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Example text
J.. k. + y.... l − y.... )2 (y. l − y. j.. l + y.... k. l + y.... ) 2 i jkl + (y. jkl − y. jk. − y. kl + y. j.. k. l − y.... )2 . i jkl The further extension to four or more factors is conceptually simple though even more typographically painful! 1 The analysis of variance calculation for a design with factorial treatment structure is illustrated for data from an experiment on the water uptake of amphibia. Frogs and toads were kept in moist or dry conditions prior to the experiment. Half of the animals were injected with a mammalian water balance hormone.
The factorial experiment is said to have ‘hidden replication’ for the p1 − p0 comparison. If there are interactions between factors, then some of this hidden replication may disappear, and this advantage of factorial experiments is diminished. However, if there are interactions, then the other two forms of design, (a) and (b), are still inferior to the factorial, because they do not permit recognition that the size of the (p1 − p0 ) effect depends on the particular combination of factor Q and factor R levels, and hence results from (a) and (b) may not be reproducible if the levels of Q and R are changed.
1) can now be recognised as main effects and interactions if we define them as follows: d j = t j. − t.. k − t.. , (dc) jk = (t jk − t j. k − t.. k ) − (t j. − t.. ) = t jk − t j. k + t.. 12 (so that t.. 53. These numerical values confirm the general pattern of conclusions discussed earlier. 37. 4 The analysis of variance identity 33 weight for zero copper; the difference between 50 and 100 is small. 86. 4 The analysis of variance identity We return now to the formal representation of the models on which the analysis of data from designed experiments may be based.