Class For Representing A (Fitted) Latent Variable Model

The lavaan class represents a (fitted) latent variable model. It contains a description of the model as specified by the user, a summary of the data, an internal matrix representation, and if the model was fitted, the fitting results.

Objects from the Class

Objects can be created via the cfa, sem, growth or lavaan functions.

Slots

version:: The lavaan package version used to create this object
call:: The function call as returned by match.call().
timing:: The elapsed time (user+system) for various parts of the program as a list, including the total time.
Options:: Named list of options that were provided by the user, or filled-in automatically.
ParTable:: Named list describing the model parameters. Can be coerced to a data.frame. In the documentation, this is called the `parameter table'.
pta:: Named list containing parameter table attributes.
Data:: Object of internal class "Data": information about the data.
SampleStats:: Object of internal class "SampleStats": sample statistics
Model:: Object of internal class "Model": the internal (matrix) representation of the model
Cache:: List of objects that are computed only once and reused many times.
Fit:: Object of internal class "Fit": the results of fitting the model. No longer used.
boot:: List. Results and information about the bootstrap.
optim:: List. Information about the optimization.
loglik:: List. Information about the loglikelihood of the model (if maximum likelihood was used).
implied:: List. Model implied statistics.
vcov:: List. Information about the variance matrix (vcov) of the model parameters.
test:: List. Different test statistics.
h1:: List. Information about the unrestricted h1 model (if available).
baseline:: List. Information about a baseline model (often the independence model) (if available).
internal:: List. For internal use only.
external:: List. Empty slot to be used by add-on packages.

Methods

coef: signature(object = "lavaan", type = "free"): Returns the estimates of the parameters in the model as a named numeric vector. If type="free", only the free parameters are returned. If type="user", all parameters listed in the parameter table are returned, including constrained and fixed parameters.
fitted.values: signature(object = "lavaan"): Returns the implied moments of the model as a list with two elements (per group): cov for the implied covariance matrix, and mean for the implied mean vector. If only the covariance matrix was analyzed, the implied mean vector will be zero.
fitted: signature(object = "lavaan"): an alias for fitted.values.
residuals: signature(object = "lavaan", type="raw"): If type = "raw", this function returns the raw (= unscaled) difference between the observed and the expected (model-implied) summary statistics. If type = "cor", or type = "cor.bollen", the observed and model implied covariance matrices are first transformed to a correlation matrix (using cov2cor()), before the residuals are computed. If type = "cor.bentler", both the observed and model implied covariance matrices are rescaled by dividing the elements by the square roots of the corresponding variances of the observed covariance matrix. If type="normalized", the residuals are divided by the square root of the asymptotic variance of the corresponding summary statistic (the variance estimate depends on the choice for the se argument). Unfortunately, these normalized residuals are not entirely correct, and this option is retained only for historical interest. If type="standardized", the residuals are divided by the square root of the asymptotic variance of these residuals. The resulting standardized residuals can be interpreted as z-scores. If type="standardized.mplus", the residuals are divided by the square root of the asymptotic variance of these residuals. However, a simplified formula is used (see the Mplus reference below) which often results in negative estimates for the variances, resulting in many NA values for the standardized residuals.
resid: signature(object = "lavaan"): an alias for residuals
vcov: signature(object = "lavaan", type = "free"): returns the covariance matrix of the estimated parameters. If type = "free" (the default), only the free parameters are included. If type = "user" (or "joint"), the joint covariance matrix of the free and the user-defined (:=) parameters is returned; the parameter names are the same as those used by coef().
predict: signature(object = "lavaan"): compute factor scores for all cases that are provided in the data frame. For complete data only.
anova: signature(object = "lavaan"): returns model comparison statistics. This method is just a wrapper around the function lavTestLRT. If only a single argument (a fitted model) is provided, this model is compared to the unrestricted model. If two or more arguments (fitted models) are provided, the models are compared in a sequential order. Test statistics are based on the likelihood ratio test. For more details and further options, see the lavTestLRT page.
update: signature(object = "lavaan", model, add, ..., evaluate = TRUE): update a fitted lavaan object and evaluate it (unless evaluate = FALSE). Note that we use the environment that is stored within the lavaan object, which is not necessarily the parent frame. The add argument is analogous to the one described in the lavTestScore page, and can be used to add parameters to the specified model rather than passing an entirely new model argument.
nobs: signature(object = "lavaan"): returns the effective number of observations used when fitting the model. In a multiple group analysis, this is the sum of all observations per group.
logLik: signature(object = "lavaan"): returns the log-likelihood of the fitted model, if maximum likelihood estimation was used. The AIC and BIC methods automatically work via logLik().
show: signature(object = "lavaan"): Print a short summary of the model fit
summary: signature(object = "lavaan", header = TRUE, fit_measures = FALSE, residuals = FALSE, estimates = TRUE, ci = FALSE, fmi = FALSE, standardized = FALSE, std = standardized, std_nox = FALSE, remove_system_eq = TRUE, remove_eq = TRUE, remove_ineq = TRUE, remove_def = FALSE, remove_nonfree = FALSE, remove_step1 = TRUE, remove_unused = TRUE, plabel = FALSE, cov_std = TRUE, rsquare = FALSE, baseline_model = NULL, h1_model = NULL, fm_args = list(standard.test = "default", scaled.test = "default", rmsea.ci.level = 0.90, rmsea.h0.closefit = 0.05, rmsea.h0.notclosefit = 0.08, robust = TRUE, cat.nonpd = "na"), modindices = FALSE, srmr_close_h0 = NULL, nd = 3L, cutoff = 0.3, dot_cutoff = 0.1): Print a nice summary of the model estimates. If header = TRUE, the header section (including fit measures) is printed. If fit.measures = TRUE, additional fit measures are added to the header section. fit.measures can also be a list, which allows one to set options related to the fit measures. See fitMeasures for more details. If residuals = TRUE, a residuals section is added (the largest residuals and the residual summary, printed as by lavResiduals with output = "text"); the underlying lavResiduals(object, output = "list") result is stored as the residuals element of the summary object. residuals can also be a list of arguments passed on to lavResiduals (for example residuals = list(type = "raw", n.largest = 10)). If both fit.measures = TRUE and residuals = TRUE and the residual summary is the SRMR (the default), the SRMR is removed from the fit measures section to avoid printing it twice. If estimates = TRUE, print the parameter estimates section. If ci = TRUE, add confidence intervals to the parameter estimates section. If fmi = TRUE, add the fmi (fraction of missing information) column, if it is available. If standardized = TRUE or a character vector, the standardized solution is also printed (see lavParameterEstimates). Note that SEs and tests are still based on unstandardized estimates. Use standardizedSolution to obtain SEs and test statistics for standardized estimates. The std.nox argument is deprecated; the standardized argument allows "std.nox" solution to be specifically requested. The standardized argument may also be a character vector of (observed) variable names (for example c("x1", "x2")); only the parameters involving these variables are then standardized, and the result is shown in a Std.usr column. This generalizes "std.nox", where the exogenous x variables are the ones left unstandardized. If remove_system_eq = TRUE, the system-generated equality constraints (using plabels) are not shown. If remove_eq = TRUE or remove_ineq = TRUE, the user-specified (in)equality constraints are not shown. If remove_def = TRUE, the user-specified parameter definitions are not shown. If remove_nonfree = TRUE, the nonfree parameters are not shown. If remove_step1 = TRUE, the parameters of the measurement part are not shown (only used when using sam().) If remove_unused = TRUE, automatically added parameters that are fixed to their default (0 or 1) values are removed. If rsquare = TRUE, the R-Square values for the dependent variables in the model are printed. The fm.args list allows one to set options related to the fit measures (see fitMeasures). baseline.model and h1.model allow one to specify user-defined baseline and/or h1 models for the fit measures. If the model is an exploratory factor analysis (EFA) model, EFA related information is printed automatically. The cutoff and dot.cutoff arguments control how the factor loadings are displayed: loadings whose absolute value is at or above cutoff are shown as numbers; loadings whose absolute value falls between dot.cutoff and cutoff are replaced by a dot; loadings whose absolute value is below dot.cutoff are left blank. If modindices = TRUE or modindices is a list, modification indices are printed for all fixed parameters. The argument nd determines the number of digits after the decimal point to be printed (currently only in the parameter estimates section.) Historically, nothing was returned, but since 0.6-12, a list is returned of class lavaan.summary for which a print function is available.

References

Yves Rosseel (2012). lavaan: An R Package for Structural Equation Modeling. Journal of Statistical Software, 48(2), 1-36. doi:10.18637/jss.v048.i02

Standardized Residuals in Mplus. Document retrieved from URL https://www.statmodel.com/download/StandardizedResiduals.pdf

Examples

HS.model <- ' visual  =~ x1 + x2 + x3
              textual =~ x4 + x5 + x6
              speed   =~ x7 + x8 + x9 '

fit <- cfa(HS.model, data = HolzingerSwineford1939)

summary(fit, standardized = TRUE, fit.measures = TRUE, rsquare = TRUE)
#> lavaan 0.7-1.3097 ended normally after 35 iterations
#> 
#>   Estimator                                         ML
#>   Optimization method                           NLMINB
#>   Number of model parameters                        21
#> 
#>   Number of observations                           301
#> 
#> Model Test User Model:
#>                                                       
#>   Test statistic                                85.306
#>   Degrees of freedom                                24
#>   P-value (Chi-square)                           0.000
#> 
#> Model Test Baseline Model:
#> 
#>   Test statistic                               918.852
#>   Degrees of freedom                                36
#>   P-value                                        0.000
#> 
#> User Model versus Baseline Model:
#> 
#>   Comparative Fit Index (CFI)                    0.931
#>   Tucker-Lewis Index (TLI)                       0.896
#> 
#> Loglikelihood and Information Criteria:
#> 
#>   Loglikelihood user model (H0)              -3737.745
#>   Loglikelihood unrestricted model (H1)      -3695.092
#>                                                       
#>   Akaike (AIC)                                7517.490
#>   Bayesian (BIC)                              7595.339
#>   Sample-size adjusted Bayesian (SABIC)       7528.739
#> 
#> Root Mean Square Error of Approximation:
#> 
#>   RMSEA                                          0.092
#>   90 Percent confidence interval - lower         0.071
#>   90 Percent confidence interval - upper         0.114
#>   P-value H_0: RMSEA <= 0.050                    0.001
#>   P-value H_0: RMSEA >= 0.080                    0.840
#> 
#> Standardized Root Mean Square Residual:
#> 
#>   SRMR                                           0.065
#> 
#> Goodness of Fit Index:
#> 
#>   Goodness of Fit Index (GFI)                    0.959
#>   90 Percent confidence interval - lower         0.939
#>   90 Percent confidence interval - upper         0.976
#> 
#> Parameter Estimates:
#> 
#>   Standard errors                             Standard
#>   Information                                 Expected
#>   Information saturated (h1) model          Structured
#> 
#> Latent Variables:
#>                    Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
#>   visual =~                                                             
#>     x1                1.000                               0.900    0.772
#>     x2                0.554    0.100    5.554    0.000    0.498    0.424
#>     x3                0.729    0.109    6.685    0.000    0.656    0.581
#>   textual =~                                                            
#>     x4                1.000                               0.990    0.852
#>     x5                1.113    0.065   17.014    0.000    1.102    0.855
#>     x6                0.926    0.055   16.703    0.000    0.917    0.838
#>   speed =~                                                              
#>     x7                1.000                               0.619    0.570
#>     x8                1.180    0.165    7.152    0.000    0.731    0.723
#>     x9                1.082    0.151    7.155    0.000    0.670    0.665
#> 
#> Covariances:
#>                    Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
#>   visual ~~                                                             
#>     textual           0.408    0.074    5.552    0.000    0.459    0.459
#>     speed             0.262    0.056    4.660    0.000    0.471    0.471
#>   textual ~~                                                            
#>     speed             0.173    0.049    3.518    0.000    0.283    0.283
#> 
#> Variances:
#>                    Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
#>    .x1                0.549    0.114    4.833    0.000    0.549    0.404
#>    .x2                1.134    0.102   11.146    0.000    1.134    0.821
#>    .x3                0.844    0.091    9.317    0.000    0.844    0.662
#>    .x4                0.371    0.048    7.779    0.000    0.371    0.275
#>    .x5                0.446    0.058    7.642    0.000    0.446    0.269
#>    .x6                0.356    0.043    8.277    0.000    0.356    0.298
#>    .x7                0.799    0.081    9.823    0.000    0.799    0.676
#>    .x8                0.488    0.074    6.573    0.000    0.488    0.477
#>    .x9                0.566    0.071    8.003    0.000    0.566    0.558
#>     visual            0.809    0.145    5.564    0.000    1.000    1.000
#>     textual           0.979    0.112    8.737    0.000    1.000    1.000
#>     speed             0.384    0.086    4.451    0.000    1.000    1.000
#> 
#> R-Square:
#>                    Estimate
#>     x1                0.596
#>     x2                0.179
#>     x3                0.338
#>     x4                0.725
#>     x5                0.731
#>     x6                0.702
#>     x7                0.324
#>     x8                0.523
#>     x9                0.442
#> 
fitted(fit)
#> $cov
#>       x1    x2    x3    x4    x5    x6    x7    x8    x9
#> x1 1.358                                                
#> x2 0.448 1.382                                          
#> x3 0.590 0.327 1.275                                    
#> x4 0.408 0.226 0.298 1.351                              
#> x5 0.454 0.252 0.331 1.090 1.660                        
#> x6 0.378 0.209 0.276 0.907 1.010 1.196                  
#> x7 0.262 0.145 0.191 0.173 0.193 0.161 1.183            
#> x8 0.309 0.171 0.226 0.205 0.228 0.190 0.453 1.022      
#> x9 0.284 0.157 0.207 0.188 0.209 0.174 0.415 0.490 1.015
#> 
coef(fit)
#>       visual=~x2       visual=~x3      textual=~x5      textual=~x6 
#>            0.554            0.729            1.113            0.926 
#>        speed=~x8        speed=~x9           x1~~x1           x2~~x2 
#>            1.180            1.082            0.549            1.134 
#>           x3~~x3           x4~~x4           x5~~x5           x6~~x6 
#>            0.844            0.371            0.446            0.356 
#>           x7~~x7           x8~~x8           x9~~x9   visual~~visual 
#>            0.799            0.488            0.566            0.809 
#> textual~~textual     speed~~speed  visual~~textual    visual~~speed 
#>            0.979            0.384            0.408            0.262 
#>   textual~~speed 
#>            0.173 
resid(fit, type = "normalized")
#> $type
#> [1] "normalized"
#> 
#> $cov
#>        x1     x2     x3     x4     x5     x6     x7     x8     x9
#> x1  0.000                                                        
#> x2 -0.493  0.000                                                 
#> x3 -0.125  1.539  0.000                                          
#> x4  1.159 -0.214 -1.170  0.000                                   
#> x5 -0.153 -0.459 -2.606  0.070  0.000                            
#> x6  0.983  0.507 -0.436 -0.130  0.048  0.000                     
#> x7 -2.423 -3.273 -1.450  0.625 -0.617 -0.240  0.000              
#> x8 -0.655 -0.896 -0.200 -1.162 -0.624 -0.375  1.170  0.000       
#> x9  2.405  1.249  2.420  0.808  1.126  0.958 -0.625 -0.504  0.000
#>