使用SAS的回归建模：方法、理论与计算实践

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"Regression Modelling. Methods, theory, and Computation with SAS" 回归分析是一种统计方法，用于研究因变量（目标变量）与一个或多个自变量（预测变量）之间的关系。在"Regression Modeling: Methods, Theory, and Computation with SAS"这本书中，作者Michael Panik深入浅出地介绍了如何使用SAS软件进行回归分析，涵盖了各种回归技术，并针对不同类型的回归问题提供了实际解决方案。书中的内容可能包括以下几个方面： 1. **基础理论**：首先，书中可能会介绍回归分析的基本概念，如线性回归、多元回归和逻辑回归等。理论部分会讲解如何建立回归模型，以及模型的假设，如线性关系、误差项的独立性、正态性和同方差性。 2. **模型选择与评估**：作者可能会讨论如何根据数据选择合适的回归模型，如逐步回归、岭回归、套索回归等。此外，还会介绍评估模型性能的指标，如R方、调整R方、残差图、AIC和BIC等。 3. **异常值处理与诊断**：回归分析中，异常值可能导致模型不准确。书中可能包含识别和处理异常值的方法，以及通过残差分析来检查模型假设是否被违反的技巧。 4. **非线性模型**：除了线性回归，书还可能涵盖非线性模型，如多项式回归、指数回归和对数回归，以及如何用SAS来实现这些模型。 5. **分类与回归树（CART）、随机森林与梯度提升**：这些现代机器学习方法也可能在书中有所提及，特别是它们如何与传统回归技术结合使用。 6. **时间序列分析**：对于具有时间序列数据的回归问题，书中可能包含ARIMA模型、季节性模型或其他时间序列模型的SAS实现。 7. **多层模型与混合效应模型**：当数据具有层次结构时，如在群体研究或重复测量设计中，多层模型或混合效应模型可能是必要的，书中可能会介绍如何在SAS中构建和估计这类模型。 8. **预测与应用**：回归模型不仅用于探索关系，还用于预测。书中的章节可能会讲解如何使用SAS进行预测，并评估预测的准确性。 9. **实例与案例研究**：为了帮助读者理解，书中可能包含了大量的实际例子和案例研究，以演示如何使用SAS解决现实世界的问题。通过"Regression Modeling: Methods, Theory, and Computation with SAS"，读者不仅可以学习到回归分析的理论知识，还能获得实践SAS编程的经验，这对于在统计分析、社会科学、医学研究、工程学和商业领域中运用回归模型的人员来说是一本宝贵的资源。

Review of Fundamentals of Statistics 3

X is said to be a continuous random variable if it assumes an in nite or uncountable num-

ber of values over some interval.

For X, a discrete random variable de ned on S, let the range of X be the set of values

R = {X

, i = 1, 2,…} and let f be a function that associates with each X

∈ R a number f(X

)

∈ [0,1] representing the probability that X = X

, i.e., P(X = X

) = f(X

), i = 1,2,… . Hence, f(X

)

depicts the probability mass at X

. In general, a function f(X), which assigns a probability

mass f(X

) to each X

within the range of X, is called a probability mass function if

a. f(X

) ≥ 0 for all i;

() ;

∑

= 1 and

c. f(X

) = 0 for all X

∉ R.

To construct a discrete probability distribution, we need to specify the random experi-

ment and de ne the random variable X. Given these ingredients, a discrete probability

distribution will be fully speci ed once the sequence of probabilities f (X

), i = 1, 2,…, is

given. Hence, a discrete probability distribution can be represented by the set of all pos-

sible pairs (X

, f (X

)), i = 1, 2,… . Once a discrete probability distribution is determined, we

may compute, for instance, probabilities such as,

Pa X b f X PX a f X

XaaX b

()();()();≤≤= ≥=

≥≤≤

∑∑

and

PX a PX a f X

() () ().<=− ≥=

∑

Whereas the probability mass function f (X

) gives P(X = X

), a related function that yields

P(X ≤ X

) is the cumulative distribution function:

FX f X

() (),=

≤

∑

(1.8)

with the X

’s placed in increasing order. Its key properties are as follows:

a. 0 ≤ F (X

) ≤ 1 for any X

;

b. F is a monotone nondecreasing function of X

;

c. P(X > X

) = 1 − F(X

);

d. P(X = X

) = F(X

) − F(X

− 1), X

an integer;

e. P(X

< X ≤ X

) = F(X

) − F(X

);

f. F(− ∞) = 0, F(+ ∞) = 1; and

g. F is continuous from the right.

Given a discrete random variable X, its mean or expected value is

EX Xf X

() ( ) ;==

∑

(1.9)

Review of Fundamentals of Statistics 7

Next, suppose

ZN Y

:(,), : ,01

and the random variables Z and Y are independent.

Then the quantity

TXYn= /

has a (Student’s) t distribution with n degrees of freedom

and is written T : t

. Thus, the t distribution is the distribution of a standard normal vari-

able divided by the square root of a χ

variable divided by its degrees of freedom, with

these variables taken to be independent. Moreover, the t distribution is symmetrical about

zero like the normal distribution, but its tails are  atter (and longer) than those of the nor-

mal distribution. As the degrees of freedom n approaches +∞, the t distribution approaches

the standard normal distribution. Quantiles of the t distribution appear in Table A.2 of the

Appendix.

Finally, if X

and X

are independent variables, then Y = (X

/n)/

/m) has an F distribution with degrees of freedom n and m and is written Y : F

n,m

(Remember that the  rst and second subscripts refer to the degrees of freedom of the

numerator and denominator, respectively.) Thus, the ratio of two independent χ

random

variables, each divided by its own degrees of freedom, is F distributed. Quantiles of the F

distribution are provided in Table A.4 of the Appendix.

Bivariate Random Variables and Probability Distributions

Suppose (X, Y) is a pair of real-valued functions de ned on a sample space S. The pair

(X, Y) is a bivariate random variable if both X and Y map elements in S into real numbers.

Thus, pair (X, Y) de nes the possible outcomes of some random experiment. Now, if the

range of (X, Y) is a discrete ordered set of points R

X,Y

= {(X

, Y

), i = 1,…,m} in the Cartesian

plane, then (X, Y) is termed a discrete bivariate random variable. However, if S consists of a

class of events representable by all open and closed rectangles, then X and Y de ned on S

are termed continuous bivariate random variables if they can assume any values within some

event A = {(X, Y)|a ≤ X ≤ b, c ≤ Y ≤ d} ⊆ S.

For the discrete case, given a random variable X : X

, i = 1,…,n (with X

< X

< … < X

) and

a second random variable Y : Y

, j = 1,…,m (with Y

< Y

< … < Y

), we may denote the joint

probability that X = X

and Y = Y

as P(X = X

, Y = Y

) = f(X

, Y

), i = 1,...,n; j = 1,...,m. Here f(X

, Y

)

is termed the probability mass at the point (X

, Y

). In general, a function f(X, Y) that assigns

a probability mass f(X

, Y

) within the ranges of the discrete random variables X and Y is

called a bivariate probability mass function if

a. f(X

, Y

) ≥ 0 for all i, j; and

fX Y

(,) .

∑∑

= 1

Then the set of nm events (X

, Y

) together with their associated probability masses

f(X

, Y

), i = 1,…,n; j = 1,…,m, constitutes the discrete bivariate probability distribution of

the random variables X and Y. Given a bivariate probability mass function f(X, Y), the

joint probability that X ≤ X

and Y ≤ Y

is provided by the bivariate cumulative distribution

function

FX Y PX X Y Y f X Y

rs r s ij

jsir

(,) ( , ) (,).=≤ ≤=

≤≤

∑∑

(1.21)

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