1. 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

2. 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

3. Analysis of Variance - Calculations

4. Analysis of Variance Table & F-Test

5. 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

6. Contrasts and Sums of Squares

7. 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

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

9. 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

10. 3 rd and 5 th Order Polynomials