Formulas for the Computation of the WILCOXON Test and other Rank Statistics

A simple program for the computation of the WILCOXON test is introduced, which is suitable for implementation on a programmable desk computer. The allocation of ranks R_i is not necessary (see BERCHTOLD, 1979). In an analogous manner any ranks respectively rank sums also can be calculated, so that any test criterion or statistical estimate called rank statistic, may be completely constituted by computer programs without explicitely ranging of values.     

An intrinsic rank test for k independent samples

Considerations on the sensitivity of the KRUSKAL-WALLIS Analysis of Variance by Ranks suggest an alternative method that is less sensitive to spurious interchange of ranks and tied ranks yet offers sufficient power. The Intrinsic Rank Test for k Independent Samples is proposed in this sense as an alternative method to the KRUSKAL- WALLIS test. The Intrinsic Rank Test is based on rank information, but the criterion ranks (intrinsic ranks) are not directly derïved from the observations, but rather indirectly, by means of a further transformation, from the ordinal ranks assigned in the usual manner. The incidence of the intrinsic ranks upon the k samples is then used to calculate a X²-like test statistic, which is readily evaluated with the Gamma distribution.     

Exact,Distribution Free Confidence Intervals for Late Effects in Censored Matched Pairs

When comparing censored survival times for matched treated and control subjects, a late effect on survival is one that does not begin to appear until some time has passed. In a study of provider specialty in the treatment of ovarian cancer, a late divergence in the Kaplan–Meier survival curves hinted at superior survival among patients of gynecological oncologists, who employ chemotherapy less intensively, when compared to patients of medical oncologists, who employ chemotherapy more intensively; we ask whether this late divergence should be taken seriously. Specifically, we develop exact, permutation tests, and exact confidence intervals formed by inverting the tests, for late effects in matched pairs subject to random but heterogeneous censoring. Unlike other exact confidence intervals with censored data, the proposed intervals do not require knowledge of censoring times for patients who die. Exact distributions are consequences of two results about signs, signed ranks, and their conditional independence properties. One test, the late effects sign test, has the binomial distribution; the other, the late effects signed rank test, uses nonstandard ranks but nonetheless has the same exact distribution as Wilcoxon's signed rank test. A simulation shows that the late effects signed rank test has substantially more power to detect late effects than do conventional tests. The confidence statement provides information about both the timing and magnitude of late effects (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A sampling study of the size and power of tests for paired survival data

The paired-t, sign, and signed rank tests were compared for samples from a bivariate exponential distribution. Each is a valid α-level test. One test was not uniformly more powerful than the others for all sample sizes, α levels, correlations, and alternative hypotheses considered, but the signed rank test did well consistently. It was always preferable to the sign test and never was appreciably worse than the paired-t test. The relative performance of the tests depends on α as well as the sample size.     

An Exact Paired Rank Test

An exact rank test for two dependent samples based on overall mid‐ranks is discussed which can be applied to metric as well as to ordinal data. The exact conditional distribution of the test statistic given the observed vector of rank differences is determined. A recursion formula is given as well as a fast shift algorithm in SAS/IML code. Moreover, it is demonstrated that the paired rank test can be more powerful than other tests for paired samples by means of a simulation study. Finally, the test is applied to a psychiatric trial with longitudinal ordinal data.     

Male and Female Reproductive Success in <Emphasis Type="Italic">Macaca sylvanus</Emphasis> in Gibraltar: No Evidence for Rank Dependence

The relationship between social rank and reproductive success is one of the key questions for understanding differences in primate social group structures. We determined the paternity of 18 infants in a social group of Barbary macaques (Macaca sylvanus) born over a period of 6 yr in the provisioned, free-ranging colony in Gibraltar. We successfully used 13 pairs of primers of variable microsatellite loci to amplify DNA from blood and hair samples and applied the computer programs CERVUS 2.0 and KINSHIP 1.3 to assign paternity to 13 candidate males. We collected data for 19 females that had given birth to 66 infants over a period of 7 yr. We used paternity analyses and female birth records to test the hypothesis that social rank is correlated with reproductive success. Results showed that numbers of paternities and maternities were equally distributed among all reproducing individuals in the social group regardless of rank. Subadult males reproduced as often as adult males. High-ranking females did not start to reproduce earlier than low-ranking females. Interestingly, there was a tendency toward a positive correlation between the ranks of mothers and the ranks of the corresponding fathers. It might be concluded either that a correlation between social rank and reproductive success is generally absent in Barbary macaques or that artificially favorable environmental conditions in Gibraltar preclude any correlation between social rank and reproductive success.     

The Matched Pairs Problem with Exponential Variates

Two classes of tests for the hypothesis of bivariate symmetry are studied. For paired exponential survival times (t_1j, t_2j), the classes of tests are those based on t_1j-t_2j and those based on log t_1j–log t_2j. For each class the sign, signed ranks, t and likelihood ratio tests are compared via Pitman's criterion of asymptotic relative efficiency (ARE). For tests based on t_1j — t_2j, it is found that: (1) the efficacy of the paired t depends on the coefficient of variation (CV) of the pair means, (2) the signed rank test has the same ARE to the sign test as for the usual location problem. For tests based on log t_1j — log t_2j, the ARE comparisons reduce to the well-known results for the one-sample location problem for samples from a logistic density. Hence, the signed rank test is asymptotically efficient. Furthermore, analyses based on log t_1j — log t_2j are not complicated by the underlying pairing mechanism.     

Seasonal rank changes for adolescent and subadult natal males in a free-ranging group of rhesus monkeys

Changes in dominance rank for adolescent and subadult natal males in a semi-free-ranging rhesus macaque group were seasonal. Three 4-year-old natal males of the highest-ranking matriline occupied high ranks in the adult male dominance hierarchy during the premating period. In the mating season they dropped in rank, and this decrement was related to a concomitant drop in their alliances. After the mating season, these males rose in rank along with their two 3-year-old kin to occupy ranks 2–5 and 7–8 in the adult male dominance hierarchy. Three-year-old natal males of matrilines lower ranking than first did not become integrated into the hierarchy at this time.     

Long-term changes in dominance ranks among ring-tailed lemurs at Berenty Reserve, Madagascar

A study was conducted between 1989 and 2001 to monitor changes in the dominance ranks among adult ring-tailed lemurs (Lemur catta) at Berenty Reserve, Madagascar. Adult females were observed to be dominant over adult males. Their rank fluctuated greatly. However, in some troops, female rank orders were fairly stable over a period of several years. In general, male ranks were more unstable than female ranks. Most young females aged 3 years occupied the lowest ranks among adult females. However, several were also observed to have attained relatively higher ranks, placing them right beneath their high-ranking mothers; this suggested the existence of dependent ranks. Mothers were dominant over their daughters. Similarly, older sisters were usually dominant over younger sisters. The mean duration of alpha status for females was 1.95 years, although considerable variation was observed in the duration of the alpha status (1–5 or more years). Most young males aged 3 years initially occupied the lowest ranks in their natal troops, and then they migrated to non-natal troops around the age of 4 years. They ascended in rank between the ages of 4 and 6 years, although there was considerable variation in the acquirement of high rank. The mean duration of alpha status for males was 2.2 years. Larger males were observed to occupy higher ranks. Occasionally, both males and females showed intense aggression (i.e., targeting aggression) towards others.     

Grooming Between Male Chimpanzees at Ngogo,Kibale National Park. II. Influence of Male Rank and Possible Competition for Partners

Allogrooming contributes to the development and maintenance of social relationships, including those that involve alliances, in many primate species. Variation in relatedness, dominance rank, and other factors can produce variation in the value of others as grooming partners. Several models have been developed to account for variation in the distribution of grooming in relation to dominance ranks. These start from the premise that individuals are attracted to high-ranking partners, but time limits, direct competition, and prior grooming engagement between high-ranking individuals can constrain access to them. Sambrook et al. (1995) formalized some of these models and showed the importance of taking group size variation into account when assessing them. Chimpanzees form multimale communities in which males are the philopatric sex. Males commonly associate and groom with each other; they also form dominance hierarchies and form alliances that influence dominance ranks and mating success. Both male rank and the rank distance between partners are significantly correlated with the distribution of grooming between males in an extremely large chimpanzee community at Ngogo, Kibale National Park, Uganda, that has more males than any other known community. High-ranking males had more grooming partners than mid- or low-ranking males. Grooming predominantly went up the dominance hierarchy, but was also concentrated among males that were close in rank. Rank and rank distance apparently both affected grooming independently of reciprocity in grooming and independently of the frequency with which males associated in temporary parties. However, the data do not clearly indicate how constraints on access to partners might have operated. Published data from a smaller chimpanzee community at Mahale show no rank or rank distance effect on male grooming. These results and earlier, conflicting findings on the association between dominance rank and grooming in male chimpanzees indicate that variation in group size, i.e., the number of males per community, probably influences the strength of any such effects, as happens for grooming between females in several cercopithecine species. Data on coalitions at Ngogo support the argument that high-ranking males are valuable social partners, and similarity in strategies of alliance formation may influence the distribution of grooming.     

Matrilineal rank Inheritance varies with absolute rank in Japanese macaques

In many cercopithecine primates, females form linear dominance hierarchies based on kinship. It is known that female rank follows the rules of matrilineal rank inheritance (MIR): (1) maternal rank inheritance, (2) maternal dominance, and (3) youngest ascendancy among sisters. Although, several determining such variation remain largely unknown. In this paper, I investigate the dominance relation-ships of 69 adult (>6 yr old) female Japanese macaques (Macaca fuscata fuscata) in a free-ranging provisioned troop living in Shiga-Heights (Nagano Prefecture, Japan) and report new evidence of intra-group variation. Dominance relationships among high-ranking females followed MRI within kin units, those among low-ranking females did not. Maternal rank inheritance and youngest ascendancy operated between mother/daughter dyads and sister dyads of high-rank, but not in the dyads of low-rank. The dominance ranks of females from low-ranking kin units were dispersed and less predictable. These findings suggest that MRI varies with absolute dominance rank, and are discussed in relation to other asymmetries between high-and low-rank     

Intertroop Transfer and Dominance Rank Structure of Nonnatal Male Japanese Macaques in Yakushima, Japan

We examined the interaction between intertroop transfer and male dominance ranks in a wild population of Japanese macaques (Macaca fuscata yakui) in Yakushima using data collected over 15 years. Intertroop transfer tended to maintain a linear, stable, and age-graded dominance rank order among nonnatal males irrespective of variation in troop size or composition. All males that joined a troop at the top of the rank order were prime adults. Among males joining at lower ranks, entry at the most subordinate position in the hierarchy was common. Males joining at lower ranks tended to join troops in which all other resident males were the same age or older. Adult males tended to join troops with few or no males. Young males tended to join troops with many resident males, and in which a relatively large proportion of males was other young ones. Intertroop transfer was responsible for most rank changes of resident males. The most common cause of males rising in rank was the emigration or death of a higher-ranking male. Males fell in rank most frequently as a result of a new male joining the troop at the top of the hierarchy. Rank reversals among resident males were rare. The cumulative effects of male transfers produce sociodemographic variation within a troop over time and sociodemographic diversity among troops in a local population. A key feature of intertroop diversity is that larger troops have a significantly greater proportion of young males than smaller troops. This diversity also creates the potential for intertroop variation in the severity of male competition and provides a range of options for transferring males.     

Is spectral distance a proxy of beta diversity at different taxonomic ranks? A test using quantile regression

Beta diversity represents a powerful indicator of ecological conditions because of its intrinsic relation with environmental gradients. In this view, remote sensing may be profitably used to derive models characterizing or estimating species turnover over an area. While several examples exist using spectral variability to estimate species diversity at several spatial scales, most of these have relied on standard correlation or regression approaches like the common Ordinary Least Square (OLS) regression which are problematic with noisy data. Moreover, very few attempts were made to derive beta diversity characterization models at different taxonomic ranks. In this paper, we performed quantile regression to test if spectral distance represents a good proxy of beta diversity considering different data thresholds and taxonomic ranks. We used plant distribution data from the North and South Carolina including 146 counties and covering a variety of vegetation formations. The dissimilarity in species composition at different taxonomic ranks (using Sørensen distance) among pairs of counties was compared with their distance in NDVI values derived from 23 yearly MODIS images. Our results indicate that (i) spectral variability represents a good proxy of beta diversity when appropriate statistics are applied and (ii) a lower taxonomic rank is important when changes in species composition are examined spatially using remotely sensed data.     

Effect of a Treatment with Naphthaleneacetic Acid on the Arrangement of Lateral Roots in Onion (Allium cepa L): General Pattern and Coordination between Ranks

Lateral roots in Allium cepa arise in longitudinal rows opposite the protoxylem poles of adventitious roots. The number, spacing within ranks and positional relationship to laterals in different ranks were analysed in control roots and those treated with an auxin (α-naphthaleneacetic acid). Treatment increased numbers of laterals per rank and distributed the more evenly in all ranks. Laterals of control roots were concentrated in two neighbouring ranks. Auxin-treated roots showed a more regular distribution of laterals between ranks and close spacing of laterals along each rank. Comparisons with theoretical random distributions suggest a dispersed spacing model for lateral root arrangement with mutual repulsion between successive lateral roots within each rank both in control and auxintreated roots. On the other hand, there is some interaction between laterals in different ranks in 0.1 mM NAA-treated roots.     

The Nonparametric Behrens‐Fisher Problem: Asymptotic Theory and a Small‐Sample Approximation

A generalization of the Behrens‐Fisher problem for two samples is examined in a nonparametric model. It is not assumed that the underlying distribution functions are continuous so that data with arbitrary ties can be handled. A rank test is considered where the asymptotic variance is estimated consistently by using the ranks over all observations as well as the ranks within each sample. The consistency of the estimator is derived in the appendix. For small samples (n₁, n₂ ≥ 10), a simple approximation by a central t‐distribution is suggested where the degrees of freedom are taken from the Satterthwaite‐Smith‐Welch approximation in the parametric Behrens‐Fisher problem. It is demonstrated by means of a simulation study that the Wilcoxon‐Mann‐Whitney‐test may be conservative or liberal depending on the ratio of the sample sizes and the variances of the underlying distribution functions. For the suggested approximation, however, it turns out that the nominal level is maintained rather accurately. The suggested nonparametric procedure is applied to a data set from a clinical trial. Moreover, a confidence interval for the nonparametric treatment effect is given.     

Early agonistic experience and rank acquisition among free-ranging infant rhesus monkeys

Several aspects of agonistic experience are described for freeranging infant rhesus monkeys (Macaca mulatta)on Cayo Santiago. Even before infants are fully integrated with peers in rank dominance based on maternal ranks,infants of highranking mothers tend to be threatened less frequently by other members of the group and are less likely to be threatened by unfamiliar individuals than are infants of lowranking mothers. There is no evidence that fearful interactions between pairs of infants are related to their mother’s ranks before 22 weeks of age. However, an imperfect hierarchy can be constructed for infants between 27 and 30 weeks of age. At this age,infants of higherranking mothers are also more likely to receive protection when threatened than are infants of lowerranking mothers. When protected, their protectors are less likely to emit fearful gestures to the infants’ threatener. Close female relatives appear to play a large role in the protection of infants and may be more directly responsible for the differences described above than the mother, other relatives, or other highranking members of the group. It is suggested that more than one mechanism, including intervention by the mother and by close female relatives,may be important in rank acquisition among peers.     

Orthogonal Components of the Kruskal-Wallis and Related K-sample Linear Rank Statistics

It is shown that the KRUSKAL -WALLIS statistic may be decomposed into components designed to detect trends in the underlying distributions. Extensions to other K sample linear rank statistics and to nonparametric regression statistics are noted.     

Social rank and sex ratio of captive long-tailed macaque females (Macaca fascicularis)

Variation in birth sex ratios in primates can be accounted for by two hypotheses: the local resource competition hypothesis [Silk: American Naturalist 121:56–66, 1983] and the hypothesis of Trivers & Willard [Science 179:90–92, 1973] concerning the maternal effect on the quality of a male. We examined the effects of female dominance rank on aspects of reproduction in three well-established captive groups of long-tailed macaques (Macaca fascicularis). High-ranking females produced a higher proportion of sons than low-ranking females, and factors other than rank did not have significant effects on birth sex ratios. Interbirth intervals following daughters were longer than those following sons, but they were independent of the mother's rank. The sons of high-ranking mothers had better survival prospects than sons of low-ranking mothers in some of the groups; no such difference was found for daughters. Overall, there was no sex difference in survival up to 5 years of age. These results support the Trivers-Willard hypothesis rather than the local resource competition hypothesis. An analysis of interbirth intervals suggested that the deviation in birth sex ratio is already established at conception.     

Comparison of social ranks based on worm-running and aggressive behaviour in young domestic fowl

Worm-running is behaviour in which a chick runs carrying a worm-like object while flock mates follow and attempt to grab the object from its beak. We hypothesised that social ranks based on worm-running frequency are stable over time and are positively correlated with social ranks based on success in aggressive interactions when older. At 8-12 days of age, we scored worm-running in 17 groups of 12 female White Leghorn chicks during three 10-min tests. Based on instantaneous scans at 5-s intervals, the bird carrying the 'worm' most often was placed in rank one and so on down the rank order. These tests were repeated at 68-70 days of age. An aggression index for each bird was calculated as the number of aggressive acts given, divided by the number given and received, during three 1-h observation periods when the birds were 68-70 days. Ranks obtained in worm-running tests were positively correlated over the two age periods (P < 0.05) but were not correlated with ranks based on the aggression index (P > 0.05). Our results indicate that worm-running ranks are not predictive of success in aggressive interactions. Instead, worm-running fits some criteria for play.     

Dominance rank predicts social network position across developmental stages in rhesus monkeys

Social network analysis is increasingly common in studying complex interactions among individuals. Across a range of primates, high-ranking adults are generally more socially connected, which results in better fitness outcomes. However, it still remains unclear whether this relationship between social network position and dominance rank emerges in infancy and whether, in species with a social transmission of dominance rank, social network positions are driven by the presence of the mother. To fill this gap, we first explored whether dominance ranks were related to social network position, measured via eigenvector centrality, in infants, juveniles, and adults in a troop of semi-free-ranging rhesus macaques (Macaca mulatta). We then examined relationships between dominance rank and eigenvector centrality in a peer-only group of yearlings who were reared with their mothers in either a rich, socially complex environment of multigenerational (MG) kin support or a unigenerational group of mothers and their infants from birth through 8 months. In Experiment 1, we found that mother's network position predicted offspring network position and that dominants across all age categories were more central in affiliative networks (social contact, social grooming, and social play). Experiment 2 showed that high-ranking yearlings in a peer-only group were more central only in the social contact network. Moreover, yearlings reared in a socially complex environment of MG kin support were more central. Our findings suggest that the relationship between dominance rank and social network position begins early in life, and that complex early social environments can promote later social competency. Our data add to the growing body of evidence that the presence/absence of the mother and kin influence how dominance rank affects social network position. These findings have important implications for the role of caregivers in the social status of developing primates, which ultimately ties to health and fitness outcomes.