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ANOVA for blocked designs: Use & misuse
(analysis of variance, randomized block, homogeneity of covariances, interaction between factors, Latin squares)
Statistics courses, especially for biologists, assume formulae = understanding and teach how to do statistics, but largely ignore what those procedures assume, and how their results mislead when those assumptions are unreasonable. The resulting misuse is, shall we say, predictable...
Use and MisuseThe randomized complete block design (and its associated analysis of variance) is heavily used in ecological and agricultural research. Blocking by age or location is also quite common in veterinary trials, but is rarely used in (human) clinical research, where very large sample sizes and (completely) randomized allocation are preferred. The exception to this is one of the repeated measures design which is analysed using a randomized block ANOVA with subjects as blocks - we cover this design under repeated measures ANOVA. In randomized complete block design, blocks are usually considered a random factor. It has been described as 'a bit of an odd duck' or 'inadequate' because of its lack of replication within blocks.
One common misuse is to use a blocked design when the blocks explain virtually none of the variation in the response variable. Whilst blocking is often essential in field work, this is seldom the case for experiments carried out under controlled conditions (as in a veterinary example we give on experimental feeds for broiler chickens) where unnecessary blocking will reduce the power of any testing. This applies especially to matched pairs data - whilst matching can reduce variability, it causes problems with missing observations (both data points lost) and greatly reduces the degrees of freedom for the error term especially for small sample sizes. However, if one has paired, then one has to live with the design through the analysis stage, even if the block factor is not significant!
All the usual ANOVA assumptions apply for randomized block ANOVA - although they seldom seem to be checked. We give a medical example on brain ventricle width and volume where variances are (wildly) heteroscedastic and data distributions are skewed. We also give analyses done on composite (ordinal) scores, pregnancy rates (proportions) and on time periods. For all such 'dodgy' data, model diagnostics should always be presented. There still seems little awareness that, if there are more than two treatments, restricted randomization introduces the further assumption of homogeneity of covariances. Whilst this assumption will usually be met if treatment allocation is randomized, this cannot be the case for observational studies. We give one medical example (on coronary flow velocity) where treatment order was not randomized and an ecological example (on the effect of different oyster aquaculture techniques on sediment microbe densities) which was an observational study where treatment could not be randomized. In both cases homogeneity of covariances (sphericity) should have been checked.
In observational studies we sometimes encounter use of the randomized block ANOVA where both factors are fixed. Such analyses have the additional assumption of no interaction between blocks and treatments. In the case of the oyster aquaculture example previously mentioned, there were strong indications of just such an interaction. This can be rectified if one uses a generalized (replicated) randomized block design and directly assesses the interaction effect. We give one example (on the effect of different diets on condition measures of guppies) where this was done. Unfortunately, when one of the interaction effects came out significant, it was ignored rather than investigated! In another example (on the effect of disturbance on species richness in a marine community) only some of the treatment levels were replicated in each block. This did allow interaction to be assessed, but at the cost of an unbalanced design and quasi F-ratios. Spatial (as opposed to temporal) Latin square designs are rarely used and, in both of the examples we give, a generalized randomized block design would have been more productive.
What the statisticians sayTwo-way ANOVA using R is covered extensively by Logan (2010) and Crawley (2007), (2005). Tamhane (2009) takes a more theoretical approach with detailed coverage of ANOVA of designed experiments - including the issue of restriction error in the randomized block design. Doncaster & Davey (2007) consider randomized block ANOVA in Chapter 4. The topic is dealt with by Quinn & Keough (2002) in Chapter 10, Underwood (1997) in Chapter 12 and Sokal & Rohlf (1995) in Chapter 11. Mead et al. (2002) is one of few texts to cover the analysis of multiple Latin squares. Jackson & Brashers (Eds) (1994) provide a useful discussion of criteria for determining whether factors in ANOVA are fixed or random.
Lew (2007) emphasizes the greater power of the randomized block ANOVA compared to simple one -way ANOVA for pharmacology experiments. St-Pierre (2006) provides an excellent review of the design and analysis of pen studies in the animal sciences showing how pens can serve either as blocks (where treatment is randomized to cows) or as experimental units (where treatments randomized to pens).
Lee et al (2008) recommend the use of mixed models for analysis of agricultural field trials with changing variance, a view countered by Piepho et al (2009) who recommends the use of transformations whenever possible. Meek et al. (2007) highlight (very) small F-ratios as red flags in the linear model possibly indicating blocks × treatment interaction in the randomized block ANOVA. Payne (2006) looks at new developments to finds alternatives to blocking by modelling spatial correlations in an experiment. Federer & Nguyen (2002) give a concise account of incomplete block designs. O'Neill & Mathews (2002) look at Levene tests of homogeneity of variance for general block and treatment designs. Newman et al. (1997) look at ANOVA when blocking factors are present focusing on the difference between fixed and random effects. Bennington & Thayne (1994) and Dutilleul (1993) address the issue of whether blocks should be specified as fixed or random factors. Samuels et al. (1988) discuss whether blocks are different from random factors.
Lentner et al. (1989) explain why the F-ratio for the block effect can be used as a measure of relative efficiency of the randomized block design versus the completely randomized design. McLean & Anderson (1980) and Anderson & McLean (1974) introduce the concept of restriction error. Perry et al. (1980) review the use and analysis of Latin Square designs in field experiments involving insect sex attractants. Schaaljea & Despaina et al. (1966) look at the robustness of homogeneity of variance tests for randomized complete block data. McHugh (1964) discusses whether the randomized block design should be replicated. The Handbook of biological statistics has a section on randomized block ANOVA. Gerard Dallal provides useful contributions on fixed versus random factors and on randomized complete block designs. University lecture notes are given by Iowa State University. Guido Wyseure gives the R code for Tukey's non additivity test.