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| Although the subject of [[sexual dimorphism]] is not in itself controversial, the measures by which it is assessed differ widely. Most of the measures are used on the assumption that a [[random variable]] is considered so that [[probability distribution]]s should be taken into account. In this review, a series of '''sexual dimorphism measures''' are discussed concerning both their definition and the probability law on which they are based. Most of them are sample functions, or [[statistics]], which account for only partial characteristics, for example the [[mean]] or [[expected value]], of the distribution involved. Further, the most widely used measure fails to incorporate an [[Inferential statistics|inferential]] support.
| | == Moncler Geneve Rue Du Rhone Mieux encore == |
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| ==Introduction==
| | Sur la péninsule Pemaquid ou à proximité, il ya un certain nombre de réserves, ou des terres mises de côté à l'usage public et protégé de développement. Parce qu'il ya tellement d'eau, chaque que nous avons fait des vents vers l'océan ou d'une rivière. Je ne serais pas surpris si ces types de conserves sont parsemées à travers l'état. <br><br>110Rs. 1003Serene272Rs. 120Rs. En 2005, Fanball a été vendue à FUN Technologies, une société cotée en bourse. Rob est resté avec FUN après la vente et a été promu président de la division des sports. De 1994 98, Rob était le fondateur et propriétaire de la première semaine le magazine de sports d'imagination, Fantasy Football Weekly. <br><br>At-il été confirmé maintenant Werrington? C'est dans le forum Rangers et ils disent qu'il sera confirmée à la fin de la semaine. Tout le monde est très excité pour cela, nous aimons un voyage en Angleterre et a pris plus de Derby, il ya quelques années. Le fait-il que le SPL, dans l'ensemble, le football sert terrible.. <br><br>Ils ouvertement acheté de l'espace publicitaire sur le forum jihadiste et insérés annonces difficiles djihadiste propaganda.Bravo! L'IC, DOD et le DOS travailler ensemble et de travailler intelligemment. C'est ce pouvoir intelligent est tout au sujet. Et un enfer de beaucoup moins cher dans le sang et le trésor de guerre de prise de vue. <br><br>Mieux encore, dans certains cas, ils vont même évangéliser en notre nom! auteurs sont certainement recherchés. "Les meilleurs arbitres sont postdocs», dit Leslie [http://www.takefive.ch/mail/redirect.asp?lang=11-Moncler-Geneve-Rue-Du-Rhone Moncler Geneve Rue Du Rhone] Sage, un rédacteur en chef de sciences physiques à la nature, à Washington DC. «Ils sont au sommet de leur jeu, bien versé dans la littérature et politiquement assez naïf pour dire la vérité.". <br><br>Vous avez en cours d'exécution. Vous avez à marcher. Pourquoi ne pas faire du jogging? avec une vitesse donnée entre marche et course? semble facile: poids juste fixés comme: 50/50 pour quelque chose exactement entre les deux, ou 10/90 pour presque pleine course. <br><br>Il est un peu plus de flexibilité pour assister à un collège privé, comme il n'y a pas de question de la résidence de l'Etat. [http://www.takefive.ch/mail/redirect.asp?lang=68-Moncler-Geneve-Boots Moncler Geneve Boots] Pour les collèges [http://www.takefive.ch/mail/redirect.asp?lang=3-Moncler-France Moncler France] privés, le moins cher de 10% (c. $ 21 500 par an, en moyenne) est un peu moins de la moitié que de 10% plus chère ($ 48 500 par an, en moyenne). <br><br>Je ne pense qu'ils pensaient des sentiments de qui que ce soit mais le leur. Ce qu'ils ont fait va à l'encontre de la Terre, la nature et tout le monde ils vivent avec leur communauté. Ils sont évidemment des gens très égoïstes.. Bien sûr, Rudolph a subi de nombreuses lignes de l'histoire avant que la version finale, et désormais souvent mémorisé à la lumière. A l'origine, Rudolph a vécu dans un village de renne entièrement distinct, toujours la victime de railleries et moqueries, mais dans un ménage chaleureuse et aimante. Le Père Noël a eu connaissance du nez rougeoyant du [http://www.takefive.ch/mail/redirect.asp?lang=16-Moncler-Outlet-Swiss Moncler Outlet Swiss] renne, jusqu'à ce qu'il a attrapé un coup d'oeil de la lumière rouge sous la porte de la chambre de Rodolphe, tout en offrant des cadeaux à Rodolphe et de sa famille.<ul> |
| It is widely known that [[sexual dimorphism]] is an important component of the morphological [[variance|variation]] in biological populations (see, e.g., Klein and Cruz-Uribe, 1983; Oxnard, 1987; Kelley, 1993). In higher Primates, sexual dimorphism is also related to some aspects of the social organization and behavior (Alexander ''et al.'', 1979; Clutton-Brock, 1985). Thus, it has been observed that the most dimorphic species tend to [[polygyny]] and a social organization based on male dominance, whereas in the less dimorphic species, [[monogamy]] and family groups are more common. Fleagle ''et al.'' (1980) and Kay (1982), on the other hand, have suggested that the behavior of extinct species can be inferred on the basis of sexual dimorphism and, e.g. Plavcan and van Shaick (1992) think that sex differences in size among primate species reflect processes of an ecological and social nature. Some references on sexual dimorphism regarding human populations can be seen in Lovejoy (1981), Borgognini Tarli and Repetto (1986) and Kappelman (1996)!
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| | | <li>[http://www.0778xzs.com/bbs/forum.php?mod=viewthread&tid=963613 http://www.0778xzs.com/bbs/forum.php?mod=viewthread&tid=963613]</li> |
| These biological facts do not appear to be controversial. However, they are based on a series of different sexual dimorphism [[Measure (mathematics)|measures]], or indices. Sexual dimorphism, in most works, is measured on the assumption that a [[random variable]] is being taken into account. This means that there is a law which accounts for the behavior of the whole set of values that compose the domain of the random variable, a law which is called [[probability distribution|distribution function]]. Because both studies of sexual dimorphism aim at establishing differences, in some random variable, between sexes and the behavior of the random variable is accounted for by its distribution function, it seems reasonably clear{{to whom?|date=January 2014}} that a sexual dimorphism study should be equivalent to a study whose main purpose is to determine to what extent the two distribution functions - one per sex - overlap (see shaded area in Fig. 1, where two [[normal distribution]]s are represented).
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| [[Image:Overlapnorm.jpg|thumb|Fig. 1. Two normal distributions.]]
| | <li>[http://vp.plexicrew.free.fr/spipplexicrew/spip.php?article20/ http://vp.plexicrew.free.fr/spipplexicrew/spip.php?article20/]</li> |
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| ==Measures based on sample means==
| | <li>[http://www.cabinsnh.com/forum.php?mod=viewthread&tid=222851 http://www.cabinsnh.com/forum.php?mod=viewthread&tid=222851]</li> |
| In Borgognini Tarli and Repetto (1986) an account of indices based on [[arithmetic mean|sample means]] can be seen. Perhaps, the most widely used is the quotient,
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| | | <li>[http://lab.nqnwebs.com/lavoz_bak/spip.php?article3099/ http://lab.nqnwebs.com/lavoz_bak/spip.php?article3099/]</li> |
| :<math>\frac {\bar{X}_m}{\bar{X}_f} ,</math>
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| | | </ul> |
| where <math>\bar{X}_m</math> is the sample mean of one sex (e.g., male) and <math>\bar{X}_f</math> the corresponding mean of the other. Nonetheless, for instance,
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| :<math>\operatorname{log}\frac {\bar{X}_m}{\bar{X}_f} ,</math> <math>100\frac {\bar{X}_m - \bar{X}_f}{\bar{X}_f} ,</math> <math>100\frac {\bar{X}_m - \bar{X}_f}{\bar{X}_f + \bar{X}_f} ,</math>
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| have also been proposed.
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| Going over the works where these indices are used, one{{who?|date=January 2014}} misses any reference to their [[parameter|parametric]] counterpart.{{OR|date=January 2014}} In other words, if we suppose that the quotient of two sample means is considered, no work can be found where, in order to make [[Inferential statistics|inferences]], the way in which the quotient is used as a point [[estimator|estimate]] of
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| :<math>\frac {\mu_m}{\mu_f} ,</math>
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| is discussed.
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| By assuming that differences between populations are the objective to analyze, when quotients of sample means are used it is important to point out that the only feature of these populations that seems to be interesting is the mean parameter. However, a population has also variance, as well as a shape which is defined by its distribution function (notice that, in general, this function depends on parameters such as means or variances).
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| ==Measures based on something more than sample means==
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| Marini ''et al.'' (1999) have illustrated that it is a good idea to consider something other than sample means when sexual dimorphism is analyzed. Possibly, the main reason is that the intrasexual variability influences both the manifestation of dimorphism and its interpretation.
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| ===Normal populations===
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| ====Sample functions====
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| It is likely that, within this type of indices, the one used the most is the well-known statistic with [[Student's t-distribution|Student's ''t'' distribution]] (see, for instance, Green, 1989). Marini ''et al.'' (1999) have observed that variability among females seems to be lower than among males, so that it appears advisable to use the form of the Student's ''t'' statistic with degrees of freedom given by the Welch-Satterthwaite approximation,
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| :<math>T = \frac {\bar{X}_1 - \bar{X}_2 - (\mu_1 - \mu_2)} {\sqrt{\frac{S^2_1}{n_1} + \frac{S^2_2}{n_2}}} : t_\nu ,</math>
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| :<math>\nu = \frac {(\frac{S^2_1}{n_1} + \frac{S^2_2}{n_2})^2} {\frac{S^2_1}{n_1(n_1 - 1)} + \frac{S^2_2}{n_2(n_2 - 1)}} ,</math>
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| where <math>S^2_i, n_i, i=1,2</math> are sample variances and sample sizes, respectively.
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| Anyway, it is important to point out the following,
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| *when this statistic is taken into account in sexual dimorphism studies, two normal populations are involved. From these populations two random samples are extracted, each one corresponding to a sex, and such samples are independent.
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| *when inferences are considered, what we are testing by using this statistic is that the difference between two mathematical expectations is a hypothesized value, say <math>\mu_0 = \mu_1 - \mu_2.</math>
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| However, in sexual dimorphism analyses, it does not appear reasonably{{according to whom?|date=January 2014}} to assume that two independent random samples have been selected. Rather on the contrary, when we sample we select some random observations - making up one sample - that sometimes correspond to one sex and sometimes to the other.
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| ====Taking parameters into account====
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| Chakraborty and Majumder (1982) have proposed an index of sexual dimorphism that is the overlapping area - to be precise, its complement - of two normal [[probability density function|density]] functions (see Fig. 1). Therefore, it is a function of four parameters <math>\mu_i,\sigma^2_i, i=1,2</math> (expected values and variances, respectively), and takes the shape of the two normals into account. Inman and Bradley (1989) have discussed this overlapping area as a measure to assess the distance between two normal densities.
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| Regarding inferences, Chakraborty and Majumder proposed a sample function constructed by considering the Laplace-DeMoivre's theorem (an application to [[binomial distribution|binomial]] laws of the [[central limit theorem]]). According to these authors, the variance of such a statistic is,
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| :<math>\operatorname{var}(\widehat{D}) = \frac {\widehat{p}_m(1 - \widehat{p}_m)}{n_m} + \frac {\widehat{p}_f(1 - \widehat{p}_f)}{n_f} ,</math> | |
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| where <math>\widehat{D}</math> is the statistic, and <math>\widehat{p}_i, n_i, i=m,f</math> (male, female) stand for the estimate of the probability of observing the measurement of an individual of the <math>i</math> sex in some interval of the real line, and the sample size of the ''i'' sex, respectively. Notice that this implies that two independent random variables with binomial distributions have to be regarded. One of such variables is ''number of individuals of the f sex in a sample of size <math>n_f</math> composed of individuals of the f sex'', which seems nonsensical.
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| ===Mixture models===
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| Authors such as Josephson ''et al.'' (1996) believe that the two sexes to be analyzed form a single population with a probabilistic behavior denominated a [[mixture model|mixture of two normal populations]]. Thus, if <math>X</math> is a random variable which is normally distributed among the females of a population and likewise this variable is normally distributed among the males of the population, then,
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| :<math>f(x) = \sum_{i=1}^n \pi_if_i(x), -\infty < x < \infty ,</math> | |
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| is the density of the mixture with two normal components, where <math>f_i, \pi_i, i=1,2</math> are the normal densities and the mixing proportions of both sexes, respectively. See an example in Fig. 2 where the thicker curve represents the mixture whereas the thinner curves are the <math>\pi_if_i</math> functions.
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| [[Image:Overlapmix.jpg|thumb|Fig. 2. A mixture of two normal components.]] | |
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| It is from a population modelled like this that a random sample with individuals of both sexes can be selected. Note that on this sample tests which are based on the normal assumption cannot be applied since, in a mixture of two normal components, <math>\pi_if_i</math> is not a normal density.
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| Josephson ''et al.'' limited themselves to considering two normal mixtures with the same component variances and mixing proportions. As a consequence, their proposal to measure sexual dimorphism is the difference between the mean parameters of the two normals involved. In estimating these central parameters, the procedure used by Josephson ''et al.'' is the one of [[Method of moments (statistics)|Pearson's moments]]. Nowadays, the [[Expectation-maximization algorithm|EM expectation maximization algorithm]] (see McLachlan and Basford, 1988) and the [[Markov chain Monte Carlo|MCMC Markov chain Monte Carlo]] [[Bayesian inference|Bayesian procedure]] (see Gilks ''et al.'', 1996) are the two competitors for estimating mixture parameters.
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| Possibly the main difference between considering two independent normal populations and a mixture model of two normal components is in the mixing proportions, which is the same as saying that in the two independent normal population model the interaction between sexes is ignored. This, in turn implies that probabilistic properties change (see Ipiña and Durand, 2000).
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| ====The MI measure====
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| Ipiña and Durand (2000, 2004) have proposed a measure of sexual dimorphism called <math>MI</math>. This proposal computes the overlapping area between the <math>\pi_1f_1</math> and <math>\pi_2f_2</math> functions, which represent the contribution of each sex to the two normal components mixture (see shaded area in Fig. 2). Thus, <math>MI</math> can be written,
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| :<math>MI = \int_R \operatorname{min}[\pi_1f_1(x), (1 - \pi_1)f_2(x)]\,dx ,</math>
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| <math>R</math> being the real line. | |
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| The smaller the overlapping area the greater the gap between the two functions <math>\pi_1f_1</math> and <math>\pi_2f_2</math>, in which case the sexual dimorphism is greater. Obviously, this index is a function of the five parameters that characterize a mixture of two normal components (<math>\mu_i, \sigma^2_i, \pi_1, i=1,2)</math>. Its range is in the interval <math>(0, 0.5]</math>, and the interested reader can see, in the work of the authors who proposed the index, the way in which an interval estimate is constructed.
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| ===Measures based on non-parametric methods=== | |
| Marini ''et al.'' (1999) have suggested the [[Kolmogorov-Smirnov test|Kolmogorov-Smirnov]] distance as a measure of sexual dimorphism. The authors use the following form of the statistic,
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| :<math>\operatorname{max}_x|F_1(x) - F_2(x)| ,</math>
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| with <math>F_i, i=1,2</math> being sample cumulative distributions corresponding to two independent random samples.
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| Such a distance has the advantage of being applicable whatever the form of the random variable distributions concerned, yet they should be continuous. The use of this distance assumes that two populations are involved. Further, the Kolmogorov-Smirnov distance is a sample function whose aim is to test that the two samples under analysis have been selected from a single distribution. If one accepts the [[null hypothesis]], then there is not sexual dimorphism; otherwise, there is.
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| ==See also==
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| *[[Bateman's principle]]
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| *[[Digit ratio]]
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| *[[Gender differences]]
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| *[[Sexual dimorphism]]
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| ==References==
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| *Alexander, R.D., Hoogland, J.L., Howard, R.D., Noonan, K.M. and Sherman, P.W. (1979) Sexual dimorphism and breeding systems in pinnipeds, ungulates, primates and humans, in ''Evolutionary Biology and Human Social Behavior: An Anthropological Perspective'', N.A. Chagnon and W. Irons, Scituate, M.A.: Duxbury Press, 402-435.
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| *Borgognini Tarli, S.M. and Repetto, E. (1986) Methodological considerations on the study of sexual dimorphism in past human populations. Hum. Evol. 1: 51-56.
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| *Chakraborty, R. and Majumder, P.P. (1982) On Bennet's measure of sex dimorphism. Am. J. Phys. Anthrop. 59: 295-298.
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| *Clutton-Brock, T.H. (1985) Size, sexual dimorphism and polygamy in primates, in ''Size and Scaling in Primate Biology'', W.L. Jungers, N. York: Plenum, 211-237.
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| *Fleagle, J.G., Kay, R.F. and Simons, E.L. (1980) Sexual dimorphism in early anthropoids. Nature 287: 328-330.
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| *Gilks, W.R., Richardson, S. and Spiegelhalter, D.J. (1996) ''Markov Chain Monte Carlo in Practice''. London: Chapman and Hall.
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| *Green, D.L. (1989) Comparison of t-tests for differences in sexual dimorphism between populations. Am. J. Phys. Anthropol. 79: 121-125.
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| *Inman, H.F. and Bradley, E.L. (1989) The overlapping coefficient as a measure of agreement between probability distributions and point estimation of the overlap of two normal densities. Commun. Statist.-Theory Meth. 18: 3851-3874.
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| *Ipiña, S.L. and Durand, A.I. (2000) A measure of sexual dimorphism in populations which are univariate normal mixtures. Bull. Math. Biol. 62: 925-941.
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| *Ipiña, S.L. and Durand, A.I. (2004) Inferential assessment of the MI index of sexual dimorphism: A comparative study with some other sexual dimorphism measures. Bull. Math. Biol. 66: 505-522.
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| *Josephson, S.C., Juell, K.E. and Rogers, A.R. (1996) Estimating sexual dimorphism by method of moments. Am. J. Phys. Anthropol. 100: 191-206.
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| *Kappelman, J. (1996) The evolution of body mass and relative brain size in fossil hominids. J. Hum. Evol. 30: 243-276.
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| *Kay, R.F. (1982) ''Sivapithecus simonsi'' a new species of Miocene hominoid with comments on the phylogenetic status of Ramapithecinae. Int. J. Primatol. 3: 113-173.
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| *Kelley, J. (1993) Taxonomic implications of sexual dimorphism in ''Lufengpithecus'', in ''Species, Species Concepts, and Primate Evolution'', W.H. Kimbel and L.B. Martin, N. York: Plenum, 429-458.
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| *Klein, R.G. and Cruz-Uribe, K. (1983) ''The Analysis of Animal Bones from Archaeological Sites''. Chicago: University of Chicago Press.
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| *Lovejoy, C.O. (1981) The origin of man. Science 211: 341-350.
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| *Marini, E. Racugno, W. and Borgognini Tarli, S.M. (1999) Univariate estimates of sexual dimorphism: the effects of intrasexual variability. Am. J. Phys. Anthrop. 109: 501-508.
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| *McLachlan, G.J. and Basford, K.E. (1988) ''Mixture Models. Inference and Applications to Clustering''. N. York: Marcel Dekker Inc.
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| *Oxnard, C.E. (1987) ''Fossils, Teeth and Sex: New Perspective in Human Evolution''. Seattle: University of Washington Press.
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| *Plavcan, J.M. and van Schaick, C.P. (1992) Intrasexual competition and canine dimorphism in anthropoid primates. Am. J. Phys. Anthropol. 87: 461-477.
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| [[Category:Applied sciences]]
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| [[Category:Sex]]
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| [[Category:Social statistics]]
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| [[Category:Sexual dimorphism]]
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Moncler Geneve Rue Du Rhone Mieux encore
Sur la péninsule Pemaquid ou à proximité, il ya un certain nombre de réserves, ou des terres mises de côté à l'usage public et protégé de développement. Parce qu'il ya tellement d'eau, chaque que nous avons fait des vents vers l'océan ou d'une rivière. Je ne serais pas surpris si ces types de conserves sont parsemées à travers l'état.
110Rs. 1003Serene272Rs. 120Rs. En 2005, Fanball a été vendue à FUN Technologies, une société cotée en bourse. Rob est resté avec FUN après la vente et a été promu président de la division des sports. De 1994 98, Rob était le fondateur et propriétaire de la première semaine le magazine de sports d'imagination, Fantasy Football Weekly.
At-il été confirmé maintenant Werrington? C'est dans le forum Rangers et ils disent qu'il sera confirmée à la fin de la semaine. Tout le monde est très excité pour cela, nous aimons un voyage en Angleterre et a pris plus de Derby, il ya quelques années. Le fait-il que le SPL, dans l'ensemble, le football sert terrible..
Ils ouvertement acheté de l'espace publicitaire sur le forum jihadiste et insérés annonces difficiles djihadiste propaganda.Bravo! L'IC, DOD et le DOS travailler ensemble et de travailler intelligemment. C'est ce pouvoir intelligent est tout au sujet. Et un enfer de beaucoup moins cher dans le sang et le trésor de guerre de prise de vue.
Mieux encore, dans certains cas, ils vont même évangéliser en notre nom! auteurs sont certainement recherchés. "Les meilleurs arbitres sont postdocs», dit Leslie Moncler Geneve Rue Du Rhone Sage, un rédacteur en chef de sciences physiques à la nature, à Washington DC. «Ils sont au sommet de leur jeu, bien versé dans la littérature et politiquement assez naïf pour dire la vérité.".
Vous avez en cours d'exécution. Vous avez à marcher. Pourquoi ne pas faire du jogging? avec une vitesse donnée entre marche et course? semble facile: poids juste fixés comme: 50/50 pour quelque chose exactement entre les deux, ou 10/90 pour presque pleine course.
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