On Determining Three Dimensional Regions of Significance

Abstract

where J& is the value of the likelihood criterion corresponding to a chosen level of significance, ?C , i.e., oC = .01, oC = .05, etc., and 5a and $* are the absolute and relative minimum sum of squares, respectively, providing two optimum estimates of the population variance, <r* , under the conditions prescribed by the properties of the of significance. These properties are that* for certain sys tems of values of two independent variables, x', y', the hypothesis, H(x', y/), should be rejected. This hypothesis in (16) referred to the difference between the best linear estimates of the means of two ran dom samples of observational values. The hypothesis could refer to other statistical functions of ob servational data as well. The distinction between this type of hypothesis and the usual null hypothesis may be made as follows: The usual null hypothesis specifies some fixed set of parameter values esti mated for the single system of values, say x? y,, provided by the experimental data, and the process of testing the hypothesis consists in setting up a critical for this specified set of values, x,, yf. If the experimental value, or sample point, falls in this critical region, the hypothesis H(x? y( ) is rejected and the generalization usually obtains for the population for which the sample is presumably represent ative. The Johnson-Neyman technique distinguishes between the systems of values of x' and y' for which the hypothesis H (x', y') should be rejected and the systems for which the hypothesis should be accepted. It thus becomes possible to specify the population in terms of the basic characteristics, x and y, for which a generalization is permissible. That is, the analysis shows first if there are values of x and y for which, say, the difference in means between two groups is significant. Should such values exist for the variables x, y, they are bounded by a conic section, i.e., an ellipse, parabola, or an hyperbola. Geometrically, then, for two independent variables, the region of significance is represented by a conic section. The section can be plotted and the hypothesis H (x', y0 represented by any point of this should be rejected at the level of significance, oc , chosen.