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Unformatted text preview: Orthogonal and Non-orthogonal Polynomial Constrasts We had carefully reviewed orthogonal polynomial contrasts in class and noted that Brian Yandell makes a compelling case for nonorthogonal polynomial contrasts. In the following example, we will revisit both methods and compare analyses. The Solution Concentration data set from Applied Linear Statistical Models, 5th ed by Kutner et al, measures concentration of a solution over time. Concentration (Y) Time in Hours (X) 0.07 9.0 0.09 9.0 0.08 9.0 0.16 7.0 0.17 7.0 0.21 7.0 0.49 5.0 0.58 5.0 0.53 5.0 1.22 3.0 1.15 3.0 1.07 3.0 2.84 1.0 2.57 1.0 3.10 1.0 A plot of the data set (Figure 1) in R shows a sharply nonlinear decreasing trend in concentration over time. Given such a trend, the natural log transformation of the response should fix both the nonlinearity and the increasing error variance. As you can see (Figure 2), the transformation works perfectly, which is likely what the textbook authors had in mind. It might be more interesting to apply polynomial models to the untransformed data, but given the violation of regression assumptions (unequal error variances), we will go ahead and model the transformed data instead. A quick inspection of the data set confirms that the independent variable, Time in Hours, is quantitative with equally-spaced levels (in increments of 2.0 hours). In addition, the design is balanced, with n = 3 replications per factor level, so this data set is appropriate for analysis with polynomial contrasts....
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- Spring '08
- Quadratic equation, orthogonal polynomial contrasts, linear term, polynomial contrasts, linear lack-of-ﬁt test, general ANOVA hypothesis