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We shall look at the diets of $N$ = 346 juvenile steelhead / rainbow trout (Oncorhynchus mykiss) obtained from 3 different rivers draining into Hood Canal, in the state of Washington, USA.¶ Some of the fish caught had been reared in a hatchery (identifiable by...

Overview described a general dissimilarity-based solution to the multivariate Behrens-Fisher problem. Their solution uses a statistic (called '$F_2$' therein) that is a modification of the original PERMANOVA pseudo F statistic (called '$F_1$' therein). This m...

Handling heterogeneity with multiple factors The most important question to answer when you are dealing with a multi-factor study design and you decide you want to account for heterogeneity in dispersions (if present) is to answer the following question: Where...

An important assumption of classical analysis of variance (ANOVA) is that the errors come from a distribution with a common variance. By this we assume, in essence, that the variability of the sampling units within each group is constant and equivalent across ...

Consider the classical one-way linear ANOVA model, as described in section 6.2 above. Specifically, we have a random variable $Y$, and we have taken a sample of size $n_i$ from each of $i = 1,\ldots, a$ groups to obtain observed values $y_{ij}$. Thus, factor A...

ANOVA is one of the most widely used statistical techniques, providing a partitioning of the measured variation of a random variable in response to one or more factors in complex experimental designs and sampling programmes. A factor is a categorical variable ...

What is meant by a 'finite' factor? Suppose, for any factor, there are a total of $A$ levels in the population. In some cases, $A$ is absolutely enormous and it may be effectively infinite in the sense of being uncountable (e.g., blades of seagrass in a large ...

The study design We consider here a study examining effects of a sewage outfall for $p$ = 151 mollusc species from subtidal habitats (3-4 m depth) in the Mediterranean Sea along the southwestern coast of Apulia, Italy (). Abundances of each species were obtain...

Comparison of results treating 'Locations' as random Historical wisdom for such a design would have treated 'Locations' as a random factor (, ). It is quite instructive to consider what the results of this analysis might have been had we done this, instead of ...

In some cases, experiments are done in a way that lacks replication, often at the smallest spatial or temporal scale in the experimental design, but sometimes at larger scales as well. Examples of designs that lack replication include (but are not limited to):...

The study design An example of a repeated-measures sampling design (Fig. 8.5) is provided in a study of Victorian avifauna by . The data consist of counts of $p$ = 27 nectarivorous bird species at each of eight sites having different levels of flowering intens...

The study design An example of a split-plot design is provided by a study of the effect of fire disturbance and grazers (excluded using fences) on the composition of plant assemblages on the central western slopes of New South Wales in south-eastern Australia ...

There are situations where it may be useful or important to include one or more quantitative co-variables in a PERMANOVA model. For example, in our study of invertebrates inhabiting holdfasts of the kelp, Ecklonia radiata, it was not possible to standardise th...

Natural cycles in biology and ecology Important situations where the treatment of multiple covariables as a single set would be desirable in PERMANOVA are cases of periodic or cyclical phenomena in biology or ecology. Examples might include: seasonal patterns...

Consider the study described by consisting of regular surveys of macroalgal cover from a rocky intertidal area at Stanley Park, Vancouver in British Columbia, Canada. Macroalgal communities have been sampled monthly (since September 2021) along each of 3 perm...

Rationale When considering the response of a whole set of variables (such as the abundances of species or taxa) simultaneously to a suite of several factors (e.g., arising from a multi-factor experiment or sampling design), it can be difficult to visualise sal...

What is a 'main effects plot'? In a main effects plot, we calculate and then show in an ordination diagram a centroid for each of the levels of each factor listed in the design file.¶ We may also (optionally) show the overall centroid as well. The centroids, a...

What is an 'interaction plot'? Although main effects plots can help us to visualise the main effects of factors and permit us to guage their relative importance in explaining overall variation, centroids based on individual main effects ignore all other factor...

To further demonstrate the utility of main effects plots and interactions for multi-way study designs, we shall look at data from visual surveys of fish assemblages along the north-eastern coast of New Zealand completed annually (during the austral summer) ove...

Rationale It is sometimes desirable to remove the effects of a factor or covariable, and examine 'what is left', i.e., to look at variation in residuals. For example, a factor of primary interest may be statistically significant in a PERMANOVA partitioning of ...