Anova and Design of Experiments

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Chapter 11

ANOVA AND DESIGN OF

EXPERIMENTS

1



General Experimental Setting

Investigator Controls One or More Independent

Variables

± Called treatment variables or factors

± Each treatment factor contains two or more groups (or

levels)

Observe Effects on Dependent Variable

± Response to groups (or levels) of independent variable

Experimental Design: The Plan Used to Test

Hypothesis

2



Design of Experiments

Completely randomized design (one way ANOVA)

Randomized block design

Factorial design ( Two way ANOVA)

3



Completely Randomized Design

Experimental Units (Subjects) are Assigned

Randomly to Groups

± Subjects are assumed homogeneous Only One Factor or Independent Variable

± With 2 or more groups (or levels)

Analyzed by One-way Analysis of Variance(ANOVA)

4



ANOVA: Assumptions

ANOVA is a parametric statistic. So, it makes

assumptions about particular parameters of the

population:

the treatment groups are independent

samples are selected at random

the populations from which the groups were

sampled are normally distributed the variances of each sample are roughly equal, or

homogeneous.

5



Factor (Training Method)

Factor Levels

(Groups)Randomly

Assigned

Units

Dependent

Variable

(Response)

21 hrs 17 hrs 31 hrs



Randomized Design Example

Training example

6



Example

Comparing magazine covers

A magazine publisher wants to compare threedifferent styles of covers for a magazine thatwill be offered for sale at supermarketcheckout lines. She assigns 60 stores atrandom to the three styles of covers andrecords the number of

magazines that are sold in a one-week period.

7



Example

Design 2 had the highest average sales

Purpose of ANOVA is to see if this observeddifference is statistically significant

8



Hypotheses of One-Way ANOVA

All population means are equal

± No treatment effect (no variation in means amonggroups)

± At least one population mean is different (others may be

the same!)

± There is a treatment effect

± Does not mean that all population means are different

9

0 1 2:c

H Q Q Q! ! !L

1 : Not all are the samei H Q



One-way ANOVA

(No Treatment Effect)

10

The Null

Hypothesis is

True

0 1 2:c

H Q Q Q! ! !L

1 : ot all are the samei H Q

1 2 3 Q Q Q! !



One-way ANOVA

(Treatment Effect Present)

11

The Null

Hypothesis isNOT True

0 1 2:c

H Q Q Q! ! !L

1 : Not all are the samei H Q

1 2 3 Q Q Q! {1 2 3 Q Q Q{ {



One-way ANOVA

(Partition of Total Variation)

Commonly referred to as:

Within Group Variation

Sum of Squares Within

Sum of Squares Error

Sum of Squares Unexplained

12

Variation Due to

Treatment SSCVariation Due to Random

Sampling SSE

Total Variation SST

Commonly referred to as:

Among Group Variation

Sum of Squares Among

Sum of Squares Between

Sum of Squares Factor

Sum of Squares Explained

Sum of Squares Treatment

= +



Total Variation

13

2

1 1

1 1

( )

: the -th observation in group: the number of observations in group

: the total number of observations in all groups

: the number of groups

the over

j

j

nc

i j

j i

i j

j

nc

i j

j i

SST X X

X i jn j

n

c

X

X

n

! !

! !

!

!

§ §

§ §all or grand mean



Total Variation

14

(continu ed)

2 2 2

11 21 cn cSS ! L

X

Response, X

Group 1 Group 2 Group 3



Among-Group Variation (Factor)

15

Variation Due to Differences Among Groups.

1

SSAM SA

c!

i Q j Q

: The sample mean o group

: The overall or grand mean

j X j

X

2

1

( )c

j j

j

SSA n X X

!

! §



Among-Group Variation

16

(continu ed)

2 2 2

1 1 2 2 c cSS n X X n X X n X X ! L

X 1 X 2 X

3 X

Response, X




Summing the variation

within each group and then

adding over all groups.

: The sample mean o group

: The -th observation in group

j

i j

X j

X i j

Within-Group Variation (Error)

17

2

1 1

( ) jnc

i j j

j i

SSW X X ! !

! § §SSW

MSW n c

!

j Q



Within-Group Variation

18

(continu ed)

22 2

11 1 21 1c

n c cSSW X X X X X X ! L

1 X 2 X

3 X

X

Response, X




Within-Group Variation

19

(continu ed)

2 2 2

1 1 2 2

1 2

( 1) ( 1) ( 1)( 1) ( 1) ( 1)

c c

c

SSW SW

n c

n S n S n S n n n

!

y y y ! y y y

F or c = 2, this is the

pooled-variance in the

t-Test.

If more than 2 groups,

use F Test.F or 2 groups, use t-Test.

F Test more limited.

j Q



One-way ANOVA

Summary Table

Source

of

Variation

Degrees

of

Freedo

m

Sum of

Squares

Mean

Squares

(Variance)

F

Statistic

Among

(Factor)c ± 1 SSA

MSA =

SSA/(c ± 1 )MSF/MSE

Within

(Err or)

n ± c SSW MSE =

SSE/(n ± c )

Total n ± 1SST =

SSA+ SSW

20



One-way Analysis of Variance

F Test Evaluate the Difference among the Mean Responses of 2 or

More (c ) Populations

± E.g. Several types of tires, oven temperature settings

Assumptions

± Samples are randomly and independently drawn

This condition must be met

± Populations are normally distributed

F Test is robust to moderate departure from normality

± Populations have equal variances Less sensitive to this requirement when samples are

of equal size from each population

21



One-way ANOVA

F Test Statistic

Test Statistic

± = MSA/MSE

MSA is mean squares among

MSW is mean squares within

Degrees of Freedom ±

±

22

M S F

M SW

!

1 1df c!

2df n c!



Features of One-way ANOVA

F Statistic The F Statistic is the Ratio of the Among

Estimate of Variance and the Within Estimate

of Variance

± The ratio must always be positive

23



Features of One-way ANOVA

F Statistic

If the Null Hypothesis is True

± df 1 = c -1 will typically be small

± df 2 = n - c will typically be large ± The ratio should be close to 1.

If the Null Hypothesis is False

± The numerator should be greater than the

denominator

± The ratio should be larger than 1

24

(continu ed)



25

Factor (Training Method)

Factor Levels

(Groups)

Randomly

Assigned

Units

Dependent

Variable

(Response)




Tr aining example



One-way ANOVA F Test

Example

As production manager, you want

to see if 3 machines have different

mean running times. You assign

15 similarly trained and

experienced workers, 5 per

machine, to the machines. At the

.05 significance level, is there a

difference in mean running times?

26

Machine1 Machine2 Machine3

25.40 23.40 20.00

26.

31 21.80 22

.2024.10 23.50 19.75

23.74 22.75 20.60

25.10 21.60 20.40



One-way ANOVA Example: Scatter

Diagram

27

27

26

25

24

23

22

21

20

19


25.40 23.40 20.00

26.31 21.80 22.20

24.10 23.50 19.7523.74 22.75 20.60

25.10 21.60 20.40

1 2

3

24.93 22.61

20.59 22.71

X X

X X

! !

! !

1 X

2 X

3 X

X



One-way ANOVA Example

Computations

28

Machine1 Machine2Machine3

25.40 23.40 20.00

26.31 21.80 22.20

24.10 23

.50 19

.75

23.74 22.75 20.60

25.10 21.60 20.40

2 2 2

5 24.93 22.71 22.61 22.71 20.59 22.71

47.164

4.2592 3.112 3.682 11.0532

/( -1) 47.16 / 2 23.5820

/( - ) 11.0532 /12 .9211

SSA

SSW

MSA SSA c

MSW SSW nc

« »! - ½

!! !

! ! !

! ! !

1

2

3

24.93

22.61

20.5922.71

X

X

X

X

!

!

!!

5

3

15

jn

c

n

!

!

!



Summary Table

29

Source of

Variation

Degrees

of

Freedom

Sum of

Squares

Mean

Squares

(Variance)

F

Statistic

Among

(Factor)3-1=2 47.1640 23.5820

MS A/MSW

=25.60

Within

(Err or) 15-3=

12 11.0532

.9211

Total 15-1=14 58.2172



One-way ANOVA Example Solution

H0: Q1= Q2 = Q3

H1: Not All Equal

E = .05df 1= 2 df 2 = 12

Critical Value(s):

30

F0 3.89

Test Statistic:

Decision:

Conclusion:

Reject at E = 0.05

There is evidence that at least

one Q i diff ers f r om the rest.

E = 0.05

Ftest MSA

MSE ! ! !

23 5820

9211 25 6

.

..



Multiple Comparison Procedures

If H0 is rejected, we conclude that the means

are different

We would like to know exactly where themeans differ

This is where Multiple Comparison Procedures

come into play.

31



Post hoc tests (such as Tukeys HSD test and

Scheffes tests) are used to make pair-by-pair

comparisons to pinpoint significant

differences.

A priori comparisons: Similar to post hoc tests,

but used only when you planned the

comparison between groups before collectingthe data.

32



TUKEYS HSD TEST

Tukeys Honestly Significant Difference test:

Limitation- it requires equal sample sizes.

HSD n

HSD = qE ,c ,N-c MSE

n

qE ,c ,N-c=critical value o f the stud entised ranged istribution (Page no.783)

33



The Tukey-Kramer Procedure

Tells which Population Means are Significantly Different

± e.g., Q1= Q2 { Q3

± 2 groups whose meansmay be significantlydifferent

Post Hoc (a posteriori) Procedure

± Done after rejection of equal means in ANOVA

Pair wise Comparisons

± Compare absolute mean differences with critical range

34

X

f (X)

Q1= Q 2

Q3



The Tukey-Kramer Procedure: Example

1. Compute absolute meandifferences:

35


25.40 23.40 20.00

26.31 21.80 22.20

24.10 23.50 19.7523.74 22.75 20.60

25.10 21.60 20.40

1 2

1 3

2 3

24.93 22.61 2.32

24.93 20.59 4.34

22.61 20.59 2.02

X X

X X

X X

! !

! ! ! !

2. Compute Critical Range:

3. All of the absolute mean diff erences are greater than thecritical range. There is a signif icance diff erence betweeneach pair of means at the 5% level of signif icance.

( , )

'

1 1Critical Range 1.6182

U c n c

j j

MSW Qn n

¨ ¸! !© ¹© ¹

ª º

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Anova and Design of Experiments