Chapter11 - Analysis of Variance

(1)

Statistics for Managers

Using Microsoft® Excel

5th Edition

Chapter 11

(2)

Learning Objectives

In this chapter, you learn:



_{The basic concepts of experimental design}



_{How to use the one-way analysis of variance}

to test for differences among the means of

several groups



_{How to use the two-way analysis of variance}

(3)

Chapter Overview

Analysis of Variance (ANOVA)

F-test

Tukey-Kramer

test

One-Way

ANOVA

Two-Way

ANOVA

(4)

General ANOVA Setting



_{Investigator controls one or more factors of interest}



_{Each factor contains two or more levels}



_{Levels can be numerical or categorical}



_{Different levels produce different groups}



_{Think of the groups as populations}



_{Observe effects on the dependent variable}



_{Are the groups the same?}



_{Experimental design: the plan used to collect the}

(5)

Completely Randomized

Design



_{Experimental units (subjects) are assigned}

randomly to the different levels (groups)



_{Subjects are assumed homogeneous}



_{Only one factor or independent variable}



_{With two or more levels (groups)}



_{Analyzed by one-factor analysis of variance}

(6)

One-Way Analysis of Variance



_{Evaluate the difference among the means of three or}

more groups

Examples: Accident rates for 1st, 2nd, and 3rd shift

Expected mileage for five brands of tires



_Assumptions



_{Populations are normally distributed}



_{Populations have equal variances}

(7)

Hypotheses: One-Way ANOVA



_{All population means are equal}



_{i.e., no treatment effect (no variation in means among groups)}



_{At least one population mean is different}



_{i.e., there is a treatment (groups) effect}



_{Does not mean that all population means are different (at}

least one of the means is different from the others)

(8)

Hypotheses: One-Way

ANOVA

All Means are the same:

The Null Hypothesis is True

(No Group Effect)

c

3

2

1

0 :

μ

H

_



_

same

the

are

μ

all

Not

:

H

₁

_j

3

2

1 μ

μ

(9)

Hypotheses: One-Way

ANOVA

At least one mean is different:

The Null Hypothesis is NOT true

(Treatment Effect is present)

c

3

2

1

0 :

μ

H

_



_

same

the

are

μ

all

Not

:

H

₁

j

3

2

1 μ

μ

_

μ

₁

_

μ

₂

_

μ

₃

(10)

Partitioning the Variation



_{Total variation can be split into two parts:}

SST = Total Sum of Squares

(Total variation)

SSA = Sum of Squares Among Groups

(Among-group variation)

SSW = Sum of Squares Within Groups

(Within-group variation)

(11)

Partitioning the Variation

Total Variation = the aggregate dispersion of the individual data

values around the overall (grand) mean of all factor levels (SST)

Within-Group Variation = dispersion that exists among the data

values within the particular factor levels (SSW)

Among-Group Variation = dispersion between the factor

sample means (SSA)

(12)

Partitioning the Variation

Among Group

Variation (SSA)

Within Group Variation

(SSW)

Total Variation (SST)

(13)

The Total Sum of Squares

SST = Total sum of squares

c = number of groups

n

_j

= number of values in group j

X

_ij

= i

th

_{value from group j}

X = grand mean (mean of all data values)

(14)

The Total Sum of Squares

(15)

Among-Group Variation

Where:

SSA = Sum of squares among groups

c = number of groups

n

_j

= sample size from group j

X

_j

= sample mean from group j

X = grand mean (mean of all data values)

2

1 )

(

X

n

SSA

c

_j

j



(16)

Among-Group Variation

2 j

c

1 j

j

(

X

)

n

SSA

_



2 c

c

2

1

1 (

X

)

n

(

X

)

...

n

(

X

)

n

SSA

_

(17)

Within-Group Variation

Where:

SSW = Sum of squares within groups

c = number of groups

(18)

Within-Group Variation

(19)

Obtaining the Mean Squares

c

n

SSW

MSW



1 c

SSA

MSA



1 n

SST

MST



Mean Squares Among

Mean Squares Within

(20)

One-Way ANOVA Table

Source of

Variation

df

SS

MS

_(Variance)

F-Ratio

Among

Groups

c-1

SSA

MSA

Within

Groups

n-c

SSW

MSW

Total

n-1

SST =

SSA+SSW

c = number of groups

n = sum of the sample sizes from all groups

df = degrees of freedom

(21)

One-Way ANOVA

Test Statistic



_{Test statistic}



_MSA

_{is mean squares among variances}



_MSW

_{is mean squares within variances}



_{Degrees of freedom}



_df

₁

_{= c – 1 (c = number of groups)}



_df

₂

_{= n – c (n = sum of all sample sizes)}

MSW

MSA

F

_

H

₀

: μ

₁

= μ

₂

= …

= μ

_c

(22)

One-Way ANOVA

Test Statistic



_{The F statistic is the ratio of the among variance to the}

within variance



_{The ratio must always be positive}



_df

₁

₌

_c

_{-1 will typically be small}



_df

₂

₌

_n

_-

_c

_{will typically be large}

Decision Rule:

Reject H

₀

if F > F

_U

,

otherwise do not reject H

₀

0 

= .05

Reject H

₀

Do not

reject H

₀

_F

(23)

One-Way ANOVA

Example

Club 1

Club 2 Club 3

254 234 200

263 218 222

241 235 197

237 227 206

251 216 204

You want to see if three

different golf clubs yield

different distances. You

randomly select five

measurements from trials on an

automated driving machine for

each club. At the .05

(24)

(25)

One-Way ANOVA

Example

X

₁

= 249.2

X

₂

= 226.0

X

₃

= 205.8

X = 227.0

n

₁

= 5

n

₂

= 5

n

₃

= 5

n = 15

c = 3

SSA = 5 (249.2 – 227)

2 _{+ 5 (226 – 227)}

2 _{+ 5 (205.8 – 227)}

2 _{= 4716.4}

(26)

One-Way ANOVA

Example

MSA = 4716.4 / (3-1) = 2358.2

MSW = 1119.6 / (15-3) = 93.3

93.3

25.275 2358.2

F

_

F

= 25.275

0 

= .05

F

_U

= 3.89

Reject H

₀

Do not

reject H

₀

Critical

Value:

(27)

One-Way ANOVA

Example



_H

₀

_{: μ}

₁

_{= μ}

₂

_{= μ}

₃



_H

₁

_{: μ}

_j

_{not all equal}



_

_{= .05}



_df

₁

_{= 2 df}

₂

_{= 12}

Decision:

Reject H

₀

at α = 0.05

Conclusion: There is

evidence that at least one

(28)

One-Way ANOVA in Excel

EXCEL:

single

factor

SUMMARY

Groups

Count

Sum

Average

Variance

Club 1

5 1246

249.2

108.2 Club 2

5 1130

226

77.5 Club 3

5 1029

205.8

94.2 ANOVA

Source of

Variation

SS

df

MS

F

P-value

F crit

Between

Groups

4716.4

2 2358.2

25.275 4.99E-05

3.89 Within

Groups

1119.6

12

93.3

(29)

The Tukey-Kramer Procedure



_{Tells which population means are significantly different}



_{e.g.: μ}

₁

_{= μ}

₂

_{≠ μ}

₃



_{Done after rejection of equal means in ANOVA}



_{Allows pair-wise comparisons}



_{Compare absolute mean differences with critical}

range

x

(30)

Tukey-Kramer Critical Range

where:

Q

_U

= Value from Studentized Range Distribution with c

and n - c degrees of freedom for the desired level

of

_

(see appendix E.9 table)

MSW = Mean Square Within

n

_j

and n

_j’

= Sample sizes from groups j and j’

(31)

The Tukey-Kramer Procedure

1. Compute absolute mean differences:

(32)

The Tukey-Kramer Procedure

5. All of the absolute mean differences are greater than

critical range. Therefore there is a significant difference

between each pair of means at the 5% level of significance.

16.285 3. Compute Critical Range:

20.2

(33)

ANOVA Assumptions



_{Randomness and Independence}



_{Select random samples from the c groups (or}

randomly assign the levels)



_Normality



_{The sample values from each group are from a}

normal population



_{Homogeneity of Variance}

(34)

ANOVA Assumptions

Levene’s Test



_{Tests the assumption that the variances of each}

group are equal.



_{First, define the null and alternative hypotheses:}



_H

₀

_{: σ}

₂₁

_{= σ}

₂₂

_{= …=σ}

_2c



_H

₁

_{: Not all σ}

_2j

_{are equal}



_{Second, compute the absolute value of the difference}

between each value and the median of each group.



_{Third, perform a one-way ANOVA on these}

(35)

Two-Way ANOVA



_{Examines the effect of}



_{Two factors of interest on the dependent}

variable



_{e.g., Percent carbonation and line speed on soft}

drink bottling process



_{Interaction between the different levels of these}

two factors



_{e.g., Does the effect of one particular carbonation}

(36)

Two-Way ANOVA



_Assumptions



_{Populations are normally distributed}



_{Populations have equal variances}

(37)

Two-Way ANOVA

Sources of Variation

Two Factors of interest: A and B

r = number of levels of factor A

c = number of levels of factor B

n

/

= number of replications for each cell

n = total number of observations in all cells

(n = rcn

/

)

(38)

Two-Way ANOVA

Sources of Variation

SST

Total Variation

SSA

Factor A Variation

SSB

Factor B Variation

SSAB

Variation due to interaction

between A and B

SSE

Random variation (Error)

Degrees of

Freedom:

r – 1

c – 1

(r – 1)(c – 1)

rc(n

/

_{– 1)}

n - 1

(39)

Two-Way ANOVA

Total Variation:

Factor A Variation:

(40)

Two-Way ANOVA

Equations

Interaction Variation:

(41)

Two-Way ANOVA

(42)

Two-Way ANOVA

Equations

factor

of

(43)

(44)

Two-Way ANOVA:

The F Test Statistic

F Test for Factor B Effect

F Test for Interaction Effect

H

₀

: μ

_1..

= μ

_2..

=

• • •

= μ

_r..

H

₁

: Not all μ

_i..

are equal

H

₀

: the interaction of A and B is

equal to zero

H

₁

: interaction of A and B isn’t zero

F Test for Factor A Effect

(45)

Two-Way ANOVA:

Summary Table

Source of

Variation

Degrees of

Freedom

Squares

Sum of

Squares

Mean

Statistic

F

Factor A

r – 1

SSA

_{= SSA}

MSA

_{/(r – 1)}

MSA/

MSE

Factor B

c – 1

SSB

_{= SSB}

MSB

_{/(c – 1)}

MSB/

MSE

AB

(Interaction) (r – 1)(c – 1)

SSAB

MSAB

= SSAB

/ (r – 1)(c – 1)

MSAB/

MSE

Error

rc(n

’

– 1)

SSE

MSE

=

SSE/rc(n’ – 1)

(46)

Two-Way ANOVA:

Features



_{Degrees of freedom always add up}



_{n-1 = rc(n}

/

-1) + (r-1) + (c-1) + (r-1)(c-1)



_{Total = error + factor A + factor B + interaction}



_{The denominator of the F}

_{Test is always the same}

but the numerator is different



_{The sums of squares always add up}



_{SST = SSE + SSA + SSB + SSAB}

(47)

Two-Way ANOVA:

Interaction



_{No Significant Interaction:}

Factor B Level 1

Factor B Level 3

Factor B Level 2

Factor A Levels

Factor B Level 1

Factor B Level 3

Factor B Level 2

Factor A Levels

M

(48)

Chapter Summary



_{Described one-way analysis of variance}



_{The logic of ANOVA}



_{ANOVA assumptions}



_{F test for difference in c means}



_{The Tukey-Kramer procedure for multiple comparisons}



_{Described two-way analysis of variance}



_{Examined effects of multiple factors}



_{Examined interaction between factors}