多元线性回归与1D模型详解

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"《多元线性与一维回归》是一篇介绍统计学中两种关键模型的文章，即多元线性回归（Multiple Linear Regression, MLR）和一维回归。在MLR中，作者David J. Olive解释了如何分析因变量Y与多个预测变量之间的关系，通过建立一个条件独立模型，Y在给定一组非零预测变量的线性组合α + βTx时变得独立，写作Y|x|(\( \alpha + \beta^T x \) 或 Y|βTx)。这种模型广泛应用于统计实践中，如广义线性模型，包括但不限于多项式回归、逻辑回归和泊松回归。文章首先介绍了多元线性回归的基本概念，强调了模型的构建方法和检验模型拟合度的重要性，如通过残差图检查模型残差的分布，以及利用方差分析（ANOVA）F测试来评估模型整体显著性。预测能力也是关键部分，讨论了如何进行预测和评估预测精度。此外，还探讨了部分变化的F检验（Partial F-test for Change in SS）和Wald t检验在模型参数估计中的应用。接着，文章关注单指数模型（Single Index Models），它们涉及一个特殊的线性结构，其中一个预测变量对响应变量的影响可以通过一个单一的函数映射。生存分析模型（Survival Regression Models）则涉及到处理时间依赖的数据，而变量选择则是优化模型复杂度和减少噪声的关键步骤。最后，文章提及其他可能遇到的问题，如模型不足检验（Lack of Fit Tests）、问题设置及解决方法，并提供了一些补充材料和练习题供读者深入学习和实践。在一维回归部分，虽然没有详细列出，但通常是指当Y与一个线性组合直接相关时的情况，这与MLR相比，模型更为简单。文章特别提到了两个重要的特殊情况：一是位置模型（Location Model），它通常用于确定平均效应；另一个是简单线性回归，这是多元线性回归的基础，但当只有一个预测变量时适用。《多元线性与一维回归》通过详细讲解这些核心概念，为理解和应用这两种重要的统计技术提供了扎实的理论基础和实践经验。"

CHAPTER 1. INTRODUCTION 3

In particular, the logistic regression model uses

ρ(α + β

x)=

exp(α + β

1+exp(α + β

In a 1D Poisson regression model,theY |x are independent

Poisson[µ(α + β

x)]

random variables. In particular, the loglinear regression model uses

µ(α + β

x)=exp(α + β

x). (1.8)

In the literature, the response variable is sometimes called the dependent

variable while the predictor variables are sometimes called carriers, covari-

ates, explanatory variables, or independent variables. The ith case (Y

, x

)

consists of the values of the response variable Y

and the predictor variables

=(x

i,1

, ..., x

i,p

)wherep is the number of predictors and i =1, ..., n. The

sample size n is the number of cases.

Box (1979) warns that “all models are wrong, but some are useful.” For

example the function g or the error distribution could be misspeciﬁed. Di-

agnostics are used to check whether model assumptions such as the form of

g and the proposed error distribution are reasonable. Often diagnostics use

residuals r

. If m is known, then the single index model (1.3) uses

= Y

−m(ˆα +

)

where (ˆα,

β)isanestimateof(α, β).

Exploratory data analysis (EDA) can be used to ﬁnd useful models when

the form of the regression or multivariate model is unknown. For example,

suppose g is a monotone function t

−1

Y = t

−1

(α + β

x + e). (1.9)

Then the transformation

Z = t(Y )=α + β

x + e (1.10)

follows a multiple linear regression model.

CHAPTER 1. INTRODUCTION 5

misspeciﬁed. The following sections show that the suﬃcient summary plot

is useful for explaining the given 1D model while the response plot can often

be used to visualize the conditional distribution of Y |(α + β

x). If there is

only one predictor x, then the plot of x versus Y is both a suﬃcient summary

plot and a response plot, but generally β is unknown and only a response

plot can be made. In Deﬁnition 1.2, since ˜α can be any constant, ˜α =0is

often used.

1.1 Multiple Linear Regression

Suppose that the response variable Y is quantitative and that at least one

predictor variable x

is quantitative. Then the multiple linear regression

(MLR) model is often a very useful model. For the MLR model,

= α+x

i,1

i,2

+···+x

i,p

= α+x

β+e

= α+β

(1.12)

for i =1,...,n. Here Y

is the response variable, x

is a p × 1 vector of

nontrivial predictors, α is an unknown constant, β is a p×1 vector of unknown

coeﬃcients, and e

is a random variable called the error.

The Gaussian or normal MLR model makes the additional assumption

that the errors e

are iid N(0,σ

) random variables. This model can also be

written as Y = α + β

x + e where e ∼ N(0,σ

), or Y |x ∼ N(α + β

x,σ

)

or Y |x ∼ N(SP,σ

). The normal MLR model is a parametric model since,

given x, the family of conditional distributions is completely speciﬁed by the

parameters α, β and σ

. Since Y |SP ∼ N(SP,σ

), the conditional mean

function E(Y |SP) ≡ M(SP)=µ(SP)=SP = α + β

x.TheMLRmodel

is discussed in detail in Chapters 2, 3 and 4.

A suﬃcient summary plot (SSP) of the suﬃcient predictor SP = α+β

versus the response variable Y

with the mean function added as a visual aid

can be useful for describing the multiple linear regression model. This plot

can not be used for real data since α and β are unknown. To make Figure 1.1,

the artiﬁcial data used n = 100 cases with k = 5 nontrivial predictors. The

data used α = −1, β =(1, 2, 3, 0, 0)

∼ N(0, 1) and x from a multivariate

normal distribution x ∼ N

(0, I).

In Figure 1.1, notice that the identity line with unit slope and zero in-

tercept corresponds to the mean function since the identity line is the line

Y = SP = α + β

x = µ(SP)=E(Y |SP). The vertical deviation of Y

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