Unequal Numbers Of Observations Research Articles

Analysis of Multifactor Classifications with Unequal Numbers of Observations

The sums of squares in the general unequal numbers analysis of variance for an n-way or n-factor classification may be obtained in general terms from standard regression theory. However, the computing formulas using the usual normal equation and solution procedures become unmanageable even with the largest computers whenever the number of factors and/or the number of levels become moderately large. With the use of the calculus of factorials as developed by Kurkjian and Zelen in 1962, and with the additional theoretical results obtained by the authors in a recent paper, it has been possible to set forth relatively simple computational procedures for obtaining sums of squares in the analysis of variance for any effect eliminating all other effects. For sums of squares of interaction to order two, the computations may be performed on an ordinary desk calculator. Higher order interactions may require the use of high speed computers. In order to ease the computational difficulty in computing sums of squares still further, an upper bound sum of squares and a sharp upper bound sum of squares were obtained. Both forms are computationally simple and appear to be closer approximations than present approximations in the literature. A numerical example of a 4 X 3 X 2 factorial with unequal numbers of observations in the subclasses is used to illustrate the application of various formulae for computing sums of squares. A convenient computational form for computing interaction sums of squares has been set forth. The results are easily extended to four factor and larger factorials with unequal numbers of observations.

The Variance of Intraclass Correlation Involving Groups with One Observation

SUMMARY An approximate formula is derived for the variance of intraclass correlation when unequal numbers of observations per group occur. The effect on the variance of t of adding groups with single observations is examined using the formula and results obtained by empirically generating data on a computer. The empirical results indicate that the approximate formula is satisfactory over the range of numbers used. Adding a group with fewer than the average number of observations per group tends to reduce Vt by increasing the degrees of freedom for groups by one, but tends to increase Vt by decreasing the average precision of estimating group means. The net effect can be either negative or positive, depeinding on t, s and the ni's. Robertson [1962] pointed out that, when the ratio of the between group mean square to the within group mean square is small, exclusion of groups below half the average size will reduce the variance of the between group component. He further suggested a method for combining estimates of the between group component when n is highly variable. Results using the formula show that the poinlt where efficiency is lost when a group of size one is added is primarily a function of the number per group, and is affected very little by the number of groups. The value of n where groups of size one should be excluded is shown graphically for varying levels of t. Increases in Vt are demonstrated using the empirical data. The empirical results suggest that the increase in Vt may be even larger than the formula indicates, especially for large values of t. Only the addition of groups of size onie is studied. Adding small groups larger than one would also tend to increase Vt wheni n and t are small.

Nestejné počty pozorování při srovnávání několika skupin s jednou kontrolní

Nestejné počty pozorování při srovnávání několika skupin s jednou kontrolní

Tests of statistical hypotheses in the case of unequal or disproportionate numbers of observations in the subclasses

General solutions of the analysis of variance in the case of unequal numbers of observations in the subclasses are presented. If we havek criteria for the classification, there will bek! solutions in making a complete analysis and 2k−1 answers, bearing different meanings, for the sum of squares between subclasses of each criterion. The sum of squares for the interaction of any order, however, will be identical in different solutions of same problem.