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New developments in structural equation modeling

Rex B Kline

Concordia University

Montréal

Set A: SCM UNL Methodology Workshop

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Topics

o Graph theory

o Mediation:

Design Conditional Causal

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Topics

o Graph theory:

Pearl’s SCM

Causal reasoning

Causal estimation

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Topics

o Mediation:

Design requirements

Conditional process modeling

Cause × mediator (SCM)

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Graph theory

o Pearl, J. (2009a). Causal inference in statistics: An overview.

Statistics Surveys, 3, 96–146. o Pearl, J. (2009b). Causality: Models, reasoning, and

inference (2nd ed.). New York: Cambridge University Press.

o Pearl, J. (2012). The causal foundations of structural equation modeling. In R. H. Hoyle (Ed.), Handbook of

structural equation modeling (pp. 68–91). New York: Guilford Press.

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Graph theory

o Bollen, K. A., & Pearl, J. (2013). Eight myths about causality

and structural equation models. In S.L. Morgan (Ed.), Handbook of causal analysis for social research (pp. 301–328). New York: Springer.

o Cole, S. R., Platt, R. W., Schisterman, E. F., Chu, H., Westreich, D., Richardson, D., & Poole, C. (2010). Illustrating bias due to conditioning on a collider. International Journal of Epidemiology, 39, 417–420

o Elwert, F. (2013). Graphical causal models. In S. L. Morgan (Ed.), Handbook of causal analysis for social research (pp. 245–273). New York, NY: Springer.

A10

Graph theory

o Elwert, F. (2014). Endogenous selection bias: The problem of

conditioning on a collider variable. Annual Review of

Sociology, 40, 31–53. o Glymour, M. M. (2006). Using causal diagrams to

understand common problems in social epidemiology. In M. Oakes & J. Kaufman (Eds), Methods in social epidemiology (pp. 387–422). San Francisco: Jossey-Bass.

o Hayduk, L., Cummings, G., Stratkotter, R., Nimmo, M., Grygoryev, K., Dosman, D., … Boadu, K. (2003). Pearl’s d-separation: One more step into causal thinking. Structural Equation Modeling, 10, 289–311.

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Graph theory

o Kenny, D. A. (2014). Mediation. Retrieved from

http://davidakenny.net/cm/mediate.htm#CI o Shipley, B. (2000). A new inferential test for path models

based on directed acyclic graphs. Structural

Equation Modeling, 7, 206–218. o Spector, P. E., & Brannick, M. T. (2011). Methodological

urban legends: The misuse of statistical control variables. Organizational Research Methods, 14, 287–

305.

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Graph theory

o Knüppel, S., & Stang, A. (2010). DAG Program: Identifying

minimal sufficient adjustment sets. Epidemiology, 21, 159. http://epi.dife.de/dag/

o Porter, K., Poole, D., Kisynski, J., Sueda, S., & Knoll, B., Mackworth, A., … Hoos, H., Gorniak, P., & Conati, C. (1999–2009). Belief and Decision Network Tool (Version 5.1.10) (computer software). http://aispace.org/bayes/

o Textor, J., Hardt, J., & Knüppel, S. (2011). DAGitty: A graphical tool for analyzing causal diagrams. Epidemiology, 5, 745. http://www.dagitty.net/

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Mediation

o Design:

Cheung, J., & MacKinnon, D. P. (2012). Mediational/indirect effects in structural equation modeling. In R. H. Hoyle (Ed.), Handbook of structural equation modeling (pp. 417–435). New York: Guilford Press.

Little, T. D. (2013). Longitudinal structural equation

modeling. New York, NY: Guilford. MacKinnon, D. P. (2011). Integrating mediators and

moderators in research design. Research on Social

Work Practice, 21, 675–681.

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Mediation

o Design:

Maxwell, S. E., & Cole, D. A. (2007). Bias in cross-sectional analyses of longitudinal mediation. Psychological

Methods, 12, 23–44. Selig, J. P., & Preacher, K. J. (2009). Mediation models for

longitudinal data in developmental research. Research

in Human Development, 6, 144–164. Wu, A. D., & Zumbo, B. D. (2008). Understanding and using

mediators and moderators. Social Indicators Research, 87, 367–392.

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Mediation

o Conditional:

Edwards, J. R., & Lambert, L. S. (2007). Methods for integrating moderation and mediation: A general analytical framework using moderated path analysis. Psychological Methods, 12, 1–22.

Hayes, A. F. (2013). Introduction to mediation, moderation,

and process control analysis: A regression-based

approach. New York: Guilford Press.

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Mediation

o Conditional:

Hayes, A. F., & Preacher, K. J. (2013). Conditional process modeling: Using structural equation modeling to examine contingent causal processes, In G. R. Hancock & R. O. Mueller (Eds.), Structural equation modeling: A

second course (pp. 219–266). Charlotte: IAP. Preacher, K. J., Rucker, D. D., & Hayes, A. F. (2007).

Addressing moderated mediation hypotheses: Theory, methods, and prescriptions. Multivariate Behavioral

Research, 42, 185–227.

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Mediation

o Causal:

Bullock, J. G., Green, D. P., & Ha, S. E. (2010). Yes, but what’s the mechanism? (Don’t expect an easy answer). Journal of Personality and Social Psychology, 98, 550–558.

Imai, K., Keele, L., & Yamamoto, T. (2010) Identification, inference and sensitivity analysis for causal mediation effects. Statistical Science, 25, 51–71.

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Mediation

o Causal:

Lange, T., Vansteelandt, S., & Bekaert, M. 2012). A simple unified approach for estimating natural direct and indirect effects. American Journal of Epidemiology, 176, 190–195.

Pearl, J. (2014). Interpretation and identification of causal mediation. Psychological Methods. Advance online publication. http://dx.doi.org/10.1037/a0036434

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Mediation

o Causal:

Petersen, M. L., Sinisi, S. E., & van der Laan, M. J. (2006). Estimation of direct causal effects. Epidemiology, 17, 276–284.

Valeri, L., & VanderWeele, T. J. (2013). Mediation analysis allowing for exposure–mediator interactions and causal Interpretation: Theoretical assumptions and implementation with SAS and SPSS macros. Psychological Methods, 18, 137–150.

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Mediation

o Hicks, R., & Tingley, D. H. (2012). MEDIATION: Stata module

for causal mediation analysis and sensitivity analysis [computer software].

http://EconPapers.repec.org/RePEc:boc:bocode:s457294

o Muthén, B. O. (2011). Applications of causally defined direct and indirect effects in mediation analysis using SEM in Mplus.

https://www.statmodel.com/download/causalmediation.pdf

o Tingley, D., Yamamoto, T., Hirose, K., Keele, L., & Imai, K. (2014). Package ‘mediation’ [computer software].

http://cran.r-project.org/web/packages/mediation/

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Intro to SCM

o Unifies:

Parametric & nonparametric

SEM and potential outcomes

Data, graphical analysis

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Intro to SCM

o Alternative to path analysis

o SEM program not needed

o But not latent variable models

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Intro to SCM

o Hayduk, L. A. & Littvay, L. (2012). Should

researchers use single indicators, best indicators, or multiple indicators in structural equation models? BMC Medical Research

Methodology, 12(159). http://www.biomedcentral.com/1471-2288/12/159

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Intro to SCM

o Multiple indicators

o Some weak

o Best indicator is better

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(a) Single endogenous indicator

A

1

X2

EX2

1

X1

EX1

1

C

1

DC

1

Y4

EY4

1

Y3

EY3

1

DY1

1

Y1

(b) rYY = .70 for Y1

.30 1

2

Ys

A

1

X2

EX2

1

X1

EX1

1

C

1

DC

1

Y4

EY4

1

Y3

EY3

1

1

Y1

EY1

1

B

1

DB

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E2 1

X2

B

1

(1 − 22

r ) 2

2s

E1 1

X1

A

1

(1 − 11r ) 2

1s

EY 1

Y

C

1

(1 − YYr ) 2

Ys

DY 1

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Intro to SCM o Bayesian networks

o Graph structure

o Dependence relations

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Intro to SCM

o Hypotheses as graphs

o Directly analyze graph (no data)

o Computer tools

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Intro to SCM

o Nonparametric causal models

o Ideas (no operationalization)

o Study planning

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Intro to SCM

o Directed acyclic graph (DAG)

o Directed cyclic graph (DCG)

o Recursive, nonrecursive

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Intro to SCM

o Causal effects identified?

o If no, what should be measured?

o If yes, how many estimates?

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Intro to SCM

o Identified model not required

o Estimate what you can

o Acknowledge info. gap for rest

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Intro to SCM

o Regression analysis

o Causal model

o Covariate selection

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Intro to SCM

o Bring on the data

o Model predictions

o Conditional independences

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Intro to SCM

o All testable hypotheses

o Vanishing partial correlations

o Vanishing tetrads

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SCM vocabulary

o Nodes, vertices (variables)

o Arcs, edges, links (paths)

o Adjacent (→), nonadjacent

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SCM vocabulary

o Parents, ancestors

o Children, descendants

o Path is any sequence of edges

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SCM vocabulary

o Directed path (causal)

o Undirected path (noncausal)

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SCM vocabulary

o Open (unblocked) path

o Closed (blocked) path

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SCM vocabulary

o Front-door path (causal)

o Back-door path (biasing)

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SCM vocabulary

o Estimate causal effect

o Block all open biasing paths

o Do not open any blocked path

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SCM vocabulary

o Ways to block or open paths:

Covariates

Sampling

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Basic graphs

o Chain:

X → W → Y

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Basic graphs

o Conditional independence:

X → W → Y

X ⊥ Y | W

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Basic graphs

o Fork:

X ← W → Y

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Basic graphs

o Conditional independence:

X ← W → Y

X ⊥ Y | W

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Basic graphs

o Inverted fork (collider):

X → W ← Y

X ⊥ Y

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Basic graphs

o Special role of colliders:

Controlling for a collider (or descendant) opens a blocked path

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Basic graphs

o Special role of colliders:

Controlling for a common outcome induces a spurious association between unrelated causes

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Basic graphs

o Special role of colliders:

Controlling for a common outcome adds a spurious component to related causes

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Basic graphs

o Inverted fork (collider):

X → W ← Y

X ⊥ Y

X Y | W ⊥

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Basic graphs

o Control for a collider (statistical):

rXY = 0 rYW = .40 rXW = .30

rXY·W = −.14

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Basic graphs

o Control for a collider (sampling):

Speed → Fatalities ← Alcohol

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Basic graphs

o Descendant of a collider:

X → W ← Y

↓ A

X Y | A ⊥

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Covariates o Achen, C. H. (2005). Let’s put garbage-can

regressions and garbage-can probits where they belong. Conflict

Management and Peace Science, 22, 327–339.

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Covariates o Regression assumes:

No causal effects between predictors Single equation

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Basic graphs

o Overcontrol bias:

X → W → Y

X ⊥ Y | W

Y on (X, W), BX = 0

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Basic graphs

o Endogenous selection bias:

X → W ← Y

X ⊥ Y

Y on (X, W), BX ≠ 0

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d-Separation

o Conditional independences

o Testable implications

o Basis for identification

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d-Separation

o Z d-separates X, Y if

1. Z closes all open paths

2. Z opens no blocked path

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X → A → B → Y

X ⊥ B | A

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X → A → B → Y

A ⊥ Y | B

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X → A → B → Y

X ⊥ Y | A

X ⊥ Y | B

X ⊥ Y | (A, B)

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X → A ← B → Y

X ⊥ B

X B | A ⊥

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X → A ← B → Y

A ⊥ Y | B

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X → A ← B → Y

X ⊥ Y

X ⊥ Y | B

X Y | A ⊥

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X B | A ⊥

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Identification

o Graphical criteria

o Sufficient (deconfounding) set

o Removes all noncausal aspects

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Identification

o Back-door criterion (total effects)

o Closes biasing (back-door) paths

o Leaves only causal

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Causal

X E Y A

C

X E Y

D

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Back-door

X C Y

X A D Y

X A D E Y

A

C

X E Y

D

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A

C

X E Y

D X C Y

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X A D Y

A

C

X E Y

D

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A

C

X E Y

D X A D E Y

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Back-door

X C Y

X A D Y

X A D E Y

Y on (X, A, C)

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Back-door

X C Y

X A D Y

X A D E Y

Y on (X, C, D)

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A

C

X E Y

D

Y on (X, C, D)

Y on (X, A, C)

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Identification

o Single-door criterion (direct)

o Recursive, linear, continuous

o Multiple estimates

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Identification

o Single-door criterion (D → Y)

o Delete →

o Sufficient set d-separates

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X

A

Y

B

C

X

A

Y

B

C

U1

U2

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d-Separation

X → A → B → Y

(X, B) (X, Y) (A, Y)

3. Z closes all open paths

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Identification

o Merchant, A. T, & Pitiphat, W. (2002). Directed

acyclic graphs (DAGs): An aid to assess confounding in dental research. Community Dentistry and Oral Epidemiology, 30: 399–404.

o Fleischer, N. L., & Diez Roux , A. V. (2008). Using directed acyclic graphs to guide analyses of neighbourhood health effects: An introduction. Journal of Epidemiology & Community Health, 62, 842-846.

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Analysis o Roth, D. L., Wiebe, D. J., Fillingim, R. B., &

Shay, K. A. (1989). Life events, fitness, hardiness, and health: A simultaneous analysis of proposed stress-resistance effects. Journal of Personality and Social

Psychology, 57, 136–142.

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Figure 8.5. A recursive path model of health factors.

Exercise

Hardiness

Illness

DIl

1

Stress

1 DSt

Fitness

1 DFi

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Variable 1 2 3 4 5

1. Exercise —

2. Hardiness −.03 —

3. Fitness .39 .07 —

4. Stress −.05 −.23 −.13 —

5. Illness −.08 −.16 −.29 .34 —

M 40.90 0.0 67.10 4.80 716.70

SD 66.50 38.00 18.40 33.50 62.48

Note. These data (correlations, means, and

variances) are from D. Roth et al. (1989); N = 373.

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Independence Conditioning set Partial correlation

Exercise ⊥ Stress Hardiness −.058

Exercise ⊥ Illness Fitness, Stress .039

Hardiness ⊥ Fitness Exercise .089

Hardiness ⊥ Illness Fitness, Stress −.081

Fitness ⊥ Stress Exercise, Hardiness −.103

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Minimally sufficient set

Direct effect ∅ Exercise Hardiness Stress Fitness

Exercise Fitness .108 (.013) .390 — — — —

Hardiness Stress −.203 (.045) −.230 — — — —

Fitness Illness — −1.036 (.183) −.305 −.951 (.168) −.280 −.849 (.162) −.250 —

Stress Illness — .628 (.091) .337 .597 (.093) .320 — .574 (.089) .307

Note. Estimates are reported as unstandardized (standard error) standardized; ∅, empty set. Values in boldface control

for the all parents of each outcome.

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Extensions

o Locate instruments

o Models → counterfactuals

o Potential outcomes (PO)

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Strengths

o Unifying model (SEM, PO)

o Supports reasoning, planning

o Local fit, not global

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Limitations

o Classical measurement

o No global fit

o Few software tools

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