Course Multiple Regression Topic Non Linear Associations 1 REPRESENTING NON LINEAR ASSOCIATIONS Thus far we have examined how we can use the OLS regression to test linear associations between variables A linear relationship between X and Y for example implies that the direction of change in Y is constant across the values of X The following figure displays such a linear association in that the value of Y constantly increases across values of X However not all associations are linear in nature Some associations may be non linear such that the direction of the relationship between X and Y changes across values of X The following figures depict 3 possible patterns of non linear association The left most figure depicts an inverted quadratic pattern in which Y increase with increasing values of X until a point at which further increases in X are associated with decreases in Y The middle figure depicts a cubic pattern in which there are two changes of direction of association between X and Y across the values of X The right most figure depicts a quartic pattern in which there are three changes of direction in Y across values of X Because patterns of association need not be linear in nature theories may specify nonlinear associations The Yerkes Dodson Law for example suggests that productivity has an inverted quadratic association with physiological arousal such that increasing levels of arousal increases productivity to a point at which further increases in arousal are detrimental to productivity Such non linear associations can be tested with OLS regression HOW MANY NON LINEAR ASSOCIATIONS CAN BE TESTED Certainly non linear patterns are not limited to the quadratic cubic and quartic patterns displayed above The direction of association between X and Y may change numerous times across the values of X Consequently there are an infinite number of non linear patterns that can be tested The number of measured X values serves as the only limiting factor to the number of patterns that can be tested For example we need to have measured at least three values of X to test a quadratic pattern With only two values of X there is no way to test whether the association between X and Y changes directions across values of X In general if A represents the number of Course Multiple Regression Topic Non Linear Associations 2 distinct values of X that we have measured we can test A 1 changes in the direction of association between X and Y across values of X Keep in mind however that having sampled enough values of X does not imply that a non linear association will be detected even if there is a non linear association in the population If the sampled values of X do not span the range of X across which the association between X and Y changes direction such a change in direction cannot be detected For example imagine that in the population Y increases as X increases to a value of 6 and increases in X beyond 6 lead to decreases in Y If we sample values of X between 1 and 5 we will detect only a linear increase and the real quadratic pattern will be missed So an adequate test of a nonlinear pattern requires that X be sampled across the range of X at which non linear patterns occur Perhaps more important than the question how many patterns can be tested is the question how many patterns should be tested Just because six distinct values of X have been sampled does not mean that a quintic pattern i e 5 direction changes should be tested Theory should dictate what patterns should be tested If a theory dictates only linear patterns then higher order trends should best be left alone SPECIFYING NON LINEAR ASSOCIATIONS WITH A LINEAR MODEL Although we are interested in testing non linear patterns we will test the non linear patterns with a linear model The model is linear in the sense that the model parameters i e betas are linear For example the following model is linear in the parameters Y Bo B1 X 1 B2 X 2 It is also possible however to test non linear associations with non linear regression The following model for example is a non linear model Y Bo B12 X 1 B 32 X 2 In this class we will focus only on the linear model The trick to testing non linear patterns with a linear model is to transform the X values That is parameters remain linear and we incorporate non linearity by transforming the measured variables There are two approaches to transforming the X variable to test for non linear trends orthogonal polynomial contrasts and polynomial powers Orthogonal Polynomial Contrasts We previously discussed orthogonal polynomial contrasts in the context of ANOVA so we won t spend much time with this approach To use this approach we transform the X variable by treating it as a nominal variable and weighting each level with the appropriate set of polynomial contrasts For example if we measured 4 values of X i e 1 2 3 4 we would treat X as 4 level nominal variable This 4 level variable would require 3 predictors i e g 1 predictors to represent a G level nominal variable in a regression equation To test the linear quadratic and cubic trends we would create 3 contrast coded predictors that are weighted with the contrast coefficients that reflect the linear quadratic and cubic trends Keep in mind that those contrasts coefficients are orthogonal and the corresponding predictors will be orthogonal when sample sizes are equal When sample sizes are not equal the contrast coded predictors will Course Multiple Regression Topic Non Linear Associations 3 not be orthogonal and the unique effects of each polynomial contrast will be revealed only when all of the predictors are fully partialled Power Polynomials The other procedure for testing non linear patterns with a linear model is to transform the X variable using polynomial powers e g X2 X3 X4 Each power represents the number of curves in the regression line or the number of times Y changes direction across X For example the following regression equation examines the linear and quadratic effects of X on Y Y Bo B1 X 1 B2 X 12 It is important to note that B2 reflects the quadratic effect of X i e X2 only when the linear effect of X is partialled Likewise if we include the cubic effect of X3 the corresponding regression parameter reflects the cubic effect only when the quadratic and linear effects are partialled So the partialling process is essential in order for the regression parameters corresponding to polynomial powers to have unique interpretations A BOGUS DATA SET TO EXPLORE
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