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10 1 From t-test to one-way anovawith n 1 + n 2 2 degrees of freedom. This takes a little thought as it may appear as though^ ^we have fitted three parameters but, in fact, there are really just the two means m 1 and m 2.Our use of the treatment effects a 1 and a 2 makes it easy to move from one model to theother (by setting them both to zero) but you can easily see that^ ^ ^ì = ( m × n + m × n ) / ( n + n )1 1 2 2 1 2^ ^ ^ ^= {( ì + a 1 ) × n 1 + ( ì + a 2 ) × n 2 }/( n 1 + n 2 )and so^^a × n = a × n .1 1 2 2The difference between these two sums of squares is known as the sum of squares dueto the treatments. This measures the effect of allowing for two different means, and hasone degree of freedom. We can assess whether this exceeds the underlying level ofvariability by comparing it with RSS 1, but first we need to divide each one by its degreesof freedom to give the treatment and residual mean squares; this takes account of thedifferent number of parameters that each one represents. By dividing the treatment meansquare by the residual mean square, we obtain a statistic known as the variance ratio. Ifwe assume that the residuals follow a Normal distribution, the variance ratio will havean F distribution on 1 and (n 1 + n 2 2) degrees of freedom. (The degrees of freedom arethe degrees of freedom for the nominator that is the sum of squares due to treatments and those for the denominator that is the residual sum of squares.) The variance ratioisIt is interesting to note that, when there are only two treatments, the variance ratio is thesquare of the t-statistic. You can verify this in the example below, or prove it from thefollowing equations:and so
1.3 One-way analysis of variance 11The variance ratio, however, can be used if there are more than two treatments. Usually,the information is all laid out in an analysis of variance table. For our example this is:Analysis of varianceVariate: yieldSource of variation d.f. s.s. m.s. v.r. F pr.method 1 36.000 36.000 8.69 0.011Residual 14 58.000 4.143Total 15 94.000Mathematically, when there are t treatments, the one-way analysis of variance can becalculated as follows:Source Sums of squares DegreesoffreedomMean squareVarianceratio^ 2Treatments i n i a i =^ 2 ^ 2 i n i m i (i n i) ì^ ^ 2Residual i j (y ij ì a i)or asTotal SS Treat SS^ 2Total i j (y ij - ì)2 ^ 2= i j y ij - (i n i)ì^ 2t 1 ( i n i a i ) / (t 1) treatmentmeansquare/ residualmeansquare^ ^ 2 i n i t {i j (yij - ì - a i) }/ ( n t )^ 2 i n i 1 {i j (y ij - ì) }/ ( n 1 )iiii0Notice that the total sum of squares in the table is RSS . Usually there is no interest inassessing whether the observations have a non-zero overall mean, and so the tablecontains the total sum of squares "corrected for the grand mean". Also notice that twopossible formulae are given for the Treatment and Total sums of squares. The secondmay be more convenient to calculate, but the first will be much more accurate if theaccuracy of the representation is limited, as on computers or calculators.
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 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
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60 4 Checking the assumptionsThe pr
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62 4 Checking the assumptionsTotal
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64 4 Checking the assumptionsMessag
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66 5 Designs with several error ter
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74 6 Design and sample size6.1 Desi
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76 6 Design and sample sizeClicking
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78 6 Design and sample sizeFigure 6
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80 6 Design and sample sizeFigure 6
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82 7 Balance and non-orthogonality7
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84 7 Balance and non-orthogonalityA
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86 7 Balance and non-orthogonalityT
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88 7 Balance and non-orthogonalityF
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90 7 Balance and non-orthogonalityf
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92 7 Balance and non-orthogonality7
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94 7 Balance and non-orthogonalityM
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96 7 Balance and non-orthogonalityM
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8 REML analysis of unbalanced desig
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100 8 REML analysis of unbalanced d
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9 Commands for analysis of variance
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116 9 Commands for analysis of vari
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118 9 Commands for analysis of vari
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IndexA2DISPLAY procedure 117A2WAY p
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122 IndexTransformation 56logarithm
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