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16种常用的数据分析方法汇总2015-11-10 分类：数据分析评论(0)经常会有朋友问到一个朋友，数据分析常用的分析方法有哪些，我需要学习哪个等等之类的问题，今天数据分析精选给大家整理了十六种常用的数据分析方法，供大家参考学习。
一、描述统计描述性统计是指运用制表和分类，图形以及计筠概括性数据来描述数据的集中趋势、离散趋势、偏度、峰度。
1、缺失值填充：常用方法：剔除法、均值法、最小邻居法、比率回归法、决策树法。
2、正态性检验：很多统计方法都要求数值服从或近似服从正态分布，所以之前需要进行正态性检验。
常用方法：非参数检验的K-量检验、P-P图、Q-Q图、W 检验、动差法。
二、假设检验1、参数检验参数检验是在已知总体分布的条件下（一股要求总体服从正态分布）对一些主要的参数(如均值、百分数、方差、相关系数等）进行的检验。
1）U验使用条件：当样本含量n较大时，样本值符合正态分布2）T检验使用条件：当样本含量n较小时，样本值符合正态分布A 单样本t检验：推断该样本来自的总体均数μ与已知的某一总体均数μ0 (常为理论值或标准值)有无差别；B 配对样本t检验：当总体均数未知时，且两个样本可以配对，同对中的两者在可能会影响处理效果的各种条件方面扱为相似；C 两独立样本t检验：无法找到在各方面极为相似的两样本作配对比较时使用。
2、非参数检验非参数检验则不考虑总体分布是否已知，常常也不是针对总体参数，而是针对总体的某些一股性假设（如总体分布的位罝是否相同，总体分布是否正态）进行检验。
适用情况：顺序类型的数据资料，这类数据的分布形态一般是未知的。
A 虽然是连续数据，但总体分布形态未知或者非正态；B 体分布虽然正态，数据也是连续类型，但样本容量极小，如10以下；主要方法包括：卡方检验、秩和检验、二项检验、游程检验、K-量检验等。
三、信度分析检査测量的可信度，例如调查问卷的真实性。
分类：1、外在信度：不同时间测量时量表的一致性程度，常用方法重测信度2、内在信度；每个量表是否测量到单一的概念，同时组成两表的内在体项一致性如何，常用方法分半信度。
统计学中的多变量分析方法统计学是一门重要的科学领域，它致力于研究如何收集、组织、分析和解释数据。
在统计学中，多变量分析方法是一种常用的技术，用于探究多个变量之间的关系和模式。
本文将介绍多变量分析方法的概念和应用场景。
一、多变量分析方法的概述在统计学中，多变量分析方法是一种通过同时考虑多个变量来研究数据集的方法。
相比传统的单变量分析方法，多变量分析方法可以更全面地探究各个变量之间的关联和影响。
为了帮助研究者更好地理解数据集中变量之间的关系，多变量分析方法提供了多种技术和模型。
其中最常用的方法包括主成分分析、因子分析、聚类分析、判别分析和回归分析。
二、主成分分析主成分分析是一种常见的多变量分析方法，用于减少数据集的维度并提取潜在的主要变量。
通过主成分分析，可以将原始数据转化为一组无关的主成分，这些主成分可以解释数据中大部分的方差。
主成分分析可用于降维、特征选择和数据可视化。
它广泛应用于生物医学、工程学、金融和市场研究等领域，有助于简化复杂数据集的分析过程。
三、因子分析因子分析是一种用于研究多个变量之间关联模式的方法。
它通过将一组观测变量转化为一组潜在的无关因子，来揭示观测变量背后的潜在结构。
因子分析可以用于探究样本中隐藏的潜在因子，如人格特征、消费者满意度和员工工作满意度等。
通过因子分析，研究者可以了解到不同变量之间的潜在关系，并进一步洞察潜在因子对观测变量的解释贡献。
四、聚类分析聚类分析是一种将样本或变量分组成类别的方法。
通过聚类分析，可以根据样本间的相似性或变量间的相关性，将数据集划分为不同的群组。
聚类分析在市场研究、社会科学和生物学等领域得到广泛应用。
它可以用于发现数据集中的隐藏模式和群组，帮助研究者识别并理解不同群体之间的相似性和差异。
五、判别分析判别分析是一种用于解释组间差异和评估变量重要性的统计方法。
它可以帮助研究者确定哪些变量对于区分不同组别的样本最具有预测性。
判别分析在医学研究、社会科学和商业决策等领域得到广泛应用。
多维分析操作方法多维分析是一种用于处理和分析多维数据的统计方法，在数据挖掘、商业智能、市场调研等领域都有广泛的应用。
多维分析的目的是通过对数据集合中的各个维度之间的关系进行探索，从而揭示出数据中存在的模式和规律。
在进行多维分析时，可以采用多种操作方法来处理数据和生成分析结果。
一、数据预处理在进行多维分析之前，必须首先进行数据预处理，以确保数据的准确性和一致性。
数据预处理的主要任务包括数据清洗、数据集成和数据变换等。
1. 数据清洗：通过去除数据中的错误、缺失和冗余等问题，保证数据的完整性和正确性。
2. 数据集成：将来自不同来源的数据进行整合，创建一个统一的数据集合，便于后续的分析和处理。
3. 数据变换：对原始数据进行变换，使其更适合进行多维分析。
常见的数据变换方法包括聚合、离散化、标准化等。
二、维度选择和维度约简在多维分析中，通常会面临维度过多的问题，因此需要对维度进行选择和约简，以减少分析的计算量和复杂度。
常见的方法包括：1. 主成分分析：通过线性变换将原始数据转换为一组新的正交变量，即主成分，用于表示原始数据的大部分变异性。
2. 因子分析：通过寻找一组潜在因子，将多个观测变量进行组合，得到一个更小的一维或二维因子空间。
3. 独立成分分析：通过寻找一组相互独立的成分，将原始数据进行解耦，找出数据中的隐藏模式和结构。
三、关联和分类分析关联和分类分析是多维分析中常用的操作方法，用于探索数据中的相关规律和潜在分类。
1. 关联分析：通过寻找数据中的关联规则和频繁项集，揭示出数据中的相互依赖和关联性。
常用的关联分析方法有Apriori算法和FP-Growth算法等。
2. 分类分析：通过将数据样本分为不同的类别，找出数据中的潜在分类结构。
常用的分类分析方法有决策树、朴素贝叶斯、支持向量机等。
四、聚类和异常检测聚类和异常检测是多维分析中常用的数据处理方法，用于发现数据中的聚类结构和异常点。
1. 聚类分析：通过将数据分为不同的聚类，找出数据中的相似性和簇结构。
方差分析和协方差分析协变量和控制变量方差分析（Analysis of Variance，简称ANOVA）是用于比较两个或多个组之间差异的一种统计方法。
它常用于实验设计中，特别是当研究者希望判断不同组别对其中一变量的均值是否存在显著差异时。
方差分析的基本思想是通过分析组间变异和组内变异的差异性，来评估不同组别之间的差异是否超出了随机误差的范围。
在执行方差分析时，我们需要计算组间平方和（Sums of Squares Between Groups, SSBG）和组内平方和（Sums of Squares Within Groups, SSWG），并以此计算F值来进行假设检验。
协方差分析（Analysis of Covariance，简称ANCOVA）则是在方差分析基础上引入了协变量（covariate）的一种分析方法。
协变量是指与主要变量（研究变量）相关的、可能对变量之间关系产生影响的另一个变量。
协方差分析旨在通过控制协变量的影响，更准确地评估主要变量对因变量的影响。
具体而言，协方差分析会使用协变量与因变量的相关性来对因变量进行线性调整，将其影响减少到最低限度。
这样可以消除协变量对因变量的干扰，使比较组之间的差异更为准确。
在研究设计中，协变量和控制变量是常用的两种概念，用于控制和修正分析过程中的干扰因素。
在实验设计中，控制变量是指研究者通过依据主要变量的研究设计，将一些可能导致干扰的因素保持恒定。
例如，在比较两种不同药物对疾病治疗效果时，研究者可以将患者的性别、年龄、体重等因素作为控制变量，确保不同组别之间的差异主要来自于药物本身的影响。
而协变量则是在非实验研究中常用的，在测量研究变量之前，研究者会对协变量进行测量和记录，并在分析过程中加以控制。
例如，研究人员可能关注不同年龄组中学生的学业成就，但同时也要控制其他因素，如家庭背景、社会经济地位等，这些因素可能会干扰到学业成就与年龄之间的关系。
总之，方差分析和协方差分析是两种常用的统计分析方法，在不同的情境下用于数据的比较和解释。
统计分析方法汇总目录基本统计分析 (1)正态性检验 (2)单变量均值检验 (3)两独立样本的均值检验 (2)配对（成对）总体均值检验 (2)回归分析 (2)方差分析 (2)列联表检验 (2)多元统计分析 (4)主成分分析 (5)因子分析 (5)聚类分析 (5)判别分析 (5)基本统计分析正态性检验许多计量资料的分析方法要求数据分布是正态或近似正态，因此对原始独立测定数据进行正态性检验是十分必要的。
正态性检验主要有三类方法:一、计算综合统计量如动差法、夏皮罗-威尔克SHAPIRO-WILK 法(W检验) 、达戈斯提诺D AGOSTINO 法(D检验) 、SHAPIRO-FRANCIA 法(W检验) .二、正态分布的拟合优度检验如皮尔逊Χ2 检验、对数似然比检验、柯尔莫哥洛夫KOLMOGOROV-SMIROV 法检验 .三、图示法(正态概率图NORMAL PROBABILITY PLOT)如分位数图(QUANTILE QUANTILEPLOT ,简称QQ图) 、百分位数(PERCENT PERCENT PLOT ,简称PP图) 和稳定化概率图(STABILIZED PROBABILITY PLOT ,简称SP 图) 等.SPSS&SAS规则：SPSS 规定:当样本含量3 ≤N ≤5000 时,结果以SHAPIRO - WILK (W 检验) 为难,当样本含量N> 5000 结果以KOLMOGOROV - SMIRNOV 为准。
而SAS 规定:当样本含量N ≤2000 时,结果以SHAPIRO - WILK (W 检验) 为准,当样本含量N>2000 时,结果以KOLMOGOROV - SMIRNOV (D 检验) 为准。
SPSS过程1、先做直方图看看是否大概符合正态分布，这个不用说了吧，GRAPH-->LEGACYDIALOGS-->HISTOGRAM-->选入变量-->OK.如果距离正态分布的样子太远了，你就不要做下面的工作啦2、ANALYZE-->DESCRIPTIVE STATISTIC-->EXPLORE-->选入变量-->选右上角的PLOTS-->打开后，选中间的NORMALLY PLOTS WITH TESTS -->OK。
统计学中的多变量分析方法多变量分析是统计学中一个重要的分析方法，用于研究多个变量之间的关系以及它们对观察结果的影响。
多变量分析可以帮助我们从多个维度来解释数据，揭示隐藏在数据背后的规律和结构。
在统计学中，常见的多变量分析方法主要包括回归分析、主成分分析、聚类分析和因子分析等。
下面将对这些方法进行详细介绍。
回归分析是一种用于研究因变量和自变量之间关系的方法。
它通过建立一个数学模型来描述这种关系，并根据数据推断模型的参数。
回归分析可以用于预测因变量的取值，也可以用于确定自变量对因变量的影响程度。
常见的回归分析方法有线性回归、多元线性回归、逻辑回归等。
主成分分析（PCA）是一种通过线性组合将多个相关变量转换为少数几个无关变量的方法。
它可以帮助我们发现数据中的主要结构和模式。
主成分分析的输出是一组新的变量，称为主成分，它们是原始变量的线性组合。
主成分分析可以用于数据降维、数据压缩和特征提取等。
聚类分析是一种将相似的个体或对象归类为一组的方法。
聚类分析基于样本之间的相似性或距离度量，将样本划分为不同的簇。
聚类分析可以用于数据分类、观察群体相似性和发现群组之间的关系等。
常用的聚类分析方法有层次聚类和k均值聚类等。
因子分析是一种用于解释变量之间关系的方法。
它通过将多个观测变量解释为少数几个潜在因子，来揭示数据背后的结构。
因子分析可以帮助我们压缩数据信息、发现共性因子和解释观测变量之间的关系。
常见的因子分析方法有主成分分析和最大似然法等。
此外，还有其他一些多变量分析方法，比如判别分析、典型相关分析、结构方程模型等，它们也在统计学的研究中得到广泛应用。
这些方法在实际研究中可以结合使用，以更全面地分析数据和解释现象。
总结来说，多变量分析是统计学中重要的分析手段，用于研究多个变量之间的关系。
常见的多变量分析方法包括回归分析、主成分分析、聚类分析和因子分析等。
这些方法可以帮助我们从多个维度来理解数据，揭示数据背后的规律和结构。
常用多变量分析方法在社会科学研究中，主要的多变量分析方法包括多变量方差分析（Multivariate analysis of variance，MANOVA）、主成分分析（Principal component analysis）、因子分析（Factor analysis）、典型相关（Canonical correlation analysis）、聚类分析（Cluster analysis）、判别分析（Discriminant analysis）、多维量表分析（Multidimensional scaling），以及近来颇受瞩目的验证性因子分析(Confirmatory factor analysis )或线性结构模型（LISREL）与逻辑斯蒂回归分析等，以下简单说明这些方法的观念和适用时机。
一、多变量方差分析MANOVA适用于同时探讨一个或多个自变量与两个以上因变量间因果关系的统计方法，依照研究者所操作自变量的个数，可以分为单因素（一个自变量）或多因素（两个以上自变量）MANOVA。
进行多变量方差分析时，自变量必须是离散的定类或定序变量，而因变量则必须是定距以上层次的变量。
二、主成分分析主成分分析的主要功能在分析多个变量间的相关，以建构变量间的总体性指标（overall indicators）。
当研究者测量一群彼此间具有高度相关的变量，则在进行显著性检验钱，为避免变量数过多，造成解释上的复杂与困扰，常会先进行主成分分析，在尽量不丧失原有信息的前提下，抽取少数几个主成分，作为代表原来变量的总体性指标，达到资料缩减（data reduction）的功能。
进行主成分分析时，并无自变量和因变量的区别，但是所有的变量都必须是定距以上层次变量。
三、因子分析因子分析与主成分分析常被研究者混用，因为二者的功能都是通过对变量间的相关分析，以达到简化数据功能。
但不同的是，主成分分析是在找出变量间最佳线性组合（linear combination）的主成分，以说明变量间最多的变异量；至于因子分析，则在于找出变量间共同的潜在结构（latent structure）或因子，以估计每一个变量在各因子上的负荷量（loading）。
市场研究中的多元统计分析方法市场研究中的多元统计分析方法是一种统计分析工具，广泛应用于市场研究中，用于研究市场上的人口统计学特征、购买行为、品牌偏好等各种因素之间的关系。
这些方法可以帮助市场研究人员深入了解消费者对产品或服务的态度和行为，为企业的市场决策提供有力的支持。
多元统计分析方法主要包括主成分分析（Principal Component Analysis，PCA）、聚类分析（Cluster Analysis）、判别分析（Discriminant Analysis）和因子分析（Factor Analysis）等。
以下将介绍其中的几种常用多元统计分析方法：1. 主成分分析（PCA）：主成分分析是一种降维技术，通过寻找原始数据中的主要信息，将大量变量转化为较少的几个主成分。
通过PCA分析，市场研究人员可以确定消费者行为中的主要因素，从而更好地理解市场细分和产品定位。
例如，PCA 可以将多个购买偏好变量转化为几个主成分，进一步揭示不同消费者群体之间的共同特征。
2. 聚类分析（Cluster Analysis）：聚类分析是将不同样本归类到相似的组中的一种方法。
通过计算各个样本之间的相似性，可以将市场中的消费者划分为不同的群体。
聚类分析可以帮助市场研究人员发现市场中的潜在市场细分，并对不同群体的特征和需求进行深入了解。
3. 判别分析（Discriminant Analysis）：判别分析是一种统计方法，用于确定哪些变量能够最好地区分不同的样本群体。
通过判别分析，市场研究人员可以了解哪些因素对于字经济特征或购买行为等方面有显著影响。
例如，判别分析可以帮助企业判断某一品牌在不同消费者群体中的影响力或市场份额。
4. 因子分析（Factor Analysis）：因子分析是一种可以揭示多个变量之间的隐藏关系的方法。
通过这种分析方法，市场研究人员可以辨别出共同维度，从而理解市场中的不同变量之间的关系。
例如，因子分析可以揭示购买行为中的主要因素，如产品价格、品牌认知、产品质量等。
Copyright © 2010, Siberian Branch of Russian Academy of Sciences, Institute of Archaeology and Ethnography of the Siberian Branch of the Russian Academy of Sciences. Published by Elsevier B.V . All rights reserved.doi:10.1016/j.aeae.2010.02.014Archaeology Ethnology & Anthropology of Eurasia 37/4 (2009) 125–136E-mail: Eurasia@archaeology.nsc.ruARCHAEOLOGY,ETHNOLOGY& ANTHROPOLOGY OF EURASIAIntroductionRoutes of the early Caucasoid migrations to Siberia and Eastern Central Asia have become a focus of scholarly interest in recent years since this issue is closely related to that of the Indo-European homelands. Certain archaeologists believe that migrants from the Near East played a major role in the origin of Southern SiberianCRANIOMETRIC EVIDENCE OF THE EARLY CAUCASOID MIGRATIONSTO SIBERIA AND EASTERN CENTRAL ASIA, WITH REFERENCETO THE INDO-EUROPEAN PROBLEM *Measurements of 220 male Neolithic and Bronze Age cranial series from Eurasia were subjected to multivariate statistical analysis. The results support the idea that people associated with the Catacomb culture played a major role in the origin of the Afanasyev culture. Okunev people of the Minusinsk Basin, those associated with Karakol, Ust-Tartas, and Krotovo cultures, and those buried in the Andronov-type cemeteries at Cherno-ozerye and Yelovka were of predominantly local Siberian origin. The Samus series resembles that from Poltavka burials. The Okunev people of Tuva and probably Yelunino people were likely descendants of the Pit Grave (Yamnaya) and early Catacomb populations of the Ukraine. The same is true of the Alakul people of western Kazakhstan, who in addition, have numerous af ¿ nities amongst Neolithic and Early Bronze Age groups of Central and Western Europe. The probable ancestors of certain Fedorov populations were the Afanasyev tribes of the Altai, whereas other Fedorov groups apparently descended from late Pit Grave and Catacomb tribes of the Northern Caucasus and the northwestern Caspian. People of Gumugou are closest to Fedorov groups of northeastern Kazakhstan and Rudny Altai, suggesting that Caucasoids migrated to Xinjiang from the north rather than from the west. Describing the gracile Caucasoids of Siberia and Eastern Central Asia as “Mediterraneans” is misleading since they display virtually no craniometric ties with the Near Eastern, Southwestern Central Asian or Transcaucasian groups. The totality of evidence suggests that they were Nordics.Keywords: Indo-Europeans, Indo-Iranians, Tocharians, Southern Siberia, Western Siberia, Central Asia, Bronze Age, craniometry.cultures of the Bronze Age (Grigoryev, 1999; Bobrov, 1994; Kiryushin, 2004), and these theories are supported by those physical anthropologists who claim that all gracile Caucasoids are Mediterraneans, i.e. southerners by origin (see especially (Khudaverdyan, 2009)). Not long ago I expressed a similar view (Kozintsev, 2000).Recently, thanks to the work of a number of craniologists, S.I. Kruts in particular, the craniometric database related to the Bronze Age steppe populations of the Ukraine and Southern Russia has grown manifold. Its statistical analysis has led to the revision of earlier*Supported by the Russian Foundation for Basic Research (Project 09-06-00184a). A.G. KozintsevMuseum of Anthropology and Ethnography, Russian Academy of Sciences,Universitetskaya Nab.3, St. Petersburg, 199034, RussiaE-mail: agkozintsev@125ANTHROPOLOGY126 A.G. Kozintsev / Archaeology Ethnology & Anthropology of Eurasia 37/4 (2009) 125–136views. A more detailed craniometric comparison of each gracile Southern Siberian group with all others suggests that there is no reason to speak of migrations to Southern Siberia from the Near East, Southwestern Central Asia or the Transcaucasia, where Southern Caucasoids (Mediterraneans) were distributed (Kozintsev, 2007, 2008).Recently, an article by a group of French geneticists was published (Keyser et al., 2009), which reported on the analysis of DNA extracted from the bone samples taken from Andronov, Karasuk, Tagar, and Tashtyk human remains. Six genes controlling eye and hair color were studied. Most individuals buried in Bronze and Iron Age mounds in Southern Siberia (15 of 23, or 65 %) had light or mixed eye color, and 8 out of 12 (67 %) had fair or chestnut hair. Given that the Bronze Age people of the Tarim Basin (the likely proto-Tocharians), whose bodies are excellently preserved thanks to natural mummi¿ cation (Mallory, Mair, 2000), had the same hair color, and that a Russian admixture alone can by no means account for the depigmentation observed in modern natives of Southern Siberia and Kazakhstan, the conclusion is obvious. The principal source of early Caucasoid migrations to Siberia and Eastern Central Asia was located not in the Near East, but in Europe, moreover not in its southern part but in areas affected by the depigmentation process. Nearly eighty years ago this conclusion was reached by G.F. Debetz (1931), who compared cranial data on the Tagar people with the evidence of Chinese written sources. Apparently, the principal migration route of Caucasoid pastoralists from Europe to the east passed mainly along the steppe belt, and judging from archaeological data, the migration process continued throughout the 3rd millennium BC (Merpert, 1982; Semenov, 1993). But where was the source located? In the Eastern European steppes? In Central Europe?According to a view shared by most specialists, archaeologists and physical anthropologists alike, the Afanasyev culture was closely related to the Pit Grave (Yamnaya) culture, and its appearance in Gorny Altai and on the Middle Yenisei was caused by a migration from the Eastern European steppes. The possible role of Poltavka and Catacomb culture elements, too, has been discussed (Tsyb, 1981, 1984). The idea is supported by new radiocarbon dates indicating the coexistence of Catacomb culture with Pit Grave culture over most of the 3rd millennium BC (Chernykh, 2008). On the other hand, very early dates of the earliest Afanasyev sites in Gorny Altai (mid-4th millennium BC) suggest that the predecessors of the Pit Grave people, speci¿ cally those associated with the Khvalynsk and Sredni Stog cultures, as well as the proto-Pit Grave (Repino) tribes, might have taken part in Afanasyev origins. This suggestion was already made by physical anthropologists (Shevchenko, 1986; Solodovnikov, 2003).With regard to the post-Afanasyev Bronze Age cultures, the traditional idea that the Okunev culture is autochthonous has given place to theories stating that the Pit Grave and Catacomb traditions (Lazaretov, 1997), or those of Afanasyev culture, which were also introduced from Europe, were critical in Okunev origins (Sher, 2006). In terms of physical anthropology however, the presumed European ancestry of the Okunev people of the Minusinsk Basin, according to A.V. Gromov (1997b), pointing to af¿ nities with the Pit Grave and Catacomb people of Kalmykia, is rather indistinct and traceable mostly at the individual level if at all. The analysis of data concerning two independent trait batteries – craniometric and cranial nonmetric – suggests that the af¿ nities of the Okunev people of the Yenisei are mostly Siberian (Gromov, 1997a, b), and the integration of these data demonstrates that the unusual trait combination observed in Okunev crania is rather archaic (plesiomorphic) and may be more ancient than both the Caucasoid and Mongoloid trait combinations (Kozintsev, 2004). According to Gromov (1997b), the Okunev people resembled the Neolithic population of the Krasnoyarsk–Kansk region. The Karakol culture of Gorny Altai is similar to Okunev culture, and craniometric parallels between people associated with these cultures were also noted. However, Karakol crania are believed to exhibit a “Mediterranean” tendency (Chikisheva, 2000; Tur, Solodovnikov, 2005).The Okunev crania from Tuva and the Yelunino crania from the Upper Ob, especially the former, are much more Caucasoid (Gokhman, 1980; Solodovnikov, Tur, 2003; Kozintsev, 2008). This agrees with archaeological facts indicating the affinities of cultures such as Yelunino and Okunev of Tuva with Early Bronze Age cultures of Central and even Western Europe (Kovalev, 2007). The possible Caucasoid ties of other pre-Andronov tribes of Southern Siberia such as Krotovo (Dremov, 1997) and Samus (Solodovnikov, 2005, 2006) have been discussed by craniologists. K.N. Solodovnikov (Ibid.) believes that in all the above pre-Andronov groups, except the Okunev group of the Yenisei, these ties are Southern Caucasoid or Mediterranean which, in his view, is especially evident in the male series.The origin of the Andronov community is one of the pivotal points in Indo-European history. The predominantly Indo-Iranian or Iranian attribution of this community is beyond doubt (Kuzmina, 2007a, b; 2008). The relationship between its two constituents, speci¿ cally the Alakul (western) and Fedorov, which spread in an eastern direction up to the Yenisei, is less clear. The Alakul variety apparently originated earlier, in the 3rd millennium BC (Chernykh, 2008) and the cultures which contributed to its origin were Poltavka, Catacomb, and Abashevo. The origin of the Fedorov variety, which originated later and coexisted with Alakul over most of the 2nd millennium BC, remains obscure (Tkacheva, Tkachev, 2008).A.G. Kozintsev / Archaeology Ethnology & Anthropology of Eurasia 37/4 (2009) 125–136 127Craniologists have discovered that the Andronov community was markedly heterogeneous. People buried in graves with Alakul or mixed Alakul-Fedorov (Kozhumberdy) ceramics in western Kazakhstan displayed a trait combination which V.V. Ginzburg (1962) described as Mediterranean, and V.P. Alekseyev (1964) as leptomorphic. Ginzburg believed that this combination evidences the af¿ nities of western Alakul people with both the Timber Grave (Srubnaya) populations of the V olga steppes and those of Southwestern Central Asia (the Amu-Darya/Syr-Darya interÀ uve). The second idea was refuted by Alekseyev, who claimed that archaeological data point solely to western (Timber Grave) af¿ nities. Ginzburg ignored the critique and repeated his conclusion in the summarizing monograph (Ginzburg, Tro¿ mova, 1972). In this case, neither he nor Alekseyev used statistical methods and relied on typological assessments.As to the Fedorov populations, most of which display the characteristically “Andronov” trait combination believed to have derived from the Cromagnon variety, G.F. Debetz (1948) claimed that they had originated in the Kazakhstan steppes from whence they moved to the Yenisei. However, V.P. Alekseyev (1961) suggested that the Fedorov people of the Yenisei had descended from the Afanasyev populations of the Altai. Fedorov groups of the Upper Ob and the Altai deviate toward a gracile variety, traditionally described as Mediterranean. The presence of alleged “Mediterraneans” in these regions was explained differently: V.A. Dremov (1997) attributed it to links with the Alakul people, whereas K.N. Solodovnikov (2005, 2007) wrote about the pre-Andronov, speci¿ cally Yelunino substratum. People buried in Andronovo-type cemeteries in the Tomsk part of the Ob Basin at Yelovka II and in the Omsk stretch of the Irtysh basin at Cherno-ozerye I, according to Dremov (1997), differed from other Andronov groups and were autochthonous. Finally, the origin of the early Caucasoid population of Xinjiang, the members of which were buried at the Bronze Age cemetery of Gumugou (Mair, 2005), is a complete mystery (Han Kangxin, 1986; Hemphill, Mallory, 2004; Kozintsev, 2008).The objective of the present article is to explore the issue of early Caucasoid migrations to Siberia and Eastern Central Asia using a large craniometric database, much of which is unpublished.Materials and methodsOnly measurements of male crania were used. The number of the Afanasyev series is nine (six from the Altai and three from the Minusinsk Basin) (Alekseyev, 1961, 1989; Solodovnikov, 2003). The post-Afanasyev material consists of four Okunev series from the Minusinsk Basin (Gromov, 1997b), one Okunev series from Tuva (Alekseyev, Gokhman, Tumen, 1987), the Karakol series (Chikisheva, 2000; Tur, Solodovnikov, 2005), the Yelunino series (Solodovnikov, Tur, 2003), the Ust-Tartas and Krotovo series from Sopka-2 (Dremov, 1997; Chikisheva, unpublished), and the Samus series (Dremov, 1997; Solodovnikov, 2005). Seven Andronov samples were used. Two of them, from western Kazakhstan (Alekseyev, 1967) and from Yermak IV near Omsk (Dremov, 1997) represent mostly the Alakul variety. Fedorov samples come from Firsovo XIV near Barnaul (Solodovnikov, 2005), from other burial grounds on the Upper Ob (Ibid.) and from northeastern Kazakhstan (Ibid.)*, from Rudny Altai (Solodovnikov, 2007), and from the Minusinsk Basin (Alekseyev, 1961; Dremov, 1997). Also, measurements of two series from “Andronoid” burial grounds in Western Siberia at Yelovka II and Cherno-ozerye I (Dremov, 1997) were used, and also those of a Bronze Age series from Gumugou (Qäwrighul), Xinjiang (Han Kangxin, 1986).Unpublished measurements of Bronze Age crania from the Ukraine were kindly provided by S.I. Kruts; sources of data on most groups published by Russian scholars are cited in my previous publications (Kozintsev, 2000, 2007, 2008). Measurements of series from Central and Western Europe and the Near East were taken from a summary compiled by I. Schwidetzky and F. Rösing (1990).The total number of male cranial series representing the Neolithic and Bronze Age populations of Eurasia and used in this analysis is 220. One hundred and twenty-eight of them, mostly from the former Soviet Union, were studied according to the craniometric program employed by Soviet and modern Russian anthropologists. Fourteen traits were taken from it: cranial length, breadth, and height, frontal breadth, bizygomatic breadth, upper facial height which was measured to the “lower prosthion”, or alveolare, nasal and orbital height and breadth, naso-malar and zygo-maxillary angles, simotic index and nasal protrusion angle. Ninety-two series from Central and Western Europe and the Near East were studied by Western anthropologists. They were measured according to a smaller program from which nine linear dimensions were taken: cranial length, breadth and height, frontal breadth, bizygomatic breadth, and nasal and orbital height and breadth. Upper facial height was not used in this case to avoid confusion between the “anterior” and the “inferior” prosthion (the difference may be considerable).Measurements were subjected to the canonical variate analysis. Groups were compared pairwise using *Solodovnikov has reduced the size of this series by demonstrating on archaeological grounds that some crania previously believed to be Andronov actually represent other populations.128 A.G. Kozintsev / Archaeology Ethnology & Anthropology of Eurasia 37/4 (2009) 125–136the Mahalanobis D2 distance corrected for sample size (Rightmire, 1969). After correction, many distances become negative and should be regarded as sample estimates of zero or of small positive values.Creating a general classi¿ cation of all groups was not among the objectives of the present study. All methods aimed at such classification result in distortions. In cluster analysis, the distortions become progressively larger as the distances increase. By contrast, in two-dimensional projections determined by canonical variates or by nonmetric scaling axes, it is the closest ties that are distorted most in order to adequately render the most general pattern of group relationships. The broader the scope of the study, and accordingly, the wider the geographical range of the analysis, the more details are sacri¿ ced for the sake of the general classi¿ cation. These distortions may have contributed to the idea that all gracile Caucasoids are close relatives. While this may be true in a bird’s-eye view, a disregard for details in such a case may lead to a serious misinterpretation.Another advantage of using pairwise distances, rather than graphic methods of dimensionality reduction, stems from the fact that the latter are sensitive to the selection of groups. Thus, the sequence of groups in terms of the expression of Caucasoid versus Mongoloid traits depends on the way these extreme combinations are represented. Distances, by contrast are independent of this factor provided the standard within-group correlation matrix is used, as in this study.ResultsListed below are the smallest corrected D2 values (normally below 1.0), based on the fourteen-trait battery and ranked in an increasing order i.e. in the order of decreasing similarity. Minimal distances based on the nine-trait battery (D2 < 0.3) are given only for those groups which reveal at least one Central or Western European or Near Eastern parallel; these parallels are numbered as in the summary (Schwidetzky, Rösing, 1990). The only exception is the Afanasyev series from Saldyar I. It is quite small and reveals numerous af¿ nities, including early Central and Western European ones, which are not in ¿ rst place.Afanasyev, Ursul, Altai: Afanasyev, Nizhni Tyumechin (–1.12); Catacomb, Don (0.59); Timber Grave, Luzanovka, V olga (0.66); early Northern Caucasian culture, Kalmykia, group II according to V.A. Safronov (0.70).Afanasyev, Saldyar I, Altai: Pit Grave, V olga–Ural area (–2.49); Pit Grave, Orenburg region (–2.47); Pit Grave – Poltavka group,Volga–Ural area (–2.42); late Catacomb, Verkhne-Tarasovka, Lower Dnieper (–2.35); Afanasyev, Afanasyeva Gora (–2.35) and Karasuk III (–2.30); early Northern Caucasian culture, Kalmykia, group II (–2.10); Afanasyev, Minusinsk Basin, pooled (–1.87); Andronov, Firsovo XIV, Upper Ob (–1.82); Pit Grave, Yuzhny Bug (–1.79); Pit Grave, Ingulets (–1.51); early Catacomb, Molochnaya (–1.36); Timber Grave, Luzanovka, V olga (–1.34); Catacomb, V olga and Kalmykia (–1.23 in both cases). Numerous remaining parallels are mostly with Catacomb and Timber Grave groups.Afanasyev, Kurota II, Altai: Poltavka (–1.38); Afanasyev, Saldyar I (0.59); Catacomb, V olga (0.96).Afanasyev, Ust-Kuyum, Altai: late Catacomb, Samara–Orel watershed (–0.42); Pit Grave, Ingulets (–0.20); Pit Grave, Stavropol area (–0.07); Andronov, Minusinsk Basin (0.51); late Catacomb, Zaporozhye (0.80).Afanasyev, southeastern Altai: Afanasyev, Nizhni Tyumechin (–0.38); Catacomb, Don (–0.18); early Catacomb, Molochnaya (–0.09); Pit Grave, Kakhovka, Lower Dnieper (–0.05); Timber Grave, Yasyrev, Lower Don (0.16); Afanasyev, Saldyar I (0.22); Pit Grave, Yuzhny Bug (0.23); Timber Grave, Luzanovka, Volga (0.24); Pit Grave, Ukraine, pooled (0.47); early Catacomb, Verkhne-Tarasovka, Lower Dnieper (0.75); Timber Grave, Krivaya Luka, Lower V olga (0.81); Timber Grave, V olga, pooled (0.86); Timber Grave, V olgograd and Astrakhan regions (0.89); Pit Grave – Poltavka, Volga–Ural area (0.91); Catacomb, Kalmykia (0.95).Afanasyev, Nizhni Tyumechin, Altai: Afanasyev, Ursul (–1.12); Catacomb, Don (–0.86); early Catacomb, Verkhne-Tarasovka, Lower Dnieper (–0.76); Timber Grave, Yasyrev, Lower Don (–0.52); Afanasyev, southeastern Altai (–0.38); Timber Grave, Krivaya Luka, Lower V olga (–0.25); Timber Grave, Luzanovka, V olga (–0.07); Pit Grave – Poltavka, V olga–Ural area (0.27); Afanasyev, Saldyar I (0.56); Pit Grave, Ingulets (0.77); Pit Grave, Ukraine, pooled (0.83).Afanasyev, Altai, pooled: Timber Grave, Luzanovka, Volga (0.23); Catacomb, Don (0.29); Timber Grave, Bashkiria (0.79); Timber Grave, Krivaya Luka, Lower V olga (0.96).Afanasyev, Karasuk III, Minusinsk Basin: Afanasyev, Saldyar I (–2.30); Pit Grave, Volga–Ural area (–0.96); Afanasyev, Afanasyeva Gora (–0.54); Timber Grave, Volga–Ural area (–0.32); late Catacomb, Verkhne-Tarasovka, Lower Dnieper (–0.25); Catacomb, Volga (–0.22); Pit Grave, Orenburg region (–0.13); Catacomb, Crimea (0.01); Pit Grave – Poltavka, Volga–Ural area (0.24); Potapovka, V olga (0.73); Timber Grave, Yasyrev, Lower Don (0.90).Afanasyev, Afanasyeva Gora, Minusinsk Basin: Afanasyev, Saldyar I (–2.35); late Catacomb, Verkhne-Tarasovka, Lower Dnieper (–1.60); Timber Grave, Volgograd and Astrakhan regions (–0.66); Afanasyev, Karasuk III (–0.54); early Catacomb, Molochnaya (–0.46); Timber Grave, Yasyrev, Lower Don (–0.13); Pit Grave, Southern Bug (0.21); early Catacomb, Kakhovka,A.G. Kozintsev / Archaeology Ethnology & Anthropology of Eurasia 37/4 (2009) 125–136 129Lower Dnieper (0.44); Pit Grave, V olga–Ural area (0.85); Timber Grave, V olga (0.88); Pit Grave, Orenburg region (0.89); Timber Grave, Krivaya Luka, Lower Volga (0.96). The analysis based on the nine-trait set revealed one Western European parallel – with the Early Bronze Age (3rd millennium BC) group from Aveyron, France, (0.04), but ties with the steppe populations of the Eastern European Bronze Age are stronger.Afanasyev, Minusinsk Basin, pooled: Afanasyev, Saldyar I (–1.87); Timber Grave, Yasyrev, Lower Don (–0.71); Timber Grave, V olgograd and Astrakhan regions (–0.12); late Catacomb, Verkhne-Tarasovka, Lower Dnieper (–0.06); Pit Grave, Orenburg region (0.22); early Catacomb, the Molochnaya (0.24); Timber Grave, Luzanovka, Volga (0.64); Pit Grave, Volga–Ural area (0.65); Catacomb, Kalmykia (0.81); Pit Grave, Yuzhny Bug (0.82); Timber Grave, Volga (0.85); Pit Grave – Poltavka, V olga–Ural area (0.93); Timber Grave, Krivaya Luka, Lower V olga (0.95); Abashevo (0.99).Okunev, Uybat group, Minusinsk Basin: the only close parallel is with Okunev of the Tas-Khaza group (–0.95). Okunev of Chernovaya (1.44), and Karasuk (1.90) rank next.Okunev, Verkhni Askiz, Minusinsk Basin: Neolithic of Krasnoyarsk–Kansk area (–0.07).Okunev, Chernovaya, Minusinsk Basin: the only close parallel is with the Neolithic of Krasnoyarsk–Kansk area (0.36). The least removed among other populations is Okunev, Uybat group (1.44).Okunev, Tas-Khaza group, Minusinsk Basin: the only close parallel is with Okunev of Uybat (–0.95). The least distant among other groups is Karasuk (1.77).Okunev, Minusinsk Basin, pooled:the only close parallel is with the Neolithic of Krasnoyarsk–Kansk area (0.15). The least removed among other groups is Karasuk (3.37).Okunev, Aimyrlyg, Tuva: Pit Grave, Ingulets (–0.21); Timber Grave, Saratov region (–0.10); early Catacomb, Molochnaya (0.41); Timber Grave, Ukraine, pooled (0.45); Sapallitepe, southern Uzbukistan (0.67). Nine-trait set: early Catacomb, Molochnaya (–1.21); Tiefstichkeramik (related to Funnel Beaker), Ostorf, Germany, Late Neolithic, late 4th millennium BC (No.106) (–1.15); Afanasyev, Afanasyeva Gora (–0.76); Pit Grave, Ingulets (–0.53); late Catacomb, Verkhne-Tarasovka, Lower Dnieper (–0.39); Afanasyev, Minusinsk Basin (–0.37); Timber Grave, Volgograd and Astrakhan regions (–0.26); Abashevo (–0.26); Sapallitepe, Southern Uzbekistan (–0.02); Yelunino (0.01); Afanasyev, Saldyar I (0.24); Timber Grave, Khryaschevka, Volga (0.29).Karakol: the only parallel is with the Neolithic and Chalcolithic of Ust-Isha and Itkul, Upper Ob (0.98). The least removed among other groups is Yelovka II (3.87), whereas Okunev is much further (7.26).Yelunino:not a single close parallel. The least removed is Okunev of Tuva (1.56), next follow Djarat and Shengavit, Kura-Araxes culture, Armenia, 4th–3rd millennia BC (2.16); the pooled series of Kura-Araxes culture from Georgia ranks third (2.65), and Gumugou, Xinjiang, fourth (3.83). Nine-trait set: Poltavka (–0.13); Okunev of Tuva (0.01); early Catacomb, Molochnaya (0.01); Timber Grave, forest-steppe Volga area (0.22); Mierzanowice, Poland, Early Bronze Age, late 3rd – early 2nd millennia BC (No.173) (0.28).Ust-Tartas, Sopka-2: Krotovo, Sopka-2 (0.72).Krotovo, Sopka-2: Ust-Tartas, Sopka-2 (0.72).Samus:not a single close parallel. The least removed is Poltavka (1.18).Alakul, western Kazakhstan: early Catacomb, Molochnaya (–1.35); Pit Grave, Ingulets (–0.36); early Catacomb, Verkhne-Tarasovka, Lower Dnieper (0.44); late Timber Grave, V olga–Ural area (0.54); Kemi-Oba, Crimea (0.88). Nine-trait set: early Catacomb, Molochnaya (–1.39); Pit Grave, Ingulets (–0.88); Timber Grave, ground burials, Ukraine (–0.79), Pit Grave, Kakhovka, Lower Dnieper (–0.67); Parkhay II, Turkmenia, Middle and Late Bronze Age (–0.61); Tiszapolgar, Hungary, Chalcolithic, 5th–4th millennia BC (No.197) (–0.38); late Timber Grave, V olga–Ural area (–0.16); Rössen, eastern France, Neolithic, 5th millennium BC (No.43) (–0.09); Globular Amphorae, Germany and Poland, Early Bronze Age (early 3rd millennium BC) (No.192)(–0.07); Timber Grave, Ukraine, pooled (–0.03); Lengyel, Hungary, Neolithic, 5th millennium BC (No.40) (0.07); Meklenburg, Germany, Early Bronze Age, 4th–3rd millennia BC (No.107) (0.07); Aveyron, France, Early Bronze Age, 3rd century BC (No.99) (0.09); Unetice, Germany and Czechia, Bronze Age, 3rd–2nd millennia BC (No.208) (0.09); Linear Band Pottery, Neolithic, 6th millennium BC (No.14) (0.11); Pit Grave, Yuzhny Bug (0.20); Veterov, Austria, Bronze Age, III–II millennia BC (No.205) (0.21).Alakul, Yermak IV, the Irtysh: not a single close parallel. The least removed is the late Pit Grave group of Kalmykia* (1.32).Fedorov, Firsovo XIV, the Upper Ob: Afanasyev, Saldyar I (–1.82); Fedorov, Rudny Altai (–0.04); Catacomb, Kalmykia (0.06); Pit Grave – Poltavka, V olga–Ural area (0.43); Timber Grave, Luzanovka, V olga (0.87); Pit Grave, Orenburg region (0.90); early Catacomb, Molochnaya (0.94).Fedorov, the Upper Ob, pooled: Catacomb, Stavropol area (0.50); late Pit Grave, Kalmykia (0.80); Pit Grave, Stavropol area (0.90).Fedorov, Rudny Altai:Samus (–0.82); Afanasyev, Saldyar I (–0.71); Timber Grave, Luzanovka, Volga (–0.12); Fedorov, Firsovo XIV (–0.04); Pit Grave – *Group III according to V.A. Safronov (Shevchenko, 1986).130 A.G. Kozintsev / Archaeology Ethnology & Anthropology of Eurasia 37/4 (2009) 125–136Poltavka, Volga–Ural area (–0.02); early Northern Caucasian culture, Kalmykia, group II (0.61); Potapovka, V olga (0.63); Fedorov, northeastern Kazakhstan (0.79); late Catacomb, Kakhovka, Lower Dnieper (0.82), Catacomb, Don (0.83); Pit Grave, Orenburg region (0.88); Poltavka (0.99).Fedorov, northeastern Kazakhstan: late Pit Grave, Kalmykia (–1.50); late Catacomb, Yuzhny Bug (–1.44); Fedorov, Minusinsk Basin (–0.67); early Northern Caucasian culture, Kalmykia, group II (–0.59); late Catacomb, Kakhovka, Lower Dnieper (–0.47); Potapovka, V olga (–0.13); Catacomb, V olga (0.08); late Catacomb, Krivoy Rog, Upper Ingulets (0.16); late Catacomb, Ukraine, pooled (0.17); Pit Grave, Kalmykia (0.47); Pit Grave, Stavropol area (0.51); late Catacomb, Samara–Orel watershed (0.54); Timber Grave, Luzanovka, V olga (0.57); late Northern Caucasian culture, Kalmykia, group IV (0.72); late Catacomb, Zaporozhye, Lower Dnieper (0.74); Fedorov, Rudny Altai (0.79); Khvalynsk, V olga–Ural area (0.88).Fedorov, Minusinsk Basin: Fedorov, northeastern Kazakhstan (–0.67); late Northern Caucasian culture, Kalmykia, group IV (–0.09); Pit Grave, Stavropol area (–0.04); late Pit Grave, Kalmykia (–0.03); late Catacomb, Krivoy Rog, Upper Ingulets (0.31); late Catacomb, Samara/Orel interÀ uve (0.39); Afanasyev, Ust-Kuyum (0.51); late Catacomb, Kakhovka, Lower Dnieper (0.57); early Northern Caucasian culture, Kalmykia, group II (0.69); Timber Grave, Luzanovka, Volga (0.72); late Catacomb, Crimea (0.83); late Catacomb, Ukraine, pooled (0.84).“Andronoid”, Cherno-ozerye I, Omsk area: late Krotovo, Sopka-2 (0.91).“Andronoid”, Yelovka II, Tomsk area: not a single close parallel. The least distant is Irmen (1.01); those ranking next are late Krotovo, Sopka-2 (1.49), Cherno-ozerye I (3.42), and Karakol (3.87).Gumugou, Xinjiang: not a single close parallel. The least distant are two Fedorov series – from Rudny Altai (1.26) and from northeastern Kazakhstan (1.28).DiscussionAfanasyevThe results challenge the traditional idea that the sole and direct ancestors of the Afanasyev people were those of Pit Grave culture. Pit Grave af¿ nities rank ¿ rst only in the cases of Saldyar I and Karasuk III. Catacomb parallels are no fewer than those with Pit Grave, and in most instances they are the most pronounced. Every Afanasyev group has close ties with Catacomb groups. By contrast, not all Afanasyev series show close Pit Grave connections: these are absent in two groups of the Altai (Ursul and Kurota II) and in the pooled Altai sample. In half of the Altai series, ties with the Catacomb people of the Don are the most distinct, and the same is true of the pooled Altai group. Afanasyeva Gora and the pooled Minusinsk series are closest to the late Catacomb of the Lower Dnieper, whereas the series from Kurota II in the Altai, is closest to Poltavka. These results are matched by archaeological facts which, according to S.V. Tsyb (1981, 1984), evidence the importance of Poltavka and Catacomb cultures in Afanasyev origins.Strangely, similarities with Timber Grave people are no less numerous. In fact, for the pooled Minusinsk group they are more distinct than those with Catacomb and Pit Grave. The Timber Grave tribes could not have played any role in Afanasyev origins because they lived later; nor do any facts point to a reverse migration of the Afanasyev people or their descendants to Europe. The results can hardly be attributed to a slightly uneven representation of the three Eastern European cultures in the database, where the Pit Grave is represented by 15 series, the Catacomb by 18, and the Timber Grave by 16. More likely, these results testify to the considerable stability and relative homogeneity of the physical type of the Eastern European steppe populations over the Bronze Age despite the succession of cultures and apparently despite microevolutionary trends such as gracilization.Attempts at tracing the origins of this type have so far been unsuccessful. On the one hand, the Pit Grave people of the Lower Dnieper (Kakhovka and Kherson areas), the Catacomb people of the same region (Verkhne-Tarasovka, early group) and those of Kalmykia are similar to the Chalcolithic Khvalynsk population (5th–4th millennia BC). Accordingly, Khvalynsk might have been ancestral for some Eurasian steppe populations of the Bronze Age. Another Chalcolithic series which represents the Sredni Stog culture is more isolated, the least removed from it being various Afanasyev groups of the Altai and the Catacomb people of the Don. All these facts may point to the deep Eastern European roots of the Pit Grave, Catacomb, and partly Afanasyev communities.On the other hand, not all the Eastern European steppe populations of the Bronze Age appear to have been autochthonous. The analysis of a larger number of groups using the reduced trait battery reveals numerous early (4th millennium BC and earlier) Central and Western European parallels for groups such as the Pit Grave from the Ingulets and early Catacomb from the Molochnaya. These ties are especially evident in four gracile early Catacomb groups of the Ukraine, which show 14 close ties with Central and Western European populations and eight with those of Transcaucasia and Southwestern Central Asia. This apparently attests to migration, since the late Catacomb people are more robust, contrary to the normal diachronic trend (Kruts, 1990) and show no such ties. Nor are these af¿ nities shown by the Afanasyev people,。
