Download pdf guide - VSN International

More documents

Recommendations

Info

82 7 Balance and non-orthogonality7.1 Confounding and efficiency factorsIn the split-plot design it is the main effect of one of the treatment factors that isestimated in the higher stratum. Statistically, we would say that this main effect isconfounded with whole plots within blocks. For the factor Variety in Section 5.1, thisis completely acceptable; the main interest in the trial was to look at the Nitrogen factorand the interaction between Nitrogen and Variety. However, on other occasions, wemay want all the main effects to be estimated with the extra precision that should beavailable in the bottom stratum, and so we may want the interactions to be estimated inthe higher strata instead.n 0 0 k n 0 0 k 0 0 0 0 n k n kThe plan above shows a design in which the interaction between the factors N and K isconfounded with blocks. The definition of the N × K interaction is that it is the differencebetween the effect of N estimated at the different levels of K. Here we have factors at twolevels 0 and n for N, and 0 and k for K. For the 0 level of K, the effect of adding N is givenby the mean of the plots with the combination (n, 0) minus the mean of the plots with(0, 0); while for K at level k, it is given by the mean of the plots with (n, k) minus themean of the plots with (0, k). So the difference between the two estimates (which givesthe interaction contrast) is{ mean of plots with (n, 0) + mean of plots with (0, k) } { mean of plots with (0, 0) + mean of plots (n, k) }The left-hand block above contains only combinations (n, 0) and (0, k), while the righthandblock contains only combinations (0, 0) and (n, k). Consequently the differencebetween the means of the plots in the two blocks also estimates the interaction: that is,the N × K interaction is confounded with blocks.Usually, in a situation like this, youwould have more than two blocks. Infact, the two blocks above are part of adesign with eight blocks, each with fourplots, that was used to study factors N, Kand D (see Yates, 1937, Design andAnalysis of Factorial Experiments, page21; also John, 1972, Statistical Designand Analysis of Experiments, page 135).The left-hand block in the plan is block3 of the design, and the right-hand blockis block 4. If we analyse just those twoblocks with treatment model N*K, theanalysis of variance table belowconfirms that the interaction isestimated in the Blocks stratum (and,Figure 7.1as we have analysed only these two blocks, there are no degrees of freedom left over forthe residual).
7.1 Confounding and efficiency factors 83Analysis of varianceVariate: Yield of potatoes in tons/acreSource of variation d.f. s.s. m.s. v.r.Blocks stratumN.K 1 0.56 0.56Blocks.*Units* stratumN 1 0.48 0.48 0.04K 1 29.86 29.86 2.25Residual 4 53.17 13.29Total 7 84.060 0 0 n k 0 n 0 d 0 k d n 0 0 0 k 0 0 0 d n k dn 0 0 0 k 0 n 0 d 0 k d 0 0 0 0 0 d n k 0 n k dn 0 0 0 0 d n k 0 0 k d 0 0 0 0 k 0 n 0 d n k d0 k 0 0 0 d n k 0 n 0 d 0 0 0 n 0 0 0 k d n k dThe plan for the whole design, above, illustrates some further sophistication. It is set upso that N.K.D is confounded in blocks 1 and 2, N.K in blocks 3 and 4, N.D in blocks 5and 6, and K.D in blocks 7 and 8. Thus, for example, N.K is estimated between blocks3 and 4, and within blocks 1, 2, 5, 6, 7 and 8. So 6/8 of the information about N.K is inthe Blocks.Plots stratum, and 2/8 is in the Blocks.Plots stratum. The main effectsof N, K and D can be estimated in every block: they are orthogonal to blocks and all theirinformation is in the Blocks.Plots stratum.The amount of information available about a term in a particular stratum is known asits efficiency factor. The efficiency factors of non-orthogonal terms (i.e. those whoseefficiency is less than one) are listed in the Information Summary, which can be obtainedby checking the Information box in the ANOVA Options menu.The whole design can be analysed using the general Analysis of Variance menu, with theDesign drop-down list box set to General Analysis of Variance, the Block Structure set toBlocks/Plots, and the Treatment Structure set to N*K*D. The analysis is shown below.
Page 1 and 2:
Anova and Design
Page 4 and 5:
ContentsIntroduction 11 From t-test
Page 6 and 7:
1 From t-test to one-way anovaIn th
Page 8 and 9:
4 1 From t-test to one-way anovaIf
Page 10 and 11:
6 1 From t-test to one-way anovaWe
Page 12 and 13:
8 1 From t-test to one-way anova1.2
Page 14 and 15:
10 1 From t-test to one-way anovawi
Page 16 and 17:
12 1 From t-test to one-way anovaAl
Page 18 and 19:
14 1 From t-test to one-way anovaYo
Page 20 and 21:
16 1 From t-test to one-way anovaFi
Page 22 and 23:
18 1 From t-test to one-way anovaTa
Page 24 and 25:
20 1 From t-test to one-way anovaTh
Page 26 and 27:
2 Blocking structuresThe blocking s
Page 28 and 29:
24 2 Blocking structuresBlock 1D4.6
Page 30 and 31:
26 2 Blocking structuresNow we repe
Page 32 and 33:
28 2 Blocking structureswould give
Page 34 and 35:
30 2 Blocking structuresAnalysis of
Page 36 and 37: 3 Treatment structureSo far we have
Page 38 and 39: 34 3 Treatment structureTables of m
Page 40 and 41: 36 3 Treatment structureFigure 3.3
Page 42 and 43: 38 3 Treatment structureThere is a
Page 44 and 45: 40 3 Treatment structureblock.*Unit
Page 46 and 47: 42 3 Treatment structureTables of c
Page 48 and 49: 44 3 Treatment structureN * Swas us
Page 50 and 51: 46 3 Treatment structureFumigant Am
Page 52 and 53: 48 3 Treatment structureFumigated S
Page 54 and 55: 50 3 Treatment structureCovariate 1
Page 56 and 57: 4 Checking the assumptionsIn this c
Page 58 and 59: 54 4 Checking the assumptions4.2 No
Page 60 and 61: 56 4 Checking the assumptions4.5 Tr
Page 62 and 63: 58 4 Checking the assumptionsFigure
Page 64 and 65: 60 4 Checking the assumptionsThe pr
Page 66 and 67: 62 4 Checking the assumptionsTotal
Page 68 and 69: 64 4 Checking the assumptionsMessag
Page 70 and 71: 66 5 Designs with several error ter
Page 78 and 79: 74 6 Design and sample size6.1 Desi
Page 80 and 81: 76 6 Design and sample sizeClicking
Page 82 and 83: 78 6 Design and sample sizeFigure 6
Page 84 and 85: 80 6 Design and sample sizeFigure 6
Page 88 and 89: 84 7 Balance and non-orthogonalityA
Page 90 and 91: 86 7 Balance and non-orthogonalityT
Page 92 and 93: 88 7 Balance and non-orthogonalityF
Page 94 and 95: 90 7 Balance and non-orthogonalityf
Page 96 and 97: 92 7 Balance and non-orthogonality7
Page 98 and 99: 94 7 Balance and non-orthogonalityM
Page 100 and 101: 96 7 Balance and non-orthogonalityM
Page 102 and 103: 8 REML analysis of unbalanced desig
Page 104 and 105: 100 8 REML analysis of unbalanced d
Page 118 and 119: 9 Commands for analysis of variance
Page 120 and 121: 116 9 Commands for analysis of vari
Page 122 and 123: 118 9 Commands for analysis of vari
Page 124 and 125: IndexA2DISPLAY procedure 117A2WAY p
Page 126: 122 IndexTransformation 56logarithm
show all

Download pdf guide - VSN International

Create successful ePaper yourself

Delete template?

Save as template?