1. Hierarchical (nested) ANOVA

2. In some two-factor experiments the level of one factor, say B, is not “cross” or “cross classified” with the other factor, say A, but is “NESTED” with it. The levels of B are different for different levels of A. –For example: 2 Areas (Study vs Control) 4 sites per area, each with 5 replicates. There is no link from any sites on one area to any sites on another area. Hierarchical ANOVA

3. That is, there are 8 sites, not 2. Study Area (A) Control Area (B) S1(A) S2(A) S3(A) S4(A) S5(B) S6(B) S7(B) S8(B) X X X X X X X X X X X X X X X X X = replications Number of sites (S)/replications need not be equal with each sites. Analysis is carried out using a nested ANOVA not a two-way ANOVA. Hierarchical ANOVA

4. A Nested design is not the same as a two- way ANOVA which is represented by: A1 A2 A3 B1 X X X X X X X X X X X X X X X B2 X X X X X X X X X X X X X X X B3 X X X X X X X X X X X X X X X Nested, or hierarchical designs are very common in environmental effects monitoring studies. There are several “Study” and several “Control” Areas. Hierarchical ANOVA

5. Objectives The nested design allows us to test two things: (1) difference between “Study” and “Control” areas, and (2) the variability of the sites within areas. If we fail to find a significant variability among the sites within areas, then a significant difference between areas would suggest that there is an environmental impact. In other words, the variability is due to differences between areas and not to variability among the sites. Hierarchical ANOVA

6. In this kind of situation, however, it is highly likely that we will find variability among the sites. Even if it should be significant, however, we can still test to see whether the difference between the areas is significantly larger than the variability among the sites with areas. Hierarchical ANOVA

7. Statistical Model Y ijk =  +  i +  (i)j +  (ij)k i indexes “A” (often called the “major factor”) (i)j indexes “B” within “A” (B is often called the “minor factor”) (ij)k indexes replication i = 1, 2, …, M j = 1, 2, …, m k = 1, 2, …, n Hierarchical ANOVA

8. Model (continue) Hierarchical ANOVA

9. Model (continue) Or, TSS = SS A + SS (A)B + SSW error = Degrees of freedom: M.m.n -1 = (M-1) + M(m-1) + Mm(n-1) Hierarchical ANOVA

10. Example M=3, m=4, n=3; 3 Areas, 4 sites within each area, 3 replications per site, total of (M.m.n = 36) data points M 1 M 2 M 3 Areas 1 2 3 4 5 6 7 8 9 10 11 12 Sites 10 12 8 13 11 13 9 10 13 14 7 10 14 8 10 12 14 11 10 9 10 13 9 7 Repl. 9 10 12 11 8 9 8 8 16 12 5 4 11 10 10 12 11 11 9 9 13 13 7 7 10.75 10.0 10.0 10.25 Hierarchical ANOVA

11. Example (continue) SS A = 4 x 3 [(10.75-10.25) 2 + (10.0-10.25) 2 + (10.0-10.25) 2 ] = 12 (0.25 + 0.0625 + 0.625) = 4.5 SS (A)B = 3 [(11-10.75) 2 + (10-10.75) 2 + (10-10.75) 2 + (12-10.75) 2 + (11-10) 2 + (11-10) 2 + (9-10) 2 + (9-10) 2 + (13-10) 2 + (13-10) 2 + (7-10) 2 + (7-10) 2 ] = 3 (42.75) = 128.25 TSS = 240.75 SSW error = 108.0 Hierarchical ANOVA

12. ANOVA Table for Example Nested ANOVA: Observations versus Area, Sites Source DF SS( 平方和 ) MS( 方差 ) F P Area 2 4.50 2.25 0.158 0.856 Sites (A)B 9 128.25 14.25 3.167 0.012** Error 24 108.00 4.50 Total 35 240.75 What are the “proper” ratios? E(MS A ) =  2 + V (A)B + V A E(MS (A)B )=  2 + V (A)B E(MSW error ) =  2 = MS A /MS (A)B = MS (A)B /MSW error Hierarchical ANOVA

13. Summary Nested designs are very common in environmental monitoring It is a refinement of the one-way ANOVA All assumptions of ANOVA hold: normality of residuals, constant variance, etc. Can be easily computed using SAS, MINITAB, etc. Need to be careful about the proper ratio of the Mean squares. Always use graphical methods e.g. boxplots and normal plots as visual aids to aid analysis. Hierarchical ANOVA

14. Sample ： Hierarchical (nested) ANOVA 58.5 1 1 59.5 1 1 77.8 2 1 80.9 2 1 84.0 3 1 83.6 3 1 70.1 4 1 68.3 4 1 69.8 1 2 56.0 2 2 54.5 2 2 50.7 3 2 49.3 3 2 63.8 4 2 65.84 2 56.6 1 3 57.5 1 3 77.8 2 3 79.2 2 3 69.9 3 3 69.2 3 3 62.1 4 3 64.5 4 3 Length mosquito cage Hierarchical ANOVA

15. Length = β 0 + β cage ╳ cage + β mosquito(cage) ╳ mosquito (cage) + error df? Hierarchical ANOVA

16. data anova6; input length mosquito cage;cards; 58.5 1 1 59.5 1 1 77.8 2 1 80.9 2 1 84.0 3 1 83.6 3 1 70.1 4 1 68.3 4 1 69.8 1 2 56.0 2 2 54.5 2 2 50.7 3 2 49.3 3 2 63.8 4 2 65.84 2 56.6 1 3 57.5 1 3 77.8 2 3 79.2 2 3 69.9 3 3 69.2 3 3 62.1 4 3 64.5 4 3 ; proc glm data=anova6; class cage mosquito; model length = cage mosquito(cage); test h=cage e=mosquito(cage); output out = out1 r=res p=pred; proc print data=out1;var res pred; proc plot data = out1;plot res*pred; proc univariate data=out1 normal plot;var res; run; Hierarchical ANOVA

17. Class Levels Values mosquito 4 1 2 3 4 cage 3 1 2 3 Number of observations 24 Hierarchical ANOVA

18. Sum of Source DFSquares Mean Square F Value Pr > F Model 11 2386.353333 216.941212 166.66 <.0001 Error 12 15.620000 1.301667 Corrected Total 23 2401.973333 Source DF Type I SS Mean Square F Value Pr > F cage 2 665.675833 332.837917 255.70 <.0001 mosquito(cage) 9 1720.677500 191.186389 146.88 <.0001 Tests of Hypotheses Using the MS for mosquito(cage) as an Error Term Source DF Type I SS Mean Square F Value Pr > F cage 2 665.675833 332.837917 1.74 0.2295 Hierarchical ANOVA

19. res res1 Hierarchical ANOVA

20. pred res Hierarchical ANOVA

21. Tests for Normality Test --Statistic--- -----p Value------ Shapiro-Wilk W 0.978828 Pr < W 0.8733 Kolmogorov-Smirnov D 0.093842 Pr > D >0.1500 Cramer-von Mises W-Sq 0.038078 Pr > W-Sq >0.2500 Anderson-Darling A-Sq 0.22057 Pr > A-Sq >0.2500 Hierarchical ANOVA

22. Two way ANOVA vs. nested ANOVA Hierarchical ANOVA