A Test for sphericity and mixed model analysis of fertilizer experiments in cultivator’s fields

Abstract

In the first part of the thesis, a test, based on the approximant distribution of the quadratic from, for testing the sphericity of a normal population, has been suggested. The distribution of the ration of the two quadratic forms is approximately Fisher. –Shedecor’s F. on which the test is based. Using C.E.P., Fox’s (1954) results, the exact power of the test may be calculated. Also using the approximate distribution of the ratio of two quadratic forces, the power of the test can be calculated. A table showing the exact and approximate power is given which shows that they are not very different and the approximation given sufficiently good results. Such a test is useful for small sample sizes only, because it is not consistent. A modification of the test has been made for large samples. Several F’s are calculated from the sample and the arithmetic mean of their square roots is found, which for large samples, follows a normal distribution with given mean and standard deviation, the values of which are different under the null and alternative hypotheses. The calculation of the power of such a test is shown.