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Design of Statistical Investigations

Design of Statistical Investigations. 11 Nested Factors. Stephen Senn. 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

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Design of Statistical Investigations

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  1. Design of Statistical Investigations 11 Nested Factors Stephen Senn SJS SDI_11

  2. 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 SJS SDI_11

  3. 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” SJS SDI_11

  4. Exp_15Nested “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 SJS SDI_11

  5. Exp_15Treatment Structure SJS SDI_11

  6. Exp_15Treatments • 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 SJS SDI_11

  7. 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 SJS SDI_11

  8. 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 SJS SDI_11

  9. 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 SJS SDI_11

  10. Exp_13As nested design SJS SDI_11

  11. Exp_13Nested 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 SJS SDI_11

  12. 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 SJS SDI_11

  13. Exp_16Clarke 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 SJS SDI_11

  14. 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 SJS SDI_11

  15. Exp_16The 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 SJS SDI_11

  16. SJS SDI_11

  17. Model SJS SDI_11

  18. Sums of Squares & Expectations SJS SDI_11

  19. SJS SDI_11

  20. ANOVA SJS SDI_11

  21. Calculations Exp_16 SJS SDI_11

  22. ANOVA Exp_16 SJS SDI_11

  23. Exp_16Components of Variance SJS SDI_11

  24. Exp_16S-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 SJS SDI_11

  25. 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 SJS SDI_11

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