Computed theoretical power for N=100 and N=200 scenarios
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Modules/ado/plus/c/cfa1.hlp
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Modules/ado/plus/c/cfa1.hlp
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{smcl}
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{.-}
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help for {cmd:cfa1} {right:author: {browse "http://stas.kolenikov.name/":Stas Kolenikov}}
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{.-}
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{title:Confirmatory factor analysis with a single factor}
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{p 8 27}
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{cmd:cfa1}
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{it:varlist}
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[{cmd:if} {it:exp}] [{cmd:in} {it:range}]
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[{cmd:aw|pw =} {it:weight}]
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[{cmd:,}
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{cmd:unitvar}
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{cmd:free}
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{cmdab:pos:var}
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{cmdab:constr:aint(}{it:numlist}{cmd:)}
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{cmdab:lev:el(}{it:#}{cmd:)}
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{cmdab:rob:ust}
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{cmd:vce(robust|oim|opg|sbentler}{cmd:)}
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{cmd:cluster(}{it:varname}{cmd:)}
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{cmd:svy}
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{cmdab:sea:rch(}{it:searchspec}{cmd:)}
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{cmd:from(}{it:initspecs}{cmd:)}
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{it:ml options}
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]
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{title:Description}
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{p}{cmd:cfa1} estimates simple confirmatory factor
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analysis model with a single factor. In this model,
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each of the variables is assumed to be an indicator
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of an underlying unobserved factor with a linear
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dependence between them:
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{center:{it:y_i = m_i + l_i xi + delta_i}}
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{p}where {it:y_i} is the {it:i}-th variable
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in the {it:varlist}, {it:m_i} is its mean,
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{it:l_i} is the latent variable loading,
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{it:xi} is the latent variable/factor,
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and {it:delta_i} is the measurement error.
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{p}The model is estimated by the maximum likelihood
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procedure.
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{p}As with all latent variable models, a number
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of identifying assumptions need to be made about
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the latent variable {it:xi}. It is assumed
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to have mean zero, and its scale is determined
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by the first variable in the {it:varlist}
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(i.e., l_1 is set to equal 1). Alternatively,
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identification can be achieved by setting the
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variance of the latent variable to 1 (with option
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{it:unitvar}). More sophisticated identification
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conditions can be achieved by specifying option
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{it:free} and then providing the necessary
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{it:constraint}.
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{title:Options}
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{ul:Identification:}
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{p 0 4}{cmd:unitvar} specifies identification by setting
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the variance of the latent variable to 1.
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{p 0 4}{cmd:free} requests to relax all identifying constraints.
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In this case, the user is responsible for provision
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of such constraints; otherwise, the estimation process
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won't converge.
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{p 0 4}{cmdab:pos:var} specifies that if one or more of the
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measurement error variances were estimated to be
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negative (known as Heywood cases), the model
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needs to be automatically re-estimated by setting
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those variances to zero. The likelihood ratio test
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is then reported comparing the models with and without
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constraints. If there is only one offending estimate,
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the proper distribution to refer this likelihood
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ratio to is a mixture of chi-squares; see
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{help j_chibar:chi-bar test}. A conservative
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test is provided by a reference to the chi-square
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distribution with the largest degrees of freedom.
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The p-value is then overstated.
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{p 0 4}{cmdab:constr:aint(}{it:numlist}{cmd:)} can be used
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to supply additional constraints. The degrees of freedom
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of the model may be wrong, then.
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{p 0 4}{cmdab:lev:el(}{it:#}{cmd:)} -- see
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{help estimation_options##level():estimation options}
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{ul:Standard error estimation:}
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{p 0 4}{cmd:vce(oim|opg|robust|sbentler}
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specifies the way to estimate the standard errors.
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See {help vce_option}. {cmd:vce(sbentler)} is an
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additional Satorra-Bentler estimator popular in
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structural equation modeling literature that relaxes
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the assumption of multivariate normality while
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keeping the assumption of proper structural specification.
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{p 0 4}{cmd:robust} is a synonum for {cmd:vce(robust)}.
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{p 0 4}{cmd:cluster(}{it:varname}{cmd:)}
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{p 0 4}{cmd:svy} instructs {cmd:cfa1} to respect the complex
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survey design, if one is specified.
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{ul:Maximization options: see {help maximize}}
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{title:Returned values}
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{p}Beside the standard {help estcom:estimation results}, {cmd:cfa1}
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also performs the overall goodness of fit test with results
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saved in {cmd:e(lr_u)}, {cmd:e(df_u)} and {cmd:e(p_u)}
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for the test statistic, its goodness of fit, and the resulting
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p-value. A test vs. the model with the independent data
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is provided with the {help ereturn} results with {cmd:indep}
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suffix. Here, under the null hypothesis,
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the covariance matrix is assumed diagonal.
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{p}When {cmd:sbentler} is specified, Satorra-Bentler
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standard errors are computed and posted as {cmd:e(V)},
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with intermediate matrices saved in {cmd:e(SBU)},
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{cmd:e(SBV)}, {cmd:e(SBGamma)} and {cmd:e(SBDelta)}.
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Also, a number of corrected overall fit test statistics
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is reported and saved: T-scaled ({cmd:ereturn} results
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with {cmd:Tscaled} suffix) and T-adjusted
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({cmd:ereturn} resuls with {cmd:Tadj} suffix;
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also, {cmd:e(SBc)} and {cmd:e(SBd)} are the
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scaling constants, with the latter also
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being the approximate degrees of freedom
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of the chi-square test)
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from Satorra and Bentler (1994), and T-double
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bar from Yuan and Bentler (1997)
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(with {cmd:T2} suffix).
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{title:References}
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{p 0 4}{bind:}Satorra, A. and Bentler, P. M. (1994)
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Corrections to test statistics and standard errors in covariance structure analysis,
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in: {it:Latent variables analysis}, SAGE.
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{p 0 4}{bind:}
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Yuan, K. H., and Bentler, P. M. (1997)
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Mean and Covariance Structure Analysis: Theoretical and Practical Improvements.
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{it:JASA}, {bf:92} (438), pp. 767--774.
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{title:Also see}
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{p 0 21}{bind:}Online: help for {help factor}
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{title:Contact}
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Stas Kolenikov, kolenikovs {it:at} missouri.edu
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