Fixed, Mixed, and Random Models*

Abstract Some explicit questions are raised regarding the adequacy of assumed linear models as a basis for the interpretation of the analysis of variance of randomized experiments. A generally applicable method for the derivation of a linear statistical model, based on the experimental situation and the design of the experiment, is exemplified. The central features of the method are the notion of “experimental unit,” the concept of “true response,” and the use of randomization in the design. A model is derived for the case where two factors having A and B levels respectively, are to be examined with respect to a population of P experimental units, where selection of levels of the factors to be tested, selection of experimental units to be used, and the allocation of selected treatment combinations to units is at random. First a linear population model is given, based on the structure of the experimental situation, whose components are (unknown) parameters of the population of (conceptual) “true” responses. Then the conditions of the design are imposed to obtain a linear statistical model whose components involve the parameters of the population model and some defined random variables reflecting the experimental procedure and design. The derived statistical model is then used to obtain expected mean squares in the analyses of variance. A second illustration of the general methodology is given for a more complex example originated by Vaurio and Daniel and discussed statistically by Scheffé. Some differences from Scheffé's results are noted.