Factorial Models

Random Effects Gauge R&R studies (Repeatability

and Reproducibility) have been an expanding area of application

Mixed Effects Models

One-way Random Effects

The one-way random effects model is quite different from the one-way fixed effects model– Yandell has a real appreciation for this

difference– We should be surprised that the

analytical approaches to the main hypotheses for these models are so similar

One-way Random Effects

In Chapter 19, Yandell considers– unbalanced designs– Smith-Satterthwaite approximations– Restricted ML estimates

We will defer the last two topics to general random and mixed effects models

One-way Random Effects

Yij Ai ij, i 1,,a, j 1,,n

ij iid N(0, 2)

Ai iid N(0,A2 )

ij, Ai indep.

One-way Random Effects

E Yij

V Yij A2 2

Cov Yij,Yij' Cov Ai, Ai V Ai A2

Yij,Yij' A2

A2 2

One-way Random EffectsE(MSTR)

Yij Ai ij

Y i. Ai iY .. A .

Y i.. Y .. 2

i

Ai A 2

i

i 2 i

One-way Random EffectsE(MSTR)

n Y i. Y .. 2

i

n Ai A 2

i

n i 2 i

En

a 1Y i. Y .. 2

i

E

n

a 1Ai A 2

i

E

n

a 1i 2

i

E MSTR nA2 2

One-way Random EffectsTesting

By a similar argument, we can show E(MSE)=2

The familiar F-test statistic for testing

Ho : i 0

will also be appropriate for testing

Ho :A2 0

One-way Testing

Under the true model,

So power analysis for balanced one-way random effects can be studied using a central F-distribution

F MSTR /(nA

2 2)

MSE / 2~

nA2 2

2F a 1,a(n 1)

One-way Random Effects

Method of Moment point estimates for 2 and

2 are available

Confidence intervals for 2 and 2/2 are available

A confidence interval for the grand mean is available

Two-way Random Effects Model

We will concentrate on a particular application—the Gauge R&R model

20.2 addresses unbalanced models– Material is accessible

Topics in 20.3 will be addressed later

20.4 and 20.5 can safely be skipped

Gauge R&RTwo-way Random Effects Model P-Part O-Operator

ijkijjiijk

ijkijjiijk

POOPY

CBAY

R R

Gauge R&R

With multiple random components, Gauge R&R studies use variance components methodology

Y2 P

2 O2 PO

2 2

Gauge R&R

Repeatability is measured by

Reproducibility is measured by

2

O2 PO

2

Gauge R&R

Unbiased estimates of the variance components are readily estimated from Expected Mean Squares (a=# parts, b=# operators, n=# reps)

E(MSE) 2

E MSParts 2 n PO2 bn P

2

E MSOperator 2 n PO2 an O

2

E MSPO 2 n PO2

Gauge R&R

Use Mean Sums of Squares for estimation

ˆ 2 MSEˆ PO

2 ?ˆ P

2 ?ˆ O

2 ?

Gauge R&R

Minitab has a Gauge R&R module– Output is specific to industrial methods

Consider an example with 3 operators, 5 parts and 2 replications

Two-way Random Effects Model

Consider results from our expected mean squares.

What would be appropriate tests for A, B, and AB?

Approximate F tests

Statistics packages may do this without your being aware of it.

Example– A, B and C random– Replication

Approximate F test

Source EMS

222

22222

22222

22222

ABABC

CABCBCAC

BABCBCAB

AABCABAC

cnn

bcnnanbn

acnnancn

bcnncnbn

AB

C

B

A

Approximate F test

Source EMS

2

22

222

222

ABC

ABCBC

ACABC

n

nan

bnn

Error

ABC

BC

AC

Approximate F test

No exact test of A, B, or C exists

We construct an approximate F test,

vu

sr

MSMSMS

MSMSMS

MSMSF

''

'

where,''/'

Approximate F test

We require E(MS’)=E(MS”) under Ho

F has an approximate F distribution, with parameters

1 MSii r

s

2

MSi2 df i

i r

s

2 MS jj u

v

2

MSj2 df j

j u

v

Approximate F test

Note that MS’, MS’’ can be linear combinations of the mean squares and not just sums

Returning to our example, how do we test

0: 2 AoH

DF for Approximate F tests

Restating the result:

d.f.

'' and

'with

ondistributi F eapproximatan has '''

then,'' and ' If

2

2

22

2

1

1

v

ujjj

s

riii

v

ujj

r

ii

dfMS

MS

dfMS

MSMSMSF

MSMSMSMS

DF for Approximate F tests

The following argument builds approximate c2 distributions for the numerator and denominator mean squares (and assumes they are independent)

We will review the argument for the numerator

The argument computes the variance of the mean square two different ways

DF for Approximate F tests

Remember that the numerator for an F random variable has the form:

Note that we already have this result for the constituent MSi

2

2

~'' want would weHence,

MSEMS

DF for Approximate F tests

For each term in the sum, we have

. where,~ 2i

22 idfi

ii MSEMSdf

i

.2ely approximat is varianceitsthen

, ddistributeely approximat is 'MSE

'MS If 2

DF for Approximate F tests

We can derive the variance by another method:

s

ri i

i

s

rii

dfMSE

MSVMSE

MSEMSV

.2

'

)('

''

4

2

2

2

2

DF for Approximate F Tests

Equating our two expressions for the variance, we obtain:

s

ri i

i

s

ri i

i

dfMSE

MSE

dfMSE

2

2

4

2

2

'

2'

2

DF for Approximate F Tests

Replacing expectations by their observed counterparts completes the derivation.

Two-way Mixed Effects Model

),0(),,0(,0

Form edUnrestrict

0,1

,0~),,0(,0

Form Restricted

22

1

1

22

1

ABijBj

a

ii

a

iijABijBj

a

ii

ijkijjiijk

NiidCNiidB

jCa

aNCNiidB

CBY

Two-way Mixed Effects Model

Both forms assume random effects and error terms are uncorrelated

Most researchers favor the restricted model conceptually; Yandell finds it outdated. It is certainly difficult to generalize.

SAS tests the unrestricted model using the RANDOM statement with the TEST option; the restricted model has to be constructed “by hand”.

Minitab tests unrestricted model in GLM, restricted model option in Balanced ANOVA.

Two-way Mixed Effects Model

2222

22222

1

2

221

2

22

AB

B11

A

EMS edUnrestrictEMS RestrictedSource

ABAB

BABB

a

ii

AB

a

ii

AB

nn

annana

bnn

a

bnn

Two-way Mixed Effects Model

The EMS suggests that the fixed effect (A) is tested against the two-way effect (AB) for both forms (F=MSA/MSAB)

The EMS suggests that the random effect (B) is tested against error (F=MSB/MSE) for the restricted model, but tested against the two-way effect (AB) for the unrestricted model (F=MSB/MSAB)

Two-way Mixed Effects Model

For the Gage R&R study, assume that Part is still a random effect, but that Operator is a fixed effect

SAS and Minitab analysis