ANALYSIS OF ANALYSIS OF VARIANCEVARIANCE(ANOVA)(ANOVA)BCT 2053CHAPTER 5

CONTENT

5.1 Introduction to ANOVA

5.2 One-Way ANOVA

5.3 Two-Way ANOVA

5.1 Introduction to ANOVA

OBJECTIVESAfter completing this chapter you should be able to:

1. Explain the purpose of ANOVA 2. Identify the assumptions that underlie the ANOVA technique 3. Describe the ANOVA hypothesis testing procedure

What is ANALYSIS OF VARIANCE (ANOVA)

the approach that allows us to use sample data to see if the values of three or more unknown population means are likely to be different

Also known as factorial experiments

this name is derived from the fact that in order to test for statistical significance between means, we are actually comparing (i.e., analyzing) variances. (so F-distribution will be used)

Example of problems Example of problems involving ANOVAinvolving ANOVA

A manager want to evaluate the performance of three (or more) employees to see if any performance different from others.

A marketing executive want to see if there’s a difference in sales productivity in the 5 company region.

A teacher wants to see if there’s a difference in student’s performance if he use 3 or more approach to teach.

The Procedural Steps for an ANOVA Test

1. State the Null and Alternative hypothesis2. Select the level of significance, α3. Determine the test distribution to use - Ftest

4. Compute the test statistic 5. Define rejection or critical region – Ftest > Fcritical

6. State the decision rule7. Make the statistical decision - conclusion

5.2 One-Way ANOVA

OBJECTIVEAfter completing this chapter you should be able to:

1. Use the one-way ANOVA technique to determine if there is a significance difference among three or more means

One-Way ANOVA Design Only one classification factor (variable) is

considered

Factor

Treatment12

i

(The level of the factor)

Response/ outcome/ dependent variable

(samples)

Replicates (1,… j) The object to a

given treatment

The resulting input grid of factorial experiment

where,  i = 1, 2, … a is the number of levels being tested.j = 1, 2, … ni is the number of replicates at each level. 

AssumptionsAssumptions

To use the one-way ANOVA test, the following assumptions must be true

The population under study have normal distribution.

The samples are drawn randomly, and each sample is independent of the other samples.

All the populations from which the samples values are obtained, have the same unknown population variances, that is for k number of populations,

2 2 21 2 k

The Null and Alternative hypothesis

1 2:o kH (All population means are equal)

1 : fori jH i j (Not all population means are equal)

2 2 21 2 1 2 and k k

If Ho is true we have k number of normal populations with

2 2 21 2 3 1 2 3 but if 3k

If H1 is true we may have k number of normal populations with

Or H1: At least one mean is different from others

Source of

Variation

Degrees of

Freedom (Df)

Sum of Squares (SS) Mean of Squares (MS) F Value

Between sample(Factor

Variation) k - 1

Within samples

(Error variation)

T - k

Total T - 1

The format of a general one-way ANOVA table

2 2..

1 1

1k n

iji j

SST x xN

2/ ( )

( )1

Treatment Factor MS TrSS Tr

k

2/

2

test

Treatment Factor

Error

F

2Error

SSEMSEN k

( )SSE SST SS Tr

2 2. ..

1

1 1( )k

ii

SS Tr x xn N

Reject Ho if , 1,test k T kF F T = k n

Example 1 The data shows the Math’s test score for 4 groups of

student with 3 different methods of study. Test the hypothesis that there is no difference between the Math’s score for 4 groups of student at significance level 0.05.

Score

Individually & Group study 80 70 85 89

Group study 60 55 58 62

Individually 65 60 62 58

Example 2 An experiment was performed to determine whether the annealing

temperature of ductile iron affects its tensile strength. Five specimens were annealed at each of four temperatures. The tensile strength (in ksi) was measured for each temperature. The results are presented in the following table. Can you conclude that there are differences among the mean strengths at α = 0.01?

Temperature(oC)

Sample Values

750 19.72 20.88 19.63 18.68 17.89

800 16.01 20.04 18.10 20.28 20.53

850 16.66 17.38 14.49 18.21 15.58

900 16.93 14.49 16.15 15.53 13.25

Example 3 Three random samples of times (in minutes) that commuters are

stuck in traffic are shown below. At α = 0.05, is there a difference in the mean times among the three cities?

Eastern Third Middle Third Western Third

48 95 29

57 52 40

24 64 68

10 64

38

Solve one-way ANOVA by EXCEL Excel – key in data (Example 1)

Solve one-way ANOVA by EXCEL

Tools – Add Ins – Analysis Toolpak – Data Analysis – ANOVA single factor – enter the data range – set a value for α - ok

Reject H0 if P-value ≤ α or F > F crit P-value < 0.05 so Reject H0

4.2 Two-Way ANOVA

OBJECTIVEAfter completing this chapter you should be able to:

1. Use the two-way ANOVA technique to determine if there is an effect of interaction between two factors experiment

Two-Way ANOVA Design Two classification factor is considered

Factor Bj = 1 2 b

Factor Ai = 1 k = 1,…n

2

a

Example A researcher whishes to test the effects of two different types of

plant food and two different types of soil on the growth of certain plant.

Some types of two way ANOVA design

B1 B2

B1 B2 B3 B1 B2 B3

B1 B2

A1

A2

A1

A2

A3

A1

A2

A3

A1

A2

A3

A4

AssumptionsAssumptions The standard two-way ANOVA tests are valid under the following

conditions:

The design must be complete • Observations are taken on every possible treatment

The design must be balanced• The number of replicates is the same for each treatment

The number of replicates per treatment, k must be at least 2

Within any treatment, the observations are a simple random sample from a normal population

The sample observations are independent of each other (the samples are not matched or paired in any way)

The population variance is the same for all treatments.

1, ,ij ijkx x

Null & Alternative Hypothesis

H0: there is no interaction effect between factor A and factor B.H1: there is an interaction effect between factor A and factor B.

H0: there is no difference in means of factor A.H1: there is a difference in means of factor A.

H0: there is no difference in means of factor B.H1: there is a difference in means of factor B.

interaction effect

Column effect

Row effect

H0: there is no effect from factor A.H1: there is effect from factor A.

H0: there is no effect from factor B.H1: there is effect from factor B.

or

or

Source (Df) Sum of Squares (SS) Mean of Squares (MS)

F Value

A(row effect)

a - 1

B(Column effect)

b - 1

Interaction(interaction

effect)(a-1)(b-1)

Error ab(n-1)

Total abn-1

The format of a general two-way ANOVA table

22 ....

1 1

1

a b

iji j

xSSAB xn abn

SSA SSB

22 .... .

1

1 b

jj

xSSB x

an abn

22 ...

1 1 1

a b n

ijki j k

xSST xabn

1 1SSABMSAB

a b

testMSAFMSE

1SSEMSE

ab n

SSE SST SSA

SSB SSAB

22 .....

1

1 a

ii

xSSA xbn abn

1

SSAMSAa

1SSBMSBb

test

MSBFMSE

testMSABFMSE

Reject if

, 1, 1

test

a ab n

FF

,( 1)( 1), 1

test

a b ab n

FF

, 1, 1

test

b ab n

FF

Procedure for Two-Way ANOVASTART

Test for an interaction between the two

factors

Is there an effect due to interaction between the

two factors?

Stop. Don’t consider the effects of either factor

without considering the effects of the other

Test for effect from column factor

Test for effect from row factor

Yes

(Reject Ho)

No (Accept Ho)

Ho: No interaction

between two factors

Ho: No effects from the column factor B (the column means are equal)

Ho: No effects from the row factor A (the row means are equal)

testMSAFMSE

testMSBFMSE

testMSABFMSE

Example 1 A chemical engineer is studying the effects of various reagents and catalyst on

the yield of a certain process. Yield is expressed as a percentage of a theoretical maximum. 2 runs of the process were made for each combination of 3 reagents and 4 catalysts.

CatalystReagent

1 2 3

A 86.8 82.4 93.4 85.2 77.9 89.6 B 71.9 72.1 74.5 87.1 87.5 82.7 C 65.5 72.4 66.7 77.1 72.7 77.8D 63.9 70.4 73.7 81.6 79.8 75.7

a) Construct an ANOVA table.b) Test is there an interaction effect between reagents and catalyst. Use α = 0.05.c) Do we need to test whether there is an effect that is due to reagents or

catalyst? Why? If Yes, test is there an effect from reagents or catalyst.

Example 2 A study was done to determine the effects of two factors on the lather

ability of soap. The two factors were type of water and glycerol. The outcome measured was the amount of foam produced in mL. The experiment was repeated 3 times for each combination of factors. The result are presented in the following table..

Water type Glycerol Foam (mL)De-ionized Absent 168 178 168

De-ionized Present 160 197 200

Tap Absent 152 142 142

Tap Present 139 160 160

Construct an ANOVA table and test is there an interaction effect between factors. Use α = 0.05.

Solve two-way ANOVA by EXCEL Excel – key in data

Solve two-way ANOVA by EXCEL

tools – Data Analysis – ANOVA two factor with replication – enter the data range – set a value for α - ok

Reject H0 if P-value ≤ α or F > F crit

Summary

The other name for ANOVA is experimental design.

ANOVA help researchers to design an experiment properly and analyzed the data it produces in correctly way.

Thank You