Continuous Orthogonal Complement Functions and Distribution-Free Goodness of Fit Tests in Moment Structure Analysis

It is shown that for any full column rank matrix X <Subscript>0</Subscript> with more rows than columns there is a neighborhood <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathcal{N}$</EquationSource> </InlineEquation> of X <Subscript>0</Subscript> and a continuous function f on <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\mathcal{N}$</EquationSource> </InlineEquation> such that f(X) is an orthogonal complement of X for all X in <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\mathcal{N}$</EquationSource> </InlineEquation>. This is used to derive a distribution free goodness of fit test for covariance structure analysis. This test was proposed some time ago and is extensively used. Unfortunately, there is an error in the proof that the proposed test statistic has an asymptotic χ <Superscript>2</Superscript> distribution. This is a potentially serious problem, without a proof the test statistic may not, in fact, be asymptoticly χ <Superscript>2</Superscript>. The proof, however, is easily fixed using a continuous orthogonal complement function. Similar problems arise in other applications where orthogonal complements are used. These can also be resolved by using continuous orthogonal complement functions. Copyright The Psychometric Society 2013