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Biometrika 1985 72(1):179-190; doi:10.1093/biomet/72.1.179
© 1985 by Biometrika Trust
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The effect of bias, variance estimation, skewness and kurtosis of the empirical logit on weighted least squares analyses

JOHN J. GART, HUGH M. PETTIGREW and DONALD G. THOMAS

Mathematical Statistics and Applied Mathematics Section, Biostatistics Branch, National Cancer Institute Bethesda, Maryland, 20205, U.S.A.

The bias and first four cumulants of the empirical logit transformation of a binomial variate are studied by means of asymptotic expansions and exact computation. The covariance of the empirical logit and its estimated variance is derived. Neither the +1/2 correction of Haldane (1955) and Anscombe (1956) nor the –1/2 suggested by Cox (1970) for weighted logit regression is universally effective in reducing bias. Other corrections are considered. The distribution of the empirical logit is shown to be considerably more skewed and to have a much larger kurtosis than the comparable binomial distribution. Methods based directly on sufficient statistics are preferred to those based on the empirical logit.

Key Words: Bias • Binomial distribution • Confidence interval • Interaction in 2x2x2 tables • Kurtosis • Logistic model • Logit • Odds ratio estimation • Simple logistic regression • Skewness


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