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Design of Statistical Investigations

Design of Statistical Investigations. 8 Factorial Designs. Stephen Senn. Introduction. So far we have been looking at complications with blocking structure However, we now introduce complications in treatment structure We now look at factorial designs

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Design of Statistical Investigations

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  1. Design of Statistical Investigations 8 Factorial Designs Stephen Senn SJS SDI_8

  2. Introduction • So far we have been looking at complications with blocking structure • However, we now introduce complications in treatment structure • We now look at factorial designs • These are designs in which there are two or more “dimensions” to the treatments SJS SDI_8

  3. Exp_8 (From Clarke and Kempson) The yield of a chemical reaction is presumed to depend on two things A: The amount (low or high) in the mixture of a certain chemical B: The presence or absence of a catalyst An experiment is run to determine the importance of these in affecting yield. SJS SDI_8

  4. Treatments in Terms of Factors SJS SDI_8

  5. Terminology • A and B are factors • “low” and “high” are levels of the factor A • “absence” and “presence” are levels of the factor B • An experiment studying combinations of factors is called a factorial experiment • If all four combinations are studied, then this is a 2 x 2 or 22 factorial. SJS SDI_8

  6. Usual Notation for 22 Factorials • A and B are the factors • a and b are the higher levels • ab = the combination of both factors at higher level • a = A at higher level B at lower level • b = A at lower level B at higher level • (1) = both factors at lower level SJS SDI_8

  7. Main Effects and Interactions The Main Effect of a factor is the average response (averaged over all levels of the other factors) to a change in the level of that factor. Thus the main effect of A is the average of the difference between a and (1) and the difference between ab and b. The interaction between two factors A and B is the difference between the effect of A at the higher level of B (ab - b) and the difference at the lower level of B (a- (1)). Sometimes, by convention, this double difference is divided by 2. SJS SDI_8

  8. 22 FactorialsDefinition of Effects SJS SDI_8

  9. Exp_9(Clarke and Kempson) • Factor S: source of supply of a particular material • Two sources • s when first is used • Factor M: the speed of running a machine • Two speeds • m whenever higher is used • Experiment run on five days • Response: Average number of defectives per batch SJS SDI_8

  10. Exp_9(Clarke and Kempson) • The days determine the block structure of the experiment • The treatment structure is that of a 2  2 factorial • S  M • (1), s, m, sm SJS SDI_8

  11. Exp_9Data SJS SDI_8

  12. Exp_9Analysis SJS SDI_8

  13. Exp_9Analysis Continued SJS SDI_8

  14. Treatment Structure • The above analysis uses a one dimensional treatment structure • Single factor with four unordered levels • We wish, however, to distinguish between constituent factors • This can be done as follows SJS SDI_8

  15. Factorial Analysis SJS SDI_8

  16. ANOVA (Factorial) SJS SDI_8

  17. Exp_9SPlus #Input data Block<-factor(rep(c(seq(1:5)),4)) Supply<-factor(rep(c(1,2),each=10)) Machine<-factor(rep(rep(c(1,2),each=5),2)) #Create new factor treatment with 4 levels Treat<-ifelse((Supply==1 & Machine==1),1,0) Treat<-ifelse((Supply==1 & Machine==2),2,Treat) Treat<-ifelse((Supply==2 & Machine==1),3,Treat) Treat<-ifelse((Supply==2 & Machine==2),4,Treat) Treat<-factor(Treat) Y<-c(5.3,5.7,5.1,5.3,5.6,11.8,13,12.6,12.1,11.5, 20,19,20.3,19.5,20.2,26.7,24.1,25.7,26,25.5) SJS SDI_8

  18. Exp_9SPlus: Treatment as Factor with 4 Levels fit1 <- aov(Y ~ Block + Treat) > summary(fit1) Df Sum of Sq Mean Sq F Value Pr(F) Block 4 0.655 0.1637 0.3350 0.8491633 Treat 3 1165.750 388.5833 795.0554 0.0000000 Residuals 12 5.865 0.4888 SJS SDI_8

  19. Exp_9SPlus: Two equivalent statements using two factors with interactions > fit2 <- aov(Y ~ Block + Supply * Machine) summary(fit2) Df Sum of Sq Mean Sq F Value Pr(F) Block 4 0.655 0.1637 0.335 0.8491633 Supply 1 966.050 966.0500 1976.573 0.0000000 Machine 1 198.450 198.4500 406.036 0.0000000 Supply:Machine 1 1.250 1.2500 2.558 0.1357512 Residuals 12 5.865 0.4888 > fit3 <- aov(Y ~ Block + Supply + Machine + Supply:Machine) > summary(fit3) Df Sum of Sq Mean Sq F Value Pr(F) Block 4 0.655 0.1637 0.335 0.8491633 Supply 1 966.050 966.0500 1976.573 0.0000000 Machine 1 198.450 198.4500 406.036 0.0000000 Supply:Machine 1 1.250 1.2500 2.558 0.1357512 Residuals 12 5.865 0.4888 SJS SDI_8

  20. Exp_9SPlus: Two equivalent statements using two factors without interactions > fit4 <- aov(Y ~ Block + Supply + Machine) > summary(fit4) Df Sum of Sq Mean Sq F Value Pr(F) Block 4 0.655 0.1637 0.299 0.8732822 Supply 1 966.050 966.0500 1765.095 0.0000000 Machine 1 198.450 198.4500 362.593 0.0000000 Residuals 13 7.115 0.5473 > fit5 <- aov(Y ~ Block + Supply * Machine - Supply:Machine) > summary(fit5) Df Sum of Sq Mean Sq F Value Pr(F) Block 4 0.655 0.1637 0.299 0.8732822 Supply 1 966.050 966.0500 1765.095 0.0000000 Machine 1 198.450 198.4500 362.593 0.0000000 Residuals 13 7.115 0.5473 SJS SDI_8

  21. Wilkinson and Roger Notation This is a common notation A = main effect of factor A, B = main effect of factor B A:B = interaction of A and B, A:B:C = three factor interaction of A, B and C + sign used to add effects - used to subtract them A*B = A + B+ A:B = main effects of A and B and their interactions A*B*C = A + B + C +A:B + A:C + B:C + A:B:C NB In their original paper Applied Statistics,1973,22,392-399, W&R used  instead of : as used in SPlus SJS SDI_8

  22. Exp_9Design Notes • The two factors and their interaction are orthogonal • consequence of treatment combinations chosen • They are also orthogonal to the blocks • This is a consequence of how they were applied • Each combination in each day of the week • This increases efficiency • Effectively treatments are compared within blocks SJS SDI_8

  23. Exp_10(Senn Example 7.1) • Cross-over comparing two formulations at two doses • Solution and Suspension • 12mg and 24mg per puff • Four periods • Four sequences in a Latin Square used • 16 Patients allocated at random • 4 to each sequence SJS SDI_8

  24. Treatment Combinations Sequences Patients ABDC 3,5,12,13 BCAD 4,6,10,16 CDBA 2,8,9,14 DACB 1,7,11,15 SJS SDI_8

  25. Exp_10Design Notes • Two dimensional block structure • 16 Patients x 4 periods • Treatment structure factorial • Formulations x doses • Treatments allocated in way that is orthogonal to block structure • Latin square (“replicated” 4 times) • Actually the patient changes SJS SDI_8

  26. Exp_10Splus Data Entry #Input data patient<-factor(rep(c(3, 5, 12, 13, 4, 6, 10, 16, 2, 8, 9, 14, 1,7, 11, 15),4)) treat<-factor(rep(c("sus12","sus24","sol12","sol24"),each=16)) form<-factor(rep(c("sus","sol"),each=32)) dose<-factor(rep(c(12,24,12,24),each=16)) period<-factor(rep(c(1,3,4,2,2,1,3,4,4,2,1,3,3,4,2,1),each=4)) fev1<-c(2.7, 2.5, 2.6, 2, 3.7, 0.9, 2.5, 2, 1.3, 2.2, 1.8, 1.9, 1.7, 2.2, 3.3, 2.2, 1.7, 2.4, 2.5, 2.2,3.6, 1.4, 2.6, 2.5, 1.3, 2.2, 1.9, 2.2, 1.7, 1.9, 3.7, 2.3, 2.2, 2.4, 2.4, 2.6, 3.7, 2.4, 2.6,2.2, 1.4, 2.3, 1, 2.2, 1.6, 1.8, 3.6, 2.4, 2.6, 2.4, 2.5, 2.6, 3.6, 1.1, 2.4, 2.7, 1.3, 2.3,2.7, 2.1, 2, 2.6, 3.3, 2.5) SJS SDI_8

  27. Exp_10SPlus Analysis #fit treat as a factor fit1<-aov(fev1~patient+period+treat) summary(fit1) model.tables(fit1, type="effects", se=T, cterms="treat") #use the factorial approach with dose and form fit2<-aov(fev1~patient+period+form*dose) summary(fit2) model.tables(fit2, type="effects", se=T, cterms=c("form","dose","form:dose")) SJS SDI_8

  28. SPlusResults 1 summary(fit1) Df Sum of Sq Mean Sq F Value Pr(F) patient 15 22.27234 1.484823 14.46822 0.0000000 period 3 0.08547 0.028490 0.27760 0.8412298 treat 3 0.36172 0.120573 1.17487 0.3307357 Residuals 42 4.31031 0.102626 Tables of effects treat sol12 sol24 sus12 sus24 0.00156 0.12031 -0.07969 -0.04219 Standard errors of effects treat 0.080088 replic. 16.000000 SJS SDI_8

  29. SPlusResults 2 Df Sum of Sq Mean Sq F Value Pr(F) patient 15 22.27234 1.484823 14.46822 0.0000000 period 3 0.08547 0.028490 0.27760 0.8412298 form 1 0.23766 0.237656 2.31574 0.1355644 dose 1 0.09766 0.097656 0.95157 0.3349054 form:dose 1 0.02641 0.026406 0.25730 0.6146314 Residuals 42 4.31031 0.102626 SJS SDI_8

  30. form sol sus 0.060938 -0.060938 dose 12 24 -0.039063 0.039063 form:dose Dim 1 : form Dim 2 : dose 12 24 sol -0.020313 0.020313 sus 0.020313 -0.020313 Standard errors of effects form dose form:dose 0.056631 0.056631 0.080088 replic. 32.000000 32.000000 16.000000 SJS SDI_8

  31. Questions According to C&K in Exp_9 the response is mean faulty items per batch based on ten batches • To what extent do you think that the model for analysis is appropriate? • What sort of distribution might number of defectives have? • How else might one analyse the data • If one knew the batch sizes? • If one did not? • What further problems might there be? SJS SDI_8

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