NONPARAME TRIC STATISTICS A

In the development of modern techniques of statistical inference, the first tests to gain prominence and wide use were those which make a good many assumptions, and rather stringent ones, about the nature of the population from which the observations were drawn. The title of these tests parametric-suggests the central importance of the population, or its parameters, in their use and interpretation. These tests also use the operations of arithmetic in the manipulation of the scores on which the inference is to be based, and therefore they are useful only with observations which are numerical. The t and F tests are the most familiar and widely used of the parametric tests, and the Pearson product-moment correlation coefficient and its associated significance test are the most familiar parametric approaches to assessing association. More recently, nonparametric or "distribution-free" statistical tests have gained prominence. As their title suggests, these tests do not make numerous or stringent assumptions about the population. In addition, most nonparametric tests may be used with non-numerical data, and it is for this reason that many of them are often referred to as "ranking tests" or "order tests." Many nonparametric tests use as their data the ranks of the observations, while others are useful with data for which even ordering is impossible, i.e., classificatory data. Some nonparametric methods, such as the X2 tests, the Fisher exact probability test, and the Spearman rank correlation, have long been among the standard tools of the statisticians. Others are relatively new, and therefore have not yet gained such widespread use. At present, however, nonparametric tests are available for all the common experimental designs. The purpose of the present paper is not to discuss the rationale and application of the various nonparametric tests. That has been done, at various levels of sophistication and with varying degrees of comprehensiveness, in other sources (2, Chap. 17; 3; 6, Chap. 16; 7; 8; 9; 10; 11; 13). Rather, the purpose is to discuss, at a non-technical level, certain issues which have arisen in connection with the use of nonparametric tests. In particular, issues are discussed which are relevant to the choice among alternative tests, parametric and nonparametric, applicable to the same experimental design. Power When alternative statistical tests are available to treat data from a given research design, as is very often the case, it is necessary for the researcher to employ some rationale in choosing among them. The criterion most often suggested is that the researcher should choose the most powerful test. The power of a test is defined as the probability that the test will reject the null hypothesis when in fact it is false and should be rejected. That is, Power -1 -probability of a Type II error Thus, a statistical test is considered a good one if it has small probability of rejecting Ho the null hypothesis when Ho is true, but a large probability of rejecting Ho when Ho is false. However, there are considerations other than power which enter into the choice of a statistical test. One must consider the nature of the population from which the sample was drawn, and the kind of measurement which was employed in the operational definitions of the variables of the research. These matters also enter into determining which statistical test is optimal for analyz-ing a particular set of data.