794 Appendix A:Tables

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1 Appendix A Tables A Table Contents Page A.1 Random numbers 794 A.2 Orthogonal polynomial trend contrast coefficients 800 A.3 Standard normal distribution 801 A.4 Student s t-distribution 802 A.5 Chi-squared distribution 803 A.6 F-distribution 804 A.7 Power of the F-test 810 A.8 Studentized range distribution 814 (Tukey s method) A.9 Multivariate t-distribution maximum 816 (Dunnett s one-sided method) A.10 Multivariate t-distribution absolute maximum 818 (Dunnett s two-sided method) A.11 Critical coefficients for the Voss Wang method 820 Springer International Publishing AG 2017 A. Dean et al., Design and Analysis of Experiments, Springer Texts in Statistics, DOI /

2 794 Appendix A:Tables Table A.1 Random numbers (Section 1) Random numbers were generated using the SAS statements retain seed ; and rn = floor(10*ranuni(seed));

3 Appendix A:Tables 795 Table A.1 Random numbers (Section 2)

4 796 Appendix A:Tables Table A.1 Random numbers (Section 3)

5 Appendix A:Tables 797 Table A.1 Random numbers (Section 4)

6 798 Appendix A:Tables Table A.1 Random numbers (Section 5)

7 Appendix A:Tables 799 Table A.1 Random numbers (Section 6)

8 800 Appendix A:Tables Table A.2 Coefficients c i for orthogonal polynomial trend contrasts v = 3 Trend c 1 c 2 c 3 Linear Quadratic v = 4 Trend c 1 c 2 c 3 c 4 Linear Quadratic Cubic v = 5 Trend c 1 c 2 c 3 c 4 c 5 Linear Quadratic Cubic Quartic v = 6 Trend c 1 c 2 c 3 c 4 c 5 c 6 Linear Quadratic Cubic Quartic Quintic v = 7 Trend c 1 c 2 c 3 c 4 c 5 c 6 c 7 Linear Quadratic Cubic Quartic Quintic Sextic

9 Appendix A:Tables 801 Table A.3 Standard normal distribution: upper α critical coefficients, z α, and upper-tail probabilities, α = P(Z > z α ) α z α z α Values were generated using the SAS statements z_alpha = probit(1-alpha); and alpha = 1 - probnorm(z_alpha);

10 802 Appendix A:Tables Table A.4 Student s t-distribution: upper α critical coefficients, t df,α,whereα = P(t df > t df,α ) df α Values t df,α were generated using the SAS statements t = tinv(1-alpha,df); for df < and t = probit(1-alpha) for df =

11 Appendix A:Tables 803 Table A.5 Chi-squared distribution: upper α critical coefficients, χ 2 df,α,whereα = P(χ 2 df >χ2 df,α ) df α Values were generated using the SAS statement chi2 = cinv(1-alpha,df);

12 804 Appendix A:Tables Table A.6 F-distribution: upper α critical coefficients, F ν1,ν 2,α, whereα = P(F ν1,ν 2 > F ν1,ν 2,α) ν 2 α ν (continued)

13 Appendix A:Tables 805 Table A.6 (F-distribution, continued) ν 2 α ν (continued)

14 806 Appendix A:Tables Table A.6 (F-distribution, continued) ν 2 α ν (continued)

15 Appendix A:Tables 807 Table A.6 (F-distribution, continued) ν 2 α ν (continued)

16 808 Appendix A:Tables Table A.6 (F-distribution, continued) ν 2 α ν (continued)

17 Appendix A:Tables 809 Table A.6 (F-distribution, continued) ν 2 α ν Values F ν1,ν2,α were generated using the SAS statement f = round(finv(1-alpha,df1,df2),0.01);

18 810 Appendix A:Tables Table A.7 Power of the F-test: π(φ) = P(F ν1,ν 2,φ > F ν1,ν 2,α) ν 1 = 1, α = 0.05 φ ν ν 1 = 2, α = 0.05 φ ν ν 1 = 3, α = 0.05 φ ν (continued)

19 Appendix A:Tables 811 Table A.7 (Power, continued) ν 1 = 4, α = 0.05 φ ν ν 1 = 5, α = 0.05 φ ν ν 1 = 6, α = 0.05 φ ν (continued)

20 812 Appendix A:Tables Table A.7 (Power, continued) ν 1 = 7, α = 0.05 φ ν ν 1 = 8, α = 0.05 φ ν (continued)

21 Appendix A:Tables 813 Table A.7 (Power, continued) ν 1 = 9, α = 0.05 φ ν ν 1 = 10, α = 0.05 φ ν Power was computed using the SAS statements nc = v*phi**2;, Falpha = finv(1-alpha,nu1,nu2);, power = 1 - probf(falpha,nu1,nu2,nc);, and power = round(power,.01); for values of the parameters v, phi, alpha, nu1 = v-1 and nu2 Falpha is the upper-α critical value of the F-distribution with nu1 and nu2 degrees of freedom, and nc is the noncentrality parameter for the corresponding noncentral F-distribution

22 814 Appendix A:Tables Table A.8 Tukey s method: upper α critical values, q v,df,α, of the studentized range distribution df α v (continued)

23 Appendix A:Tables 815 Table A.8 (Tukey s method, continued) df α v Values q v,df,α were generated using the SAS statement qt = probmc( range,.,prob,df,v);, where prob = 1 α, and df=. for df =

24 816 Appendix A:Tables Table A.9 Dunnett s one-sided method: upper α critical coefficients w D1 = t (0.5) v 1,df,α df α v (continued)

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Table A.1 Random numbers (section 1)

Table A.1 Random numbers (section 1) A Tables Table Contents Page A.1 Random numbers 696 A.2 Orthogonal polynomial trend contrast coefficients 702 A.3 Standard normal distribution 703 A.4 Student s t-distribution 704 A.5 Chi-squared distribution