{smcl}
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help for {cmd:cfa1} {right:author: {browse "http://stas.kolenikov.name/":Stas Kolenikov}}
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{title:Confirmatory factor analysis with a single factor}

{p 8 27}
{cmd:cfa1}
{it:varlist}
[{cmd:if} {it:exp}] [{cmd:in} {it:range}]
[{cmd:aw|pw =} {it:weight}]
[{cmd:,}
     {cmd:unitvar}
     {cmd:free}
     {cmdab:pos:var}
     {cmdab:constr:aint(}{it:numlist}{cmd:)}
     {cmdab:lev:el(}{it:#}{cmd:)}
     {cmdab:rob:ust}
     {cmd:vce(robust|oim|opg|sbentler}{cmd:)}
     {cmd:cluster(}{it:varname}{cmd:)}
     {cmd:svy}
     {cmdab:sea:rch(}{it:searchspec}{cmd:)}
     {cmd:from(}{it:initspecs}{cmd:)}
     {it:ml options}
]

{title:Description}

{p}{cmd:cfa1} estimates simple confirmatory factor
analysis model with a single factor. In this model,
each of the variables is assumed to be an indicator
of an underlying unobserved factor with a linear
dependence between them:

{center:{it:y_i = m_i + l_i xi + delta_i}}

{p}where {it:y_i} is the {it:i}-th variable
in the {it:varlist}, {it:m_i} is its mean,
{it:l_i} is the latent variable loading,
{it:xi} is the latent variable/factor,
and {it:delta_i} is the measurement error.

{p}The model is estimated by the maximum likelihood
procedure.

{p}As with all latent variable models, a number
of identifying assumptions need to be made about
the latent variable {it:xi}. It is assumed
to have mean zero, and its scale is determined
by the first variable in the {it:varlist}
(i.e., l_1 is set to equal 1). Alternatively,
identification can be achieved by setting the
variance of the latent variable to 1 (with option
{it:unitvar}). More sophisticated identification
conditions can be achieved by specifying option
{it:free} and then providing the necessary
{it:constraint}.


{title:Options}

{ul:Identification:}

{p 0 4}{cmd:unitvar} specifies identification by setting
       the variance of the latent variable to 1.

{p 0 4}{cmd:free} requests to relax all identifying constraints.
       In this case, the user is responsible for provision
       of such constraints; otherwise, the estimation process
       won't converge.

{p 0 4}{cmdab:pos:var} specifies that if one or more of the
       measurement error variances were estimated to be
       negative (known as Heywood cases), the model
       needs to be automatically re-estimated by setting
       those variances to zero. The likelihood ratio test
       is then reported comparing the models with and without
       constraints. If there is only one offending estimate,
       the proper distribution to refer this likelihood
       ratio to is a mixture of chi-squares; see
       {help j_chibar:chi-bar test}. A conservative
       test is provided by a reference to the chi-square
       distribution with the largest degrees of freedom.
       The p-value is then overstated.

{p 0 4}{cmdab:constr:aint(}{it:numlist}{cmd:)} can be used
       to supply additional constraints. The degrees of freedom
       of the model may be wrong, then.

{p 0 4}{cmdab:lev:el(}{it:#}{cmd:)} -- see
       {help estimation_options##level():estimation options}

{ul:Standard error estimation:}

{p 0 4}{cmd:vce(oim|opg|robust|sbentler}
       specifies the way to estimate the standard errors.
       See {help vce_option}. {cmd:vce(sbentler)} is an
       additional Satorra-Bentler estimator popular in
       structural equation modeling literature that relaxes
       the assumption of multivariate normality while
       keeping the assumption of proper structural specification.

{p 0 4}{cmd:robust} is a synonum for {cmd:vce(robust)}.

{p 0 4}{cmd:cluster(}{it:varname}{cmd:)}

{p 0 4}{cmd:svy} instructs {cmd:cfa1} to respect the complex
       survey design, if one is specified.

{ul:Maximization options: see {help maximize}}

{title:Returned values}

{p}Beside the standard {help estcom:estimation results}, {cmd:cfa1}
also performs the overall goodness of fit test with results
saved in {cmd:e(lr_u)}, {cmd:e(df_u)} and {cmd:e(p_u)}
for the test statistic, its goodness of fit, and the resulting
p-value. A test vs. the model with the independent data
is provided with the {help ereturn} results with {cmd:indep}
suffix. Here, under the null hypothesis,
the covariance matrix is assumed diagonal.

{p}When {cmd:sbentler} is specified, Satorra-Bentler
standard errors are computed and posted as {cmd:e(V)},
with intermediate matrices saved in {cmd:e(SBU)},
{cmd:e(SBV)}, {cmd:e(SBGamma)} and {cmd:e(SBDelta)}.
Also, a number of corrected overall fit test statistics
is reported and saved: T-scaled ({cmd:ereturn} results
with {cmd:Tscaled} suffix) and T-adjusted
({cmd:ereturn} resuls with {cmd:Tadj} suffix;
also, {cmd:e(SBc)} and {cmd:e(SBd)} are the
scaling constants, with the latter also
being the approximate degrees of freedom
of the chi-square test)
from Satorra and Bentler (1994), and T-double
bar from Yuan and Bentler (1997)
(with {cmd:T2} suffix).


{title:References}

{p 0 4}{bind:}Satorra, A. and Bentler, P. M. (1994)
  Corrections to test statistics and standard errors in covariance structure analysis,
  in: {it:Latent variables analysis}, SAGE.

{p 0 4}{bind:}
    Yuan, K. H.,  and Bentler, P. M. (1997)
    Mean and Covariance Structure Analysis: Theoretical and Practical Improvements.
    {it:JASA}, {bf:92} (438), pp. 767--774.


{title:Also see}

{p 0 21}{bind:}Online:    help for {help factor}

{title:Contact}

Stas Kolenikov, kolenikovs {it:at} missouri.edu
