Journal of Consulting and Clinical Psychology 1995, Vol. 63. No. 1,52-59
In the public domain
METHODOLOGICAL DEVELOPMENTS
The Trait-State-Error Model for Multiwave Data
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David A. Kenny University of Connecticut
Alex Zautra Arizona State University
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.
Increasingly, researchers in clinical psychology are gathering multiwave data to address the effects of variables over brief intervals of time. In the past, multiwave clinical research was performed with only case studies (e.g., Kazdin, 1981), and statistical procedures were developed to evaluate a series of measurements on single individuals (e.g., McCleary & Hay, 1980). It is becoming increasingly common, however, for researchers to take repeated measurements on a relatively large number of participants but with many fewer assessments than the 50 or more necessary to use formal time-series statistical procedures (West & Hepworth, 1991). Behavioral medicine researchers, for example, regularly measure individuals daily, weekly, monthly, or quarterly at four or more occasions (Ormel & Schaufeli, 1991). There are no established procedures for handling these short time series of psychological measures taken on a relatively large number of participants. In this article, we introduce a new approach to the assessment of relationships between two variables measured over time that uses structural equation modeling. Consider the example of positive events and negative events measured each month for a year. Of interest is whether a person's current status on both variables is caused by the previous month's variables. Do negative events lead to positive events, or do positive events lead to more or fewer negative events? It is not clear how the data from such multiwave data sets should be analyzed. Most researchers faced with such data have chosen two-wave analysis strategies; that is, they treat multiwave
data as sets of two-wave data. So for a given time, positive events and negative events would be treated as outcome variables, and both events from the previous wave would be treated as predictor variables. One drawback of this strategy is that it results in many partially redundant analyses. For instance, if there were 12 waves of data, there would be 12 X 11 /2 = 66 sets of two waves. For multiwave data, two-wave models are less than practical. Another possibility is growth-curve models (McArdle & Epstein, 1987). However, those methods were not developed to measure cross-variable lagged effects and, therefore, are not useful for the problem discussed in this article. Currently, work is underway to estimate causal effects with growth-curve models, but that work is not completed. Another alternative for the analysis of multiwave data is pooled time-series regression analysis (Dielman, 1989). There are two steps in this analysis. In the first step, participant effects are removed by subtracting from each score the mean of the participant. So for both positive and negative events, its mean would be computed for each participant across the multiple time points. Then from each positive and negative event score, the mean would be subtracted. Next, an analysis on the data is conducted in which the degrees of freedom are reduced by the number of participants minus one. Therefore, one treats Person X Time (or score) as the unit of analysis, and one would estimate two regression equations using the scores adjusted for the mean. In the first, negative events would be the dependent variable and it would be regressed on the previous month's negative and positive events. A parallel regression equation is estimated for positive events. The pooled time-series analysis can be done in one step by creating participant dummy variables and then entering in the appropriate predictors. In this analysis, the effect of participant would be removed by the use of dummy variables and not by subtraction. Two regression equations would be estimated: Negative events would be regressed on the previous month's positive and negative events, and positive events would be regressed on the previous month's positive and negative events.
David A. Kenny, Department of Psychology, University of Connecticut; Alex Zautra, Department of Psychology, Arizona State University. This research was supported in part by Grant BNS-9008077 from the National Science Foundation and by Grant AG4924-06 from the National Institute on Aging. We thank Niall Bolger for helpful comments on a draft of this article. Correspondence concerning this article should be addressed to David A. Kenny, Department of Psychology, Box U-20, Room 107, University of Connecticut, 406 Babbidge Road, Storrs, Connecticut 06269-1020. Electronic mail may be sent via Bitnet to
[email protected].
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TRAIT-STATE-ERROR MODEL
There is much to recommend the pooled time-series approach over the two-wave approach. It views the variance in scores as being due to two sources: First there is a stable, unchanging individual difference; second, there is a changing variable component. The former component is removed by either subtraction or dummy variables leaving the latter component to be analyzed. Second, within this approach, the results of the different two-wave analyses can be pooled into a single analysis. Instead of dozens of analyses, there is just one two-wave analysis. Although superior to the two-wave approach, the conventional pooled time-series approach has a serious limitation. It neglects to consider the problem of measurement error in the variables. It has been long known that regression analyses with nonexperimental data, in which there are errors of measurement in the predictor variables, yield biased estimates of effect estimates ( Kenny, 1979). The bias is not always benign in the sense that estimates are only attenuated. It can happen that coefficients that are positive are mistakenly estimated as negative, and coefficients that are truly zero are estimated to be significant ( Bollen, 1989). For this reason, a latent variable approach is required to obtain unbiased estimates of regression coefficients. The approach that we take to model trait, state, and error is a latent variable approach. Such an approach has been applied to multiwave data by others ( Hertzog & Nesselroade, 1 987; Ormel & Schaufeli, 1991; Steyer, Ferring, & Schmitt, 1992), but these procedures all require multiple measures of the construct at each time. What makes our approach unique is that it has no such requirement. This article has four sections. In the first, we present the traitstate-error model for a single variable measured at four or more waves. In the second, we develop a bivariate model and so allow for lagged causal effects. In the third section, we illustrate the model. In the fourth section, we explore the implications of the model for analysis of interventions, two-wave analysis methods, and diary data.
are implicitly one. The state source has a correlation that is less than one but greater than zero. Its correlational structure is a first-order, autoregressive model (Kenny & Campbell, 1989). The error component has zero correlations over time. The three key assumptions of the TSE are as follows: stationarity, independence, and autoregressive structure for the state factor. The stationarity assumption is that the variance explained by each source is the same at all n times. Independence is that the sources of variance (trait, state, and error) are not correlated with each other at the same time or across time. A first-order autoregressive structure for the state variable implies that the correlation over time is ak where k is the lag length or the difference between waves (Kenny & Campbell, 1989). In terms of an equation, 5, = aSt - 1 + u,. For stationarity to be ensured, of the state variance or V( S,) = V( St - 1 ), it follows that V( Ut) = ( I - a2)V( 5, - 1 ) because V( S,) = aiV( St - I )
Identification If there are four or more waves of data, the TSE model can, in principle, be estimated. Let us define the elements of the variance-covariance matrix. The variances of the measures can be denoted as Vx. The covariance between x2 and x4 would be a lag-2 ( 4 - 2 ) covariance, or C2. If there are four time points, there are three types of covariances: lag 1, lag 2, and lag 3. The stationarity assumption implies that covariances of the same lag are equal to each other. There are then four pieces of information that can be used to estimate the model: one variance and three lag covariances. Given stationarity, independence, and first-order autoregressive process for the state factor, the equations for the variances and covariances are as follows: VX=VT+VS+VK
C3= VT+a*Vs, Univariate Model Imagine variable x measured at n equally spaced periods of measurement. So Xy refers to the score of person / measured at time./. The trait-state-error model is as follows: The terms are defined as follows. First, M} is a constant that is added to each score. The other terms in the model are assumed to have zero means. The term Tt is the trait component. It represents variance in a person's score that does not change over time. The term Sy represents the state or changing portion of variance. It is assumed that there is a correlation over time between state components but that the correlation is less than 1 . The final source of variance Ey represents random variance or variance that does not correlate across time. The model is presented in Figure 1 . Therefore, the model has three sources of variation; trait, state, and error, and so is called the trait-state-error model or TSE. These three sources differ in their correlations over time. The trait source does not change and so all of its correlations
where K symbolizes variance. We have then four equations with four unknowns, VT, Vs, VE, and a . The system of equations is nonlinear. For there to be an in-range solution (i.e., nonnegative variances and an a greater than 0 and less than 1 ), it follows that Cj/C 2 must be greater than C2/C3. Interestingly, both Kenny and Campbell (1989) and Campbell and Reichardt (1990) have noted that such correlational structures are quite common in multiwave data. Estimation The estimate of the autoregressive parameter is
~ C 2 -C 3
' The researcher may wish to allow that the state variance to be nonstationary but still assume that disturbance variance is stationary. For such a model, V(St)* V ( V ) / ( { -a2).
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DAVID A. KENNY AND ALEX ZAUTRA
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e
Figure 1. Univariate trait-state-error model. Numerals 1 through 4 indicate the four lags; e = error; u = disturbance.
One can solve for Vsby [C, - C2]/(a - a2) or [C2 - C 3 ] / ( a 2 - a3). The estimate of the variance of the trait, or VT, equals C| — Vs, and the estimate of the error variance for each wave is VX-VT-VS. The trait-state-error model can be estimated by a structural equation modeling computer program (e.g., LISREL, EQS, EASYPATH, or CALIS). Whatever program is used, it is essential, given the number of equality constraints that are made in the model, that the covariance and not the correlation matrix be analyzed (Cudeck, 1989). The parameters of the model are as follows: V(T), the trait variance; V(U), the disturbance variance of the state factor; a, the autoregressive coefficient for the state factor; and V(E), the error variance. Given stationarity, the state variances are the same at each wave. It follows that the state variance is determined by the disturbance variance of the state factor and the autoregressive parameter. To see this, the state variables equal the state at the previous occasion multiplied by the autoregressive coefficient plus a disturbance, or S, = aS, — 1 + U,. It follows that Vs = a2Vs + Vv. Solving through for Vs yields Vv/( 1 - a2). (One could use the phantom variable approach [ Rindskopf, 1984 ] to force this nonlinear constraint.) A simple iterative approach is to fix V ( S i ) to some value (e.g., 1.0) and estimate the model. Then, re-estimate the model using Vv/( 1 — a 2 ) from the previous model as the fixed value for V( S\) in the next model. This is repeated until V( S) is Basically the same (e.g., within .005) at all waves. Convergence can be slow and may take as many as 10 iterations. If this strategy does not lead to a stable solution after 20 or more iterations, the investigator needs to reconsider whether the stationarity assumption has been met. The model has four free parameters and 10 elements of the variance-covariance matrix: four variances and 4 X 3/2 = six covariances. There are 6 degrees of freedom and these 6 degrees of freedom evaluate the stationarity assumption. Three of the six evaluate the equality of the four variances, two evaluate the equality of the three lag-1 covariances, and the last degree of freedom evaluates the equality of the two lag-2 covariances. If there are more than four waves of data, the model contains no
additional parameters and so the chi-square has more degrees of freedom. In general, for the univariate model, there are n(n + 1 )/2 — 4 degrees of freedom, where n is the number of waves. We have found the following approach to estimation works well with LISREL 7. Estimation should proceed with user-provided starting values and unweighted least squares (ULS). LISREL's generated starting values do not appear to work well. Once a solution has been obtained, the ULS estimates can be used as starting values for maximum likelihood estimation. Moreover, because the number of factors is greater than the number of variables, the admissibility test in LISREL should be turned off.2 Complications and Extensions If the autoregressive parameter is very near one or zero, there are likely to be estimation difficulties. If it is near one and therefore the state disturbance variance is near zero, the trait factor should be dropped. If it is near zero, the error variances should be dropped. It is generally advisable to compare the fit of the trait-state-error model to a model that contains only the trait factor and errors and to a model that contains only the state factors and errors. If either model fits as well as the combined trait-state-error model, then the simpler model should be preferred. If the state factor is dropped, one can allow the loadings of the trait factor to vary across waves and the model is still identified. If the trait factor is dropped, then the autoregressive parameter can vary between adjacent waves. The error variances, or VE, can be different at each wave and the model is still identified. Also, the factor loadings for each variable can vary across time, but within time the trait and state factors must still load equally for each variable. The loading of one reference variable must be fixed to 1. This allows for differential variance at each wave and this type of stationarity has been referred to as quasi-stationarity (Kenny & Campbell, !
The LISREL setup is available from David A. Kenny.
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TRAIT-STATE-ERROR MODEL
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e.
Figure 2. Bivariate trait-state-error model. Paths from state factors (a, b, c, and d) are denoted as ax Oyx, axy, and ayy in the text. Numerals 1 through 4 indicate the four lags; e = error; u = disturbance.
1989). For four-wave data, the resulting model with both of these modifications would have zero degrees of freedom because an additional six degree parameters (three error variances and three loadings) are estimated. Such a model is just-identified and there is no chi-square test. Bivariate Model The major interest in this article is the causal relationship between two variables. There are two variables, X and Y, each of which is measured at each wave. Both X and /are assumed to be caused by a trait, Tx or Ty, a state factor, Sx or Sy, and by measurement error, Ex or Ey. So we have
X = Tx + Sx + Ex and Y=
Ty
+
Sy
+
Ey.
For each variable, there are the four parameters described in the previous section: trait variance V( Tx) and V( Ty); state disturbance variance V(UX) and V(Uy)', error variance V(EX) and V(Ey); and two autoregressive parameters (a^) and (ayy). There are five additional bivariate parameters or parameters that describe the association between X and Y. The first three involve correlations between the X and Y variables at the trait C( Tx, Ty), state C( Ux, Uy), and error C(EX, Ey) levels. The next two are lag-1 regression coefficients of state variables: the effect of Y on X (a xy ) and the effect of X on Y (ayx). There are a total of 13 parameters, and the model's degrees of freedom are n(2n+ 1) — 13, where n is the number of waves. The model is presented in pictorial form in Figure 2. In principle, one could allow for lag-2 effects; that is, one's current standing is affected by the previous time and the time
before that. This article does not explore this complexity because it seems theoretically unlikely. There are three nonlinear constraints on the parameters. Each state variance at Time 1 equals the disturbance variance of the state divided by 1 minus the multiple correlation squared of the state factor. The state covariance at Time 1 equals a function of the covariance of the state disturbances and the lag coefficients.3 A practical estimation strategy is to re-estimate the model and constrain the parameters to their value in the previous solution. The estimation stops when the parameters do not change very much.
Covariates If there are covariates that operate at the trait level (e.g., gender or ethnicity), their correlation can be estimated and tested at that level. Although one could introduce them into the structural equation model, there is another alternative. One first computes the mean score for each variable (Xand Y) across the n time points. One then correlates a covariate with this mean score. However, this correlation is attenuated because of the fact that the mean score estimates but does not equal the trait factor. To disattenuate the correlation, one multiplies the previous correlation by the trait standard deviation (taken from the structural equation model run) divided by the standard deviation of the mean of the measures. (This ratio of standard deviations equals the square root of the reliability of mean score.) 3
That function is as follows: c- x _ a»fl*yV(Ux) + •
y) + C( U*,
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DAVID A. KENNY AND ALEX ZAUTRA
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Illustration A bivariate example is provided from a study of the everyday life events of 160 older adults (Zautra, Reich, & Guarnaccia, 1990). Everyday events can depict both routinely pleasant transactions between the person and his or her environment, and also minor stressful disruptions in routine. Stable characteristics of the person, such as age and social status, affect the likelihood of these events (Zautra, Finch, Reich, & Guarnaccia, 1991). In addition, changes in person-environment relations can influence everyday life. Thus, we expected to find "traits" that describe stable individual differences in events, and changing event frequencies coinciding with changes in person-environment relations, which we labeled as event "states." The original study was designed to assess the patterns of small stressors and desirable experiences among older adults in widely divergent life circumstances. Adults between the ages of 60 and 80 were recruited for the study if they were disabled or if their spouse had died within the past 4 to 6 months; they were recruited as controls by being neither disabled nor conjugally bereaved. Everyday life events were assessed using the Inventory of Small Life Events, modified to suit older adults (see Zautra et al. [1990] for a full description of the measure). This inventory was administered monthly to the older adults for 10 months. Separate scores for desirable and undesirable small events were obtained by summing reports of the occurrences of events across 12 different life concerns: education, religion, financial, recreation, social life, love and marriage, transportation, immediate family, extended family, work, crime and legal matters, and household events. Events on this scale were required to have properties that would distinguish them from measures of personality or states of psychological well-being and distress: (a) The event had to identify a specific occurrence for which there was a discrete beginning; Changes as "more" or "less" of a quantity were removed or rewritten to reduce ambiguity and lower subjectivity in the rating of these events, (b) Only events that were unambiguous as to desirability were retained on the basis of consensus ratings of experts in this research, (c) Events had to refer to observable occurrences and, therefore, thoughts and feelings were excluded from the inventory, (d) The events represented a small change in everyday lifestyle routine and not a major event. Thus, these events could not have a magnitude readjustment score greater than 250 units on the basis of ratings made with standard magnitude estimation procedures (Holmes & Rahe, 1967). For each event, respondents were asked if the event happened and, if the answer was "yes," if it happened more than twice in the past month? Scores ranged from 0 to 2 on each event, and the event scores were summed separately to yield a total score for desirable and undesirable events. Each respondent was interviewed 10 times at monthly intervals. With only a few exceptions, all interviews for each participant were conducted by the same female interviewer with the first and last interviews conducted in the participant's home and the middle eight interviews conducted by telephone. For the purposes of this illustration, the participant groups were combined and only the last 9 months of data were used. More desirable and undesirable events were reported at the initial interview because of telescoping: reporting events that hap-
pened longer than 1 month ago as occurring in the past month. The result of this reporting tendency was increased variance and covariance between measures at the initial assessment, violating the stationarity assumption. Respondents reported many more desirable events than undesirable events, with scores ranging from 11.4 to 17.9 desirable events and from 2.8 to 4.4 undesirable events each month. Undesirable event reports were not normally distributed showing a positive skew, so both event scores were subjected to square root transformations prior to analysis. Results Ordinarily, researchers would expect life events not to exhibit "trait-like" properties. Indeed, event measures often are used to assess change, particularly change in environmental conditions. Although major events that occur rarely may be defined as indicators of "change," repeated assessments of small events may reveal stable patterns of daily life, as well as fluctuations in life experience. Initial correlational analyses suggested, in fact, that it would be valuable to analyze the data using the trait-state-error model. First, there were substantial correlations of .53 and higher between scores on measures taken 1 month apart suggesting either strong autoregressive factors or stable individual differences that could form a trait. These stable patterns suggested also the likelihood of state variables that are moderately correlated with the next month's scores on the same measure. One of the major challenges of the model is, therefore, to estimate the variance that is due to traits for desirable and undesirable events, and the degree of autocorrelation between the monthly latent state variables. There were also consistent positive correlations between desirable and undesirable events for each month. These correlations ranged from .3 to .4. Consistent with past research in reports of events, respondents with many desirable events also tended to report many undesirable events. There are a number of possible explanations for such relationships, and the traitstate-error model can examine the likelihood of these alternatives. The positive correlations may derive from correlated measurement errors caused by the shared methods of administration that can be modeled as correlations between errors of measurements across methods. Alternatively, an unmeasured third variable may be responsible as a cause of both and this can be modeled as a significant parameter relating structural disturbances of the state variables. It may also be that stable individual differences in activity level may be responsible for persistent positive correlations between the frequency of positive and negative events (Zautra et al., 1991). Personality features, enduring health, and social status differences may also affect the occurrence of desirable and undesirable events that would be revealed in correlations between trait factors and correlations between event traits and exogenous variables. It is also possible that increases in desirable events may cause an increase in undesirable events. In this sample of older adults, high numbers of desirable events may constitute "overdoing it," leading to negative outcomes. Previous analysis of the first 2 months of this data set suggested that such was the case. Such
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TRAIT-STATE-ERROR MODEL effects would be observed as a lagged "causal" path between the state factor for desirable events at one month and the state factor for undesirable events the next. In addition, combinations of these effects are possible, including the presence of negative lagged effects, which are obscured because of positive correlations between method factors or other relationships. Only by partitioning the variance in trait-state-error model is it possible to identify such relationships. We first estimated the multivariate model, and we found that the trait correlation between the positive and negative events, trait factors was 1.00. To force a perfect correlation between trait factors we had the positive trait factor (7» cause the negative trait factor with no disturbance. Only 12 parameters were estimated in this model.4 The parameters estimated were the variance of the latent trait, the disturbance variance (set equal for the latent states, Months 2-9), correlations between disturbances (set equal for Months 2-9), the errors for desirable and undesirable event measures (set equal for Months 1-9), correlations between errors (also set equal), a causal parameter between the desirable event trait and the undesirable event trait, two autoregressive parameters, a 1-month lagged effect of latent state of desirable events on undesirable events, and a 1-month lagged effect of latent state of undesirable events on desirable events. The fit of the model is x 2 ( 159) = 171.48, p < .237, and an impressive Tucker-Lewis index of .993. The 12 parameter estimates are contained in Table 1. It is possible to evaluate the assumption that the loadings of the measured variables on the trait variables are equivalent across the 9 months by fixing the proportions of variance to state and trait across months and freeing the loadings for each month but one. The overall test of equal loadings suggested that the loadings did not vary significantly, x 2 (16) = 6.63. There was also no evidence that the error variances differed over time. Thus, there was no need to modify the model. Because there are estimates of trait, state, and error variances,
Table 1 Parameter Estimates and Their Z Tests Parameter V(Tf) V(Tn) V(UP) V(Un)
at/,, un) V(E ) P
V(En) €(£,,£„) aap apa am anp V(SP) Y(Sn) QS^SJ
Estimate .802 .256' .052 .068 .036 .397 .367 .016 .728 -.064 .890 -.160 .109 .319 .015
7.490 7.703" 1.518 2.345 1.376 11.667 12.601 0.658 3.287 -0.945 13.022 -0.852 — —
Note. V = variance; T = trait; U = disturbance; C = covariance; E = error; a = autoregressive coefficient; S = state. Dashes indicate that data were not significant. "Estimated by V(TP) multiplied by the squared path from Tp to Tn. b Test of the path from Tp to Ta.
Table 2 Variance Partitioning for the Variables Variable Desirable Undesirable
Trait
State
Error
.61** .27**
.08* .34**
.31** .39**
.08
.04
1.00**
r
p
