1. SJS SDI_111 Design of Statistical Investigations Stephen Senn 11 Nested Factors

2. SJS SDI_112 Crossed Factors So far the treatment and blocking factors we have considered have been crossed. In principle every level of one could be observed with every level of the other. –Every treatment in each block Or at least the same treatments in various blocks –Each level of a factor in combination with each of another

3. SJS SDI_113 Nested Factors Sometimes some factors can only appear within other factors Blocks with sub-blocks –Example: Patients within given group allocated a particular sequence Episodes of treatment within patients Treatments with sub-treatments Such factors are nested

4. SJS SDI_114 Exp_15 Nested Treatments Suppose that we wish to compare two beta- agonists in asthma, formoterol and salmeterol Formoterol has three formulations solution, single-dose dry-powder inhaler, multi- dose dry-powder inhaler Salmeterol has two suspension, multi-dose dry powder inhaler

5. SJS SDI_115 Exp_15 Treatment Structure

6. SJS SDI_116 Exp_15 Treatments From one point of view we have five treatments –defined by combination of molecule and formulation We may have a hierarchy of interest –primarily to compare molecules then to compare formulations within molecules –possibly delivery type within formulations

7. SJS SDI_117 Exp_15 Possible factors (levels) A: Treatments ( Formoterol, Salmeterol) B: Formoterol formulation (Solution, Powder) B*: Salmeterol formulation (Suspension, Powder) C: Formoterol powder device (Single, Multi) Note that B* is not really the same as B and each of the lower level factors only has meaning in the context of the higher level

8. SJS SDI_118 Wilkinson and Roger Notation We encountered this in connection with factorial designs Now we add an operator / for nested designs A/B = A + A:B Not that if B is a factor nested within A, it has no meaning on its own. Hence the main effect B does not exist on its own. NB In their original papers Applied Statistics,1973,22,392-399, W&R used instead of : as used in S-PLUS

9. SJS SDI_119 Exp_13 We encountered this example before We could regard this as an example of a nested design Treatments, placebo, ISF, MTA Doses within treatments

10. SJS SDI_1110 Exp_13 As nested design

11. SJS SDI_1111 Exp_13 Nested Analysis > #As before but treat as nested factors fit2 <- aov(AUC ~ Patient + Period + Active/Formul/Dose, na.action = na.exclude) > summary(fit2, corr = F) Df Sum of Sq Mean Sq F Value Pr(F) Patient 157 80.29301 0.511420 70.5027 0.0000000 Period 4 0.02092 0.005230 0.7210 0.5777861 Active 1 1.63959 1.639591 226.0286 0.0000000 Formul %in% Active 1 0.66308 0.663078 91.4097 0.0000000 Dose %in% (Active/Formul) 4 0.22666 0.056664 7.8115 0.0000038 Residuals 603 4.37411 0.007254

12. SJS SDI_1112 Random Treatment Effects We now pick up a theme we alluded to in lecture 10 Cases where our principle interest is in random effects –not random blocks –random treatments This example has nesting

13. SJS SDI_1113 Exp_16 Clarke and Kempson Example 13.1 Four labs, A,B,C,D. Six samples of uniform batch given to each. However a sample intended for A is sent to B by mistake

14. SJS SDI_1114 Fixed or Random? If we are interested in the performance of these four labs, we can consider them as fixed However we may be interested in using them to tell us how measurements vary in general from lab to lab If they are a sample of such labs, we could consider the effects as random

15. SJS SDI_1115 Exp_16 The Data Lab Sample Result 1 A 1 16.0 2 A 2 17.1 3 A 3 16.9 4 A 4 17.2 5 A 5 17.0 6 B 1 17.0 7 B 2 17.3 8 B 3 16.2 9 B 4 17.1 10 B 5 16.0 11 B 6 17.2 12 B 7 17.0 Lab Sample Result 13 C 1 16.9 14 C 2 16.1 15 C 3 16.4 16 C 4 16.1 17 C 5 16.6 18 C 6 16.3 19 D 1 15.0 20 D 2 15.9 21 D 3 16.0 22 D 4 15.9 23 D 5 16.2 24 D 6 15.9

16. SJS SDI_1116

17. SJS SDI_1117 Model

18. SJS SDI_1118 Sums of Squares & Expectations

19. SJS SDI_1119

20. SJS SDI_1120 ANOVA

21. SJS SDI_1121 Calculations Exp_16

22. SJS SDI_1122 ANOVA Exp_16

23. SJS SDI_1123 Exp_16 Components of Variance

24. SJS SDI_1124 Exp_16 S-PLUS Analysis > is.random(one.frame) <- T > varcomp.1 <- varcomp(Result ~ Lab, data = one.frame, method = "reml") > summary(varcomp.1) Call: varcomp(formula = Result ~ Lab, data = one.frame, method = "reml") Variance Estimates: Variance Lab 0.2000226 Residuals 0.1927181 Method: reml Approximate Covariance Matrix of Variance Estimates: Lab Residuals Lab 0.03612192 -0.00063555 Residuals -0.00063555 0.00379463

25. SJS SDI_1125 Exp_14 Revisited > #Variance components analysis Subject.ran <- data.frame(Subject) > is.random(Subject.ran) <- T > varcomp(lAUC ~ Subject + Formulation, data = Subject.ran) Variances: Subject Residuals 0.0766226 0.003424223 > varcomp(lAUC ~ Subject * Formulation, data = Subject.ran) Variances: Subject Subject:Formulation Residuals 0.07679968 -0.0005244036 0.003764744