We describe a simple general formula for approximating the p-value for testing a scalar parameter in the presence of nuisance parameters. The formula covers both frequentist and Bayesian contexts and does not require explicit nuisance parameterisation. Implementation is discussed in terms of computer algebra packages. Examples are given and the relationship to Barndor-Nielsen's approximation is discussed.

Bayes [Philos. Trans. R. Soc. Lond. 53 (1763) 370--418; 54 296--325] introduced the observed likelihood function to statistical inference and provided a weight function to calibrate the parameter; he also introduced a confidence distribution on the parameter space but did not provide present justifications. Of course the names likelihood and confidence did not appear until much later: Fisher [Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 222 (1922) 309--368] for likelihood and Neyman [Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 237 (1937) 333--380] for confidence. Lindley [J. Roy. Statist. Soc. Ser. B 20 (1958) 102--107] showed that the Bayes and the confidence results were different when the model was not location. This paper examines the occurrence of true statements from the Bayes approach and from the confidence approach, and shows that the proportion of true statements in the Bayes case depends critically on the presence of linearity in the model; and with departure from this linearity the Bayes approach can be a poor approximation and be seriously misleading. Bayesian integration of weighted likelihood thus provides a first-order linear approximation to confidence, but without linearity can give substantially incorrect results.Comment: Published in at http://dx.doi.org/10.1214/11-STS352 the Statistical Science (http://www.imstat.org/sts/) by the Institute of Mathematical Statistics (http://www.imstat.org

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We consider inference on a vector-valued parameter of interest in a linear exponential family, in the presence of a finite-dimensional nuisance parameter. Based on higher order asymptotic theory for likelihood, we propose a directional test whose p-value is computed using one-dimensional integration. For discrete responses this extends the development of Davison et al. (2006), and some of our examples concern testing in contingency tables. For continuous responses the work extends the directional test of Cheah et al. (1994). Examples and simulations illustrate the high accuracy of the method, which we compare with the usual likelihood ratio test and with an adjusted version due to Skovgaard (2001). In high-dimensional settings, such as covariance selection, the approach works essentially perfectly, whereas its competitors can fail catastrophically.