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1.37 Asymmetrical designs (Mediterranean molluscs)
Although a previous section has been devoted to the analysis of unbalanced designs, there are some special cases of designs having missing cells which deserve extra attention. Such designs are commonly referred to as asymmetrical designs, and consist essential...
1.38 Environmental impacts
Some further comments are appropriate here regarding experimental designs to detect environmental impact (, , , ). These designs generally include measurements of a response variable of interest before and after a potential impact from one or more control site...
2.1 General description
Key reference Method: PERMDISP is a routine for testing the homogeneity of multivariate dispersions on the basis of any resemblance measure. The test is a dissimilarity-based multivariate extension of Levene’s test (), following the ideas of , and , who...
2.2 Rationale
There are various reasons why one might wish to perform an explicit test of the null hypothesis of no differences in the within-group multivariate dispersion among groups. First, such a test provides a useful logical complement to the test for location differe...
2.3 Multivariate Levene’s test (Bumpus’ sparrows)
proposed doing an analysis of variance (ANOVA) on the absolute values of deviations of observations from their group mean. A multivariate analogue was described by and given in , based on distances. Let $y _{ij}$ be a vector of p response variables for the j...
2.4 Generalisation to dissimilarities
Of course, in many applications that we will encounter (especially in the case of community data), the Euclidean distance may not be the most appropriate measure for the analysis. What we require is a test of homogeneity of dispersions that will allow any rese...
2.5 $P$-values by permutation
The other hurdle that must be cleared is to recognise that, in line with the philosophy of all of the routines in the PERMANOVA+ add-on, we have no particular reason to assume that the distribution of the z’s will be necessarily normal. Yet, to use tabled P-va...
2.6 Test based on medians
Levene’s test (for univariate data) can be made more robust (i.e. less affected by outliers) by using deviations from medians rather than deviations from means (, ). However, for multivariate data, there is more than one way to define a median (, , ). One defi...
2.7 Ecological example (Tikus Island corals)
An ecological example of the test for homogeneity is provided by considering a study by on coral assemblages from South Tikus Island, Indonesia. Percentage cover was measured along 10 transects for 75 species of coral in each of several years (1981-1988) whic...
2.8 Choice of measure
An extremely important point is that the test of dispersion is going to be critically affected by the transformation, standardisation and resemblance measure used as the basis of the analysis. It is pretty well appreciated by most practitioners that transformi...
2.9 Dispersion as beta diversity (Norwegian macrofauna)
When used on species composition (presence/absence) data in conjunction with certain resemblance measures, the test for homogeneity of multivariate dispersions is directly interpretable as a test for similarity in beta diversity among groups (). Whittaker (, )...
2.10 Small sample sizes
There is one necessary restriction on the use of PERMDISP, which is that the number of replicate samples per group must exceed n = 2. The reason is that, if there are only two replicates, then, by definition, the distance to the centroid for those two samples ...
2.11 Dispersion in nested designs (Okura macrofauna)
In many situations, the experimental design is not as simple as a one-way analysis among groups. For more complex designs, several tests of dispersion may be possible and relevant at a number of different levels. The sort of tests that will be logical to do in...
2.12 Dispersion in crossed designs (Cryptic fish)
When two factors are crossed with one another, there may be several possible hypotheses concerning relative dispersions among groups. An important aspect of analysing multi-factorial designs is to keep straight when you are focusing on dispersion effects and w...
2.13 Concluding remarks
PERMDISP is designed to test the null hypothesis of no differences in dispersions among a priori groups. Although this may be the only goal in some cases, PERMDISP can also provide a useful companion test to PERMANOVA in order to clarify the nature of multivar...
3.1 General description
Key references Method: , PCO is a routine for performing principal coordinates analysis () on the basis of a (symmetric) resemblance matrix. PCO is also sometimes referred to as classical scaling, with origins in the psychometric literature (). PCO place...
3.2 Rationale
It is difficult to visualise patterns in the responses of whole sets of variables simultaneously. Each variable can be considered a dimension, with its own story to tell in terms of its mean, variance, skewness, etc. For most sets of multivariate data, there a...
3.3 Mechanics of PCO
To construct axes that maximise fitted variation (or minimise residual variation) in the cloud of points defined by the resemblance measure chosen, the calculation of eigenvalues (sometimes called “latent roots”) and their associated eigenvectors is required. ...
3.4 Example: Victorian avifauna
As an example, consider the data on Victorian avifauna at the level of individual surveys, in the file vicsurv.pri of the ‘VictAvi’ folder in the ‘Examples add-on’ directory. For simplicity, we shall focus only on a subset of the samples: Select > Samples > •F...
3.5 Negative eigenvalues
The sharp-sighted will have noticed a conundrum in the output given for the Victorian avifauna shown in Fig. 3.2. The values for the percentage variation explained for PCO axes 10 through 15 are negative! How can this be? Variance, in the usual sense, is alway...