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1 From t-test to one-way anovaIn this chapter you will learn• how to use the t-test to compare two treatments• the mathematical equations that lie behind the t-test • how to calculate a t-test by hand • the T-Test menu• how to use one-way analysis of variance to compare several treatments• the model fitted in one-way anova• the mathematical equations that lie behind one-way anova • the statistical philosophy behind one-way anova• the relationship between one-way anova and the t-test for two treatments• how to use the One- and two-way ANOVA menu for one-way anova• how to plot the means from one-way anova• how to fit polynomial contrasts to quantitative treatments • how to do multiple-comparison tests Note: the topics marked are optional.
1.1 Comparing two treatments: the two-sample t-test 31.1 Comparing two treatments: the two-sample t-testSuppose we have two sets of units, each of which standard 23has received a different treatment. For example,they might be animals that have been fed two new 24different diets, or plots that have been given new 21different fertilisers, or subjects with differentdrugs, or plants with different fungicides, or standard 22widgets that have been formed by differentmanufacturing methods, and so on.new 22In this first section, we assume that the units do standard 19not have any special structure for example thatthe animals are all of the same breed, or that thestandard 21plots are in a fairly uniform field, or that the new 20subjects are of similar ages, weights and heights,and so on. So we have two sets of observationsnew 25(one for each treatment), and we want to know if standard 20they differ by more than random variation.The table shows data from an (unstructured)standard 17experiment to study yields from two different new 26manufacturing methods.We want to know whether the yields of the twostandard 18methods differ by more than we would expect new 24from the random variability in the experiment. Wewould also like to estimate the likely yields fromnew 22each method. Data like this are often analysed standard 20using a two-sample t-test.The assumption for the t-test is that each grouphas a Normal distribution. It is generally assumedExample 1.1that the distributions both have the same variance(this can be checked) and that they may have different means.We estimate the means by the averages of the observations with each treatment.standard: ( 23 + 22 + 19 + 21 + 20 + 17 + 18 + 20 ) / 8 = 20new: ( 24 + 21 + 22 + 20 + 25 + 26 + 24 + 22 ) / 8 = 23i.e. in mathematical notation, the mean of the distribution of the data {y ij : j = 1...n i } ingroup i is estimated byusually written as
Page 1 and 2: Anova and Design
Page 4 and 5: ContentsIntroduction 11 From t-test
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Page 26 and 27: 2 Blocking structuresThe blocking s
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Page 34 and 35: 30 2 Blocking structuresAnalysis of
Page 36 and 37: 3 Treatment structureSo far we have
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4 Checking the assumptionsIn this c
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54 4 Checking the assumptions4.2 No
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56 4 Checking the assumptions4.5 Tr
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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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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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