Blue crabs (Callinectes sapidus) are of great commercial value in North Carolina and the appearance of diseased crabs in 1986 in a specific region of the Pamlico River caused great concern. Gemperline et al (1992) cited the hypothesis that ''environmental stress weakens the organism so that its normal immunological response is unable to ward off opportunistic infection by chitinoclastic bacteria''. These bacteria were considered to be the cause of penetration of the carapace with lesions of 5-25 mm. To investigate whether trace element levels might be associated with the occurrence of the disease, Gemperline and his colleagues collected gill, hepatopancreas, and muscle tissue samples taken from blue crabs in each of three categories: Albemarle Sound, non-diseased Pamlico and diseased Pamlico. Twenty-five trace elements were used for the main analysis, so that a 48 (tissue samples) by 3 (tissue types) by 25 (trace elements) three-mode array was available for analysis.
Copyright image: Mike Oesterling - Courtesy of the Virginia Institute of Marine Science. Please respect this copyright by not using this image without permission.
Samples from three individual crabs were pooled, giving a total of 16 pooled tissue samples in each category and a grand total of 48 pooled tissue samples. To facilitate the discussion, we will refer to a ``blue crab'' rather than as ``the combined tissue of three individual crabs collected from the same location''.
Thus there is a (between-subject) design on the first mode, because each crab belongs to one category:
A three-way data array X = (x(i,j,k)) has the following form
|-----|i=1 |-----| |i=2 |-----| | |.. | | | |.. | | |____|i=I=48 k=K=25 | |____| k=2 |_____| k=1 j=1,.,J=3
The actual data file has the following form:
j=1,.,J=3 |-----|i=1 | |i=2 | |.. k= 1 | |.. |_____|i=I=48 |-----|i=1 | |i=2 | |.. k= 2 | |.. |_____|i=I=48 |-----|i=1 | |i=2 | |.. k=K=25 | |.. |_____|i=I=48
Thus the first mode (i) is nested in the third mode (k) so that there are 1200 rows, i.e. I=48 (Crabs) times K=25 (Trace elements) rows and there are J=3 (Tissue types) columns. The first 16 crabs are the Albermarle ones, the next 16 the healthy Pamlico ones and the final 16 crabs are the diseased Pamlico crabs.
In the Kroonenberg et al. principal component analyses, the data were centred per column (tissue type) x(j,k) so that the columns were in deviation of their mean. After this the frontal slices (trace elements) Xk were normalised to sum of squares equal to 1, so that the total sum of squares of the preprocessed data was 25. For a three-mode cluster analyses no preprocessing is necessary because means and covariances are explicitly modelled. For details see Kroonenberg (2008). Applied multiway data analysis. Hoboken NJ: Wiley (Chapters 6 and 16).
[Download the zipped 1989 Blue Crabs Data]