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Chapter 15 Introduction to the Analysis of Variance IThe Omnibus Null Hypothesis H 0 :  1 =  2 = . . . =  p H 1 :  j =  j ’. A.Answering General Versus Specific Research Questions 1. Population contrast,  i , and sample contrast.

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Chapter 15 Introduction to the Analysis of Variance IThe Omnibus Null Hypothesis

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Chapter 15 introduction to the analysis of variance i the omnibus null hypothesis

Chapter 15

Introduction to the Analysis of Variance

IThe Omnibus Null Hypothesis

H0: 1 = 2 = . . . = p

H1: j = j’


Chapter 15 introduction to the analysis of variance i the omnibus null hypothesis

A.Answering General Versus Specific Research

Questions

1. Population contrast, i, and sample contrast

2.Pairwise and nonpairwise contrasts


Chapter 15 introduction to the analysis of variance i the omnibus null hypothesis

B.Analysis of Variance Versus Multiple t Tests

1.Number of pairwise contrasts among p means

is given by p(p– 1)/2

p = 3 3(3 – 1)/2 = 3

p = 4 4(4 – 1)/2 = 6

p = 5 5(5 – 1)/2 = 10

2.If C = 3 contrasts among p = 3 means are tested

using a t statistic at  = .05, the probability of

one or more type I errors is less than


Chapter 15 introduction to the analysis of variance i the omnibus null hypothesis

3.As C increases, the probability of making one or

more Type I errors using a t statistic increases

dramatically.


Chapter 15 introduction to the analysis of variance i the omnibus null hypothesis

4.Analysis of variance tests the omnibus null

hypothesis, H0: 1 = 2 = . . . = p , and controls

probability of making a Type I error at, say,

 = .05 for any number of means.

5.Rejection of the null hypothesis makes the

alternative hypothesis, H1: j ≠ j’, tenable.


Chapter 15 introduction to the analysis of variance i the omnibus null hypothesis

IIBasic Concepts In ANOVA

A.Notation

1.Two subscripts are used to denote a score, Xij.

The i subscript denotes one of the i = 1, . . . , n

participants in a treatment level. The j subscript

denotes one of the j = 1, . . . , p treatment levels.

2. The jth level of treatment A is denoted by aj.


Chapter 15 introduction to the analysis of variance i the omnibus null hypothesis

a1 a2 a3 a4

X11 X12 X13 X14

X21 X22 X23 X24

Xn1 Xn2 Xn3 Xn4


Chapter 15 introduction to the analysis of variance i the omnibus null hypothesis

B.Composite Nature of a Score

1. A score reflects the effects of four variables:

independent variable

characteristics of the participants in the

experiment

chance fluctuations in the participant’s

performance

environmental and other uncontrolled

variables


Chapter 15 introduction to the analysis of variance i the omnibus null hypothesis

2.Sample model equation for a score

3.The statistics estimate parameters of the model

equation as follows


Chapter 15 introduction to the analysis of variance i the omnibus null hypothesis

4.Illustration of the sample model equation using the

weight-loss data in Table 1.

Table 1. One-Month Weight Losses for Three Diets

a1 a2 a3

71012

91311

8915

6714


Chapter 15 introduction to the analysis of variance i the omnibus null hypothesis

5.Let X11 = 7 denote Joan’s weight loss. She used

diet a1. Her score is a composite that tells a story.

6.Joan used a less effective diet than other girls

(8 – 9.67 = –1.67), and she lost less weight than

other girlson the same diet (8 – 9 = –1).


Chapter 15 introduction to the analysis of variance i the omnibus null hypothesis

C.Partition of the Total Sum of Squares (SSTO)

1.The total variability among scores in the diet

experiment

also is a composite that can be decomposed into

between-groups sum of squares (SSBG)

 within-groups sum of squares (SSWG)


Chapter 15 introduction to the analysis of variance i the omnibus null hypothesis

D.Degrees of Freedom for SSTO, SSBG, and

SSWG

1.dfTO = np – 1

2.dfBG = p – 1

3.dfWG = p(n – 1)

E.Mean Squares, MS, and F Statistic


Chapter 15 introduction to the analysis of variance i the omnibus null hypothesis

F.Nature of MSBG and MSWG

1.Expected value of MSBG and MSWG when the

null hypothesis is true.

2.Expected value of MSBG and MSWG when the

null hypothesis is false.


Chapter 15 introduction to the analysis of variance i the omnibus null hypothesis

3.MSBG represents variation among participants

who have been treated differently—received

different treatment levels.

4.MSWG represents variation among participants

who have been treated the same—received

the same treatment level.

5.F = MSBG/MSWG values close to 1 suggest that

the treatment levels did not affect the dependent

variable; large values suggest that the treatment

levels had an effect.


Chapter 15 introduction to the analysis of variance i the omnibus null hypothesis

IIICompletely Randomized Design (CR-p Design)

A.Characteristics of a CR-p Design

1.Design has one treatment, treatment A, with p

levels.

2.N = n1 + n2 + . . . + np participants are randomly

assigned to the p treatment levels.

3.It is desirable, but not necessary, to have the same

number of participants in each treatment level.


Chapter 15 introduction to the analysis of variance i the omnibus null hypothesis

B.Comparison of layouts for a t-test design for

independent samples and a CR-3 design

Participant1a1Participant1a1

Participant2a1Participant2a1

Participant10a1Participant10a1

Participant11a2Participant11a2

Participant12a2Participant12a2

Participant20a2Participant20a2

Participant21 a3

Participant22a3

Participant30 a3


Chapter 15 introduction to the analysis of variance i the omnibus null hypothesis

C.Descriptive Statistics for Weight-Loss Data

In Table 1

Table 2. Means and Standard Deviations for Weight-Loss Data

Diet

a1a2a3

8.009.0012.00

2.212.21 2.31


Chapter 15 introduction to the analysis of variance i the omnibus null hypothesis

Figure 1. Stacked box plots for the weight-loss data. The

distributions are relatively symmetrical and have similar

dispersions.


Chapter 15 introduction to the analysis of variance i the omnibus null hypothesis

Table 3. Computational Procedures for CR-3 Design

a1 a2 a3

71012

91311

8915

6714


Chapter 15 introduction to the analysis of variance i the omnibus null hypothesis

D.Sum of Squares Formulas for CR-3 Design


Chapter 15 introduction to the analysis of variance i the omnibus null hypothesis

Table 4. ANOVA Table for Weight-Loss Data

SourceSS df MS F

1.Between86.667p – 1 = 243.3348.60*groups (BG)

Three diets

2.Within 136.000p(n – 1) = 275.037

groups (WG)

3. Total222.667np – 1 = 29

*p < .002


Chapter 15 introduction to the analysis of variance i the omnibus null hypothesis

E.Assumptions for CR-p Design

1.The model equation,

reflects all of the sources of variation that affect

Xij.

2.Random sampling or random assignment

3.The j = 1, . . . , p populations are normally

distributed.

4.Variances of the j = 1, . . . , p populations are

equal.


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