Chapter 13 ANOVAThe Design and Analysis of Single Factor Experiments - Part II

Chapter 13B

Class will begin in a few minutes.

Reaching out to the internet

Today’s Special My gosh, this does look promising.

Fixed versus Random Factors

Fixed Effects Model – conclusions only apply to factor levels included in the study

Random-Effects Model – conclusions extend to the population of factor levels where the levels in the study are a sample assume population of factor levels is either infinite or large

enough to be treated as infinite

almost the same model as before

2 2ijVar Y but this is different

20,NID 20,NID

The Model

For a random-effects model, the appropriate hypotheses to test are:

The ANOVA decomposition of total variability is still valid:

Expected Values of the Mean Squares

0Treatments

E

MSF

MSunder H0:

The Estimators of the Variance Components

The Problem (Example 13-4)DOE in the Textile Industry

The Data and ANOVA

The Variances

Figure 13-8

The distribution of fabric strength. (a) Current process, (b) improved process.

LSL – Lower Specification Limit

Randomized Block Designs (RBD)

Variability among subjects may mask or obscure treatment effect of interest

Nuisance variable of individualized differences can be minimized using a RBD

Appropriate when one treatment with r > 2 levels assignment of subjects to blocks so that

variability among subjects within any block is less than the variability among the blocks

random assignment of treatments to subjects within blocks

Our Very First Blocking Example

The factor levels: 4 different cockpit designs The experimental units: 12 pilots with varying years

of experiences The response variable: Number of pilot errors in

flying a simulated program Block on years experience

1-2 3-4 5 or more

Randomly assign treatment levels (i.e. cockpit design) to pilots in each block.

CRD versus RBD

RBD yrs 1-2 yrs 3-4 yrs 5+ DesignA Joe Tom BobB Jill Ted BarbaraC John Terrie BillD Jim Teresa Betty

CRD Design subjectsA Joe Ted TerrieB Teresa Bill BobC Jim Barbara JillD John Tom Betty

Complete Confounding

CRD yrs 1-2 yrs 3-4 yrs 5+ Design A B C

Joe Tom BobJill Ted BarbaraJohn Terrie Bill

The devil made me do it this way.

13-4 Randomized Complete Block Designs

The randomized block design is an extension of the paired t-test to situations where the factor of interest has more than two levels.

Figure 13-9 A randomized complete block design.

13-4.1 Design and Statistical Analyses

For example, consider the situation of Example 10-9, where two different methods were used to predict the shear strength of steel plate girders. Say we use four girders as the experimental units.

13-4.1 Design and Statistical Analyses

General procedure for a randomized complete block design:

The Linear Statistical Model

We assume

• treatments and blocks are initially fixed effects

• blocks do not interact

•

The Hypothesis

The partitioning of the sums of squares

SST = SSTreatements + SSBlocks + SSE

d.f.’s ab – 1 = (a – 1) + (b – 1) + (a – 1)(b – 1)

Mean Squares

The effect of the blocks may or may not be interesting in and of itself.

Notice that the effect of blocks is to reduce the sums of squares left as error. This means that the calculation of F0 will always produce a larger quantity than otherwise.

The test on the i becomes more sensitive.

SSE = SSTotals - SSTreatments - SSBlocks

Expected Values Of the Mean Squares

If the i are zero, this is an

unbiased estimate of 2.

A Mean Square

Computational Formulae

ANOVA

Reject H0 if 0 , 1,( 1)( 1)a a bF f

Example 13-5

SS for Example 13-5

More SS for Example 13-5

ANOVA for Example 13-5

Minitab Output for Example 13-5

Fisher’s Least Significant Difference for Example 13-5

Figure 13-10 Results of Fisher’s LSD method.

.05/2,12

2 2(.0792)2.1788 .3878

5EMS

LSD tn

CI on chemical

Individual 95% CIChemical Mean ----------+---------+---------+---------+-

1 1.14 (--*---)

2 1.76 (--*--)

3 1.38 (--*---)

4 3.56 (---*--)

----------+---------+---------+---------+-

.05/2,12

2 2(.0792)2.1788 .3878

5EMS

LSD tn

CI on samples

Individual 95% CI

sample Mean -+---------+---------+---------+---------+

1 2.30 (----*----)

2 2.53 (----*----)

3 0.87 (-----*----)

4 2.20 (----*----)

5 1.90 (----*----)

-+---------+---------+---------+---------+

0.60 1.20 1.80 2.40 3.00

.05/2,12

2 2(.0792)2.1788 .4336

4EMS

LSD tn

An Excel Comparison

A computer model simulates depot overhaul of jet engines in order to estimate repair cycle times in days

Three probability distributions are used to model task times gamma lognormal Weibull

Blocking occurs by using the same random number stream with each treatment (distribution)

Go to the next slide to observe the highly interesting results

Highly Interesting Resultsrandom number seed

treatment 1 2 3 4gamma 13 22 18 39lognormal 16 24 17 44Weibull 5 4 1 22

ANOVASource of Variation SS df MS F P-value F crit

Rows 703.5 2 351.75 40.71704 0.000323 5.143253Columns 1106.917 3 368.9722 42.71061 0.000192 4.757063Error 51.83333 6 8.638889

Total 1862.25 11

engine overhaul days

CI on distributions

Individual 95% CI

distribution Mean ---+---------+---------+---------+--------

gamma 23.0 (-----*-----)

lognorma 25.2 (-----*-----)

Weibull 8.0 (-----*-----)

---+---------+---------+---------+--------

6.0 12.0 18.0 24.0

.05/2,12

2 2(8.639)2.4469 5.0855

4EMS

LSD tn

CI on Random Number Seeds

Individual 95% CI

seed Mean --+---------+---------+---------+---------

1 11.3 (----*----)

2 16.7 (----*----)

3 12.0 (----*----)

4 35.0 (----*----)

--+---------+---------+---------+---------

8.0 16.0 24.0 32.0

.05/2,12

2 2(8.639)2.4469 5.8722

3EMS

LSD tn

Even Higher Interesting Results

One-Way ANOVA without blockingANOVA

Source of Variation SS df MS F P-value F critBetween Groups 703.5 2 351.75 2.732039 0.118245 4.256495Within Groups 1158.75 9 128.75

Total 1862.25 11

F.01,2,9 = 8.02F.05,2,9 = 4.26

Why it appears that the variability introduced by

the random number stream is masking the effect of the task time distributions. This is nothing but garbage.

Randomized Complete Block Design is Now Over

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