A Modified Satterthwaite (1941,1946) Effective Degrees of Freedom Approximation

TL;DR

This paper proposes a theoretical correction to Satterthwaite's effective degrees of freedom approximation, improving accuracy for small component degrees of freedom by leveraging moments of normal variables.

Contribution

It introduces a new correction based on analytical moments, extending and partially validating Johnson & Rust's empirical adjustment for complex variance estimates.

Findings

Enhanced approximation accuracy for small degrees of freedom

Theoretical foundation replaces empirical constants

Applicable to complex variance estimation scenarios

Abstract

This study introduces a correction to the approximation of effective degrees of freedom as proposed by Satterthwaite (1941, 1946), specifically addressing scenarios where component degrees of freedom are small. The correction is grounded in analytical results concerning the moments of standard normal random variables. This modification is applicable to complex variance estimates that involve both small and large degrees of freedom, offering an enhanced approximation of the higher moments required by Satterthwaite's framework. Additionally, this correction extends and partially validates the empirically derived adjustment by Johnson & Rust (1992), as it is based on theoretical foundations rather than simulations used to derive empirical transformation constants.