西格玛回归分析

回归分析 Regression Analysis,目的 Objectives,介绍相关性及回归的基本概念 Introduce The Basic Concepts of Correlation and Regression 把回归与六西格玛路线图结合起来 Link Regression To The Six Sigma Roadmap 学习多元回归的使用 Review the use of Multiple Regression,介绍相关性及回归的基本概念 Introduce The Basic Concepts of Correlation and Regression 把回归与六西格玛路线图结合起来 Link Regression To The Six Sigma Roadmap 学习多元回归的使用 Review the use of Multiple Regression,介绍相关性及回归的基本概念 Introduce The Basic Concepts of Correlation and Regression 把回归与六西格玛路线图结合起来 Link Regression To The Six Sigma Roadmap 学习多元回归的使用 Review the use of Multiple Regression,项目跟踪图,第五版,项目开始日期,21/01/2004,项目类别,“Y”变量数据,采集计划,制定项目,日程,启动项目书,DMAIC改善,定义,确定”Y”变量和,起草项目书,项目书,得以批准,流程图,C&E矩阵或,故障树分析FTA,第三十天 MBB审阅,FMEA,或 故障树分析FTA,测量系统分析 MSA,关键”X”变量,数据采集计划,MBB审阅,测量,21/01/2004,04/02/2004,11/02/2004,25/02/2004,09/03/2004,09/03/2004,09/03/2004,初始能力研究,多元变量流程分析,MBB审阅,合同批准,分析,22/03/2004,15/04/2004,15/04/2004,15/04/2004,15/04/2004,单因子或多因子测试,实验设计(DOE),MBB审阅,改善,31/05/2004,31/05/2004,31/05/2004,控制计划,最终能力研究,控制阶段FMEA回顾,重新修订RPN,MBB审阅,项目最终汇报,及报告,项目审核,及项目收尾,控制,21/06/2004,29/06/2004,29/06/2004,05/07/2004,09/07/2004,09/07/2004,19/07/2004,(根据需要使用),客户心声/业务之声调查 VOC/VOB,需求分析,流程再造,解决方案设计,流程再造,在这里输入开始日期,确定改善方案,由项目发起人在备选项目数据库中完成,在6西格玛,数据库,查找相似项目,实施改善,移交培训/,流程所有人签准,再造路线图的日程是独立计算的,与以上DMAIC的日期不相关,实际完成日期,计划完成日期,图例,2/1/02002,2/3/02,完成,画钩,分析路线图 Analyze Roadmap 单一因子 X -单一因子 Y Single X - Single Y,输入变量 X X Data,离散Discrete,连续Continuous,输出变量 Y Y Data,离散Discrete,连续Continuous,卡方相关性分析 Chi-Square,逻辑回归 Logistic Regression,方差分析， 均值/中位数测试 ANOVA Means / Medians Tests,回归Regression,什么是 Y ? _ 数据类型? _,什么是 X ? _ 数据类型 ? _,应该使用何种工具? _,案例 #1 Scenario #1,管理者想知道接线员的经验(以月为单位衡量) 是否会对接听顾客热线电话需要的时间有影响,相关性 Correlation,什么是相关性 ? What is correlation? 你是否有过如此经验:测量某些产品并送至顾客处，但他们回来告诉你的产品不符规格? Have you ever measured something and then shipped to your customer only for them to tell you it doesnt meet spec? 在奥林匹克溜冰比赛上，你认为两个裁判成绩之相关性有多高? How well correlated do you think two ice skating judges are at the Olympics?,“+”相关性的强度及趋向 Strength and Direction of “+” Correlation,“-”相关性的强度及趋向 Strength and Direction of “-” Correlation,r应该多大？ How Big Should r Be?,依你的样本大小，若所得的相关性比表中的值大，则可视为 “重要” 或统计显著。 By going to the Sample Size of your sample, any correlation that is greater than the table value is considered to be “important” or statistically significant.,r应该多大？ How Big Should r Be?,勿需担心此表，Minitab 可以帮助我们! But dont worry about the tables, Minitab can help us out! 在相关性程序中选取“Display p-values”选项， Minitab 将会显示出是否显著 If you identify the Display p-values option in the Correlation procedure, Minitab will give you the indication if it is significant 找寻比 0.05 小的 p-值 Look for a p-value less than 0.05,回归的定义 Regression Definition,回归分析是一种用以分析变量间相关性的统计工具。统计课程中通常称其为“最佳拟合线” Regression analysis is a statistical technique used to analyze the correlation between variables. This is sometimes taught in statistical courses as computing the “line of best fit”. 本周课程将讨论一元回归分析，其探讨对象为一连续Y与一连续X 之关系, 之后,我们还将讨论多元回归分析- 对象为单一Y与多个连续X之间的关系. In this module we will look at both Simple Regression which compares one continuous Y to one continuous X and Multiple Regression - the comparison of one continuous Y with more than one continuous X,例子: X=发动机重量 Y=每加仑公里数,回归案例 Regression Example,你认为两变量间是什么关系?What Can You Say About The Relationship Between The 2 Variables ?,例子: X= 司机的经验 Y=每加仑公里数,回归案例 Regression Example,你认为两变量间是什么关系?What Can You Say About The Relationship Between The 2 Variables ?,例子: X= 燃料添加剂量 Y=每加仑公里数,回归案例 Regression Example,你认为两变量间是什么关系?What Can You Say About The Relationship Between The 2 Variables ?,回归的定义 Regression Definitions,0 1 2 4 5 X=司机的经验,Y=MPG,正相关性Positive Correlation,负相关性Negative Correlation,无相关性 No Correlation,回归Regression,路线分析图 Analyze Roadmap,规划分析内容,收集数据,在Minitab里绘制拟合线图,评估 R2 及 P值的显著性,评估残差,制定决策,路线分析图 Analyze Roadmap,需要评估的案例: Project: 032 REGRESSION 011404.mpj, worksheet: Brake.mtw,进行了对21种速度的测试. 你认为数据表明了什么?,Minitab 回归 Regression,Minitab 输出,你会做何决策?,拟合线从何而来 Where Does This Fitted Line Come From?,Minitab 找到一条线,使各点至线段之距离为最小 Minitab finds a line which will minimize the distances from the plotted points to the line,* * * * * * * * * * *,拟合线Fitted Line,实际数据点Actual Data Point,实际数据点与拟合线间的距离Distance Between Actual Point and Line,Input Variable (X),Output Variable (Y),任何拟合线的一般等式如下 The general equation of any line is as follows.,Y intercept,Slope of the line,(,),Y,m,X,b,=,+,高中代数 所提之方程式,在回归中，等式以 b0 and b1,Y-截距b0 是当X=0时,Y的高. 换句话说,是拟合线与Y轴的交点.,输入变量Input Variable (X),响应变量 Response Variable (Y),斜率是指:Slope of the line means “Y的变化” “change in Y” b1 = “X里1个单位的变化” “1 unit change in X”,b1,Minitab 输出,Y = 182.8 + .4763速度 这个等式給我们一个对流程 行为的估计This equation gives us an estimation of process behavior. 注意R2 = 69.5%, 我们一会儿回将讲到.,Minitab 输出,Y = 182.8 + .4763x,Ex: 如果速度为400,刹车大概需要多大的距离? If the speed measures 400, what approximate value should we expect in distance? Ex: 如果速度为1000,刹车大概需要多大的距离? If the speed measures 1000, what approximate value should we expect in distance?,回归的等式为The regression equation is 刹车距离Braking Distance = 182.8 + 0.4763 速度Speed S = 13.5571 R-Sq = 69.5% R-Sq(adj) = 67.9% 方差分析Analysis of Variance Source DF SS MS F P Regression 1 7955.9 7955.91 43.29 0.000 Error 19 3492.1 183.79 Total 20 11448.0,Minitab 更多输出,R2 (Same one as before),R2 - 有何意义?,R2与P值，有助我们以统计做决策。R2被称为 判断系数 R2 and P , help us put some statistical backing behind our decisions. The R2 is called the coefficient of determination R2 值代表“多少”输出变异总量可由回归模式所解释，其值介于 0 到 1 (0% 到 100%)。此值越高代表对该模式的可信度越高. R2 is a measure of the amount of variation in the output that is explained by the regression model. It will always be a value between 0 and 1 (0% to 100%). The higher this amount, the greater confidence we have in the model itself.,R2,100%,0%,R2 - 有何意义?,The R2 = 69.5% 这表明有69.5%的Y(刹车距离)的变差可以由X(速度)来解释.This means 69.5% of the variation in Y (Braking Distance) can be explained by the X (Speed). 30.5% 是由其他因素引起的.30.5% is due to something else.,你的决策是什么?,R2-该为多大值? How Big Should It Be ?,视分析对象而定 如对安全系统或回纹针 That answer “depends” on what you are studying, e.g. safety systems or paper clips. 如果你在实验一个新的安全保障系统, 你的数据将由交通部审查.你的数值该需要有多“好”？ If you are experimenting with a new safety restraint system, your numbers will probably be reviewed by the Department of Transportation. How “good” should you be ? 不同的课题会有不同的决策标准 (通常为 +80%)。重要的是我们必须认识到 R2 越高,统计模式越好。 Different texts suggest different decision criteria (usually +80%). The important thing to realize is that the higher the R2 the better the model.,回归分析: 刹车距离v. 速度Regression Analysis: Braking Dist versus Speed 回归的等式为The regression equation is 刹车距离Braking Distance = 182.8 + 0.4763 速度Speed S = 13.5571 R-Sq = 69.5% R-Sq(adj) = 67.9% 方差分析Analysis of Variance Source DF SS MS F P Regression 1 7955.9 7955.91 43.29 0.000 Error 19 3492.1 183.79 Total 20 11448.0,P值里怎么了? What Is Going On Here ?,Another P Value !,零假设: 线段斜率=0 (无相关性) Ho: Slope of The Line = 0 (No correlation) 备择假设: 线段斜率 = 0 (有相关性) Ha: Slope of The Line 0 (There is correlation),记住 P要小, Ho要倒 When P is low, Ho must go !,Minitab 回归- 残差&拟合数 Regression - Residuals & Fits,Speed Distance RESI1 FITS1 336 325 -17.8392 342.839 418 375 -6.8948 381.895 355 367 15.1113 351.889 445 385 -9.7546 394.755 365 375 18.3484 356.652 455 395 -4.5175 399.517 395 395 24.0598 370.940 405 365 -10.7031 375.703 346 355 7.3979 347.60 . . . . . . . . . . . .,Minitab 更多输出 More Output,速度Speed 距离Distance 残差1 RESI1 拟合数1 FITS1 336 325 -17.8392 342.839,残差&拟合数- 它们是什么? Residual & Fit - What Are They ?,拟和线Fitted Line,336,325,实际点Actual Point,残差距离Residual Distance (-17.8392),理论拟合点Theoretical Fit,342,速度Speed 距离Distance 残差1 RESI1 拟合数1 FITS1 336 325 -17.8392 342.839 残差- 点到拟合线的垂直距离 在线下方为负, 在线上方为正. Residual - The vertical distance to the fitted line Negative is below , positive is above 拟合数- Y值在拟合线上的理论值 Fits - The theoretical y value on the fitted line,残差&拟合数- 它们是什么? Residual & Fit - What Are They ?,回归- 残差&拟合数- 图表总结 Regression - Residuals & Fits Graphical Summary,数据应该通过 “胖铅笔测试” “Fat Pencil Test”,残差分析 Residual Analysis,数据应该像钟型分布 Data Should Fit A Bell Shaped Curve,比较P值与残差正态分布测试的结果 Check P value with Normality test on Residuals,数据应在控制线内, 调查异常点 Data Should Be In Control Investigate Outliers,残差分析 Residual Analysis,数据应无任何规律 Data Should Exhibit No Patterns,其他案例 Other Examples,使用Minitab Project: 032 regression 011404.mpj 练习 #1: Analyze worksheet Paint.mtw Y = 油漆厚度Paint Thickness X1 = 气压Air Pressure X2 = 黏度Viscosity 练习 #2: Analyze worksheet Cust.mtw Y = 客户回应时间Customer Response Time X1 = 代理人有经验程度Experience Level of Agent X2 = 与客户的距离Distance From Customer Site 练习 #3: Analyze Mystery Data.mtw,注意陈述中的因果关系 Beware of Stating Causality,即使我们建立了Y与X之相关性，但并不能确定X之变异将一定导致Y之变异。 If we establish a correlation between Y and a X, that doesnt necessarily mean variation in X caused variation in Y. 其它潜藏的变量，可能造成X与Y之改变。 Other variables may be lurking that cause both X and Y to vary.,研究指出当医院规模增加，病人死亡率亦显著提升。 这么说来，我们应该避免去大型医院就诊吗？ Research has consistently shown that as the hospital size increases, the death rate of patients dramatically increases. So, should we avoid large hospitals?,回归问题探讨：Xs 缺失 Regression Issues - Missing Xs,有关一个城市的数据显示，当城市里鹳的数量增加时，城市人口也增加鹳真的影响城市人口吗？ Data on a city showed that as population density of storks increased, so did the towns population. Did storks influence the population ?,回归问题探讨：Xs 缺失 Regression Issues - Missing Xs,回归问题探讨 Regression Issues 研究范围太狭窄 Range Of Study Too Small,$ 车值Value of Car,车龄Age of Car,现在的数据看来如何？What Would This Look Like Now ?,0 1 5 10 15 20 25 30 35 40 45 50,回归问题探讨 Regression Issues 研究范围太狭窄 Range Of Study Too Small,分析路线图 Analyze Roadmap,输入变量 X X Data,单一因子 X Single X,多因子 Xs Multiple Xs,输出变量 Y Y Data,单一输出 Y Single Y,多元输出 Y Multiple Ys,多变量分析 Multivariate Analysis (注意: 这与多元变量分析不同) (Note: This Is Not The Same As Multi-Vari Analysis),输入变量 X X Data,离散 Discrete,连续 Continuous,输出变量 Y Y Data,卡方相关性分析 Chi-Square,逻辑回归 Logistic Regression,T 测试，方差分析， 均值/中位数测试 T-test, ANOVA Means/ Medians Tests,回归 Regression,多元回归 Multiple Regression,Medians Tests,Multiple Logistic Regression,多元逻辑回归,离散 Discrete,连续 Continuous,离散 Discrete,连续 Continuous,离散 Discrete,连续 Continuous,2, 3, 4 因子方差分析 中位数测试,多元逻辑回归,Multiple Logistic Regression,输入变量 X X Data,输出变量 Y Y Data,多元回归分析 Multiple Regression Analysis,两个或多个流程变量（Xs)可能对流程表现产生影响(Y). Two or more process variables (Xs) may have an influence upon process performance (Y). 多元回归应用于有两个或多个可能的预测变量的情况Multiple regression is used whenever there are two or more possible predictor variables. 多元回归的一般等式为The general form of the multiple regression equation is,n,n,X,b,X,b,X,b,b,Y,+,+,+,+,=,.,2,2,1,1,0,案例：刹车板销售量 Example: Brake Sales,例中对刹车板销售量进行次的观察已知有五个流程变量和一个表现变量，： Twenty observations regarding Brake Sales are given. There are Five known process variables and one performance variable, Y: X1 = 年度Year X2 = 市场营销费用Mktg$ X3 = 今年销售人员数Sales Rep X4 = 去年（销售人员）数LY(Sales Rep) X5 = 产品Product Y = 销售Sales,利用数据找出可能影响”销售量”的”重要的几个”流程变量. Use the data to mine for the “vital few” process variables that may influence “Sales”.,刹车板销售量数据,Year Mktg$ SalesRep LY(SalesRep) Product Sales 1 9.6 30 20 18 130 2 10.3 20 30 17 157 3 10.2 15 20 19 129 4 10.4 25 15 22 129 5 10.6 30 25 24 162 6 10.7 15 30 18 154 7 10.5 25 15 17 132 8 10.9 35 25 16 172 9 11.0 40 35 14 207 10 11.1 20 40 18 204 11 11.2 25 20 22 144 12 11.2 35 25 25 175 13 11.4 5 35 27 167 14 11.2 12 5 28 97 15 11.6 16 12 18 122 16 11.7 21 16 16 139 17 11.8 22 21 15 153 18 11.8 24 22 16 156 19 11.8 26 24 10 172 20 12.1 28 26 18 178,032 regression 011404.mpj Worksheet “Sales.MTW”,刹车板销售量,我们的目的是找到适用于下列形式的多元回归： Our goal is to fit a multiple regression of the following form,这个问题便于阐明下列多元回归的其他方面： This problem will illustrate the following additional aspects of multiple regression 去掉没有解释能力的变量 elimination of X-variables that have no explanatory power; 残差分析 residual analysis,留在模式里的变量是能控制的在西格玛里，我们的目标就是 要控制少数变量 What stays in the model must have controls. In Six Sigma, goal is to control a few.,多元回归 Multiple Regression,路线分析图,规划分析内容,收集数据,利用回归或最佳子集分析Analyze Using Regression or Best Subsets,评估残差,制定决策,评估 R2 及 P值的显著性,多元共线性分析（相关性） Multicollinearity “X” Check (correlation),使用多元回归简化模式 Run Multiple Regression Reduced Model,因为有多条线， 就不再使用拟合线图， No longer fitted line plot due to multiple lines,相关的预测变量（多元共线性） Correlated Predictor Variables (Multicollinearity),n,n,X,b,X,b,X,b,b,Y,+,+,+,+,=,.,2,2,1,1,0,流程结果（）与预测变量（s)间的相关性是有用的它可以帮助我们找出可能的因果关系 Correlation between the process output (Y) and the predictor variables (Xs) is good - helps us identify possible cause and effect relationships. 相反，预测变量间的相关性却是一个问题 Correlation between predictors, in contrast, is a problem. 计算里的正负符号和预测变量间的相关性大小可能有错误.Calculated signs and magnitudes of correlated predictors may be wrong. 计算出的P值可能偏大.Calculated P-values may be large.,预测变量间的高相关性被称为”共线性” High correlation between predictor variables is called “collinearity”,多元共线性：刹车板销售量 Multicollinearity: Brake Sales,左侧是前刹车板销售量 预测变量：Predictor Variables: (1) 年度Year; (2)市场营销费用Marketing $; (3) 今年销售人员数量How many Sales Reps this year; (4)去年销售人员数量How many Sales Reps last year. (5) 产品Product,032 regression 011404.mpj Worksheet “Sales.MTW”,Year Mktg$ SalesRep LY(SalesRep) Product Sales 1 9.6 30 20 18 130 2 10.3 20 30 17 157 3 10.2 15 20 19 129 4 10.4 25 15 22 129 5 10.6 30 25 24 162 6 10.7 15 30 18 154 7 10.5 25 15 17 132 8 10.9 35 25 16 172 9 11.0 40 35 14 207 10 11.1 20 40 18 204 11 11.2 25 20 22 144 12 11.2 35 25 25 175 13 11.4 5 35 27 167 14 11.2 12 5 28 97 15 11.6 16 12 18 122 16 11.7 21 16 16 139 17 11.8 22 21 15 153 18 11.8 24 22 16 156 19 11.8 26 24 10 172 20 12.1 28 26 18 178,多元共线性：刹车板销售量,选择所有五个预测变量和响应变量 Select all five predictor variables and the response variable.,使用 Minitab 菜单, STAT BASIC STATS CORRELATION. 不选择p值选项Uncheck p value,年度和市场营销费用有着很高的相关性！我们必须只能选择一个作为预测变量在回归拟合中使用市场营销费用可能受年度影响，因此我们保留市场营销费用，而去掉年度变量The Year and Marketing$ Variables are highly correlated! We will have to choose one or the other of the correlated predictor variables (but not both) to use in a regression fit. Possible that marketing$ is a function of the year - so keep the marketing $ and eliminate year.,基本原则, 如果相关性 0.8 or - 0.8时则去掉变量,相关性：年度，市场营销费用，销售人员数，去年销售人员数，产品 Correlations: Year, Mktg$, Sales Rep, LY(SalesRep), Product Year Mktg$ Sales Rep LY(Sales Mktg$ 0.973 Sales Re -0.095 -0.097 LY(Sales -0.111 0.002 0.143 Product -0.172 -0.164 -0.353 -0.181 Cell Contents: Pearson correlation,多元共线性：刹车板销售量,多元回归 Multiple Regression,路线分析图,规划分析内容,收集数据,利用回归或最佳子集分析Analyze Using Regression or Best Subsets,评估残差,制定决策,评估 R2 及 P值的显著性,多元共线性分析（相关性） Multicollinearity “X” Check (correlation),使用多元回归简化模式 Run Multiple Regression Reduced Model,因为有多条线， 就不再使用拟合线图， No longer fitted line plot due to multiple lines,最佳子集回归 Best Subsets Regression,“最好的模式是尽可能简单的模式，没有比此模式更简单的” “The best model is the one that is as simple as possible, but no simpler.” - Albert Einstein,最佳子集回归：刹车板销售 Best Subsets: Brake Sales,选择所剩四个预测变量和响应变量.(去掉”年度”) Select the four remaining predictor variables and the response variable. (Exclude “year”),使用 Minitab 菜单, STAT Regression Best Subsets.,最佳子集回归：刹车板销售,注意”年度”从模式中去掉了.,Best Subsets Regression: Sales versus Mktg$, Sales Rep, . Response is Sales S L a Y P l ( r M e S o k s a d t l u g R e c Vars R-Sq R-Sq(adj) C-p S $ e s t 1 79.0 77.8 156.0 12.841 X 1 20.9 16.6 631.3 24.910 X 2 90.1 89.0 66.8 9.0570 X X 2 85.2 83.5 107.0 11.084 X X 3 98.2 97.8 3.0 4.0222 X X X 3 90.5 88.7 65.8 9.1570 X X X 4 98.2 97.7 5.0 4.1540 X X X X,多元回归 Multiple Regression,路线分析图,规划分析内容,收集数据,利用回归或最佳子集分析Analyze Using Regression or Best Subsets,评估残差,制定决策,评估 R2 及 P值的显著性,多元共线性分析（相关性） Multicollinearity “X” Check (correlation),使用多元回归简化模式 Run Multiple Regression Reduced Model,因为有多条线， 就不再使用拟合线图， No longer fitted line plot due to multiple lines,回归:刹车板销售 Regression: Brake Sales,选择所有四个预测变量和响应变量. Select all four predictor variables and the response variable.,使用 Minitab 菜单, STAT Regression Regression,回归分析:刹车板销售 Regression Analysis: Brake Sales,零假设 = 变量间没有任何关系 备择假设= 变量间有一些关系 Ho = No relationship between variables Ha = Some relationship exists between variables,Regression Analysis: Sales versus Mktg$, Sales Rep, . The regression equation is Sales = - 66.6 + 11.8 Mktg$ + 1.18 Sales Rep + 2.70 LY(SalesRep) - 0.007 Product Predictor Coef SE Coef T P Constant -66.64 19.17 -3.48 0.003 Mktg$ 11.838 1.494 7.92 0.000 Ha Sales Re 1.1751 0.1224 9.60 0.000 Ha LY(Sales 2.7023 0.1154 23.42 0.000 Ha Product -0.0068 0.2337 -0.03 0.977 Ho S = 4.154 R-Sq = 98.2% R-Sq(adj) = 97.7%,回归/简化模式:刹车板销售 Regression/Reduced Model: Brake Sales,选择所剩三个预测变量和响应变量. Select the three remaining predictor variables and the response variable.,Using Minitab Menu, STAT Regression Regression,记住检查残差图 Remember to check your residual plots,回归分析:刹车板销售 Regression Analysis: Brake Sales,零假设 = 变量间没有任何关系 备择假设= 变量间有一些关系 Ho = No relationship between variables Ha = Some relationship exists between variables,回归分析:销售量v.市场营销费用,销售人员数,去年销售人员数 Regression Analysis: Sales versus Mktg$, Sales Rep, LY(SalesRep) The regression equation is Sales = - 66.9 + 11.8 Mktg$ + 1.18 Sales Rep + 2.70 LY(SalesRep) Predictor Coef SE Coef T P Constant -66.91 16.22 -4.12 0.001 Mktg$ 11.847 1.414 8.38 0.000 Ha Sales Re 1.1764 0.1106 10.64 0.000 Ha LY(Sales 2.7027 0.1106 24.44 0.000 Ha S = 4.022 R-Sq = 98.2% R-Sq(adj) = 97.8%,刹车板销售案例的其他MiniTab 输出 The Rest of Mini Tab Output Brake Sales,Analysis of Variance Source DF SS MS