Estimating the Degrees of Freedom for Linear Combinations of Mean Squares by Satterthwaite's Formula

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Abstract
In calculating the variance of a mean or in constructing an approximate F-test, it is frequently necessary, particularly with unbalanced data, to form a linear function of mean squares, MS = Σa i, where the ai are known constants. Satterthwaite [1] suggests that MS is approximately distributed as x 2 f E(MS)/ft where the degrees of freedom is estimated by, f = (MS)2/Σ[(aiMSi )2/fi ], where fi is the degrees of freedom associated with MSi . Satterthwaite remarks that caution should be used in applying this formula where some of the ai are negative. The primary purpose of this paper is to investigate the validity of Satterthwaite's procedure when some MSi are subtracted. Due to the mathematical complexity of the distribution of MS, the problem is studied by a computer simulation.

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