Body size estimation of small‐bodied humans: Applicability of current methods

Body size (stature and mass) estimates are integral to understanding the lifeways of past populations.Body size estimation of an archaeological skeletal sample can be problematic when the body size or proportions of the population are distinctive. One such population is that of the Holocene Later Stone Age (LSA) of southern Africa, in which small stature (mean femoral length = 407 mm, n = 52) and narrow pelves (mean bi‐iliac breadth = 210 mm, n = 50) produce a distinctive adult body size/shape, making it difficult to identify appropriate body size estimation methods. Material culture, morphology, and culture history link the Later Stone Age people with the descendant population collectively known as the Khoe‐San. Stature estimates based on skeletal “anatomical” linear measures (the Fully method) and on long bone length are compared, along with body mass estimates derived from “morphometric” (bi‐iliac breath/stature) and “biomechanical” (femoral head diameter) methods, in a LSA adult skeletal sample (n = 52) from the from coastal and near‐coastal regions of South Africa. Indices of sexual dimorphism (ISD) for each method are compared with data from living populations. Fully anatomical stature is most congruent with Olivier's femur + tibia method, although both produce low ISD. McHenry's femoral head body mass formula produces estimates most consistent with the bi‐iliac breadth/staturemethod for the females, although the males display higher degrees of disagreement among methods. These results highlight the need for formulae derived from reference samples from a wider range of body sizes to improve the reliability of existing methods. Am J Phys Anthropol, 2010. © 2009 Wiley‐Liss, Inc.     

The Determination of Sample Sizes in the Comparison of Two Multinomial Proportions from Ordered Categories

We consider sample size determination for ordered categorical data when the alternative assumption is the proportional odds model. In this paper the sample size formula proposed by Whitehead (Statistics in Medicine, 12 , 2257–2271, 1993) is compared with the methods based on exact and asymptotic linear rank tests with Wilcoxon and trend scores. We show that Whitehead's formula, which is based on a normal approximation, works well when the sample size is moderate to large but recommend the exact method with Wilcoxon scores for small sample sizes. The consequences of misspecification in models are also investigated.     

Evaluation of Noether's Method of Sample Size Determination for the Wilcoxon-Mann-Whitney Test

NOETHER (1987) proposed a method of sample size determination for the Wilcoxon-Mann-Whitney test. To obtain a sample size formula, he restricted himself to alternatives that differ only slightly from the null hypothesis, so that the unknown variance o² of the Mann-Whitney statistic can be approximated by the known variance under the null hypothesis which depends only on n. This fact is frequently forgotten in statistical practice. In this paper, we compare Noether's large sample solution against an alternative approach based on upper bounds of σ² which is valid for any alternatives. This comparison shows that Noether's approximation is sufficiently reliable with small and large deviations from the null hypothesis.     

A Note on Measures of Multivariate Kurtosis

We propose a measure of multivariate kurtosis suggested from Mardia's measure of multivariate skewness b₁,p, and examine its relationship both to Mardia's measure of multivariate kurtosis b₂,p, and to a smooth test of multivariate kurtosis ǔ₄².     

Inconstancy of Taylor'sb: Simulated sampling with different quadrat sizes and spatial distributions

Taylor 's power law, s²=am^b, provides a precise summary of the relationship between sample variance (s²) and sample mean (m) for many organisms. The coefficient b has been interpreted as an index of aggregation, with a characteristic value for a given species in a particular environment, and has been thought to be independent of the sample unit. Simulation studies were conducted that demonstrate that the value of b may vary with the size of the sample unit in quadrat sampling, and this relationship, in turn, depends on the underlying spatial distribution of the population. For example, simulated populations with hierarchical aggregation on a large scale produced values of b that increased with the size of the sample unit. In contrast, for a simulated population with randomly distributed clusters of individuals, the value of b eventually decreased with increasing quadrat size, as sample counts became more uniform. A single value ofTaylor 's b, determined with a particular sample unit, provides neither a fixed index of aggregation nor a complete picture of a species' spatial distribution. Rather, it describes a consistent relationship between sample variance and sample mean over a range of densities, on a spatial scale related to the size of the sample unit. This relationship may reflect, but not uniquely define, density-dependent population and behavioral processes governing the spatial distribution of the organism. Interpretation ofTaylor 'sb for a particular organism should be qualified by reference to the sample unit, and comparisons should not be made between cases in which different sample units were used. Whenever possible, a range of sample units should be used to provide information about the pattern of distribution of a population on various spatial scales.     

Properties and Application of the Location-Test V (n,k)

The present paper is concerned with the properties of a test statistic V(n, k) to test location differences in the one-sample case with known hypothetical distribution G(x). The test is similar to the WILCOXON two-sample statistic after replacement of the second sample by quantiles of the hypothetical distribution. A comparison with the exact distribution of V(n, k) shows that an approximation by means of the normal distribution provides good results even for small sample sizes. The V-test is unbiased against one-tailed alternatives and it is consistent with a restriction which is hardly relevant in practical applications. With regard to the application we are interested especially in the power and robustness against extreme observations for small sample size n. It is shown that in a normal distribution with known standard deviation V(n, k) is more powerful than STUDENT's t for small n and more robust in the sense considered here. The test statistic is based on grouping of the observations into classes of equal expected frequency. A generalization to arbitrary classes provides an essential extension of applicability such as to discrete distributions and to situations where only relative frequencies of G(x) in fixed classes are known.     

Mixed Moment Estimation in Hermite Distribution and its Comparison with Other Estimation Procedures from Sample Size Considerations

In this paper (1) expressions (correct to n^?2 terms) for biases, variances, and covariances of the estimators a and b of Hermite distribution with probability generating function Exp[a(t–1) + b(t–1)] are obtained for two mixed moment estimates; (2) for the biases and variance-covariances, approximate regions of the parameter space (a>0, b>0) have been outlined where a sample of size 100 can be considered as “safe” in the sense that contribution of second order terms in them is 5% of that from the first order term; (3) comparison of the biases and variance-covariances of these two sets of estimators are made with those for the moment estimators, maximum likelihood estimates and the even point estimators for a sample of size 100 using the terms up to order n^?2; (4) the comparisons based on n^?2 terms in (3) have not only provided information on the estimation procedures included in the Hermite distribution, but also demonstrated the importance of higher order terms in the sampling properties of the various alternative techniques for the Hermite distribution.     

Spatial pattern indices based on distances between individuals on a line-segment with finite length

Let us consider a strip-wise habitat of line-segment, like a corridor, to simplify the subject mathematically, and assume that the length of the habitat is γ and there are n individuals. Here, we assume that the spatial pattern of the individuals is random if the n distances from the left end of the habitat to each individual follow a uniform distribution on the strip. Under such an assumption, the variance of the distances between any two neighbors is represented by the formula nθ²(n+1)⁻²(n+2)⁻¹ and the variance between n+1 distances between n individuals from the left end to the right end to the strip, is represented by the formula nθ²(n+1)⁻²(n+2)⁻¹. These two kinds of variances can be used for determining (1) the spatial pattern of a population on the strip and (2) the spatial structure within the population, by comparison with the variances calculated from the data. Two examples cited from the literature, a cattle population on a pasture and an aphid population on a sycamore leaf, are presented.     

The effectiveness of using co-dominant polymorphic allelic series for (1) checking pedigrees and (2) distinguishing full-sib pair members

Formulae express the effectiveness of parentage exclusion tests and differences separating full-sib pairs by compounding genotypic information on discrete examples of co-dominant alleles segregating at gene loci on different autosomes. Such polymorphisms occur among structural genes and polymorphic DNA sequences. Two general formulae state the theoretical effectiveness of using co-dominant alleles for (1) testing parentage and (2) distinguishing sibs. The formula for parentage exclusion tates the probability (P_E) that a given series of co-dominant alleles of known frequency should detect a falsely recorded father (or mother). The other formula describes how genetic polymorphism can distinguished closely related individuals. It states the probability (P_S) that alleles distinguish the members of full-sib pairs, dizygotic twins and tissue chimeras. To derive the two general formulae, particular formulae were calculated for n = 2,3 and 4 co-dominant alleles. By increasing the numbers of alleles, the formulae were seen to contain recurrent patterns which were then expressed in the two general formulae for n alleles. Some examples demonstrate applications of the two formulae in problems concerning parentage and sibship.     

Nucleosome unit repeat size in aneuploid mice

Male 8-day-old mice that have part of chromosome 7 translocated to an X chromosome [T(X;7)1Ct] and that are chromosomally unbalanced for chromosome 7, and consequently trisomic for that part of chromosome 7, were found to have a smaller nucleosome repeat unit size than normal littermate males (Rake, A. V., and Edwards, R. H.,Biochem. Genet. 25:671, 1987). This smaller nucleosome size is maintained in adult trisomic males. Males with a balanced chromosomal translocation [T(X;7)1Ct] had a normal nucleosome size compared to their littermates. The nucleosome unit size is not altered in two other types of aneuploid mice studied (XO vs XX, 2n=39 and 40, respectively; and Ts12¹⁷ vs normal, 2n=40 and 41, respectively).     

Using a Normal Approximation to Test for the Binomial Distribution

The commonly used method to test for the binomial distribution is the x²-test. In this paper, we introduce an alternative method to test for the binomial distribution. Suppose N is the number of sample groups with n individuals each, x_ij is the jth sample in ith group, a Bernoulli variable with parameter and VVI=s²/[m(1 - m)/n]. Then it is well know that the asymptotic distribution of the statistic (N - 1) VVI is x²(N - 1) under the hypothesis p₁ = p₂ = … = p_N. Here we find that VVI has an asymptotic normal distribution N(1, 2(1 - 1/n)/(N - 1)). Unlike the x²-statistic, the variance of the normal test statistic is a function of n. This method is convenient in detecting spatial patterns and dispersion in the study of diseased organisms (e.g., plants) in field samples.     

Longitudinal somatometrical study on the growth patterns of newborn Japanese monkeys

Growth of Japanese macaques during their first year was analyzed longitudinally, using body measurements. Measurements of 44 somatometrical characters were taken on seven animals. Work with the newborn data produced a formula which fits well. The formula is: y=a (x+b) ⁿ (y=size, x=age). At first, growth was analyzed, character by character, using birth sizes and incremental increases. The results show a major growth pattern for many characters: increments vary inversely with birth size. Application of the growth formula produced two further insights: (1) growth pattern is not so simple as imagined from the birth size-increment pattern; and (2) the characters which deviate from the birth size-increment pattern have large growth only at the earliest period (in few months), in spite of their small size. Sex differences were clear within the first year, especially for characters which differ greatly between adults.     

Multiple Comparisons of Polynomial Distributions

Asymptotically correct 90 and 95 percentage points are given for multiple comparisons with control and for all pair comparisons of several independent samples of equal size from polynomial distributions. Test statistics are the maxima of the X²-statistics for single comparisons. For only two categories the asymptotic distributions of these test statistics result from DUNNETT'S many-one tests and TUKEY'S range test (cf. MILLER, 1981). The percentage points for comparisons with control are computed from the limit distribution of the test statistic under the overall hypothesis H₀. To some extent the applicability of these bounds is investigated by simulation. The bounds can also be used to improve Holm's sequentially rejective Bonferroni test procedure (cf. HOLM, 1979). The percentage points for all pair comparisons are obtained by large simulations. Especially for 3×3-tables the limit distribution of the test statistic under H₀ is derived also for samples of unequal size. Also these bounds can improve the corresponding Bonferroni-Holm procedure. Finally from SKIDÁK's probability inequality for normal random vectors (cf. SKIDÁK, 1967) a similar inequality is derived for dependent X²-variables applicable to simultaneous X²-tests.     

On A Statistical Test to Detect Spatial Pattern

In the analysis of spatial patterns, the most extensively used index of dispersion is s²/m in which s² and m are respectively the sample variance and sample mean of the count x in each sample unit. For statistical testing, the statistic I' = (n — 1) s²/m has been introduced since it has an approximate X distribution under the null hypothesis that individuals are distributed randomly. The main problem with the use of index I' is that the random distribution and the superdispersed distribution are not distinguished. In this paper, we have tried to induce a new statistical method for the detection of spatial pattern, based on the special characteristic of Poisson distribution that both of the m and s² are the unbiased estimates of the parameter.     

Further Study on the Statistical Test to Detect Spatial Pattern

In the study of spatial patterns, the statistic I' = (n — 1)s²/x was commonly used. In this paper, we found that x — s² has an approximated normal distribution with zero mean if the x_i's (i = 1 to n) are independent identically distributed Poisson variables. Based on this conclusion, the hypothesis that a point pattern is completely random can be tested directly. And a method for the test of spatial patterns was proposed which can be sued as an alternative to the Chi-square based dispersion index test.     

A Comparative Statistical Analysis of Genetic Distances. I - Estimation of Distances and of their Variances: Distribution of the Estimators

We present a Monte-Carlo simulation analysis of the statistical properties of absolute genetic distance and of Nei's minimum and standard genetic distances. The estimation of distances (bias) and of their variances is analysed as well as the distributions of distance and variance estimators, taking into account both gamete and locus samplings. Both of Nei's statistics are non-linear when distances are small and consequently the distributions of their estimators are extremely asymmetrical. It is difficult to find theoretical laws that fit such asymmetrical distributions. Absolute genetic distance is linear and its distributions are better fit by a normal distribution. When distances are medium or large, minimum distance and absolute distance distributions are close to a normal distribution, but those of the standard distance can never be considered as normal. For large distances the jack-knife estimator of the standard distance variance is bad; another standard distance estimator is suggested. Absolute distance, which has the best mathematical properties, is particularly interesting for small distances if the gamete sample size is large, even when the number of loci is small. When both distance and gamete sample size are small, this statistic is biased.     

Robustness of Statistical Methods II. Methods for the one-Sample Problem

This paper presents a new exact method of the investigation of the robustness in the one-sample-case of the u-, t- and χ²-statistics for discrete alternatives to the underlying normal distribution. First results for sample sizes up to n = 230 are given.     

One-Way Classification ANOVA Model for the Mixture of Two Multivariate Normal Populations

When a sample of size n is available from a mixture of two normal populations with different mean vectors and a common covariance matrix, SRIVASTAVA and AWAN (1982) develop one-way ANOVA analysis for testing a certain composite linear hypothesis. They show that the error and hypothesis sum of products matrices have independent noncentral Wishart densities of rank unity each. However, they do not obtain the necessary Wilks' λ for testing the desired hypothesis. The present paper obtains the density of λ. This density is doubly noncentral multivariate beta density. The derivation is based on generalized Sverdrup's lemmas, KABE (1965), (1974).     

K‐Sample Test and Sample Size Calculation for Comparing Slopes in Data with Repeated Measurements

Sample size calculations based on two‐sample comparisons of slopes in repeated measurements have been reported by many investigators. In contrast, the literature has paid relatively little attention to the design and analysis of K‐sample trials in repeated measurements studies where K is 3 or greater. Jung and Ahn (2003) derived a closed sample size formula for two‐sample comparisons of slopes by taking into account the impact of missing data. We extend their method to compare K‐sample slopes in repeated measurement studies using the generalized estimating equation (GEE) approach based on independent working correlation structure. We investigate the performance of the sample size formula since the sample size formula is based on asymptotic theory. The proposed sample size formula is illustrated using a clinical trial example. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)