Article

A Critique of the Cross-Lagged Panel Model

Abstract

The cross-lagged panel model (CLPM) is believed by many to overcome the problems associated with the use of cross-lagged correlations as a way to study causal influences in longitudinal panel data. The current article, however, shows that if stability of constructs is to some extent of a trait-like, time-invariant nature, the autoregressive relationships of the CLPM fail to adequately account for this. As a result, the lagged parameters that are obtained with the CLPM do not represent the actual within-person relationships over time, and this may lead to erroneous conclusions regarding the presence, predominance, and sign of causal influences. In this article we present an alternative model that separates the within-person process from stable between-person differences through the inclusion of random intercepts, and we discuss how this model is related to existing structural equation models that include cross-lagged relationships. We derive the analytical relationship between the cross-lagged parameters from the CLPM and the alternative model, and use simulations to demonstrate the spurious results that may arise when using the CLPM to analyze data that include stable, trait-like individual differences. We also present a modeling strategy to avoid this pitfall and illustrate this using an empirical data set. The implications for both existing and future cross-lagged panel research are discussed. (PsycINFO Database Record (c) 2015 APA, all rights reserved).
Cross-lagged panel model 1
Running head: CROSS-LAGGED PANEL MODEL
A critique of the cross-lagged panel model
E. L. Hamaker1,R.M.Kuiper
1andR.P.P.P.Grasman
2
1. Methodology and Statistics, Faculty of Social and Behavioural Sciences, Utrecht
University
2. Psychological Methodology, University of Amsterdam
Author Note:
This study was supported by the Netherlands Organization for Scientific Research
(NWO; VIDI Grant 452-10-007).
Correspondence concerning this paper should be addressed to E. L. Hamaker,
Methodology and Statistics, Faculty of Social and Behavioural Sciences, Utrecht
University, P.O. Box 80140, 3508 TC, Utrecht, The Netherlands. Email:
e.l.hamaker@uu.nl.
Published in 2015, in Psychological Methods, 20(1), 102-116.
doi: 10.1037/a0038889.
Cross-lagged panel model 2
Abstract
The cross-lagged panel model is believed by many to overcome the problems
associated with the use of cross-lagged correlations as a way to study causal
influences in longitudinal panel data. The current paper however shows that if
stability of constructs is to some extent of a trait-like, time-invariant nature, the
autoregressive relationships of the cross-lagged panel model fail to adequately
account for this. As a result, the lagged parameters that are obtained with the
cross-lagged panel model do not represent the actual within-person relationships
over time, and this may lead to erroneous conclusions regarding the presence,
predominance, and sign of causal influences.
We present an alternative model that separates the within-person process from
stable between-person differences, and discuss how this model is related to existing
structural equation models that include cross-lagged relationships. Furthermore, we
derive the analytical relationship between the cross-lagged parameters from this
alternative model and those from the cross-lagged panel model. Through
simulations we demonstrate the spurious results that may arise when using the
cross-lagged panel model to analyze data that include stable, trait-like individual
differences. This is followed by the presentation of a modeling strategy to avoid this
pitfall, which we illustrate using an empirical data set. The implications for existing
and future cross-lagged panel research are discussed.
Cross-lagged panel model 3
A critique of the cross-lagged panel model
In 1980, Rogosa’s seminal paper A critique of the cross-lagged correlation was
published, which successfully conveyed the message that comparing cross-lagged
correlations from longitudinal panel data is an inappropriate basis for making causal
inferences.1One of the key insights stemming from Rogosa’s paper is that, if two
constructs are characterized by different degrees of stability, the comparison of
cross-lagged correlations may lead to spurious conclusions regarding the causal
mechanism. Since then, most researchers interested in causality in panel data have
abandoned cross-lagged correlations and endorsed what we will referred to in this
paper as the cross-lagged panel model (CLPM) instead. In the CLPM stability of
the constructs is controlled for through the inclusion of autoregressive relationships,
and it is therefore often believed that the cross-lagged regression parameters
obtained with this model are the most appropriate measures for studying causality
in longitudinal correlational data (e.g., Deary, Allerhand, & Der, 2009; Soenens,
Luyckx, Vansteekiste, Duriez, & Goossens, 2008; Wood, Maltby, Gillett, Linley, &
Joseph, 2008). Specifically, it is common practice to standardize the cross-lagged
regression coefficients and compare their relative strength to determine which
variable has a stronger causal influence on the other (Bentler & Speckart, 1981).
The current paper forms a sequel to the warning given by Rogosa (1980), in
that it will be argued that not only should we account for stability, but we also need
to account for the right kind of stability. It will be shown that if stability of the
constructs is to some extent of a trait-like, time-invariant nature, the inclusion of
autoregressive parameters will fail to adequately control for this. As a result the
Cross-lagged panel model 4
estimates of the cross-lagged regression coefficients will be biased, which may lead to
erroneous conclusions regarding the underlying causal pattern. This message is not
novel in itself: In fact, it has been recognized repeatedly that the “omitted variable
problem” may affect the estimation of the cross-lagged coefficients (e.g., Dwyer,
1983; Finkel, 1995; Heise, 1970), and diverse modeling strategies have been
proposed to account for unobserved variables that influence the observed variables.
However, given the popularity of the CLPM, it seems that either this warning has
been lost on a large group of substantive researchers, or many researchers are simply
not convinced that this could form a serious problem.
In the current paper, we therefore present a closely related alternative
structural equation modeling (SEM) approach that is inspired by considering
cross-lagged panel data from a multilevel perspective, implying we need to
distinguish between the within-person and the between person level. We show that
this alternative SEM approach can lead to very different conclusions than the
traditional CLPM when considering the three major objectives of cross-lagged panel
research, that is: a) whether or not variables influence each other; b) which of the
variables is causally dominant; and c) what the sign of influence is. In doing so we
hope to raise awareness about the limitations of the traditional CLPM, and to
stimulate researchers to consider alternative SEM approaches.
This paper is organized as follows. In the first section, two models for
investigating cross-lagged effects are presented: the traditional CLPM and an
extension of this model based on taking a multilevel perspective. We discuss the
meaning of each model, the way they predict change, and the minimum number of
waves needed for identification. In the second section, we discuss four other SEM
Cross-lagged panel model 5
approaches that include cross-lagged relationships and discuss how these are related
to the model we propose. In doing so, we sketch the broader context of the current
account and point the reader in the direction of other alternatives. The third
section consists of a more in-depth comparison of the traditional CLPM and the
proposed alternative. In the fourth section, a modeling strategy is proposed to
ensure that – if present – both forms of stability are accounted for and we illustrate
this using an empirical data set. The paper ends with summarizing the most
important findings of the present study, discussing the implications for longitudinal
research, and providing guidelines for future cross-lagged panel research.
Two models for studying reciprocal influences
Cross-lagged panel research is concerned with the effect of two or more
variables on each other over time. To give an impression of the kinds of questions
researchers have tried to tackle using the CLPM, consider the following anthology:
Do maternal warmth and praise reduce internalizing and externalizing problems in
children with autism (Smith, Greenberg, Mailick Seltzer, & Hong, 2008)? Is the
relationship between parenting and adolescent delinquency bidirectional
(Gault-Sherman, 2012)? Does gratitude foster social support or vice versa (Wood et
al., 2008)? What is the direction of causality between intelligence and academic
achievement (Watkins, Lei, & Canivez, 2007)? Is processing speed a foundation for
successful cognitive aging (Deary et al., 2009)? What is the role of a pessimistic
explanatory style on developing and maintaining social support networks in
adolescents (Ciarrochi & Heaven, 2008)? What is the directional nature of the
relationship between the quality of the parent-child relationship and a child’s
Cross-lagged panel model 6
ADHD symptoms (Lifford, Harold, & Thapar, 2008)? And – at a macro
social-economic level – what is the direction of causality between intelligence and
economic welfare of nations (Rindermann, 2008)?
In this section the traditional CLPM is presented, which is the most typical
modeling approach for this kind of research. In addition, an alternative model is
presented, which we refer to as the random intercepts cross-lagged panel model
(RI-CLPM), that accounts for trait-like, time-invariant stability through the
inclusion of a random intercept (i.e., a factor with all loadings constrained to 1).
This random intercept partials out between-person variance such that the lagged
relationships in the RI-CLPM actually pertain to within-person (or within-dyad)
dynamics. We discuss how these models predict change, how many measurement
waves are needed for identification, and how they are related to each other.
The CLPM
The CLPM can be used if two or more variables have been measured at two or
more occasions, and if the interest is in their influences on each other over time. Let
xand ydenote two distinct variables which were measured multiple times, and
which will be analyzed with the CLPM. While this approach typically consists of
modeling the covariance structure only, the means are included here as well; note
however that no constraints are imposed on them, which is equivalent to analyzing
the centered data.
A graphical representation of this model is given in the left panel of Figure 1
(see Appendix 1 for the corresponding SEM specification). The measurement
Cross-lagged panel model 7
equations can then be expressed as
xit =μt+ξit (1a)
yit =πt+ηit (1b)
where ξit and ηit represent the individual’s temporal deviations from the temporal
group means μtand πtrespectively. These temporal deviations are modeled with
the structural equations
ξit =αtξi,t1+βtηi,t1+uit (1c)
ηit =δtηi,t1+γtξi,t1+vit.(1d)
The autoregressive parameter αtand δtare included to account for the stability of
the constructs: The closer these autoregressive parameters are to one, the more
stable the rank order of individuals is from one occasion to the next. However, even
when the stability coefficients are very high, when enough time passes, the original
rank order will be lost. Hence, it is not stability of a trait-like nature, and it is
therefore often referred to as temporal stability instead (e.g., Heise, 1970).
Insert Figure 1 about here
The cross-lagged parameters βtand γtform the key to investigating reciprocal
causal effects in this model (Rogosa, 1980): Through standardizing these
parameters, a comparison of the relative effects of xand yon each other can be
made, which can then be used to determine causal predominance (Bentler &
Speckart, 1981). These parameters are often interpreted in terms of predicting
Cross-lagged panel model 8
change (e.g., Finkel, 1995; Ribeiro et al., 2011; Rindermann, 2008). To show the
reasoning behind this interpretation, we write
yit yi,t1=πt+ηitπt1+ηi,t1
=πtπt1+δt1ηi,t1+γtξi,t1+vit,(2)
which shows that the cross-lagged parameter γtindicates the extent to which the
change in ycan be predicted from the individual’s prior deviation from the group
mean on x(i.e., ξi,t1=xi,t1μt1), while controlling for the structural change in
y(i.e., πtπt1), and one’s prior deviation from the group mean on y(i.e.,
ηi,t1=yi,t1πt1).
The CLPM is just identified with only two waves of data, which makes it an
appealing modeling approach from a practical point of view: In fact, we found that
45% of the datasets published in 2012, which were used to estimate this model,
consisted of only two waves of data. In Figure 2 the distribution of all 117 datasets
from 2012 is given.2This is noteworthy, because it implies that in almost half of the
applications, the parameters of the CLPM and their standard errors can be
estimated, but it is not possible to evaluate whether the model provides a proper
description of the actual underlying mechanism (as the model is just identified and
will yield a perfect fit, which is really not meaningful).
Insert Figure 2 about here
Cross-lagged panel model 9
The RI-CLPM
As described above, the CLPM only accounts for temporal stability through
the inclusion of autoregressive parameters. This implies that in this model it is
implicitly assumed that every person varies over time around the same means μt
and πt, and that there are no trait-like individual differences that endure. At closer
consideration, this is a rather problematic assumption, as it is difficult to imagine a
psychological construct – whether behavioral, cognitive, emotional or
psychophysiological – that is not to some extent characterized by stable individual
differences (if not for the entire lifespan, then at least for the duration of the study).
Longitudinal data can actually be thought of as multilevel data, in which
occasions are nested within individuals (or other systems, like dyads). When
considering this perspective, it becomes clear that we need to separate the
within-person level from the between-person level. This idea motivated the
development of the alternative model we present here, which can be thought of as
an extension of the CLPM that accounts not only for temporal stability, but also for
time-invariant, trait-like stability through the inclusion of a random intercept. This
alternative model can be expressed as
xit =μt+κi+ξ
it (3a)
yit =πt+ωi+η
it (3b)
where μtand πtare again the temporal group means. The additional terms κiand
ωiare the individual’s trait-like deviations from these means: They can be thought
of as latent variables or factors whose factor loadings are all constrained to 1, as in
case of random intercepts in latent growth curve (LGC) modeling (with the
Cross-lagged panel model 10
difference that here the group means are allowed to vary freely over time). We have
added an asterisk to the temporal deviation terms ξ
it and η
it, to emphasize these
terms are different from the individual deviation terms in the traditional CLPM: In
the current model they represent the individual’s temporal deviations from their
expected scores (i.e., μt+κiand πt+ωi), rather than from the group means (i.e., μt
and πt).
Subsequently these deviations are model as
ξ
it =α
tξ
i,t1+β
tη
i,t1+u
it (3c)
η
it =δ
tη
i,t1+γ
tξ
i,t1+v
it,(3d)
where the autoregressive and cross-lagged regression parameters differ from the ones
in the CLPM, as indicated by the asterisks. That is, the autoregressive parameters
α
tand δ
tdo not represent the stability of the rank order of individuals from one
occasion to the next, but rather the amount of within-person carry-over effect (cf.,
Hamaker, 2012; Kuppens, Allen, & Sheeber, 2010; Suls, Green, & Hillis, 1998): If it
is positive, it implies that occasions on which a person scored above his/her
expected score are likely to be followed by occasions on which he/she still scores
above the expected score again, and vice versa.3
The main interest here is however in the cross-lagged parameters β
tand γ
t,
which indicate the extent to which the two variables influence each other.
Specifically, γ
tindicates the degree by which deviations from an individual’s
expected score on y(i.e., η
it =yit −{πt+ωi}) can be predicted from preceding
deviations from one’s expected score on x(i.e., ξ
i,t1=xi,t1−{μt+κi}), while
controlling for the individual’s deviation of the preceding expected score on y(i.e.,
η
i,t1=yi,t1−{πt1+ωi}). The cross-lagged relationships pertain to a process
Cross-lagged panel model 11
that takes place at the within-person level and they are therefore of key interest
when the interest is in reciprocal influences over time within individuals or dyads. A
graphical representation of this model is given in the right panel of Figure 1 (see
Appendix 1 for the corresponding SEM specification).
Expressing change in the RI-CLPM, we can write
yit yi,t1=πt+ωi+η
itπt1+ωi+η
i,t1
=πtπt1+δ
t1η
i,t1+γ
tξ
i,t1+v
it,(4)
which shows that the cross-lagged parameter indicates the extent to which the
change in ycan be predicted from the individual’s prior deviation from his/her
expected score on the other variable (i.e., ξ
i,t1=xi,t1−{μt+κi}), while
controlling for the structural change in y(i.e., πtπt1)and the prior deviation
from one’s expected score on y(i.e., η
i,t1=yi,t1−{πt1+ωi}).
The expressions in Equations 2 and 4 are similar, but unless κiand ωiare
zero, the CLPM predicts change from other aspects than the RI-CLPM. In fact, it is
easy to see that the traditional CLPM is nested under the current model, as it can
be obtained from the latter by fixing the variances and covariance of κiand ωito
zero. To compare the two models statistically, a chi-square bar test should be used,
as it requires two parameters to be fixed at the boundaries of the parameter space
(see for details: Stoel, Galindo Garre, Dolan, & Wittenboer, 2006).
While the CLPM requires only two waves of data, the RI-CLPM requires at
least three waves of data, in which case there is 1 degree of freedom (df).4If the
intervals are of the same size, and if we assume that the effects the variables have on
each other remain stable over time, we could decide to constrain the lagged
parameters over time, giving us an additional 4 df (i.e., 5 df in total). Furthermore,
Cross-lagged panel model 12
we could investigate whether the means can be constrained over time, such that we
obtain another 4 df (resulting in 9 df in total). If on the other hand, we are not
willing to make these assumptions, and we are not sure whether the effect of the
time-invariant stability components κiand ωiare equal over time, we may wish to
remove the constraint on the factor loadings. This relaxation may especially be of
interest when the observations are made further apart in time, and we expect that
we are also measuring some structural changes. However, this would imply that κi
and ωino longer represents random intercepts (as in multilevel modeling), but
rather represent latent variables or traits (as common in SEM). Even more so, it
would imply we need more waves of data to estimate this model.
Conclusion
The CLPM is nested under the RI-CLPM. The latter is an attempt to
disentangle the within-person process from stable between-person differences while
the former does not differentiate between these two levels that are likely to be
present in the data. The question thus rises what happens if the data were
generated by the RI-CLPM, but are analyzed using the CLPM: Most likely this will
lead to a contamination of the estimated within-person reciprocal effects, but to
obtain more insight into this matter, we need to take a closer look at the
relationship between the cross-lagged parameters from both models.
However, before doing this, we consider how the RI-CLPM is connected to
other longitudinal SEM approaches that include cross-lagged relationships: In doing
so we aim to present a broader context for the current exposition and provide some
reference points for readers already familiar with (some of ) these SEM approaches.
Cross-lagged panel model 13
Relatedness to other existing SEM approaches
There are several other longitudinal SEM approaches that can be used for
bivariate data and which include cross-lagged relationships. Here we consider four of
these, that is: a) the Stable Trait Autoregressive Trait and State (STARTS) model
(Kenny & Zautra, 2001; Kenny & Zautra, 1995); b) the Autoregressive Latent
Trajectory (ALT) model (Bollen & Curran, 2006; Curran & Bollen, 2001); c) the
Latent Change Score (LCS) model (Hamagami & McArdle, 2001; McArdle &
Hamagami, 2001); and d) a modification of the Latent State-Trait (LST) model
(Schmitt & Steyer, 1993; Steyer, Schwenkmezger, & Auer, 1990). In this section we
discuss the relatedness between the RI-CLPM and these four alternatives, focussing
on the substantive and methodological similarities and differences. Note that this
section is decidedly not meant as an in depth evaluation of these diverse alternatives:
The interested reader is referred to the included citations for further details.
STARTS model by Kenny and Zautra
The STARTS model by Kenny and Zautra (2001), is also known as the Trait
State Error (TSE) model (Kenny & Zautra, 1995). It allows the user to decompose
observed variance into three components: a) the stable trait, which does not change;
b) the autoregressive trait, which changes according to an autoregressive process;
and c) the state or error, which is unique to the occasion. Originally, Kenny and
Zautra (1995) included constraints over time in their model, such that the relative
contributions of these three components remains stable over time, but these
constraints may be relaxed if enough measurement waves are available (cf. Lucas &
Donnellan, 2007).
Cross-lagged panel model 14
Most applications of this model are based on univariate repeated
measurements, but Kenny and Zautra (1995) also present a bivariate extension of
their model. The RI-CLPM proposed in this paper differs from the bivariate
STARTS model in that it does not include measurement error: The RI-CLPM can
thus be thought of as a special case of the STARTS model (without the constraints
on the lagged relationships over time), in which the observations are modeled
without measurement error.
Clearly, the inclusion of measurement error in itself is recommendable, as we
know that measurement error is likely to be present in psychological measurements.
However, Kenny and Zautra (2001) indicate that the model is often difficult to
estimate, and that it may require 10 or more waves of data. Cole, Martin, and
Steiger (2005) performed a simulation study and concluded that the (univariate)
STARTS model frequently led to improper solutions that were difficult to interpret
(i.e., negative variance estimates, or problems with convergence in the form of
singularity of the approximate Hessian matrix). They also discuss some of the
reasons for this: For instance, when the autoregressive parameter is very close to
zero, it becomes difficult to distinguish between variance that is due to measurement
error, and variance that is the stochastic input of the autoregressive process. Thus,
while extending the model with measurement error may be preferable from a
theoretical point of view, the practical consequences (i.e., having to have many more
measurement waves), make it a less attractive alternative for the traditional CLPM.
ALT model by Curran and Bollen
The ALT model was developed by Curran and Bollen (2001; see also Bollen &
Curran, 2006), to “combine the best of two worlds”: It allows people to be
Cross-lagged panel model 15
characterized by their own trajectory over time (as in the LGC model), while their
observations may also exhibit some carry-over effect from one occasion to the next
(as in the autoregressive or simplex model). In the bivariate extension of the ALT
model presented by Curran and Bollen (2001), the random effects that describe the
individual trajectories may be correlated to each other across the variables (as is the
case in a bivariate LGC model), and there may also be cross-lagged influences
between the observations (as in the CLPM).
While this hybrid model seems to have a lot of potential, applying and
interpreting the ALT model is not as straight forward as one may be inclined to
think at first: Because the lagged relationships are included in this model between
the observations, there is a recursiveness in the model, which has some adverse
effects. First, it implies the process needs to be “started up”, for which Curran and
Bollen (2001) propose two solutions: Either the first observation is treated as
exogenous, or nonlinear constraints are imposed on the loadings for the first
occasion. While treating the first occasion as exogenous is relatively easy, Jongerling
and Hamaker (2011) show that this may lead to rather unexpected growth curves:
For instance, in an ALT model with a random constant only (i.e., no linear trend
parameter), one may actually be modeling an increasing or decreasing trend over
time. Such undesirable effects are not encountered when using the nonlinear
constraints to start up the process, but these require the assumption that the lagged
effects are constant over time,5and are more difficult to impose, especially in the
bivariate case.
Second, the recursiveness in the ALT model implies that the random constant
and the random change parameter no longer have the original role of individuals’
Cross-lagged panel model 16
intercepts and slopes (cf. Hamaker, 2005). For instance, the random constant not
only affects an observation directly, but also indirectly through (all) previous
occasions. Hamaker (2005) has shown that under the assumption that the lagged
effects are invariant over time, the ALT model can be rewritten as a LGC model
with autoregressive residuals, with the advantage that the random parameters in
this reparametrization serve as the random intercept and slope that describe the
underlying individuals’ deterministic trends. This result has also been extended to
multivariate processes, meaning that the bivariate ALT models can be rewritten as
a bivariate LGC model with residuals that are characterized by autoregressive and
cross-lagged regressive relationships (cf. Hamaker, 2005).
Considering this latter parametrization, the RI-CLPM is related to a bivariate
ALT model with only random intercepts and no random slopes. However, in the
RI-CLPM we do not constrain the mean structure, meaning that there may be
changes–possibly, but not necessarily linear–over time, which are identical for all
individuals. If the group means can be constrained to be equal over time, the
RI-CLPM is nested under the ALT model with only a random intercept and no
slope (using the parametrization proposed by Hamaker, 2005, to avoid the
recursiveness in the model).
LCS model by McArdle and Hamagami
The LCS model, also known as the Latent Difference Score (LDS) model, was
proposed by McArdle and Hamagami (2001; Hamagami & McArdle, 2001), and
forms a rather general modeling framework that includes many longitudinal SEM
approaches as special cases. What is characteristic of the LCS model is that latent
changes (i.e., the differences scores corrected for measurement error), from one
Cross-lagged panel model 17
occasion to the next are modeled as a function of a constant change parameter and a
proportional change parameter that depends on the preceding score: For this reason
the model is also referred to as the Dual Change Score model (McArdle, 2009).
In the bivariate extension of this model, change is not only a function of a
constant change parameter and the proportional change parameter, but also of the
preceding score on the other variable. The cross-lagged paths, going from one
variable to the change in the other, are referred to as coupling parameters,rather
than cross-lagged regression parameters. The interpretation is the same however, in
that significant coupling parameters imply that one variable has the tendency to
impact changes in the other variable (McArdle & Grimm, 2010). But instead of
comparing standardized coefficients in order to determine which variable is causally
dominant, the coupling parameters are used to set up a vector field which depicts
the expected changes from one occasion to the next on both variables as a function
of the current state (see Boker & McArdle, 1995; McArdle, 2009; McArdle &
Grimm, 2010). This plot is then used to make statements like: “The resulting flow
shows a dynamic process, where scores on Non-Verbal abilities have a tendency to
impact score changes on the Verbal scores, but there is no notable reverse effect.”
(p. 348, McArdle, 2005).
The LCS model has been extended with what has been referred to as “dynamic
error”, to distinguish it from measurement error (see for instance McArdle, 2001):
While measurement error only affects the observation at the current occasion,
dynamic error feeds forward through the lagged relationships, affecting the
trajectory of the system and making it a stochastic rather than deterministic
process. The RI-CLPM can be thought of as closely related to the LCS model with
Cross-lagged panel model 18
dynamic error, but without measurement error or a constant change parameter.
However, the LCS model is characterized by a similar recursiveness as is present in
the ALT model, and therefore the random intercept term, which directly affects the
first latent score, also influences future occasions indirectly. Because the process is
not “started up” as is done in ALT modeling, the recursiveness is not dealt with in
such a way that we can ensure the process is stable in the absence of a constant
change parameter. As a result, the RI-CLPM is not a special case of the LCS
model, although they may be closely related in certain situations.
The LST model by Steyer and colleagues
The LST model was originally developed to distinguish between measurement
error and the true score (i.e., a latent variable), and to further decompose the true
score into a trait-like and a state-like part (Schmitt & Steyer, 1993; Steyer et al.,
1990). In practice this typically implies that it is assumed that there is an
underlying construct, which is measured by multiple indicators. This underlying
construct at a particular occasion is referred to as the state, which is then
decomposed into a trait-like part and an occasion-specific part: The trait-like part is
included as a second-order factor, relating the states, which are represented by the
first-order factors, to each other. The occasion-specific part is the residual part of
the state factor, which was not accounted for by the trait.
The LST model has been extended with autoregressive relationships either
between the state factors (introducing a similar recursiveness as exists in the ALT
model and the LCS model), or between the occasion-specific components (to avoid
the detrimental recursiveness in the model): The latter has been coined the Trait
State Occasion (TSO) model (Cole et al., 2005). Recently, the TSO has been
Cross-lagged panel model 19
modified by Luhmann, Schimmack, and Eid (2011) to handle single indicator data.
In this modified model, the measurement error term is omitted, the trait factor is
modeled as a separate factor with free factor loadings over time (rather than a
second-order factor), and second-order autoregressive relationships are included.
Note that if the measurement error term had been kept (and the second-order
autoregressive relationships were omitted), the model would be identical to the
STARTS model.
Luhmann et al. (2011) also propose a bivariate version of the model, which
includes cross-lagged regression paths between the occasion-specific components
(and no second-order autoregressive relationships). The RI-CLPM can be seen as a
special case of this bivariate single indicator LST model, in which the factor
loadings for the traits are constrained to 1 over time. In applying this model to
empirical data, Luhmann et al. focus on decomposing the variance into separate
parts, as is also the goal in applying the STARTS model and the original LST
model. Furthermore, they decompose the covariance between the two variables into
a part accounted for by the traits, a part accounted for by the autoregressive and
cross-lagged regressive relationships, and a part due to the relationship between the
residuals of the occasion-specific factors.
Conclusion
Clearly, the models discussed above show some overlap with each other and
with the RI-CLPM presented in the current paper. When considering these diverse
modeling strategies, two observations seem of key importance. First, if researchers
are specifically interested in decomposing the variance into trait-like and state-like
components and the means are not of interest, the STARTS model and the models
Cross-lagged panel model 20
based on the LST model are most relevant. In contrast, if the interest is in
individual developmental trajectories, the ALT model and the LCS model are more
appropriate, as they are based on modeling both the mean structure and the
covariance structure and allow for individuals to have their own growth curves.
Second, the STARTS model, the ALT model and the LST model are most typically
applied to univariate data (even though the original LST model uses multiple
indicators); while bivariate (or multivariate) extensions are possible, they do not
form the core focus and the cross-lagged regression parameters are not the key
interest. In contrast, the LCS model is most typically used to investigate how two
variables influence each other (based on the expected change described with the
vector field), although it can also be applied to univariate data.
The above observations are relevant, because they help pitting the RI-CLPM
against these alternatives. The main inspiration for proposing the RI-CLPM is that
we want to obtain estimates of cross-lagged regression parameters that truly reflect
the underlying reciprocal process that takes place at the within-person level. The
model thus requires bivariate (or multivariate) data, the mean structure is not
(necessarily) of interest, and the focus is on how (i.e., positive or negative
cross-lagged coefficients), and how much (i.e., compare standardized absolute values
of cross-lagged coefficients) the variables influence each other. Hence, because the
focus is on the covariance structure rather than the mean and covariance structures,
we could say that the RI-CLPM is more closely related to the STARTS model and
the LST and TSO models. However, the goal is not to decompose the variance and
covariance into trait-like and state-like parts, but to determine how the variables
influence each other through the cross-lagged relationships at the within-person,
Cross-lagged panel model 21
state-like level, while controlling for trait-like differences at the between-person
level. With this goal in mind, the RI-CLPM can be thought of as more closely
related to the bivariate ALT model or the LCS model, although there is no inherent
interest in individual developmental trajectories.
In sum, it can be stated that all models discussed in this section could serve as
alternatives to the CLPM: Each model forms an attempt to separate between-person
trait-like differences from the within-person reciprocal process. While some of these
models include desirable properties such as measurement error and/or differences in
developmental trajectories, the advantage of the RI-CLPM is that it is most closely
related to the CLPM and requires only three waves of data. Since two or three
waves of data are currently the norm in cross-lagged panel research, the RI-CLPM
is more likely to be considered by researchers as a feasible alternative than models
that require (many) more waves. In the following sections we focus on the CLPM
and the RI-CLPM, but we return to the issue of other alternatives in the discussion.
Comparing the cross-lagged parameters
Cross-lagged panel research is characterized by three major objectives: first,
the aim is to determine whether the variables have a significant effect on each other;
second, the question is which variable is causally dominant; and third, researchers
want to know whether a variable has a positive or negative influence on the other
variable. If researchers use the CLPM when the data were actually generated by the
RI-CLPM, the question is whether this alters their conclusions with respect to these
three objectives. In this section we focus on these issues through considering the
cross-lagged regression parameters from both models analytically and in simulations.
Cross-lagged panel model 22
Analytical comparison
In Appendix 2 we show that the standardized cross-lagged regression
parameter in the CLPM from variable xto variable ycan be expressed as a function
of the parameters of the RI-CLPM, that is
γt
SD(xi,t1)
SD(yit)=1cov(ωi
i)+cov(η
i,t1
i,t1)21
×cov(ωi
i)+δ
tcov(η
i,t1
i,t1)+γ
tvar(ξ
i,t1)
cov(ωi
i)+cov(η
i,t1
i,t1)
×var(ωi)+δ
tvar(η
i,t1)+γ
tcov(η
i,t1
i,t1),(5)
which shows that it is a complex function of: a) the cross-lagged regression
coefficient from variable xto variable y,thatisγ
t; b) the within-person
autoregressive parameter of variable y,thatisδ
t; c) the covariance between the
within-person deviations at the previous time point, that is cov(η
i,t1
i,t1); d) the
variance of the within-person deviation at the preceding occasion, that is var(η
i,t1);
e) the variance of the trait-like component, that is var(ωi); and f ) the covariance
between the trait-like components, that is cov(ωi
i).
Considering the first objective of cross-lagged panel research, that is, is there a
significant effect of one variable on the other, the relationship in Equation 5 is not
very informative, although it may be expected that the two models will not
necessarily lead to same conclusion regarding the presence of a cross-lagged
relationship.
With respect to the second objective, the question is whether the difference in
absolute values of the standard cross-lagged coefficients is of the same sign across
Cross-lagged panel model 23
the two models. That is, the question is whether
|γSD(xi,t1)
SD(yit)|−|βSD(yi,t1)
SD(xit)|and |γSD(ξ
i,t1)
SD(η
it)|−|βSD(η
i,t1)
SD(ξ
it)|.
are either both positive, leading to the conclusion that xis causally dominant, or
both negative, leading to the conclusion that yis causally dominant. If these
differences are not of the same sign, this implies that using one model leads to the
conclusion that xis causally dominant, while the other model leads to the
conclusion that yis causally dominant. Clearly, that is not a desirable situation.
For instance, when investigating the reciprocal influences of mothers’ harshness and
children’s behavioral problems, the RI-CLPM may indicated that the mothers are
causally dominant and form the driving force in this potentially negative spiral,
while the CLPM may point to the children as being the instigator of maladaptive
patterns. However, due to the rather complex relationships between the models’
differences of absolute standardized cross-lagged parameters, it is difficult to
evaluate when these models will lead to conflicting conclusions, although in general
we may expect that larger trait-like differences are likely to have a stronger effect
than in case of small between-person differences.
The third objective concerns the sign of the cross-lagged parameters. Thus the
question is: If γ>0, will γ>0, and when γ<0, will γ<0? Naturally, the same
question applies to βand β. Although this is not immediately apparent from the
expression in Equation 5, the many unrelated terms from the two levels strongly
suggest that γand γnot necessarily have the same sign. This is again quite
disturbing, as it suggests that using the CLPM may lead to the conclusion that
mothers’ harshness has a damping effect on children’s behavioral problems, while
the RI-CLPM may indicate that mothers’ harshness actually exacerbates the
Cross-lagged panel model 24
children’s behavioral problems.
Simulations
In order to further investigate the effect of using the CLPM instead of the
RI-CLPM with respect to the three objectives of cross-lagged panel research
identified above, we performed a series of simulations based on four models. We
emphasize that the models used here were handpicked, to illustrate several specific
situations that can arise, and we do not claim that these are necessarily reflecting
realistic scenarios. Specifically, we used Mplus (Muth´en & Muth´en, 1998-2012), to
simulate two-wave bivariate data according to a RI-CLPM, which were subsequently
used to estimate the traditional CLPM. For each model, 1000 replications were
generated, of N= 200 each. Saving the parameter estimates in a separate file,
which we then imported into R (R Core Team, 2012), we computed the
standardized cross-lagged parameters (as Mplus does not allow for the computation
of standardized parameters in case of Monte Carlo simulations).
In the first model, we had autoregressive parameters of .5 and no cross-lagged
regression coefficients. The within-person variances of both variables was set to 1,
and the covariance between the two variables was .4. Since we made sure the
process was stationary (meaning the variances and covariances are stable over time;
cf. Hamilton, 1994), this implies that the residual variances at the second wave were
.75 and the residual covariance was .3. The between-person variances were set to 3
for each variable, and the covariance at this level was set to -2. Hence, this
represents a process which is characterized by a negative correlation at the
between-person level, while there is a positive correlation at the within-person
level,6which can be seen as an instance of Simpson’s paradox (cf., Kievit,
Cross-lagged panel model 25
Frankenhuis, Waldorp, & Borsboom, 2013). In the upper-left panel of Figure 3, the
standardized cross-lagged parameter estimates of this model are plotted. It clearly
shows that the point estimates are far from the generating values (indicated by the
diamond). The average βestimate was -.118 (SD=.036, average SE=.036), and the
average γestimate was -.120, (SD=.037, average SE=.036). Considering whether
the 95% confidence intervals of these parameter estimates contained zero, we
obtained coverage rates of .105 for the βparameter, .103 for the γparameter, which
implies that in about 90% of the cases, the CLPM would lead to the conclusion that
there is at least one significant negative cross-lagged parameter, although no
cross-lagged relationships were present in the model that generated the data.
Insert Figure 3 about here
The second model is based on autoregressive parameters of .5 and cross-lagged
regression parameters of .3. The within-person variances were set to 1, and the
within-person covariance to .5, which through the additional constraint of
stationarity implies that the innovations variances were .51 and the covariance
between the innovations was .03. The between-person variances were set to 2, and
the between-person covariance was set to -1. In the upper-right panel of Figure 3
the standardized cross-lagged parameter estimates are plotted. Based on 1000
replications, the average βestimate was -.003 (SD=.032, average SE=.034), and the
average γestimate was -.003 (SD=.034, average SE=.034). Coverage rates for the
95% confidence interval containing zero were .951 and .945 respectively, indicating
that in about 95% of the cases it would be concluded that these parameters are
Cross-lagged panel model 26
non-significantly different from zero, although there were substantial cross-lagged
relationships in the model that generated the data.
The third model is based on autoregressive parameters of .5, and a
cross-lagged parameter βof -.3 (from variable yto variable x), and a cross-lagged
parameter γof .1 (from xto y). The within-person variances were both set to 1,
and the covariance to -.5, such that the innovation variances were .51 for the
x-variable and .79 for the y-variable, while their covariance was -.29 (to ensure
stationarity). The between-person variances were set to 2 and their covariance was
set to 1. The standardized cross-lagged parameter estimates are given in the
lower-left panel of Figure 3. It shows that while the original combination of
parameter values is in the area that is characterized by a standardized |γ|that is
smaller than the standardized |β|, indicating that variable yis causally dominant,
most point estimates fall in the area in which the standardized |γ|is actually larger
than the standardized |β|, leading to the opposite conclusion that variable xis
causally dominant. The average estimate for βis .002 (SD=.039, average SE=.040),
and for γit is .151 (SD=.034, average SE=.033). For β(which equalled -.3 in the
generating model), the coverage rate of the 95% confidence interval containing zero
was .958, which implies that in about 95% of the cases the conclusion would be that
there is a nonsignficant relationship from yto x. The coverage rate γ(where true γ
is .1) was .010, which implies that in 90% of the cases a significant relationship from
variable xto ywould be detected. This further shows that the CLPM may result in
the wrong variable being identified as being causally dominant.
Finally, in the fourth model the autoregressive parameters were set to .5, the β
parameter (from variable ytox)to.3andaγparameter of .1. The within-person
Cross-lagged panel model 27
variances are both 1, the covariance is .5, which in combination with the restriction
of stationarity leads to the innovation variances of variables xand ybeing .72 and
.60 respectively, and a covariance between the innovations of -.056. The
between-person variances ware set to 3 and their covariance to -2. The standardized
point estimates of the cross-lagged parameters are presented in the lower-right panel
of Figure 3, showing that, while the generating cross-lagged parameters implied that
variable ywas causally dominant, the parameter estimates almost always lead to
the conclusion that variable xis causally dominant. The average point estimate for
βwas -.023 (SD=.037, average SE=.036), and for γit was -.093 (SD=.033, average
SE=.033). The 95% confidence intervals included zero with a rate of .897 for β,and
.192 for γ, meaning that in almost 90% of the cases we would fail to detect the
relationship from variable yto x(which in reality was .3), while in more than 80%
of the cases we would detect a significant negative relationship from variable xto y
(which in reality was .1). This illustrates another disturbing fact: The CLPM may
result in a significant estimate of a cross-lagged parameter that actually has a
different sign than the corresponding cross-lagged parameter in the generating
model.
Conclusion
While the algebraic relationship in Equation 5 shows that the cross-lagged
parameters from the two models are not necessarily identical, it is not easy to see
how they will differ, especially in the light of the three objectives of cross-lagged
panel research. The simulations we presented here show however that the CLPM
can lead to spurious results regarding all three objectives in this line of research,
that is, it can be misleading with respect to: a) the presence of causal relationships
Cross-lagged panel model 28
(models 1 and 2); b) the causal priority of two variables (models 3 and 4); and c)
the sign of the causal relationship (model 4).
The simulations here were designed to illustrate these specific situations,
without the intention to represent typical psychological processes. The fact is that
we do not know what would be typical values for the parameter of the RI-CLPM,
because this is not a model that is currently used in practice. In the simulations here
the between-person variance was relatively large (i.e., two or three times as large as
the within-person variance), and in general it can be stated that the results from
the CLPM deviated more from the generating RI-CLPM when the between-person
variances increased. Furthermore, the correlation at the between-person level also
influences the results, especially if it is of the opposite sign of the correlation that
exists at the within-person level (i.e., in the presence of Simpson’s paradox).
Finally, sample size affects the variability in estimates and their standard errors
(i.e., both are inversely related to sample size), but the bias resulting from
estimating a model that does not distinguish between within-person dynamics and
between-person trait-like differences does not vanish when sample size increases.
Modeling strategy
To avoid the pitfall exposed above, we propose a modeling strategy that allows
us to investigate whether there are trait-like, time-invariant individual differences
present in the constructs that are studied, which should be accounted for through
the inclusion of a random intercept. This strategy is based on the fact that the
CLPM is nested under the RI-CLPM, such that if three or more waves of data are
available, both models can be fitted to the data and can be compared using a
Cross-lagged panel model 29
chi-bar-square test for the difference in chi-squares (Stoel et al., 2006). We illustrate
this strategy using data that are reported in Soenens et al. (2008), concerning the
effect of diverse aspects of parenting style on depressive symptoms of adolescents
and vice versa. The data were obtained from 396 students and consist of three
waves, with intervals of one year, starting in the fall of the first year in college.
We begin with considering the relationship between Parental Psychological
Control (based on items like “My parents are less friendly to me if I don’t see things
like they do”), and Adolescents’ Depressive Symptomatology.First,wefitamodelin
which the means of each variable are constrained over time (i.e., μt=μand πt=π),
while the covariance structure is unconstrained: Models in which the group means
do not change over time facilitate interpretation, although it time-invariant means
are no prerequisite for the models considered here. The fit of this model is not
satisfactory (chi-square is 13.75, 3 df, p=.008; RMSEA = .078; CFI = .990; SRMR
=.024), and inspection of the means shows that especially the mean of Adolescents’
Depressive Symptomatology at the first wave is higher than at the other two waves:
This measurement is from the first semester that the participants are in college, and
the elevated average may thus reflect the difficulties associated with getting
adjusted to these new circumstances. Freeing this mean leads to appropriate model
fit (χ2(3) = 3.33, p=.344; RMSEA= .017; CFI= 1.000; SRMR= .011). Although
constraining this first mean does not affect our results for the lagged parameters in
a substantive way, the results reported below are based on the model in which this
first mean for Adolescents’ Depressive Symptomatology is not constrained to be
equal to the means at subsequent waves.
Second, we model the covariance structure using the RI-CLPM, while keeping
Cross-lagged panel model 30
the constraints on the means (except for the first mean of Adolescents’ Depressive
Symptomatology ), and time-invariant lagged parameters. This model fits well
(chi-square is 9.85, 8 df, p=.276; RMSEA = .024; CFI = .998; SRMR = .025).
Finally, we fit the CLPM, with the same constraints on the means and lagged
parameters as used in the previous model. This model does not fit well (chi-square
is 66.18, 11 df, p<.001; RMSEA = .113; CFI = .943; SRMR = .042), although
some SEM users may claim on the basis of the CFI and the SRMR that the model
fits approximately. Note that since the null-model here consists of fixing two
parameters on the boundary of the parameter space (i.e., two variances fixed to
zero), the standard chi-square difference test will be too conservative (see Stoel et
al., 2006). The chi-square difference is 66.18 9.85 = 56.33, with 3 df, which is
significant at an αof .05 (that is, p<.01).
To show that the substantive interpretation of the underlying process depends
on the model one uses, we consider the standardized cross-lagged regression
parameter estimates from both models presented in Figure 4. It shows that both
models lead to significant positive cross-lagged parameters. However, while the
RI-CLPM indicates that the effect of Parental Psychological Control on Adolescents’
Depressive Symptomatology is only slightly larger than the reverse effect (i.e., .240
versus .212 and .265 versus .205 between wave 1 and wave 2), the CLPM leads to
the conclusion that the effect of parents on adolescents is much larger than that of
adolescents on their parents (i.e., .239 versus .139 and .248 versus .134 between
wave 1 and wave 2). Hence, using the CLPM would lead to the conclusion that
parents are causally dominant, while the RI-CLPM leads to the conclusion that the
reciprocal process is much more symmetric.
Cross-lagged panel model 31
Insert Figure 4 about here
We apply the same procedure for the variables Parental Responsiveness (based
on items like “My parents make me feel better after I discussed my worries with
them”), and Adolescents’ Depressive Symptomatology. Here, both the first mean of
the adolescents’ variable, and the last mean of the parents’ variable were estimated
freely, in order to obtain a fitting model (chi-square is .933, 2 df, p=.627; RMSEA
=.000; CFI = 1.000; SRMR = .006): The last mean of Parental Responsiveness was
significantly lower than that at the other two measurement waves, which may reflect
the increasing independence of the adolescents in the third year of college. The
RI-CLPM fitted well (chi-square is 11.86, 7 df, p=.105; RMSEA = .042; CFI
=.996; SRMR = .031), while the CLPM did not lead to a well fitting model
(chi-square is 76.01, 10 df, p<.001; RMSEA = .129; CFI = .939; SRMR = .048),
although again, some SEM users may claim this model is not too far off. The
chi-square difference is 76.01 11.86 = 64.15, with 3 df, which is significant at an α
of .05 (that is, p<.01).
Comparing the standardized lagged parameter estimates from both models
given in Figure 4, the RI-CLPM leads to the conclusion that there are no reciprocal
influences between Parental Responsiveness and Adolescents’ Depressive
Symptomatology, whereas the CLPM leads to the conclusion that there is a
significant negative effect from Parental Responsiveness to subsequent Adolescents’
Depressive Symptomatology (and while there is no significant effect from adolescents
to parents, it would be concluded that parents are causally dominant here).
Cross-lagged panel model 32
In conclusion, the modeling strategy illustrated above shows that it is possible
to investigate whether the constructs are characterized by time-invariant, trait-like
individual differences, and that using the traditional CLPM can lead to erroneous
conclusions regarding the pattern of mutual influences. Hence, researchers should
make sure to use an alternative that decomposes the variance into between-person
differences and the within-person process. If the constructs are not characterized by
time-invariant, trait-like individual differences, running the RI-CLPM will not affect
the results substantially, although in that case one can also use the simpler CLPM
instead.
Discussion
Rogosa summarized his critique on the cross-lagged correlation
methodology–which he referred to as CLC–saying: “CLC may indicate the absence
of direct causal influence when important causal influences, balanced or unbalanced,
are present. Also, CLC may indicate a causal predominance when no causal effects
are present. Moreover, CLC may indicate a causal predominance opposite to that of
the actual structure of the data; that is, CLC may indicate that Xcauses Ywhen
the reverse is true.” (p. 246, Rogosa, 1980). In the current paper, similar problems
have been exposed in the context of the CLPM. That is, the CLPM may indicate
there are reciprocal effects when these do not exist (model 1), and may fail to detect
them when they do exist (model 2). Furthermore, the CLPM may identify one
variable as being causally dominant, when in fact the other variable is (models 3
and 4). Finally, the CLPM may indicate a negative influence from one variable on
another, while in reality the effect is positive (model 4).
Cross-lagged panel model 33
The source of these problems is the failure to adequately separate the
within-person and the between-person level in the presence of time-invariant,
trait-like individual differences. As a result, the estimates of lagged parameters are
confounded by the relationship that exists at the between-person level (see
Hamaker, 2012 for other situations in which this confounding may occur). As it is
reasonable to assume that most psychological constructs that are studied with
cross-lagged panel designs are to some extent characterized by time-invariant
stability reflecting a trait-like property (at least for the duration of the study), it
follows that many lagged parameters reported in the literature will not reflect the
actual within-person (causal) mechanism.
This is especially problematic if one wishes to use the results from cross-lagged
panel research as a basis for future interventions. For instance, the results obtained
with the traditional CLPM for adolescent depression and parental responsiveness in
this paper, would lead the researcher to conclude that increasing parental
responsiveness should result in a reduction in depressive symptoms on part of the
adolescent; however, the RI-CLPM shows that this result is an artefact, and that
there is actually no lagged effect from parental responsiveness to adolescents’
depression. Note that this does not imply that the two variables are unrelated: In
fact, the trait-like individual differences are negatively correlated (estimated
correlation is -.443, SE= .067, p<.001), indicating that parents who tend to be
more responsive on average, tend to have adolescents who suffer less from depressive
symptomatology on average. However, we cannot derive a causal mechanism from
these results, which explains this relationship and that can be used as the
foundation for an intervention. This shows that “getting it right” with respect to
Cross-lagged panel model 34
the cross-lagged relationships is not just an academic concern.
We found that 45% of the studies that make use of the CLPM are based on
only two waves of data. In these cases, the CLPM is saturated, and hence no
statements regarding model fit can be made: That is, the model will always fit
perfectly, and the interest in estimating this model is simply in obtaining estimates
of the cross-lagged regression parameters which are corrected for the temporal
stability of the constructs. This implies that to date, it is impossible to tell what
portion of the results reported in the literature based on the CLPM provide truthful
reflections of the actual reciprocal mechanisms, and what portion is flawed and if so,
how serious these errors are.
Researchers interested in studying lagged relationships are therefore well
advised to employ the following approach. First, a minimum of three measurement
waves are required: Only then can the within-person process be controlled for stable
between-person differences through the inclusion of a random intercept. Second,
start with a model in which only the means are constrained over time, while the
covariance structure is estimated freely: This allows one to determine whether there
are structural changes over time. If this model proves tenable, subsequent models
can be specified for the covariance structure, while leaving the means constrained
over time. If the first model proves untenable however, the researcher should
identify the source of misfit, and consider freeing certain means (as we did in the
empirical applications included in this paper), or refrain to an alternative modeling
approach, such as LGC or ALT modeling (Hamaker, 2005). If there is no need to
refrain to an alternative model based on the mean structure, the researcher is
advised to continue with the extended models which were proposed in this paper,
Cross-lagged panel model 35
and compare the CLPM to the RI-CLPM in order to determine whether the
constructs are characterized by trait-like between-person differences, or that it can
be assumed that all individuals vary around the same mean or trend.
In addition, if researchers expect their measurements to contain measurement
error, they are advised to either obtain a large number of repeated measurements
(say >10) such that they can estimate a bivariate STARTS model, or to obtain
multiple indicators (e.g., test halves) such that they can fit a bivariate TSO model.
In both cases they will be able to distinguish between the within-person process and
stable trait-like between-person differences, while controlling for measurement error.
Finally, although we restricted our focus here on an alternative model in which
the effect of the between-person differences is constant over time (such that it can
be modeled as a random intercept and there is a clear connection with multilevel
modeling), we recognize that this will not always be a tenable assumption. That is,
when observations are taken further apart in time, the effect of stable individual
differences may vary over time, such that the constraints on the factor loadings
should be relaxed. This implies that more waves of data are necessary, and that
other alternatives such as discussed in this paper should also be considered. Our
intention is not to try to convince researchers that the RI-CLPM is necessarily the
best alternative for the CLPM – there will be many instances where one of the other
alternatives is more suited – but rather, to raise awareness regarding the limitation
of the CLPM for uncovering within-person reciprocal processes.
Cross-lagged panel model 36
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Appendix 1
Specifying a CLPM for three occasions can be done with the measurement
equation
xi1
yi1
xi2
yi2
xi3
yi3
=
μ1
π1
μ2
π2
μ3
π3
+
ξ1i
η1i
ξ2i
η2i
ξ3i
η31
,(6a)
and structural equation
ξ1i
η1i
ξ2i
η2i
ξ3i
η31
=
000000
000000
α2β20000
γ2δ20000
00α3β300
00γ3δ300
ξ1i
η1i
ξ2i
η2i
ξ3i
η31
+
ξ1i
η1i
u2i
v2i
u3i
v31
,(6b)
where the covariance matrix of the latter residual vector is
Ψ=
σ2
x1
σx1y1σ2
y1
00σ2
u2
00σu2v2σ2
v2
0000σ2
u3
0000σu3v3σ2
v3
.(6c)
Cross-lagged panel model 43
Note that the variances and covariance between ξi1and ηi1are identical to those of
xi1and yi1in this model.
Specifying the RI-CLPM for three waves of data in SEM software is based on
the measurement equation
xi1
yi1
xi2
yi2
xi3
yi3
=
μ1
π1
μ2
π2
μ3
π3
+
10000010
01000001
00100010
00010001
00001010
00000101
ξ
1i
η
1i
ξ
2i
η
2i
ξ
3i
η
31
κi
ωi
,(7a)
and structural equation
ξ
1i
η
1i
ξ
2i
η
2i
ξ
3i
η
3i
κi
ωi
=
00000000
00000000
α
2β
20 0 0000
γ
2δ
20 0 0000
00α
3β
30000
00γ
3δ
30000
00000000
00000000
ξ
1i
η
1i
ξ
2i
η
2i
ξ
3i
η
3i
κi
ωi
+
η
1i
ξ
1i
u
2i
v
2i
u
3i
v
3i
κi
ωi
,(7b)
Cross-lagged panel model 44
where the covariance matrix of the latter residual vector is
Ψ=
σ2
ξ
1
σξ
1η
1σ2
η
1
00σ2
u
2
00σu
2v
2σ2
v
2
0000σ2
u
3
0000σu
3v
3σ2
v
3
000000σ2
κ
000000σκ,ω σ2
ω
.(7c)
Note that in contrast to the previous model, here the variances and covariance of ξ
1i
and η
1iare not identical to those of xi1and yi1(unless κi=ωi=0foralli).
Cross-lagged panel model 45
Appendix 2
The standardized cross-lagged parameters in the traditional CLPM can be
expressed as partial correlations (e.g., Heise, 1970). Focussing on the cross-lagged
parameter γtfrom ξi,t1to ηit, and making use of the fact that ξi,t and ηit are the
group mean centered variablesxit and yit , we can write
γt
σ(xi,t1)
σ(yit)=ρ(xi,t1yit)ρ(yi,t1xi,t1)ρ(yi,t1yit)
1ρ(yi,t1xi,t1)2.(8)
In order to see how the cross-lagged parameter γfrom the traditional CLPM
is related to the cross-lagged parameters γof the RI-CLPM, we need to express the
correlations used on the righthand side of Equation 8 in terms of the parameters of
the latter model. If we assume that all the observed variables are standardized, the
correlation between yi,t1and yit can be expressed as
ρ(yi,t1yit)=E
ωi+η
i,t1ωi+η
it
=E
ω2
i+E
η
i,t1η
it
=var(ωi)+E
η
i,t1δ
tη
i,t1+γ
tξ
i,t1+v
it
=var(ωi)+E
δ
tη2
i,t1+E
γ
tη
i,t1ξ
i,t1
=var(ωi)+δ
tvar(η
i,t1)+γ
tcov(η
i,t1
i,t1),(9)
while the correlation between yi,t1and xi,t1can be expressed as
ρy1x1=E
ωi+η
i,t1κi+ξ
i,t1
=E
ωiκi+E
η
i,t1ξ
i,t1
=cov(ωi
i)+cov(η
i,t1
i,t1),(10)
Cross-lagged panel model 46
and the correlation between yit and xi,t1can be expressed as
ρ(xi,t1yit)=E
κi+ξ
i,t1ωi+η
it
=E
κiωi+E
ξ
i,t1η
it
=cov(ωi
i)+E
ξ
i,t1δ
tη
i,t1+γ
tξ
i,t1+v
it
=cov(ωi
i)+E
δ
tξ
i,t1η
i,t1+E
γ
tξ2
i,t1
=cov(ωi
i)+δ
tcov(η
i,t1
i,t1)+γ
tvar(ξ
i,t1) (11)
Using these expressions for the correlations in Equation 8, we can now write
γt
SD(xi,t1)
SD(yit)=cov(ωi
i)+δ
tcov(η
i,t1
i,t1)+γ
tvar(ξ
i,t1)
1cov(ωi
i)+cov(η
i,t1
i,t1)2
cov(ωi
i)+cov(η
i,t1
i,t1)var(ωi)+δ
tvar(η
i,t1)+γ
tcov(η
i,t1
i,t1)
1cov(ωi
i)+cov(η
i,t1
i,t1)2.
Similarly, the relationship between βtand β
tcan be derived.
Cross-lagged panel model 47
Foot notes
1While the omitted variable problem implies that we cannot make strong
causal statements based on correlational data, it does not prohibit the use of the
concept of Granger causality (Granger, 1969). However, many researchers using
cross-lagged regression refrain from using the term causal, and use terms like
reciprocal relationship (Erickson, Wolfe, King, King, & Sharkansky, 2001; Lindwall,
Larsman, & Hagger, 2011), role (Ribeiro et al., 2011), cross-domain effects (Burt,
Obradovi´c, Long, & Masten, 2008), exposure (Cole, Nolen-Hoeksma, Girgus, &
Paul, 2006), impact (Gault-Sherman, 2012), or influence (Green, Furrer, &
McAllister, 2011), instead. It may be argued however, that these alternative terms
also imply a causal mechanism, and even more so, that an interest in causality is
actually the driving force behind these studies. Therefore, we decided to use the
terms causal and causality in the current paper, although we acknowledging that
strong causal statements can only be based on experimental designs, and we should
confine ourselves to the concept of Granger causality.
2We used PsychINFO and searched for peer reviewed papers that appeared in
2012 and which made reference to the term “cross-lagged” in either the title, the
abstract or the key words. We found 115 peer reviewed publications of which two
were on time series analysis, one on multilevel modeling, and one did not include an
application. The 111 remaining publications reported on 117 datasets.
3One could also say these autoregressive parameters indicate the stability of the
rank-order of individual deviations, but this is less appealing from a substantive
viewpoint.
4The number of observed statistics in the covariance matrix is equal to
Cross-lagged panel model 48
(6*7)/2=21, while the number of parameters for the covariance structure equals 20,
that is: 2 variance and 1 covariance for the between-person structure (i.e., the
random intercepts), 2 variances and 1 covariance for the first occasion at the
within-person level, 4 lagged parameters for the first interval, 4 lagged parameters
for the second interval, 2 residual variances and 1 residual covariance at the second
occasion at the within-person level, and 2 residual variances and 1 residual
covariance at the third occasion at the within-person level.
5Actually, one only has to assume the lagged relationships were invariant before
the observations started, which is rather abstract when considering the model as a
local description instead of an everlasting truth; hence, this is not a very restrictive
assumption in practice.
6A possible example could be the relationship between number of words typed
per minute and the number of typos: At the within-person level there is a positive
relationship, as a person tends to make more mistakes when (s)he types faster, while
at the between-person level there is a negative relationship as people who have more
experience tend to type faster while making fewer mistakes, and vice versa.
Cross-lagged panel model 49
Figure Captions
Figure 1. Two bivariate models for three waves of data: the standard CLPM, and
the alternative RI-CLPM. Squares denote observed variables; circles represent latent
variables; triangles represent means.
Figure 2. Histogram of number of waves per data set from 111 peer reviewed
publications referring to cross-lagged research in 2012.
Figure 3. Standardized cross-lagged parameter estimates obtained with the
traditional CLPM. Generating values from the RI-CLPM are denoted by the
diamond. Areas A indicate solutions in which |β|<|γ|such that variable xis
causally dominant; areas B indicate solutions in which β|>|γ|such that variable y
is causally dominant. Only 250 estimates (of the 1000 replications) per model are
plotted for reasons of clarity.
Figure 4. Standardized parameter estimates for Soenens data obtained with the
RI-CLPM (above the arrows) and the CLPM (below the arrows). Standard errors
are given between parentheses. indicates significant at α=.05; ∗∗ indicates
significant at α=.01; ∗∗∗ indicates significant at α=.001.
y1 y2 y3
η2 η3
δ2 δ3
v2
η1
1
1
π1 π2 π3
1 1
1 1
CLPM with means RI-CLPM with means
ξ2 ξ3
α2 α3
ξ1
v3
u2 u3
x1 x2 x3
1 1 1
1
μ2
μ1 μ3
β2
γ2
β3
γ3
y1 y2 y3
η2 η3
δ2 δ3
v2
η1
1 1 1
1 1
ξ2 ξ3
α2 α3
ξ1
v3
u2 u3
x1 x2 x3
1 1 1
β2
γ2
β3
γ3
κ
1
1 1
1
ω
1
1
1
1
π1
π2
π3
μ2
μ3
μ1
0
10
20
30
40
50
60
123456
Number of datasets
2 3 4 5 6 7
Number of waves
-0.4 -0.2 0.0 0.2 0.4
-0.4 -0.2 0.0 0.2 0.4
Model 1
Standardized beta
Standardized gamma
-0.4 -0.2 0.0 0.2 0.4
-0.4 -0.2 0.0 0.2 0.4
Model 4
Standardized beta
Standardized gamma
A
A
BB
-0.4 -0.2 0.0 0.2 0.4
-0.4 -0.2 0.0 0.2 0.4
Model 3
Standardized beta
Standardized gamma
A
A
BB
-0.4 -0.2 0.0 0.2 0.4
-0.4 -0.2 0.0 0.2 0.4
Model 2
Standardized beta
Standardized gamma
P1 .230 (.073)*
.657 (.024)*** P2
A1 A2
.122 (.077)
.368 (.033)***
P3
A3
.265 (.094)** *
.675 (.025)***
.113 (.076)
.357 (.035)***
Parental Control
P1 .132 (.130)*
.780 (.180)*** P2
A1 A2
.073 (.088)
.424 (.033)***
P3
A3
.101 (.119)*
.772 (.020)***
.063 (.079)
.414 (.035)***
Parental Responsiveness
Adolescent Depressive Symptomatology
Adolescent Depressive Symptomatology
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The status-legitimacy hypothesis proposes that those who are most disadvantaged by unequal social systems are even more likely than members of more advantaged groups to provide ideological support for the very social system that is responsible for their disadvantages. Li, Yang, Wu, and Kou (2020) sought to expand the generalizability of this hypothesis by testing it in China, addressing inconsistencies surrounding the empirical support for this hypothesis by postulating that the construct of status should be separated into an objective and subjective status marker. They reported that objective SES (income & education) negatively predicted system justification, while subjective SES positively predicted system justification. In the present study we attempt to replicate and extend the work of Li et al. in a cross-cultural comparison of demographic stratified quota online samples in China and the United States. We test the status-legitimacy hypothesis using objective and subjective SES to predict system justification using cross-sectional and cross-lagged regression analyses. We received partial support for Li et al.’s findings. Specifically, subjective SES positively predicted system justification for both societies during cross-sectional and cross-lagged longitudinal analyses. However, we failed to replicate Li et al.’s findings surrounding objective SES in China during cross-sectional and cross-lagged analyses.
... Auto-regressive cross-lagged models (ARCL) are often used to examine time-specific and reciprocal relations. However, this approach aggregates and confounds the within-patient and between-patient components of change (Curran & Hancock, 2021;Hamaker et al., 2015). In the LCM-SR elements known from the ARCL, models are placed on the time-specific residuals within the LCM, rather than on the measured variables, as is usually done in the ARCL. ...
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Previous research suggests that common relationship factors are composed of two overarching factors, "Confidence in the therapist" and "Confidence in the treatment." The aim of this naturalistic process-outcome study was to investigate the reciprocal relationships between these two constructs and patients' symptom level across treatment. The sample consisted of 587 patients who were admitted to an inpatient program and treated with psychotherapy for a range of mental health disorders, such as chronic depression, anxiety disorders, and eating disorders. Our data consisted of weekly measures of symptomatic distress (Patient Health Questionnaire) and the common relationship factors were measured weekly using a newly developed scale. Latent curve modeling with structured residuals was used to investigate the between- and within effects of week-to-week changes in the two components as predictors of subsequent symptom level. An increase in both relationship factors predicted a decrease in subsequent levels of symptoms at the within-patient level, and the other way around, but the two relationship factors did not systematically relate to one another at the within-patient level over the course of treatment. Our findings indicate that patients' perceptions of the therapist as a person and their appraisal of the treatment, are important, different predictors of therapeutic change. Furthermore, they support prior research demonstrating a reciprocal relationship between common relationship factors and symptomatic distress and add to existing common factor theory by exploring the role of two central relationship dimensions and using a method which examines reciprocal relationships and within-patient effects simultaneously. (PsycInfo Database Record (c) 2022 APA, all rights reserved).
... Traditionally, CLPM examines the prospective effect of between-person differences in one construct on the overtime change in between-person differences in the other construct, and vice versa, by controlling for the constructs' autoregressive effects over time. But in recent years, CLPM has been criticized for being unable to distinguish the between-and within-person variance when estimating the crosslagged associations (Hamaker et al., 2015). In comparison, RI-CLPM additionally captures the stable between-person variance in each construct using random intercept factors when yielding cross-lagged effects, namely the prospective effect of a within-person deviation from the trait level of one construct on the overtime change in the within-person deviation from the trait level of the other construct. ...
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This study examined the reciprocal associations between academic contingent self-worth and life satisfaction at the between- and within-person level among college students. 1183 freshmen from eight Chinese universities (736 females, Mage = 18.61 at Wave 1) reported on their academic contingent self-worth and life satisfaction three times over a year, and the data were analyzed using both cross-lagged panel model and random intercept cross-lagged model. Findings of both approaches consistently revealed negative bidirectional cross-lagged associations between academic contingent self-worth and life satisfaction both at the between- and within-person level. Academic contingent self-worth predicted decreased life satisfaction from Wave 2 to Wave 3, but not from Wave 1 to Wave 2, whereas life satisfaction predicted decreased academic contingent self-worth across three waves. The findings suggested that college students’ tendency to base their self-worth on academic performance may gradually impair their life satisfaction, and promoting life satisfaction may in turn help to alleviate academic contingent self-worth among college students.
... These approaches determine individuals' and/or couples' rank-order position for a given construct and evaluate associations with their rank-order position in the same or other constructs at later time points (Lee et al., 2021). However, because these models only consider the relative position of individuals on a given construct-rather than the absolute level-within-person (or within-couple) change is confounded with between-person differences (Hamaker et al., 2015). Despite this, such models can be useful for those seeking to understand factors that contribute to individuals' or couples' relative rank-order position on outcomes of interest. ...
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Longitudinal dyadic research provides significant benefits for our understanding of romantic couple relationships. In this systematic review, we begin by providing a broad overview of topical trends and approaches in longitudinal couple relationships research from 2002 through 2021. Then, we narrow our review to dyadic relationship quality articles, highlighting key themes as well as noting important gaps in the research. Using an intersectional perspective that acknowledges multiple ways that disadvantage, power, and oppression may be seen in both research and in couples’ lived experience, we note prominent paradigms used in examining couple relationships, what types of questions have been most valued, and what groups and approaches are underrepresented in the literature. Most longitudinal couple relationships research is quantitative, relies on self-report approaches from American couples in the early-to-middle years of their relationships, concentrates more on negative aspects of relationships than positives, and takes a communication-satisfaction paradigm in studying couples. We see a clear need to increase the use of methodologies beyond self-report measures, conduct more studies with within-group minority, older adult, culturally-diverse, and context-specific samples to explore the diversity of relationships and fully consider both strengths and positive processes in relationships as well as the challenges couples experience.
... Specifically, further examining the directionality of perceived psychosocial learning environment and students' intentions to quit would be of great theoretical and practical interest. Other theories that more explicitly assume transactional associations could, for instance, guide a cross-lagged panel model with random intercept (Hamaker et al., 2015), which is well suited to test bidirectional associations. ...
Book
Background: National and international research has repeatedly shown that many late adolescents have poor motivation for school. Moreover, the fact that a considerable proportion of youth do not complete upper secondary education is an insistent challenge with severe costs for the individual and society. This thesis concentrates on upper secondary students’ intentions to quit school, which is considered an indicator of a negative motivational process that can lead to dropout from school. From a motivation theory perspective (self-determination theory, in particular), intentions to quit school is considered a persistence-related academic outcome. A theoretical rationale based on self-determination theory (SDT) and achievement goal theory (AGT) of how and why perceptions of the psychosocial learning environment may contribute to the development of such intentions is proposed. Emanating from this theoretical ground and previous evidence, research questions considering how the following aspects of the psychosocial learning environment are related to intentions to quit school were posed: perceived teacher support (emotional support, autonomy granting, and feedback quality), loneliness among peers, and perceived mastery climate. Thus, while decades of research on school dropout have focused on demographic factors and students’ academic achievement level, the current approach scrutinizes the potential in the learning environment on a process that do not limit itself to the final “pass or fail” (dropout vs. completion) yet acknowledges the broader and gradual process of the individual’s more or less prominent intentions to quit school. Enhanced knowledge regarding this process can be vital from a dropout preventive perspective, but also for increased understanding of how the psychosocial learning environment in upper secondary school is related to student motivation. Aims: The overall aim was to empirically investigate how students’ perceptions of the psychosocial learning environment in upper secondary school are related to their intentions to quit school. Three separate studies had specific aims subordinate to this. Hopefully, knowledge derived from this work can contribute to inform measures to optimize students’ motivation and increase their likelihood of completing upper secondary education. Methodology: The thesis has a quantitative approach, and all three studies were empirical investigations of a sample of 1379 students in upper secondary schools in Rogaland, Norway. The main data source was self-reports from these students on three occasions during upper secondary school: T1 in the second semester of the first year, T2 in the first semester of the second year, and T3 in the second semester of the second year, giving a total timespan of 13 months. In addition to self-reports, register data on students’ previous academic achievement, gender, and study track in upper secondary were obtained from county administration, which were applied as control variables in the structural models. Study I had a cross sectional design, and Study Ⅱ and Study Ⅲ had longitudinal panel designs. To investigate the specific research questions, different statistical methods were applied, primarily types of structural equation modeling (SEM) in Mplus. This included confirmatory factor analyses (CFA), mediation models, multigroup testing of moderation, latent growth curve models (LGCM), and growth mixture models (GMM). Results: In the cross-sectional design of Study Ⅰ, the main aim was to investigate the degree to which three aspects of perceived teacher support (i.e., emotional support, autonomy granting, and feedback quality) were related to intentions to quit school, directly, and/or indirectly via emotional engagement and academic boredom. Relevant individual background variables (gender, prior academic achievement, immigrant background, as well as study track) were accounted for. The SEM results showed that all three aspects of perceived teacher support were indirectly negatively associated with intentions to quit school. In addition, emotional support showed a direct negative association with intentions to quit and thus appeared to be a particularly important aspect of perceived teacher support. In Study Ⅱ, the main aim was to investigate intentions to quit school longitudinally, and specifically scrutinize how individual change in intentions to quit was related to initial levels and changes in perceived emotional support from teachers and loneliness among peers at school. Initially, unconditional latent growth curve models indicated an average increase in intentions to quit school and loneliness among peers during the study period, and no average change in emotional support from teachers. However, substantial individual differences were found in the trajectories of all these three concepts. A multivariate latent growth curve model with the rate of change in intentions to quit as the final outcome showed no significant prediction from initial levels of either emotional support or loneliness; however, a substantial inverse associated change with perceived emotional support from teachers and a strong positive association with change in loneliness among peers was found. In Study Ⅲ, individual change in intentions to quit school was kept as the focal outcome yet investigated from the outset of potential trajectory subgroups of perceived emotional support from teachers. The substantial between-student differences in individual trajectories of perceived emotional support detected in Study Ⅱ served as an important ground for this person-centered approach. Furthermore, change in perceived mastery climate was theorized to function as an intermediate variable in a hypothesized association with change in intentions to quit school. Three distinct trajectory subgroups of perceived emotional support from teachers were identified: stable-high (84.9%; the normative group), decreasing (7.8%), and low-increasing (7.3%). Compared to the normative group, membership in the decreasing emotional support trajectory subgroup was indirectly associated with more increase in intentions to quit, and this association was fully mediated by a more negative development in perceived mastery climate. Membership in the low-increasing group was associated with more positive development in mastery climate, but no significant indirect association with change in intentions to quit was found. Conclusion: Prominent in all three studies, was the central role of perceived emotional support from teachers as negatively associated with students’ intentions to quit school. This was also persistent when accounting for background variables, and predominantly when investigating longitudinal relationships. Students with decreasing trajectories of perceived emotional support during the first and second years of upper secondary school were more likely to have steeper increase in intentions to quit school during this phase. However, the opposite route was not supported and requires further research. In addition to emotional support from teachers, individual trajectories of loneliness among peers were closely related to individual trajectories of intentions to quit school, and these results add to previous research conducted in cross-sectional designs. In sum, the current work contributes to empirical support for psychosocial factors in school having a substantial potential to keep students motivated to continue upper secondary school, and this should be considered in all efforts to enhance late adolescents’ academic motivation and to increase upper secondary completion rates.
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Longitudinal data analysis is gaining attention from researchers because it enables us to examine both between- and within-person effects simultaneously. Traditionally, centering has been used with multilevel models to estimate these two effects. However, recent studies found that centering could not disaggregate the between-person and within-person effects when a time-varying predictor shows time trends. This article develops methods for disaggregating the two effects in the presence of time trends using the latent curve model. The proposed methods reveal the link between centering and detrending, which are often seen as different preprocessing for different purposes. Two simulations are conducted to assess and compare the performance of the proposed and existing models. The results show that models with a slope factor behind a predictor can unbiasedly estimate the between- and within-person effects. Also, models with latent between-person predictors can unbiasedly estimate the between-person effect, while those with observed ones suffer from bias.
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Although researchers in clinical psychology routinely gather data in which many individuals respond at multiple times, there is not a standard way to analyze such data. A new approach for the analysis of such data is described. It is proposed that a person's current standing on a variable is caused by 3 sources of variance: a term that does not change (trait), a term that changes (state), and a random term (error). It is shown how structural equation modeling can be used to estimate such a model. An extended example is presented in which the correlations between variables are quite different at the trait, state, and error levels.
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Intensive longitudinal data provide rich information, which is best captured when specialized models are used in the analysis. One of these models is the multilevel autoregressive model, which psychologists have applied successfully to study affect regulation as well as alcohol use. A limitation of this model is that the autoregressive parameter is treated as a fixed, trait-like property of a person. We argue that the autoregressive parameter may be state-dependent, for example, if the strength of affect regulation depends on the intensity of affect experienced. To allow such intra-individual variation, we propose a multilevel threshold autoregressive model. Using simulations, we show that this model can be used to detect state-dependent regulation with adequate power and Type I error. The potential of the new modeling approach is illustrated with two empirical applications that extend the basic model to address additional substantive research questions. Electronic supplementary material The online version of this article (doi:10.1007/s11336-014-9417-x) contains supplementary material, which is available to authorized users.
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We present a revision of latent state-trait (LST-R) theory with new definitions of states and traits. This theory applies whenever we study the consistency of behavior, its variability, and its change over time. States and traits are defined in terms of probability theory. This allows for a seamless transition from theory to statistical modeling of empirical data. LST-R theory not only gives insights into the nature of latent variables but it also takes into account four fundamental facts: Observations are fallible, they never happen in a situational vacuum, they are always made using a specific method of observations, and there is no person without a past. Although the first fact necessitates considering measurement error, the second fact requires allowances for situational fluctuations. The third fact implies that, in the first place, states and traits are method specific. Furthermore, compared to the previous version of LST theory (see, e.g., Steyer et al. 1992, 1999), our revision is based on the notion of a person-at-time-t. The new definitions in LST-R theory have far-reaching implications that not only concern the properties of states, traits, and the associated concepts of measurement errors and state residuals, but also are related to the analysis of states and traits in longitudinal observational and intervention studies. Expected final online publication date for the Annual Review of Clinical Psychology Volume 11 is March 28, 2015. Please see http://www.annualreviews.org/catalog/pubdates.aspx for revised estimates.
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The authors examined the relationship over time of posttraumatic stress disorder (PTSD) and depression symptoms in a sample of Gulf War veterans. A large sample (N = 2,949) of Gulf War veterans was assessed immediately following their return from the Gulf region. and 18-24 months later. Participants completed a number of self-report questionnaires including the Mississippi Scale for Combat-Related PTSD (T. M. Keane, J. M. Caddell, & K. L. Taylor, 1988) and the Brief Symptom Inventory (L. R. Derogatis & N. Melisaratos, 1983) at both time points and an extended and updated version of the Laufer Combat Scale (M. Gallops, R. S. Laufer, & T. Yager, 1981) at the initial assessment. A latent-variable, cross-lag panel model found evidence for a reciprocal relation between PTSD and Depression. Follow-up models examining reexperiencing, avoidance-numbing, and hyperarousal symptoms separately showed that for reexperiencing and avoidance-numbing symptoms, the overall reciprocal relation held. For hyperarousal symptoms, however, the association was from early hyperarousal to later depression symptoms only.
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An effective technique for data analysis in the social sciences The recent explosion in longitudinal data in the social sciences highlights the need for this timely publication. Latent Curve Models: A Structural Equation Perspective provides an effective technique to analyze latent curve models (LCMs). This type of data features random intercepts and slopes that permit each case in a sample to have a different trajectory over time. Furthermore, researchers can include variables to predict the parameters governing these trajectories. The authors synthesize a vast amount of research and findings and, at the same time, provide original results. The book analyzes LCMs from the perspective of structural equation models (SEMs) with latent variables. While the authors discuss simple regression-based procedures that are useful in the early stages of LCMs, most of the presentation uses SEMs as a driving tool. This cutting-edge work includes some of the authors' recent work on the autoregressive latent trajectory model, suggests new models for method factors in multiple indicators, discusses repeated latent variable models, and establishes the identification of a variety of LCMs. This text has been thoroughly class-tested and makes extensive use of pedagogical tools to aid readers in mastering and applying LCMs quickly and easily to their own data sets. Key features include: • Chapter introductions and summaries that provide a quick overview of highlights • Empirical examples provided throughout that allow readers to test their newly found knowledge and discover practical applications • Conclusions at the end of each chapter that stress the essential points that readers need to understand for advancement to more sophisticated topics • Extensive footnoting that points the way to the primary literature for more information on particular topics With its emphasis on modeling and the use of numerous examples, this is an excellent book for graduate courses in latent trajectory models as well as a supplemental text for courses in structural modeling. This book is an excellent aid and reference for researchers in quantitative social and behavioral sciences who need to analyze longitudinal data.
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A latent state-trait model has been used to assess the extent to which egoistic and moralistic self-enhancement represent: (a) stable individual differences and (b) systematic effects of the situation and/or the person-situation interaction. Analyses were conducted on a sample of 187 adults (64% females). Findings revealed that both self-enhancement tendencies mostly capture stable interindividual differences, although significant occasion-specific effects were observed. Egoistic self-enhancement presents a higher proportion of trait variance than moralistic self-enhancement. The egoistic dimension was mostly related with the stable (trait) components of conscientiousness and emotional stability. The moralistic dimension, on the contrary, was mostly related with the transient component of emotional stability. Potential explanations for the observed differences between egoistic and moralistic self-enhancement were discussed and interpreted in terms of their implications for personality assessment.