Lecture 9_Simple Linear Regression 第九章 简单线性回归分析
应用回归分析含定性变量的回归模型第九章课后答案
第9章 含定性变量的回归模型思考与练习参考答案9.1 一个学生使用含有季节定性自变量的回归模型，对春夏秋冬四个季节引入4个0-1型自变量，用SPSS 软件计算的结果中总是自动删除了其中的一个自变量，他为此感到困惑不解。
出现这种情况的原因是什么？ 答：假如这个含有季节定性自变量的回归模型为：其中含有k 个定量变量，记为x i 。
对春夏秋冬四个季节引入4个0-1型自变量，记为D i ，只取了6个观测值，其中春季与夏季取了两次，秋、冬各取到一次观测值，则样本设计矩阵为：显然，(X,D)中的第1列可表示成后4列的线性组合，从而(X,D)不满秩，参数无法唯一求出。
这就是所谓的“虚拟变量陷井”，应避免。
当某自变量x j 对其余p-1个自变量的复判定系数2j R 超过一定界限时，SPSS 软件将拒绝这个自变量x j 进入回归模型。
称Tol j =1-2j R 为自变量x j 的容忍度（Tolerance ），SPSS 软件的默认容忍度为0.0001。
也就是说，当2j R ＞0.9999时，自变量x j 将被自动拒绝在回归方程之外，除非我们修改容忍度的默认值。
而在这个模型中出现了完全共线性，所以SPSS 软件计算的结果中总是自动删除了其中的一个定性自变量。
9.2对自变量中含有定性变量的问题，为什么不对同一属性分别建立回归模型，而采取设虚拟变量的方法建立回归模型？答：原因有两个，以例9.1说明。
一是因为模型假设对每类家庭具有相同的斜率和误差方差，把两类家庭放在一起可以对公共斜率做出最佳估计；二是对于其他tt t t kt k t t D D D X X Y μαααβββ++++++=332211110 ⎪⎪⎪⎪⎪⎪⎪⎪⎭⎫⎝⎛=000110010110001010010010100011)(616515414313212111k k k k k k X X X X X X X X X X X XD X,⎪⎪⎪⎪⎪⎭⎫ ⎝⎛=k βββ 10β⎪⎪⎪⎪⎪⎭⎫ ⎝⎛=4321ααααα统计推断，用一个带有虚拟变量的回归模型来进行也会更加准确，这是均方误差的自由度更多。
linear regression知识点
linear regression知识点1.引言1.1 概述引言部分是文章的开头，用来介绍文章的背景和重要性。
在"概述"部分，我们可以对linear regression（线性回归）的基本概念和作用进行简单介绍。
概述：线性回归是机器学习领域中最简单且最常用的回归方法之一。
它是一种建立输入变量（自变量）和输出变量（因变量）之间线性关系的统计学模型。
线性回归可以帮助我们探索和理解数据，预测未知的因变量值，并在实际问题中做出决策。
线性回归的基本思想是基于已知的训练数据，通过拟合一条直线（或超平面）来近似描述输入和输出之间的关系。
这条直线可以用来做预测和回答各种问题。
线性回归的关键是通过最小化预测值与实际观测值之间的差距，找到最佳拟合直线。
线性回归不仅可以用于预测连续性数值型数据，还可以用于分类问题，例如将输出变量划分为两个或多个不同的类别。
尽管线性回归在实际问题中很常见，但它也有一些局限性，例如对于非线性关系的建模能力较弱。
为了克服这些局限性，研究人员还提出了各种改进方法。
本文将深入探讨线性回归的基本概念和原理，介绍线性回归模型的建立与求解过程，并探讨线性回归在实际应用中的场景和局限性，同时提出一些改进方法。
通过阅读本文，读者将能够全面了解线性回归的知识和应用，从而在实际问题中更好地应用和理解线性回归方法。
下面我们将详细介绍本文的结构和目的。
1.2 文章结构文章结构部分的内容可以描述整篇文章的组织和安排，可以按照以下内容进行阐述：在本篇文章中，我们将从引言、正文和结论三个部分来组织和阐述关于Linear Regression（线性回归）的知识点。
首先，在引言部分，我们将对线性回归进行概述，介绍其基本概念和原理。
同时，我们将阐明本篇文章的目的，即通过介绍线性回归的知识点，让读者对线性回归有一个全面的了解。
接着，在正文部分，我们将分为两个小节来详细讲解线性回归的知识点。
首先，我们将介绍线性回归的基本概念，包括线性回归的定义、特点以及模型表示等。
应用回归分析第九章部分答案-最新年文档
第9章 非线性回归9.1 在非线性回归线性化时，对因变量作变换应注意什么问题？答：在对非线性回归模型线性化时，对因变量作变换时不仅要注意回归函数的形式， 还要注意误差项的形式。
如：(1) 乘性误差项，模型形式为e y AK L αβε=， (2) 加性误差项，模型形式为y AK L αβε=+。
对乘法误差项模型（1）可通过两边取对数转化成线性模型，（2）不能线性化。
一般总是假定非线性模型误差项的形式就是能够使回归模型线性化的形式，为了方便通常省去误差项，仅考虑回归函数的形式。
9.2为了研究生产率与废料率之间的关系，记录了如表9.14所示的数据，请画出散点图，根据散点图的趋势拟合适当的回归模型。
表9.14 生产率x （单位/周） 135 000 废品率y （%） 5.2 6.5 6.8 8.1 10.2 10.3 13.0解：先画出散点图如下图：从散点图大致可以判断出x 和y 之间呈抛物线或指数曲线，由此采用二次方程式和指数函数进行曲线回归。
（1）二次曲线SPSS 输出结果如下：从上表可以得到回归方程为：72ˆ 5.8430.087 4.4710yx x -=-+⨯ 由x 的系数检验P 值大于0.05，得到x 的系数未通过显著性检验。
由x 2的系数检验P 值小于0.05，得到x 2的系数通过了显著性检验。
（2）指数曲线从上表可以得到回归方程为：0.0002t ˆ 4.003ye = 由参数检验P 值≈0<0.05，得到回归方程的参数都非常显著。
从R2值，σ的估计值和模型检验统计量F值、t值及拟合图综合考虑，指数拟合效果更好一些。
9.3 已知变量x与y的样本数据如表9.15，画出散点图，试用αeβ/x来拟合回归模型，假设：(1)乘性误差项，模型形式为y=αeβ/x eε(2)加性误差项，模型形式为y=αeβ/x+ε。
表9.15解：散点图：(1)乘性误差项，模型形式为y=αeβ/x eε线性化：lny=lnα+β/x +ε令y1=lny, a=lnα,x1=1/x .做y1与x1的线性回归，SPSS输出结果如下：从以上结果可以得到回归方程为：y1=-3.856+6.08x1F检验和t检验的P值≈0<0.05，得到回归方程及其参数都非常显著。
Lecture 9_Simple Linear Regression 第九章 简单线性回归分析
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Y i β0β1X iεi
Observed Value of Y for Xi
Predicted Value of Y for Xi
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Random Error
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Chap 12-10
Business Statistics: A First Course
Fifth Edition
Chapter 12
Simple Linear Regression
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Learning Objectives
In this chapter, you learn:
How to use regression analysis to predict the value of a dependent variable based on an independent variable
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Simple Linear Regression Model
Only one independent variable, X Relationship between X and Y is
described by a linear function Changes in Y are assumed to be related
A scatter plot can be used to show the relationship between two variables
Correlation analysis is used to measure the strength of the association (linear relationship) between two variables
线性回归
Simple Linear RegressionChapter TopicsTypes of Regression ModelsDetermining the Simple Linear RegressionEquationMeasures of VariationAssumptions of Regression and CorrelationResidual AnalysisMeasuring AutocorrelationInferences about the Slope© 2003 Prentice-Hall, Inc. Chap 10-2Chapter Topics(continued)Correlation -Measuring the Strength of theAssociationEstimation of Mean Values and Prediction ofIndividual ValuesPitfalls in Regression and Ethical Issues© 2003 Prentice-Hall, Inc. Chap 10-3Purpose of Regression AnalysisRegression Analysis is Used Primarily to Model Causality and Provide PredictionPredict the values of a dependent (response)variable based on values of at least oneindependent (explanatory) variableExplain the effect of the independent variables onthe dependent variablePositive Linear Relationship Negative Linear Relationship Relationship NOT Linear No RelationshipSimple Linear Regression ModelRelationship Between Variables is Describedby a Linear FunctionThe Change of One Variable Causes the OtherVariable to ChangeA Dependency of One Variable on the Other© 2003 Prentice-Hall, Inc. Chap 10-6Linear Regression Equationand are obtained by finding the values of and that minimizes the sum of the squared residualsprovides an estimate ofprovides and estimate of 0b 1b 0b 1b 0b β01b β1(continued)()2211ˆnni iii i Y Y e==−=∑∑Simple Linear Regression:ExampleYou wish to examine the linear dependency of the annual sales of produce stores on their sizes in square footage. Sample data for 7 stores were obtained. Find the equation of the straight line that fits the data best.Annual Store Square SalesFeet($1000)1 1,7263,6812 1,5423,395 32,8166,653 45,5559,543 51,2923,318 62,2085,563 71,3133,760Interpretation of Results:ExampleThe slope of 1.487 means that each increase of oneunit in X, we predict the average of Y to increase by an estimated 1.487 units.The equation estimates that for each increase of 1 square foot in the size of the store, the expected annual sales are predicted to increase by $1487.ˆ1636.415 1.487i iY X =+Simple Linear Regression inPHStatIn Excel, use PHStat| Regression | Simple Linear Regression …EXCEL Spreadsheet of Regression Sales on FootageMicrosoft ExcelWorksheetMeasures of Variation:The Sum of Squares SST=SSR+ SSETotal Sample Variability =ExplainedVariability+UnexplainedVariabilityLinear Regression Assumptions NormalityY values are normally distributed for each XProbability distribution of error is normal2.Homoscedasticity (Constant Variance)3.Independence of ErrorsResidual Analysis PurposesExamine linearityEvaluate violations of assumptionsGraphical Analysis of Residuals Plot residuals vs. X and timeResidual Analysis for HomoscedasticityHeteroscedasticity9HomoscedasticitySRXSRXYXXYDurbin-Watson Statistic inPHStatPHStat| Regression | Simple LinearRegression …Check the box for Durbin-Watson Statistic© 2003 Prentice-Hall, Inc. Chap 10-36Accept H 0(no autocorrelatin)Using the Durbin-WatsonStatistic: No autocorrelation (error terms are independent): There is autocorrelation (error terms are notindependent)0H 1H 042d L 4-d Ld U4-d U Reject H 0(positive autocorrelation)Inconclusive Reject H 0(negative autocorrelation)Relationship between a t Testand an F TestNull and Alternative HypothesesH 0: β1= 0(No Linear Dependency)H 1: β1≠ 0(Linear Dependency)()221,2n n t F −−=Purpose of Correlation AnalysisCorrelation Analysis is Used to MeasureStrength of Association (Linear Relationship)Between 2 Numerical VariablesOnly Strength of the Relationship is ConcernedNo Causal Effect is Implied© 2003 Prentice-Hall, Inc. Chap 10-47Purpose of Correlation Analysis(continued)Population Correlation Coefficient ρ(Rho) isUsed to Measure the Strength between theVariablesSample Correlation Coefficient r is anEstimate of ρ and is Used to Measure theStrength of the Linear Relationship in theSample Observations© 2003 Prentice-Hall, Inc. Chap 10-48Sample of Observations from Various r ValuesYX YXYXYX YXr= -1r= -.6r= 0r= .6r= 1Features of ρ and rUnit FreeRange between -1 and 1The Closer to -1, the Stronger the NegativeLinear RelationshipThe Closer to 1, the Stronger the PositiveLinear RelationshipThe Closer to 0, the Weaker the LinearRelationship© 2003 Prentice-Hall, Inc. Chap 10-50。
线性回归(Linear Regression)
1. #!/usr/bin/python2. # -*- coding: utf-8 -*-3.4. """5. author ： duanxxnj@6. time : 2016-06-19-20-487.8. 这个是线性回归的示例代码9. 使用一条直线对一个二维数据拟合10.11. """12. print(__doc__)13.14.15. import matplotlib.pyplot as plt16. import numpy as np17. from sklearn import datasets, linear_model18.19. # 加载用于回归模型的数据集20. # 这个数据集中一共有442个样本，特征向量维度为1021. # 特征向量每个变量为实数，变化范围（-.2 ，.2）22. # 目标输出为实数，变化范围（25 ，346）23. diabetes = datasets.load_diabetes()24.25. # 查看数据集的基本信息26. print diabetes.data.shape27. print diabetes.data.dtype28. print diabetes.target.shape29. print diabetes.target.dtype30.31. # 为了便于画图显示32. # 仅仅使用一维数据作为训练用的X33. # 这里使用np.newaxis的目的是让行向量变成列向量34. # 这样diabetes_X每一项都代表一个样本35. diabetes_X = diabetes.data[:, np.newaxis,2]36.37. # 此时diabetes_X的shape是(442L, 1L)38. # 如果上面一行代码是：diabetes_X = diabetes.data[:, 2]39. # 则diabetes_X的shape是(442L，)，是一个行向量40. print diabetes_X.shape41.42. # 人工将输入数据划分为训练集和测试集43. # 前400个样本作为训练用，后20个样本作为测试用44. diabetes_X_train = diabetes_X[:-20]45. diabetes_X_test = diabetes_X[-20:]46. diabetes_y_train = diabetes.target[:-20]47. diabetes_y_test = diabetes.target[-20:]48.49. # 初始化一个线性回归模型50. regr = linear_model.LinearRegression()51.52. # 基于训练数据，对线性回归模型进行训练53. regr.fit(diabetes_X_train, diabetes_y_train)54.55. # 模型的参数56. print'模型参数：', regr.coef_57. print'模型截距：', regr.intercept_58.59. # 模型在测试集上的均方差（mean square error）60. print("测试集上的均方差: %.2f"61. % np.mean((regr.predict(diabetes_X_test)- diabetes_y_test)**2))62. # 模型在测试集上的得分，得分结果在0到1之间，数值越大，说明模型越好63. print('模型得分: %.2f'% regr.score(diabetes_X_test, diabetes_y_test))64.65. # 绘制模型在测试集上的效果66. plt.scatter(diabetes_X_test, diabetes_y_test, color='black')67. plt.plot(diabetes_X_test, regr.predict(diabetes_X_test), color='blue',68. linewidth=3)69.70. plt.grid()71. plt.show()。
各种线性回归模型原理
各种线性回归模型原理线性回归是一种广泛应用于统计学和机器学习领域的方法，用于建立自变量和因变量之间线性关系的模型。
在这里，我将介绍一些常见的线性回归模型及其原理。
1. 简单线性回归模型（Simple Linear Regression）简单线性回归模型是最简单的线性回归模型，用来描述一个自变量和一个因变量之间的线性关系。
模型方程为：Y=α+βX+ε其中，Y是因变量，X是自变量，α是截距，β是斜率，ε是误差。
模型的目标是找到最优的α和β，使得模型的残差平方和最小。
这可以通过最小二乘法来实现，即求解最小化残差平方和的估计值。
2. 多元线性回归模型（Multiple Linear Regression）多元线性回归模型是简单线性回归模型的扩展，用来描述多个自变量和一个因变量之间的线性关系。
模型方程为：Y=α+β1X1+β2X2+...+βnXn+ε其中，Y是因变量，X1,X2,...,Xn是自变量，α是截距，β1,β2,...,βn是自变量的系数，ε是误差。
多元线性回归模型的参数估计同样可以通过最小二乘法来实现，找到使残差平方和最小的系数估计值。
3. 岭回归（Ridge Regression）岭回归是一种用于处理多重共线性问题的线性回归方法。
在多元线性回归中，如果自变量之间存在高度相关性，会导致参数估计不稳定性。
岭回归加入一个正则化项，通过调节正则化参数λ来调整模型的复杂度，从而降低模型的过拟合风险。
模型方程为：Y=α+β1X1+β2X2+...+βnXn+ε+λ∑βi^2其中，λ是正则化参数，∑βi^2是所有参数的平方和。
岭回归通过最小化残差平方和和正则化项之和来估计参数。
当λ=0时，岭回归变为多元线性回归，当λ→∞时，参数估计值将趋近于0。
4. Lasso回归（Lasso Regression）Lasso回归是另一种用于处理多重共线性问题的线性回归方法，与岭回归不同的是，Lasso回归使用L1正则化，可以使得一些参数估计为0，从而实现特征选择。
线性回归LinearRegression
线性回归LinearRegression 成本函数（cost function）也叫损失函数（loss function），⽤来定义模型与观测值的误差。
模型预测的价格与训练集数据的差异称为残差（residuals）或训练误差（test errors）。
我们可以通过残差之和最⼩化实现最佳拟合，也就是说模型预测的值与训练集的数据最接近就是最佳拟合。
对模型的拟合度进⾏评估的函数称为残差平⽅和（residual sum of squares）成本函数。
就是让所有训练数据与模型的残差的平⽅之和最⼩。
我们⽤R⽅（r-squared）评估预测的效果。
R⽅也叫确定系数（coefficient of determination），表⽰模型对现实数据拟合的程度。
计算R⽅的⽅法有⼏种。
⼀元线性回归中R⽅等于⽪尔逊积矩相关系数（Pearson product moment correlation coefficient 或Pearson's r）的平⽅。
这种⽅法计算的R⽅⼀定介于0～1之间的正数。
其他计算⽅法，包括scikit-learn中的⽅法，不是⽤⽪尔逊积矩相关系数的平⽅计算的，因此当模型拟合效果很差的时候R⽅会是负值。
SStot是⽅差平⽅和 SSres是残差的平⽅和⼀元线性回归X_test = [[8], [9], [11], [16], [12]]y_test = [[11], [8.5], [15], [18], [11]]model = LinearRegression()model.fit(X, y)model.score(X_test, y_test）score⽅法计算R⽅多元线性回归最⼩⼆乘的代码from numpy.linalg import lstsqprint(lstsq(X, y)[0])多项式回归⼀种特殊的多元线性回归⽅法，增加了指数项（x 的次数⼤于1）。
现实世界中的曲线关系都是通过增加多项式实现的，其实现⽅式和多元线性回归类似。
Lecture 9_Simple Linear Regression 第九章 简单线性回归分析
Population Slope Coefficient
Independent Variable
Random Error term
Y i β0β1X iεi
Linear component
Random Error component
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House Price in $1000s (Y) 245 312 279 308 199 219 405 324 319 255
Square Feet (X) 1400 1600 1700 1875 1100 1550 2350 2450 1425 1700
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Price = 98.2 + 0.110 Square Feet
Predictor Coef SE Coef T P Constant 98.25 58.03 1.69 0.129 Square Feet 0.10977 0.03297 3.33 0.010
S = 41.3303 R-Sq = 58.1% R-Sq(adj) = 52.8%
450 400 350 300 250 200 150 100
Estimate of the regression intercept
Estimate of the regression slope
Value of X for
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The Least Squares Method
ChapterSimpleLinearRegression商务统计教学
b0 y b1 x
where: xi = value of independent variable for ith observation yi = value of dependent variable for ith _ observation x = mean value for independent variable _ y = mean value for dependent variable n = total number of observations
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where: yi = observed value of the dependent variable for the ith observation ^ y i = estimated value of the dependent variable for the ith observation
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Estimated Regression Equation Sample Statistics b0, b1
Slide 8
ˆ b0 b1 x y
Least Squares Method
Least Squares Criterion
The simple linear regression equation is:
E(y) = b0 + b1x
• Graph of the regression equation is a straight line. • b0 is the y intercept of the regression line. • b1 is the slope of the regression line. • E(y) is the expected value of y for a given x value.
09离散因变量模型
09离散因变量模型⽬录离散因变量模型要考察⼈们做出某种具体选择的情况及其影响因素时，可把这些离散的定性变量作为因变量进⾏分析，把影响因素作为⾃变量，这样建⽴的模型称之为离散选择模型。
如出⾏交通⼯具选择的情况。
还有⼀种是因变量是以离散计数的⽅式描述的，分析⾃变量对计数因变量的影响所建⽴的模型，称之为计数模型。
如发⽣交通事故的次数。
线性概率模型离散选择模型在⼴义线性模型（generalized linear model）的框架下展开，并依赖结果是两个或多个选择将模型分位⼆项选择、多项选择模型和受限因变量模型离散选择模型主要研究选择结果的概率与影响因素之间的关系，即Prob(事件i发⽣) = Prob(Y=i)=F(影响因素)其中，影响因素可能包含做出选择的主体属性和选择⽅案属性。
如选择何种交通⼯具出⾏，既受到选择主体收⼊程度、⽣活习惯等属性的影响，也收到交通⼯具的价格、便捷性等属性的影响。
⽰例：对影响⼿机购买意向的因素进⾏分析购买意向为定性变量，有两种选择：0表⽰不购买，1表⽰购买。
其影响因素可能有性别、年龄、收⼊、职位、⾏业等诸多因素。
设因变量y表⽰是否购买⼿机，则有y= \begin{cases} 0 & 不购买 \\ 1 & 购买 \end{cases}影响y的因素记为x=(x_1,x_2,\cdots, x_n)，根据多元回归的思想，可得y = \beta_0 + \beta_1 x_1+\beta_2 x_2+\cdots +\beta_n x_n + \varepsilon其中，(\beta_1,\beta_1,\cdots, \beta_n)^T=\beta表⽰回归模型中的参数即回归系数，则简化为y = \beta_0 + \beta x + \varepsilon在因变量是离散变量的情况下，不能把\beta_i(i=1,2,\cdots,n)理解为保持其他因素不变的情况下对y的边际影响，因为y的取值为1或0。
scikit-learn linearregression 公式表达式
scikit-learn linearregression 公式表达式Scikit-learn是一个流行的Python机器学习库，其中包含了许多常用的机器学习算法。
其中之一就是线性回归算法，它是一种用于建立线性模型的监督学习算法。
在Scikit-learn中，线性回归模型可以通过LinearRegression类来实现。
线性回归模型的目标是通过拟合一条直线来建立输入特征和输出目标之间的关系。
具体而言，我们希望找到一条直线，使得通过这条直线预测的输出值与实际输出值之间的差异最小。
线性回归模型的公式表达式可以表示为：y = β0 + β1x1 + β2x2 + ... + βn*xn其中，y是要预测的输出目标，x1, x2, ..., xn是输入特征，β0, β1, β2, ..., βn是模型的参数。
在Scikit-learn中，线性回归模型的参数可以通过最小二乘法来估计。
最小二乘法的目标是最小化预测值与实际值之间的平方误差和。
线性回归模型的训练过程可以分为几个步骤：1.导入LinearRegression类：首先，我们需要导入Scikit-learn库中的LinearRegression类。
2.创建线性回归对象：然后，我们可以创建一个线性回归对象，通过调用LinearRegression构造函数。
3.拟合模型：接下来，我们可以使用fit方法来拟合模型。
fit方法接受输入特征和输出目标作为参数，并根据最小二乘法来估计模型的参数。
4.预测：一旦模型被拟合，我们就可以使用predict方法来进行预测。
predict方法接受输入特征作为参数，并返回对应的输出目标的预测值。
线性回归模型的性能可以通过多种指标来评估，例如均方误差（Mean Squared Error）和决定系数（R-squared）。
在Scikit-learn中，我们可以使用mean_squared_error和r2_score函数来计算这些指标。
回归直线法应用的原理
回归直线法应用的原理1. 简介回归直线法（Linear Regression）是统计学中常用的一种分析方法，用于研究自变量与因变量之间的关系。
该方法可以通过拟合一条直线来描述自变量和因变量之间的线性关系，从而进行预测、推断和探索性分析。
2. 原理回归直线法的原理基于最小二乘法，通过寻找一条直线使得观测值与预测值之间的残差平方和最小化。
该直线由两个参数确定：截距（intercept）和斜率（slope）。
直线的方程可以表示为：Y = a + bX，其中Y表示因变量，X表示自变量，a表示截距，b表示斜率。
3. 应用步骤回归直线法的应用通常包括以下几个步骤：3.1 数据收集首先需要收集自变量和因变量的相关数据。
数据的质量和完整性对回归分析的结果具有重要影响，因此保证数据的准确性和可靠性是非常重要的。
3.2 数据预处理在进行回归分析之前，需要对数据进行预处理。
这包括数据清洗、缺失值处理、异常值处理、数据标准化等步骤，以确保数据的合理性和一致性。
3.3 模型拟合模型拟合是回归直线法的核心步骤。
通过最小二乘法，寻找最佳的直线拟合数据，使得观测值和预测值之间的残差平方和最小化。
计算得到的拟合直线的截距和斜率可以描述自变量和因变量之间的线性关系。
3.4 模型评估模型评估是判断模型质量的重要步骤。
常见的评估指标包括决定系数（R^2）、均方误差（MSE）和标准误差（SE）等。
这些指标可以用于评估模型的拟合程度和预测能力。
3.5 结果解释和应用最后，根据拟合的回归直线和模型评估结果，进行结果解释和应用。
通过该直线可以进行因变量的预测、因素分析、趋势预测等应用。
4. 注意事项在进行回归直线法分析时，需要注意以下几点：4.1 线性关系假设回归直线法的前提是自变量和因变量之间存在线性关系。
在进行分析前，需要先验证自变量和因变量之间的线性关系假设。
4.2 多重共线性多重共线性是指自变量之间存在高度相关性的情况。
当存在多重共线性时，会影响回归分析的结果和可靠性。
Simple
Simple Linear Regression 笔记Simple Linear RegressionCreating the Regression LineCalculating b1 & b0, creating the lineand testing its significance with a t-test.DEFINITIONS:b1 - This is the SLOPE of the regression line. Thus this is the amount that the Y variable (dependent) will change for each 1 unit change in the X variable.b0 - This is the intercept of the regression line with the y-axis. In otherwords it is the value of Y if the value of X = 0.Y-hat = b0 + b1(x) - This is the sample regression line. You must calculate b0 & b1 to create this line. Y-hat stands for the predicted value of Y, and it can be obtained by plugging an individual value of x into the equation and calculating y-hat. EXAMPLE:A firm wants to see if there is sales is explained by the number of hours overtime that their salespeople work. Using a spreadsheet containing 25 months of sales & overtime figures, the following calculations are made; SSx = 85, SSy = 997 and SSxy = 2,765, X-bar = 13 and Y-bar = 67,987, also s(b1) = 21.87. Create the regression line.(1) find b1 - One method of caluating b1 is b1 = SSxy/SSx = 2765/85 = 32.53. This is the slope of the line - for every unit change in X, y will increase by 32.53. It is a positive number, thus its a direct relationship - as X goes up, so does Y. However, if b1 = -32.53, then we would know the relationship between X & Y is an inverse relationship - as X goes up, y goes down)(2) find b0 - again the formula is on pg. 420 and is b0 = Y-bar - b1(x-bar) = 67,987 - 32.53(13) = 67,987 - 422.89 = 67,564, this is the intercept of the line and the Y-axis, and can be interpreted as the value of Y if zero hours of overtime (x=0) are worked.(3) Create Line - Y-hat = b0 + b1(x) or Y-hat = 67,564 + 32.53(x), This line quantifies the relationship between X & Y. Under the normal error model, b1 is unbiased for Beta1 with:Sy/x is the residual standard deviation:But is this Relationship "Significant"Since it is based on a sample and we wish to generalize to a population, it must be tested to see if it is "significant," meaning would the relationship we found actually exist in the population or is the result due to sampling error (our sample did not represent the true population). The specific test we use is a t-test to test to see if b1 is different from 0. Since B1 would be the slope of the regression line in the population, it makes sense to test to see if it is different from zero. If it is zero, then our slope is 0, meaning if we graphed the relationship between X & Y we would end up with a horizontal (flat) line. And if this line is flat then we know that no matter what value the X variable takes on, the Y variable's value will not change. This means there is no linear relationship between the two variables. This also means that the regression line we calculated is useless for explaining or predicting the dependent variable.TESTING B1 We use our standard five step hypothesis testing procedure.Hypotheses: H0: B1 = 0, H1: B1 not = 0Critical value: a t-value based on n-2 degrees of freedom. Also divide alpha by 2 because it is a 2-tailed test. In this case n = 25 (25 months data used) thus n-2 = 23. With alpha = .05 we have alpha/2 = .025 and then t = 2.069 (from t-table inside front cover of book).Calculated Value: The formula is on page 442 and is simply t = b1/s(b1) = 32.53/21.87 = 1.49. s(b1) is the standard error of b1 and is given in the problem)Compare: t-calc < t-crit and thus accept H0.Conclusion: B1 = 0, the population slope of our regression is a flat line, thus there is no linear relationship between sales and overtime worked, and the sample regression line we calculated is not useful for explaining or predicting sales dollars from overtime worked.CorrelationCorrelation is a measure of the degree of linear association between two variables. The value of a correlation can range from -1, thru 0, to +1. A correlation = 0 means there is no LINEAR association between the two variables, a value of -1 or +1 means there is a perfect linear association between the two variables, the difference being that -1 indicates a perfect inverse relationship and +1 a perfect positive relationship. The sample notation for a correlation is "r" while the population correlation coefficient is represented by the greek letter "Rho" (which looks like a small "p").We often want to find out if a calculated sample correlation would be "significant." Again this would mean we would test to see if Rho = 0 or not. If Rho=0 then there would be no linear relationship between the two variables in the population.AN EXAMPLE:Based on a sample of 42 days, the correlation between sales and number of sunny hours in the day is calculated for the Sunglass Hut store in Meridian Mall. The r = .56. Is this a "significant" correlation?This is a basic hypothesis test.....Hypotheses: H0: Rho = 0, H1: Rho not = 0.Critical Value: The t-test for the significance of Rho has n-2 degrees of freedom, and alpha will need to be divided by 2, thus n-2 = 40 and alpha (.05/2) = .025 ... from the table we find: 2.021.Calculated Value: The formula on page 438 is t = r / sqr root of (1-r-sqrd)/(n-2). In this case that equals .56 / the square root of (1-.56-squared)/(40) = .56/.131 = 4.27Compare: The t-calc is larger than the t-crit thus we REJECT Ho.Conclusion: Rho does not equal zero and thus there is evidence of a linear association between the two variables in the population.The F-test in RegressionEXAMPLEUsing the information given, construct the ANOVA table and determine whether there is a regression relationship between years of car ownership (Y) and salary (X). n= 47, SSR = 458 and SSE = 1281.ANOVA Table: The anova table is on page 451, and is basically the same as a one-way ANOVA table. The first thing we need is the df, and by definition the df for the regression = 1, the df for the error = n-2 or 45, and the total df = n-1 or 46. Next we need the MS calculations. MSR = SSR/df for the regression = SSR/1 = SSR or 458. MSE = SSE/n-2 = 1281/45 = 28.47. Finally, the F-calc = MSR/MSE or 458/28.47 = 16.09.Hypotheses: H0: There is no regression relationship,i.e, B1 =0. H1: There is a regression relationship, i.e, B1 is not = 0. Critical Value: F(num. df, den. df) = F(1, 45) at alpha = .05 = 4.08Calculated Value: from above ANOVA table = 16.09Compare: F-calc larger than F-crit thus REJECTConclusion: There is a regression (linear) relationship between years of car ownership and salary.The Coefficient of Determination - r-sqrdWe can also test the significance of the regression coefficient using an F-test. Since we only have one coefficient in simple linear regression, this test is analagous to the t-test. However, when we proceed to multiple regression, the F-test will be atest of ALL of the regression coefficients jointly being 0. (Note: b0 is not a coefficient and we generally do not test its significance although we could do so with a t-test just as we did ofr b1.r-sqrd is always a number between 0 and 1. The closer it is to 1.0 the better the X-Y relationship predicts or explains the variance in Y. Unfortunately there are no set values that allow you to say that is a "good" r-sqrd or "bad" r-sqrd. Such a determination is subjective and is determined by the research you are conducting. If nobody has ever explained more that 15% of the variance in some Y variable before, and you design a study that explains 25% of variance, then this might be considered good r-sqrd, even though the actual number, 25%, is not very high.EXAMPLE:What is the r-sqrd if SSR = 345 and SSE = 123?r-sqrd = SSR/SST. We don't have SST, but we know that SSR + SSE = SST, thus SST = 345 + 123 = 468, thus r-sqrd = 345/468 = .737. This means that the regression relationship between X & Y explains 73.7% of the variance in the Y variable. Under most circumstances this would be a high amount, but again we would have to know more about our research varaibles.。
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Chap 12-17
Simple Linear Regression Example: Using Excel
Business Statistics: A First Course, 5e © 2009 Prentice-Hall, Inc..
Learning Objectives
In this chapter, you learn:
How to use regression analysis to predict the value of a dependent variable based on an independent variable The meaning of the regression coefficients b0 and b1 How to evaluate the assumptions of regression analysis and know what to do if the assumptions are violated To make inferences about the slope and correlation coefficient
X
Chap 12-10
Simple Linear Regression Equation (Prediction Line)
The simple linear regression equation provides an estimate of the population regression line
Business Statistics: A First Course
Fifth Edition
Chapter 12 Simple Linear Regression
Business Statistics: A First Course, 5e © 2009 Prentice-Hall, Inc.
Chap 12-1
Business Statistics: A First Course, 5e © 2009 Prentice-Hall, Inc..
Chap 12-14
Simple Linear Regression Example
A real estate agent wishes to examine the relationship between the selling price of a home and its size (measured in square feet)
Simple Linear Regression Model
Population Slope Coefficient Random Error term
Population Y intercept Dependent Variable
Independent Variable
Yi β0 β1Xi εi
Business Statistics: A First Course, 5e © 2009 Prentice-Hall, Inc..
Chap 12-13
Interpretation of the Slope and the Intercept
b0 is the estimated mean value of Y when the value of X is zero b1 is the estimated change in the mean value of Y as a result of a one-unit change in X
Linear component Random Error component
Chap 12-9
Business Statistics: A First Course, 5e © 2009 Prentice-Hall, Inc..
Simple Linear Regression Model
(continued)
X
X
Y
Y
X
Business Statistics: A First Course, 5e © 2009 Prentice-Hall, Inc..
X
Chap 12-7
Types of Relationships
(continued) No relationship Y
X
Y
X
Business Statistics: A First Course, 5e © 2009 Prentice-Hall, Inc.. Chap 12-8
Chap 12-15
Simple Linear Regression Example: Data
House Price in $1000s (Y) 245 312 279 308 199 219 Square Feet (X) 1400 1600 1700 1875 1100 1550
405
324 319 255
A random sample of 10 houses is selected Dependent variable (Y) = house price in $1000s Independent variable (X) = square feet
Business Statistics: A First Course, 5e © 2009 Prentice-Hall, Inc..
X
Y
Y
X
Business Statistics: A First Course, 5e © 2009 Prentice-Hall, Inc..
X
Chap 12-6
Types of Relationships
(continued) Strong relationships Y Y Weak relationships
Y
Observed Value of Y for Xi
Yi β0 β1Xi εi
εi
Slope = β1
Random Error for this Xi value
Predicted Value of Y for Xi Intercept = β0
Xi
Business Statistics: A First Course, 5e © 2009 Prentice-Hall, Inc..
Dependent variable:
the variable we wish to predict or explain Independent variable: the variable used to predict or explain the dependent variable
Business Statistics: A First Course, 5e © 2009 Prentice-Hall, Inc.. Chap 12-4
A scatter plot can be used to show the relationship between two variables
Correlation analysis is used to measure the strength of the association (linear relationship) between two variables
2350
2450 1425 1700
Business Statistics: A First Course, 5e © 2009 Prentice-Hall, Inc..
Chap 12-16
Simple Linear Regression Example: Scatter Plot
House price model: Scatter Plot
Business Statistics: A First Course, 5e © 2009 Prentice-Hall, Inc..
Chap 12-5
Types of Relationships
Linear relationships Y Y Curvilinear relationships
X
ˆ )2 min (Y (b b X ))2 min (Yi Yi i 0 1 i
Business Statistics: A First Course, 5e © 2009 Prentice-Hall, Inc..
Chap 12-12
Finding the Least Squares Equation
Estimated (or predicted) Y value for observation i
Estimate of the regression intercept
Estimate of the regression slope
Value of X for observation i
ˆ Yi b0 b1Xi
To estimate mean values and predict individual values
Business Statistics: A First Course, 5e © 2009 Prentice-Hall, Inc.. Chap 12-2
Correlation vs. Regression
Correlation is only concerned with strength of the relationship
No causal effect is implied with correlation
Scatter plots were first presented in Ch. 2 Correlation was first presented in Ch. 3