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

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

Analysis of Variance - Calculations

Analysis of Variance Table & F-Test

Dunnett’s Pairwise Comparisons with a Control 1-Sided Tests: H0i: mi-m1 = 0 HAi: mi-m1 > 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

Contrasts and Sums of Squares

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

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

Fitted Polynomial Regression Model To Obtain the kth order fitted Polynomial, we multiply the estimated “Contrasts” for P0,...,Pk by the corresponding Coefficients of the Contrasts for each Dose. Note that P0 is not a contrast, but a linear function of the means

3rd and 5th Order Polynomials