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• Uncorrected critical value: a = 0.05, v = (3 - 1)(5 - 1) = 8, critical %a v = 15.5. Bcritical = 15.5/(2 x 16) = 0.484. ,

• Bonferroni correction for 7 simultaneous tests: a' = a/(n/2 - 1) = 0.05/7, v = 8, critical x2',v = 21.0. Bcritical = 21.0/32 = 0.656.

• Progressive Bonferroni correction. Example for the 4th test: a' = a/4 = 0.05/4, v = 8, critical x2',v = 19.5. Bcritical = 19.5/32 = 0.609.

Thus, the only significant period in the data series is Tk = 5.

The contingency periodogram can be directly applied to qualitative descriptors. Quantitative or semiquantitative descriptors must be divided into states before analysis with the contingency periodogram. A method to do so is described in Legendre et al. (1981).

In their paper, Legendre et al. (1981) established the robustness of the contingency periodogram in the presence of strong random variations, which often occur in ecological data series, and its ability to identify hidden periods in series of nonquantitative ecological data. Another advantage of the contingency periodogram is its ability to analyse very short data series.

One of the applications of the contingency periodogram is the analysis of multivariate series (e.g. multi-species; Ecological application 12.4b). Such series may be transformed into a single qualitative variable describing a partition of the observations, found through a clustering method. With the contingency periodogram, it is possible to analyse the data series, now transformed into a single nonordered variable corresponding to the partition of the observations. The only alternative approach would be to carry out the analysis on the multivariate distance matrix among observations, using the Mantel correlogram described in Subsection 13.1.5.

Ecological application 12.4b

Phytoplankton was enumerated in a series of 175 water samples collected hourly at an anchor station in the St. Lawrence Estuary (Québec). Using the contingency periodogram, Legendre et al. (1981) analysed the first 80 h of that series, which corresponded to neap tides. The original data consisted of six functional taxonomic groups. The six-dimensional quantitative descriptor was transformed into a one-dimensional qualitative descriptor by clustering the 80 observations (using flexible clustering; Subsection 8.5.10). Five clusters of "hours" were obtained; each hour of the series was attributed to one of them. Each cluster thus defined a state of the new qualitative variable resulting from the classification of the hourly data.

When applied to the qualitative series, the contingency periodogram identified a significant period T = 3 h, which suggested rapid changes in surface waters at the sampling site. The integer multiples (harmonics) of the basic period (3 h) in the series also appeared in the contingency periodogram. Periods T = 6 h, 9 h, and so on, had about the same significance as the basic period, so that they did not indicate the presence of additional periods in the series.

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