CORRESPONDENCE-ANALYSIS OF DATA ARRAYS - STUDY ON THE CHEMOTAXY OF PARASITIZED LARVAE

A study on the effect of mucus on the movement of infective copepodites of Caligus minimus, buccal parasite of sea bass, Dicentrarchus labrax, leads to a data array formed by 42 bar charts describing the distributions of these larvae depending on length of the experiment and the type and quantity of mucus. A standard application of correspondence analysis to this data array, followed by the usual technique of supplementary points for the means of the experiments with the same length or the same mucus suggests an effect of length and mucus on the behavior of the copepodites: when the length of the experiment increases, the spread of the bar charts increases too; the central tendency of the bar charts moves with the attracting strength of the mucus. In order to proceed from the qualitative result given by correspondence analysis to a quantification of the effects, we develop the following hypothesis testing approach. First, the total dispersion measure, called inertia I-I of the study is divided into two parts, I, explained for example by the mucus classes, and I-2 = I - I-1, the residual part. To do that, consider F, the table of available frequencies. D-I and D-J are respectively the diagonal matrices constructed with the row and column margins of F. ($) over tilde F is the centered table obtained from F using the weights given by the diagonal of D-I. The total inertia of the study is: I = Tr(($) over tilde F'D-I ($) over tilde FDJ-1). Let U-T be the table of the indicator variables of the mucus classes and P-T = U-T(U-T'DIUT)(-1) U'D-T(I) the D-I-orthogonal projector associated with it. I-1 = Tr((P-T ($) over tilde F)'D-I(P-T ($) over tilde F)D-J(-1)) and I-2 is the corresponding quantity for ($) over tilde F - P-T ($) over tilde F. Under the null hypothesis of no effect of mucus on the behavior of copepodites, I-1 can be compared with similar quantities obtained after random partitions of the copepodites with the same number of classes and the same number of elements in the classes. The generation of such random partitions gives the distribution curve of I-1 under the null hypothesis. An extremal position for the observed I-1 is interpreted as grounds for rejection of the null hypothesis. For the data array studied, the test rejects the null hypothesis of no effect for both mucus and length of the experiments. The approach is used in this paper in a two-factor context. The paper extrapolates to how this approach can be extended to more complicated designs.