Fit Structural Equation Models using the SAM approach

Fit a Structural Equation Model (SEM) using the Structural After Measurement (SAM) approach.

Usage

sam(model = NULL, data = NULL, aux = NULL, cmd = "sem", se = "twostep",
    mm_list = NULL, mm_args = list(bounds = "wide.zerovar"), 
    struc_args = list(estimator = "ML"), 
    sam_method = "local", ..., 
    local_options = list(M.method = "ML", lambda.correction = TRUE, 
                         alpha.correction = 0L, twolevel.method = "h1"), 
    global_options = list(), 
    bootstrap = list(R = 1000L, type = "ordinary", show.progress = FALSE),
    output = "lavaan",
    bootstrap_args = bootstrap
)

Arguments

model: A description of the user-specified model. Typically, the model is described using the lavaan model syntax. See model.syntax for more information. Alternatively, a parameter table (e.g., the output of the lavParTable() function) is also accepted.
data: A data frame containing the observed variables used in the model.
aux: Character vector. Names of auxiliary observed variables, forwarded to the underlying measurement and structural model fits. See the lavaan function for more details.
cmd: Character. Which command is used to run the sem models. The possible choices are "sem", "cfa" or "lavaan", determining how we deal with default options.
se: Character. The type of standard errors that are used in the final (structural) model. If "twostep" (the default), the standard errors take the estimation uncertainty of the first (measurement) stage into account. If "standard", this uncertainty is ignored, and we treat the measurement information as known. If "none", no standard errors are computed. The experimental "twostep.huber.white" variant computes the two-step correction as a stacked (Murphy-Topel type) sandwich with casewise-score based (first-order) meat – the two-step analogue of se = "robust.huber.white" in sem; it is currently available for continuous complete or missing = "ml" data (single- or multigroup, single level, no equality constraints), and falls back to "twostep.robust" or "twostep" otherwise. If the data are clustered (a cluster= argument is provided, without multilevel model syntax), se = "twostep" is automatically replaced by se = "twostep.robust", and the standard errors (also for se = "local" and se = "bootstrap", which resamples whole clusters) account for the clustering. For two-level models, se = "local" is available for the single-group continuous complete-data setting; it uses the cluster-sandwich covariance of the saturated (h1) estimates, and also enables the corrected structural test statistic. For estimator = "PML", se = "local" is the only available two-step correction (single group, complete data, all-ordinal indicators): it combines the casewise influences of the saturated thresholds/polychoric correlations with the casewise pairwise-likelihood influences of the measurement-block estimates, and also enables the corrected structural test statistic.
mm_list: List. Define the measurement blocks. Each element of the list should be either a single name of a latent variable, or a vector of latent variable names. If omitted, a separate measurement block is used for each latent variable.
mm_args: List. Optional arguments for the fitting function(s) of the measurement block(s) only. See lavOptions for a complete list.
struc_args: List. Optional arguments for the fitting function of the structural part only. See lavOptions for a complete list.
sam_method: Character. Can be set to "local", "global" or "fsr". In the latter case, the results are the same as if Bartlett factor scores were used, without any bias correction.
...: Many more options can be specified, using 'name = value'. See lavOptions for a complete list. These options affect both the measurement blocks and the structural part.
local_options: List. Options specific for local SAM method (these options may change over time). If lambda.correction = TRUE, we ensure that the variance matrix of the latent variables (VETA) is positive definite. The alpha.correction option must be an integer. Acceptable values are in the range 0 to N-1. If zero (the default), no small sample correction is performed, and the bias-correction is the same as with local SAM. When equal to N-1, the bias-correction is eliminated, and the results are the same as naive FSR. Typical values are 0, P+1 (where P is the number of predictors in the structural model), P+5, and (N-1)/2.
global_options: List. Options specific for global SAM method (not used for now).
bootstrap: List. Only used when se = "bootstrap". Typical elements of this list are R: the number of bootstrap samples, and type, which can be set to "ordinary" (the default) or "parametric". A single number is also accepted (as in sem()) and is interpreted as R, the number of bootstrap draws.
output: Character. If "lavaan", a lavaan object is returned. If "list", a list is returned with all the ingredients from the different stages.
bootstrap_args: List. Deprecated, use bootstrap instead.

Details

The sam function automates the SAM approach by first estimating the measurement part of the model and then the structural part of the model. See the reference for more details.

Note that in the current implementation, all indicators of latent variables must be observed. As a result, second-order factor structures are not yet supported.

If sam.method is "local", "fsr" or "cfsr", the structural part of the model is fitted to the estimated latent (co)variances (and means) from step 1. This step-2 structural model is stored in the @internal slot of the returned object, and can be retrieved by lavInspect(fit, "sam.struc.fit.object"). The functions fitMeasures (and summary(fit, fit.measures = TRUE)), lavResiduals (and summary(fit, residuals = TRUE)), modindices and lavTestLRT (when all compared models are local SAM models) operate on this structural model: the reported quantities describe the fit of the structural part only, conditional on the (fixed) measurement model of step 1. When available, the reported (scaled) test statistic and the robust fit measures are corrected for the estimation of the measurement model in step 1.

If sam.method is "global", the returned object is an ordinary (joint) lavaan object, and fitMeasures returns the usual fit measures of the joint model. Note, however, that the parameters were estimated using a two-step procedure; the behavior of fit measures in this setting has not been studied, and they should be interpreted with caution.

Exogenous covariates and conditional.x

Observed exogenous covariates may enter the structural part of the model. With continuous data, they are simply treated as (pseudo) latent variables (the default conditional.x = FALSE), but conditional.x = TRUE is supported as well (single-group, single-level models): the structural part is then fitted to the latent intercepts, the latent-on-covariate slopes, and the residual latent (co)variances, all conditional on the covariates.

With categorical (ordered) data, conditional.x = TRUE is the lavaan default (when exogenous covariates are present), and this is also the recommended setting: the covariates are conditioned on (as in a probit regression), rather than being absorbed into the joint polychoric correlation matrix – which would (wrongly) treat them as jointly normal, a real concern for binary/dummy covariates. In this setting, the measurement blocks of step 1 are estimated with conditional.x = TRUE as well (each block latent variable is regressed on all exogenous covariates, MIMIC style), so that the measurement parameters live on the same conditional latent-response scale (Var(y*|x) = 1, delta parameterization) as the joint model. An ordered endogenous variable in the structural part is supported too (with or without covariates): it enters the structural model on its latent-response scale, while its thresholds are fixed at their sample-based values.

When conditional.x = TRUE is combined with a local sam.method, the default se = "twostep" is automatically replaced by se = "twostep.robust": the classic two-step formula is built on the joint information matrix, which does not describe the local step-2 estimator in this setting (it underestimates the sampling variability, especially for binary covariates). The robust variant is computed as a structural-space sandwich here, and coincides with se = "local".

Not (yet) available in combination with conditional.x = TRUE: multiple groups (for the local standard errors and the corrected test statistic), multilevel models, latent interactions, and the Yuan-Chan scaled global test statistic of sam_method = "global" (the standard, unscaled test is reported instead, with a warning).

Value

If output = "lavaan", an object of class lavaan, for which several methods are available, including a summary method. If output = "list", a list.

References

Rosseel and Loh (2021). A structural-after-measurement approach to Structural Equation Modeling. Psychological Methods. Advance online publication. https://dx.doi.org/10.1037/met0000503

Examples

## The industrialization and Political Democracy Example 
## Bollen (1989), page 332
model <- ' 
  # latent variable definitions
     ind60 =~ x1 + x2 + x3
     dem60 =~ y1 + a*y2 + b*y3 + c*y4
     dem65 =~ y5 + a*y6 + b*y7 + c*y8

  # regressions
    dem60 ~ ind60
    dem65 ~ ind60 + dem60

  # residual correlations
    y1 ~~ y5
    y2 ~~ y4 + y6
    y3 ~~ y7
    y4 ~~ y8
    y6 ~~ y8
'

fit.sam <- sam(model, data = PoliticalDemocracy,
               mm.list = list(ind = "ind60", dem = c("dem60", "dem65")))
summary(fit.sam)
#> This is lavaan 0.7-1.3097 -- using the SAM approach to SEM
#> 
#>   SAM method                                     LOCAL
#>   Mapping matrix M method                           ML
#>   Number of measurement blocks                       2
#>   Estimator measurement part                        ML
#>   Estimator  structural part                        ML
#> 
#>   Number of observations                            75
#> 
#> Summary Information Measurement + Structural:
#> 
#>   Block      Latent Nind Chisq Df
#>       1       ind60    3  0.00  0
#>       2 dem60,dem65    8 15.32 16
#> 
#>   Model-based reliability latent variables:
#> 
#>   ind60 dem60 dem65
#>   0.966 0.868  0.87
#> 
#>   Summary Information Structural part:
#> 
#>   chisq df pvalue cfi rmsea srmr
#>       0  0     NA   1     0    0
#> 
#> Parameter Estimates:
#> 
#>   Standard errors                              Twostep
#>   Information                                 Expected
#>   Information saturated (h1) model          Structured
#> 
#> Regressions:
#>                    Estimate  Std.Err  z-value  P(>|z|)
#>   dem60 ~                                             
#>     ind60             1.454    0.389    3.741    0.000
#>   dem65 ~                                             
#>     ind60             0.558    0.225    2.480    0.013
#>     dem60             0.871    0.076   11.497    0.000
#> 
#> Variances:
#>                    Estimate  Std.Err  z-value  P(>|z|)
#>     ind60             0.446    0.087    5.135    0.000
#>    .dem60             3.766    0.848    4.439    0.000
#>    .dem65             0.189    0.224    0.843    0.399
#>