应用统计方法：方差分析、设计与回归解析

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"《Analysis of Variance, Design, and Regression: Applied Statistical Methods》是一本针对统计学硕士研究生的教材，主要探讨了分析方差、实验设计以及回归分析等基础统计方法，并对计数数据进行了一些讨论。这本书适合那些数学基础相对较弱但对回归课程感兴趣的非统计学专业学生。书中虽然照顾到数学基础知识较弱的读者，但仍提供了完整的矩阵形式的回归理论阐述。" 在本书中，作者Ronald Christensen从概率论和随机变量的基础概念出发，详细讲解了统计学中的关键主题： 1. 概率：介绍概率的基本概念，是理解统计推断的基础。 2. 随机变量与期望：讨论随机变量的期望值和方差，以及Chebyshev不等式，帮助读者理解变量的集中趋势和离散程度。 3. 连续分布：涵盖连续概率分布，如正态分布，这对分析方差和回归分析至关重要。 4. 二项分布和多项分布：这些离散概率分布常用于处理有限次试验的成功次数，是实验设计中的常见模型。接着，作者介绍了单样本的统计分析： 1. 示例和介绍：通过实例引入统计分析的基本思路。 2. 关于总体均值的推断：包括置信区间和假设检验，这是统计推断的核心内容。 3. 预测区间：提供对未来观测值的预测范围，适用于实际应用。 4. 正态性检验：在进行统计分析时，检查数据是否符合正态分布是必要的步骤。 5. 变换：当原始数据不符合模型假设时，可能需要对数据进行转换。 6. 关于总体方差的推断：对于方差的估计和检验也是数据分析的重要组成部分。然后，书中的第三章深入到一般性的假设检验和置信区间的理论： 1. 置信区间理论：解释如何构建和理解置信区间。 2. 假设检验理论：详细说明了假设检验的过程和标准。 3. 测试和置信区间的有效性：讨论了这些统计方法在不同条件下的可靠性。 4. 置信区间与假设检验的关系：这两者在统计推断中密切关联，提供了对结果的双重理解。 5. 理论的发展：作者可能进一步探讨了这些方法的理论基础和发展历史。通过以上内容，读者不仅可以掌握分析方差、实验设计和回归分析的基本技能，还能理解统计推断背后的理论框架，为他们在实际研究和工作中应用这些统计方法打下坚实的基础。这本书特别适合对统计学有一定兴趣，尤其是对回归分析有需求的硕士生或专业人士。

1.2 RANDOM VARIABLES AND EXPECTATIONS 5

Second toss

Heads Tails Total

First toss Heads 1/4 1/4 1/2

Tails 1/4 1/4 1/2

Total 1/2 1/2 1

The probability of heads on each toss is 1/2. The probability of tails is 1/2. We will deﬁne two

random variables:

(r, s) =

{

1 if r = H

0 if r = T

(r, s) =

{

1 if s = H

0 if s = T

Thus, y

is 1 if the ﬁrst toss is heads and 0 otherwise. Similarly, y

is 1 if the second toss is heads

and 0 otherwise.

The event y

= 1 occurs if and only if we get heads on the ﬁrst toss. We get heads on the ﬁrst toss

by getting either of the outcome pairs (H, H) or (H, T ). In other words, the event y

= 1 is equivalent

to the event {(H,H), (H,T )}. The probability of y

= 1 is just the sum of the probabilities of the

outcomes in {(H, H), (H, T )}.

Pr(y

= 1) = Pr(H,H) + Pr(H, T )

= 1/4 + 1/4 = 1/2.

Similarly,

Pr(y

= 0) = Pr(T, H) + Pr(T,T )

= 1/2

Pr(y

= 1) = 1/2

Pr(y

= 0) = 1/2 .

Now deﬁne another random variable,

W (r,s) = y

(r, s) + y

(r, s).

The random variable W is the total number of heads in two tosses:

W (H,H) = 2

W (H,T ) = W (T, H) = 1

W (T, T ) = 0.

Moreover,

Pr(W = 2) = Pr(H,H) = 1/4

Pr(W = 1) = Pr(H,T ) + Pr(T, H) = 1/2

Pr(W = 0) = Pr(T,T ) = 1/4 .

These three equalities deﬁne a probability on the outcomes 0, 1, 2. In working with W , we can

ignore the original outcomes of head-tail pairs and work only with the new outcomes 0, 1, 2 and

their associated probabilities. We can do the same thing for y

and y

. The probability table given

earlier can be rewritten in terms of y

and y

6 1. INTRODUCTION

1 0 y

totals

1 1/4 1/4 1/2

0 1/4 1/4 1/2

totals 1/2 1/2 1

Note that, for example, Pr [(y

) = (1, 0)] = 1/4 and Pr(y

= 1) = 1/2. This table shows the

distribution of the probabilities for y

and y

both separately (marginally) and jointly.

For any random variable, a statement of the possible outcomes and their associated probabilities

is referred to as the (marginal) probability distribution of the random variable. For two or more

random variables, a table or other statement of the possible joint outcomes and their associated

probabilities is referred to as the joint probability distribution of the random variables.

All of the entries in the center of the distribution table given above for y

and y

are independent.

For example,

Pr[(y

) = (1, 0)] ≡ Pr(y

= 1 and y

= 0) = Pr(y

= 1)Pr(y

= 0).

We therefore say that y

and y

are independent. In general, two random variables y

and y

are

independent if any event involving only y

is independent of any event involving only y

Independence is an extremely important concept in statistics. Observations to be analyzed are

commonly assumed to be independent. This means that the random aspect of one observation con-

tains no information about the random aspect of any other observation. (However, every observation

tells us about ﬁxed aspects of the underlying population such as the population center.) For most

purposes in applied statistics, just this intuitive understanding of independence is sufﬁcient.

1.2.1 Expected values and variances

The expected value (population mean) of a random variable is a number characterizing the middle

of the distribution. For a random variable y with a discrete distribution (i.e., one having a ﬁnite or

countable number of outcomes), the expected value is

E(y) ≡

∑

all r

rPr(y = r).

EXAMPLE 1.2.2. Let y be the result of picking one of the numbers 2, 4, 6, 8 at random. Because

the numbers are chosen at random,

1/4 = Pr(y = 2) = Pr(y = 4) = Pr(y = 6) = Pr(y = 8).

The expected value in this simple example is just the mean (average) of the four possible outcomes.

E(y) = 2

(

)

+ 4

(

)

+ 6

(

)

+ 8

(

)

= (2 + 4 + 6 + 8)/4

= 5.

EXAMPLE 1.2.3. Five pieces of paper are placed in a hat. The papers have the numbers 2, 4, 6,

6, and 8 written on them. A piece of paper is picked at random. The expected value of the number

drawn is the mean of the numbers on the ﬁve pieces of paper. Let y be the random variable that

8 1. INTRODUCTION

But don’t forget that I’m taking a loss on the ice-water I serve to players and I also have to pay for

the pictures of my extended family that I’ve decorated my ofﬁce with.

Although the games z

, z

, and z

have the same expected value, the games (random variables)

are very different. Game z

has the same outcomes as z

, but much more of its probability is placed

farther from the middle value 15. The extreme observations 12 and 18 are much more probable

under z

than z

. If you currently have $16, need $18 for your grandmother’s bunion removal, and

anything less than $18 has no value to you, then z

is obviously a better game for you than z

Both z

and z

are much more tightly packed around 15 than is z

. If you needed $25 for the

bunion removal, z

is the game to play because you can win it all in one play with probability .25.

In either of the other games you would have to win at least ﬁve times to get $25, a much less likely

occurrence. Of course you should realize that the most probable result is that Grandma will have

to live with her bunion. You are unlikely to win either $18 or $25. While the ethical moral of this

example is that a fool and his money are soon parted, the statistical point is that there is more to a

random variable than its mean. The variability of random variables is also important. 2

The (population) variance is a measure of how spread out a distribution is from its expected

value. Let y be a random variable having a discrete distribution with E(y) =

, then the variance of

y is

Var(y) ≡

∑

all r

(r −

)

Pr(y = r).

This is the average squared distance of the outcomes from the center of the population. More tech-

nically, it is the expected squared distance between the outcomes and the mean of the distribution.

EXAMPLE 1.2.6. Using the random variables of Example 1.2.5,

Var(z

) = (12 −15)

(.25) + (14 −15)

(.25)

+ (16 −15)

(.25) + (18 −15)

(.25)

= 5

Var(z

) = (12 −15)

(.4) + (14 −15)

(.1)

+ (16 −15)

(.1) + (18 −15)

(.4)

= 7.4

Var(z

) = (5 −15)

(.25) + (10 −15)

(.25)

+ (20 −15)

(.25) + (25 −15)

(.25)

= 62.5

The increasing variances from z

through z

indicate that the random variables are increasingly

spread out. However, the value Var(z

) = 62.5 seems too large to measure the relative variabilities

of the three random variables. More on this later. 2

EXAMPLE 1.2.7. Consider the coin tossing random variables of Examples 1.2.1 and 1.2.4.

Var(y

) =

(

1 −

)

(

0 −

)

Var(y

) =

Var(W ) = (2 −1)

(

)

+ (1 −1)

(

)

+ (0 −1)

(

)

A problem with the variance is that it is measured on the wrong scale. If y is measured in meters,

1.2 RANDOM VARIABLES AND EXPECTATIONS 9

Var(y) involves the terms (r −

)

; hence it is measured in meters squared. To get things back on

the original scale, we consider the standard deviation of y

Std. dev. (y) ≡

√

Var(y).

EXAMPLE 1.2.8. Consider the random variables of Examples 1.2.5 and 1.2.6.

Std. dev. (z

) =

√

= 2.236

Std. dev. (z

) =

√

7.4

= 2.720

Std. dev. (z

) ≡

√

62.5

= 7.906

The standard deviation of z

is 3 to 4 times larger than the others. From examining the distribu-

tions, the standard deviations seem to be more intuitive measures of relative variability than the

variances. The variance of z

is 8.5 to 12.5 times larger than the other variances; these values seem

unreasonably inﬂated. 2

Standard deviations and variances are useful as measures of the relative dispersions of different

random variables. The actual numbers themselves do not mean much. Moreover, there are other

equally good measures of dispersion that can give results that are somewhat inconsistent with these.

One reason standard deviations and variances are so widely used is because they are convenient

mathematically. In addition, normal (Gaussian) distributions are widely used in applied statistics

and are completely characterized by their expected values (means) and variances (or standard devi-

ations). Knowing these two numbers, the mean and variance, one knows everything about a normal

distribution.

1.2.2 Chebyshev’s inequality

Another place in which the numerical values of standard deviations are useful is in applications

of Chebyshev’s inequality. Chebyshev’s inequality gives a lower bound on the probability that a

random variable is within an interval. Chebyshev’s inequality is important in quality control work

(control charts) and in evaluating prediction intervals.

Let y be a random variable with E(y) =

and Var(y) =

. Chebyshev’s inequality states that

for any number k > 1,

Pr[

−k

< y <

+ k

] ≥ 1 −

Thus the probability that y will fall within k standard deviations of

is at least 1 −(1/k

The beauty of Chebyshev’s inequality is that it holds for absolutely any random variable y. Thus

we can always make some statement about the probability that y is in a symmetric interval about

. In many cases, for particular choices of y, the probability of being in the interval can be much

greater than 1 −k

−2

. For example, if k = 3 and y has a normal distribution as discussed in the next

section, the probability of being in the interval is actually .997, whereas Chebyshev’s inequality

only assures us that the probability is no less than 1 −3

−2

= .889. However, we know the lower

bound of .889 applies regardless of whether y has a normal distribution.

1.2.3 Covariances and correlations

Often we take two (or more) observations on the same member of a population. We might observe

the height and weight of a person. We might observe the IQs of a wife and husband. (Here the

population consists of married couples.) In such cases we may want a numerical measure of the

relationship between the pairs of observations. Data analysis related to these concepts is known as

regression analysis and is discussed in Chapters 7, 13, 14, and 15. These ideas are also brieﬂy used

for testing normality in Section 2.4.

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