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The coherent framework in the back of stable perform; for operating statisticians, complex undergraduates, starting graduate students.

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T then it is reasonable to go on and test the null hypothesis H0 : τ1 = φ against the alternative hypothesis H1 : τ1 = φ. We obtain the sum of squares for this by pretending that all the new treatments are a single treatment: it is equal to (sumT =1 )2 ∑ti=2 (sumT =i )2 sum2 + − . 4. Let C be the ‘control’ factor on Ω: it is defined by C(ω) = 1 2 if T (ω) = 1 otherwise. Define the subspace VC of V to consist of all those vectors v for which vα = vβ whenever C(α) = C(β). Then dimVC = 2 and V0 ⊂ VC ⊂ VT .

In such circumstances, comparisons between new treatments are more informative then any comparison between a new treatment and the control. This suggests that the new treatments should have higher replication than the control. However, the person who wants to include the control treatment probably wants to compare all new treatments with it, rather than among themselves, so will want higher replication for the control. You may have to compromise on equal replication. 2 Comparing new treatments in the presence of a control Suppose that treatment 1 is a control treatment and that treatments 2, .

There are three reasons why I do so. In the first place, I think that the calculations are more transparent when the ‘sum of squares’ and ‘degrees of freedom’ columns add up to the genuine totals rather than to adjusted totals. In the second place, fitting V0 as a submodel of VT is a first taste of what we shall do many times with structured treatments: fit submodels and see what is left over. 4 shows the chain of vector subspaces {0} ⊆ V0 ⊆ VT according to the usual convention that a smaller space is shown below a larger space, with a vertical line to indicate that the smaller space is contained in the larger.

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