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Probing causal mechanisms and strengthening causal inference by means of mixture mediation modeling SESSION 3.5: Modeling Treatment and Causal Effects Modern Modeling Methods Conference, Storrs, CT, May 20-21, 2014 Emil Coman 1 , Judith Fifield 1 , Suzanne Suggs 2 , Deborah Dauser-Forrest 1 , Martin-Peele Melanie 1 1 Ethel Donahue TRIPP Center, U. of Connecticut Health Center, 2 Università della Svizzera italiana, Switzerland The 1st author thanks David Kenny for introducing him to the causal modeling world through the path analysis backdoor, for his constant mentoring and extensive generous discussions and advice, and to SEMNET mentors who so generously shared their time and expertise and taught Latent Variable modeling to many students of applied statistics across the globe for many years.

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Probing causal mechanisms and

strengthening causal inference by

means of mixture mediation modeling

SESSION 3.5: Modeling Treatment and Causal Effects

Modern Modeling Methods Conference, Storrs, CT, May 20-21, 2014

Emil Coman1, Judith Fifield1, Suzanne Suggs2, Deborah Dauser-Forrest1, Martin-Peele Melanie1

1Ethel Donahue TRIPP Center, U. of Connecticut Health Center, 2 Università della Svizzera italiana, Switzerland

The 1st author thanks David Kenny for introducing him to the causal modeling

world through the path analysis backdoor, for his constant mentoring and extensive

generous discussions and advice, and to SEMNET mentors who so generously

shared their time and expertise and taught Latent Variable modeling to many

students of applied statistics across the globe for many years.

Origin of structural models – Sewall Wright

“Accounting for correlations” using the Method of Path Coefficients

"Senior Animal Husbandman in Animal Genetics, Bureau of Animal Industry, United States Department of Agriculture“

2

Wright, S. (1921). Correlation and causation. Part I Method of path coefficients. Journal of agricultural research, 20(7), 557-585. [CITED BY 2119, GOOGLE.SCOLAR.COM ]

MMM 2014 Mixture Mediation

Origin of Latent Variables– Sewall Wright

“Unknown cause’

all Figures online

3

Wright, S. (1921). Correlation and causation. Part I Method of path coefficients. Journal of agricultural research, 20(7), 557-585. [CITED BY 2119, GOOGLE.SCOLAR.COM ]

MMM 2014 Mixture Mediation

Research Questions about the intervention

RQ 1. Is the intervention showing benefits to all

participants?

RQ 2. Do some participants respond better/worse

to the intervention?

RQ 3. Are there mechanisms of change operating

differently in classes of participants?

4 MMM 2014 Mixture Mediation

Current SisterTalk study description

MMM 2014 Mixture Mediation 5

A community-based intervention conducted in Black churches (SisterTalk) aimed to achieve weight loss among women through lifestyle behavioral changes. Some unique features:

1. The protocol and program content were developed with the church community.

2. The principles of weight loss and weight control were translated into faith-based messages. URL

3. Faith-based messages were written and delivered by church leaders and health messages delivered by African American and Black professionals.

Gans, K.M., Kumanyika, S. K., Lovell, H. J., Risica, P. M., Goldman, R., Odoms-Young, A., Lasater, T. M. (2003). The development of SisterTalk: A cable TV-delivered weight control program for black women. Preventive Medicine, 37(6), 654-667. Gans, K.M., Risica, P.M., Kirtania, U., Jennings, A., Strolla, L. O., Steiner-Asiedu, M., . . . Lasater, T.M. (2009). Dietary behaviors and portion sizes of black women who enrolled in SisterTalk and variation by demographic characteristics. Journal of Nutrition Education and Behavior, 41(1), 32-40. Fifield et al (2014, Under Review). “Praying to lose”: Results from SisterTalk Hartford, a collaborative translation and randomized controlled trial of a theoretically-based weight loss program for the Black and African American Church

BMI and ‘Bad’ Food Preparation means with 95% confidence intervals by

intervention condition

MMM 2014 Mixture Mediation 6

BMI and Food Preparation Behaviors means (from RANOVAs)

MMM 2014 Mixture Mediation 7

Methodological approach to analyze changes

Simple modeling options considered here:

1. Effect of intervention on post BMI

1. Add auto-regressive paths (AR1)

2. Effects on true scores

3. Mediation through ‘Bad’ Food Preparation skills

4. Mixture classic mediation

2. Effect of intervention on BMI changes

1. Mediation through Food Preparation changes

2. Mixture LCS mediation

8

Rogosa, D., Brandt, D., & Zimowski, M. (1982). A growth curve approach to the measurement of change. Psychological Bulletin, 92(3), 726-748. doi: 10.1037/0033-2909.92.3.726

MMM 2014 Mixture Mediation

Brief LCS review 1. AR1 model

Notes: Not labeled regression coefficients are equal to 1 (unity).

Y2 Y1

error

µY1*

αY2*

αe@0

β

9

σ2e*

σ2Y1*

MMM 2014 Mixture Mediation

2. LCS: Latent Change Score model

Notes: All un-labeled regression coefficients are equal to 1 (unity).

See URL for a brief history.

Y2 Y1

LCS(1->2)

µY1*

αY2@0

αLCS*

eY2 αe@0

γ = β-1

10

σ2Y1*

σ2e@0

σ2LCS*

Coman, E. N., Picho, K., McArdle, J. J., Villagra, V., Dierker, L., & Iordache, E. (2013). The paired t-test as a simple latent change score model. Frontiers in Quantitative Psychology and Measurement, 4, Article 738. doi: 10.3389/fpsyg.2013.00738

MMM 2014 Mixture Mediation

Food Preparation LCS model

Notes: All un-labeled regression coefficients are equal to 1 (unity). Age feeds into FoodP1.

Fit χ2 (2) = 8.96, p(χ2) = .011 , CFI = .961

FoodP2 FoodP1

LCS(1->2)

1.26

0

.208

eY2 0

+.683.001

11

.100

0

.020

Tx 0/1 -.17.001

MMM 2014 Mixture Mediation

BMI LCS model

Notes: All un-labeled regression coefficients are equal to 1 (unity). Age feeds into BMI1.

Fit χ2 (2) = 4.93, p(χ2) = .085 , CFI = .996; residual LCS variance < 0 though.

BMI2 BMI1

LCS(1->2)

41.2

0

0

eY2 0

+1.00.001

12

55.8

0

-4.5

Coman, E. N., Picho, K., McArdle, J. J., Villagra, V., Dierker, L., & Iordache, E. (2013). The paired t-test as a simple latent change score model. Frontiers in Quantitative Psychology and Measurement, 4, Article 738. doi: 10.3389/fpsyg.2013.00738

Tx 0/1 -.75.001

MMM 2014 Mixture Mediation

Software for mixture mediation models

1. AMOS 1. The (previously) free v.5 did not have ‘mixture’

2. V. 16 and on have ‘mixture’ modeling implemented with Bayesian estimation.

3. Advantage: can ‘see’ your own hypothesized structural relations

2. Mplus 1. The ‘gold standard’ of course.

2. LCS models need to be ‘watched’ for unexpected behavior: Mplus correlates LCS score residuals across time, sometimes; it doesn’t estimate covariance between exogenous, and intercepts of LCSs, unless directly specified.

13

Coman, E. , Fifield, J. , McArdle, J.J. , Davis-Smith, M. (2014). Adding nuance and practical meaning to comparing the effectiveness of diabetes prevention interventions: uncovering classes of comparable patients who follow distinct trajectories if change (poster). Diabetes Symposium - “Innovative Approaches to Diabetes Research and Therapies”, Yale Diabetes Research Symposium, May 14, 2-14. Coman, E. , Fifield, J. , McArdle, J.J. , Davis-Smith, M.(2014). Investigating differential changes using Mixture Latent Change Scores (MLCS) modeling. Talk - Modern Modeling Methods (M3) Conference, U. of Connecticut, Storrs, CT, May 20-21, 2014. Coman, E. , Fifield, J. , McArdle, J.J. , Davis-Smith, M. (2014). Investigating differential changes using Mixture Latent Change Scores (MLCS) modeling. Joint meeting of the American Statistical Association, August 2 - 7, 2014.

MMM 2014 Mixture Mediation

Mixture modeling

14

Ding, C. (2006). Using regression mixture analysis in educational research. Practical Assessment Research and Evaluation, 11(11), 1-11. McLachlan, G., & Peel, D. (2000). Finite mixture models. New York: Wiley.

Model based mixture modeling (finite mixture models, McLachlan& Peel, 2000, or causal models with presumed latent classes, Ding, 2006) are categorical latent variable models that attempt to recover latent classes.

Muthen : “Mixture modeling refers to modeling with categorical latent variables that represent subpopulations where population membership is not known but is inferred from the data. (2009, Mplus Guide: 131).

Arbuckle : “Mixture regression modeling (Ding, 2006) is appropriate when you have a regression model that is incorrect for an entire population, but where the population can be divided into subgroups in such a way that the regression model is correct in each subgroup.” (2008, AMOS 16 Guide: 559).

MMM 2014 Mixture Mediation

Mixture modeling

(MM) – Jeff Harring

15

Harring, J. R. (December 4, 5 & 6, 2013). Introduction to Finite Mixture Models. Details: http://www.cilvr.umd.edu/Workshops/CILVRworkshoppageFMM2013.html College Park, Maryland. Pearson, K. (1894). Contributions to the mathematical theory of evolution. Philosophical Transactions of the Royal Society of London. A, 185, 71-110.

MM is old. Karl Pearson showed in 1894 that if data are the result of a mixture of probability distributions (like two normal distributions with different means and variances), the resulting distribution will appear as one asymmetric and bimodal, when in fact it represents two homogenous normal distributions, or two subpopulations.

MMM 2014 Mixture Mediation

Mixture modeling (MM) – Jeff Harring

16 Harring, J. R. (December 4, 5 & 6, 2013). Introduction to Finite Mixture Models. Details: http://www.cilvr.umd.edu/Workshops/CILVRworkshoppageFMM2013.html. College Park, Maryland.

The general mixture specification assumes a composite (mixed)

distribution as the sum of K classes pdf’s (probability density

functions) distributions, each with class-specific parameters θk, and

weights or mixing proportions φi, which specify in fact the class

proportions.

The likelihood for one patient’s observations is then obtained as the

probability of his/her data as a function of the parameters, and then a

global likelihood function L is calculated as the product of individual

patients’ likelihoods.

Using Maximum Likelihood (ML) or Bayesian estimation, one can then

obtain the parameter values that maximize the loglikelihood L.

MM software use in practice is an Expectation-Maximization (EM)

algorithm, which acts as an optimizer (not estimator) by generating

starting values for the parameters θk and then, given posterior

probabilities for φik, obtains new estimates for φik and θk, and

continues in such steps until the change in the likelihood from

successive iterations is sufficiently small. (Harring, 2013).

MMM 2014 Mixture Mediation

Classic mediation –no measurement error

0

bfhq

0

t2fhq0, eVarM2 eM2

MnAge, VarAge

age

0, eVarM1

eM1

0

bbmi

0

t2bmi0, eVarY2 eY2

0, eVarY1

eY1

1

1

1

1

PropT, VarInterv

interv

MMM 2014 Mixture Mediation 17

Mediation with measurement error 0

seff1

0

seff2

0, meVarM2

meM2

MnAge_1, VarAge_1

age

0, meVarM1

meM1

0

bbmi

0

t2bmi

0, meVarY2

eY2

0, meVarY1

eY1

IntLY2_1

LY2

IntLY1_1

LY1

IntLM1_1

LM1

IntLY2_1

LM2

1

1

0, eVarLX1_1

eLM1

0, VarM2_1eLM2

0, VarY2_1

eLY2

0, eVarLY1_1

eLY1

1

1

1

1

1

1

rhoM_1

rhoY_1

1

1

1

1

PropT_Eq, VarInterv_1

interv

a_1

cPrime_1

b_1

MMM 2014 Mixture Mediation 18

Classic 15%&5% measurement errors mediation results

Amos provides bootstrap confidence intervals, and BC p values for all

effects, direct, indirect, total.

MMM 2014 Mixture Mediation 19

FoodPrep2

Tx 0/1

BMI2

Classic errors-in-variables mediation results

-.11*.014 =

Indirect = a·b

In subscripts two tailed bootstrap confidence significance; 25.4% of

total effect is through mediator. Age feeds into w1 outcomes. Fit bad.

-.07*.015 = a

-.32*.045 = c’

+1.5*.011 = b

-.432*.011 = Total effect = a·b + c’

MMM 2014 Mixture Mediation 20

BMI1

FoodPrep1

Age

FoodPrep2

Tx 0/1

BMI2

AMOS Classic 15%M and 5%Y measurement errors

mediation results

-.19*.009 =

Indirect = a·b

In subscripts two tailed bootstrap confidence significance; 26% of total

effect is through mediator. Age feeds into w1 outcomes. Fit much

better, still bad. Mplus yields similar estimates.

-.17*.015 = a

-.53*.009 = c’

+1.13*.013 = b

-.717*.009 = Total effect = a·b + c’

MMM 2014 Mixture Mediation 21

LCS Mediation with measurement error

IntTCM21

TC_M21

0

bfhq

0

t2fhq

0, meVarM2 meM2

0, vrTCM21

eTC_M

MnAge, VarAge

age

0, meVarM1 meM1

IntTCY21

TC_Y21

0

bbmi

0

t2bmi

0, meVarY2 eY2

0, vrTCY21

eTC_Y

0, meVarY1 eY1

1

1

0

LY20

LY1

0

LM1

0

LM2

Y->TCY

1

1M->TCM

11

0, eVarLX1

eLM1 0, 0 eLM2

0, 0 eLY2

0, eVarLY1

eLY1

1 1

1

1

1 1

1

1

11

11

PropT, VarInterv

interv

a

bcPrime

MMM 2014 Mixture Mediation 22

Δ12FoodPrep

Tx 0/1

Δ12BMI

LCS mediation 1 results

+.36NS.006

= Indirect = a·b

In subscripts two tailed bootstrap confidence significance. Mplus

yields similar results.

-.16*.026 = a

-1.12*.007 = c’

-2.25NS.010 = b

-.76.008 = Total effect = a·b + c’

MMM 2014 Mixture Mediation 23

FAT intake

behaviors2

0/1 BMI2

Classic Mixture Modeling (MM) mediation illustration

a·b

The C=Class (categorical) variable can point to (cause differences in)

several paths and variable means (intercepts).

c’ a·b + c’

b a C

MMM 2014 Mixture Mediation 24

AMOS setup for MM mediation

25

AMOS (like Mplus) literally sets up a 2 (or more) group model, in which the grouping variable is a STRING variable present in the data with no values (unless training data is specified: like ‘improver’ &‘decliner’).

MMM 2014 Mixture Mediation

Mplus MM example

26

Illustration of a mixture Tx effect regression

CLASSES = whohow (2);

ANALYSIS: TYPE = MIXTURE;

MODEL:

%OVERALL%

OutcomeWave2 on Tx;

%whohow#1%

OutcomeWave2 on Tx@0;

[OutcomeWave2(intC1)]; !intercept class 1

%whohow#2%

OutcomeWave2 on Tx;

[OutcomeWave2(intC2)]; !intercept class 2

YTX CE

Classes

HTE

MMM 2014 Mixture Mediation

FoodPrep2

0/1

BMI2

Classic mixture mediation illustration

±.xx*/±.yyNS

= Indirect = a·b

The TOTAL effect is another parameter that can be held equal/allowed

to differ across latent classes.

±.xx*/±.yyNS= a

±.xx*/±.yyNS= c’

±.xx*/±.yyNS= b

±.xx*/±.yyNS = Total effect = a·b + c’

27 MMM 2014 Mixture Mediation

AMOS MM output excerpt

AMOS MM output excerpt Additional Estimands

MMM 2014 Mixture Mediation

FoodPrep2

Tx 0/1

BMI2

AMOS Classic mediation 2-class MM results

+.10*.009/-.27.016

= Indirect = a·b

AMOS provides ‘Additional Estimands’; in subscripts standard errors

of parameters; convergence statistic = 1.012; Posterior predictive p=.95, class 1: 22%.

-.08*.002/-.33*.012= a

-.78*.011/-1.07*.037= c’

-1.21.088/+.81.041= b

-.68*.013/-1.34*.031 = Total effect = a·b + c’

MMM 2014 Mixture Mediation 30

Classic Mediation Mixture modeling results

31

The classic mediation MM seem to extract

I. 2 similar classes in terms of the total effects, but

II. distinct in indirect effects: leading to decline and

improvement, respectively (AMOS results).

III. The Mplus results suggest no significant indirect

effects in either class.

MMM 2014 Mixture Mediation

Δ12FoodPrep

0/1

Δ12BMI

LCS mixture mediation illustration

Indirect = a·b

The TOTAL effect is another parameter that can be held equal/allowed

to differ across latent classes.

a

c’

b

C

MMM 2014 Mixture Mediation 32

a·b + c’

Δ12FoodPrep

0/1

Δ12BMI

AMOS LCS mixture mediation 1 results

+12.6.618/+.56131

= Indirect = a·b

In subscripts standard errors of parameters; convergence statistic bad =

1.025; Posterior predictive p=.99; class 1: 61%.

-.09.004/-.28.006= a

-13.3.603/-.64.155=

c’ -1486.34/1.87.45= b

-.63.051/-.09.088 = Total effect = a·b + c’

MMM 2014 Mixture Mediation 33

LCS Mediation Mixture modeling results

34

The LCS MM yield a different picture in AMOS and

Mplus:

I. significant and non-significant total effects in the 2

classes

II. no direct effect in either class, and

III. only the a leg significant in AMOS in both classes,

or either the a or b leg significant in the Mplus

results.

MMM 2014 Mixture Mediation

Mediation Mixture modeling Conclusions

35

Kenny, D.A. and C.M. Judd, Power anomalies in testing mediation. Psychological Science, 2013: p. 0956797613502676.

While the MM mediation is not aiming directly at uncovering

classes of cases for which specific indirect effects elements

(the a X-to-M, the b M-to-Y, the c X-to-Y, and the c' total

effect) are/not significant, the procedure does reveal such

classes. Specific input conditions can be entered for unique

expected scenarios, like no a path in one class, but no b

path in another, leading to no indirect effect in either.

There are many moving parts in mixture mediation, let alone

dynamic (LCS) mixture mediation models. Substantive

knowledge needs to be fed into expectations about class

parameters, so that meaningful classes can be extracted.

For example, one should not let the optimizing algorithm

search blindly for total effect parameters, but feed in starting

values (in Mplus) like c1=0 & c2>0.

MMM 2014 Mixture Mediation

DONE!

36

Thanks

[email protected]

MMM 2014 Mixture Mediation