Q&A ON CAUSAL INDICATORS IN STRUCTURAL EQUATION MODELS

Transcription

1 Q&A ON CAUSAL INDICATORS IN STRUCTURAL EQUATION MODELS Kenneth Bollen University of North Carolina at Chapel Hill Presented at Consortium for the Advancement of Research Methods and Analysis (CARMA), Webcast from University of Nebraska, April 6, 2018.

2 REFLECTIONS ON CAUSAL INDICATORS Introduction Six questions 1. What are causal indicators? 2. Are causal and formative indicators the same thing? 3. Do causal indicators create a construct? 4. Is interpretational confounding a necessary consequence of using causal indicators? 5. Can we consider causal indicators as measures? 6. How can we distinguish between causal and effect (reflective) indicators? Implications & Discussion

3 INTRODUCTION Origins in Social & Behavioral Sciences Blalock (1963, AJS) Causal Inferences for Unmeasured Variables Indicators as causes and effects of unmeasured variables Focus on single indicator causing each latent variable Minority race (indicator), Exposure to discrimination (latent variable) Later considers >1 causal indicator Latent variable: Frustration Causal indicators: withhold sleep, withhold food

4 INTRODUCTION Little attention to causal indicators Researchers continue to assume effect (reflective) indicators Virtually all measurement tools (factor analysis, Cronbach s alpha, item response theory) implicitly assume effect indicators Last years more interest Research suggesting that causal indicators part of measures Discussions of consequences of ignoring causal indicators More attempts to model and discuss causal indicators Last 5-10 years backlash period Critics suggest problems inherent to causal indicators Interpretational confounding Not really measures of latent variables Identification issues

5 INTRODUCTION Backlash papers and original papers on causal indicators leave many in confusion Numerous questions arise from these clashes See Bollen & Diamantopoulos (2017) for full discussion Six questions I consider 1. What are causal indicators? 2. Are causal and formative indicators the same thing? 3. Do causal indicators create a construct? 4. Is interpretational confounding a necessary consequence of using causal indicators? 5. Can we consider causal indicators as measures? 6. How can we distinguish between causal and effect (reflective) indicators?

6 1. WHAT ARE CAUSAL INDICATORS? Figure 1 Five Effect (Reflective) Indicators, One Latent Variable η y 1 y 2 y 3 y 4 y 5 ε 1 ε 2 ε 3 ε 4 ε 5

7 1. WHAT ARE CAUSAL INDICATORS? Figure 2 Five Causal Indicators, One Latent Variable η ζ x 1 x 2 x 3 x 4 x 5

8 1. WHAT ARE CAUSAL INDICATORS? Causal indicators η = α η + γ η1 x 1 + γ η2 x 2 + γ η 3 x 3 + γ η 4 x 4 + γ η5 x 5 +ζ X causal indicators Conceptual unity (measure same thing) Need not correlate with each other Disturbance captures all other omitted influences on latent variable Examples: Latent variable: extent of social interaction Xs are time w/ family, time w/ friends, time w/ coworkers, time with strangers Latent variable: exposure to stress Xs are lose job, get married, get divorced, move houses, etc. Latent variable: SES Xs are income, education, occupation

9 2. ARE CAUSAL & FORMATIVE INDICATORS THE SAME THING? Early definition of formative indicators: When constructs are conceived as explanatory combinations of indicators (such as population change or marketing mix ) which are determined by a combination of variables, their indicators should be formative Fornell & Bookstein, (1982, p.442). Latent variable is linear combination of observed indicators. C 1i = w 10 + w 11 x 1i + w 12 x 2i +!+ w 1Q x Qi Weighted sum of indicators No disturbance or error Conceptual unity not required

10 2. ARE CAUSAL & FORMATIVE INDICATORS THE SAME THING? Formative indicators: contemporary usage ambiguous Sometimes refers to causal indicators η = α η + γ η1 x 1 + γ η2 x 2 + γ η 3 x 3 + γ η 4 x 4 + γ η5 x 5 +ζ Sometimes refers to composite indicators C 1i = w 10 + w 11 x 1i + w 12 x 2i +!+ w 1Q x Qi

11 2. ARE CAUSAL & FORMATIVE INDICATORS THE SAME THING? Two Different usages of Formative Indicators x 1 γ 11 ζ 1 x 1 w 11 x 2 x Q γ 12 γ 1Q η 1 x 2 x Q w 12 w 1Q C 1 Causal Indicators Composite Indicators

12 2. ARE CAUSAL & FORMATIVE INDICATORS THE SAME THING? Formative indicators: contemporary usage ambiguous Creates problems because causal indicators and composite indicators have different properties Composite can be arbitrary combination of observed variables relationships to various outcomes might differ & lead to different weights Causal indicators Indicators should measure same thing Coefficients of indicators stable across outcomes

13 2. ARE CAUSAL & FORMATIVE INDICATORS THE SAME THING? Formative indicators: contemporary usage ambiguous What to do? Replace formative with causal indicators or composite indicators depending on intended meaning. Will do this for remainder of this presentation Use causal-formative indicators or composite-formative indicators Bollen & Diamantopoulos (2017) follow 2 nd strategy

14 3. Do causal indicators create a construct? Widespread belief that constructs with causal indicators do not exist independently of the indicators Operationalist perspective Indicators are the constructs Quotes: Yet, the whole point of a formative LV like F is that its meaning should be determined by its predictors, not by its consequences. (Treiblmaier, Bentler, & Mair, 2011, p.5) This lack of distinction between the indicators and the construct is explicit in the conceptual definition of formative measurement (Lee, Cadogan & Chamberlain, 2013, p. 7). Such measures are termed formative, meaning the construct is formed or induced by its measures (Fornell & Bookstein, 1982).

15 3. Do causal indicators create a construct? My viewpoint: Concepts Ideas that have some unity or something in common Meaning of concept spelt out in theoretical definition Dimensionality of concept determined by number of distinct components contained in concept Each dimension represented by 1 latent variable Indicators Observed variables that measure a latent variable Researcher hypothesizes either: Observed variableàlatent or latentàobserved variable Concepts precede indicators for either causal or effect indicators Indicators do not create constructs

16 3. Do causal indicators create a construct? My viewpoint: Example: Face-to-face social interaction Definition: face-to-face social interaction is the amount of time that an individual spends relating to another person or group who is in the same physical space. Dimension: One primary (though could elaborate for other dimensions) Latent variable: social interaction Indicators: time with family, time with friends, time with coworkers, time with strangers. Indicatorsàsocial interaction Concept of Face-to-face social interaction exists as a theoretical idea, prior to development of indicators. Indicators are developed to measure concept. Causal indicators do not create concept. Concept devised first. Indicators are then created or sought.

17 4. Is interpretational confounding a necessary consequence of using causal indicators? Interpretational confounding: the assignment of the other than a priori assigned empirical meaning of an unobserved variable. Burt (1976, SMR) Number of researchers claim that interpretational confounding is inherent in the use of causal indicators Quotes: The nature (empirical meaning) of the latent construct depends on the dependent variables or constructs included in the model. Formative measurement models are subject to interpretational confounding Howell, Breivik & Wilcox (2007, 208) The parameters relating the observed variables to their purported formative latent variable are functions of the number and nature of endogenous latent variables and their measures. Bagozzi (2007, p.236)

18 4. Is interpretational confounding a necessary consequence of using causal indicators? Claims: Meaning of latent variable shifts according to outcomes, either: Effect indicators Other latent variables Mixture of indicators and latent variables Same causal indicators in 2 studies lead to different latent variables if outcomes of that latent variable not the same Causal indicators are especially susceptible to this problem Effect indicators have interpretational confounding, but less likely than causal indicators

19 4. Is interpretational confounding a necessary consequence of using causal indicators? My viewpoint: Interpretational confounding potential issue for both causal and effect (reflective) indicators Burt s (1976) original arguments were for effect indicators Interpretational confounding is due to misspecification not to causal indicators [see Bollen (2007) and Bainter & Bollen (2014)] If different sets of outcomes are changing latent variable, then simultaneous fitting of ALL indicators will lead to decline in model fit

20 Four causal indicators for η 1 with four latent variable outcomes (η 2 to η 5 ) see Bainter & Bollen (2014)

21 4. Is interpretational confounding a necessary consequence of using causal indicators? If critics are right, then analyzing subset of outcomes should change coefficients of causal indicators Changes should be beyond sampling fluctuations If Bollen (2007) is right, then should not change after taking into account sampling fluctuations or scaling differences Bollen (2007) and Bainter & Bollen (2014) examine with simulation data Use Figure from previous slide to generate data Analyze full model from figure, then analyze only first two outcomes and finally only last two outcomes Interpreting confounding should lead causal indicator coefficients to change If coefficients stable, contradicts numerous claims that causal indicator coefficients change with outcome

22 Four causal indicators for η 1 with four latent variable outcomes (η 2 to η 5 ) see Bainter & Bollen (2014)

23 Table 4 from Bainter & Bollen (2014)

24 4. Is interpretational confounding a necessary consequence of using causal indicators? If model correct, then analyzing subset of outcomes does not change coefficients of causal indicators beyond that expected by sampling fluctuations Claim of inherent interpretation confounding is not supported Structural misspecification will be accompanied by shift in coefficients of causal indicators or effect indicators

25 5. Can we consider causal indicators as measures? Variety of researchers argue that causal indicators are not measures & should not be treated as such. QUOTES: One wonders whether it is appropriate to view formative models as measurement models in the first place. They might be better conceptualized as models for indexing or summarizing the indicators or as causal (Borsboom, Mellenbergh & van Heerden, 2004, p. 1069). Early literature conceptualizes causal indicators as causes of unmeasured constructs rather than as measures of latent constructs as is their common interpretation by formative measurement proponents today (Hardin & Marcoulides, 2011, p. 759).

26 5. Can we consider causal indicators as measures? My Viewpoint: Measurement is the process by which a concept is linked to one or more latent variables, and these are linked to observed variables. Bollen (1989, p.180) Theoretical definition gives meaning to concept and reveals dimensionality. Each dimension, represented by latent variable. Each latent variable has indicator(s) Indicators should conform to theoretical definition Indicators of same latent variable have conceptual unity Causal indicators should conform to theoretical definition A measurement model specifies a structural model connecting latent variables to one or more measures or observed variables. Bollen (1989, p.182) Causal indicators have direct relation to latent variable Causal indicatoràlatent variable

27 5. Can we consider causal indicators as measures? My Viewpoint: A measurement model specifies a structural model connecting latent variables to one or more measures or observed variables. Bollen (1989, p.182) Does not require latent variableàindicator Permits indicatoràlatent variable Definition permits causal indicators or effect indicators as measures Measurement model s task is to specify nature of relationship Example: Face-to-face social interaction Definition: face-to-face social interaction is the amount of time that an individual spends relating to another person or group who is in the same geographical space. Latent variable: social interaction Indicators: time with family, time with friends, time with coworkers, time with strangers. Indicatorsàsocial interaction

28 5. Can we consider causal indicators as measures? Definition of what it means to be a measure answers this question My definition permits causal or effect indicators or mixture of the two. Indicator must correspond to theoretical definition of the concept

29 5. Can we consider causal indicators as measures? Some definitions define a measure as depending on the latent variable Tautological argument: if you define a measure as an observed variable that depends on a latent variable, then by definition causal indicators do not qualify

30 5. Can we consider causal indicators as measures? Why not just consider causal indicators as covariates? Covariates need not have conceptual unity, causal indicators must. Effects of covariates differ by outcome, causal indicators do not unless model is misspecified Why not just consider causal indicators as compositeformative indicators? Composites need not have conceptual unity, causal indicators must Composite indicators form an exact weighted sum composite, causal indicators affect latent variable but so does error term Composite indicators can be arbitrary grouping of variables and need not keep the same weights as outcome differs. Causal indicators not the same as covariates or composite indicators (Bollen & Bauldry, 2011)

31 6. How can we distinguish between causal and effect (reflective) indicators?

32 6. How can we distinguish between causal and effect (reflective) indicators? Generally, effect (reflective) indicators each positively related to the same latent variable, should correlate with each other. Causal indicators can have +, -, or no association with each other ML Likelihood Ratio tests cannot compare Effect indicators & causal indicators not nested models Causal indicators alone are not identified

33 6. How can we distinguish between causal and effect (reflective) indicators?

34 6. How can we distinguish between causal and effect (reflective) indicators? Vanishing tetrad test for causal indicators [Bollen & Ting, 2000, Psych. Methods] τ 1234 = σ 12 σ 34 - σ 13 σ 24 = 0, τ 1342 = σ 13 σ 42 - σ 14 σ 32 =0 τ 1423 = σ 14 σ 23 - σ 12 σ 43 = 0

35 6. How can we distinguish between causal and effect (reflective) indicators? Vanishing tetrad test for causal indicators [Bollen & Ting, 2000, Psych. Methods] τ 1234 = σ 12 σ 34 - σ 13 σ 24 0, τ 1342 = σ 13 σ 42 - σ 14 σ 32 0 τ 1423 = σ 14 σ 23 - σ 12 σ 43 0

36 6. How can we distinguish between causal and effect (reflective) indicators? Effect indicators imply all tetrads = 0; Causal indicators imply no tetrads need be zero Nested relation: tetrads implied to be zero for causal indicators are subset of those for effect indicators. Test statistic: Bollen (1990, SMR) asymp. Chi square; df= # zero tetrads Stata procedure (Bauldry & Bollen, 2016) SAS macro (Hipp, Bauer, Bollen, 2005, SEMing) Tests of mixtures of causal and effect indicators

37 DISCUSSION & IMPLICATIONS Six questions 1. What are causal indicators? 2. Are causal and formative indicators the same thing? 3. Do causal indicators create a construct? 4. Is interpretational confounding a necessary consequence of using causal indicators? 5. Can we consider causal indicators as measures? 6. How can we distinguish between causal and effect (reflective) indicators? Presentation gave my response to each Many misunderstandings and odd claims about causal indicators. Backlash against nontraditional type of measures has generated false assertions

38 DISCUSSION & IMPLICATIONS Researchers should not seek causal indicators instead of effect indicators No claim that we should abandon reflective/effect indicators in favor of causal indicators Rather researchers should be alert to the presence of causal indicators in many existing scales Treating causal indicators as effect indicators is a misspecification Causal indicators behave differently from effect indicators and might mislead researchers who assume they are effect indicators.

39 DISCUSSION & IMPLICATIONS Current publications on causal-formative indicators are disappointing Critiques are rarely rigorous and formally based Ambiguity and differences in the way same terms are used Attempts at censorship Some authors argue that there should be a moratorium on causalformative indidators Strong, emotional appeals not to use causal indicators

40 DISCUSSION & IMPLICATIONS Compared to effect (reflective) indicators, the idea of causal indicators is relatively new An open discussion about causal indicators and contrasting their properties with those of effect indicators is needed Attempts to close off discussions is counterproductive Backlash hopefully will give way to reasoned exploration A scientific approach to measurement demands this

41 REFERENCES Bainter, Sierra and K. A. Bollen. (2014). "Interpretational Confounding or Confounded Interpretations of Causal Indicators." Focus Article, Measurement: Interdisciplinary Research & Perspectives 12: Bauldry, Shawn and K. A. Bollen. (2016). "tetrad: A Set of Stata Commands for Confirmatory Tetrad Analysis." Structural Equation Modeling 23: Bollen, K.A. (2011). Evaluating Effect, Composite, and Causal Indicators in Structural Equation Models (SEMs). Management Information Systems Quarterly (MISQ) 35: Bollen, K.A. (2007). Interpretational confounding is due to misspecification, not to type of indicator. Psychological Methods 12: Bollen, K.A. (1990). "Outlier Screening and a Distribution Free Test for Vanishing Tetrads." Sociological Methods and Research 19: Bollen, K.A. (1984). "Multiple Indicators: Internal Consistency or No Necessary Relationship. Quality and Quantity 18: Bollen, K.A. and S. Bauldry. (2011). Three Cs in Measurement Models: Causal Indicators Composite Indicators, and Covariates. Psychological Methods 16(3):

42 REFERENCES Bollen, K. A. and W. R. Davis. (2009). Causal Indicator Models: Identification, Estimation, and Testing. Structural Equation Modeling 16: Bollen, K.A. and Adamantios Diamantopoulos. (2017). "In Defense of Causal-Formative Indicators: A Minority Report." Psychological Methods 22: Bollen, K.A. and R. Lennox. (1991). "Conventional Wisdom on Measurement: A Structural Equation Perspective." Psychological Bulletin, 110: Bollen, K.A., R.D. Lennox, and D.L. Dahly. (2009). Practical Application of the Vanishing Tetrad Test for Causal Indicator Measurement Models: An Example from Health-Related Quality of Life. Statistics in Medicine 28: Bollen, K.A. and Kwok-fai Ting. (2000). A Tetrad Test for Causal Indicators. Psychological Methods 5:3-22. Hipp, J., D.J. Bauer, and K. A. Bollen. (2005). Conducting Tetrad Tests of Model Fit and Contrasts of Tetrad-Nested Models: A New SAS Macro. Structural Equation Modeling 12:76-93.