Inference on covariance matrices under rank restrictions

In reduced-rank regression, a matrix of expectations is modeled as a lower rank matrix. In factor analysis, a covariance matrix is modeled as a linear function of a diagonal matrix and a lower rank matrix. A common set of test procedures can be applied to both models when the following conditions are met: the rank of the lower rank matrix is small, the diagonal matrix in the factor model is an identity, and the reduced-rank regression model is not replicated. This paper gives three results for making inferences under these conditions: (1) It is shown that locally best invariant test of sphericity in the factor analysis model is identical to the locally best invariant test of rank-O against rank-r in the reduced-rank model, provided that r is small. (2) An extended table of null percentiles of the likelihood ratio test statistic for rank-1 alternatives in both models is computed. (3) Confidence intervals for parameters in a rank-1 alternative are given. The performance of the proposed methods is evaluated in a simulation study. The methods are illustrated on two-way fixed and mixed effects classifications.