note that is rejected for large values of the testing procedure thus proceeds as follows compute the outcomet sayof the test statistic in ( evaluate the -value ( )with ( kn reject if this -value is too smallsay less than for nested models [ xi ] das in section the test statistic in theorem can now be used to test whether certain xi are needed or not in particularsoftware packages will report the outcomes of fi analysis of variance ky (iy ( - /pi ky (dk /( ( in the order under the null hypothesis that (iand ( - have the same expectation (that isadding xi to xi- has no additional effect on reducing the approximation error)the test statistic fi has an (pi pdistributionand the corresponding -values quantify the strength of the decision to include an additional variable in the model or not this procedure is called analysis of variance (anovanote that the output of an anova table depends on the order in which the variables are considered example (crop yield (cont )we continue examples and decompose the linear model as |{ |{zc |{ |{ { |{zb is the crop yield dependent on treatment levels as well as blockswe first test whether we can remove block as factor in the model against it playing significant role in explaining the crop yields specificallywe test versus using theorem now the vector ( is the projection of onto the ( )-dimensional space spanned by the columns of [ ]and ( is the projection of onto the ( )-dimensional space spanned by the columns of :[ the test statistict sayunder has an ( distribution the python code below calculates the outcome of the test statistic and the corresponding -value we find which gives -value - this shows that the block effects are extremely important for explaining the data using the extended model (including the block effects)we can test whether or notthat iswhether the treatments have significant effect on the crop yield in the presence of the block factor this is done in the last six lines of the code below the outcome of