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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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So far... We have been estimating differences caused by application of various treatments, and determining the probability that an observed difference.

So far... We have been estimating differences caused by application of various treatments, and determining the probability that an observed difference.

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