Statistical methods, like those of Chambers, Duncombe-Yinger, and Reschovsky-Imazeki, appear to have a greater level of precision than the judgments of professional educators about what spending is necessary to achieve acceptable results. (Duncombe and Yinger, Chapter 8 in this volume, refer to professionals' judgments as "ad hoc, with no demonstrated connection between the environmental factors and educational costs" [p. 271].) Yet in reality, the precision implied by statistical modeling may be misleading because each of the definitions of data used in these equations, and rationales for their use, requires assumptions and judgments that are not necessarily more precise than those of professionals operating without statistical models. For example, the Duncombe and Yinger control for "efficiency" assumes that districts are more efficient to the extent they rely on local, rather than state funds, but this assumption requires greater examination than the authors can give it. As we discuss below, the percentage of students receiving free or reduced price lunches is a very poor proxy for at-risk children, yet most modelers must use this indicator because none other is available. Most models control for teacher experience in defining what a teacher "costs," yet few education researchers believe that there is a simple relationship between teacher quality and experience. (To the extent the relationship is not simply linear, then pay for experience may incorporate some inefficiency, not a cost of adequacy.) Similarly, Chamber's analysis shows that teacher salaries are higher where more teachers graduated from colleges whose entering freshmen had higher SAT scores, yet few professional educators would be willing to conclude that such teachers are necessarily higher quality teachers, and so the higher prices paid for them might be an inefficient expenditure rather than a district choice to purchase higher quality.

Or consider the assumptions implicit in the Duncombe and Yinger attempt to estimate adequacy by an "indirect" voter preference-tax price methodology. In effect (to paraphrase Justice Stewart's oft-repeated observation about pornography), the indirect method acknowledges that we can't define the elements of an adequate education, but insists that, nonetheless, voters know it when they see it. While this is an interesting hypothesis, and worthy of extensive empirical investigation, it is itself as imprecise as professional judgment. Even if tax-price behavior reflects something about the value of education, voters may value education based on incomplete or inaccurate information. Indeed, one of the dilemmas confronting advocates of school choice as a competitive means of improving school quality is that, when choice has actually been implemented, parents have chosen schools for a wide variety of reasons, quality not necessarily being paramount. Some parents choose schools from inadequate information or unequal access to it (Carnegie Foundation, 1992); some because they wish to associate with others from their racial or ethnic subgroup (Wells, 1993); some because they believe schools are of better quality if they enroll children from more affluent neighborhoods (Willms and Echols, 1993). Yet the "indirect" method of postulating outcomes must assume, if it is to be used as a basis for calculating true



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