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Design of Statistical Investigations

Design of Statistical Investigations. 5. Orthogonal Designs Randomised Blocks. Stephen Senn. Blocks. So far we have ignored differences in experimental units Some subsets of units may be similar to each other but different from other subsets Such similar subsets are called blocks

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Design of Statistical Investigations

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  1. Design of Statistical Investigations 5. Orthogonal Designs Randomised Blocks Stephen Senn SJS SDI_5

  2. Blocks • So far we have ignored differences in experimental units • Some subsets of units may be similar to each other but different from other subsets • Such similar subsets are called blocks • The presence of blocks can be exploited • By design • And by analysis SJS SDI_5

  3. Randomised Block Design • We identify blocks of experimental material • We allocate treatments to block at random in such a way that • each treatment appears in every block • if a treatment appears m times in one block it appears m times in all blocks • but subject to no further restriction • Referred to as a randomised block design SJS SDI_5

  4. # to create randomized blocks • n.b<-6 # number of blocks • n.t<-3 # number of treatments • n.r<-2 # number of replicates • #create vector of treatments • treat<-c(rep(seq(1,n.t),n.r)) • #creat vector of blocks • block<-seq(1,n.b) • #create one permuted block • unit<-sample(treat) • #create other permuted blocks • #and join them • for(i in 1:(n.b-1)){ • unit<-rbind(unit,sample(treat)) • } • design.frame<-data.frame(block,unit) • design.frame #print design Note use of sample function SJS SDI_5

  5. Randomised BlocksSPlus Output • > design.frame • block unit.1 unit.2 unit.3 unit.4 unit.5 unit.6 • 1 1 1 3 3 2 1 2 • 2 2 1 2 3 3 2 1 • 3 3 2 3 3 2 1 1 • 4 4 3 2 1 3 2 1 • 5 5 2 3 2 1 1 3 • 6 6 1 3 3 2 1 2 SJS SDI_5

  6. Exp_5 Graff-Lonnevig and Browaldh (1990), Senn and Auclair (1990) Cross-over trial of single doses of 12 g formoterol compared with 200 g salbutamol in 13 asthmatic children. Main outcome measure peak expiratory flow (PEF) 8 hours after treatment. Two sequences used with wash-out in between. SJS SDI_5

  7. Design Points • Treatments are given in two periods • Washout is used to allow possible carry-over to disappear • Two sequences were used • Permits blinding • A voids associating particular treatment with particular period • We shall assume patients were randomised to the two sequences SJS SDI_5

  8. Exp_5The Data We shall ignore the sequence information for the moment. If we have assigned patients at random to the two possible sequences, this is a randomised blocks design SJS SDI_5

  9. Questions • What do we note about the precision of measurement? • What possible explanation is there? • What do we note about the patient numbers? • What possible explanation is there? SJS SDI_5

  10. Blocks in a Cross-over • In this design the units are episodes of treatment • As the graphs that follow will show, there is a correlation between results from the same patient • Patients form the blocks of the experiment • Naturally • And by design SJS SDI_5

  11. SJS SDI_5

  12. SJS SDI_5

  13. Points and Questions • The graph plots the salbutamol reading against the formoterol reading • Each point represents a patient • triangles salbutamol/formoterol sequence • squares formoterol/salbutamol sequence • All the points except one are to the right of the line of equality • What does this suggest? SJS SDI_5

  14. Blocking • From field trials in agriculture • A block was a set of plots of presumed similar fertility • Design trick was to use each treatment within a given block • Compare like with like • Eliminate a source of variation • Now used to describe any set of similar units use in design SJS SDI_5

  15. Blocks - Examples • Centres in a multi-centre trial • Units are patients • Cars in a fuel consumption experiment • Units are runs • Patients in a cross-over trial • Units are episodes of treatment • Fermentation tanks in a plant • Units are runs SJS SDI_5

  16. Model for Randomised Blocks Assume for simplicity every treatment appears once in each block Basic model Quantity to be minimised Normal equations obtain by differentiating sum of squares with respect to unknown parameters and setting equal to zero SJS SDI_5

  17. Some Notation Total of all observations Number of observations Total on treatment i Total in block j SJS SDI_5

  18. Solutions SJS SDI_5

  19. Identifiability • (5.1) Is over-parameterised • Not all effects are identifiable • However contrasts of the form below are uniquely identifiable These are of particular interest These are not SJS SDI_5

  20. Indentifiability continued • Predictions are also identifiable SJS SDI_5

  21. Exp_5: Means SJS SDI_5

  22. Exp_5: Means SJS SDI_5

  23. Predicted Value and ResidualPatient 7, Formoterol Note that the data are laid out in columns for treatments and rows for blocks ( patients) for convenience but that our notation suggested rows for treatments columns for blocks. Our subscripts reflect this latter convention. Note also that since patient 8 is missing we have a potential ambiguity regarding subscripts for patients 9 onwards. SJS SDI_5

  24. Exp_5Predicted Values SJS SDI_5

  25. Exp_5Residuals SJS SDI_5

  26. Sums of Squares Expanding we get…. SJS SDI_5

  27. Hence we get... SJS SDI_5

  28. ANOVA Identity SJS SDI_5

  29. ANOVA Table SJS SDI_5

  30. Computational Approaches Reminder In general SJS SDI_5

  31. Computational Approaches (cont) Thus to calculate the Total Sum of Squares we may proceed as follows SJS SDI_5

  32. Computational Approaches (cont) Sum of squares between blocks SJS SDI_5

  33. Computational Approaches (cont) Sum of squares between treatments SJS SDI_5

  34. Exp_5: Calculation 1 SJS SDI_5

  35. Exp_5: Calculation 2 SJS SDI_5

  36. Exp_5: Calculation 3 SJS SDI_5

  37. Exp_5 Analysis using Excel This uses the data analysis menu of Excel SJS SDI_5

  38. Exp_5ANOVA Analysis using SPlus • (Data input details omitted) • #ANOVA just fitting treat • fit1<-aov(pef~treat) • summary(fit1) • #ANOVA fitting treat and patient • fit2<-aov(pef~patient+treat) • summary(fit2) SJS SDI_5

  39. Exp_5SPlus Results • > summary(fit1) • Df Sum of Sq Mean Sq F Value Pr(F) • treat 1 13388.5 13388.46 2.56853 0.1220902 • Residuals 24 125100.0 5212.50 • > #ANOVA just fitting treat and patient • fit2 <- aov(pef ~ patient + treat) • > summary(fit2) • Df Sum of Sq Mean Sq F Value Pr(F) • patient 12 115213.5 9601.12 11.65357 0.000079348 • treat 1 13388.5 13388.46 16.25053 0.001665618 • Residuals 12 9886.5 823.88 SJS SDI_5

  40. Questions • Has the treatment sum of squares changed in fitting “patient”? • Are the degrees of freedom for treatment different? • What has changed? • Why has it changed? • What is the net effect? SJS SDI_5

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