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2 Blocking structuresThe blocking structure of an experiment is used to describe the underlying structure ofthe "experimental units", which are the smallest items on which the experiment is done.For example, the experimental units might be the subjects in a medical experiment, theplots of a field experiment, or the individual plants in a glasshouse experiment.In this chapter you will learn• how to improve the precision of an experiment by grouping the units into similarsets called "blocks"• how randomization can avoid bias by guarding against unforeseen differencesamongst the units• how to design and analyse a complete randomized block design• how to recognise situations that may require more than one type of blocking• how to design and analyse a Latin square design Note: the topics marked are optional.
2.1 Completely randomized designs2.1 Completely randomized designs 23In the simplest case, no formal structure is imposed on the units and treatments are justallocated to units at random (we will look later at how this is done in practice). This iscalled a completely randomized design.One of the assumptions behind a completely randomized design is that the set of unitsto which the treatments are applied are effectively identical. For example:• in a field experiment, that there are no systematic differences in the underlyingfertility, drainage etc. of the plots;• in a glasshouse, it assumes that the light and temperature are the same for each rowof pots;• in a factory, that the workforce behaves in essentially the same way at differenttimes of day, days of the week and so on;• in educational studies, that children in different schools are approximately thesame, or students studying different subjects at Universities, or in different yeargroups etc.Many of the designs that people use in practice are of this type. However, as we shall see,we can often obtain substantial improvements in precision and efficiency by studying thestructure of the experimental units, and defining the block structure accordingly.2.2 Randomized block designsThe are some situations where it is obvious that the units are non uniform, For example,if a field experiment is laid out on a slope, plots at the top of the slope may be "better"than plots at the bottom. Several problems can then arise.1. The random allocation of treatments to plots may not seem "fair". For example, allthe replicates of treatment A may be allocated to "good" plots whilst all replicatesof treatment B might be allocated to "bad" plots. If there was no difference betweenA and B, this allocation of plots could lead to treatment A appearing to be muchbetter than treatment B.2. The differences between plots will increase the residual sum of squares, and hence2the estimate of the random variability (the variance ó ). This means that thetreatment differences must be larger to give a significant F-test and standard errorsof differences between treatments will be larger, i.e. the experiment will give lessprecise results.When you know that there are differences between units, you can avoid bias and improveprecision by grouping (or blocking) the units into homogenous groups i.e. groups of unitsthat are effectively identical. The simplest situation is the complete randomized-blockdesign. Here• there is a single grouping factor, usually known as blocks;• each block has the same number of units, usually one for each treatment;• within each block, the treatments are allocated randomly to the units.Consider the field experiment described above. Suppose this experiment is designed totest the effect of four treatments A, B, C and D on the yield of winter wheat. Theexperiment is laid out in three rows along the side of a hill.
Page 1 and 2: Anova and Design
Page 4 and 5: ContentsIntroduction 11 From t-test
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IndexA2DISPLAY procedure 117A2WAY p
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122 IndexTransformation 56logarithm
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