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In search of golden rules: Comment on hypothesis-testing approaches to setting cutoff values for fit indexes and dangers in overgeneralizing Hu and Bentler's (1999) findings.

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Many biological objects possess bilateral symmetry about a midline or midplane, up to a "noise" term. This paper uses landmark-based methods to measure departures from bilateral symmetry, especially for the two-group problem where one group is more asymmetric than the other. In this paper, we formulate our work in the framework of size-and-shape analysis including registration via rigid body motion. Our starting point is a vector of elementary asymmetry features defined at the individual landmark coordinates for each object. We introduce two approaches for testing. In the first, the elementary features are combined into a scalar composite asymmetry measure for each object. Then standard univariate tests can be used to compare the two groups. In the second approach, a univariate test statistic is constructed for each elementary feature. The maximum of these statistics lead to an overall test statistic to compare the two groups and we then provide a technique to extract the important features from the landmark data. Our methodology is illustrated on a pre-registered smile dataset collected to assess the success of cleft lip surgery on human subjects. The asymmetry in a group of cleft lip subjects is compared to a group of normal subjects, and statistically significant differences have been found by univariate tests in the first approach. Further, our feature extraction method leads to an anatomically plausible set of landmarks for medical applications.

Summary Directional data analysis is emerging as an important area of statistics. Within the past two decades, various new techniques have appeared, mostly to meet the needs of scientific workers dealing with directional data. The paper first introduces the two basic models for the multi‐dimensional case known as the von Mises–Fisher distribution and the Bingham distribution. Their sampling distribution theory depends heavily on the isotropic case and some developments are discussed. An optimum property of an important test for the von Mises–Fisher case is established. A non‐parametric test is proposed for the hypothesis of independence for observations on a torus. In addition to some numerical examples on the preceding topics, five case studies are given which illuminate the power of this new methodology. The case studies are concerned with cancer research, origins of comets, arrival times of patients, navigational problems and biological rhythms. Some unsolved problems are also indicated.