课件：利用多发病例家族之单倍体资料进行遗传相关性研究83730.ppt

迴 歸 分 析,相關係數（Correlation）,描述兩個變數X、Y之間的線性相關 Example: data1中的身高及體重,2,如何量化這樣的線性關係呢？ Correlation! Linear correlation!,相關係數（Correlation）,By definition, the correlation between X and Y is Its estimate, Pearsons correlation coefficient,3,相關係數（Correlation）,ro: positively correlated r0: negatively correlated r=0: no linear correlation r=0不代表、Y之間沒有關係，有可能只是他們之間的關係不是線性的 畫圖還是必要的,4,相關係數（Correlation）,R程式：cor(x,y,method = c(“pearson“, “kendall“, “spearman“) ) x: 數值向量或是矩陣 y: 數值向量，當x是矩陣的時候，可以不需輸入,5,相關係數（Correlation）,若想進一步檢定 vs. 檢定統計量 95% confidence interval:,6,相關係數（Correlation）,R程式：cor.test(x, y, alternative = c(“two.sided“, “less“, “greater“), method = c(“pearson“, “kendall“, “spearman“), exact = NULL, conf.level = 0.95, continuity = FALSE, .) x: 數值向量 y: 數值向量 exact: T或F，表示是否計算exact p-value continuity: 是否需要進行連續校正,7,所以身高與體重有統計顯著的正相關,Practice,8,請畫出在Surgical data中，liver與clot的散佈圖。請問由圖中，可以看出liver與clot的關係嗎？ 請計算liver與clot的相關係數。 請檢定liver與clot之相關係數是否為0。 Q: 除了看相關性的強度，能不能看彼此如何影響？Regression!,Linear Regression,Step1： 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； Step3：該線性關係如何描述； Step4：如何描述血壓和體重、性別、等等的關係。 Y: response variable, dependent variable (say, bp) X: covariate, explanatory variable, independent variable (say, weight),9,Linear Regression,Q: how does X affect Y? Can we fit a line in the scatter plot? In fact, we should say , where is called error, is normal with zero mean and variance 2.,10,Regression model -simple linear regression,11,直線上的點是估的,叫fitted values, 這是已知體重X之後,期望的血壓值,是期望值,故人稱 regress toward the mean; 這和觀察值不同,有sampling variation,Estimate coefficients,How to find (intercept) and (slope)? Least Squares! Minimize residual sum of squares Take derivative,12,“residual” is the difference between fitted and observed values; Y軸的差,Estimate coefficients,Rearrange the terms, get normal equations Solving the normal equations, we get estimates,13,Are these LSE good?,Are they unbiased? Standard errors of these estimates?,14,Unbiased,Are these LSE good?,In statistics, to ask “Are these estimates good?” is the same as asking “Are they close to the true values?” They are good in the sense that they are unbiased. They are best linear unbiased estimators (BLUE) Gauss-Markov theorem: Under the conditions of regression model (mean, constant variance, uncorrelated errors), the least squares estimators are unbiased and have minimum variance among all unbiased linear estimators.,15,Estimation of variance,can be estimated by Therefore,16,Linear regression using R,R程式：lm(formula, data, .) formula: yx，其中y是response，x是covariate,17,3.943=70.8432/17.9663,Linear regression,Confidence interval of and ? Use t-distribution with df=n-2 Testing if the coefficient =0? If =0? Use t with df=n-2 An increase of 1kg in Weight leads to an increase of 0.7167 in Bp. If someone weighs 70kg, then his/her bp is estimated by 70.84 + 0.7270 = 121.24 - interpolation,18,Linear regression,Meaningful when estimating bp with 120kg? not really, outside the range of the data, dangerous extrapolation Regression does not imply causality. It simply reflects the regression relation between X (weight) and Y (bp). This regression does not say X causes Y. Can we use bp to predict weight? yes, if weight is the variable of interest,19,Practice,想知道在Surgical data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？,20,Homework,想知道在Surgical data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？,21,How good is the regression？,How good does the line explain all the variation in y? How good does the fitted correlation of (X,Y) explain Y? 因為 定義判斷係數（coefficient of determination）: Pearsons correlation coefficient In simple linear regression,22,total deviation in responses around the grand mean,deviation of observations around fitted line,deviation of fitted values around the grand mean,SSTO,SSE,SSR,percentage of variation explained by regression line,Example,23,R20.4149,AVOVA table of regression,24,SSE,SSR,Practice,在Surgical data中，模式為liverclot 請問在此模型中，判斷係數為多少,25,Diagnostics,26,基本假設：殘差平均為0，相差變異數相同，殘差之間不相關,看看殘差的分佈情況,看殘差和index的關係（應該要沒關係）,殘差應該要和fitted value無關,殘差應該要與解釋變數無關,Diagnostics,If,27,Randomly scattered around zero!,From minus to positive! Model may not be proper. Time effect? (If x=time),Linearity 有問題試試polynomial 或transform X?,Constant var有問題;若X值大則var大;試試加別的X或是weighted LS?,Example,28,Q-Q plot,如果殘差服從常態分配，那麼除了它的長條圖像常態之外，它的排名的值和實際母體同排名的值像不像呢？ The quantile of the residual versus the normal quantile:,29,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%,對常態來說第0.26分位的quantile是-0.64; 即P(Z-0.64) =26%,Plot these two columns,Q-Q plot,If close to a X=Y straight line, then residuals close to normality! 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-Q plot,31,Y is right skewed,Y is left skewed,Diagnostics in R,32,Diagnostics,plots to examine The linear effect of each predictor: or Constant variance: Independence of samples: or Normality assumption: Q-Q plot Other important predictors? Say : Are there outliers: , scatter plot, If Yes, examine if it is true outlier, or gross error. If Yes, more data near this point. If No, delete the data point before regression analysis. 6fitted model23145,33,Practice,在Surgical data中，模式為liverclot 請問此模式符合迴歸的假設嗎？,34,Multiple linear regression,Extension of SLR, including more than one predictors in the model,35,Linear?,Linear?,Difference?,Multiple linear regression,Model: : regression coefficients : observed data are independent In matrix form,36,Multiple linear regression,哪些term可以放到X中呢？ Predictors: 如例子中的weight, age, sex Transformations of predictors Polynomials: and Dummy variables and factors Interactions and other combinations of predictors:,37,Example,38,Inference of regression coefficients,和SLR時一樣，用最小平方法 satisfy Gauss-Markov Thm,39,Inference of regression coefficients,和在SLR中相同，我們想要估計 的confidence interval, 或是進行檢定，需要先估計出 Recall,