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In statistical terms, a correlation is a measure of the extent to which two series of observations vary together. There are a number of different types of correlation that may be calculated for a data series:

1. Pearson's correlation coefficient(r): Probably the best known measure of correlation, it is computed for two vectors (series of observations) provided that the two variables are quantitative and that the relationship between them is linear. It is simply calculated as:

rxy=(sxy)/(sx*sy)

which is the covariance divided by the standard deviation of the two variables.

2. Spearman's rank correlation coefficient (rs): Again, calculated on two variables. It is used in place of the Pearson coeffcient when the assumptions of the latter are not fulfilled. It is calculated as:

rs=sum(y'1*y'2)/(sum(y'12)*sum(y'22))0.5

where y' represents the value of the variable after centering on the mean.

3. Kendall's rank correlation coefficient (tau): Preferred to the Spearman coefficient because of its greater power. It is calculated by transforming the data into ranks, and counting the number of times an observation in the first variable has a greater value than an observation in the second variable.
4. Mantel's multivariate correlation (rM): The Mantel coefficient is used when more than two variables are of interest. Rather than computing the correlation on the raw values of the variables, Mantels' correlation coefficient is calculated on two distance matrices. The basic procedure is as follows:

1. Identifiy the two groups of variables of interest, and place them in two matrices X and Y.
2. Transform X and Y into two distance (or similarity) matrices using and index of association.
3. Center and reduce each distance matrix (meaning, subtract from each observation in the matrix the mean value, and then divide by the standard deviation).
4. Multiply the two distance matrices together.
5. Compute the sum of all the non-diagonal values in the matrix containing the results of the preceeding step, and divide this sum by n-1 where n is the number of values in one distance matrix (excluding the diagonal).

While this may seem like a complicated procedure, it is simply the Pearson's correlation coefficient calculated on multivariate data which has been transformed into a distance matrix.

Cor`re*la"tion (-l?"sh?n), n. [LL. correlatio; L. cor- + relatio: cf. F. corr'elation. Cf. Correlation.]

Reciprocal relation; corresponding similarity or parallelism of relation or law; capacity of being converted into, or of giving place to, one another, under certain conditions; as, the correlation of forces, or of zymotic diseases.

Correlation of energy, the relation to one another of different forms of energy; -- usually having some reference to the principle of conservation of energy. See Conservation of energy, under Conservation. -- Correlation of forces, the relation between the forces which matter, endowed with various forms of energy, may exert.

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