Reverse arrow dynamics - Concordia University

40 Pages·2013·1.24 MB·English

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Chapter 2 v
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ReveRse ARRow DynAmics
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Feedback Loops and Formative
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m1easurement
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0 Rex B. Klin e
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Life can only be understood backwards; but it must be lived forwards.
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—Soren Kierkegaard (1843; quoted in Watkin, 2004, para. 31)
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©This chapter is about two types of special covariance structure models
where some arrorws (paths) point backwards compared with more standard
models. The first kind is nonrecursive structural models with feedback

loops where sets of variables are specified as causes and effects of each
l
other in a cross-sectional design. An example of a feedback loop is the
specificlation V1 ⇄ V2 where V1 is presumed to affect V2 and vice-versa;
that Ais, there is feedback. It is relatively easy to think of several real world
causal processes, especially dynamic ones that may be based on cycles of
mutual influence, including the relation between parental behavior and
child behavior, rates of incarceration and crime rates, and violence on the
part of protestors and police. In standard, or recursive, structural models
Structural Equation Modeling: A Second Course (2nd ed.), pages 39–77
Copyright © 2013 by Information Age Publishing
All rights of reproduction in any form reserved. 39
40  r. B. KLINe
estimated in cross-sectional designs, all presumed causal effects aPre speci- P
fied as unidirectional only so that no two variables are causes of each other.
Thus, it is impossible to estimate feedback effects when analyzing recur-
sive structural models in cross-sectional designs. For example, a recursive
model for such a design could include either the path V1 → V2 or the path
d d
A A
V2 → V1, but not both.
The second type of special model considered in this chapter are for-
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mative measurement models—also known as emergent variable systems—
where observed variables (indicators) are specified as causes of underlying
factors (e.g., V1 → F1). The latter are referred to as latent composvites be- v
I I
cause they are represented as the consequence of their indicators plus error
variance. This directionality specification is reversed comparerd with stan- r

dard, or reflective, measurement models where factors are conceptualized
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as latent variables (constructs) that affect observed scores (e.g., F1 → V1)
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plus measurement error. It is reflective measurement that is based on clas-
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sical measurement theory, but there are certain research contexts where
the assumption that causality “flows” from factors to indicator is untenable.
e e
The specification that m1easurement is formative instead of reflective may 1
be an option in such cases.
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The specification of either nonrecursive structural models with feedback
loops or formativ0e measurement mode ls potentially extends the range of 0
hypotheses that can be tested in stsructural equation modeling (SEM). s
There are special considerations in the analysis of both kinds of models,
t t
however, and the failure to pay heed to these requirements may render the
results 2meaningless. Thus, the main goal of this presentation is to help read- 2
h h
ers make informed decisions about the estimation of reciprocal causation
in cross-sectional designs or the specification of formative measurement in
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SEM. Specifically, assumptions of both nonrecursive structural models with
feedback loops and formative measurement models are explained, and an
example of the anialysis of each type of model just mentioned is described. i
© ©
Readers can dorwnload all syntax, data, and output files for these examples r
for three different SEM computer tools, EQS, LISREL, and Mplus, from a

freely-accessible Web page.1 The computer syntax and output files are all
l l
plain-text (ASCII) files that require nothing more than a basic text editor,
such als Notepad in Microsoft Windows, to view their contents. For readers l
whAo use programs other than EQS, LISREL, or Mplus, it is still worthwhile A
to view these files because (1) there are common principles about pro-
gramming that apply across different SEM computer tools and (2) it can
be helpful to view the same analysis from somewhat different perspectives.
Contents of EQS, LISREL, and Mplus syntax files only for both examples
are listed in chapter appendices.
reverse arrow Dynamics  41
P NoNrecursive Models with Feedback loops P
There are two different kinds of feedback loops that can be specified in a
nonrecursive structural model, direct and indirect. The most common are
direct feedback loops where just two variables are specified as reciprocally af-
d d
A fecting one another, such as V1 ⇄ V2. Indirect feedback loAops involve three
or more variables that make up a presumed unidirectional cycle of influence.
e In a path diagram, an indirect feedback loop with V1, V2, and V3 would be e
represented as a “triangle” with paths that connect them in the order speci-
fied by the researcher. Shown without error terms or other variables in the
v v
I model, an example of an indirect feedback looIp with three variables is:
r r
V1
e e
3 3
s s
e e
V3 V2
1 1
r Because each variable in the feedback loop jurst illustrated serves as a me-
diator for the other two variables (e.g., V2 mediates the effect of V1 on V3,
0 0
and so on), feedback is thus indirect. A structural model with an indirect
s feedback loop is automatically nonrecurssive. Both direct and indirect feed-
back effects are estimated among variables measured concurrently instead
t t
of at different times; that is, the design is cross-sectional, not longitudinal.
2 Although2 there are theoretical models that would seem to map well onto
h h
indirect feedback loops—for instance, Carson (1982) described maladap-
tive self-fulfilling prophesies in psychopathology as an “unbroken causal
g g
loop between social perception, behavioral enactment, and environmental

reaction” (p. 576)—there are few reports of the estimation of models with
i indirect loops in thei behavioral science literature. This is probably due to
© ©
technical challenges in the analysis of such models. These same obstacles
r r
(elaborated later) also apply to models with direct feedback loops, but they
are somewhat less vexing when there are only two variables involved in cy-
l cles of preslumed mutual causation. Accordingly, only nonrecursive models
with direct feedback loops are considered next.
l l
Estimating reciprocal causality in a cross-sectional design by analyzing a
A A
nonrecursive model with a feedback loop can be understood as a proxy for
estimating such effects in a longitudinal design by analyzing a cross-lagged
panel model (Wong & Law, 1999). A feedback loop between V1 and V2 is
presented in Figure 2.1a without disturbances or other variables. Note that
(1) Figure 2.1a represents a fragment within a larger nonrecursive mod-
el and (2) it is not generally possible to estimate direct feedback without
42  r. B. KLINe
(a) Direct Feedback Loop (b) Panel Model P P
V1 V1 V1
1 2
d d
A A
V2 V2 V2
1 2 e e
Figure 2.1 Reciprocal causal effects between V1 and V2 represented with (a) a
v v
direct feedback loop based on a cross-sectional Idesign and (b) cross-lagged effects I
based on a longitudinal design (panel model), both models shown without distur-
bances or other variables. r r

e e
other variables in the model due to identification. Variables V1 and V2 in
3 3
Figure 2.1a are measured concurrently in a cross-sectional design, which
implies the absence of temporal precedence, or the mseasurement of a pre- s
sumed cause before that of the presumed effect. Evidence for reciprocal
e e
causation is indicated wh1en estimates of both direct effects in a feedback 1
loop, or V1 → V2 and V2 → V1 in Figure 2.1a, are of appreciable magni-
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tude, given the research problem.
Presented in F0igure 2.1b is a fragmen t from a cross-lagged panel model 0
shown without disturbances or other variables and where V1 and V2 are
s s
each measured at different times in a longitudinal design. For example,
V1 is measured at time 1 and agtain at time 2 designated by, respectively, t
V1 and V1 in Figure 2.1b; variable V2 is likewise measured twice (V2 and
1 22 1 2
V2 ). The possibility to collehct additional measurements of V1 and V2 at h
2
later times is also represented in Figure 2.1b. Presumed reciprocal causa-
tion is represented in tghe panel model of Figure 2.1b by the cross-lagged g
direct effects between V1 and V2 measured at different times, specifically,
V1  → V2 and V2  → V1 . Evidence for reciprocal causation is indicated
1 2 i1 2 i
© when estimates of both cross-lagged direct effects in a panel model are of ©
appreciable margnitude, again depending on the research area.2 r
Although it is theoretically and mathematically possible to estimate re-

ciprocal causation in cross-sectional designs when analyzing nonrecursive
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models with direct feedback loops, there is controversy about the adequacy
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of those estimates (Wong & Law, 1999). The main reason is the absence of
temAporal precedence in cross-sectional designs. Causal effects are typically A
understood as taking place within a finite period, that is, there is a latency
or lag between changes in causal variables and subsequent changes in out-
come variables. With concurrent measurement there are no lags whatso-
ever, so in this sense the measurement occasions in cross-sectional designs
are always wrong. This implies that the two-way paths that make up a direct
reverse arrow Dynamics  43
P feedback loop, such as V1 ⇄ V2 in Figure 2.1a, represent an instantaPneous
cycling process, but in reality there is no such causal mechanism (Hunter
& Gerbing, 1982).3 This may be true, but a feedback can still be viewed as
a proxy or statistical model for a longitudinal process where causal effects
occur within some definite latency. We do not expect statistical models to
d d
A A
exactly mirror the inner workings of a complex reality. Instead, a statistical
model is an approximation tool that helps researchers to structure their
e e
thinking (i.e., generate good ideas) in order to make sense of a phenom-
enon of interest (Humphreys, 2003). If the approximation is too coarse,
v then the model will be rejected. v
I I
In contrast to model with feedback for cross-sectional data, definite
r latencies (lags) for presumed causal effects are explicitly reprersented in

panel models by the measurement periods in a longitudinal design. For
e e
example, variables V1 and V2 in Figure 2.1b represent the measurement
3 1 2 3
of V1 at time 1 and V2 at some later time 2. This temporal precedence in
s s
measurement is consistent with the interpretation that the path V1  → V2
1 2
corresponds to a causal effect. But it is critical to correctly specify the lag
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1 interval when estimating 1cross-lagged direct effects in longitudinal de-
signs. This is because even if V1 actually causes V2, the observed magni-
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tude of the direct effect V1  → V2 may be too low if the measurement
1 2
0 interval is either to0o short (causal effects take time to materialize) or too
s long (temporary effects have dissipateds). The same requirement for a cor-
rect measurement interval applies to the other cross-lagged direct effect in
t t
Figure 2.1b, V2  → V1 .
1 2
2 The ab2sence of time precedence in cross-sectional designs may not al-
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ways be a liability in the estimation of reciprocal causation. Finkel (1995)
argued that the lag for some causal effects is so short that it would be
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im practical to measure them over time. An example is reciprocal effects
of the moods of spouses on each other. Although the causal lags in this
i example are not zerio, they may be so short as to be virtually synchronous.
© ©
r If so, then the asrsumption of instantaneous cycling for feedback loops in
nonrecursive designs would not be indefensible. Indeed, it may even be

more appropriate to estimate reciprocal effects with very short lags in a
l l
cross-sectional design even when longitudinal panel data are available
l (Wong &l Law, 1999). The true length of causal lags is not always known.
A In thAis case, longitudinal data collected according to some particular mea-
surement schedule are not automatically superior to cross-sectional data.
There is also the reality that longitudinal designs require more resources
than cross-sectional designs. For many researchers, the estimation of re-
ciprocal causation between variables measured simultaneously is the only
viable alternative to a longitudinal design.
44  r. B. KLINe
identification requirements of Models P P
with Feedback loops
Presented in Figure 2.2a is the most basic type of nonrecursive structural
model with a direct feedback loop that is identified. This model includes
d d
observed variables only (i.e., it is path model), but theA same basic configu- A
ration is required when variables in the structural model are latent each
measured with multiple indicators (i.e., standard reflective measuremenet) e
and the whole model is a latent variable path model, not a measured vari-
able path model (Kline, 2010, chap. 6). The two-headed curved arrows that
v v
exit and re-enter the same variable ( ) inI Figure 2.2a represent the vari- I
ances of exogenous variables, which are generally free model parameters
r r
in SEM. The disturbances of the two v ariables that make up the feedback
loop in Figure 2.2a, V3 and V3, are assumed to covary (D3   D4). This
e e
specification makes sense for two reasons: (1) if variables are presumed
3 3
to mutually cause one another, then it is plausible that there are common
s s
omitted causes of both; (2) some of the errors in predicting one variable in
AU: There is no part a direct feedback loop, such as V3 in Figure 2.1c,e are due to the other vari- e
(c) in Figure 2.1 1 1
(a) All possible disturbance correlations
r r
0 0
1 D3
s s
V1 V3
t t
2 2
h h
1 D4
V2 g V3 g

i i
© ©
(b) All possible disturbance correlations within recursively related blocks
r r
1 D3 1 D5
l l
l V1 V3 V5 l
A A
1 D4 1 D6
V2 V4 V6
Figure 2.2 Two examples of nonrecursive path models with feedback loops.
reverse arrow Dynamics  45
P able in that loop, or V4 in the figure, and vice versa (Schaubroeck, P1990).
Although there is no requirement for correlated disturbances for variables
involved in feedback loops, the presence of disturbance correlations in par-
ticular patterns in nonrecursive models helps to determine their identifica-
tion status, a point elaborated next.
d d
A The model in Figure 2.2a satisfies the two necessary Aidentification re-
quirements for any type of structural equation model: (1) all latent vari-
e ables are assigned a metric (scale), and (2) the number of free model pa-e
rameters is less than or equal to the number of observations (i.e., the model
degrees of freedom, df , are at least zero). In path models, disturbances
v M v
I can be viewed as latent (unmeasured) exogeInous variables each of which
requires a scale. The numerals (1) that appear in Figure 2.2a next to paths
r r
that point from disturbances to correspo nding endogenous variables, such
e as D3 → V3, are scaling constants that assign to the disturbeances a scale
3 related that of the unexplained var3iance of its endogenous variable. With
four (4) observed variables in Figure 2.2a, there are a total of 4(5)/2 = 10
s s
observations, which equals the number of variances and unique covarianc-
e es among the four variables. The number of free paerameters for the model
1 1
in Figure 2.2a is 10, which includes the variances of four exogenous vari-
ables (of V1, V2, D3, and D4), two covariances between pairs of exogenous
r r
variables (V1   V1, D3   D4), and four direct effects, or
0 0
s V1 → V3, V2 →V4, V3 →s V4, and V4 → V3.
t Because df = 0 (i.e., 10 observationts, 10 free parameters), the model in Fig-
M
ure 2.2a would, if actually identified, perfectly fit the data (i.e., the observed
2 2
h h
and predicted covariance matrices are identical), which means that no par-
ticular hypothesis would be tested in the analysis. It is usually only more
g complex nonrecursive mogdels with positive degrees of freedom (df > 0)—
M
which allow for the possibility of model-data discrepancies—that are ana-
lyzed in actual studies. But the basic pattern of direct effects in Figure 2.2a
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© ©from external variables (those not in the feedback loop, or V1 and V2) to
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variables involved in the feedback loop (V3, V4) respect both of the require-
ments for ide ntifying certain kinds of nonrecursive models outlined next.
If all latent variables are scaled and df ≥ 0, then any recursive structural
l l M
model is identified (e.g., Bollen, 1989, pp. 95–98). This characteristic of
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recursive structural models simplifies their analysis. Cross-lagged panel
A A
models are typically recursive, but they can be nonrecursive depending on
their pattern of disturbance covariances (if any). Unfortunately, the case
for nonrecursive models is more complicated. There are algebraic means
to determine whether a nonrecursive model is identified (Berry, 1984), but
these methods are practical only for very simple models. But there are two
heuristics (rules) that involve determining whether a nonrecursive model
46  r. B. KLINe
meets certain requirements for identification that are straightfoPrward to P
apply (Kline, 2010). These rules assume that df  ≥ 0 and all latent variables
M
are properly scaled in a nonrecursive model. The first rule is for the order
condition, which is a necessary requirement for identification. This means
that satisfying the order condition does not guarantee identification, but
d d
failing to meet this condition says that the model is Anot identified. The A
second rule is for the rank condition, which is a sufficient requirement for
identification, so a model that meets this requirement is in fact identifiede. e
The heuristics for the order condition and the rank condition apply to
either (1) nonrecursive models with feedback loops and all possible distur-
v v
bance covariances or (2) nonrecursive moIdels with ≥ 2 pairs of feedback I
loops with unidirectional paths between the loops but all possible distur-
r r
bance covariances within each loop. T he latter models are referred to as
block recursive by some authors even though the whole emodel is nonre- e
cursive. Consider the two nonrec3ursive path models in Figure 2.2. For both 3
models, df ≥ 0 and every latent variable is scaled, but these facts are not
M s s
sufficient to identify either model. The model of Figure 2.2a has an indirect
feedback loop that involves V3–V4 and all possiblee disturbance correlations e
1 1
(1), or D3   D4. The model of Figure 2.2b is a block recursive model with
two direct feedback loops, one that involves V3 and V4 and another loop
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made up of V5 and V6. Each block of these variable pairs in Figure 2.2b
contains all possi0ble disturbance correla tions (1)— D3   D4 for the first 0
block, D5   D6 for the second—but sthe disturbances across the blocks are s
independent (e.g., D3 is uncorrelated with D5). The pattern of direct effects
within each block in Figure 2.2b tare nonrecursive (e.g., V3 V4), but effects t
betwee2n the blocks are unidirectional (recursive) (e.g., V3 → V5). Thus, the 2
h h
two blocks of endogenous variables in the model of Figure 2.2b are recur-
sively related to each other even though the whole model is nonrecursive.
Order condition. Thge order condition is a counting rule applied to g

each variable in a feedback loop. If the order condition is not satisfied,
the equation for that variable is under-identified, which implies that the
i i
© whole model is not identified. One evaluates the order condition by tallying ©
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the number of variables in the structural model (except disturbances) that
have direct effects on each variable in a feedback loop versus the number
that do not; the latter are excluded variables. The order condition requires
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for models with all possible disturbance correlations that the number of
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variables excluded from the equation of each variable in a feedback loop
A A
exceeds the total number of endogenous in the whole model variables mi-
nus 1. For example, the model in Figure 2.2a with all possible disturbance
correlations has two variables in a feedback loop, V3 and V4. These two vari-
ables are the only endogenous variables in the model, so the total number
of endogenous variables is 2. Therefore, variables V3 and V4 in Figure 2.2a
must each have a minimum of 2 − 1 = 1 other variable excluded from each
reverse arrow Dynamics  47
P of their equations, which is here true: V1 is excluded from the equatiPon for
V4, and V2 is excluded from the equation for V3. Because there is 1 variable
excluded from the equation of each endogenous variable in Figure 2.2a,
the order condition is satisfied.
The order condition is evaluated separately for each feedback loop in
d d
A a block recursive model. For example, variables V3 and VA4 are involved in
the first feedback loop in the block recursive model of Figure 2.2b. For the
e moment we ignore the second feedback loop comprised of V5 and V6. The e
total number of endogenous variables for the first feedback loop is 2, or V3
and V4. This means that at least 2 – 1 = 1 variables must be excluded for
v v
I each of V3 and V4, which is true here: V1 is oImitted from the equation for
V4, and V2 is the excluded from the equation for V3. But the total number
r r
of endogenous variables for the second feedback loop comprised of V5
e and V6 in Figure 2.2b is 4, or V3 through V6. This is becausee the first feed-
3 back loop with V3 and V4 in Figure 32.2b is recursively related to the second
feedback loop with V5 and V6. Therefore, the order condition requires a
s s
minimum of 4 – 1 = 3 excluded variables for each of V5 and V6, which is
e true here: V1, V2, and V4 are omitted from the equaetion for V5, and V1, V2,
1 1
and V3 are excluded from the equation for V6. Thus, the block recursive
model of Figure 2.2b satisfies the order condition.
r r
Rank condition. Because the order condition is only necessary, it is still
0 uncertain whether t0he models in Figure 2. 2 are actually identified. Evalua-
s tion of the sufficient rank condition, hoswever, will provide the answer. The
rank condition is usually described in the SEM literature in matrix terms
t (e.g., Bollen, 1989, pp. 98–103). Bterry (1984) devised an algorithm for
2 checking2 the rank condition that does not require extensive knowledge of
h h
matrix operations. A non-technical description follows. The rank condition
can be viewed as a requirement that each variable in a feedback loop has a
g unique pattern of direct egffects on it from variables outside the loop. Such

a pattern of direct effects provides a “statistical anchor” so that the free
parameters of variables involved in feedback loops, including
i i
© ©
r r
V3 → V4, V4 → V3, Var(D3), Var(D4), and D3   D4

for the model in Figure 2.2a where “Var” means “variance,” can be esti-
l l
mated distinctly from one another. Each of the two endogenous variables
l l
in Figure 2.2a has a unique pattern of direct effects on it from variables
A A
external to the feedback loop; that is, V1 → V3 and V2 → V4. If all direct
effects on both endogenous variables in a direct feedback loop are from the
same set of external variables, then the whole nonrecursive model would
fail the rank condition.
Because the analogy just described does not hold for nonrecursive struc-
tural models that do not have feedback loops (e.g., Kline, 2010, pp. 106–
48  r. B. KLINe
108), a more formal means of evaluating the rank condition is needPed. Such P
a method that does not require knowledge of matrix algebra is described in
Kline (2010). The application of this method to the model in Figure 2.2a is
summarized in Appendix A of this chapter. The outcome of checking the
rank condition is the conclusion that the model in Figure 2.2a is identified;
d d
specifically, it is just-identified because df = 0. See KlineA (2010, p. 153) for a A
M
demonstration that the block recursive model in Figure 2.2b also meets the
rank condition. Thus, the model is identified, specifically, it is also just-ideneti- e
fied because df = 0. See Rigdon (1995), who described a graphical technique
M
for evaluating whether nonrecursive models with direct feedback loops are
v v
identified and also Eusebi (2008), who deIscribed a graphical counterpart I
of the rank condition that requires knowledge of undirected, directed, and
r r
directed acyclic graphs from graphical models theory.
There may be no sufficient conditions that are straightfeorward to apply e
in order to check the identificatio3n status of nonrecursive structural models 3
with disturbance covariances (including none) that do not match the two
s s
patterns described earlier. There are some empirical checks, however, that
can be conducted to evaluate the uniqueness ofe a converged solution for e
1 1
such models (see Bollen, 1989, pp. 246–251; Kline, 2010, p. 233). These
tests are only necessary conditions for identification. That is, a solution that
r r
passes them is not guaranteed to be unique.
Technical prob0lems in the analysis a re common even for nonrecursive 0
models proven to be identified. For exsample, iterative estimation of models s
with feedback loops may fail to converge unless start values are quite accu-
rate. This is especially true for thte direct effects and disturbance variances t
and co2variances of endogenous variables involved in feedback loops; see 2
h h
Kline (2010, pp. 172–175, 185) for an example of how to specify start values
for a feedback loop. Even if estimation converges, the solution may be inad-
missible, that is, it contagins Heywood cases such as a negative variance esti- g

mate, an estimated correlation with an absolute value > 1.00, or other kinds
of illogical result such as an estimated standard error that is so large that
i i
© no interpretation is reasonable. Possible causes of Heywood cases include ©
r r
specification errors, nonidentification of the model, bad start values, the
presence o f outliers that distort the solution, or a combination of a small
sample size and only two indicators when analyzing latent variable models
l l
(Chen, Bollen, Paxton, Curran, & Kirby, 2001).
l l
A A
special assumptions of Models with Feedback loops
Estimation of reciprocal effects by analyzing a nonrecursive model with a
feedback loop is based on two special assumptions. These assumptions are
required because data from a cross-sectional design give only a “snapshot”