The effect of bias, variance estimation, skewness and kurtosis of the empirical logit on weighted least squares analyses

SUMMARY 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 + 2 correction of Haldane (1955) and Anscombe (1956) nor the -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.

[1]  E. Bedrick.  Estimating the Variance of Empirical Logits and Contrasts in Empirical Log Probabilities , 1984 .

[2]  S. S. Wilks,et al.  The Advanced Theory of Statistics. I. Distribution Theory , 1959 .

[3]  J. Gart.  Alternative Analyses of Contingency Tables , 1966 .

[4]  S. Hitchcock,et al.  A note on the estimation of the parameters of the logistic function, using the minimum logit X 2 method , 1962 .

[5]  G. Koch,et al.  Analysis of categorical data by linear models. , 1969, Biometrics.

[6]  R. Plackett.  A Note on Interactions in Contingency Tables , 1962 .

[7]  M. Kendall,et al.  The advanced theory of statistics , 1945 .

[8]  Modification of the empirical logit to reduce bias in simple linear logistic regression , 1985 .

[9]  Jerome Cornfield,et al.  A Statistical Problem Arising from Retrospective Studies , 1956 .

[10]  J. Berkson.  Maximum Likelihood and Minimum x 2 Estimates of the Logistic Function , 1955 .

[11]  J. Grizzle.  A NEW METHOD OF TESTING HYPOTHESES AND ESTIMATING PARAMETERS FOR THE LOGISTIC MODEL , 1961 .

[12]  C. Odoroff.  A Comparison of Minimum Logit Chi-Square Estimation and Maximum Likelihood Estimation in 2×2×2 and 3×2×2 Contingency Tables: Tests for Interaction , 1970 .

[13]  B. Haldane.  THE ESTIMATION AND SIGNIFICANCE OF THE LOGARITHM OF A RATIO OF FREQUENCIES , 1956, Annals of human genetics.

[14]  David R. Cox.  The analysis of binary data , 1970 .

[15]  F. Yates.  Marginal percentages in multiway tables of quantal data with disproportionate frequencies , 1961 .

[16]  J. Gart,et al.  On the bias of various estimators of the logit and its variance with application to quantal bioassay. , 1967, Biometrika.

[17]  L. A. Goodman.  Interactions in Multidimensional Contingency Tables , 1964 .

[18]  Norman E. Breslow,et al.  Odds ratio estimators when the data are sparse , 1981 .

[19]  M. Bartlett,et al.  APPROXIMATE CONFIDENCE INTERVALS , 1953 .

[20]  B. Woolf.  ON ESTIMATING THE RELATION BETWEEN BLOOD GROUP AND DISEASE , 1955, Annals of human genetics.

[21]  D G Thomas,et al.  The performance of three approximate confidence limit methods for the odds ratio. , 1982, American journal of epidemiology.

[22]  J. Berkson.  Application of the Logistic Function to Bio-Assay , 1944 .

[23]  F. J. Anscombe,et al.  On estimating binomial response relations , 1956 .