Polynomials

In social-behavioral sciences, the term polynomial concerns the use of exponents or special coefficients to assess trends in linear models, with applications in ordinary least squares (OLS) regression, general linear model (GLM) analysis of variance (ANOVA), and other advanced multivariate applications such as hierarchical linear modeling, logistic regression, and generalized estimating equations. In more formal mathematics, polynomials are algebraic expressions that include exponents and have other specific properties. The simplest case of a polynomial is the OLS linear regression equation Y = A + bX1 (in OLS regression, X1 is typically represented as X). One of the earliest examples of polynomials to assess trends can be traced to Karl Pearson's 1905 work [Page 1049]addressing “skew correlation,” where he used an exponent approach to assess nonlinear association.

The following discussion addresses the use of powered polynomials in OLS regression and orthogonal polynomials derived from special coefficients in both OLS regression and GLM ANOVA (these approaches may be used in other multivariate analyses as well). Both nonorthogonal and orthogonal polynomials are also discussed.

Polynomials can take the form of powered coefficients in OLS linear equations used for trend analysis. The linear equation of illustrates the use of powered polynomials, with b1X as the linear term (not shown is X taken to the first power), and the squared term b2X2 producing a quadratic term. When powered polynomials are used for trend analysis, the resulting trends are nonorthogonal, meaning nonindependent. When the trends are created using powered polynomials, the quadratic and higher order terms are built from the linear term (for example, X is used to create X2), and thus are highly correlated. To make the terms orthogonal (i.e., independent), the data require mean-centering as noted by Patricia Cohen, Jacob Cohen, Stephen G. West, and Leona S. Aiken. Otherwise, hierarchical regression is used in assessing the trends to account for nonorthogonality.

Orthogonal polynomials are formed through special coefficients called coefficients of orthogonal polynomials, a term used by William Hays, and are provided in most advanced statistics textbooks. They are constructed to be independent of each other. In using orthogonal polynomials for trend analysis, the derived linear and quadratic terms are independent of each other when the group or condition sizes are equal. If group or condition sizes are unequal, the resulting polynomials will not be orthogonal.

The special coefficients for orthogonal polynomials may be used in both OLS regression and GLM ANOVA designs. In regression, the orthogonal polynomial coefficients are used directly as predictor variables for each trend, whereas in ANOVA, the coefficients are used to multiply the group or condition mean values in a fashion similar to those used to assess planned comparisons.

An illustration using orthogonal polynomials in OLS regression is offered below. Assume Y is a behavioral outcome, and X is an ordered grouping variable with three levels representing drug dosage (1 = 0 mg, 2 = 10 mg, 3 = 20 mg). Each group has an n of 30 (N = 90). With three dosage levels, a linear and quadratic trend may be fit. Two sets of orthogonal polynomial coefficients taken from David C. Howell are appliedtofitthese trends:–101 for the linear trend; and 1 −2 1 for the quadratic trend. In OLS regression, the resulting regression equation would be Y = A + bXlinear + bXquadratic. The linear trend Xlinear is formed by recoding the dosage variable using the orthogonal polynomial coefficients for the linear trend. Those receiving 0 mg of the drug are assigned “–1,” those receiving10 mg are assigned “0,” and those with 20 mg of the drug are assigned “1.” The same is done for the quadratic trend Xquadratic, with those receiving 0 mg assigned “1,” those receiving 10 mg assigned “–2,” and those receiving 20 mg assigned “1.” If this analysis was performed in GLM ANOVA, the same orthogonal polynomial coefficients would be used as planned comparisons to repartition the treatment variance into linear and quadratic components.

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