Testable hypotheses in singular fixed linear models

Hypotheses that restrict nonestimable parameters in singular (or overparameterized) fixed linear models are considered nontes table by most aothors and are not allowed by most computer packages. In this article, a different approach is taken and hypotheses are classified as completely testable, partially testable, or nontestable on the basis of the number of degrees of freedom associated with them. The convenience of this approach is illustrated with examples and by developing a related general theory of equivalent hypotheses, reparameterizations, and restrictions. A method of transforming partially testable hypotheses into equivalent completely testable hypotheses is described.