多重線性回歸與相關(guān)

1、衛(wèi)生統(tǒng)計學(xué)多重線性回歸與相關(guān)仇玉蘭上海交通大學(xué)醫(yī)學(xué)院公共衛(wèi)生學(xué)院QIU YulanShanghai Jiao Tong University School of Medicine多重線性回歸（multiple linearregression）與多重相關(guān)（multiplecorrelation）是研究續(xù)型因變量和多個自變量之間線性關(guān)系的統(tǒng)計學(xué)分析方法。QIU YulanShanghai Jiao Tong University School of Medicine多重線性回歸的概念及其統(tǒng)計描述一、數(shù)據(jù)與模型例：為了研究空氣中一氧化碳（NO）的濃度與汽車流量等因素的關(guān)系，有人測定了某城市交通點在

2、時間內(nèi)過往的汽車數(shù)、氣溫、空氣濕度、風(fēng)速以及空氣中的NO的濃度，數(shù)據(jù)如下表：QIU YulanShanghai Jiao Tong University School of Medicine空氣中 NO 濃度與相關(guān)因素的監(jiān)測數(shù)據(jù)多重線性回歸分析的基本目的是用一個以上的自變量（x1，x2，xp）的數(shù)值估計另一個反應(yīng)變量（Y）的平均水平及其變異性的統(tǒng)計學(xué)分析方法。QIU YulanShanghai Jiao Tong University School of Medicine多重線性回歸數(shù)學(xué)模型：Y的平均數(shù) = b0 + b1 x1 + ××× + b p xpY為應(yīng)

3、變量，x為自變量；0 為常數(shù)項， 即截距，表示當(dāng)所有自變量為0時反應(yīng)變量Y的平均值；j （1，×××p）是待定參數(shù)，為自變量Xj的偏回歸系數(shù)（partialregression coefficient）。QIU YulanShanghai Jiao Tong University School of Medicinej（j=1,2, ×××，p）表示在其它自變量固定不變的情況下，自變量Xj每改變一個時，單獨引起應(yīng)變量Y的平均改變量。QIU YulanShanghai Jiao Tong University School of Med

4、icine當(dāng)取得一組資料后，可以用參數(shù)估計法求出偏回歸系數(shù)的估計值b0,b1, ×××,bp， 相應(yīng)由樣本估計而得到的多重線性回歸方程為：Y = b0 + b1 X1 + b2 X 2 + ×××bp X pY為在各X j取一組定值條件下應(yīng)變量Y的平均估計值或平均預(yù)測值。QIU YulanShanghai Jiao Tong University School of Medicine比較各自變量對應(yīng)變量相對貢獻(xiàn)大小時，由于各自變量的和變異度不同，不能直接用偏回歸系數(shù)的大小作比較，須用標(biāo)準(zhǔn)化偏回歸系數(shù)（standardized part

5、ial regression coefficient）來作比較。標(biāo)準(zhǔn)化偏回歸系數(shù)較大的自變量對反應(yīng)變量Y的貢獻(xiàn)較大。QIU YulanShanghai Jiao Tong University School of Medicine標(biāo)準(zhǔn)化偏回歸系數(shù)：對原始變量的觀察值作標(biāo)準(zhǔn)正態(tài)化變換后，再配合回歸方程得到的偏回歸系數(shù)即為標(biāo)準(zhǔn)化偏回歸系數(shù)。自變量X原始觀察數(shù)據(jù)標(biāo)準(zhǔn)化： i iQIU YulanShanghai Jiao Tong University School of Medicine二、偏回歸系數(shù)的估計多重線性回歸分析前提條件：（1）線性（linear）因變量Y與自變量X呈線性關(guān)系通過散點圖觀

6、察X與Y的分布形態(tài)是否有直線趨勢QIU YulanShanghai Jiao Tong University School of Medicine（2）（independent）利用專業(yè)知識來間是否有關(guān)聯(lián)性。任意兩個觀察值之（3）正態(tài)（normal distribution）在一定范圍內(nèi)，任意給定X值，對應(yīng)的隨量Y都服從正態(tài)分布?？衫脤I(yè)知識和殘差散點圖來。QIU YulanShanghai Jiao Tong University School of Medicine殘差分析（residual analysis）：殘差分析通過標(biāo)準(zhǔn)化殘差圖來進(jìn)行QIU YulanShanghai Jiao

7、Tong University School of Medicine殘差ei = Yi - Yi標(biāo)準(zhǔn)化殘差e* = ei - e（殘差值均數(shù)）iS（殘差值標(biāo)準(zhǔn)差）e標(biāo)準(zhǔn)化殘差圖：以應(yīng)變量Y（或因變量X）為橫坐標(biāo)，以標(biāo)準(zhǔn)化殘差為縱坐標(biāo)，標(biāo)準(zhǔn)化殘差圖。的散點圖即為當(dāng)標(biāo)準(zhǔn)化殘差圖中的散點絕大部分在±2倍標(biāo)準(zhǔn)差之間、在以0參考線的上下隨機(jī)均勻地散布時，則模型與數(shù)據(jù)擬合的較好QIU YulanShanghai Jiao Tong University School of Medicine（4）等方差（equal variance）在一定范圍內(nèi)，不同的X值所對應(yīng)的隨機(jī)變量Y的方差相等。可用（X，

8、Y）的散點圖或殘差散點圖來。QIU YulanShanghai Jiao Tong University School of Medicine建立回歸方程的過程就是對回歸模型中的參數(shù)即偏回歸系數(shù)進(jìn)行估計的過程。求參數(shù)估計值的常用方法是最小二乘法原則（least squares method），即使殘差平方和達(dá)到最小的方法，使得這個模型的理論值和觀察值之間的離差平方之和盡可能地小。QIU YulanShanghai Jiao Tong University School of Medicine上例題NO濃度與相關(guān)因素的回歸方程：Y =- 0.14166 + 0.00011619 X1 + 0.0

9、0449 X 2- 0.00000655X 3 - 0.03468X 4計算依靠統(tǒng)計軟件包完成QIU YulanShanghai Jiao Tong University School of Medicine多重線性回歸的假設(shè)檢驗建立回歸方程后，需考慮：1. 方程是否符合資料特點？2. 各個自變量對應(yīng)變量的影響是否具有統(tǒng)計學(xué)意義？3. 每一觀察點是否都能用這一方程得到很好的預(yù)報？QIU YulanShanghai Jiao Tong University School of Medicine一、整體回歸效應(yīng)的假設(shè)檢驗（方差分析）確定所得的回歸方程是否有意義，用方差分析對回歸方程進(jìn)行檢驗。檢驗結(jié)

10、論與方差分析相同。QIU YulanShanghai Jiao Tong University School of MedicineSS總 = SS回 + SS殘n 總 = n -1，n回 = p（自變量個數(shù)），n 殘 = n - p -1MS= SS回 ，MS= SS殘 ；F = MS回回n殘nMS回殘殘例1：QIU YulanShanghai Jiao Tong University School of MedicineQIU YulanShanghai Jiao Tong University School of Medicine檢驗回歸方程整體意義的方差分析表QIU YulanShan

11、ghai Jiao Tong University School of Medicine變異來源SSdfMSFP總變異2425.30129回歸模型1773.3434443.33617.00<0.0001殘差651.9582526.078總變異SS總表示沒有利用X的信息時，Y的觀察值的變異，其自由度=n-1；殘差SS殘差表示回歸未能解釋的那部分變異， 其自由度=n-自變量個數(shù)p-1；回歸模型SS回表示回歸模型的貢獻(xiàn)，其自由度=自變量個數(shù)p；MS為相應(yīng)的平方和SS與其自由度之商；F=MS回/MS殘差。P<0.05表示回歸方程具有統(tǒng)計學(xué)意義。QIU YulanShanghai Jiao

12、Tong University School of MedicineQIU YulanShanghai Jiao Tong University School of MedicineMSum maryMRR SquareAdjusted R SquareStd. Error of the Estimate12.811a.841b.657.707.645.6855.450915.13104a. Predictors: (Constant), 瘦素b. Predictors: (Constant), 瘦素, 體重指數(shù)QIU YulanShanghai Jiao Tong University Sc

13、hool of MedicineAN OVAcSum of SquaresdfMean SquareFSig.M1RegressionResidual Total1593.353831.9482425.301128291593.35329.71253.626.000a2RegressionResidual Total1714.458710.8442425.30122729857.22926.32832.560.000ba. Predictors: (Constant), 瘦素b. Predictors: (Constant), 瘦素, 體重指數(shù)c. Dependent Variable: 脂聯(lián)

14、素例2：QIU YulanShanghai Jiao Tong University School of Medicined.Dependent Variable: 一氧化氮YQIU YulanShanghai Jiao Tong University School of MedicineMSum marydMRR SquareAdjusted R SquareStd. Error of the EstimateChange StatisticsR Square ChangeF Changedf1df2Sig. F Change123.808a.851b.887c.653.725.787.63

15、7.698.755.035801.032640.029387.653.072.06341.3765.4685.907111222120.00.02.02095a. Predictors: (Constant), 車流X1b. Predictors: (Constant), 車流X1, 風(fēng)速X4c. Predictors: (Constant), 車流X1, 風(fēng)速X4, 氣 2ANOVAdSum ofMSquaresdfMeanSquareFSig.1RegressionResidual Total.000a.053.028.08112223.053.00141.3762RegressionRe

16、sidual Total.000b.059.022.08122123.029.00127.623.000c3RegressionResidual Total.064.017.08132023.021.00124.687a.b.Predictors:Predictors:(Constant),(Constant),車流X1車流X1,風(fēng)速X4c.Predictors:(Constant),車流X1,風(fēng)速X4,氣2d.Dependent Variable: 一氧化氮YQIU YulanShanghai Jiao Tong University School of Medicine二、偏回歸系數(shù)的 t

17、 檢驗偏回歸系數(shù)j的假設(shè)檢驗：偏回歸系數(shù)的 t 檢驗是在回歸方程具有統(tǒng)計學(xué)意義的情況下，檢驗?zāi)硞€總體偏回歸系數(shù)等于零的假設(shè)，以是否相應(yīng)的那個自變量對回歸確有貢獻(xiàn)。QIU YulanShanghai Jiao Tong University School of MedicineQIU YulanShanghai Jiao Tong University School of MedicineH0 : b j= 0, H1 : b j¹ 0t= bj bjSbjSbj為第j個偏回歸系數(shù)的標(biāo)準(zhǔn)誤a.b.c.Predictors in the MPredictors in the M Depe

18、ndent Variable: (Constant), 瘦素: (Constant), 瘦素, 體重指數(shù)脂聯(lián)素QIU YulanShanghai Jiao Tong University School of Medicinea. Dependent Variable: 脂聯(lián)素CoefficientsaExcluded VariablecsMBeta IntSig.Partial CorrelationCollinearity StatisticsTolerance1體重指數(shù)病程空腹血糖-.362a-.027a-.185a-2.145-.237-1.707.041.814.099-.382-.0

19、46-.312.381.998.9742病程空腹血糖-.082b-.147b-.756-1.385.457.178-.147-.262.947.937MUnstandardized CoefficientsStandardized CoefficientstSig.95% Confidence Interval for BBStd. ErrorBetaLower BoundUpper Bound1(Constant)瘦素30.528-1.1611.882.159-.81116.219-7.323.000.00026.672-1.48634.383-.8372(Constant)瘦素體重指數(shù)53

20、.481-.753-1.08710.848.242.507-.525-.3624.930-3.112-2.145.000.004.04131.223-1.249-2.12775.739-.256-.047上述四個變量中，變量 X1、X2 和X4 的偏回歸系數(shù)在 0.05 概率水平具有統(tǒng)計學(xué)意義而氣濕（ X3 ）對 NO 濃度的影響無統(tǒng)計學(xué)意義。QIU YulanShanghai Jiao Tong University School of MedicineCo effi cie ntsaUnstandardized CoefficientsStandardized Coefficients95

21、% Confidence Interval for BCorrelationsMBStd. ErrorBetatSig.Lower BoundUpper BoundZero-orderPartialPart1(Constant)車流X1-.135.000.035.000.808-3.8296.432.001.000-.209.000-.062.000.808.808.2(Constant)車流X1 風(fēng)速X4-.050.000-.025.049.000.011.623-.325-1.0274.476-2.338.316.000.029-.151.000-.048.051.000-.003.808

22、-.680.699-.455.-.3(Constant)車流X1 風(fēng)速X42-.142.000-.035.004.058.000.010.002.592-.448.273-2.4524.699-3.3162.430.024.000.003.025-.263.000-.057.001-.021.000-.013.008.808-.680.017.724-.596.477.-.251a. Dependent Variable: 一氧化氮Y808513268485342氣d.Dependent Variable: 一氧化氮YQIU YulanShanghai Jiao Tong University

23、 School of MedicineExcluded VariabledsM Beta IntSig.Partial CorrelationCollinearity StatisticsTolerance1氣2濕X3 速X4.134a-.048a-.325a1.059-.344-2.338.302.734.029.225-.075-.455.980.844.677氣風(fēng)2氣2.273b-.018b2.430-.141.025.889.477-.032.844.835氣濕X33氣濕X3-.001c-.009.993-.002.832a. Predictors in the M: (Constan

24、t), 車流X1b. Predictors in the M: (Constant), 車流X1, 風(fēng)速X4c. Predictors in the M: (Constant), 車流X1, 風(fēng)速X4, 氣2復(fù)相關(guān)系數(shù)與偏相關(guān)系數(shù)一、確定系數(shù)、復(fù)相關(guān)系數(shù)與調(diào)整確定系數(shù)復(fù)相關(guān)系數(shù)的平方稱為確定系數(shù)(coefficient of determination) ，或決定系數(shù)，記為R2 ，用以反映線性回歸模型能在多大程度上解釋反應(yīng)變量 Y 的變異性。QIU YulanShanghai Jiao Tong University School of Medicine確定系數(shù)R2，其定義是回歸平方和SS回歸

25、占總離均差平方和SS總的比例QIU YulanShanghai Jiao Tong University School of Medicine2SS回(回歸平方和) R=SS(總平方和)總回歸平方和即對回歸模型的貢獻(xiàn)R2 用以反映線性回歸模型能在多大程度上解釋反應(yīng)變量 Y 的變異性，或者說回歸方程使反應(yīng)變量Y的總變異減少了的百分比， 其取值范圍為0R21。R2接近于1，表示樣本數(shù)據(jù)很好的擬合了所選用的線性回歸模型。用R2可定量評價在 Y 的總變異中，由 x 變量組建立的線性回歸方程所能解釋的比例。QIU YulanShanghai Jiao Tong University School of

26、MedicineR2=0.707，表示包含瘦素和體重指數(shù)兩個自變量的回歸方程可解釋脂聯(lián)素的變異性為70.7%。QIU YulanShanghai Jiao Tong University School of MedicineMSum maryMRR SquareAdjusted R SquareStd. Error of the Estimate12.811a.841b.657.707.645.6855.450915.13104a. Predictors: (Constant), 瘦素b. Predictors: (Constant), 瘦素, 體重指數(shù)確定系數(shù)的平方根即R稱為復(fù)相關(guān)系數(shù)（mu

27、ltiple correlation coefficient）， 它表示p個自變量共同對應(yīng)變量線性相關(guān)的密切程度， 0R1QIU YulanShanghai Jiao Tong University School of MedicineR =R2復(fù)相關(guān)系數(shù)亦等于Y與其回歸估計值Y 的相關(guān)系數(shù)R = Cor(Y ,Y)QIU YulanShanghai Jiao Tong University School of Medicine上例：R =R2 =0.7072 = 0.84表示瘦素、體重指數(shù)這兩個變量的線性組合與脂聯(lián)素水平的復(fù)相關(guān)系數(shù)為0.84。調(diào)整的R2 (adjusted R-square

28、) ：當(dāng)回歸方程中包含有很多自變量，即使其中有一些自變量對解釋反應(yīng)變量變異的貢獻(xiàn)極小，隨著回歸方程的自變量的增加，R2值表現(xiàn)為只增不減，這是確定相關(guān)系數(shù)R2 的缺點。調(diào)整的礦記為R2 ，旨在對回歸方程中自變量個數(shù)實施懲罰。QIU YulanShanghai Jiao Tong University School of Medicinen為樣本量，p為方程中自變量的個數(shù)。較大的p會使值降低，從而實施對回歸方程中自變量個數(shù)的懲罰。QIU YulanShanghai Jiao Tong University School of MedicineR2a22p(1- R2 )2p(1- R2 )Ra=

29、R- n - p -1 = R-æ n1 öpç-1-÷è pp ø=2 -(1- R2 )Ræ n1 öç-1-÷è pp ød.Dependent Variable: 一氧化氮YQIU YulanShanghai Jiao Tong University School of MedicineMSum marydMRR SquareAdjusted R SquareStd. Error of the EstimateChange StatisticsR Square Cha

30、ngeF Changedf1df2Sig. F Change123.808a.851b.887c.653.725.787.637.698.755.035801.032640.029387.653.072.06341.3765.4685.907111222120.00.02.02095a. Predictors: (Constant), 車流X1b. Predictors: (Constant), 車流X1, 風(fēng)速X4c. Predictors: (Constant), 車流X1, 風(fēng)速X4, 氣 2二、偏相關(guān)系數(shù)（partial correlation coefficient）實例：兄弟倆大明

31、和小明暑期勤工儉學(xué)， 大明在超市賣冷飲，小明在游泳池收門 票。大明冷飲賣得多，小明的門票也收 得多，大明冷飲賣得少，小明的門票也 收得少。問題：吃冷飲與游泳有關(guān)系嗎？QIU YulanShanghai Jiao Tong University School of Medicine對冷飲銷售量、游泳人數(shù)以及當(dāng)天的氣溫進(jìn)行分析，發(fā)現(xiàn)冷飲銷售量和游泳人數(shù)呈正相關(guān)；冷飲銷售量與氣溫相關(guān)性更強(qiáng)。但當(dāng)扣除氣溫的影響之后，冷飲銷售量與游泳人數(shù)的相關(guān)性幾乎不存在。所以，扣除其他變量的影響后，變量Y與X的相關(guān)，稱為Y與X的偏相關(guān)系數(shù)。QIU YulanShanghai Jiao Tong University

32、School of MedicineQIU YulanShanghai Jiao Tong University School of MedicineQIU YulanShanghai Jiao Tong University School of MedicineCorrelations*. Correlation is significant at the 0.01 level (2-tailed).冷飲銷售量游泳人數(shù)氣溫冷飲銷售量 Pearson CorrelationSig. (2-tailed) N111.972*.00011.989*.00011游泳人數(shù)Pearson Correla

33、tionSig. (2-tailed) N.972*.00011111.976*.00011氣溫Pearson CorrelationSig. (2-tailed) N.989*.00011.976*.00011111QIU YulanShanghai Jiao Tong University School of MedicineCo rrel ati onsControl Variables冷飲銷售量游泳人數(shù)氣溫冷飲銷售量 CorrelationSignificance (2-tailed) df1.000. 0.215.5518游泳人數(shù)CorrelationSignificance (2-

34、tailed) df.215.55181.000. 0QIU YulanShanghai Jiao Tong University School of MedicineCorrelationsControl Variables冷飲銷售量氣溫游泳人數(shù)冷飲銷售量 CorrelationSignificance (2-tailed) df1.000. 0.787.0078氣溫CorrelationSignificance (2-tailed) df.787.00781.000. 0QIU YulanShanghai Jiao Tong University School of MedicineCor

35、relationsControl Variables氣溫游泳人數(shù)冷飲銷售量氣溫CorrelationSignificance (2-tailed) df1.000. 0.419.2298游泳人數(shù) CorrelationSignificance (2-tailed) df.419.22981.000. 0QIU YulanShanghai Jiao Tong University School of MedicineCorrelations*. Correlation is significant at the 0.01 level (2-tailed).脂聯(lián)素體重指數(shù)脂聯(lián)素Pearson Co

36、rrelationSig. (2-tailed) N130-.776*.00030體重指數(shù)Pearson CorrelationSig. (2-tailed) N-.776*.00030130QIU YulanShanghai Jiao Tong University School of MedicineCorrelations脂聯(lián)素病程脂聯(lián)素Pearson CorrelationSig. (2-tailed) N130.011.95430病程Pearson CorrelationSig. (2-tailed) N.011.95430130QIU YulanShanghai Jiao Tong

37、 University School of MedicineCorrelations*. Correlation is significant at the 0.01 level (2-tailed).脂聯(lián)素瘦素脂聯(lián)素Pearson CorrelationSig. (2-tailed) N130-.811*.00030瘦素Pearson CorrelationSig. (2-tailed) N-.811*.00030130QIU YulanShanghai Jiao Tong University School of MedicineCorrelations脂聯(lián)素空腹血糖脂聯(lián)素Pearson

38、CorrelationSig. (2-tailed) N130-.049.79830空腹血糖Pearson CorrelationSig. (2-tailed) N-.049.79830130Co rrel ati onsQIU YulanShanghai Jiao Tong University School of MedicineControl Variables脂聯(lián)素體重指數(shù)病程 & 瘦素 & 空腹血糖脂聯(lián)素CorrelationSignificance (2-tailed) df1.000. 0-.362.06325體重指數(shù)CorrelationSignificance

39、 (2-tailed) df-.362.063251.000. 0CorrelationsQIU YulanShanghai Jiao Tong University School of MedicineControl Variables脂聯(lián)素病程瘦素 & 空腹血糖 & 體重指數(shù)脂聯(lián)素CorrelationSignificance (2-tailed) df1.000. 0-.123.54025病程CorrelationSignificance (2-tailed) df-.123.540251.000. 0CorrelationsQIU YulanShanghai Jiao

40、Tong University School of MedicineControl Variables脂聯(lián)素瘦素空腹血糖 & 體重指數(shù) & 病程脂聯(lián)素CorrelationSignificance (2-tailed) df1.000. 0-.540.00425瘦素CorrelationSignificance (2-tailed) df-.540.004251.000. 0Co rrel ati onsQIU YulanShanghai Jiao Tong University School of MedicineControl Variables脂聯(lián)素空腹血糖體重指數(shù) &a

41、mp; 病程 & 瘦素脂聯(lián)素CorrelationSignificance (2-tailed) df1.000. 0-.250.20825空腹血糖CorrelationSignificance (2-tailed) df-.250.208251.000. 0當(dāng)偏相關(guān)系數(shù)與偏回歸系數(shù)發(fā)生時，應(yīng)仔細(xì)分析發(fā)生的這個變量。如脂聯(lián)素相關(guān)因素中，多重線性回歸分析結(jié)果表示瘦素和體重指數(shù)有意義，而偏相關(guān)系數(shù)結(jié)果顯示體重指數(shù)沒有意義。因此對體重指數(shù)對脂聯(lián)素的究竟有沒有影響就要慎重考慮，慎下結(jié)論。QIU YulanShanghai Jiao Tong University School of Medic

42、ineQIU YulanShanghai Jiao Tong University School of MedicineCorrelations*. Correlation is significant at the 0.01 level (2-tailed).一氧化氮Y車流X1一氧化氮YPearson CorrelationSig. (2-tailed) N124.808*.00024車流X1Pearson CorrelationSig. (2-tailed) N.808*.00024124QIU YulanShanghai Jiao Tong University School of Me

43、dicineCo rrel ati ons一氧化氮Y氣 2一氧化氮Y Pearson CorrelationSig. (2-tailed) N氣 2Pearson CorrelationSig. (2-tailed) N124.017.93624.017.93624124QIU YulanShanghai Jiao Tong University School of MedicineCo rrel ati ons一氧化氮Y氣濕X3一氧化氮Y Pearson CorrelationSig. (2-tailed) N124.279.18824氣濕X3Pearson CorrelationSig.

44、(2-tailed) N.279.18824124QIU YulanShanghai Jiao Tong University School of MedicineCorrelations*. Correlation is significant at the 0.01 level (2-tailed).一氧化氮Y風(fēng)速X4一氧化氮YPearson CorrelationSig. (2-tailed) N124-.680*.00024風(fēng)速X4Pearson CorrelationSig. (2-tailed) N-.680*.00024124氣2&氣濕X3&一氧化氮YCorrel

45、ationSignificance df1.000.696風(fēng)速X4(2-tailed).000019車流X1CorrelationSignificance df.6961.000(2-tailed).000.190QIU YulanShanghai Jiao Tong University School of MedicineCo rrel ati onsControl Variables一氧化氮Y車流X1Co rrel ati onsControl Variables一氧化氮Y氣2氣濕X3 & 風(fēng)速X4 & 車流X1一氧化氮YCorrelationSignificance (

46、2-tailed) df1.000. 0.477.02919氣2Correlation.4771.000QIU YulanShanghai Jiao Tong University School of MedicineSignificance (2-tailed).029.df190Co rrel ati onsQIU YulanShanghai Jiao Tong University School of MedicineControl Variables一氧化氮Y氣濕X3風(fēng)速X4 & 車流X1 & 氣 2一氧化氮Y CorrelationSignificance (2-ta

47、iled) df1.000. 0-.002.99319氣濕X3CorrelationSignificance (2-tailed) df-.002.993191.000. 0車流X1 & 氣2 &一氧化氮YCorrelation1.000-.593QIU YulanShanghai Jiao Tong University School of Medicine氣濕X3Significance (2-tailed).005 df019風(fēng)速X4Correlation-.5931.000Significance (2-tailed).005.df190Co rrel ati onsC

48、ontrol Variables一氧化氮Y風(fēng)速X4空氣中NO濃度與各自變量的相關(guān)系數(shù)和偏相關(guān)系數(shù)偏相關(guān)系數(shù)偏相關(guān)系數(shù)P值自變量相關(guān)系數(shù)車流X10.808*0.017#0.279#-0.68*0.6960.477-0.002-0.5930.00050.02890.99250.0046氣2氣濕X3風(fēng)速X4注：*p<0.05；#p>0.05QIU YulanShanghai Jiao Tong University School of Medicine自變量篩選對多元線性回歸方程中的自變量要進(jìn)行選擇，使回歸方程中只包含對應(yīng)變量有統(tǒng)計學(xué)意義的自變量，即所謂的“最優(yōu)” 方程。QIU YulanShanghai Jiao Tong University School of Medicine一、自變量篩選的標(biāo)準(zhǔn)與原則1. 殘差平方和縮小或決定系數(shù)增大2. 殘差均方縮小或調(diào)整決定