EIGHTMYTHSABOUT
CAUSALITYAND
STRUCTURALEQUATION
MODELS*
KennethA.Bollen,UniversityofNorthCarolinaatChapelHill
JudeaPearl,UniversityofCaliforniaatLosAngeles
*TheauthorswouldliketothankShawnBauldry,StephenCole,KeithMarcus,CameronMcIntosh,Stan
Mulaik,JohannesTextor,andotherresearchersfromSEMNETfortheircommentsonandcritiquesof
ourpaper.B
ollen’sworkwaspartiallysupportedbyNSFSES061
7276.Pearl’sworkwaspartially
supportedbygrantsfromNIH#1R01LM009961‐01,NSF#IIS‐0914211and#IIS‐1018922,andONR
#N000‐14‐09‐1‐0665and#N00014‐10‐0933.
In S.L. Morgan (Ed.), Handbook of Causal Analysis for Social Research, Chapter 15, 301-328, Springer 2013.
TECHNICAL REPORT
R-393
April 2012
1
ABSTRACT
CausalitywasatthecenteroftheearlyhistoryofStructuralEquationModels(SEMs)whichcontinueto
serveasthemostpopularapproachtocausalanalysisinthesocialsciences.Throughdecadesof
developmentcriticsanddefensesofthecapabilityofSEMstosupportcausalinferencehave
accumulated.Avarietyofmisundersta
ndingsandmythsaboutthenatureofSEMsandtheirrolein
causalanalysishaveemergedandtheirrepetitionhasledsometobelievetheyaretrue.Ourpaperis
organizedbypresentingeightmythsaboutcausalityandSEMsinthehopethatthiswillleadtoamore
accurat
eunderstanding.Morespecifically,theeightmythsare:(1)SEMsaimtoestablishcausal
relationsfromassociationsalone,(2)SEMsandregressionareessentiallyequivalent,(3)Nocausation
withoutmanipulation,(4)SEMsarenotequippedtohandlenonlinearcausalrelationships,(5)A
potentialoutcomeframeworkismoreprincipledthanSEMs,(6)SEMsarenotappli
cabletoexperiments
withrandomizedtreatments,(7)MediationanalysisinSEMsisinherentlynoncausal,and(8)SEMsdo
nottestanymajorpartofthetheoryagainstthedata.Wepresentthefactsthatdispellthesemyths,
describewhatSEMscanandcannotdo,andbrieflypresentourcri
tiqueofcurrentpracticeusingSEMs.
WeconcludethatthecurrentcapabilitiesofSEMstoformalizeandimplementcausalinferencetasksare
indispensible;itspotential todomoreisevengreater.
EIGHT MYTHS ABOUT CAUSALITY AND STRUCTURAL EQUATION MODELS
Social scientists’ interest in causal effects is as old as the social sciences. Attention to
the philosophical underpinnings and the methodological challenges of analyzing
causality has waxed and waned. Other authors in this volume trace the history of the
concept of causality in the social sciences and we leave this task to their skilled hands.
But we do note that we are at a time when there is a renaissance, if not a revolution in
the methodology of causal inference, and structural equation models play a major role
in this renaissance.
Our emphasis in this chapter is on causality and structural equation models (SEMs). If
nothing else, the pervasiveness of SEMs justifies such a focus. SEM applications are
published in numerous substantive journals. Methodological developments on SEMs
regularly appear in journals such as Sociological Methods & Research, Psychometrika,
Sociological Methodology, Multivariate Behavioral Research, Psychological Methods,
and Structural Equation Modeling, not to mention journals in the econometrics literature.
Over 3,000 subscribers belong to SEMNET, a listserv devoted to SEMs. Thus interest
in SEMs is high and continues to grow (e.g., Hershberger 2003; Schnoll, Fang, and
Manne 2004; Shah and Goldstein 2006).
Discussions of causality in SEMs are hardly in proportion to their widespread use.
Indeed, criticisms of using SEMs in analysis of causes are more frequent than
2
explanations of the role causality in SEMs. Misunderstandings of SEMs are evident in
many of these. Some suggest that there is only one true way to attack causality and
that way excludes SEMs. Others claim that SEMs are equivalent to regression analysis
or that SEM methodology is incompatible with intervention analysis or the potential
outcome framework. On the other hand, there are valid concerns that arise from more
thoughtful literature that deserve more discussion. We will address both the distortions
and the insights from critics in our chapter.
We also would like to emphasize that SEMs have not emerged from a smooth linear
evolution of homogenous thought. Like any vital field, there are differences and
debates that surround it. However, there are enough common themes and
characteristics to cohere, and we seek to emphasize those commonalities in our
discussion.
Our paper is organized by presenting eight myths about causality and SEMs in the hope
that this will lead to a more accurate understanding. More specifically, the eight myths
are: (1) SEMs aim to establish causal relations from associations alone, (2) SEMs and
regression are essentially equivalent, (3) No causation without manipulation, (4) SEMs
are not equipped to handle nonlinear causal relationships, (5) A potential outcome
framework is more principled than SEMs, (6) SEMs are not applicable to experiments
with randomized treatments, (7) Mediation analysis in SEMs is inherently noncausal,
and (8) SEMs do not test any major part of the theory against the data.
In the next section we provide the model and assumptions of SEM. The primary section
on the eight myths follows and we end with our conclusion section.
MODEL AND ASSUMPTIONS OF SEMs
Numerous scholars across several disciplines are responsible for the development of
and popularization of SEMs. Blalock (1960, 1961, 1962, 1969), Duncan (1966, 1975),
Jöreskog (1969, 1970, 1973), and Goldberger (1972; Goldberger and Duncan 1973)
were prominent among these in the wave of developments in the 1960s and 1970s. But
looking back further and if forced to list just one name for the origins of SEMs, Sewall
Wright (1918, 1921, 1934), the developer of path analysis, would be a good choice.
Over time this model has evolved in several directions. Perhaps the most popular
general SEM that takes account of measurement error in observed variables is the
LISREL model proposed by Jöreskog and Sörbom (1978). This model simplifies if
measurement error is negligible as we will illustrate below. But for now we present the
general model so as to be more inclusive in the type of structural equations that we can
handle. We also note that this model is linear in the parameters and assumes that the
coefficients are constant over individuals. Later when we address the myth that SEMs
cannot incorporate nonlinearity or heterogeneity, we will present a more general
nonparametric form of SEMs which relaxes these assumptions. But to keep things
simpler, we now stay with the widely used linear SEM with constant coefficients.
3
This SEM consists of two major parts. The first is a set of equations that give the causal
relations between the substantive variables of interest, also called “latent variables”,
because they are often inaccessible to direct measurement (Bollen 2002). Self-esteem,
depression, social capital, and socioeconomic status are just a few of the numerous
variables that are theoretically important but are not currently measured without
substantial measurement error. The latent variable model gives the causal relationships
between these variables in the absence of measurement error. It is
1
iiii
ζ
Γ
ξα=η
+
++ ηB
η
(1)
The second part of the model ties the observed variables or measures to the
substantive latent variables in a two equation measurement model of
iiyyi
ε
ηΛα=
+
+y
(2)
iixxi
δ
Λα=
+
+ ξx (3)
In these equations, the subscript of i stands for the ith case,
i
η
is the vector of latent
endogenous variables,
η
α
is the vector of intercepts,
B
is the matrix of coefficients that
give the expected effect
2
of the
i
η on
i
η
where its main diagonal is zero
3
,
i
ξ is the
vector of latent exogenous variables,
Γ
is the matrix of coefficients that give the
expected effects of
i
ξ on
i
η , and
i
ζ
is the vector of equation disturbances that consists
of all other influences of
i
η that are not included in the equation. The latent variable
model assumes that the mean of the disturbances is zero [E(
i
ζ
)=0] and that the
disturbances are uncorrelated with the latent exogenous variables [COV(
i
ζ
,
i
ξ )=0]. If
on reflection a researcher’s knowledge suggests a violation of this latter assumption,
then those variables correlated with the disturbances are not exogenous and should be
included as an endogenous latent variable in the model.
The covariance matrix of
i
ξ is Φ and the covariance matrix of
i
ζ
is Ψ. The researcher
determines whether these elements are freely estimated or are constrained to zero or
some other value.
In the measurement model,
i
y is the vector of indicators of
i
η
,
y
α
is the vector of
intercepts,
y
Λ
is the factor loading matrix that gives the expected effects of
i
η on
i
y ,
1
ThenotationslightlydepartsfromtheLISRELnotationinitsrepresentationofintercepts.
2
Theexpectedeffectreferstotheexpectedvalueoftheeffectofoneη
onanother.
3
Thisrulesoutavariablewithadirecteffectonitself.
4
and
i
ε is the vector of unique factors (or disturbances) that consists of all the other
influences on
i
y that are not part of
i
η
. The
i
x is the vector of indicators of
i
ξ ,
x
α
is the
vector of intercepts,
x
Λ is the factor loading matrix that gives the expected effects of
i
ξ
on
i
x , and
i
δ is the vector of unique factors (or disturbances) that consists of all the
other influences on
i
x that are not part of
i
ξ . The measurement model assumes that
the means of disturbances (unique factors) [E(
i
ε
), E(
i
δ )] are zero and that the different
disturbances are uncorrelated with each other and with the latent exogenous variables
[i.e., COV(
i
ε ,
i
ξ ),COV(
i
δ ,
i
ξ ),COV(
i
ε
,
i
ζ
),COV(
i
δ ,
i
ζ
) are all zero]. Each of these
assumptions requires thoughtful evaluation. Those that are violated will require a
respecification of the model to incorporate the covariance. The covariance matrix for
i
δ
is ϴ
δ
and the covariance matrix for
i
ε
is ϴ
ε
. The researcher must decide whether these
elements are fixed to zero, some other constraint, or are freely estimated.
The SEM explicitly recognizes that the substantive variables represented in
i
η and
i
ξ
are likely measured with error and possibly measured by multiple indicators. Therefore,
the preceding separate specification links the observed variables that serve as
indicators to their corresponding latent variables. Indicators influenced by single or
multiple latent variables are easy to accommodate. Researchers can include correlated
disturbances from the latent variable or measurement model by freely estimating the
respective matrix entries in the covariance matrices of these disturbances mentioned
above (i.e., Ψ , ϴ
δ
, ϴ
ε
). If it happens that an observed variable has negligible
measurement error, it is easy to represent this by setting the observed variable and
latent variable equal (e.g., =
i
x
3 i
ξ
3
).
Now we focus on the “Structural” in Structural Equation Models. By structural we mean
that the researcher incorporates causal assumptions as part of the model. In other
words, each equation is a representation of causal relationships between a set of
variables, and the form of each equation conveys the assumptions that the analyst has
asserted.
To illustrate, we retreat from the general latent variable structural equation model
presented above and make the previously mentioned simplifying assumption that all
variables are measured without error. Formally, this means that the measurement
model becomes
ii
η=y and
ii
ξ
=x . This permits us to replace the latent variables with
the observed variables and our latent variable model becomes the well-known
simultaneous equation model of
iiiii
ζ
Γ
α=
+
+
+ xyBy
η
(4)
We can distinguish weak and strong causal assumptions. Strong causal assumptions
are ones that assume that parameters take specific values. For instance, a claim that
