## U.S. Department of Agriculture: Agricultural Research Service, Lincoln, Nebraska

## Document Type

Article

## Date of this Version

December 1964

## Abstract

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, depending 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 point 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 *V t* may be even larger than the formula indicates, especially for large values of *t*. Only the addition of groups of size one is studied. Adding small groups larger than one would also tend to increase *Vt*when n and *t* are small.

## Comments

Published in BIOMETRICS, DECEMBER 1964.