Analysis of Variance

Outlines: Designing Engineering Experiments Completely Randomized Single-Factor

Experiment Random Effects Model Randomized Complete Block Design

Designing Engineering Experiments Factor: Parameter of interest Levels: Possible value of Factors. Analysis of Variance (ANOVA): Analysis the effects of

different factor levels to the response. Randomization: Random running order. Controllable variables: Other parameters that involve in the

experiment Experiment activities:

1. Conjecture

2. Experiment

3. Analysis

4. Conclusion

Completely Randomized Single-Factor Experiment Ex: A manufacturer of paper is interested in improving the tensile

strength of the product. The product engineer thinks that tensile strength is a function of the hardwood concentration in the paper. The range of hardwood concentrations of practical interest is between 5 and 20% . The engineers design to investigate four levels of hardwood concentration: 5% 10% 15% and 20%. They design to make up six test specimens at each concentration level.

Completely Randomized Single-Factor Experiment Randomizing order: randomly select the order for each run

to reduce the effect of nuisance variable such as the warm-up effect.

Box plot: Represent the variability within a treatment and the variability between treatments.

Analysis of Variance

Typical data for single-factor experiment

Linear Statistical Model

For each treatment

nj

aiEY ijiij ,...,2,1

,...,2,1

iiijiij wherenj

aiEY

,,...,2,1

,...,2,1For each

treatment, yij is normal

distribution with µi and

Eij has a normal distribution with mean 0 and sd

=

Analysis of Variance

We are interested in the equality of the a treatment means µ1, µ2,…, µa

If H0 is true, each observation consists of the overall mean µ plus the random error Eij => Changing the level of the factor has no effect on the mean response.

ANOVA partitions the total variability in the sample data into two components 1. The variation between treatments 2. The variation within treatment

0:1

0...: 3210

i

a

H

H

If there are no differences between

treatments => 1=2

Analysis of Variance

Total variation is described by the total sum of squares (SST)

Analysis of Variance

Degree of freedom

Mean square for treatment

Mean square for error

To verify hypothesis, we compare By using F test statistic

)1(11 naaan

SSSSSS EtreatmentT

)1/( aSSMS treatmenttreatment

)]1(/[ naSSMS EE

Etreatment MsandMS

Analysis of Variance

If H0 is reject => MStreatment > MSE => H1 should be upper-tail test

Reject H0 if Computing formulas for ANOVA

ANOVA Table

)1(,1,0 naaff

Analysis of Variance Ex. Consider the tensile strength. We can use ANOVA to

test the hypothesis that different hardwood concentrations do not affect the mean tensile strength of the paper. Use α=0.01

0:

0:

1

43210

iH

H

For at least one i

Analysis of Variance

f0 = 19.60 >f0.01,3,20 =4.94 , reject H0

Conclusion: hardwood concentration significantly affects the mean strength of the paper

Analysis of Variance

Confidence interval on the mean of the ith treatment

Ex. Find the 95% CI of the mean strength of 20% hardwood concentration?

34.2319

6/51.6)086.2(17.216/51.6)086.2(17.21

4

4

20,025.0.4420,025.0.4

n

MSty

n

MSty EE

Analysis of Variance

Confidence interval on a difference in treatment means

Ex. Find a 95% CI on the difference in means µ3-µ2

Multiple comparisons following ANOVA When is rejected in ANOVA, we know

that some of the treatment are different. ANOVA doesn’t identify which means are different. Methods for investigating this issue is called multiple

comparisons methods. Fisher LSD : compares all pairs of the means (µi and µj)

H0: µi = µj with test-statistic

The pair of means µi and µj would be declared significantly different if

0...: 3210 aH

nMS

yyt

E

ji

2..

0

LSDyy ji || ..

Multiple comparisons following ANOVA Ex. Apply the Fisher LSD to the hardwood concentration

experiment. a= 4 levels, n=6 replicates, and t0.025,20 = 2.086

The treatment means are: The value of LSD, Compare the difference for every pairs of treatments and

LSD,

psiypsiypsiypsiy 17.21,00.17,67.15,00.10 .4.3.2.1

07.36/)51.6(2086.2/220,025.0 nMStLSD E

07.367.500.1067.151.2

*07.333.167.1500.172.3

07.3700.1000.171.3

07.317.400.1717.213.4

07.350.567.1517.212.4

07.317.1100.1017.211.4

vs

vs

vs

vs

vs

vs

0 5 10 15 20 25

5% 10% 15% 20%

Model Adequacy Checking

Residual Analysis and Model Checking Residual VS time: Test independence assumption of

error Residual VS fitted values: Test constant variance

assumption of error Normality Plot: test the normal distribution

assumption of error.

Randomized Complete Block Design An extension of the paired t-test but with more than 2

treatments. Reduce the nuisance factor. Ex. 3 methods could be used to evaluate the strength

reading on steel plate girders. If there are 4 plates and each plate is large enough to hold all the treatment, the experimental design would be appear as Figure.

Randomized Complete Block Design

ANOVA Sums of square

Randomized Complete Block Design Computing Formulas for ANOVA randomized block

Randomized Complete Block Design ANOVA for a Randomized Complete Block Design

Randomized Complete Block Design Ex. Fabric Strength Data from a randomized complete

block design can be shown in table. We want to test the effect of chemical type to the strength of fabric by using α=0.01

H0:all types provide identical strengthH1: not equal strength

Randomized Complete Block Design

f0=75.13 > f0.01,3,12 =5.95, reject Ho and conclude that there is a significant difference in the chemical types

Randomized Complete Block Design Multiple comparison Similar to simple ANOVA but

Ex. Refer to previous example, use Fisher’s LSD method to analyze the difference between each pair of treatment.

b

MStLSD E

ba

2)1)(1(,2/

39.05

)08.0(2179.2

5

)08.0(2)4*3(,025.0 tLSD

type 4 results in significantly different strengthsthan the other three types . types 2 and 3 do not differ, and types 1 and 3 do notdiffer. There may be a small difference in strength between types 1 and 2.

Exercise

1. A civil engineer is interested in determining whether four different methods of estimating flood flow frequency produce equivalent estimates of peak discharge when applied to the same watershed. Each procedure is used six times on the watershed and the resulting discharge data are shown in table

Estimation method

Observation

1 0.34 0.12 1.23 0.70 1.75 0.12

2 0.91 2.94 2.14 2.36 2.86 4.55

3 6.31 8.37 9.75 6.09 9.82 7.24

4 17.15

11.82

10.95 17.20 14.35 16.82

Exercise

2. An experiment was conducted to investigate leaking current in a SOS MOSFETS device. The purpose of the experiment was to investigate how leakage current varies as the channel length changes. Four channel lengths were selected. For each channel length, five different widths were also used, and width is to be considered a nuisance factor. The data are as follows:

Exercise

3. An article in the American Industrial Hygiene Association Journal (Vol. 37, 1976, pp. 418–422) describes a field test for detecting the presence of arsenic in urine samples. The test has been proposed for use among forestry workers because of the increasing use of organic arsenics in that industry. The experiment compared the test as performed by both a trainee and an experienced trainer to an analysis at a remote laboratory. Four subjects were selected for testing and are considered as blocks. The response variable is arsenic content (in ppm) in the subject’s urine. The data are as follows:

Exercise

4. An article in the Food Technology Journal (Vol. 10, 1956, pp. 39–42) describes a study on the protopectin content of tomatoes during storage. Four storage times were selected, and samples from nine lots of tomatoes were analyzed. The protopectin content (expressed as hydrochloric acid soluble fraction mg/kg) is in the following table.

Exercise

5. An article in the IEEE Transactions on Components, Hybrids, and Manufacturing Technology (Vol.15, No. 2, 1992, pp. 146–153) describes an experiment in which the contact resistance of a brake-only relay was studied for three different materials (all were silver-based alloys). The data are as follows.

Exercise

6. An article in Lubrication Engineering (December 1990) describes the results of an experiment designed to investigate the effects of carbon material properties on the progression of blisters on carbon face seals. The carbon face seals are used extensively in equipment such as air turbine starters. Five different carbon materials were tested, and the surface roughness was measured. The data are as follows:

Exercise

7. article in Communications of the ACM (Vol. 30, No. 5, 1987) studied different algorithms for estimating software development costs. Six algorithms were applied to eight software development projects and the percent error in estimating the development cost was observed. The data are in the table at the bottom of the page.