Layman’s abstract for paper on a directional look at F‐tests

Every few days, we will be publishing layman’s abstracts of new articles from our prestigious portfolio of journals in statistics. The aim is to highlight the latest research to a broader audience in an accessible format.

The article featured today is from the Canadian Journal of Statistics, with the full article now available to read here.

Mccormack, A., Reid, N., Sartori, N. and Theivendran, S.‐A. (2019), A directional look at F‐tests. Can J Statistics, 47: 619-627. doi:10.1002/cjs.11515

Directional testing is an approach to inference about vector parameters that was proposed in the 1980s by Fraser and Massam for regression models and by Skovgaard for exponential families. They did not find wide use, partly because the computations required high-dimensional integration, and partly perhaps because their properties were not well understood. More recently Davison, Fraser, Reid and Sartori, and Fraser, Reid and Sartori showed how higher order asymptotic theory could be used to enable computation of directional p-values by one-dimensional integration. In essence directional tests create a one-parameter sub-model by focusing on the conditional distribution of a statistic that measures the distance of a vector parameter from an hypothesized value, given its direction from the null value. This requires an underlying vector space, which is straightforward in the case of linear regression models, and in the case of linear exponential family models. In more general models an initial approximations is used to create this vector space.

In this paper the authors study models where analytic expressions for the integrals can be obtained, and show that the directional test is essentially equivalent to an F-tests in these models.

The models studied in this paper include comparison of rates in two exponential distributions, comparison of variances in normal theory problems, tests of hypotheses on regression components in a normal theory linear regression, and Hotelling’s test for the expected value of a multivariate normal distribution. The first two models involve just a scalar parameter of interest, but the calculations involved illuminate the two-sided nature of directional tests and suggest an analytic approach to the latter two models. In the linear regression model the directional test turns out to be equivalent to the omnibus F-test, so although the construction is conditional, the resulting inference is unconditional. This somewhat surprising result helps to shed light on the inferential properties of directional tests.