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1、Biostatistics Xingming Zhao, Department of Computer Science, Tongji University Outline Multiple linear regression Logistic regression Partial correlation and multiple correlation 1. Multiple linear regression Why multiple linear regression? Simple linear regression weight?height? weight?y = a+bxheig
2、ht? y = a+b 1x1 +b2x2+b3x3 Multiple linear regression weight?height?gender?nationality? Why multiple linear regression? y = a+b 1x1 +b2x2+b3x3 ? y = a+b3x3 y = a+b2x2 y = a+b 1x1 Multiple linear regression y = a+b 1x1 +b2x2+.+bnxn y = a+b i xi i=1 n + intercept Slopes Partial regression coefficients
3、 Random Error ? Multiple linear regression Case 1? Number of children /family? Education level? Family income? Multiple linear regression observation? Case 1? #children /family? Education level? Family income? #children /family? Multiple linear regression Case 1? Null hypothesis 1: no association be
4、tween education level and number of children Null hypothesis 2: no association between family income and number of children Multiple linear regression ?4?7: ?$?(?)?$? ?(?)?$?$?$?$?$?$? ?89:6?7? 3? ?$?(?)?)?$? ?(?)?$?( ?6?54?9=? 2?: ?$?(?$?$?(?$?$?$? ?(?$?$? 0?5=?7?1? 2$?$? ?$?(?$? Case 1? Multiple l
5、inear regression y = a+bx Bivariate regression is fitting a line to the data in a two-dimensional space y = a+b 1x1 +b2x2 Trivariate regression is fitting a plane to the data in a three-dimensional space Multiple linear regression ?4?7?$?(?)? ?89:6?7? 3?$?(?)?)? ?6?54?9=? 2?$?(?$?$?(?$? 0?5=?7?1? 2$
6、? ?$? ? ? ? ? y =11.80.36x10.40x2 Multiple linear regression Mathematically, the plane can be described as: y = a+b 1x1 +b2x2 y(a+b 1x1 +b2x2) ? ? Expected number of Children = 11.8 - 0.36*Educ - 0.40*Income Multiple linear regression Case 2 Blood pressure? Birth weight? age? Multiple linear regress
7、ion ? ? ? ? ? ? The REG Procedure Model: MODEL1 Dependent Variable: sysbp Number of Observations Read 16 Number of Observations Used 16 Analysis of Variance Sum of Mean Source DF Squares Square F Value Pr F Model 2 591.03564 295.51782 48.08 |t| Estimate Corr Type II Intercept 1 53.45019 4.53189 11.7
8、9 .0001 0 . brthwgt 1 0.12558 0.03434 3.66 0.0029 0.35208 0.50715 agedys 1 5.88772 0.68021 8.66 ff summary(ff) Multiple linear regression The regression model y = 53.45+0.126*x 1 +5.89*x2 Which one is more important? Multiple linear regression Variable ranking xx+ s x y bs x bs x /s y Multiple linea
9、r regression standardized regression coefficient bs= b*(s x /s y ) Partial regression coefficient standard regression coefficient Multiple linear regression For results in case 2, sx1=18.75 sx2= 0.946 sy= 6.69 (1) the average increase in SBP is 0.352 standard-deviation units of blood pressure per st
10、andard-deviation increase in birthweight (2) the average increase in SBP is 0.833 standard-deviation units of blood pressure per standard-deviation increase in age (3) age appears to be more important variable Goodness of fit of multiple linear regression Question: Is the multiple regression model s
11、ignificant to explain the data? Null hypothesis: H0: 1 = 2=. = k =0 H1: At least one of i 0 Goodness of fit of multiple linear regression ? ? ? ? ? ? We would like to test various hypotheses concerning the data in Table 11.9. First, we would like to test the overall hypothesis that birthweight and a
12、ge when considered together are signifi cant predictors of blood pressure. How can this be done? Specifi cally, we will test the hypothesis vs. H1: at least one of ?1, ?k ? 0. The test of signifi cance is similar to the F test in Section 11.4. The test procedure for a level ? test is given as follow
13、s. ? ? ?(1) Estimate the regression parameters using the method of least squares, and compute Reg SS and Res SS, where Res SS Reg SSTotal SSRes SS ? ? ? ? yy ii i n 2 1 T Total SS ? ? ? ? ? ? yy yab x i i n ijij j k 2 1 1 xij ? jth independent variable for the ith subject, j ? 1, k; i ? 1, n ?(2) Compute Reg MS ? Reg SS/k, Res MS ? Res SS/(n k 1). ?(3) Compute the test statistic F = Re
