questions?
A couple of helpful sources on LTA:
• Collins, L.M., & Lanza, S.T. (2010). Latent Class and Latent
Transition Analysis: With applications in the social, behavioral,
and health sciences. Hoboken, NJ: John Wiley & Sons, Inc.
• Lanza, S.T., Patrick, M.E., & Maggs, J.L. (2010). Latent
transition analysis: Benefits of a latent variable approach to
modeling transitions in substance use. Journal of Drug Issues,
40, 93-120.
• Nylund (2007) Doctoral dissertation: Latent Transition Analysis:
Modeling Extensions and an Application to Peer Victimization
(Chapter 2): available at
https://www.statmodel.com/download/nylunddis.pdf
References:
• Adler-Baeder, F., Kerpelman, J. L., Schramm, D. G., Higginbotham, B., & Paulk,
•
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•
•
•
•
A. (2007). The impact of relationship education on adolescents of diverse
backgrounds. Family Relations, 56(3), 291-303. doi: 10.1111/j.17413729.2007.00460.x
Alvord, M. K. & Grados, J. J. (2005). Enhancing resilience in children: a proactive
approach. Professional Psychology: Research and Practice, 36(3), 238-245.
doi:10.1037/0735-7028.36.3.238
Barber, B. K. (1996). Parental psychological control: Revisiting a neglected
construct. Child Development, 67(6), 3296-3319. doi:10.2307/1131780
Masten, A.S. & Coatsworth, J.D. (1998). The development of competence in
favorable and unfavorable environments: lessons from research on successful
children. American Psychologist, 53(2), 205-220. doi:10.1037/0003066X.53.2.205
Patterson, G. R., DeBaryshe, B. D., & Ramsey, E. ( 1989). A developmental
perspective on antisocial behavior. American Psychologist, 44, 329– 335.
Steeger, C. M., & Gondoli, D. M. (2013). Mother–adolescent conflict as a mediator
between adolescent problem behaviors and maternal psychological control.
Developmental Psychology, 49(4), 804-814. doi:10.1037/a0028599
Vinita, T., & Saroj, V. (2012). General health of adolescents in relation to
perceived parental psychological control. Social Science International, 28(1), 5973.