應用統(tǒng)計學-第八章因子分析-新

1、2022-2-231第八章 因子分析n因子分析的目的與類型因子分析的目的與類型n探索性因子分析的模型探索性因子分析的模型n因子模型的求解因子模型的求解n因子旋轉(zhuǎn)因子旋轉(zhuǎn)n因子得分因子得分n因子分析的因子分析的SPSS實現(xiàn)實現(xiàn)n實例分析實例分析n確認性因子分析確認性因子分析Origins of Factor AnalysisIn conjunction with his famous two-factor theory of intelligenceCharles Spearman1863-1945一、因子分析的目的與類型Origins of Factor AnalysisnWanted to

2、estimate intelligence of 24 children in a village school.nRealized way of measuring intelligence was imperfect and that the correlation between any two variables (say, ones score on a mathematics exam and on a classics exam) would be underestimated.nNoticed that the observed correlations between the

3、 variables he was interested in were all positive and followed a pattern.nSpearman wanted to develop a model that would reflect the pattern he saw.What did Spearman notice?Correlations Between Examination ScoresNotice the trend across each row on the upper diagonal認為存在著“general intelligence”，影響著個體在所

4、有智力活動中的表現(xiàn)(解釋各變量間的高度相關(guān)），而個體在不同智力活動中表現(xiàn)的差異則是由另一些“specific factors”決定的（解釋相關(guān)程度差異）。區(qū)分這兩類因素可以更準確地預測出某個人在某項工作中的表現(xiàn)。two-factor theory of intelligence nCould model each test score as having two types of components: one common to all the scores and one specific to the particular testvf: available to the same in

5、dividual to the same degree for all intellectual acts v : varies in strength from one act to another nIf one knows how a person performs on one task that is highly saturated with “f, one can safely predict a similar level of performance for a another highly “f saturated task. nthe most important inf

6、ormation to have about a persons intellectual ability is an estimate of their “f Schematicallyclassicsfrenchenglishmathdiscrmusic123456fGoals of Factor Analysisnmodel correlation v通過對多個變量的相關(guān)系數(shù)矩陣的研究，找出同時影響或支配多個變量的共性因素。nallow for contextual of the new variablesnevaluate the original data in light of t

7、he new variables實例1n考查人體的五項生理指標：收縮壓、舒張壓、考查人體的五項生理指標：收縮壓、舒張壓、心跳間隔、呼吸間隔和舌下溫度。心跳間隔、呼吸間隔和舌下溫度。n從生理學知識可知，這五項指標是受植物神經(jīng)從生理學知識可知，這五項指標是受植物神經(jīng)支配的，植物神經(jīng)又分為交感神經(jīng)和負交感神支配的，植物神經(jīng)又分為交感神經(jīng)和負交感神經(jīng)，因此這五項指標至少受到兩個公共因子的經(jīng)，因此這五項指標至少受到兩個公共因子的影響，也可用因子模型去處理。影響，也可用因子模型去處理。n五項指標均可觀測，而兩個公共因子是不可直五項指標均可觀測，而兩個公共因子是不可直接觀測的：通過指標與公共因子的關(guān)系診病。

8、接觀測的：通過指標與公共因子的關(guān)系診病。舒張壓舒張壓心跳間隔心跳間隔呼吸間隔呼吸間隔舌下溫度舌下溫度收縮壓收縮壓交感神經(jīng)交感神經(jīng)負交感神經(jīng)負交感神經(jīng)實例2 n林登根據(jù)他收集的來自139名運動員的比賽數(shù)據(jù)，對第二次世界大戰(zhàn)以來奧林匹克十項全能比賽的得分作了因子分析研究。這十個全能項目為：v100米跑x1、跳遠x2、鉛球x3、跳高x4、400米跑x5、110米跨欄x6、鐵餅x7、撐桿跳x8、標槍x9、1500米跑x10n對10個變量標準化后的因子分析表明，十項得分基本上可歸結(jié)于他們的短跑速度、爆發(fā)性臂力、爆發(fā)性腿力和耐力這四個方面，每一方面都稱為一個公共因子。因子分析的類型n探索性因子分析探索性因

9、子分析exploratory Factor Analysis v根據(jù)變量間相關(guān)關(guān)系探索因子結(jié)構(gòu)根據(jù)變量間相關(guān)關(guān)系探索因子結(jié)構(gòu)v實例實例2n確認性因子分析確認性因子分析Confirmatory Factor Analysis v檢驗對因子結(jié)構(gòu)的先驗認識是否合理，評估因子模檢驗對因子結(jié)構(gòu)的先驗認識是否合理，評估因子模型的擬合程度型的擬合程度v實例實例1二、探索性因子分析模型n正交因子模型正交因子模型n重要假設(shè)重要假設(shè)n因子載荷陣的統(tǒng)計意義因子載荷陣的統(tǒng)計意義1. 正交因子模型ijmjijifix1aobserved variablesfactor loadingscommon factorsspe

10、cific factors設(shè)：可觀測隨機變量xi，E(xi)=i，i=1,2, p,不可觀測正交隨機變量fj，j=1,2,m, E(fj)=0，(fj)=1，一般因子模型：反映了各變量與公共因子的關(guān)系pmpmpppmmmmfffxfffxfffxaaaaaaaaa2211d22222121d211212111d1mpFAxAFxTmppmpfffxxxd2121dd2d1d),(),(),(則：設(shè)：因子載荷陣因子分析：求出各因子載荷量aij，并在此基礎(chǔ)上計算各樣本的因子得分，據(jù)此評價樣本，預測。與線性回歸模與線性回歸模型的區(qū)別？型的區(qū)別？因子載荷量中中心心化化變變量量), 2 , 1(2211

11、mjxxxfpsjpsjsjj因子模型x1x2x3xixp123iP Pf1f2fm十項全能例104103102101103343332312242322211141312111500100耐力爆發(fā)性腿力爆發(fā)性臂力短跑速度米耐力爆發(fā)性腿力爆發(fā)性臂力短跑速度鉛球耐力爆發(fā)性腿力爆發(fā)性臂力短跑速度跳遠耐力爆發(fā)性腿力爆發(fā)性臂力短跑速度米跑，aaaaaaaaaaaaaaaa因子模型因子模型因子得分計算公式因子得分計算公式ssssssssssssxxxxxxxxxxxx10104242141101032321311010222212110101212111，耐力爆發(fā)性腿力爆發(fā)性臂力短跑速度2. Impor

12、tant Assumptionsnf1, f2, , fm are independent, with identical distributions having a mean of 0 and a variance of 1n 1、 2 、 pare independent, with distributions having a mean of 0 and variances i2nfi and j are independent for all i, j combinations0)()cov(),()var(,)(,)var(,)(22221FF,0IF0FTpEdiagEE即：Un

13、der the assumptions aboveActually, the goal of “factor analysis” is to try to decompose the covariance matrix (or correlation matrix for standardized data) into two parts each in the form dictated above.22221d)var()var()var()var()var(pTTTAAxAFAFAxTT1/21/2FF)(U(UURRUUZTTU,)var(主成分分析：當m=p時，var(x x)=AA

14、T然而只有當m0.2Measure Reliabilityntells us whether a particular variable Xi does a good job of measuring the true underlying factor that it purports to measurevRule of thumb: i0.7X1XiXk1i3f11ikReliability of the composite indexntells us whether the measurement model is good enough to measure the latent

15、factorX1XiXk1i3f11ik83. 0)()()(1221212kiikiikiic因子分析：止痛藥例Communalities.453.569.509.652.517.652.499.634.487.637.479.609NSTOMACHNSIDEEFFSTOPPAINWKSQUICKKPAWAKELIMRELIEInitialExtractionExtraction Method: Principal Axis Factoring.Rotated Factor Matrixa.141.7411.440E-02.808.8019.787E-02.794-5.53E-023.900

16、E-02-.797-.777-7.38E-02NSTOMACHNSIDEEFFSTOPPAINWKSQUICKKPAWAKELIMRELIE12FactorFactor1:有效性Factor2:和緩性極大似然估計結(jié)果83. 0)()()(1221212kiikiikiieffectiveRotated Factor Matrixa.141.7411.440E-02.808.8019.787E-02.794-5.53E-023.900E-02-.797-.777-7.38E-02NSTOMACHNSIDEEFFSTOPPAINWKSQUICKKPAWAKELIMRELIE12Factor比較模型

17、80. 0)()()(1221212kiikiikiieffective本章小結(jié)n因子分析：因子分析：n探索性因子模型探索性因子模型FAxTd因子模型求解：因子模型求解：因子旋轉(zhuǎn)：使每個變量的載荷都盡可能集中在某個因子上，因子旋轉(zhuǎn)：使每個變量的載荷都盡可能集中在某個因子上，以使公因子易于解釋。因子旋轉(zhuǎn)不改變變量共同度和特殊以使公因子易于解釋。因子旋轉(zhuǎn)不改變變量共同度和特殊因子方差。因子方差。因子模型建立之后，通過因子得分評價各樣本在各個公共因因子模型建立之后，通過因子得分評價各樣本在各個公共因子上的表現(xiàn)子上的表現(xiàn)確認性因子分析n先驗因子結(jié)構(gòu)先驗因子結(jié)構(gòu)n因子間可相關(guān)因子間可相關(guān)n可估計標準誤差，做參數(shù)檢驗、擬合優(yōu)可估計標準誤差，做參數(shù)檢驗、擬合優(yōu)度檢驗、測量指標可靠度度檢驗、測量指標可靠度n便于模型比較便于模型比較因子分析與主成分分析的區(qū)別n主成分分析只是通常的變量變換，不能做模型主成分分析只是通常的變量變換，不能做模型v將一組具有相關(guān)關(guān)系的變量變換為一組互不相關(guān)的變量將一組具有相關(guān)關(guān)系的變量變換為一組互不相關(guān)的變量v主成分是可觀測的原始變量的線性組合主成分是可觀測的原始變量的線性組合v主成分個數(shù)主成分個數(shù)=變量數(shù)（應用時只取前幾個主成分）變量數(shù)（應用時只取前幾個主成分）v主成分不一定可以解釋主成分不一定可以解釋n因子分析需要構(gòu)造因子模型因子分析需要構(gòu)造因子模型v用盡可能少的公