The rank transformation as a method of discrimination with some examples

The procedure of statistical discrimination Is simple in theory but so simple in practice. An observation x₀possibly uiultivariate, is to be classified into one of several populations π₁,…,π_k which have respectively, the density functions f₁(x), ? ? ? , f_k(x). The decision procedure is to evaluate each density function at X₀ to see which function gives the largest value f_i(X₀) , and then to declare that X₀ belongs to the population corresponding to the largest value. If these den-sities can be assumed to be normal with equal covariance matricesthen the decision procedure is known as Fisher’s linear discrimi-nant function (LDF) method. In the case of unequal covariance matrices the procedure is called the quadratic discriminant func-tion (QDF) method. If the densities cannot be assumed to be nor-mal then the LDF and QDF might not perform well. Several different procedures have appeared in the literature which offer discriminant procedures for nonnormal data. However, these pro-cedures are generally difficult to use and are not readily available as canned statistical programs.

Another approach to discriminant analysis is to use some sortof mathematical trans format ion on the samples so that their distribution function is approximately normal, and then use the convenient LDF and QDF methods. One transformation that:applies to all distributions equally well is the rank transformation. The result of this transformation is that a very simple and easy to use procedure is made available. This procedure is quite robust as is evidenced by comparisons of the rank transform results with several published simulation studies.