A Multilevel Structural Equation Model for Dyadic Data



A Multilevel Structural Equation Model for Dyadic Data

Jason T. Newsom

Portland State University

RUNNING HEAD: Multilevel SEM for Dyadic Data

Submitted for publication.

Draft: 8/2/01

Address correspondence to Jason T. Newsom, Ph.D., Institute on Aging, School of Community Health, Portland State University, P.O. Box 751, Portland, OR 97207-0751, email: newsomj@pdx.edu. Partial support for preparation of this article was provided by AG 15159 and AG14130 from the National Institute on Aging. The author wishes to thank Joop Hox and David Morgan for helpful discussions and Patrick Curran and Tor Neilands for comments on an earlier draft.

A Multilevel Structural Equation Model for Dyadic Data

Abstract

Dyadic data involving couples, twins, or parent-child pairs are common in the social sciences, but available statistical approaches are limited in the types of hypotheses that can be tested with dyadic data. A novel structural modeling approach, based on latent growth curve model specifications, is proposed for use with dyadic data. The approach allows researchers to test more sophisticated causal models, incorporate latent variables, and estimate more complex error structures than is currently possible using hierarchical linear modeling or multilevel structural equation models. A brief introduction to multilevel regression and latent growth curve models is given, and the equivalence of the statistical model for nested and longitudinal data is explained. Possible expansion of the strategy for application with small groups and with unbalanced data is briefly discussed.

Running Head: Multilevel SEM for Dyadic Data

Social scientists commonly study dyadic data, as found in research on marital couples (Raudenbush, Brennen, & Barnett, 1995), twins (Kendler, Karkowski, & Prescott, 1999), employees and supervisors (Fleenor, McCauley, & Brutus, 1996), or doctors and patients (Goldberg, Cohen, & Rubin, 1998). When data are collected from both individuals of the dyad, scores from each individual in a dyad are usually dependent. Data from each member of the dyad cannot be treated as independent observations without the underestimation of standard errors, resulting in increased Type I error. In addition, many measures, such as household income, are measured at the dyadic level rather than the individual level. Other variables, although measured at the individual level, partially reflect experiences or circumstances that are common to both members of the couple (e.g., marital quality), and, thus, containing individual-level and group-level components. For these reasons, dyadic data can be considered "hierarchically structured."

Multilevel regression (MLR; also known as hierarchical linear models or HLM; Bryk & Raudenbush, 1992; Kreft & de Leeuw, 1998; Snijders & Bosker, 1999) and multilevel structural equation models (multilevel SEM: Muthen,1989,1994; Muthen & Satorra, 1989) are two approaches to analyzing hierarchically structured data in which individuals are nested within groups.[i] MLR can be conceptualized as a two-level regression model, with the level-1 model represented by the usual regression equation generated for each group. The level-2 model involves prediction of intercept and slope coefficients obtained from the level-1 models. These models are now commonly used throughout the social sciences and have been applied to dyadic data by a number of researchers (Barnett, Marshall, Raudenbush, & Brennan, 1993; Ozer, Barnett, Brennan, & Sperling, 1998; Segal & Hershberger, 1999).

The multilevel SEM approach to hierarchically structured data is a recently developed approach (e.g., Muthen, 1989, 1997a; Muthen & Satorra, 1995; McArdle, & Hamagami, 1996) that can estimate measurement, path, or full structural models at the individual and group-level (usually termed within and between levels, respectively). The multilevel SEM approach involves analyzing separate covariance matrices at the within (level 1) and between level (level 2). The ability to estimate measurement error is an important advantage over regression-based methods, but the approach is limited to separate models at the within and between levels and hypotheses involving causal relations between the two levels cannot be tested. Moreover, slope variability across groups cannot be estimated with the technique (Kaplan, 2000). Multilevel SEM can also be cumbersome to implement because separate within and between covariance matrices must be estimated and read into an SEM software package. The within and between covariance matrices are then tested using a multigroup structural model with special model specifications (Muthen, 1997a; McArdle, & Hamagami, 1996). Although matrix estimation has been automated and model estimation has been simplified with Mplus software (Muthen & Muthen, 2001), other software packages require more laborious procedures to estimate multilevel models.

Growth curve analysis, another common application of MLR, is statistically equivalent to MLR with hierarchically structured data, because repeated measures are nested within individuals (Bryk & Raudenbush, 1992). Although latent growth curve analysis has an analogous statistical relationship to multilevel SEM, the actual implementation or model specification of the two types of models is quite different. The latent growth curve specification does not have the same limitations as multilevel SEM in that there are no restrictions on relationships between level-1 and level-2 variables and it is convenient to implement in any software package.

The statistical equivalence of the multilevel and growth curve models allows for the possibility of using a latent growth curve approach to model specification with certain types of nested data. Based on this rationale, I describe below a novel approach to the analysis of dyadic data that employs a latent growth curve model specification. The approach has advantages over the MLR approach to dyadic data without the limitations of multilevel SEMs, because latent variables can be used, more complex error structures can be estimated, more sophisticated causal hypotheses can be tested, and model specification is convenient with any SEM package.

I begin by giving a brief overview of latent growth models and hierarchical linear models. A general familiarity with structural equation models is assumed, but familiarity with multilevel regression models is not assumed. After illustrating the equivalence of MLR growth models and latent growth models, I describe a new approach to analysis of dyadic data using a growth curve formulation. Finally, I discuss some possibilities for how this approach may be expanded for use with small groups and in situations in which group sizes are not equal.

Latent Growth Curve Models

Both MLR and SEM can be used to analyze longitudinal data by estimating individual growth curves. The statistical models for growth curve analysis and hierarchical analysis are identical, because repeated measures can be considered to be nested within individuals (Bryk & Raudenbush, 1992). Thus, growth curve models, in general, are a special case of the multilevel model. The SEM approach, called latent growth curve modeling (McArdle, 1988; Meredith & Tisak, 1990), has advantages compared to multilevel regression because of the ability to estimate measurement error when multiple indicators are used and the ability to specify complex error structures (Willet, 1994). Although, statistically, multilevel structural models and latent growth curve models are identical, their implementation with structural modeling packages is quite different. Implementation of the latent growth model does not require estimation of separate covariance matrices nor does it require a multigroup structural model. Latent growth models estimate latent intercepts and slopes representing an individuals initial level and change in the dependent variable over time. The variability in intercepts or slopes across individuals and the factors that explain the variability are often of interest to researchers. Most importantly, latent growth curve models afford considerable flexibility for researchers, because the slopes and intercepts can be incorporated into more complex models as predictors or outcomes.

Latent growth curve models estimate individual growth curves or "trajectories" by using repeated measures as indicators of two latent variables, an intercept variable (η0) and a slope variable (η1). The interpretation of the intercept variable depends on how the loadings on the slope factor are fixed. For instance, one approach to defining the slope variable is to fix loadings to values 0, 1, 2, 3, 4, … , t-1, sometimes referred to as "time codes." In this case, the intercept latent variable represents the initial value, because the first loading on η1 is set to 0. One can also “center “ these loadings by setting the middle time point to 0 (e.g., -3.,-2,-1,0,1,2,3), giving the intercept factor the average value of y at the middle time point, t0. Figure 1 illustrates a simple growth curve model with four time points.

Algebraically, the latent growth curve model is represented by the following formulas. For simplicity, the Lisrel "all-y" notation is used throughout (Hayduk, 1987):

level-1 equation (measurement model):

|[pic] |(1.1) |

level-2 equations (structural model):

|[pic] |(1.2) |

| [pic] |(1.3) |

In equation (1.1), yti is the dependent variable. The subscripts i and t indicate a measurement within an individual, i, for each time point, t. [pic]is a latent variable that represents the level-1 intercept, [pic]is a latent variable that represents the relationship between the time code and the dependent variable (i.e., the growth trajectory), λti are the loadings for each time point on the intercept latent variable (η0) and the slope latent variable (η1). The measurement intercept, ν, associated with each loading (τ matrix) is assumed to be zero as it is in most traditional structural models and in latent growth curve models (e.g., Muthen, 1997b; Willet & Sayer, 1994), and therefore is not shown above. To simplify, no level-2 predictors are presented in (1.2) or (1.3), but predictors of the intercepts or slopes could be included. In the level-2 equations, α0 and α1 are the intercepts or the average value of η0 and η1 for all individuals. ζ0 and ζ1 are residuals. The variance for residuals is found in the ψ matrix.

More traditionally, the structural model is represented by grouping each variable into matrices:

|[pic] |(1.4) |

|[pic] |(1.5) |

In equations (1.4) and (1.5), Λ is a 2 X t matrix representing the relationship between 2 latent variables, η0 and η1, and t indicators, one for each time point. The first column in Λ, which corresponds to η0, is comprised of all 1s, because each loading for this variable is set equal to 1 to define it as the intercept. θε is the matrix of residual errors for each indicator at each time point. α is a 2 X 1 vector containing latent means for the intercept and slope, representing the average intercept across individuals and the average slope (i.e., trajectory) across individuals, respectively. ψ is a matrix of error terms (i.e., the ζ elements) and provides information about the variances of the intercepts and slopes, η0 and η1, and their covariances. The variance of the intercepts and slopes, η0 and η1, are obtained by estimation of the ψ matrix.

Although the formulae will not be presented here, it is possible within the latent growth curve framework, to use multiple indicators of each construct at each time point (e.g., McArdle, 1988; Duncan, Duncan, Strycker, Li, & Alpert, 1999). In this formulation, latent variables at each time point are the used as indicators of second-order latent intercept and slope variables. The resulting model accounts for residual variation at each time point that is not accounted for by the growth parameters and measurement error. In addition, correlated error structures across time points are possible.

Multilevel Regression Models

Multilevel regression models (or hierarchical linear models) estimate predictive relationships when the data are nested or hierarchically structured, as in the case of students nested within schools. The statistical model used for hierarchically structured data is the same statistical model used for longitudinal analysis of individual growth curves. With growth curve models, longitudinal data measurements are considered to be nested within individuals. In general, a multilevel regression with a single level-1 predictor and no level-2 predictors can be written with two sets of equations:

Level-1 equation:

|[pic] |(1.6) |

Level-2 equations:

|[pic] |(1.7) |

|[pic] |(1.8) |

Equation (1.6) is the familiar regression equation, with r representing error or unexplained variance. The subscripts i and g indicate whether the value is for each individual or each group. In equation (1.7), the intercept values for each group serve as the dependent variable. For simplicity sake, there are no predictors in equation (1.7) or (1.8). In (1.7), γ00 is the intercept (mean of all group intercepts), and u0 is the error or remaining variance. The variance of [pic]across the groups gives an estimate of the variability of the intercept values, [pic], across groups. Because the intercepts represent adjusted means for each group (i.e., adjusting or controlling for the effects of xi), [pic]is the variance of the adjusted means for each group. In MLR texts, [pic] is typically referred to as [pic].

In the third equation, (1.8), the slope estimates, β1g, obtained from the level-1 regression model for each group serve as values of the dependent variable. γ10 is the intercept in this equation, and represents the average of all slopes, β1g, interpreted as the average effect of xi on the dependent variable across all groups. u1 is the error term and its variance, [pic], represents the variability of the slopes across groups (i.e., the variability in the relationship between x and y across the groups). [pic] is customarily referred to as [pic]. One can also examine the covariances or the correlations between the slopes and intercepts as they covary across groups, [pic].

By substituting equations (1.7) and (1.8) into equation (1.6), the MLR model can be expressed as a single regression equation,

|[pic] |(1.9) |

or, by rearranging the terms,

|[pic] |(1.10) |

If growth curve models are tested, the level-1 x-variable is replaced by time codes, xt (e.g., 0, 1, 2, 3, . . ., t-1). The dependent variable at each time point is regressed on the time code at level-1. Instead of individuals nested within groups, repeated measures are nested within individuals. In other words, level 2 consists of individuals rather than groups. In growth models, [pic] (i.e., the[pic]) represents the variability of the initial or baseline value of y across individuals, [pic], or [pic], is the variation of the growth across individuals, and [pic], or [pic], is the covariation of the initial value and the growth in y across individuals.

Comparing the SEM and MLR growth models

With a single measure at each time point, the two approaches to growth models are essentially identical. The parallels between the SEM and MLR approaches can be seen by comparing their algebraic formulas (see Table 1).

These formulas are fully equivalent, although this may not be apparent at first glance. In SEM, loadings are used in place of level-1 regression coefficients. In equation (1.1), the level-1 intercept is represented by the product term λtiη0i, which refers to the loadings for latent intercept variable and the intercept variable itself. The Λ matrix is analogous to the X matrix in matrix regression in which the first column is a vector of 1's used to produce the intercept. By setting the loadings for the intercept (η0i) to 1, the product of the loadings and the intercept (λtiη0i ) of (1.1) is simply equivalent to the intercept term, β0, of equation (1.6). The next term in equation (1.1), λtiη1i, representing the slope factor, can also be considered identical to the slope in equation (1.6) as long as the loadings in the Λ matrix are set to values that would be used as predictors in growth curve analysis, such as 0, 1, 2, 3, . . . t-1. Here, λti in (1.1) is equivalent to xt in (1.6). Because λ's are equivalent to x's and the η's are equivalent to β's, it would make more sense to re-express equation (1.1) as,

|[pic] |(1.11) |

By estimating the means and variances of η0i and η1i , we can obtain estimates of the average latent intercept and average latent slope and the extent to which they vary across individuals.

A Multilevel Structural Equation Model for Dyadic Data

Because growth models and two-level hierarchical regression models are identical statistical models, as illustrated above, it is possible to specify a multilevel SEM for certain hierarchical data situations that use the same model specifications as those used in latent growth models. I start by describing the data requirements necessary with dyadic data (e.g., couples, twins, mother-child dyads). I then describe two model specification options, a single-indicator model and a second-order multiple indicator model, giving an example of each. The second-order multiple indicator model is then used in an example illustrating the use of a level-2 (dyad-level) predictor.

Data characteristics

At minimum, a single dependent measure obtained from each member of the dyad is needed.[ii] Multiple indicators for each individual can also be used, with a minimum of three indicators for each member of the dyad. Dyads will be assumed to be non-exchangable. That is, there is a basis for distinguishing members of each couple in an identical manner in all groups. Examples might include husbands and wives, mother and child, first born and second born, or caregiver and care recipient. The data set should be configured in the so-called "repeated measures" format, in which each case in the data matrix contains information about both members of the dyad. For example, each record contains information about the husband and the wife, recorded under different variable names (e.g., y1h, y2h, y3h, y4w, y5w, y6w). This configuration is analogous to that used for latent growth curve analysis.

Example data set

To illustrate, I will use an example from a study I conducted recently examining interactions between spousal caregivers and care recipients. There are 116 couples (232 individuals), in which each member of the couple was interviewed separately. I examine five items from the Veit and Ware (1983) positive affect subscale of the Mental Health Inventory. Items such as "How much of the time have you felt the future look hopeful and promising?" were rated on a 6-point scale of frequency of occurrence. Thus the analysis is based on 10 variables—5 items for caregivers and 5 items for care recipients. In the single-indicator model, illustrated first, the five items are averaged for caregivers and for care recipients.

Single-indicator models

Intercept-only model. I first take the simplest case in which there is only one measure (i.e., indicator) of positive affect for caregivers and for care recipients (the measure was computed by averaging the five items for each). This model specification follows that of the growth curve model described above in the case in which there is only two time points tested and is depicted in Figure 2. The basic model, which I will call the intercept-only model, includes no level-1 or level-2 predictors. At level 2, there is only one equation, because there is no slope obtained from level 1. Using the multilevel regression notation, this is the model given by the following separate equations:

Level-1 equation:

|[pic] |(1.12) |

Level-2 equation:

|[pic] |(1.13) |

This model can be shown to be equal to a random effects ANOVA (Raudenbush, 1993), in which β0 is the mean score for each dyad and the estimate of the variance of the residual, rig, is the within-group variation, usually designated by σ2. γ00, the level-2 intercept, represents the average of the dyad means and the estimate of the variance of u0g, known as τ00, is the between group variation of the means.

Using the equivalent structural modeling notation, these equations would be:

Level-1 equation:

|[pic] |(1.14) |

Level-2 equation:

|[pic] |(1.15) |

One can calculate the ratio of between-dyad variation relative to the total variation using the intraclass correlation coefficient. The intraclass correlation provides information about the degree to which dyad members have similar scores on the dependent variable. In HLM notation, intraclass correlation coefficient is expressed as a ratio of between to within plus between variation:

|[pic] |(1.16) |

Using the SEM approach, the measurement residual represents the within-dyad variation and the variance of the latent intercept, η0, represents the between group variation. So, the intraclass correlation coefficient is given by:

|[pic] |(1.17) |

where ψ00 represents the variance of η0.

Results for the intercept-only model. Parallel analyses were conducted using Mplus (Version 2, Muthen & Muthen, 2001) and HLM 5 (Raudenbush, Bryk, Cheong, & Congdon, 2000). The results reported in Table 2 indicate nearly identical findings with the two statistical packages. The average positive affect score for dyads was 4.189. The between-dyad variance, given by the random effect for the intercept using HLM, and the variance of the intercept variable in Mplus was approximately .28 and was significantly different from zero in either case. A significant variance indicates that the average positive affect score for each couple varies across the dyads. The within-dyad variance was .570, indicating there was greater variation within-dyads than between dyads. The intraclass correlation coefficient was approximately .33, indicating couples had positive affect scores that were moderately related. The association between caregiver and care recipient positive affect indicates that an OLS regression assuming couples were independent cases would not be appropriate and would provide underestimates of standard errors in statistical tests.

Difference model. A predictor at level 1 can be added by incorporating a slope variable to represent the difference between dyad members (e.g., gender). In the context of the care recipient study, a variable designating whether an individual was a caregiver or care recipient was used. A mean slope significantly different from zero would indicate a significant difference between caregivers and care recipients on positive affect. The difference model is depicted in Figure 3. Two latent variables are defined: a latent intercept, η0, and a latent slope, η1,. There is only one indicator for each of these latent variables. The intercept variable, η0, is defined by fixing loadings on each of the two indicators to 1. The slope variable, η1, is defined by fixing loadings on the same two indicators to 0 and 1. The average intercept and average slope across couples (i.e., γ00 and γ01 in MLR notation) are obtained by estimating the mean structures of each (not estimated by default in SEM software packages). The variances of the slopes and intercepts across couples are indicated by the variances of η0 and η1 (i.e., the ψ matrix).

Results for the difference model. As above, parallel analyses were conducted using Mplus and HLM 5. With SEM or MLR, it is not possible to estimate variances (i.e., random effects) for the slopes and intercepts simultaneously with dyadic data.[iii] One can obtain variance estimates by running separate models fixing the random effect (variance) of the intercept, the slope, or the correlation between them to a fixed value (most typically, zero). Separate analyses suggested that the slope variance was not significantly different from zero, and, therefore, the variance of the slope (and the covariance between the slope and intercept) was fixed to zero.

Analyses are presented for dummy coding of the caregiving variable (0 and 1) and for group-mean centering (-.5 and +.5). When dummy coding is used, the intercept represents the average score for caregivers (because they were coded as 0). When group-centering is used, the average intercept represents the grand mean for all couples (caregivers and care recipients combined). In multilevel regression, group-mean centering is achieved by subtracting each individual’s score from the mean of the dyad (this can be done automatically in the HLM software). To obtain the group-mean centered solution using the SEM approach, however, one simply sets the loadings of the slope variable to -.5 and +.5, rather than 0 and 1. For the single indicator model with no level-2 predictors, the overall model fit cannot be evaluated because the model is just identified. However, slope and intercept estimates, their variance estimates, and their significance tests are available.

As can be seen in Table 3, the means for the intercept and slope variables and their standard errors are identical in the SEM method and the MLR method. The average intercept in the dummy coded example of 4.222 represents the average positive affect for caregivers, and the average intercept in the centered example of 4.189 represents the average positive affect for caregivers and recipients combined. Notice that the average intercept obtained with centering differs little from that obtained with dummy coding. This will not always be true, but, in this case, it is because there is very little difference between caregivers and care recipients on positive affect scores. The average slope of -.067, which is not significantly different from zero, represents the difference between caregivers and care recipients on positive affect.

Second-order factor model

The above approach may be useful if only one item or measure is available for each individual, but the single-indicator model assumes no measurement error. A second-order factor model approach, similar to the second-order growth curve approach described by McArdle (1988) referred to as the curve-of-factors-scores (CUFFS) model, is also possible. In this model, two parallel first-order latent variables for each member of the dyad, each defined by multiple indicators. To define the intercept and slope, fixed paths are set leading from two second-order factors (which, for continuity, I will continue to refer to as η0 and η1, respectively) to the two first order factors (referred to here as ηcg and ηcr). Both second-order loadings are set to 1 for the intercept factor, η0. As with the single-indicator model, a choice of uncentered, dummy coding (0,1) or group-mean centered coding (-.5,+.5) for the slope, η1, provides different interpretations for the intercept. With dummy coding, the intercept represents the mean of dyad members who are assigned 0 for the slope variable. With, centering, the intercept represents the average of the means of both dyad members. Figure 4 illustrates the model using dummy coding.

Several details of the specification are important. First, it is important to investigate the measurement model and ensure that the fit of the first-order factors is adequate before proceeding. Second, the intercepts for indicators of the first-order measurement equation, ν, are set to zero as is typical in most structural equation models This is a common specification in most traditional structural models (Bollen, 1987), but, in some software packages, they may be estimated by default because mean structures are requested. Third, loadings should be constrained to be equal for parallel items in the first order factors. This is an assumption of factorial invariance across members of the dyad and is similar to the latent growth curve assumption of longitudinal factorial invariance (Meredith & Tisak, 1982; Nesselroade, 1983) and can be tested using chi-square difference tests. If the constraints are not imposed and factor patterns are allowed to differ, the second-order factors may be confounded with dyad differences in measurement. Fourth, disturbances for the first order factors are set equal to one another. This provides a single estimate of the within-group variance. The intraclass correlation can be obtained using the estimate for the first order disturbances and the estimate of the variance of η0 in an intercept-only model and equation (1.17).[iv]

Fifth, the second-order factor model will be theoretically identified when three or more indicators are used. However, when the variance estimate for the intercept or slope is near zero in the population, estimation difficulties may arise due to empirical underidentification. When a population value is near zero, the sample estimate may be negative due to sampling error. This leads to difficulties with estimation in SEM, because the estimated covariance matrix cannot be inverted. In larger samples, however, there will be less risk of empirical underidentification problems, because the sample estimate will be nearer to the population estimate. Under these circumstances, the researcher may wish to set this variance to zero (or, rather, a very small positive value) in order to obtain an estimate provided there are no other difficulties with the model specification. Within the structural modeling literature, setting a variance or other parameter to a fixed value in order to obtain a solution is controversial, but setting random effects (i.e., variance estimates) to zero is fairly common practice within the MLR literature. One important reason for the difference in these standards involves the interpretation of the variance estimate. In SEM, it is unusual to expect zero variance of a latent variable in the population, and therefore a zero or negative variance typically indicates a problem with model specification. However, in the context of multilevel analysis, nonsignificant or near zero variance simply means there is little difference among groups in the level-1 intercepts or slopes. For example, it is not unreasonable to assume the difference in affect between caregivers and recipients is of approximately equal magnitude in all dyads.

Example

Measurement model fit. A single-factor measurement model of positive affect with 5 indicators fit the data well. The fit was significantly improved by the inclusion of a correlation between two measurement errors ("How much of the time has your daily life been full of things that were interesting to you?" and " How happy, satisfied, or pleased have you been with your personal life?") for both caregiver and care recipient factors. These two items are the only items concerning daily life satisfaction rather than hope for the future or a positive mood. The resulting two-factor model had a nonsignificant chi-square (χ2 (36, N=116) = 40.20, p=.29) and alternative fit indices (TLI=.990, RMSEA=.032) that indicated a good fit.

Intercept only model. To obtain an estimate of the intraclass correlation with the second-order model, a single, intercept-factor was defined, setting the loadings to 1. This model also fit the data well, although the chi-square was significant (χ2 (46, N=116) =67.03, p = .02, TLI=.960, RMSEA = .063). The mean of η0= 4.435, representing the average latent-variable affect score. The variance estimate obtained for η0 was .311, representing variance between dyads. The within-group variance estimate is obtained from the estimate of the disturbances of ηcg and ηcr (set equal) and was .478. Using these values, the estimate of the intraclass correlation is .394. Note that this value is somewhat larger than the estimate obtained with the single-indicator model (.33). The difference is due to measurement error attenuation in the single-indicator model.

Difference model. Next, a model estimating the slope with centered coding was tested. This model provides a test of whether, on average, there is a difference between caregivers' and care recipients' affect (i.e., mean of the slope variable, η0) and whether this difference varies significantly across couples (i.e., variance of the slope variable, η1). An initial test of the model in which both the intercept and the slope variance were estimated resulted in estimation difficulties. Sensitivity tests suggested that this was a result of minimal variance in the slope across dyads (as also suggested by the single-indicator model) indicating similar caregiver-recipient differences in affect. Consequently, the model was estimated setting the value of the slope variance to near zero (.0001) to identify the model. The model fit the data well (χ2 (44, N=116) =65.08, p = .02, TLI=.958, RMSEA = .064). The mean intercept value was significantly different from zero (α0=4.434, p ................
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