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ANOVA for blocked designsPrinciples Randomized complete block ANOVA Formulae Pooling Very small Fratios Latin square ANOVA Formulae AssumptionsPrinciplesWe introduced the principles of blocked designs in The commonest design, known as the randomized complete block design (RCBD), is to have one unit assigned to each treatment level per block. Providing block is a truly random factor  and there really is no interest in comparing blocks  this can be the most efficient design. The alternative is to have several replicates of each treatment per block (sometimes termed a generalized randomized block design). The advantage of having replicated treatments in each block is that any interaction between blocks and treatments can be evaluated (see below), and is strongly recommended if the blocks represent a clear environmental gradient (for example soil moisture content). However, precision usually decreases as the number of experiment units (or size of units) per block increases. We deal with analysis of the generalized randomized block design in the More Information page on Factorial ANOVA If there are two blocking factors, then the Latin square design may be appropriate. However, they are much less used than randomized block designs and make additional (sometimes highly questionable) assumptions. If there are a very large number of treatment levels (often the case in agricultural variety trials), it may not even be possible to have every treatment level within each block. Instead a carefully selected set of treatment levels are put in each block giving an randomized incomplete block design. Such designs are not recommended unless unavoidable. Randomized complete block ANOVAModel & expected mean squaresWe will assume a mixed model for the randomized block design  with the treatment
effect fixed and the block effect random  but see the discussion on this issue in the core
text
Examination of the expected mean squares shows that we can obtain an unbiased test
of the treatment effect using the residual mean square as the denominator in the F
ratio. The Fratio for the treatment effect is therefore obtained by dividing MS_{A}
by MS_{Res}. The Pvalue for this Fratio is obtained for
There is no unbiased test of the block effect unless we assume there is no restriction error and no treatment × block interaction. If we make those assumptions, an approximate Fratio for the block effect is obtained by dividing MS_{S} by MS_{Res}. The Pvalue for this Fratio is obtained for s− 1 and Great care must be taken when analyzing randomized block designs with statistical packages. The widely used general linear model cannot accommodate random factors  it assumes all factors are fixed. This produces what are called narrow sense estimates of the standard errors. These represent variation over repetitions of the experiment only if one uses exactly the same blocks and simply rerandomizes the assignment of treatments to the experimental units. Fortunately the standard error of a difference between two least squares means is the same whichever model is used because differences between two means does not involve the blocking factor. Thus, inferences for pairwise differences are unaffected. But estimated confidence intervals of means are much too narrow. If blocks are random, we really need broad sense estimates of the standard error which would correspond to repetitions of the experiment with another sample of blocks. In recent years some statistical packages (including SAS and R) can analyze mixed model ANOVAs by fitting the random effects using maximum likelihood techniques. Computational formulaeWe will take a balanced experiment with 'a' group (= treatment) levels, each replicated once in 's' blocks. Group (treatment) totals are denoted as TA_{1} to TA_{a}, block totals as TS_{1} to TS_{s} and the grand total as G. The total, block, group and residual sums of squares are calculated as follows: If blocks are taken as a fixed factor, the standard error of a treatment mean is given by
If blocks are taken as a random factor, the standard error of a treatment mean is given
by
If blocks fixed or random, the standard error of the difference between means is given
by:
PoolingAfter using a randomized block design, it is not unusual to find that the block effect is not only not significant, but so small that it would have been better to have not blocked in the first place. It might then be tempting to reanalyze the data using a completely randomized design in order to gain degrees of freedom. In fact this approach is specifically recommended by some statisticians when analyzing matched pairs cluster randomized trials. Statisticians (as usual) do not agree on this issue, but the predominant view is that pooling would represent another case of pseudoreplication. Treatments have clearly not been allocated at random overall, but only within blocks. Hence it would be incorrect to ignore the blocks in the analysis of the experiment. Moreover, if blocks are left in the model, the resulting Pvalues closely approximate randomization test Pvalues. Conceptually, therefore, the Pvalues are tied directly to the chance mechanism involved in randomization. We have said the aim is to minimize the variance among units within blocks relative to the variance among blocks. But that does not necessarily mean we should try to maximize differences between blocks. If there is a strong interaction between treatment and blocks, then maximizing differences between blocks may make the situation worse. We consider this again below in relation to the Latin square ANOVA. Very small FratiosSince the main interest is in whether Fratios are significantly large, it is not
surprising that little attention is usually paid to an Fratio that is unusually small. If
the model is correct and all assumptions are satisfied, then the ratios of the
Latin square ANOVAModel & expected mean squaresWe will assume for the Latin square design that the treatment effect is fixed, whilst the row and column effects are random.
The Fratio for the treatment effect (assuming no interaction effects) is
obtained by dividing MS_{A} by MS_{Res}. The Pvalue
for this Fratio is obtained for Approximate Fratio for the row and column effects (assuming no restriction
error and no interaction effects ) are obtained by dividing MS_{R} and
MS_{C} by MS_{Res}. The Pvalues for these F
ratios are obtained for a−1 and Computational formulaeIn a Latin square design the number of group (= treatment) levels (a) will be the same as the number of rows and the number of columns. Group (treatment) totals are denoted as TA_{1} to TA_{a}, row totals as TR_{1} to TR_{a}, column totals as TC_{1} to TC_{a} and the grand total as G. The total, group, row, column, and residual sums of squares are calculated as follows: If rows and columns are both taken as a fixed factors, the standard error of a treatment mean is given by
If rows and columns are taken as a random factors, then presumably one would use If rows and columns are fixed or random, the standard error of the difference between means is given
by:
AssumptionsThe same assumptions as for a onefactor ANOVA must also hold for blocked ANOVA, namely:
But in addition, if there are more than two treatment levels, the restricted allocation of treatments to plots within a block introduces a further assumption  namely
There is one further assumption that must be made for an unbiased assessment of the treatment effect if 'blocks' is a fixed rather than a random factor:
