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1、2008-9-31Simple Linear RegressionSimple Linear Regression林燧恒 卫生统计教研室 复旦大学公共卫生学院林燧恒 卫生统计教研室 复旦大学公共卫生学院HistoryHistoryMethod of least squares published by Legendre (1805) and Gauss (1809)Gauss-Markov Theorem (1821)Galton (1877) studied the relationship G()p between parental characteristics and their pr
2、ogenys characteristics (law of reversion)Galton (1885) first referred to regressionPearson (1895) product moment correlation coefficientSimple Linear RegressionSimple Linear RegressionPopulation model Yi= x+ i= 0+ 1Xi+ i, where E(i) = 0,2i=2,i,j = 0, i jE(i) 0,i, i,j 0, i j - Xiis a fixed known cons
3、tant - E(Yi) is a linear function of the predictor Xiwith intercept 0and slope 1-2Yi constant for all Xi(homoscedasticity)Simple Linear RegressionSimple Linear RegressionSimple Linear RegressionSimple Linear RegressionRandom sample of n pairs of observations, (x1, y1), , (xn, yn) used to estimate re
4、gression model by estimating 0and 1Least squares finds regression line that minimizes the sum of squares of the vertical distances of the data points from the line 最小二乘法依据点到回归线的垂直距离 平方和的最小值来确定回归线Simple Linear RegressionSimple Linear RegressionLeast squares predicted regression equation = b0+ b1Xi= +
5、 b1(Xi), where b0= b1and b1= sxy/XYiY2 xsYX011xy sx= sample standard deviation of Xsxy= sample covariance between X and Y)( )(111YYXXninii= =2008-9-32Simple Linear RegressionSimple Linear Regression200250300per 100,000)Yi ei50100150CHD (Deaths 0246810 Wine Alcohol (litres per person)Y= b0+b1XGaussGa
6、uss- -Markov Theorem Markov Theorem For the linear regression model with Yi= 0+ 1Xi+ i,E(i) = 02i=2(homoscedasticity)i,j = 0, ij (uncorrelated errors) The LS estimators are BLUE (Best Linear Unbiased Estimators), i.e. minimum mean square error among all linear unbiased estimatorsMaximum Likelihood M
7、aximum Likelihood 最大似然法最大似然法最大似然法最大似然法Yinormally distributed, i iid N(0,2)R. A. Fisher 1912-22Find the most likely parameter value based upon the data f (x xx |)upon the data, f(x1,x2, xn| 0, 1) Sufficient and invariant充分与不变Maximum Likelihood Maximum Likelihood 最大似然法最大似然法最大似然法最大似然法Consistent 一至性(con
8、verges to parameter as n) but need not be unbiasedMinimum variance as n(attains Cramr- Rao lower bound)Asymptotically normally distributedLeast Squares Regression LineLeast Squares Regression Linegives the predicted mean value of Y for a given X valueSlope measures the change in for every 1 unit inc
9、rease in X Standardized slope estimate = b s /s=YYStandardized slope estimate = b1sx /sy= average increase in Y (in SD units) per SD increase in XLeast squares slope is a rescaled version of the Pearson correlation coefficient r b1= rsy /sxInference on Inference on 1 1Test H0: 1 = 0 using test stati
10、stict*= b1/ sb1which has a t(n2) distribution under H02-sided P value = P(|t(n2)| t*|H0:1 = 0) 100(1)% confidence interval for 1 b1t (1/2; n2)sb12008-9-33Interpretation of Confidence IntervalInterpretation of Confidence IntervalA 95% C.I. is an interval that will include 1 in 95% of all samples (i.e
11、. if 95% C.I.s are calculated for many samples then 95% of those C.I.s will include 1) 个95%可信区间就是个在95%所有样一个95%可信区间就是一个在95%所有样 本中会包含1的区间：如果在很多样本里 计算每一个样本的95%可信区间，那么在 这些可信区间中有95%会包含1Interpretation of Confidence IntervalInterpretation of Confidence IntervalThe 95% C.I. calculated from any one sample, s
12、ay (30.3, 8.9), will either contain or exclude 1To say “we are 95% confident that 1lies b t( 30 38 9)” dtthbetween (30.3, 8.9)” does not mean there is 95% probability that 1is within the limits 当我们说有95%的可信度1是在某一个样 本计算得来的可信区间中，这并不意味着 1有95%的概率在该区间内Interpretation of Hypothesis TestsInterpretation of Hy
13、pothesis TestsHypothesis testing is a pre-planned inferential method where and are pre- specified based on the sample size and expected effect size under H1假设检验是一个要预先设计的推断方法， 根据样本量和备择假设指定下的期望效应 大小，预先指定类和类错误Interpretation of Hypothesis TestsInterpretation of Hypothesis TestsBy fixing the error rates,
14、 we have a decision rule (to reject or not reject H0) that “in the long run of experience shall not often be wrong” (e.g. if level tests are done for many samples then when H0is true we will yp0falsely reject H0100% of the time) 当犯错误的概率被设定后，决策规则（ 拒绝或不拒绝H0）在以后的推断中就不 会经常出错。（如果以检验水准在很多 的样本分别做检验，当H0为真时，
15、有 100%次是错误拒绝H0的）Interpretation of Interpretation of P P ValuesValuesP value measures the discrepancy between the sample data and H0(i.e. probability that a sample statistic as or more extreme than that observed would randomly occur by hif Ht)chance if H0was true) P值测量现有样本数据和H0的差异(就是 在以H0为真相时，能随机取得一个与当 前样本统计量相同或更加极端结果的概 率)Interpretation of Interpretation of P P ValuesValuesP value does not give P(H0is true) P值不等于H0为真的概率P value should not be interpreted as a type I error rate P值不应理解成犯类错误的概率2008-9-34Inference for Mean Predicted ValueInference for Mean Predicted ValuePredict h= E(Yh
