衛(wèi)生統(tǒng)計(jì)學(xué)課件12多重線性回歸分析研

1、引言,多重線性回歸（多元線性回歸,logistic,回歸,Cox,回歸,判別分析、聚類分析,主成分分析、因子分析,多因素分析是研究多種因素互相聯(lián)系、互相制約,的規(guī)律性的一個(gè)重要而活躍的統(tǒng)計(jì)學(xué)分支,70,年,代后在醫(yī)學(xué)領(lǐng)域應(yīng)用廣泛，常用的方法有,多重線性回歸分析,用途,探討多個(gè)自變量與應(yīng)變量之間的依存關(guān)系以及各,個(gè)自變量對(duì)應(yīng)變量的相對(duì)貢獻(xiàn)大小，從而探討應(yīng),變量的主要影響因素,人的,體重,與身高、胸圍,血壓值,與年齡、性別、勞動(dòng)強(qiáng)度、飲食習(xí)慣,吸煙狀況、家族史,糖尿病人的,血糖,與胰島素、糖化血紅蛋白、血清,總膽固醇、甘油三脂,應(yīng)用條件,應(yīng)變量,為定量變量,自變量,可以是定量變量，也,可以是分類變

2、量，但分類變量的個(gè)數(shù)不宜超過(guò)自,變量個(gè)數(shù)的,1/3,各自變量彼此獨(dú)立,各個(gè)自變量取不同值的組合時(shí)應(yīng)變量服從,正態(tài)分,布且方差齊,m,m,x,b,x,b,b,y,1,1,0,式中,b,0,為截距,b,j,j=1,2,m,為偏回歸系數(shù),偏回歸系數(shù),b,j,表示在其他自變量固定不變的,情況下，自變量,X,j,每改變一個(gè)單位時(shí)，單獨(dú)引,起應(yīng)變量,y,的平均改變量,模型的構(gòu)造,參數(shù)估計(jì),求參數(shù)估計(jì)值的常用方法是,最小二乘法,即使,殘差平,方和達(dá)到最小,的方法,假設(shè)檢驗(yàn),對(duì)整個(gè)回歸方程進(jìn)行假設(shè)檢驗(yàn),對(duì),偏回歸系數(shù),進(jìn)行假設(shè)檢驗(yàn),t,bj,b,j,s,bj,誤差,回歸,回歸,誤差,回歸,回歸,MS,MS,n

3、,SS,SS,F,1,例,15-1,為探討女大學(xué)生的,體重、胸圍,與胸圍呼吸差,對(duì),肺活量,的影響，某研究者調(diào),查了,20,名女大學(xué)生的相關(guān)資料，見(jiàn)表,15-1,并分別用,體重、胸圍與胸圍呼吸差,對(duì),肺活量,進(jìn)行線性回歸分析,多重線性回歸分析的步驟,一）估計(jì)各項(xiàng)參數(shù)，建立多重線性回歸方程模型,二）對(duì)整個(gè)模型進(jìn)行假設(shè)檢驗(yàn)，模型有意義的前提,下，再分別對(duì)各,偏回歸系數(shù),進(jìn)行假設(shè)檢驗(yàn),三）計(jì)算相應(yīng)指標(biāo)，對(duì)模型的擬合效果進(jìn)行評(píng)價(jià),多重線性回歸方程的建立,Analyze,Regression,Linear,Dependent :Y,Independent(s):X1,X2,X3,Method:Enter

4、,OK,R,復(fù)相關(guān)系數(shù),0.884,R Square,決定系數(shù),0.781,Adj R-Sq,校正決定系數(shù),0.740,Std.Error of the Estimate,剩余,標(biāo),準(zhǔn)差,216.0570680,Model S,um,mary,8,84,a,7,81,7,40,216.0570,680,Model,1,R,R Square,Adju,sted R Square,Std. Error of,the Estim,ate,Predictors: (Constant), X3, X2, X1,a,ANO,VA,b,2664,484.4,94,3,8881,61.49,8,19.026,

5、0,00,a,7468,90.50,6,16,4668,0.657,3411,375.0,00,19,Regression,Residual,Total,Model,1,Sum of Squ,ares,df,Mean Sq,uare,F,Sig,Predictors: (Constant), X3, X2, X1,a,Dependent Variable: Y,b,C,oe,ffi,ci,en,ts,a,2262.081,1081.870,2.09,1,0,53,48.135,22.058,3,42,2.182,0,44,38.550,13.346,4,44,2.889,0,11,104.58

6、5,74.361,2,60,1.406,1,79,Constant,X1,X2,X3,Model,1,B,Std. Error,U,nstand,ardized,Co,eff,icients,Beta,Standardized,Co,eff,icients,t,Sig,Dependent Variable: Y,a,3,2,1,585,104,550,38,135,48,081,2262,X,X,X,Y,回歸方程的假設(shè)檢驗(yàn)與評(píng)價(jià),一）回歸方程的假設(shè)檢驗(yàn),二）偏回歸系數(shù)的假設(shè)檢驗(yàn),三）有關(guān)評(píng)價(jià)指標(biāo),一）回歸方程的方差分析,H0,所有回歸系數(shù)為,0,H1,至少有一個(gè)回歸系數(shù)不為,0,ANO,VA,

7、b,2664,484.4,94,3,8881,61.49,8,19.026,0,00,a,7468,90.50,6,16,4668,0.657,3411,375.0,00,19,Regression,Residual,Total,Model,1,Sum of Squ,ares,df,Mean Sq,uare,F,Sig,Predicto,rs: (Co,nstant), X3, X2, X1,a,Dependent Variable: Y,b,二）偏回歸系數(shù)的假設(shè)檢驗(yàn)及其評(píng)價(jià),各偏回歸系數(shù)的,t,檢驗(yàn),Coe,fficients,a,2262.081,1081.870,2.091,053,48

8、.135,22.058,342,2.182,044,38.550,13.346,444,2.889,011,104.585,74.361,260,1.406,179,Constant,X1,X2,X3,Model,1,B,Std. Error,Unstandardized,Coeff,icients,Beta,Standardized,Coeff,icients,t,Sig,Dependent Variable: Y,a,標(biāo)準(zhǔn)化回歸系數(shù),可說(shuō)明各自變量相對(duì)貢獻(xiàn)大小,Coe,fficients,a,2262.081,1081.870,2.091,053,48.135,22.058,342,2.1

9、82,044,38.550,13.346,444,2.889,011,104.585,74.361,260,1.406,179,Constant,X1,X2,X3,Model,1,B,Std. Error,Unstandardized,Coeff,icients,Beta,Standardized,Coeff,icients,t,Sig,Dependent Variable: Y,a,Y,j,j,YY,jj,j,YY,jj,j,j,S,S,b,n,l,n,l,b,l,l,b,b,1,1,三）有關(guān)評(píng)價(jià)指標(biāo),R,復(fù)相關(guān)系數(shù),0.884,R Square,決定系數(shù),0.781,Adj R-Sq,校正

10、決定系數(shù),0.740,Std.Error of the Estimate,剩余,標(biāo),準(zhǔn)差,216.0570680,Std.Error of the Estimate,剩余標(biāo)準(zhǔn)差,反映了回歸方程的精度，其值越小說(shuō)明回歸,效果越好,057,216,657,46680,1,1,2,12,殘,殘,MS,m,n,SS,m,n,Y,Y,S,m,Y,決定系數(shù),determination coefficient,說(shuō)明所有自變量能解釋,Y,變化的百分比,取,值,0,1,，越接近,1,模型擬合越好,781,0,000,3411375,494,2664484,1,2,總,殘,總,回,SS,SS,SS,SS,R,復(fù)相

11、關(guān)系數(shù),multiple correlation,coefficient,說(shuō)明所有自變量與,Y,間的線性相關(guān)程度,如果只有一個(gè)自變量，此時(shí),884,0,781,0,2,R,R,Y,r,R,校正決定系數(shù),Adjusted,determination coefficient,740,0,19,000,3411375,16,506,746890,1,1,1,1,1,1,1,1,1,2,2,總,殘,總,殘,MS,MS,n,SS,p,n,SS,p,n,n,R,R,c,Y,響,考慮了自變量個(gè)數(shù)的影,2,2,R,R,c,擬合效果好,擬合效果一般,方程擬合效果較差,7,0,7,0,0.4,4,0,2,2,2,

12、c,c,c,R,R,R,多重線性回歸自變量的選擇,Coe,fficients,a,2262.081,1081.870,2.091,053,48.135,22.058,342,2.182,044,38.550,13.346,444,2.889,011,104.585,74.361,260,1.406,179,Constant,X1,X2,X3,Model,1,B,Std. Error,Unstandardized,Coeff,icients,Beta,Standardized,Coeff,icients,t,Sig,Dependent Variable: Y,a,為什么要篩選自變量,變量多增加了

13、模型的復(fù)雜度,計(jì)算量增大,估計(jì)和預(yù)測(cè)的精度下降,模型應(yīng)用費(fèi)用增加,篩選自變量的常用方法,全局擇優(yōu)法,校正決定系數(shù),Cp,準(zhǔn)則,AIC,準(zhǔn)則,逐步選擇法,后退法,前進(jìn)法,逐步回歸法,一）全局擇優(yōu)法,根據(jù)一些準(zhǔn)則建立,最優(yōu)”回歸模型,校正決定系數(shù)（考慮了自變量的個(gè)數(shù),Cp,準(zhǔn)則,C,即,criterion,p,為所選模型中變量的個(gè),數(shù),Cp,接近,p+1,模型為最優(yōu),AIC (Akaikes Information Criterion,準(zhǔn)則,AIC,越小越好,二）逐步選擇法,1,前進(jìn)法,forward selection,2,后退法,backward elimination,3,逐步回歸法,ste

14、pwise regression,它們的共同特點(diǎn)是每一步只引入或剔除一,個(gè)自變量。決定其取舍則基于對(duì)偏回歸平,方和的,F,檢驗(yàn),1,1,1,2,1,p,n,p,n,SS,SS,SS,F,j,j,殘,回,回,向后剔除法,先建立一個(gè)包含全部自變量的回歸,方程，然后每次剔除一個(gè)無(wú)統(tǒng)計(jì)學(xué)意義的自變量,直到不能剔除時(shí)為止。此法計(jì)算量大，有時(shí)不能,實(shí)現(xiàn),向前引入法,由一個(gè)自變量開(kāi)始，每次引入一個(gè),有統(tǒng)計(jì)學(xué)意義的自變量，由少到多，直到無(wú)自變,量可以引入為止。此法建立的方程有時(shí)不夠精煉,逐步篩選法,取上述兩種方法的優(yōu)點(diǎn)，引入和剔,除交替進(jìn)行，直到無(wú)變量可以引入，同時(shí)也無(wú)自,變量可以剔除為止。目前比較常用,SP

15、SS,操作,Analyze,Regression,Linear,Dependent :Y,Independent(s):X1,X2,X3,Method:Stepwise,OK,ANO,VA,e,2042,821.830,1,2042,821.830,26.868,0,00,a,1368,553.170,18,7603,0.73,2,3411,375.000,19,2442,190.549,2,1221,095.274,21.419,0,00,b,9691,84.4,51,17,5701,0.85,0,3411,375.000,19,2664,484.494,3,8881,61.4,98,19.

16、026,0,00,c,7468,90.5,06,16,4668,0.65,7,3411,375.000,19,2572,146.452,2,1286,073.226,26.052,0,00,d,8392,28.5,48,17,4936,6.38,5,3411,375.000,19,Regression,Residual,Total,Regression,Residual,Total,Regression,Residual,Total,Regression,Residual,Total,Model,1,2,3,4,Sum of Sq,uares,df,Mean Sq,uare,F,Sig,Pre

17、dicto,rs: (Co,nstant), X3,a,Predicto,rs: (Co,nstant), X3, X2,b,Predicto,rs: (Co,nstant), X3, X2, X1,c,Predicto,rs: (Co,nstant), X2, X1,d,Dependent Variable: Y,e,C,oe,ffi,ci,e,n,ts,a,1596.573,154.786,10.315,0,00,311.834,60.159,7,74,5.183,0,00,542.691,819.307,.662,5,17,194.562,68.389,4,83,2.845,0,11,3

18、9.031,14.747,4,49,2.647,0,17,2262.081,1081.870,2.09,1,0,53,104.585,74.361,2,60,1.406,1,79,38.550,13.346,4,44,2.889,0,11,48.135,22.058,3,42,2.182,0,44,3331.970,791.085,4.21,2,0,01,48.498,11.639,5,58,4.167,0,01,65.336,18.877,4,64,3.461,0,03,Const,ant,X,3,X,2,X,1,Const,ant,X,3,X,2,X,1,Const,ant,X,3,X,2,X,1,Const,ant,X,3,X,2,X,1,Model,1,2,3,4,B,St,d. Error,U,nstand,a,rdi,zed,Co,e,ff,i,ci,ents,Bet,a,St,andardized,Co,e,ff,i,ci,ents,t,Si,g,Dependent Variable: Y,a,2,1,498,48,336,65,970,3331,X,X,Y,多重線性回歸的應(yīng)用,及其注意事項(xiàng),一）應(yīng)用,影響因素分析，控制混雜因素,預(yù)測(cè)：由自變量值推出應(yīng)變量,Y,的值,控制：指定應(yīng)變量,Y,的值查看自變量的改變量,1,變量的數(shù)量化,1,自變量為連