Composites • INLAvaan

What is a composite?

A composite is a weighted sum of observed variables. In lavaan syntax, C <~ x1 + x2 + x3 defines $C_i = w_1 x_{i1} + w_2 x_{i2} + w_3 x_{i3} = \mathbf w^\top \mathbf x_i ,$ with the weights estimated and the first fixed at 1 to set the scale. Unlike a common factor, a composite has no disturbance term. That is, it summarises its indicators instead of explaining them.

A composite is not a defined parameter either. A := parameter such as ind := a*b is a function of the parameters—one value per posterior draw, with no effect on the likelihood. A composite has a value for every person, and it changes the model-implied covariance matrix $\boldsymbol\Sigma(\boldsymbol\theta)$ through its weights.

One outcome or several

With a single outcome, the composite model is a regression in disguise: C <~ x1 + x2 + x3; x4 ~ C fits exactly as x4 ~ x1 + x2 + x3 does, with slope $\gamma_1$ on $C$ and weights $w_j = \gamma_j / \gamma_1$ . Further outcomes see the indicators only through $C$ , so their regression coefficients on the indicators must be proportional. Each extra outcome adds $p - 1$ degrees of freedom for $p$ indicators, and the model below has two.

Not PLS-SEM

Partial least squares path modelling (PLS-SEM) is Wold’s iterative estimation algorithm, which has no likelihood. lavaan instead fits the composite model of Dijkstra (2017) by maximum likelihood, the same model that the Henseler–Ogasawara specification (Schuberth 2023) reaches through latent variables. INLAvaan puts priors on this likelihood, so the result is Bayesian composite SEM, not Bayesian PLS.

An example

We regress two verbal tests of the Holzinger and Swineford data (x4 and x5) on a composite of the three visual tests (x1 to x3).

model <- "
  C <~ x1 + x2 + x3
  x4 ~ C
  x5 ~ C
  x4 ~~ x5
"
utils::data("HolzingerSwineford1939", package = "lavaan")

fit <- asem(model, HolzingerSwineford1939)
#> ℹ Mode finding and Hessian computation.
#> ✔ Posterior mode and Hessian. [578ms]
#> 
#> ℹ Performing VB correction.
#> ✔ VB correction; mean |δ| = 0.150σ. [1.6s]
#> 
#> ⠙ Fitting 0/7 skew-normal marginals.
#> ⠹ Fitting 1/7 skew-normal marginals.
#> ✔ Fit 7/7 skew-normal marginals. [1s]
#> 
#> ⠙ Posterior sampling and summarising.
#> ⠹ Computing fit indices (PPP/DIC).
#> ✔ Summarise 1000 posterior draws. [3s]
#> 
#> ℹ Fit measures: PPP, DIC.
summary(fit)
#> INLAvaan 0.3.2.9003 ended normally after 29 iterations
#> 
#>   Estimator                                      BAYES
#>   Optimization method                           NLMINB
#>   Number of model parameters                         7
#> 
#>   Number of observations                           301
#> 
#> Model Test (User Model):
#> 
#>    Marginal log-likelihood                   -2232.252 
#>    PPP (Chi-square)                              0.449 
#> 
#> Information Criteria:
#> 
#>    Deviance (DIC)                             4419.930 
#>    Effective parameters (pD)                     6.972 
#> 
#> Parameter Estimates:
#> 
#>    Marginalisation method                     SKEWNORM
#>    VB correction                                  TRUE
#> 
#> Composites:
#>                    Estimate       SD     2.5%    97.5%     NMAD    Prior       
#>   C <~                                                                         
#>     x1                1.000                                                    
#>     x2                0.169    0.167   -0.132    0.523    0.007    normal(0,10)
#>     x3               -0.069    0.175   -0.387    0.300    0.006    normal(0,10)
#> 
#> Regressions:
#>                    Estimate       SD     2.5%    97.5%     NMAD    Prior       
#>   x4 ~                                                                         
#>     C                 0.351    0.059    0.236    0.466    0.009    normal(0,10)
#>   x5 ~                                                                         
#>     C                 0.313    0.066    0.186    0.444    0.008    normal(0,10)
#> 
#> Covariances:
#>                    Estimate       SD     2.5%    97.5%     NMAD    Prior       
#>  .x4 ~~                                                                        
#>    .x5                0.946    0.098    0.766    1.149    0.002       beta(1,1)
#>   x1 ~~                                                                        
#>     x2                0.407                                                    
#>     x3                0.580                                                    
#>   x2 ~~                                                                        
#>     x3                0.451                                                    
#> 
#> Variances:
#>                    Estimate       SD     2.5%    97.5%     NMAD    Prior       
#>     x1                1.358                                                    
#>     x2                1.382                                                    
#>     x3                1.275                                                    
#>    .x4                1.184    0.098    1.008    1.391    0.002 gamma(1,.5)[sd]
#>    .x5                1.535    0.127    1.307    1.804    0.003 gamma(1,.5)[sd]
#>     C                 1.521    0.262    1.141    2.136

The marker x1 is fixed at 1. The variance of C is not a free parameter but the sample variance of the scores $\mathbf w^\top \mathbf x_i$ , so its posterior comes from that of the weights, much as the Bayesian $R^2$ of Gelman et al. (2019) is computed draw by draw with the predictors held fixed. With the indicators’ covariances fixed at their sample values, it reflects uncertainty in the weights only. The posterior means sit close to lavaan’s maximum likelihood estimates.

fit_ml <- sem(model, HolzingerSwineford1939)
round(cbind(INLAvaan = coef(fit), lavaan = coef(fit_ml)), 3)
#>        INLAvaan lavaan
#> C<~x2     0.169  0.151
#> C<~x3    -0.069 -0.086
#> x4~C      0.351  0.367
#> x5~C      0.313  0.328
#> x4~~x5    0.946  0.928
#> x4~~x4    1.184  1.160
#> x5~~x5    1.535  1.508

Practical notes

Order the indicators. The other weights are ratios to the marker, so list first the indicator with the largest expected weight. A marker with a weight near zero leaves the rest poorly determined.
Higher-order means. A composite’s mean is fixed by the means of its indicators, so a latent variable measured only through composites needs its mean fixed at zero; INLAvaan stops with an error otherwise. For growth models on composites, use asem() with meanstructure = TRUE instead of agrowth().
Model comparison. As in lavaan, the covariances of each composite’s indicators are fixed at their sample values, so marginal likelihoods and the DIC compare only models with the same composites.
Limits. Composites are supported in single-level models with continuous data, in one or more groups. Two-level models, ordinal data, composites.cov = "free" and covariances between the indicators of a composite and other variables stop with an error.

References

Dijkstra, Theo K. 2017. “A Perfect Match Between a Model and a Mode.” In Partial Least Squares Path Modeling. Springer International Publishing. https://doi.org/10.1007/978-3-319-64069-3_4.

Gelman, Andrew, Ben Goodrich, Jonah Gabry, and Aki Vehtari. 2019. “R-Squared for Bayesian Regression Models.” The American Statistician 73 (3): 307–9. https://doi.org/10.1080/00031305.2018.1549100.

Schuberth, Florian. 2023. “The Henseler-Ogasawara Specification of Composites in Structural Equation Modeling: A Tutorial.” Psychological Methods 28 (4): 843–59. https://doi.org/10.1037/met0000432.