Chapter 10

The Analysis of

Variance

10.1

Single-Factor ANOVA

The analysis of variance (ANOVA), refers to a collection of experimental situations and statistical procedures for the analysis of quantitative responses from experimental units.

The Analysis of Variance

Terminology

The characteristic that differentiates the treatments or populations from one another is called the factor under study, and the different treatments or populations are referred to as the levels of the factor.

Single-Factor ANOVA

Single-factor ANOVA focuses on a comparison of more than two population or treatment means. Let

I = the number of treatments (populations)

1 the mean of population 1 or the true average response when treatment 1 is applied

being compared.

I ..

the mean of population I or the true average response when treatment I is applied

Then the hypotheses of interest are

0 1 2: ... IH

versus

a :H at least two of the are different

'si

Notation

,i jX

,i jx

The random variable that denotes the jth measurement taken from the ith population, or the measurement taken on the jth experimental unit that receives the ith treatment

The observed value of Xi,j when the experiment is performed

Assumptions

The I population or treatment distributions are all normal with the same varianceEach Xi,j is normally distributed with

2.

2, ,( ) ( )i j i i jE X V X

The Mean Square for Treatments and Error

Mean square for treatments:2 2

1MSTr [( ..) ... ( ..) ]1 I

JX X X X

I

2( ..)1 i

i

JX X

I

Mean square for error:2 2 21 2 ...

MSE IS S S

I

The Test Statistic

The test statistic for single-factor ANOVA is F = MSTr/MSE.

Expected Value

When H0 is true,

2(MSTr) (MSE)E E

When H0 is false,2(MSTr) (MSE)E E

F Distributions and Test

Let F = MSTr/MSE be the statistic in a single-factor ANOVA problem involving I populations or treatments with a random sample of J observations from each one. When H0 is true (basic assumptions true) , F has an F distribution with v1= I – 1 and v2= I(J – 1). The rejection region

, 1, ( 1)I I Jf F specifies a test with significance level .

Formulas for ANOVA

Total sum of squares (SST)

Treatment sum of squares (SSTr)

Error sum of squares (SSE)

2 2..

1 1

1SST

I J

iji j

x xIJ

2 2. ..

1

1SSTr

I

ii

ix x

J IJ

2.

1 1

SSEI J

ij ii j

x x

Fundamental Indentity

SST = SSTr + SSE

Mean Squares

SSTrMSTr =

1I SSE

MSE = 1I J

MSTr =

MSEF

ANOVA Table

Source of Variation

df Sum of squares

Mean Square

f

Treatments I – 1 SSTr MSTr MSTr/MSE

Error I(J – 1) SSE MSE

Total IJ – 1 SST

10.2

Multiple Comparisons in

ANOVA

Studentized Range Distribution and Pairwise Differences

With probability 1 ,

. . , , ( 1) MSE /i j I I J i jX X Q J

. . , , ( 1) MSE /i j I I JX X Q J

for every i and j with .i j

The T Method for Identifying Significantly Different 'i s

1. Select extract

2. Calculate

3. List the sample means in increasing order, underline those that differ by more than w. Any pair not underscored by the same line corresponds to a pair that are significantly different.

, , ( 1) MSE /I I Jw Q J

, , ( 1).I I JQ

Confidence Intervals for Other Parametric Functions

Let .i ic Xij’s are normally distributed.

2

2ˆi i i

i i

V V c X cJ

Estimating by MSE and forming results in a t variable leads to

2 ˆˆˆ

ˆ ˆ( ) 2

/ 2, ( 1)MSE i

i i I Jc

c x tJ

10.3

More on Single-Factor ANOVA

ANOVA Model

ij i ijX

The assumptions of a single-factor ANOVA can be modeled by

ij.i

represents a random deviation from the population or true treatment mean

MSTr

2 2(MSTr)1 i

JE

I

Note that when H0 is true 2 0.i

for the F Test

Consider a set of parameter values1,..., n for which H0 is not true.

The probability of a type II error, is the probability that H0 is not rejected when that set is the set of true values.

,

2 2..

1 1

1SST df 1

iJI

iji j

X X nn

Single-Factor ANOVA When Sample Sizes are Unequal

2 2. ..

1

1 1SSTr df 1

I

iii

X X IJ n

SSE = SST – SSTr df n I

Single-Factor ANOVA When Sample Sizes are Unequal

MSTr SSTr where MSTr =

MSE 1f

I

SSE

MSE = n I

Rejection region:

Test statistic value:

, 1,I n If F

Multiple Comparisons (Unequal Sample Sizes)

Let , , 1MSE 1 1

2ij I ni j

w QJ J

Then the probability is approximately1 that

. . . .i j ij i j i j ijX X w X X w

for every i and j with .i j

Data Transformation

If a known function of then a transformation h(Xij) that “stabilizes the variance” so that V[h(Xij)] is approximately the same for each i is given by

,i( ) ( ),ij iV X g

1/ 2( ) ( ) .h x g x dx

A Random Effects Model

with ( ) ( ) 0ij i ij i ijX A E A E

2 2( ) ( )ij i AV V A

All Ai’s and are normally distributed and independent of one another.

'sij