The decision as to which analysis to employ should depend on these considerations that will probably lead to the most appropriate analysis, and will also enable the researcher to interpret correctly the results of the analysis. Unlike many previous researchers who strongly recommended one single (or very few) method(s), the point of view advocated in this chapter is that there is a very large number of potential methods, and that several considerations (some of which are to a certain extent "subjective") have to be taken into account for the final choice.
In the introduction to this chapter we stated a nonexistence theorem, namely that there is no way to partition the response variance uanmbiguously when unequal and disproportionate cell frequencies are involved. The fact that several methods can be employed for the same set of data and yet yield different solutions is a manifestation of this statement. In fact, two researchers may analyze the same set of data by using two different methods and consequently obtain different results. Such a state of affairs is likely to occur if the two experimenters differ in their evaluations on some of the considerations listed earlier, and the question who employed the "correct" method is in many instances unanswerable. Thus the purpose of the present chapter was not to suggest the "correct" method for dealing with unbalanced designs, but rather to present the reader with a set of possible methods and the relevant considerations to be taken into account in order to reduce the ambiguity that has developed around this topic.
It might be argued that the discrepancies between the different solutions, as dictated by different methods, are rather small and insignificant, and that for all practical purposes any method will do. For example, Chilag ( 1975) conducted a simulation study in which she compared the three different methods of Overall and Speigel ( 1960) and the unweighted means method ( Winer, 1971). Chilag ( 1975) found that: "For the populations and design studied, all four methods will have the same power and the same significance level even though the sums of squares for the effets are different" (p. 1). Whether this conclusion can be generalized to other populations and designs beyond those that were studied is questionable. In a more recent study, Milligan, Wong, and Thompson ( 1987) reported the results of a Monte Carlo simulation study from which they concluded that standard computational routines for unequal cell sizes are non-robust. They proposed that the lack of robustness can, at least partly, be accounted for by the unequal cell frequencies and thus warned that such studies should be interpreted with care. Given that unbalanced designs may frequently occur and often cannot be avoided, robustness properties should indeed be further studied.
To construct a coding system which will maintain the requirement of orthogonality for proportional n's, we start by defining dummy variables for a