Testing for normality in truncated anthropometric samples

Abstract

Anthropometric historical analysis depends on the assumption that human characteristics—such as height—are normally distributed. I propose and evaluate a metric entropy, based on nonparametrically estimated densities, as a statistic for a consistent test of normality. My first test applies to full distributions for which other tests already exist and performs similarly. A modified version applies to truncated samples for which no test has been previously devised. This second test exhibits correct size and high power against standard alternatives. In contrast to the distributional prior of Floud et al. (1990), the test rejects normality in large parts of their sample; the remaining data reveal a downward trend in height, not upward as they argue.

Contribution

This paper adds two useful tools to the researchers’ arsenal of tests aimed at detecting departures from normality. In contrast to the existing parametric tests, the tests proposed here are consistent tests building on a metric entropy based on nonparametrically estimated densities. Importantly, their performance is quite remarkable in simulated data.

The first test applies to full distributions and is shown to have a performance in line with the performance of its parametric counterparts. The second test is alone in its class as it is the first to apply to truncated samples that are commonplace in the field. Size and power investigations show again reasonably good behaviour of the test.

The classic data set of Floud et al. (1990) is re-analysed in the light of these new tests. It is shown that the normal distributional prior adopted by these authors, and the current literature is an inappropriate description of the recruits’ height distribution in most cases. This is particularily true for the youngest individuals. The consequence of these tests is quite dramatic. The upward secular trend drawn by the Floud et al. (1990) estimates turns—if one restricts calculations to validly inferred estimated average heights—into a downward secular trend also previously obtained by other scholars like Komlos (1993).