迴 歸 分 析相關(guān)係數(shù)（Correlation）

描述兩個變數(shù)X、Y之間的線性相關(guān)Example:data1中的身高及體重

2如何量化這樣的線性關(guān)係呢？Correlation!Linearcorrelation!相關(guān)係數(shù)（Correlation）

Bydefinition,thecorrelationbetweenXandYisItsestimate,Pearson’scorrelationcoefficient3相關(guān)係數(shù)（Correlation）

r>o:positivelycorrelatedr<0:negativelycorrelatedr=0:nolinearcorrelationr=0不代表、Y之間沒有關(guān)係，有可能只是他們之間的關(guān)係不是線性的→畫圖還是必要的4相關(guān)係數(shù)（Correlation）

R程式：cor(x,y,method=c("pearson","kendall","spearman")))x:數(shù)值向量或是矩陣y:數(shù)值向量，當x是矩陣的時候，可以不需輸入5相關(guān)係數(shù)（Correlation）

若想進一步檢定vs.檢定統(tǒng)計量95%confidenceinterval:6相關(guān)係數(shù)（Correlation）

7所以身高與體重有統(tǒng)計顯著的正相關(guān)Practice8請畫出在Surgicaldata中，liver與clot的散佈圖。請問由圖中，可以看出liver與clot的關(guān)係嗎？請計算liver與clot的相關(guān)係數(shù)。請檢定liver與clot之相關(guān)係數(shù)是否為0。

Q:除了看相關(guān)性的強度，能不能看彼此如何影響？Regression!LinearRegressionStep1：

血壓的分布，該分布是否男女有別；Step2：血壓是否和體重有線性相關(guān)；Step3：該線性關(guān)係如何描述；Step4：如何描述血壓和體重、性別、等等的關(guān)係。Y:responsevariable,dependentvariable(say,bp)X:covariate,explanatoryvariable,independentvariable(say,weight)9LinearRegressionQ:howdoesX

affectY?Canwefitalineinthescatterplot?Infact,weshouldsay,where

iscallederror,

isnormalwithzeromeanandvariance

2.10Regression

model-simplelinearregression

11直線上的點是估的,叫fittedvalues,這是已知體重X之後,期望的血壓值,是期望值,故人稱regresstowardthemean;這和觀察值不同,有samplingvariationEstimatecoefficientsHowtofind(intercept)and(slope)?LeastSquares!Minimizeresidualsumofsquares

Takederivative12“residual”isthedifferencebetweenfittedandobservedvalues;Y軸的差EstimatecoefficientsRearrangetheterms,getnormalequationsSolvingthenormalequations,wegetestimates13AretheseLSEgood?Aretheyunbiased?Standarderrorsoftheseestimates?14UnbiasedAretheseLSEgood?Instatistics,toask“Aretheseestimatesgood?”isthesameasasking“Aretheyclosetothetruevalues?”Theyaregoodinthesensethattheyareunbiased.Theyarebestlinearunbiasedestimators(BLUE)Gauss-Markovtheorem:Undertheconditionsofregressionmodel(mean,constantvariance,uncorrelatederrors),theleastsquaresestimatorsareunbiasedandhaveminimumvarianceamongallunbiasedlinearestimators.15EstimationofvariancecanbeestimatedbyTherefore16LinearregressionusingRR程式：lm(formula,data,...)formula:y~x，其中y是response，x是covariate

173.943=70.8432/17.9663Linearregression18LinearregressionMeaningfulwhenestimatingbpwith120kg?notreally,outsidetherangeofthedata,dangerousextrapolationRegressiondoesnotimplycausality.ItsimplyreflectstheregressionrelationbetweenX(weight)andY(bp).ThisregressiondoesnotsayXcausesY.Canweusebptopredictweight?yes,ifweightisthevariableofinterest19Practice想知道在Surgicaldata中，clot如何影響liver，請建立liver與clot之迴歸模式。如何解釋此模型呢？請問clot對liver的影響是顯著的嗎？20Homework想知道在Surgicaldata中，enzyme如何影響SVtime，請建立enzyme與SVtime之迴歸模式。如何解釋此模型呢？請問enzyme對SVtime的影響是顯著的嗎？21Howgoodistheregression？Howgooddoesthelineexplainallthevariationiny?Howgooddoesthefittedcorrelationof(X,Y)explainY?因為定義判斷係數(shù)（coefficientofdetermination）:

Pearson’scorrelationcoefficientInsimplelinearregression,22totaldeviationinresponsesaroundthegrandmeandeviationofobservationsaroundfittedlinedeviationoffittedvaluesaroundthegrandmeanSSTOSSESSRpercentageofvariationexplainedbyregressionlineExample

23R2＝0.4149AVOVAtableofregression

24SSESSRPractice在Surgicaldata中，模式為liver~clot請問在此模型中，判斷係數(shù)為多少25Diagnostics26基本假設(shè)：殘差平均為0，相差變異數(shù)相同，殘差之間不相關(guān)看看殘差的分佈情況看殘差和index的關(guān)係（應(yīng)該要沒關(guān)係）殘差應(yīng)該要和fittedvalue無關(guān)殘差應(yīng)該要與解釋變數(shù)無關(guān)DiagnosticsIf…27Randomlyscatteredaroundzero!Fromminustopositive!Modelmaynotbeproper.Timeeffect?(Ifx=time)Linearity有問題試試polynomial或transformX?Constantvar有問題;若X值大則var大;試試加別的X或是weightedLS?Example

28Q-Qplot如果殘差服從常態(tài)分配，那麼除了它的長條圖像常態(tài)之外，它的排名的值和實際母體同排名的值像不像呢？Thequantileoftheresidualversusthenormalquantile:29OrderedStudentdizedResiduals,Q(quantiles)Rank/total(cumulativeprob.)c.p.FindzwhereP(Z

z)=c.p.Usemodifiedc.p.*=(rank-0.375)/(n+0.25)getz*s.t.P(Z

z*)=c.p.*

Qc.p.zc.p.*

z*-1.881/6=17%-0.9710%-1.28-1.332/6=33%-0.4326%-0.64-0.9

3/6=50%042%-0.20.34/6=67%0.4358%0.20.75/6=83%0.9774%0.641.76/6=100%inf90%1.28將殘差標準化,再排序,第2/6(=0.33)分位的quantile是-1.33即P(ei

-1.33)=2/6算出排序的名次對常態(tài)來說，第2/6(=0.33)分位的quantile是-0.43;即P(Z

-0.43)=2/6=33%對常態(tài)來說第0.26分位的quantile是-0.64;即P(Z

-0.64)=26%PlotthesetwocolumnsQ-QplotIfclosetoaX=Ystraightline,thenresidualsclosetonormality!R程式：qqnorm(model1$”residuals”)30殘差中排名4/6的殘差值和N(0,1)中累積機率為4/6的值Q-Qplot

31YisrightskewedYisleftskewedDiagnosticsinR

32Diagnostics33Practice在Surgicaldata中，模式為liver~clot請問此模式符合迴歸的假設(shè)嗎？34MultiplelinearregressionExtensionofSLR,includingmorethanonepredictorsinthemodel35Linear?Linear?Difference?MultiplelinearregressionModel::regressioncoefficients:observeddataareindependent

Inmatrixform36Multiplelinearregression哪些term可以放到X中呢？Predictors:如例子中的weight,age,sexTransformationsofpredictorsPolynomials:andDummyvariablesandfactorsIn