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3.6 Covariates 49To do an analysis of covariance,you simply need to check theCovariates box in the Analysis ofVariance menu, and enter thecovariate in the box immediately tothe right, as shown in Figure 3.14. Ifyou have several covariates, you canenter them as a list (separated bycommas). You can even enter amodel formula: for example, youcould put Lnpriorcount.Blocks Figure 3.14to fit a different regressioncoefficient in each block.Clicking on Run in Figure 3.14 produces an analysis-of-variance table that containsextra lines to assess how much the final (log) counts depend on the initial counts, afterremoving the effects of treatments. The treatment effects (and s.s.) are also adjusted totake account of the fact that the plots with the various treatments had different numbersof nematodes before the experiment. This adjustment causes some loss of efficiency inthe treatment estimation. The remaining efficiency is measured by the covarianceefficiency factor, shown for each treatment term in the "cov. ef." column of theanalysis-of-variance table. The values are in the range zero to one. A value of zeroindicates that the treatment contrasts are completely correlated with the covariates: afterthe covariates have been fitted there is no information left about the treatments. A valueof one indicates that the covariates and the treatment term are orthogonal. Usually thevalues will be around 0.8 to 0.9. A low value should be taken as a warning: either themeasurements used as covariates have been affected by the treatments, which can occurwhen the measurements on covariates are taken after instead of before the experiment;or the random allocation of treatments has been unfortunate in that some treatments areon units with generally low values of the covariates while others are on generally highones.For a residual line in the analysis of variance, the value in the "cov. ef." columnmeasures how much the covariates have improved the precision of the experiment. Thisis calculated by dividing the residual mean square in the unadjusted analysis (whichexcludes the covariates) by its value in the adjusted analysis.To assess the full effect of the covariate on the estimation of a treatment term, youshould multiply its covariance efficiency factor by the covariance efficiency factor of theresidual with which it is to be compared. For Fumigant in the example, the calculationwould be 0.99 × 2.48. So fitting the covariate has improved the precision with whichFumigant is estimated.Analysis of variance (adjusted for covariate)Variate: LncountCovariate: LnpriorcountSource of variation d.f. s.s. m.s. v.r. cov.ef. F pr.Blocks stratum