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1-Way ANOVA with Numeric Factor – Dose-Response Dose Response Studies in Laboratory Animals S.J. Ruberg (1995). “Dose Response Studies. II. Analysis and Interpretation,” Journal of Biopharmaceutical Statistics, 5(1), 15-42

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Data Description N=60 animals tested g=10 doses (0.0 to 4.5 by 0.5) n i = 6 animals per dose Data given as mean and standard deviation by dose

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Analysis of Variance - Calculations

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Analysis of Variance Table & F-Test

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Dunnett’s Pairwise Comparisons with a Control 1-Sided Tests: H 0i : i - 1 = 0 H Ai : i - 1 > 0 i=2,…,10 Overall Experiment-wise error rate = 0.05 Number of Comparisons = 9 Critical Value (50 Error DF, 9 Comparisons) = 2.49 Std Error of difference in pairs of means =SQRT(60.08(2/6))=SQRT(20.0267)=4.48 Minimum Significant Difference = 2.49(4.48) = 11.14

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Contrasts and Sums of Squares

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Orthogonal Polynomials Coefficients of Dose Means that describe the structure of means in polynomial form: Linear, Quadratic, Cubic,… (up to order g-1=9 for this example) Squared Coefficients Sum to 1 Products of Coefficients Sum to 0 for Different Polynomial Contrasts (Orthogonal) Note: P0 is not a contrast, but is used to get the intercept in regression

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Estimated Contrasts, Sums of Squares, ANOVA Based on the F-tests, we will consider the Orders 5 and 3 Polynomials

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Fitted Polynomial Regression Model To Obtain the k th order fitted Polynomial, we multiply the estimated “Contrasts” for P 0,...,P k by the corresponding Coefficients of the Contrasts for each Dose. Note that P 0 is not a contrast, but a linear function of the means

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3 rd and 5 th Order Polynomials

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Orthogonal Linear Contrasts A technique for partitioning ANOVA sum of squares into individual degrees of freedom.

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