alexandre j.s. morin, simon a. houle, & david litalien...panel discussion: increasing our...

Panel Discussion: Increasing our Analytic Sophistication

Person-Centered AnalysesAlexandre J.S. Morin, Simon A. Houle, &

David Litalien

2017 Conference on Commitment

Focus: Mixture Modeling / Generalized Structural Equation Modeling.

Meyer, J.P., & Morin, A.J.S. (2016). A person-centered approach

to commitment research: Theory, research, and

methodology. Journal of Organizational Behavior, 37, 584-612.

Morin, A.J.S. (2016). Person-centered research strategies in

commitment research. In J.P. Meyer (Ed.), The Handbook of

Employee Commitment (p. 490-508). Cheltenham, UK: Edward

Elgar

Morin, A.J.S., & Wang, J.C.K. (2016). A gentle introduction to

mixture modeling using physical fitness data. In N. Ntoumanis, &

N. Myers (Eds.), An Introduction to Intermediate and Advanced

Statistical Analyses for Sport and Exercise Scientists (pp. 183-210).

London, UK: Wiley.

http://smslabstats.weebly.com/

https://www.statmodel.com/

https://www.statisticalinnovations.com/

https://www.r-project.org/

http://www.gllamm.org/

Psychological Constructs Can Be Categorical

The latent variable

is categorical

C

S1 S2 S3 SX…

S1 S2 S3 SX…

F

Factor Analyses

Vs. latent profile analyses

S1 S2 S3 SX…

F

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Person 1 Person 2 Person 3 Sample Mean

Factor (or group of competencies) 1

Factor (or group of competencies) 2

S1 S2 S3 SX…

F1 F2

The latent variable

is categorical C

S1 S2 S3 SX…

0

1

2

3

4

5

6

Profile 1 Profile 2 Profile 3 Profile 4

Competency 1

Competency 2

Competency 3

Competency 4

Factor Analyses

Vs. latent profile analyses

3 Characteristics of Person-Centered Models

(Mixture Models)

• Typological

• Prototypical

• Exploratory

Typological: Resulting in a classification system designed to help

categorize individuals more accurately into qualitatively and

quantitatively distinct subpopulations.

0

1

2

3

4

5

6


Competency 1

Competency 2

Competency 3

Competency 4

• Prototypical: With all participants having a probability of

membership in all profiles based on their degree of similarity

with the profiles’ specific configurations.

0

1

2

3

4

5

6


Competency 1

Competency 2

Competency 3

• Exploratory: Although exceptions exist. For this reason, a

class-enumeration process typically needs to be conducted –

and multiple random starts need to be integrated. Model LL #fp AIC CAIC BIC ABIC Entropy

Organizational Commitment

1 Profile -4704.605 6 9421.211 9457.135 9451.135 9432.078 Na

2 Profiles -4586.674 13 9199.347 9274.185 9264.185 9222.894 0.556

3 Profiles -4467.097 20 8974.193 9087.943 9073.943 9010.419 0.688

4 Profiles -4401.543 27 8857.086 9009.748 8991.748 8905.990 0.705

5 Profiles -4364.052 34 8796.105 8987.679 8965.679 8857.688 0.681

6 Profiles -4320.230 41 8722.460 8952.947 8926.947 8796.722 0.712

7 Profiles -4301.966 48 8699.932 8969.331 8939.331 8786.873 0.726

8 Profiles -4292.316 55 8694.632 9002.944 8968.944 8794.253 0.753

9 Profiles -4267.496 62 8658.992 9006.217 8968.217 8771.292 0.721

10 Profiles -4258.751 69 8655.501 9041.638 8999.638 8780.480 0.732

7000

7500

8000

8500

9000

9500

1 2 3 4 5 6 7 8 9 10

Number of Latent Profiles

AIC

CAIC

BIC

ABIC

Method Vs. Application

By exploratory, I am referring to the method itself which, like

EFA (or ESEM) is exploratory in nature (all items are associated

with all latent factors or all latent categories). In addition,

absolute fit indices are not available with mixture models.

However, the applications of both methods (e.g., using target

rotation or Bayes prior for EFA/ESEM) can be confirmatory and

theory-driven.

C

S1 S2 S3 SX…

X

XC

Y

YC

C

S1 S2 S3 SX…

C

S1 S2 S3 SX…

FC

Generalized SEM

Latent Profile Analysis

K

k

ykky

K

k

ykky1

22

1

2)(

C

S1 S2 S3 SX…

Mixture Indicators?• Anything goes, and any form of combination.

• Mixture models are not affected by the scale of the

indicators (contrary to cluster analyses).

• In theory, fully latent mixture models can be

estimated where items are used to estimate factors,

which are themselves used to estimate profiles.

• But mixture models are very complex, and

computer-intensive. So they are more typically

estimated using scale scores.

Scale Scores ? • What about measurement errors???

• Why not use factor scores:

o Provide a partial control for measurement error

o Forces you to demonstrate that the measurement model fits

the data well.

o Preserves the nature of the measurement model: invariance,

cross loadings, method factors, etc.

o Naturally standardized. Morin, A.J.S., Meyer, J.P., Creusier, J., & Biétry, F. (2016). Multiple-group analysis of similarity in latent profile

solutions. Organizational Research Methods, 19, 231-254.

Morin, A.J.S., & Marsh, H.W. (2015). Disentangling Shape from Levels Effects in Person-Centred Analyses:

An Illustration Based University Teacher Multidimensional Profiles of Effectiveness. Structural Equation

Modeling, 22, 39-59.

Morin, A.J.S., Boudrias, J.-S., Marsh, H.W., Madore, I., & Desrumaux, P. (2016). Further reflections on

disentangling shape and level effects in person-centered analyses: An illustration aimed at exploring the

dimensionality of psychological health. Structural Equation Modeling, 23, 438-454

Morin, A.J.S., Boudrias, J.-S., Marsh, H.W., McInerney, D.M., Dagenais-Desmarais, V., & Madore, I. (In Press).

Complementary variable- and person-centered approaches to exploring the dimensionality of

psychometric constructs: Application to psychological wellbeing at work. Journal of Business and

Psychology. EarlyView DOI 10.1007/s10869-016-9448-7

Basic Mplus input:DATA:

FILE IS Commitment.dat;

VARIABLE:

NAMES = ID Status unit Pred1 Pred2 Cor1 Cor2 Out1 Out2 AC1

NC1 CC1 AC2 NC2 CC2;

USEVARIABLES = AC1 NC1 CC1;

MISSING = all (999);

IDVARIABLE = ID;

CLASSES = c (3);

Analysis:

TYPE = MIXTURE;

ESTIMATOR = MLR;

PROCESS = 3;

STARTS = 3000 100;

STITERATIONS = 100;

CLASSES: serves to identify the number of latent profiles to be estimated.

STARTS: serves to identify the number of random starts (first number) and the number retained for final optimization (second number)

STITERATIONS : Identifies the number of iterations for each random starts

Basic Mplus input:DATA:


VARIABLE:

NAMES = ID Status unit Pred1 Pred2 Cor1 Cor2 Out1 Out2 AC1




IDVARIABLE = ID;

CLASSES = c (3);

Analysis:

TYPE = MIXTURE;

ESTIMATOR = MLR;

PROCESS = 3;

STARTS = 3000 100;

STITERATIONS = 100;

PROCESS: Your best friend with mixture models… Allows you to use multiple processors in parallel.

RANDOM STARTS RESULTS RANKED FROM THE BEST TO THE WORST

LOGLIKELIHOOD VALUES

15 perturbed starting value run(s) did not converge.

Final stage loglikelihood values at local maxima, seeds, and initial stage start numbers:

-6095.887 991399 1433

-6095.887 165268 2436 The best loglikelihood needs to be

-6095.887 551639 55 replicated. This is the case here.

-6095.887 58353 1723

-6095.887 168648 1788

-6095.887 445692 2796

-6096.828 762461 425

-6096.828 62715 2599

-6104.647 264951 1130

-6104.647 930872 277

-6104.647 670998 1876

-6104.698 98068 998

-6105.491 361131 2563

-6105.491 293428 2473

-6105.491 18741 2970

…

Model section for LPAMODEL:

%OVERALL%

%c#1%

[AC1 NC1 CC1];

AC1 NC1 CC1;

%c#2%

[AC1 NC1 CC1];

AC1 NC1 CC1;

%c#3%

[AC1 NC1 CC1];

AC1 NC1 CC1;

%OVERALL%This section is used to describe parts of the model that are common to all profiles.

%C#1% to %C#k% This section is used to describe parts of the model that are specific to each profiles.

MODEL:

%OVERALL%

%c#1%

[AC1 NC1 CC1];

AC1 NC1 CC1;

%c#2%

[AC1 NC1 CC1];

AC1 NC1 CC1;

%c#3%

[AC1 NC1 CC1];

AC1 NC1 CC1;

MODEL:

%OVERALL%

%c#1%

[AC1 NC1 CC1];

%c#2%

[AC1 NC1 CC1];

%c#3%

[AC1 NC1 CC1];

MODEL:

%OVERALL%

%c#1%

[AC1 NC1 CC1];

AC1 NC1 CC1 (v1-v3);

%c#2%

[AC1 NC1 CC1];


%c#3%

[AC1 NC1 CC1];


Peugh, J. & Fan, X. (2013). Modeling unobserved heterogeneity using latent profile analysis: A Monte Carlo simulation. Structural Equation Modeling, 20, 616-639.

To request the free estimation of the means and variances of the indicators in each profile

Mplus default: Variances are specified as equal across classes. Right: short-cut syntaxLeft: Complete syntax

Unequal variances Equal Variances

0

1

2

3

4

5

6

7

0

1

2

3

4

5

6

7

Suggested end of the inputOUTPUT:

SAMPSTAT CINTERVAL RESIDUAL SVALUES TECH1 TECH7 TECH11 TECH14;

! To save the probabilities of class membership for each participants according to the ID

variable in a new data set. Name this data set as you want but make sure to change

the name from one model to the next.

SAVEDATA:

FILE IS LPA_5_final.dat;

FORMAT IS FREE;

SAVE=CPROBABILITIES;

Morin, A.J.S., Morizot, J., Boudrias, J.-S., & Madore, I. (2011). A multifoci person-centered perspective on workplace affective commitment: A latent profile/factor mixture Analysis. OrganizationalResearch Methods, 14 (1), 58-90.

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

Profile 1 Profile 2 Profile 3 Profile 4 Profile 5

OrganizationSupervisorGroupJobCareerWorkCustomers

7%workplace-committed

31%supervisor-committed

17%career-committed

19%Uncommited

5%committed

Class Enumeration• Mixture models are essentially exploratory.

• The typical approach to determine whether profiles are

needed, and the number of profiles that is needed is called the

class enumeration process.

• Essentially, solutions varying from 1 to some number of

profiles are estimated, and contrasted.

• “Some” number: This is typically determined based on the

following considerations (with no absolute rules):

o What is the range of profiles that you can expect based on

the literature: Add at least 2-3 to this number.

o Sample size: smaller samples makes it possible to identify

fewer profiles.

Class Enumeration

Criteria to guide the selection of the optimal solution:

o Theoretical Conformity and Value

o Statistical adequacy

oDoes adding one profile adds something

meaningful to the solution ? OR does it simply

split an existing profiles into two with a similar

shape that is simply more or less extreme?

o The size of the added profiles: Tricky criteria.

• Some mention 1% or 5%.

• But this really depends on the sample size.

• You want a solution that is stable… Profiles smaller

than 25, or perhaps ideally 50, participants are

generally desirable.

Class Enumeration

o Statistical indicators

• Akaike Information Criteria (AIC)

• Consistent AIC (CAIC)

• Bayesian Information Criterion (BIC)

• Sample size adjusted BIC (ABIC)

• A lower value on these indicators suggest a better

fitting model. How much lower ? No clear guideline

exists here.

• Simulation studies shows that the AIC does not

work.

• The CAIC is calculated by hand: BIC + number of

free parameters.

Class Enumeration


• The model log likelihood could be used to calculate a

chi-square difference test (as in SEM), to compare

nested models including the same number of profiles,

but not models including a different number of profiles.

• The Lo-Mendell-Rubin (LMR) and the Adjusted LMR

(aLMR) likelihood ratio test. Typically both give the

same consultation and only one is reported. TECH 11

• The Bootstrap Likelihood Ratio Rest (BLRT).TECH 14

• A non-significant LMR, aLMR, BLRT supports the

model with one fewer class.

• Simulation studies shows that the LMR-aLMR do not

work well.

Class Enumeration


• Entropy: Varies between 0 and 1 and summarizes the

classification accuracy of participants into profiles.

• i.e. when most participants have a probability close

to 1 of membership into their most likely profile and

a probability close to 0 of membership into the other

profiles, the entropy will be close to 1.

• This is also an indicator of class separation.

• The entropy should never be used to select the final

solution. Mixture models can be fully proper even

when the classification is more “fuzzy”, it just means

that correspondences to prototypes are not as clear.

Class Enumeration

LMR, aLMR, BLRT• These tests compare a k-profile solution with a k-1-profile

solution.

• As such, the LogLikelihood associated with the k-1 profile

solution should correspond to the true k-1 profile solution,

and at least some of the BLRT (bootstrapped) replications

should converge properly.

• When this does not happen, you may want to increase the

number of random starts and iterations specifically used for

these tests.

Analysis:

TYPE = MIXTURE;

ESTIMATOR = MLR;

PROCESS = 3;

STARTS = 3000 100;

STITERATIONS = 100;

K-1STARTS = 3000 100;

LRTSTARTS = 3000 100 3000 100;

LRTBOOTSTRAP = 100;

OUTPUT:

SAMPSTAT CINTERVAL RESIDUAL SVALUES TECH1 TECH7 ;

TECH11 TECH14;

K-1STARTS : For LMR/ALMR and BLRT. Specify the number of random starts for the model with one less profile.

LRTSTARTS : For BLRT. Specify the number of random starts for the model itself (second) and the model with one less profile (first).

LRTBOOTSTRAP : For BLRT. Specify the number of bootstrap draws

Shortcut?Conduct the class enumeration process without requesting

LMR/aLMR and BLRT. Then pick the best final solution for each

number of profiles and use either the seed, or the starts values,

to re-estimate this solution while turning the random start

function to 0 and only focusing on the k-1 starts,

1. Using the seed

RANDOM STARTS RESULTS RANKED FROM THE BEST TO THE WORST

LOGLIKELIHOOD VALUES

15 perturbed starting value run(s) did not converge.

Final stage loglikelihood values at local maxima, seeds, and initial stage start numbers:

-6095.887 991399 1433

-6095.887 165268 2436

-6095.887 551639 55

-6095.887 58353 1723

-6095.887 168648 1788

-6095.887 445692 2796

-6096.828 762461 425

-6096.828 62715 2599

-6104.647 264951 1130

-6104.647 930872 277

-6104.647 670998 1876

-6104.698 98068 998

-6105.491 361131 2563

-6105.491 293428 2473

-6105.491 18741 2970

…

Analysis:

TYPE = MIXTURE;

ESTIMATOR = MLR;

PROCESS = 3;

STARTS = 0;

OPTSEED = 991399;

K-1STARTS = 3000 100;

2. Using the starts values

MODEL COMMAND WITH FINAL ESTIMATES USED AS STARTING VALUES

%OVERALL%

[ c#1*-0.93515 ];

[ c#2*0.49113 ];

%C#1%

[ CC1*0.39664 ];

[ NC1*-0.93155 ];

[ AC1*-1.59367 ];

CC1*1.59137;

NC1*2.12542;

AC1*0.53789;

%C#2%

[CC1*-0.15201 ];

[NC1*-0.29812 ];

[AC1*-0.18248 ];

CC1*0.72513;

NC1*0.43116;

AC1*0.42376;

%C#3%

[CC1*0.08333 ];

[NC1*0.86526 ];

[AC1*0.94111 ];

…

AND

Analysis:TYPE = MIXTURE; ESTIMATOR = MLR;PROCESS = 3; STARTS = 0; K-1STARTS = 3000 100;

Class Enumeration: In Practice• The various indicators will very seldom converge on the same

solution, hence the importance of considering all of them, as

well as the other criteria (theoretical and statistical adequacy,

added-value, size, etc.).

• Often, the indicators will even keep on decreasing (AIC,

CAIC, BIC, ABIC) or being significant (LMR, aLMR, BLRT)

when profiles are added, up until the maximum number of

profiles that you have decided to estimate, and even when

solutions stop being proper statistically (revealing even

“empty” profiles.

• The information criteria can then be represented by an elbow

plot. In these plots, the key is not to locate the first angle (as

in the scree plot for EFA), but the point after which a plateau

is reached (decreases becomes negligible).

Model LL #fp AIC CAIC BIC ABIC Entropy

Organizational Commitment 1 Profile -4704.605 6 9421.211 9457.135 9451.135 9432.078 Na

2 Profiles -4586.674 13 9199.347 9274.185 9264.185 9222.894 0.556

3 Profiles -4467.097 20 8974.193 9087.943 9073.943 9010.419 0.688 4 Profiles -4401.543 27 8857.086 9009.748 8991.748 8905.990 0.705

5 Profiles -4364.052 34 8796.105 8987.679 8965.679 8857.688 0.681


8 Profiles -4292.316 55 8694.632 9002.944 8968.944 8794.253 0.753


7000

7500

8000

8500

9000

9500

1 2 3 4 5 6 7 8 9 10

Number of Latent Profiles

AIC

CAIC

BIC

ABIC

Model LL #fp AIC CAIC BIC ABIC Entropy aLMR BLRT

1 -9582 12 19188 19262 19250 19212 Na Na Na

2 -8824 25 17827 17852 17827 17747 0.784 ≤ 0.001 ≤ 0.001

3 -8556 38 17189 17422 17383 17263 0.832 ≤ 0.001 ≤ 0.001

4 -8430 51 16964 17276 17225 17063 0.821 0.0237 ≤ 0.001

5 -8353 64 16834 17225 17161 16958 0.844 ≤ 0.001 ≤ 0.001

6 -8301 77 16757 17228 17151 16906 0.792 0.013 ≤ 0.001

7 -8277 90 16733 17283 17193 16908 0.784 0.024 ≤ 0.001

8 -8227 103 16660 17290 17187 16859 0.741 1.000 ≤ 0.001

Output1. Is the log likelihood replicated ?

Final stage loglikelihood values at local maxima, seeds, and initial stage start

numbers:

-2217.891 804821 3936

-2217.891 743830 1368

-2217.891 50983 834

-2217.891 816796 1728

-2217.891 619846 3528

-2217.891 554395 2195

-2217.891 822698 621

-2217.891 674419 2870

-2217.891 657909 2912

-2217.891 968846 970

-2217.891 714455 476

-2217.891 112441 3926

-2217.891 786248 2076

…

Output2. Model fit

THE MODEL ESTIMATION TERMINATED NORMALLY

MODEL FIT INFORMATION

Number of Free Parameters 20

Loglikelihood

H0 Value -2217.891

H0 Scaling Correction Factor 0.853

for MLR

Information Criteria

Akaike (AIC) 4475.783

Bayesian (BIC) 4559.752

Sample-Size Adjusted BIC 4496.272

(n* = (n + 2) / 24)

Output2. Model fit

GO TO THE END OF THE OUTPUT

TECHNICAL 11 OUTPUT

Random Starts Specifications for the k-1 Class Analysis Model

Number of initial stage random starts 5000

Number of final stage optimizations 200

VUONG-LO-MENDELL-RUBIN LIKELIHOOD RATIO TEST FOR 2 (H0) VERSUS 3 CLASSES

H0 LoglikelihoodValue -2717.569

2 Times the Loglikelihood Difference 999.355

Difference in the Number of Parameters 7

Mean 29.905

Standard Deviation 54.016

P-Value 0.0000

LO-MENDELL-RUBIN ADJUSTED LRT TEST

Value 976.842

P-Value 0.0000

Output2. Model fit

TECHNICAL 14 OUTPUT

Random Starts Specifications for the k-1 Class Analysis Model

Number of initial stage random starts 5000

…Number of bootstrap draws requested 100

PARAMETRIC BOOTSTRAPPED LIKELIHOOD RATIO TEST FOR 2 (H0) VERSUS 3 CLASSES

H0 LoglikelihoodValue -2717.569

2 Times the Loglikelihood Difference 999.355

Difference in the Number of Parameters 7

Approximate P-Value 0.0000

Successful Bootstrap Draws 0WARNING: THE BEST LOGLIKELIHOOD VALUE WAS NOT REPLICATED IN 93 OUT OF 100

WARNING: 100 OUT OF 100 BOOTSTRAP DRAWS DID NOT CONVERGE.

THE P-VALUE MAY NOT BE TRUSTWORTHY.

INCREASE THE NUMBER OF RANDOM STARTS USING THE LRTSTARTS OPTION.

Output3. Class Counts

FINAL CLASS COUNTS AND PROPORTIONS FOR THE LATENT CLASSES

BASED ON THE ESTIMATED MODEL

1 30.99528 0.06300

2 429.02931 0.87201

3 31.97540 0.06499

FINAL CLASS COUNTS AND PROPORTIONS FOR THE LATENT CLASS PATTERNS

BASED ON ESTIMATED POSTERIOR PROBABILITIES

1 30.99528 0.06300

2 429.02931 0.87201

3 31.97540 0.06499

CLASSIFICATION QUALITY Entropy 1.000

CLASSIFICATION OF INDIVIDUALS BASED ON THEIR MOST LIKELY LATENT CLASS

MEMBERSHIP

1 31 0.06301

2 429 0.87195

3 32 0.06504

Output3. Class Counts

FINAL CLASS COUNTS AND PROPORTIONS FOR THE LATENT CLASSES

BASED ON THE ESTIMATED MODEL1 30.99528 0.06300

2 429.02931 0.87201

3 31.97540 0.06499

FINAL CLASS COUNTS AND PROPORTIONS FOR THE LATENT CLASS PATTERNS

BASED ON ESTIMATED POSTERIOR PROBABILITIES1 30.99528 0.06300

2 429.02931 0.87201

3 31.97540 0.06499

CLASSIFICATION QUALITY Entropy 1.000

CLASSIFICATION BASED ON MOST LIKELY CLASS MEMBERSHIP1 31 0.06301

2 429 0.87195

3 32 0.06504

Fully model based estimate of the size of each profile (n = 30.999; 6.3%)

Based probability of class membership. Typically equal to model-based.

Based on most likely class assignment. Less accurate.

Output4. Classification Summary

Average Latent Class Probabilities for Most Likely Latent Class Membership (Row)

by Latent Class (Column)

1 2 3

1 1.000 0.000 0.000

2 0.000 1.000 0.000

3 0.000 0.001 0.999

Output5. Results

For LPA, stick with the unstandardized results.

For estimates of correlations or regression, then also look at the STDYX results.

MODEL RESULTS

Two-Tailed

Estimate S.E. Est./S.E. P-Value

Latent Class 1

Means

FAC 1.139 0.002 460.118 0.000

FCC 4.183 0.314 13.324 0.000

FNC 1.213 0.016 74.366 0.000

Variances

FAC 0.000 0.000 7.395 0.000

FCC 3.052 0.373 8.187 0.000

FNC 0.008 0.001 7.307 0.000

…

Reporting• Raw scores: Impossible to compare across scales or studies.

• Normed scores: Ideal, but seldom available.

• Standardized scores (no need to transform if using factor

scores): Easily interpretable, but also sample specific.

Recommendation:

• Histogram (do not use line graphs, these are impossible to

interpret) using standardized scores for the means

• Table for the exact mean and variance results, with confidence

intervals.

Profile 1 Profile 2 Profile 3 Profile 4 Profile 5 Profile 6 Profile 6

Mean [CI] Mean [CI] Mean [CI] Mean [CI] Mean [CI] Mean [CI] Mean [CI]

Affective Commitment -.028

[-.284; .228]

-.792

[-1.065; -.519]

-.435

[-.632; -.239]

-1.623

[-1.753; -1.492]

.393

[.102; .684]

.632

[.525; .740]

.945

[.863; 1.027]

Normative Commitment .088

[-.093; .268]

-.854

[-1.083; -.625]

-.406

[-.526; -.286]

-1.445

[-1.587; -1.304]

-.010

[-.251; .232]

.555

[.424; .685]

1.019

[.855; 1.184]

Continuance Commitment:

High Sacrifice

1.050

[.898; 1.203]

-.960

[-1.053; -.866]

-.081

[-.183; .020]

-.702

[-.923; -.481]

-.604

[-.927; -.282]

.496

[.364; .627]

1.639

[1.533; 1.744] Continuance Commitment:

Low Alternatives

1.427

[1.277; 1.577]

-.364

[-.466; -.261]

.379

[.216; .543]

.777

[.535; 1.020]

-.811

[-1.094; -.529]

.172

[-.007; .350]

1.289

[1.076; 1.501]

Profile 1 Profile 2 Profile 3 Profile 4 Profile 5 Profile 6 Profile 6 Variance

[CI]

Variance

[CI]

Variance

[CI]

Variance

[CI]

Variance

[CI]

Variance

[CI]

Variance

[CI]

Affective Commitment .277

[.058; .495]

.286

[.198; .374]

.245

[.159; .331]

.277

[.202; .351]

.281

[.126; .436]

.156

[.129; .182]

.065

[.036; .095]

Normative Commitment .317

[.201; .433]

.227

[.155; .300]

.205

[.168; .242]

.300

[.207; .394]

.393

[.243; .544]

.207

[.146; .267]

.216

[.126; .306]

Continuance Commitment: High Sacrifice

.033 [-.016; .082]

.028 [.004; .051]

.139 [.107; .171]

.598 [.353; .843]

.241 [.152; .330]

.140 [.099; .181]

.105 [.074; .135]

Continuance Commitment:

Low Alternatives

.082

[-.086; .251]

.060

[.009; .111]

.214

[.168; .259]

.833

[.577; 1.089]

.214

[.144; .284]

.274

[.198; .350]

.408

[.276; .540]

BUT

HOF

C

S1 S2 S3 SX…

Factor mixture analysis.

Shape Versus Level• Profiles should present qualitative (shape) differences.

• Ordered profiles, showing only quantitative level differences,

would be better represented by variable-centered methods.

• In some cases both level and shape effects can be strong,making the identification of qualitatively distinct profilesharder since strong level effects tend to create equally strongquantitative differences.

• For instance, some employees may be globally good, or badacross dimensions of effectiveness when compared to others.

• However these profiles may still present relative areas ofstrength and weaknesses worthy of consideration.

• Unfortunately, these specific areas will be harder to identifygiven that even these weaknesses may be relatively strongcompared to the levels observed in weaker profiles.

-2

-1.5

-1

-0.5

0

0.5

1

1.5

Class 1 Class 2 Class 3 Class 4

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

Class 1 Class 2 Class 3 Class 4

Morin, A.J.S., & Marsh, H.W. (2015). Disentangling Shape from

Levels Effects in Person-Centred Analyses: An Illustration Based

University Teacher Multidimensional Profiles of Effectiveness.

Structural Equation Modeling, 22, 39-59.

Model 1C

S1 S2 S3 SX…

X

Model 1: Classical latent profile analysis.

MODEL:

%OVERALL%

VAR1 VAR2 VAR3 VAR4 VAR5 VAR6 VAR7 VAR8 VAR9;

[VAR1 VAR2 VAR3 VAR4 VAR5 VAR6 VAR7 VAR8 VAR9];

VAR1-VAR9 ON CONT;

! Due to the control, conditional independence needs to be specified.

VAR1 WITH VAR2-VAR9@0; VAR2 WITH VAR3-VAR9@0;



VAR7 WITH VAR8-VAR9@0; VAR8 WITH VAR9@0;

%c#1%



%c#2%



[…]

Model 3

HOF

C

S1 S2 S3 SX…

X

Model 3: Factor mixture analysis with a class

invariant higher order latent factor.

MODEL:

%OVERALL%



VAR1-VAR9 ON CONT;




VAR7 WITH VAR8-VAR9@0; VAR8 WITH VAR9@0;

HOF BY VAR1* VAR2 VAR3 VAR4 VAR5 VAR6 VAR7 VAR8 VAR9;

[HOF@0]; HOF@1;

%c#1%



%c#2%



[…]

-2

-1.5

-1

-0.5

0

0.5

1

1.5


F1 = Learning/value

F2 = Enthusiasm

F3 = Group interaction

F4 = Organization/Clarity

F5 = Individual Rapport

F6 = Workload/difficulty

F7 = Breadth of Coverage

F8 = Exams/grading

F9 = Assignments/readings

Results from the latent profiles models based on 9 factors (Model 1)

-1.5

-1

-0.5

0

0.5

1


F1 = Learning/value

F2 = Enthusiasm

F3 = Group interaction

F4 = Organization/Clarity

F5 = Individual Rapport

F6 = Workload/difficulty

F7 = Breadth of Coverage

F8 = Exams/grading

F9 = Assignments/readings

Results from the factor mixture models based on 9 factors (Model 3)

Poor (11%)

Good (25%)Cognitive

Performance Oriented

(25%)

Relations and Performance

Oriented (20%)

Cognitive and Mastery

Oriented (19%)

BUTThe Factor Mixture approach also assumes that the mean level

on the global factor is equal across all profiles, and corresponds

to the sample average.

Morin, A.J.S., Boudrias, J.-S., Marsh, H.W., Madore, I., & Desrumaux, P. (2016). Further reflections on disentangling shape and level effects in person-centered analyses: An illustration aimed at exploring the dimensionality of psychological health. Structural Equation Modeling, 23, 438-454

Morin, A.J.S., Boudrias, J.-S., Marsh, H.W., McInerney, D.M., Dagenais-Desmarais, V., & Madore, I. (In Press). Complementary variable- and person-centered approaches to exploring the dimensionality of psychometric constructs: Application to psychological wellbeing at work. Journal of Business and Psychology. Early View DOI 10.1007/s10869-016-9448-7

The World Health Organization (2014) defines psychological

health as a state characterized not only by the absence of

psychological distress, but also by the presence of psychological

wellbeing.

Wellbeing

Health

+ -

Distress

Psychological Health at Work

Preliminary results showed that a ESEM model clearly fitted the

data better than a CFA model, and that a bifactor-ESEM model

was even better.

Profiles were calculated from factor scores saved from the ESEM

model, as well as from the Bifactor-ESEM model.

Psychological Wellbeing:• Harmony (BHAR)• Serenity (BSER)• Involvement (BIMP)

Psychological Distress:• Irritability/ aggression (DIRR)• Anxiety/depression (DANX)• Withdrawal (DDES)

Latent Profile Model

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0


Harmony

Serenity

Involvement

Irritability

Anxiety/Depression

Distance

PURE LEVELArgues for a single dimension, and for the idea that wellbeing and distress are opposite ends

of the same continuum

Factor Mixture Model

Latent Profile Model (Bifactor)

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

Class 1 Class 2 Class 3 Class 4 Class 5

G-Factor

BHAR

BSER

BIMP

DIRR

DANX

DDES

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0


Global Psychological Health

Harmony

Serenity

Involvement

Irritability

Anxiety/Depression

Distance

Normative Profile: 61% of the sample for which health is best represented as a single dimension (with inter-individual variability)

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0



Harmony

Serenity

Involvement

Irritability

Anxiety/Depression

Distance

Adapted Profile: 11% of the sample who have high level of global health, do not feel any distress, but are also not really thriving

Flourishing Profile: Only 1.38% of the sample who really thrive. Suggesting that not feeling any distress is critical to being in good health, but that high level of wellbeing are necessary to thrive.

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0



Harmony

Serenity

Involvement

Irritability

Anxiety/Depression

Distance

Stressfully Involved Profile: 14% of the sample who appear to cope with work stress by increasing involvement

Harmoniously Distanced Profile: 12% of the sample who appear to cope with work stress by distancing themselves, thus restoring some level of harmony and serenity.

Factor Mixture Models

C

S1 S2 S3 SX…

F

HOF

C

S1 S2 S3 SX…

Factor Mixture Models• Can be used as a multiple-group CFA model to test for the

invariance of a measure across unobserved groups of

participants.

• I.e. the profiles are used to locate “response sets”.

• Carter, N.T., Dalal, D.K., Lake, C.J., Lin, B.C., & Zickar, M.J. (2011). Using

mixed-model item response theory to analyze organizational survey

responses: An illustration using the Job Descriptive Index. Organizational

Research Methods, 14, 116-146.

• Tay, L., Diener, E., Drasgow, F., & Vermunt, J.K. (2011). Multilevel mixed-

measurement IRT analysis: An explication and application to self-reported

emotions across the world. Organizational Research Methods, 14, 177-207.

• Tay, L., Drasgow, D.A., & Vermunt, J.K. (2011). Using mixed-measurement

item response theory with covariates (MM-IRT-C) to ascertain observed

and unobserved measurement equivalence. Organizational Research

Methods, 14, 147-176.

Factor Mixture Models• Can be used as an overarching framework to explore the

“true” underlying nature of psychological constructs as

continuous, ordinal, nominal, etc.

• Lubke, G.H., & Muthén, B.O (2005). Investigating population heterogeneity

with factor mixture models. Psychological Methods, 10, 21-39.

• Borsboom, D., Rhemtulla, M., Cramer, A.O.J., van der Maas, H.L.J., Scheffer,

M., & Dolan, C.V. (2016). Kinds versus continua: a review of psychometric

approaches to uncover the structure of psychiatric constructs. Psychological

Medicine, 46, 1567-1579.

• Clark, S.L., Muthén, B.O., Kaprio, J., D’Onofrio, B.M., Viken, R., & Rose, R.J.

(2013). Models and strategies for factor mixture analysis: An example

concerning the structure of underlying psychological disorders. Structural

Equation Modeling, 20, 681-703.

• Masyn, K., Henderson, C., & Greenbaum, P. (2010). Exploring the latent

structures of psychological constructs in social development using the

Dimensional-Categorical Spectrum. Social Development, 19, 470–493.

CFA Vs. latent profile analyses

Choosing between CFA and LPA is not easy since a

k-class LPA model has identical covariance

implications than a k-1 common factor model and

thus represents an equivalent model (Bauer &

Curran, 2004; Steinley & McDonald, 2007).

Construct Validation

Construct validation is the only way to demonstrate

the meaningfulness of latent profile solutions

oHeuristic Value






oHeuristic Value



o Shows meaningful and well-differentiated

relations to key covariates:

• Predictors

• Correlates

• Outcomes.

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5


Age

Tenure

Relations colleagues

Relations supervisor

Justice

Satisfaction


17%career-committed


19%Uncommited

5%committed

-1.0

-0.5

0.0

0.5

1.0


In-role performance

OCB tasks

OCB group

OCB organization

OCB supervisor

OCB customers

Intent to quit


17%career-committed


19%Uncommited

5%committed

Predictors and Outcomes

C

S1 S2 S3 SX…

X

XC

Y

YC

Indicator

Multinomial logistic

regressionThis is a key advantage of mixture models, relative to

cluster analyses.


C

S1 S2 S3 SX…

X

XC

Y

YC

But these should not change the nature of the profiles.

Indicator


regression

Always conduct the class enumeration process before adding covariates (Diallo et

al., 2016).

Solution 1. Use the start values from the retained solution and

turn off the random start function


C

S1 S2 S3 SX…

X

XC

Y

YC

But these should not change the nature of the

profiles.

Solution 2: Auxiliary functions: (E) For simple correlates(DCON) (DCAT) for outcomes(R3STEP) for predictors(DU3STEP) (DE3STEP) for continuous outcomes(BCH) for continuous outcomes

Indicator


regression

Integrating covariates to the final retained

solution

Using the starts values

MODEL COMMAND WITH FINAL ESTIMATES USED AS STARTING VALUES

%OVERALL%

[ c#1*-0.93515 ];

[ c#2*0.49113 ];

%C#1%

[ CC1*0.39664 ];

[ NC1*-0.93155 ];

[ AC1*-1.59367 ];

CC1*1.59137;

NC1*2.12542;

AC1*0.53789;

%C#2%

[CC1*-0.15201 ];

[NC1*-0.29812 ];

[AC1*-0.18248 ];

CC1*0.72513;

NC1*0.43116;

AC1*0.42376;

%C#3%

[CC1*0.08333 ];

[NC1*0.86526 ];

[AC1*0.94111 ];

…

AND

Analysis:TYPE = MIXTURE; ESTIMATOR = MLR;PROCESS = 3; STARTS = 0;

%OVERALL%

[ c#[email protected] ];

[ c#[email protected] ];

%C#1%

[ [email protected] ];



[email protected];

[email protected];

[email protected];

%C#2%

[[email protected] ];



[email protected];

[email protected];

[email protected];

%C#3%




…

AND

Analysis:TYPE = MIXTURE; ESTIMATOR = MLR;PROCESS = 3; STARTS = 0;

DATA: FILE IS Commitment.dat;

VARIABLE:

NAMES = ID unit Pred1 Pred2 Cor1 Cor2 Out1 Out2 AC1NC1 CC1 AC2 NC2 CC2;

USEVARIABLES = AC1 NC1 CC1 Pred1 Pred2 ;

MISSING = all (999); IDVARIABLE = ID; CLASSES = c (3);

Analysis: […]MODEL:

%OVERALL%

c#1-c#2 ON Pred1 Pred2 ;

%c#1%

[AC1 NC1 CC1];

AC1 NC1 CC1;

%c#2%

[AC1 NC1 CC1];

AC1 NC1 CC1;

%c#3%

[AC1 NC1 CC1];

AC1 NC1 CC1;

Always one less class statement than the total number of profiles (c#1-c#2 with 3 profiles, c#1-c#3 with 4 profiles, etc.)

Approaches 1-2: Predictors

Multinomial logistic regressionCategorical Latent Variables

C#1 ON

Pred1 1.122 0.409 2.745 0.006

Pred2 -0.012 0.076 -0.153 0.879

C#2 ON

Pred1 0.806 0.390 2.067 0.039

Pred2 0.133 0.082 1.620 0.105

LOGISTIC REGRESSION ODDS RATIO RESULTS

Categorical Latent Variables

C#1 ON

Pred1 3.072

Pred2 0.988

C#2 ON

Pred1 2.238

Pred2 1.142

• All of these coefficients refer to the likelihood of being

member of the target profile relative to the referent profile

(always the last one). Pairwise comparisons.

• These coefficients can be converted to easier to interpret

odds ratio by taking the antilogarithm of the unstandardized

regression coefficient OR = ln-1(coef) = ecoef. Easy to calculate

in Excel using the EXP function.

• An odds ratio indicates, for each unit increase in the

coefficient, how much the likelihood of membership into the

target group has increased (recovery) relative to the

comparison group.

• OR = 1.5: 1.5 times more likely (50% more likely)

• OR = -.5: 50% less likely; -.33 = 3 times less likely.

ALTERNATIVE PARAMETERIZATIONS FOR THE

CATEGORICAL LATENT VARIABLE REGRESSION

Parameterization using Reference Class 1

C#2 ON

Pred1 -0.317 0.393 -0.806 0.420

Pred2 0.145 0.059 2.449 0.014

C#3 ON

Pred1 -0.812 0.553 -1.469 0.142

Pred2 0.215 0.069 3.102 0.002

Parameterization using Reference Class 2

C#1 ON

Pred1 0.317 0.393 0.806 0.420

Pred2 -0.145 0.059 -2.449 0.014

C#3 ON

Pred1 -0.495 0.533 -0.928 0.353

Pred2 0.070 0.061 1.156 0.248

You just need to calculate the

OR

Approaches 1-2: OutcomesDATA: FILE IS Commitment.dat;

VARIABLE:

NAMES = ID unit Pred1 Pred2 Cor1 Cor2 Out1 Out2 AC1 NC1 CC1 AC2 NC2 CC2;

USEVARIABLES = AC1 NC1 CC1 Out1 Out2 ;


Analysis: […]

MODEL:

%OVERALL%

%c#1%

[AC1 NC1 CC1]; AC1 NC1 CC1;

[Out1 Out2] (o11-012);

%c#2%


[Out1 Out2] (o21-022);

%c#3%


[Out1 Out2] (o31-032);

MODEL CONSTRAINT:

NEW (ye12); ye12 = o11-o21;

NEW (ye13); ye13 = o11-o31;

NEW (ye23); ye23 = o21-o31;

NEW (yp12); yp12 = o12-o22;

NEW (yp13); yp13 = o12-o32;

NEW (yp23); yp23 = o22-o32;

MODEL RESULTS

Two-Tailed


Latent Class 1

Means

FAC 1.139 0.002 460.118 0.000

FCC 4.183 0.314 13.324 0.000

FNC 1.213 0.016 74.366 0.000

OUT1 -1.460 0.157 -9.293 0.000

OUT2 -1.438 0.122 -11.823 0.000

Variances

FAC 0.000 0.000 7.395 0.000

FCC 3.052 0.373 8.187 0.000

FNC 0.008 0.001 7.307 0.000

OUT1 0.226 0.013 18.047 0.000

OUT2 0.161 0.006 24.915 0.000

…

New/Additional Parameters

YE12 -1.287 0.091 -14.163 0.000

YE13 -2.533 0.168 -15.116 0.000

…

Integrating covariates to the final retained

solution

3. Auxiliary

Auxiliary: Predictors DATA: FILE IS Commitment.dat;

VARIABLE:

NAMES = ID unit Pred1 Pred2 Cor1 Cor2 Out1 Out2 AC1NC1 CC1

AC2 NC2 CC2;



AUXILIARY = Pred1 (R3STEP) Pred2 (R3STEP);

Analysis: […]

MODEL:

%OVERALL%

%c#1%


%c#2%


%c#3%


TESTS OF CATEGORICAL LATENT VARIABLE MULTINOMIAL

LOGISTIC REGRESSIONS USING THE 3-STEP PROCEDURE

Two-Tailed


C#1 ON

Pred1 1.646 0.616 2.673 0.008

Pred1 0.221 0.231 0.958 0.338

C#2 ON

Pred1 1.562 0.606 2.575 0.010

Pred2 0.317 0.229 1.388 0.165

Multinomial logistic regression

Including all alternative referent profiles.

You only need to calculate the OR.

Warnings

Sometimes the nature of the profiles changes when the

R3STEP approach is applied. In this situation, you will

get a warning. This is the main limitation of this

approach.

Auxiliary: CorrelatesDATA: FILE IS Commitment.dat;

VARIABLE:


AC2 NC2 CC2;



AUXILIARY = Cor1 (e) Cor2 (e);

Analysis: […]

MODEL:

%OVERALL%

%c#1%


%c#2%


%c#3%


EQUALITY TESTS OF MEANS ACROSS CLASSES USING POSTERIOR

PROBABILITY-BASED MULTIPLE IMPUTATIONS WITH 4 DEGREE(S) OF

FREEDOM FOR THE OVERALL TEST AND 1 DEGREE OF FREEDOM FOR

THE PAIRWISE TESTS

Cor1

Mean S.E. Mean S.E.

Class 1 0.744 0.037 Class 2 0.740 0.035

Class 3 0.598 0.064

Chi-Square P-Value Chi-Square P-Value

Overall test 4.473 0.346 Class 1 vs. 2 0.008 0.930

Class 1 vs. 3 3.783 0.052 Class 2 vs. 3 3.628 0.057

Cor2

Mean S.E. Mean S.E.

Class 1 29.797 0.267 Class 2 30.549 0.287

Class 3 32.491 0.661



Class 1 vs. 3 14.094 0.000 Class 2 vs. 3 7.019 0.008

Auxiliary: OutcomesDATA: FILE IS Commitment.dat;

VARIABLE:


AC2 NC2 CC2;



AUXILIARY = Out1 (DCON) Out1 (DCAT);

AUXILIARY = Out1 (DU3STEP) Out1 (DE3STEP);

AUXILIARY = Out1 (BCH) Out1 (BCH);

DCON/DCAT: Continuous/Categorical outcomes. Generally works well. DU3STEP/ DE3STEP: Continuous outcomes specified as having either unequal or equal variance across profiles. Works well, but same instability than R3STEP. BCH: Works very well, for continuous outcomes. No clear winner. I use DCAT for categorical outcomes and BCH otherwise.

EQUALITY TESTS OF MEANS/PROBABILITIES ACROSS CLASSES

OUT1

Prob S.E. Odds Ratio S.E. 2.5% C.I. 97.5% C.I.

Class 1

Category 1 0.236 0.041 1.000 0.000 1.000 1.000

Category 2 0.764 0.041 4.834 2.838 0.000 10.397

Class 2

Category 1 0.254 0.036 1.000 0.000 1.000 1.000

Category 2 0.746 0.036 4.367 2.530 0.000 9.326

Class 3

Category 1 0.429 0.066 1.000 0.000 1.000 1.000

Category 2 0.571 0.066 1.984 1.210 0.000 4.354

Chi-Square P-Value Degrees of Freedom

Overall test 14.692 0.005 4

Class 1 vs. 2 0.106 0.745 1

Class 1 vs. 3 6.001 0.014 1

Class 2 vs. 3 5.072 0.024 1

DCAT

EQUALITY TESTS OF MEANS/PROBABILITIES ACROSS CLASSES

OUT2

Approximate Approximate

Mean S.E. Mean S.E.

Class 1 29.674 0.252 Class 2 30.539 0.272

Class 3 32.817 0.619

Approximate

Chi-Square P-Value Degrees of Freedom

Overall test 29.957 0.000 4

Class 1 vs. 2 5.457 0.019 1

Class 1 vs. 3 22.122 0.000 1

Class 2 vs. 3 11.350 0.001 1

DCON

EQUALITY TESTS OF MEANS ACROSS CLASSES USING THE 3-STEP

PROCEDURE WITH 4 DEGREE(S) OF FREEDOM FOR THE OVERALL TEST

OUT1

Mean S.E. Mean S.E.

Class 1 999.000 999.000 Class 2 999.000 999.000

Class 3 999.000 999.000



Class 1 vs. 3 999.000 999.000 Class 2 vs. 3 999.000 999.000

Final Class Counts and Proportions for the Latent Class Patterns Based On

Estimated Posterior Probabilities for TURN: Step 1 vs. Step 3

Latent

Classes Step 1 Step 3

1 83.09193 0.31474 9.58211 0.03630

2 85.35850 0.32333 80.36276 0.30440

3 17.59572 0.06665 23.75866 0.08999

4 45.63659 0.17287 147.37306 0.55823

5 32.31726 0.12241 2.92341 0.01107

DU3STEP/DE3STEP

EQUALITY TESTS OF MEANS ACROSS CLASSES USING THE BCH

PROCEDURE WITH 4 DEGREE(S) OF FREEDOM FOR THE OVERALL TEST

OUT1

Mean S.E. Mean S.E.

Class 1 0.767 0.039 Class 2 0.739 0.036

Class 3 0.562 0.066



Class 1 vs. 3 6.910 0.009 Class 2 vs. 3 5.283 0.022

OUT2

Mean S.E. Mean S.E.

Class 1 29.659 0.289 Class 2 30.530 0.293

Class 3 32.865 0.700



Class 1 vs. 3 17.183 0.000 Class 2 vs. 3 9.046 0.003

BCH




oHeuristic Value



o Shows meaningful and well-differentiated relations

to key covariates: Predictors – Correlates –

Outcomes.

oGeneralizes across samples and time points.




oHeuristic Value





Outcomes.


Testing the Generalizability of LPA

Solutions




oHeuristic Value





Outcomes.


Morin, A.J.S., Meyer, J.P., Creusier, J., & Biétry, F. (2016).

Multiple-group analysis of similarity in latent profile

solutions. Organizational Research Methods, 19, 231-254.

Stability and Change Across Samples and

Time

Within-Person Profile Similarity (Longitudinal)

Within-Sample (longitudinal) or Across Sample

Profile Similarity: o Configural

o Structural

o Dispersion

o Distributional

o Predictive

o Explanatory

Profile Similarity Across Samples

• Configural: o Tests if the same number of latent profiles can be identified across groups.

o Conduct the class enumeration process separately in each group.

• Structural

• Dispersion

• Distributional

• Predictive

• Explanatory

Group 1

Group 2

0

1

2

3

4

5

6

7

AC (Profile 1)NC (Profile 1)CC (Profile 1) AC (Profile 2)NC (Profile 2)CC (Profile 2)

0

1

2

3

4

5

6

7

8


DATA:


VARIABLE:

NAMES = ID Country unit Pred1

Pred2 Cor1 Cor2 Out1 Out2 AC1




IDVARIABLE = ID;

KNOWCLASS = cg (Status = 1 Status = 2);

CLASSES = cg (2) c (3);

Analysis:

TYPE = MIXTURE;

ESTIMATOR = MLR;

PROCESS = 3;

STARTS = 3000 100;

STITERATIONS = 100;

KNOWCLASS: identify a “known class” as one additional latent categorical variables in which the membership is “exact” rather than probabilistic. The model includes 2 latent categorical variables.

Model

%OVERALL%

c#1 on cg#1; c#2 on cg#1;

%cg#1.c#1%

[AC1 NC1 CC1] (m1-m3);


%cg#1.c#2%

[AC1 NC1 CC1] (m4-m6);


%cg#1.c#3%

[AC1 NC1 CC1] (m7-m9);


To indicate that class sizes differ across group (always one statement less than the number of groups).

Profile 1, Group 1

Label each parameter to indicate inequality across groups and profiles.

%cg#2.c#1%[AC1 NC1 CC1] (mm1-mm3); AC1 NC1 CC1 (vv1-vv3); %cg#2.c#2%[AC1 NC1 CC1] (mm4-mm6); AC1 NC1 CC1 (vv4-vv6); %cg#2.c#3%[AC1 NC1 CC1] (mm7-mm9); AC1 NC1 CC1 (vv7-vv9);


• Configural

• Structural: o Prerequisite: Configural similarity.

o Tests whether the indicators’ within-profile levels are the same across groups.

• Dispersion

• Distributional

• Predictive

• Explanatory

Group 1

Group 2

0

1

2

3

4

5

6

7


0

1

2

3

4

5

6

7

AC (Profile 1) NC (Profile 1) CC (Profile 1) AC (Profile 2) NC (Profile 2) CC (Profile 2)

Model

%OVERALL%


%cg#1.c#1%

[AC1 NC1 CC1] (m1-m3);


%cg#1.c#2%

[AC1 NC1 CC1] (m4-m6);


%cg#1.c#3%

[AC1 NC1 CC1] (m7-m9);


Identify equal means across groups.

%cg#2.c#1%[AC1 NC1 CC1] (m1-m3); AC1 NC1 CC1 (vv1-vv3);




• Configural

• Structural

• Dispersion:o Prerequisite: Configural and structural similarity.

o Tests whether the indicators’ within-profile variability is the same across groups.

o Latent profiles do not assume that all members of a profile share the exact same configuration of indicators. This step tests whether within-profile inter-individual differences are stable across groups.

• Distributional

• Predictive

• Explanatory

Group 1

Group 2

0

1

2

3

4

5

6

7


0

1

2

3

4

5

6

7


Model

%OVERALL%


%cg#1.c#1%

[AC1 NC1 CC1] (m1-m3);


%cg#1.c#2%

[AC1 NC1 CC1] (m4-m6);


%cg#1.c#3%

[AC1 NC1 CC1] (m7-m9);


Identify equal variances across groups.

%cg#2.c#1%[AC1 NC1 CC1] (m1-m3); AC1 NC1 CC1 (v1-v3);




• Configural

• Structural

• Dispersion

• Distributional:o Prerequisite: Configural and structural similarity.

o Tests whether the relative size of the profiles is similar across groups.

• Predictive

• Explanatory

Profile 1 Profile 2 Profile 3

Profile 1 Profile 2

Profile 3Profile 1 Profile 2 Profile 3

Group 1

Group 2

Model

%OVERALL%


%cg#1.c#1%

[AC1 NC1 CC1] (m1-m3);


%cg#1.c#2%

[AC1 NC1 CC1] (m4-m6);


%cg#1.c#3%

[AC1 NC1 CC1] (m7-m9);


Take out the %overall% part





• Configural

• Structural

• Dispersion

• Distributional

• Predictive:o Prerequisite: Configural and structural similarity.

o This step includes predictors (direct inclusion) of profile membership to the most similar (2-4) model.

o Tests if the predictors-profiles relations are the same across groups.

o Use starts values from the retained (configural, structural, dispersion, distribution) solution to ensure stability

• Explanatory

C

S1 S2 S3 SX…

X

XC

Y

YC

Model

%OVERALL%

c#1-c#2 ON Pred1@0 Pred2@0;

%cg#1.c#1%

[AC1 NC1 CC1] (m1-m3);


%cg#1.c#2%

[AC1 NC1 CC1] (m4-m6);


%cg#1.c#3%

[AC1 NC1 CC1] (m7-m9);





MODEL cg:%cg#1%c#1-c#2 ON Pred1 Pred2;%cg#2%c#1-c#2 ON Pred1 Pred2;

Constrain predictions to 0 in the %Overall% section to freely estimate them in both groups

FREE

Model

%OVERALL%

c#1-c#2 ON Pred1* Pred2*;

%cg#1.c#1%

[AC1 NC1 CC1] (m1-m3);


%cg#1.c#2%

[AC1 NC1 CC1] (m4-m6);


%cg#1.c#3%

[AC1 NC1 CC1] (m7-m9);





MODEL cg:%cg#1%c#1-c#2 ON Pred1 Pred2;%cg#2%c#1-c#2 ON Pred1 Pred2;

SIMILAR


• Configural

• Structural

• Dispersion

• Distributional

• Predictive

• Explanatory: o Prerequisite: Configural and structural similarity.

o This step includes outcomes (direct inclusion) of profile membership to the most similar (2-4) model.

o Tests if the profiles-outcomes relations are the same across groups or time points.

o Use starts values from the retained (configural, structural, dispersion, distribution) solution to ensure stability

C

S1 S2 S3 SX…

X

XC

Y

YC

Model

%OVERALL%

%cg#1.c#1%

[AC1 NC1 CC1] (m1-m3);


[Out1] (oa1);

[Out2] (ob1);

%cg#1.c#2%

[AC1 NC1 CC1] (m4-m6);


[Out1] (oa2);

[Out2] (ob2);

%cg#1.c#3%

[AC1 NC1 CC1] (m7-m9);


[Out1] (oa3);

[Out2] (ob3);

%cg#2.c#1%[AC1 NC1 CC1] (m1-m3); AC1 NC1 CC1 (v1-v3); [Out1] (oaa1);[Out2] (obb1);



FREE

MODEL CONSTRAINT:

NEW (y12); y12 = oa1-oa2;

NEW (y13); y13 = oa1-oa3;

NEW (y23); y23 = oa2-oa3;

NEW (z12); z12 = ob1-ob2

NEW (z13); z13 = ob1-ob3;

NEW (z23); z23 = ob2-ob3;

NEW (yy12); yy12 = oaa1-oaa2;



NEW (zz12); zz12 = obb1-obb2

NEW (zz13); zz13 = obb1-obb3;

NEW (zz23); zz23 = obb2-obb3;

Model

%OVERALL%

%cg#1.c#1%

[AC1 NC1 CC1] (m1-m3);


[Out1] (oa1);

[Out2] (ob1);

%cg#1.c#2%

[AC1 NC1 CC1] (m4-m6);


[Out1] (oa2);

[Out2] (ob2);

%cg#1.c#3%

[AC1 NC1 CC1] (m7-m9);


[Out1] (oa3);

[Out2] (ob3);

%cg#2.c#1%[AC1 NC1 CC1] (m1-m3); AC1 NC1 CC1 (v1-v3); [Out1] (oa1);[Out2] (ob1);



SIMILAR

MODEL CONSTRAINT:

NEW (y12); y12 = oa1-oa2;

NEW (y13); y13 = oa1-oa3;

NEW (y23); y23 = oa2-oa3;

NEW (z12); z12 = ob1-ob2

NEW (z13); z13 = ob1-ob3;

NEW (z23); z23 = ob2-ob3;

France versus North America

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

Profile 1 (CC-

Dominant)

Profile 2 (Moderately

Committed)

Profile 3 (AC/NC-

Dominant)

Profile 4 (NC-

Dominant)

Profile 5 (AC-

Dominant)

Affective Commitment (AC)

Normative Commitment (NC)

Continuance Commitment (CC)

Identical profiles were found in both cultures, with evidence of distributional differences

France versus North America

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

Profile 1 (CC-

Dominant)

Profile 2 (Moderately

Committed)

Profile 3 (AC/NC-

Dominant)

Profile 4 (NC-

Dominant)

Profile 5 (AC-

Dominant)

Affective Commitment (AC)

Normative Commitment (NC)

Continuance Commitment (CC)

Confirmatory?Finch, W.H., & Bronk, K.C. (2011). Conducting confirmatory

latent class analysis using Mplus. Structural Equation Modeling, 18,

132-151.

• Using starts values and parameter constraints (equality

restrictions, deterministic restrictions [equality to a specific

value], and inequality restrictions) it is possible to rely on a

confirmatory approach to the estimation of LPA (as long as

prior research and theory is sufficiently advanced to provide

such hypotheses).

• The model constraint approaches even makes it possible to

implement proportionality constraints (e.g., twice as high), and

range restrictions constraints (e.g., between 1 and 3).

• The constrained model is then compared to the

unconstrained model in terms of fit.

Longitudinal Mixture Models

Latent Transition Analysis

C1

X1 X2 X3

C2

X1 X2 X3

Time t Time t + 1

Xi

… Xi

…

Kam, C., Morin, A.J.S., Meyer, J.P., & Topolnytsky, L. (2016).

Are commitment profiles stable and predictable? A latent

transition analysis. Journal of Management, 42 (6), 1462-1490.

Latent Transition Analysis• Latent transition analyses (LTA) are a very broad class of models

that can be used to assess the connections between any number

of latent categorical variables.

• Due to modern computer limitations, these models are typically

limited to 3, sometimes 4, latent categorical variables.

• These models can be used to assess the connections between

any type of latent categorical variable (LPA, MRA, growth

mixtures) based on the same, or different, sets of indicators

assessed at different, or similar, time points.

• The typical application of LTA is longitudinal, and assess the

connection between two LPA models based on the same set of

indicators measured at different time points.

• LTA can thus be used to assess the similarity of LPA solutions

over time.

CLASSES = c1 (4) c2 (4);

Analysis:

type = mixture;

estimator = MLR;

Process = 3;

STARTS = 10000 400;

STITERATIONs = 2000;

MODEL:

%OVERALL%

c2 on c1;

MODEL C1:

%c1#1%

[PAR8 TEA8 PEER8](ma1-ma3); PAR8 TEA8 PEER8 (va1-va3);

%c1#2%


%c1#3%


%c1#4%


MODEL C2:

%c2#1%

[PAR11 TEA11 PEER11](mb1-mb3); PAR11 TEA11 PEER11 (vb1-vb3);

%c2#2%

[PAR11 TEA11 PEER11](mb4-mb6); PAR11 TEA11 PEER11(vb4-vb6);

%c2#3%


%c2#4%


FINAL CLASS COUNTS AND PROPORTIONS FOR THE LATENT CLASS

PATTERNS BASED ON THE ESTIMATED MODEL

Latent Class

Pattern

1 1 5.07928 0.00202

1 2 464.55651 0.18508

1 3 0.00000 0.00000

1 4 0.00000 0.00000

2 1 105.76553 0.04214

2 2 89.60346 0.03570

2 3 623.25736 0.24831

2 4 0.00000 0.00000

3 1 8.21090 0.00327

3 2 433.19442 0.17259

3 3 57.47310 0.02290

3 4 495.70411 0.19749

4 1 43.44788 0.01731

4 2 27.33755 0.01089

4 3 156.36990 0.06230

4 4 0.00000 0.00000

FINAL CLASS COUNTS AND

PROPORTIONS FOR EACH LATENT CLASS

VARIABLE BASED ON THE ESTIMATED

MODEL

Latent Class

Variable Class

C1 1 469.63580 0.18711

2 818.62634 0.32615

3 994.58252 0.39625

4 227.15533 0.09050

C2 1 162.50359 0.06474

2 1014.69189 0.40426

3 837.10034 0.33351

4 495.70410 0.19749

• BASED ON ESTIMATED POSTERIOR PROBABILITIES• BASED ON THEIR MOST LIKELY LATENT CLASS

PATTERN

LATENT TRANSITION PROBABILITIES BASED ON THE ESTIMATED MODEL

C1 Classes (Rows) by C2 Classes (Columns)

1 2 3 4

1 0.011 0.989 0.000 0.000

2 0.129 0.109 0.761 0.000

3 0.008 0.436 0.058 0.498

4 0.191 0.120 0.688 0.000

MODEL RESULTS

Two-Tailed


Latent Class Pattern 1 1

Means

PAR8 -0.777 0.067 -11.608 0.000

TEA8 -0.956 0.128 -7.487 0.000

PEER8 -0.664 0.077 -8.612 0.000

PAR11 1.124 0.007 151.806 0.000

TEA11 0.785 0.126 6.215 0.000

PEER11 1.026 0.003 306.602 0.000

LTA with predictors• In tests of multi-group LPA similarity, we contrasted two models

for tests of predictive similarity:

1. One in which the effects of the predictors were freely

estimated in all samples.

2. One in which the effects of the predictors were constrained

to equality across samples.

• In LTA, we can contrast three models:


estimated at all time points, and across profiles estimated at

the preceding time points (to predict specific profile-to-

profile transitions).


estimated at all time points, but not initial profiles.

3. One in which the effects of the predictors were constrained

to equality across times points and initial profiles.

LTA with outcomes• Similar to tests of multi-group LPA similarity.

• Two models are contrasted for tests of explanatory similarity:

1. One in which the outcome levels are freely estimated in all

samples or time points.

2. One in which the outcome levels are constrained to equality

across samples or time points.

For applications, see:

Morin, A.J.S., & Litalien, D. (2017).Webnote: Longitudinal Tests of

Profile Similarity and Latent Transition Analyses. Montreal, QC:

Substantive Methodological Synergy Research Laboratory.

http://smslabstats.weebly.com/uploads/1/0/0/6/100647486/lta_dis

tributional_similarity_v02.pdf

Growth Mixture AnalysesGMA/GMM

Individual trajectories

Latent curve models allow one to synthesizeindividual trajectories with few latentparameters through a restricted factor model.

An average trajectory with

a random intercept and

slope is estimated

Y1 Y2 Y3 Y4

αy βy

1 1 1

1 1 2 3 0

Y5

1 4


Random: Each individual has its own trajectory (intercept and

slope variance).Model-based

time-structured (trait) variability


Residuals:Few persons

follow a perfectly linear

(curvilinear, etc.) trajectory.

Residual state-like variability.

Y1 Y2 Y3 Y4

αy βy

1 1 1

1 1 2 3 0

Y5

1 4

Y1 Y2 Y3 Y4

αy βy

1 1 1 1

1 0

*

Yt

1

itk itk itk itk itk

* *

C Predictors

Y1 Y2 Y3 Y4

αy βy

1 1 1 1

1 0

*

Yt

1

itk itk itk itk itk

* *

C Predictors

Growth Mixture Analyses combine Latent Profile Analyses and Latent Curve models by the addition of a latent class/profile variable to a latent curve model to allow for the identification of k subgroups following different trajectories.

In theory, all parameters can be freely estimated across classes.

Y1 Y2 Y3 Y4

αy βy

1 1 1 1

1 0

*

Yt

1

itk itk itk itk itk

* *

C Predictors

Time codes: Reflect the passage of time.

Linear: 0-1-2-3-4-5 (or some other reflecting the true passage of time, e.g. 0-1-3-4-6).

Quadratic: Addition of a second slope factor with squared time codes (0-1-4-9-16-25).

Piecewise: Two slopes with loadings reflecting the transition point: • 0-1-2-3-3-3-3• 0-0-0-0-1-2-3

Multibase: Free loadings reflecting the % of change between 0 and 1.

Morin, A.J.S., Maïano, C., Nagengast, B., Marsh, H.W., Morizot, J., &

Janosz, M. (2011). Growth Mixture Modeling of adolescents

trajectories of anxiety across adolescence: The impact of

untested invariance assumptions on substantive interpretations.

Structural Equation Modeling.

~ 1000 Montreal adolescents measured 5 times over a 4 year

period.

GMA Vs LCGALCGA (no inter-individual variability)

GMA (class specific variability that can either be invariant or freely estimated)

© Morin et al., 2010

Results: LCGA

LCGAMODEL:

%OVERALL%

i s q | ANXT1@-1 ANXT3@0 ANXT4@1 ANXT5@2 ANXT6@3;

i@0 s@0 q@0;

!No need for class-specific statements

NOTE: add the following for graphs of the trajectoriesPLOT:TYPE IS PLOT3;SERIES = ANXT1- ANXT6(*);


Results: GMA, Mplus default

GMA-MDMODEL:

%OVERALL%


!No need for class-specific statements


Results: GMA, free estimation

GGMA-LVMODEL:

%OVERALL%


! To specify the effects of predictors on class membership

c#1-c#4 ON SEX FAM SES FEELSE EB1 EB3 VICT1 WITN1;

! To specify the effects of predictors on the intercept factor (same thing for s and q)

I ON SEX FAM SES FEELSE EB1 EB3 VICT1 WITN1;

%c#1%

i s q; [I S Q];

i WITH s;

s WITH q;

i WITH q;

! To specify that the effects of predictors on the intercept factor are freely estimated in all classes (same thing for s and q)

I ON SEX FAM SES FEELSE EB1 EB3 VICT1 WITN1;

! Use same specifications for %c#2%, %c#3%, %c#4%, and %c#5%

Morin, A.J.S., Rodriguez, D., Fallu, J.-S., Maïano, C., & Janosz, M.

(2012). Academic Achievement and Adolescent Smoking: A

General Growth Mixture Model. Addiction.

741 non-smoking adolescents. Official GPA ratings obtained 8

times (beginning and end of school years).

And what about ?yitk


Residuals:Few persons

follow a perfectly linear

(curvilinear, etc.) trajectory.

Residual state-like variability.

• Latent curve models consider these time-specific deviations from

the overall trajectories as random “errors” to be controlled

rather than substantively meaningful deviations from the generic

trajectory.

• Such deviations may indeed represent state-like “shocks” to the

overall trajectories. These “shocks” may result from meaningful

situation-specific perturbations (e.g. the death of a loved one) or

successes (e.g. admission into a highly competitive program). Such

time-specific, state-like, deviations may sometimes be quite strong

and/or vary across time and/or subpopulations,

• These deviations, when considered over time, provide a direct

estimate of state-like instability over time.

MODEL:%OVERALL%i s q | ngen1b1@0 ngen2b1@1 ngen1b2@2ngen2b2@3 ngen1b3@4 ngen2b3@5 ngen1b4@6ngen2b4@7 ;%c#1%i s q; [i s q];i WITH s q; s WITH q;ngen1b1 ngen2b1 ngen1b2 ngen2b2 ngen1b3ngen2b3 ngen1b4 ngen2b4;! Use same specifications for %c#2% and %c#3

Smoking onsets: 7.1%



A simulation study• Diallo, T.M.O, Morin, A.J.S. & Lu, H. (2016). Impact of

misspecifications of the latent variance-covariance and residual

matrices on the class enumeration accuracy of growth mixture

models. Structural Equation Modeling, 23 (4), 507-531. DOI:

10.1080/10705511.2016.1169188

• In a series of 4 studies, we contrasted population models

corresponding to the LCGA, Mplus default, or distinct latent

variance-covariance matrices, while also considering presence of

class-invariant or distinct residual structures.

• Overall, our results clearly show the advantages of relying on

freely estimated latent variance-covariance and residual

structures in each latent classes.

But this also depends on sample

size and convergence

Solinger, O.N., Van Olffen, W., Roe, R.A., & Hofmans, J. (2013) On Becoming (Un)Committed: A

Taxonomy and Test of Newcomer Onboarding Scenarios. Organization Science, 24, 1640–1661.

N = 72

Realism?• Start with a theoretically optimal model.

o The model converges

o The model converges but includes out-of-bound estimates: Constrain/fix them

o The model does not converge, or converge on a degenerate solution (many improper

estimates, empty profiles): Then you need to simplify the model.

• Possibly go with a model with equal residual and equal latent

variance-covariances to ensure that the best fitting solution

includes more than 1 profile.

• Then freely estimate either the residuals, or the latent

variance-covariances, depending on objectives.

• Models with homoscedastic residuals (equal over time) can

also be considered.

Mixture Regression / Mixture SEM

C

• Mixture regression analysis (MRA) identifies profiles of

participants differing from one another at the level of

estimated relations (regressions) between constructs.

• LPAs regroups participants based on their configuration on a

set of indicators

• MRAs identify subpopulations in which the relationships

among constructs differ.

Simply:

𝒀 = 𝒂𝒚𝒙 + 𝒃𝒚𝒙 ∗ 𝑿 + 𝜹

?

Mixture Regression

𝒀 = 𝒂𝒚𝒙 + 𝒃𝒚𝒙 ∗ 𝑿 + 𝜹

• Outcomes’ means and variance (representing the intercepts

and residuals of the regression) need to be freely estimated

across profiles to obtain profile-specific regression equations.

• More complex applications are possible, such as combining

LPAs and MRAs to create profiles based on both their

configuration of indicators, and relationships between

indicators.

• The estimation of this type of MRAs requires the free

estimation of the predictors’ means and variances in the

extracted profiles. Such models also reveal potential

interactions among predictors, resulting in profiles in which

the relations among constructs may differ as a function of

predictors’ levels.

%OVERALL%

AC2 ON AC1 NC1 CC1;

%c#1%

AC2 ON AC1 NC1 CC1;

AC2;

[AC2];

%c#2%

AC2 ON AC1 NC1 CC1;

AC2;

[AC2];

%c#3%

AC2 ON AC1 NC1 CC1;

AC2;

[AC2];

%OVERALL%

AC2 ON AC1 NC1 CC1;

%c#1%

AC2 ON AC1 NC1 CC1;

AC2 AC1 NC1 CC1;

[AC2 AC1 NC1 CC1];

%c#2%

AC2 ON AC1 NC1 CC1;

AC2 AC1 NC1 CC1;

[AC2 AC1 NC1 CC1];

%c#3%

AC2 ON AC1 NC1 CC1;

AC2 AC1 NC1 CC1;

[AC2 AC1 NC1 CC1];

Gillet, N., Morin, A.J.S., Sandrin, E., & Houle, S.A.

Upcoming !

Workaholism

Work Engagement

Sleep Difficulties

Work-Family Conflicts

Engaged

Workaholic

Engaged-Workaholic

Profile 1 Profile 2 Profile 3

Engagement –> Work-Family

Conflicts.041 (.086) .004 (.241) -.015 (.171)

Engagement –> Sleeping

Difficulties.023 (.095) .155 (.154) .417 (.316)

Workaholism –> Work-Family

Conflicts .550 (.059)** .440 (.153)* .582 (.097)**

Workaholism –> Sleeping

Difficulties.244 (.121)* .287 (.171) -.293 (.257)

Chénard-Poirier, L.-A., Morin, A.J.S., & Boudrias, J.-S. (In

Press). On the merits of coherent leadership

empowerment behaviors: A mixture regression

approach. Journal of Vocational Behavior.

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2


Delegation

Coaching

Recognition

Efficacy-Tasks

Improvement-Tasks

Collaboration-Group

Improvement-Group

Involvement-Organization

Efficacy-Tasks Improvement-Tasks Involvement-Org.

Predictor b (s.e) β b (s.e) β b (s.e) β

Profile 1

Delegation 0.05 (0.01)** .71 -0.08 (0.05) -.27 0.131 (0.24) .13 Coaching 0.01 (0.01) .15 0.15 (0.07)* .50 -0.057 (0.37) -.05

Recognition -0.02 (0.01)* -.28 -0.03 (0.03) -.11 0.100 (0.21) .10

Profile 2 Delegation 0.27 (0.13)* .51 0.25 (0.19) .34 0.09 (0.09) .32

Coaching -0.08 (0.12) -.15 0.22 (0.20) .31 0.06 (0.07) .21

Recognition -0.13 (0.13) -.24 -0.37 (0.15)* -.49 -0.14 (0.06)* -.48

Profile 3

Delegation 0.33 (0.11)** .51 0.13 (0.08) .28 0.19 (0.12) .31

Coaching -0.24 (0.09)** -.37 0.01 (0.08) .02 0.08 (0.11) .13 Recognition 0.12 (0.08) .20 0.02 (0.07) .04 -0.09 (0.12) -.15

Profile 4

Delegation 0.36 (0.17)* .31 -0.12 (0.18) -.12 0.01 (0.10) .02 Coaching 0.61 (0.18)** .46 0.54 (0.21)** .48 -0.12 (0.15) -.15

Recognition -0.47 (0.17)** -.39 -0.23 (0.22) -.22 0.22 (0.13)* .30

Efficacy-Tasks Improvement-Tasks Involvement-Org.

Predictor b (s.e) β b (s.e) β b (s.e) β

Profile 1

Delegation 0.05 (0.01)** .71 -0.08 (0.05) -.27 0.131 (0.24) .13 Coaching 0.01 (0.01) .15 0.15 (0.07)* .50 -0.057 (0.37) -.05

Recognition -0.02 (0.01)* -.28 -0.03 (0.03) -.11 0.100 (0.21) .10

Profile 2 Delegation 0.27 (0.13)* .51 0.25 (0.19) .34 0.09 (0.09) .32

Coaching -0.08 (0.12) -.15 0.22 (0.20) .31 0.06 (0.07) .21

Recognition -0.13 (0.13) -.24 -0.37 (0.15)* -.49 -0.14 (0.06)* -.48

Profile 3

Delegation 0.33 (0.11)** .51 0.13 (0.08) .28 0.19 (0.12) .31

Coaching -0.24 (0.09)** -.37 0.01 (0.08) .02 0.08 (0.11) .13 Recognition 0.12 (0.08) .20 0.02 (0.07) .04 -0.09 (0.12) -.15

Profile 4

Delegation 0.36 (0.17)* .31 -0.12 (0.18) -.12 0.01 (0.10) .02 Coaching 0.61 (0.18)** .46 0.54 (0.21)** .48 -0.12 (0.15) -.15

Recognition -0.47 (0.17)** -.39 -0.23 (0.22) -.22 0.22 (0.13)* .30

%OVERALL%

CONSC AMELT COLL AMELG IMPL ON DELEG COACH RECON ;

CONSC AMELT COLL AMELG IMPL DELEG COACH RECON;

[CONSC AMELT COLL AMELG IMPL DELEG COACH RECON];

%c#1%




%c#2%




%c#3%




%c#4%




Organizational Research Methods: Special

Issue on Person-Centered Analyses

• Confirmatory LPA

• Multilevel LPA

• Extended Capabilities through Manual Auxiliary Approaches

(mediation, moderation, etc.).

• K-Centres functional clustering for longitudinal data

• Among others

alexandre j.s. morin, simon a. houle, & david litalien...panel discussion: increasing our...

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