Statistical Inference: Likelihood to Significance

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Abstract
The concepts of likelihood and significance were defined and initially developed by R. A. Fisher, but followed almost separate and distinct routes. We suggest that a central function of statistical inference is in fact the conversion of the first, likelihood, into the second, significance: a linking of the Fisher concepts. A first-order asymptotic route for this is incorporated into most statistical packages. It uses the standardized maximum likelihood estimate, the standardized score, or the signed square root of the likelihood ratio statistic as arguments for the standard normal distribution function, thus giving approximate tail probabilities or observed levels of significance. Recent third-order asymptotic methods provide a substantial increase in accuracy but need the first derivative dependence of likelihood on the data value as an additional input. This can be envisaged as the effect on the likelihood function of dithering the data point. Extensions to the multivariate, multiparameter context are surveyed, indicating major areas for continuing research.

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