Considered are balanced factorial models for one-way analysis of variance with continuous data, fixed effects, and at least four replications. Special definitions, with a median basis, are given for the constant term and for contrasts in the effects. For normality models and many other important kinds of models, however, the resulting parameters are the same as those occurring for these models. The only assumption necessarily made is that of independence among replications. Then, approximate tests, confidence intervals, and median estimates can be obtained for the constant term and for specified contrasts in effects. Exact tests, intervals, and estimates can be obtained for special cases which are themselves of a moderately general nature. Some of the exact results are of a sign test type while others can use methodology developed for investigating symmetry about zero. These models and methods are a generalization of median analysis of variance through use of concepts developed for the linearized nonlinear regression approach.