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48 3 Treatment structureFumigated Single 5.483 5.280 5.818 5.371rep. 4 4 4 4Double 5.575 5.026 5.707 5.570rep. 4 4 4 4Standard errors of differences of meansTable Fumigant Fumigant Fumigant FumigantAmount Type AmountTyperep. unequal 16 unequal unequald.f. 36 36 36 36s.e.d. 0.2439 0.3450 min.rep0.1494 0.1725 0.2113 0.2727 max-min0.1725X 0.1725X max.rep(No comparisons in categories where s.e.d. marked with an X)Notice that, when tables of means have unequal replication, the general Analysis ofVariance menu provides three standard errors of difference for each table:• to compare a pair of means each with the minimum replication of those in the table,• to compare a mean with minimum replication with one with maximum replication,• and to compare a pair of means that both have the maximum replication.The "X" beside the standard errors of difference for maximum replication indicates thatthere is actually only one mean in the table with the maximum replication. So this is anunavailable comparison.3.6 CovariatesCovariates incorporate additional quantitative information into an analysis. Sometimesyou may have measurements made on the units before the experiment was carried out.This can be used to allocate the units to blocks but, even after this grouping, they maycontain additional useful information. Analysis of covariance incorporates quantitativeinformation of this sort into the analysis providing a further way of decreasingvariability.In the example in Section 3.5, nematode counts were done prior to the experiment aswell as afterwards. Analysis of covariance includes the (transformed) initial counts as alinear term in the model, rather like a regression analysis except that here we have thefactors for blocks and treatments as well.y = ì + â + f + ft + fl + ftl + b × (x x) + åijkl i j jk jl jkl ijkl ijklwhere y and x are the logarithms of the counts.ijklijkl