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# Parameters, Confounding, and Least Squares

DOI link for Parameters, Confounding, and Least Squares

Parameters, Confounding, and Least Squares book

# Parameters, Confounding, and Least Squares

DOI link for Parameters, Confounding, and Least Squares

Parameters, Confounding, and Least Squares book

## ABSTRACT

The subscript “nut* helps to distinguish nuisance variates from other types of variates. In particular, suppose that X nui, Ζχ, Z2, and Z3 are four

Y — β\Χ\ + p2%2 4-PnuiXnui θΖ\.

are both related to the value assumed by the nuisance variate X nui· (The reason why the yj 1 — p\ and y/l — p| terms are included in these equations will be considered in the next section in detail.) For the purposes of this section it suffices to note that Z2 designates the part of the key variate, X i, that is unrelated to the nuisance variate Xnui· Correspondingly, Z3 plays this same role for the key variate X2 , while variates Z2 and Z3 are independent, and hence unrelated, both to each other and, again thanks to assumed independence, to X nu%·

W ithout y jl — p\ and y/l — p|, even were it the case that some process discloses all four parameter values, β\,βν,βηή and σ , of the model Y =

The symbol X nui represents an observed, not an ideal, quantity. It is simplistic to assume that somehow the βηηίΧηηί term of model Y = β\Χ\ + β2Χ2 + βηηίΧηηί + σΖχ draws all nuisance contamination out of the quantities Χχ and X 2 . Omission of the βηηίΧηηί term within Y = β\Χ\ + /32X 2 + βηηίΧηηί + σΖχ can, of course, also cause difficulties. (W ithout this term, the only way that X nui is left to affect Y is through the variates Χχ and X 2.)

The mere inclusion of βηηίΧηηί does not guarantee that parameter es timation procedures can winnow the effects of X nu% from those parts of X i and X2 that are attributable to X nui · Unless, X nui , X i , and X2 are adjusted to form quantities like Ζχ, Z 2, and Z3, which for multivariate Nor mal variates are ideal in the sense of being independent (and in this sense pure), the model Y = β\Χ\ + #2X2 + finuiXnui + provides three in terconnected avenues of X nui influence: (1) via β\Χ\; (2) via #2X2 or; (3 ) through the more direct but not necessarily exclusive route via the term β η η ίΧ η η ί ·

if not more, subject to these imperfections, as are ANOVA representa tions. No amount of computing strategy or computer time can patch all flaws of an incautious ANOVA design. Similarly, confounding control via μΥ\χ = (jjQ 4-ω\Χ\ + cj2 £ 2 + j . . . , +ωυχν is a research pathway of last resort that should be regarded as a computational substitute that will hopefully yield the same results that could also be obtained by implementing the partial correlation procedures described in Chapter 12.