lavaan.survey: Complex survey analysis of structural equation models (SEM)
In lavaan.survey: Complex Survey Structural Equation Modeling (SEM)

Description Usage Arguments Details Value Note Author(s) References See Also Examples

Takes a lavaan fit object and a complex survey design object as input and returns a structural equation modeling analysis based on the fit object, where the complex sampling design is taken into account.

The structural equation model parameter estimates are "aggregated" (Skinner, Holt & Smith 1989), i.e. they consistently estimate parameters aggregated over any clusters and strata and no explicit modeling of the effects of clusters and strata is involved. Standard errors are design-based. See Satorra and Muthen (1995) and references below for details on the procedure.

Both the pseudo-maximum likelihood (PML) procedure popular in the SEM world (e.g. Asparouhov 2005; Stapleton 2006) and weighted least squares procedures similar to aggregate regression modeling with complex sampling (e.g. Fuller 2009, chapter 6) are implemented.

It is possible to give a list of multiply imputed datasets to svydesign as data. lavaan.survey will then apply the standard Rubin (1987) formula to obtain point and variance estimates under multiple imputation. Some care is required with this procedure when survey weights are also involved, however (see Notes).

lavaan.survey(lavaan.fit, survey.design, 
	     estimator=c("MLM", "MLMV", "MLMVS", "WLS", "DWLS", "ML"),
	     estimator.gamma=c("default","Yuan-Bentler"))

`lavaan.fit`	A `lavaan` object resulting from a lavaan call. Since this is the estimator that will be used in the complex sample estimates, for comparability it can be convenient to use the same estimator in the call generating the lavaan fit object as in the `lavaan.survey` call. By default this is "MLM".
`survey.design`	An `svydesign` object resulting from a call to `svydesign` in the survey package. This allows for incorporation of clustering, stratification, unequal probability weights, finite population correction, and multiple imputation. See the survey documentation for more information.
`estimator`	The estimator used determines how parameter estimates are obtained, how standard errors are calculated, and how the test statistic and all measures derived from it are adjusted. See `lavaan`. The default estimator is MLM. It is recommended to use one of the ML estimators.
`estimator.gamma`	Whether to use the usual estimator of Gamma as given by `svyvar` (the variance-covariance matrix of the observed variances and covariances), or apply some kind of smoothing or adjustment. Currently the only other option is the Yuan-Bentler (1998) adjustment based on model residuals.

The user specifies a complex sampling design with the survey package's svydesign function, and a structural equation model with lavaan.

lavaan.survey follows these steps:

The covariance matrix of the observed variables (or matrices in the case of multiple group analysis) is estimated using the svyvar command from the survey package.
The asymptotic covariance matrix of the variances and covariances is obtained from the svyvar output (the "Gamma" matrix)
The last step depends on the estimation method chosen:
1. [MLM, MLMV, MLMVS] The lavaan model is re-fit using Maximum Likelihood with the covariance matrix as data. After normal-theory ML estimation, the standard errors (vcov matrix), likelihood ratio ("chi-square") statistic, and all derived fit indices and statistics are adjusted for the complex sampling design using the Gamma matrix. I.e. the Satorra-Bentler (SB) corrections are obtained ("MLM" estimation in lavaan terminology). This procedure is equivalent to "pseudo"-maximum likelihood (PML).
2. [WLS, DWLS] The lavaan model is re-fit using Weighted Least Squares with the covariance matrix as data, and the Moore-Penrose inverse of the Gamma matrix as estimation weights. If DWLS is chosen only the diagonal of the weight matrix is used.

An object of class lavaan, where the estimates, standard errors, vcov matrix, chi-square statistic, and fit measures based on the chi-square take into account the complex survey design. Several methods are available for lavaan objects, including a summary method.

1) Some care should be taken when applying multiple imputation with survey weights. The weights should be incorporated in the imputation, and even then the variance produced by the usual Rubin (1987) estimator may not be consistent (Kott 1995; Kim et al. 2006).

If multiple imputation is used to deal with unit nonresponse, calibration and/or propensity score weighting with jackknifing may be a more appropriate method. See the survey package.

2) Note that when using PML or WLS, the Gamma matrix need not be positive definite. Preliminary investigations suggest that it often is not. This may happen due to reduction of effective sample size from clustering, for instance. In itself this need not be a problem, depending on the restrictiveness of the model. In such cases lavaan.survey checks explicitly whether the covariance matrix of the parameter estimates is still positive definite and produces a warning otherwise.

3) Currently only structural equation models for continuous variables are implemented.

Daniel Oberski - http://daob.nl/ - daniel.oberski@gmail.com

Asparouhov T (2005). Sampling Weights in Latent Variable Modeling. Structural Equation Modeling, 12(3), 411-434.

Bollen, K, Tueller, S, Oberski, DL (2013). Issues in the Structural Equation Modeling of Complex Survey Data. In: Proceedings of the 59th World Statistics Congress 2013 (International Statistical Institute, ed.), Hong Kong. http://daob.nl/publications/

Fuller WA (2009). Sampling Statistics. John Wiley & Sons, New York.

Kim J, Brick J, Fuller WA, Kalton G (2006). On the Bias of the Multiple-Imputation Variance Estimator in Survey Sampling. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 68(3), 509-521.

Oberski, D.L. (2014). lavaan.survey: An R Package for Complex Survey Analysis of Structural Equation Models. Journal of Statistical Software, 57(1), 1-27. http://www.jstatsoft.org/v57/i01/.

Oberski, D. and Saris, W. (2012). A model-based procedure to evaluate the relative effects of different TSE components on structural equation model parameter estimates. Presentation given at the International Total Survey Error Workshop in Santpoort, the Netherlands. http://daob.nl/publications/

Satorra, A., & Bentler, P. M. (1994). Corrections to test statistics and standard errors in covariance structure analysis.

Satorra, A., and Muthen, B. (1995). Complex sample data in structural equation modeling. Sociological methodology, 25, 267-316.

Skinner C, Holt D, Smith T (1989). Analysis of Complex Surveys. John Wiley & Sons, New York.

Stapleton L (2006). An Assessment of Practical Solutions for Structural Equation Modeling with Complex Sample Data. Structural Equation Modeling, 13(1), 28-58.

Stapleton L (2008). Variance Estimation Using Replication Methods in Structural Equation Modeling with Complex Sample Data. Structural Equation Modeling, 15(2), 183-210.

Yuan K, Bentler P (1998). Normal Theory Based Test Statistics in Structural Equation Modelling. British Journal of Mathematical and Statistical Psychology, 51(2), 289-309.

pval.pFsum

cardinale ess.dk ess4.gb liss pisa.be.2003

svydesign lavaan

###### A single group example #######

# European Social Survey Denmark data (SRS)
data(ess.dk)

# A saturated model with reciprocal effects from Saris & Gallhofer
dk.model <- "
  socialTrust ~ 1 + systemTrust + fearCrime
  systemTrust ~ 1 + socialTrust + efficacy
  socialTrust ~~ systemTrust
"
lavaan.fit <- lavaan(dk.model, data=ess.dk, auto.var=TRUE, estimator="MLM")
summary(lavaan.fit)

# Create a survey design object with interviewer clustering
survey.design <- svydesign(ids=~intnum, prob=~1, data=ess.dk)

survey.fit <- lavaan.survey(lavaan.fit=lavaan.fit, survey.design=survey.design)
summary(survey.fit)



###### A multiple group example #######

data(HolzingerSwineford1939)

# The Holzinger and Swineford (1939) example - some model with complex restrictions
HS.model <- ' visual  =~ x1 + x2 + c(lam31, lam31)*x3
              textual =~ x4 + x5 + c(lam62, lam62)*x6
              speed   =~ x7 + x8 + c(lam93, lam93)*x9 
             speed ~ textual 
             textual ~ visual'

# Fit multiple group per school
fit <- lavaan(HS.model, data=HolzingerSwineford1939,
              int.ov.free=TRUE, meanstructure=TRUE,
              auto.var=TRUE, auto.fix.first=TRUE, group="school",
              auto.cov.lv.x=TRUE, estimator="MLM")
summary(fit, fit.measures=TRUE)

# Create fictional clusters in the HS data
set.seed(20121025)
HolzingerSwineford1939$clus <- sample(1:100, size=nrow(HolzingerSwineford1939), replace=TRUE)
survey.design <- svydesign(ids=~clus, prob=~1, data=HolzingerSwineford1939)

summary(fit.survey <- lavaan.survey(fit, survey.design))


# For more examples, please see the Journal of Statistical Software Paper, 
#  the accompanying datasets ?cardinale ?ess4.gb ?liss ?pisa.be.2003
#  and my homepage http://daob.nl/