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A suitable criterion for a control chart designed to detect shifts in the population mean vector for multivariate normal data is Hotelling's $T^2$, the (normalised) deviation of a sample vector (or a sample mean vector) measured at time $t$ from some known (or...

To demonstrate the use of Hotelling's $T^2$ in a multivariate control-chart setting, it is useful to examine the method in 2 dimensions in Euclidean space (a bivariate system), which can be easily drawn and visualised. We shall examine bivariate patterns for t...

Essential steps Suppose we have an $(N \times p)$ data matrix, $\bm{Y}$, and we can capture the important relationships among the $N$ sampling units in this matrix by calculating some chosen dissimilarity measure (e.g., Bray-Curtis) to yield an $(N \times N)$ ...

We offer here a few additional notes regarding the implementation of control charts in real applications. The control-chart dialog in PRIMER 8 offers many options. It is especially important to pay close attention to all of the choices that can affect the null...

We shall implement a control chart on data from the North American Breeding Bird Survey (BBS) (). We will specifically look at abundances of $p$ = 156 breeding birds from a single route in Grand Forks, British Columbia, Canada in a time series that includes 38...

PRIMER 8 offers a host of new options for standardising data (either samples or variables), via the menu item: Pre-treatment > Standardise. The new 'Standardise' dialog window in PRIMER 8, by comparison with that in PRIMER 7, is shown below (Fig. 13.1). Fig. ...

Rationale Suppose we have data where the variables consist of different size classes of mussels (as we shall shortly see in a real example). In such cases, where the variables have a natural order, we may, of course, simply treat the variables multivariately j...

To implement the new standardisation routine in PRIMER 8 and (simultaneously) demonstrate the utility of analysing cumulative percentages, we shall examine a study by , who measured the lengths of mussels (Mytilus trossulus) at two glacially influenced estuari...

There are many situations where the standardisation of samples is required as a pre-treatment prior to analysis, but which needs to be done separately within groups of variables that may be identified by an indicator.¶ Here, we shall consider a dataset compris...

There is a new tool in PRIMER 8 that permits the end-user to create a set of ordered groups, based on the numerical values of a given variable. Rationale Below we provide a few examples of this tool's potential utility. There are many more! Binning and consoli...

To demonstrate the creation and use of ordered groups from a continuous variable, we will look at data comprised of an excerpt from the West Coast Groundfish Bottom Trawl (Slope and Shelf Combination) Survey, conducted annually by the National Oceanic and Atmo...

Accessibility It is important to make graphics accessible to those with color vision deficiencies. We have therefore re-vamped the colour palette for P8 to achieve distinctive colours for plots (by default) that carefully accommodate the most common forms of c...

In PRIMER 8, the options available for selecting samples or selecting variables have been expanded considerably from what they were in PRIMER 7. Selecting Samples When you click Select > Samples... in P8, you will see the new dialog shown at right in Fig. 15.2...

There are many situations where it would be very handy to be able to change the names of levels of a factor (or to change the names of groups for an indicator). We often need to tweak the names of levels of factors. For example, The names are too long and you...

In PRIMER 8, there is a new tool that allows us to add customised values and labels to coordinate axes in graphics. Consider the following scatter plot of species richness ($S$) vs. structural complexity of the substratum (measured using a chain-and-tape metho...

In PRIMER 8 there is a new tool, accessed by clicking Tools > Split Data..., which allows you to split a data sheet into several separate data sheets, corresponding to: separate groups of samples, based on a factor; or separate groups of variables, based on a...

There is a new facility in PRIMER 8 to create Line plots in two different ways: with one line for every variable (across all samples); or with one line for every sample (across all variables). This is a considerable improvement on the line plot dialog in PRI...

In PRIMER 8, you can now output more detailed statistical information from either dispersion-weighting or variability weighting to a worksheet. You can see this additional option in the comparison of the 'Dispersion Weighting' dialog in PRIMER 8, compared to t...

In PRIMER 8, it is now possible to output diagnostic plots from the CAP routine (i.e., canonical analysis of principal coordinates, an analysis obtained from a resemblance matrix by clicking PERMANOVA+ > CAP). You simply tick the new option to '$\checkmark$Do ...

Background Consider a cloud of $N$ points (sampling units) in a $p$-dimensional multivariate space. Principal components analysis (PCA) is an ordination method that will find an axis through that cloud of points so as to maximise the total variance of the poin...