ENGR 610Applied Statistics

Fall 2007 - Week 9

Marshall University

CITE

Jack Smith

Overview for Today Review Design of Experiments, Ch 10

One-Factor Experiments Randomized Block Experiments

Go over homework problems: 10.27, 10.28 Design of Experiments, Ch 11

Two-Factor Factorial Designs Factorial Designs Involving Three or More Factors Fractional Factorial Design The Taguchi Approach

Homework assignment

Design of Experiments R.A. Fisher (Rothamsted Ag Exp Station)

Study effects of multiple factors simultaneously Randomization Homogeneous blocking

One-Way ANOVA (Analysis of Variance) One factor with different levels of “treatment” Partitioning of variation - within and among treatment groups Generalization of two-sample t Test

Two-Way ANOVA One factor against randomized blocks (paired treatments) Generalization of two-sample paired t Test

One-Way ANOVA ANOVA = Analysis of Variance

However, goal is to discern differences in means One-Way ANOVA = One factor, multiple treatments (levels) Randomly assign treatment groups Partition total variation (sum of squares)

SST = SSA + SSW SSA = variation among treatment groups SSW = variation within treatment groups (across all groups)

Compare mean squares (variances): MS = SS / df Perform F Test on MSA / MSW

H0: all treatment group means are equal H1: at least one group mean is different

Partitioning of Total Variation Total variation

Within-group variation

Among-group variation

SST (X ij X)2

i1

n j

j1

c

SSW (X ij X j )2

i1

n j

j1

c

SSA n j (X j X)2

j1

c

X 1

nX ij

i1

n j

j1

c

X j 1

n jX ij

i1

n j

(Grand mean)

(Group mean)

c = number of treatment groupsn = total number of observationsnj = observations for group jXij = i-th observation for group j

Mean Squares (Variances) Total mean square (variance)

MST = SST / (n-1) Within-group mean square

MSW = SSW / (n-c) Among-group mean square

MSA = SSA / (c-1)

F Test F = MSA / MSW Reject H0 if F > FU(,c-1,n-c) [or p<]

FU from Table A.7

One-Way ANOVA SummarySource Degrees of

Freedom (df)Sum of Squares (SS)

Mean Square (MS) (Variance)

F p-value

Among groups

c-1 SSA MSA = SSA/(c-1) MSA/MSW

Within groups

n-c SSW MSW = SSW/(n-c)

Total n-1 SST

Tukey-Kramer Comparison of Means

Critical Studentized range (Q) test

qU(,c,n-c) from Table A.9

Perform on each of the c(c-1)/2 pairs of group means Analogous to t test using pooled variance for

comparing two sample means with equal variances

X i X j qUMSW

2

1

ni

1

n j

One-Way ANOVA Assumptions and Limitations Assumptions for F test

Random and independent (unbiased) assignments Normal distribution of experimental error Homogeneity of variance within and across group

(essential for pooling assumed in MSW) Limitations of One-Factor Design

Inefficient use of experiments Can not isolate interactions among factors

Randomized Block Model Matched or repeated measurements assigned to a

block, with random assignment to treatment groups Minimize within-block variation to maximize treatment

effect Further partition within-group variation

SSW = SSBL + SSE SSBL = Among-block variation SSE = Random variation (experimental error) Total variation: SST = SSA + SSBL + SSE

Separate F tests for treatment and block effects Two-way ANOVA, treatment groups vs blocks, but the

focus is only on treatment effects

Partitioning of Total Variation Total variation

Among-group variation

Among-block variation

SST (X ij X)2

i1

r

j1

c

SSAr (X j X)2

j1

c

X 1

rcX ij

i1

r

j1

c

X j 1

rX ij

i1

r

(Grand mean)

(Group mean)

SSBLc (X i X)2

i1

r

X i 1

cX ij

j1

c

(Block mean)

Partitioning, cont’d Random error

SSE SST SSA SSBL (X ij X i X j X)2

i1

r

j1

c

c = number of treatment groupsr = number of blocks n = total number of observations (rc)Xij = i-th block observation for group j

Mean Squares (Variances) Total mean square (variance)

MST = SST / (rc-1) Among-group mean square

MSA = SSA / (c-1) Among-block mean square

MSBL = SSBL / (r-1) Mean square error

MSE = SSE / (r-1)(c-1)

F Test for Treatment Effects F = MSA / MSE Reject H0 if F > FU(,c-1,(r-1)(c-1))

FU from Table A.7

Two-Way ANOVA SummarySource Degrees of

Freedom (df)Sum of Squares (SS)

Mean Square (MS) (Variance)

F p-value

Among groups

c-1 SSA MSA = SSA/(c-1) MSA/MSE

Among blocks

r-1 SSBL MSBL = SSBL/(r-1) MSBL/MSE

Error (r-1)(c-1) SSE MSE = SSE/(r-1)(c-1)

Total rc-1 SST

F Test for Block Effects F = MSBL / MSE Reject H0 if F > FU(,r-1,(r-1)(c-1))

FU from Table A.7

Assumes no interaction between treatments and blocks

Used only to examine effectiveness of blocking in reducing experimental error

Compute relative efficiency (RE) to estimate leveraging effect of blocking on precision

Estimated Relative Efficiency Relative Efficiency

Estimates the number of observations in each treatment group needed to obtain the same precision for comparison of treatment group means as with randomized block design. nj (without blocking) RE*r (with blocking)

RE (r 1)MSBL r(c 1)MSE

(rc 1)MSE

Tukey-Kramer Comparison of Means

Critical Studentized range (Q) test

qU(,c,(r-1)(c-1)) from Table A.9 Where group sizes (number of blocks, r) are equal

Perform on each of the c(c-1)/2 pairs of group means Analogous to paired t test for the comparison of two-

sample means (or one-sample test on differences)

X i X j qUMSE

r

Factorial Designs Two or more factors simultaneously Includes interaction terms Typically

2-level: high(+), low(-) 3-level: high(+), center(0), low(-)

Replicates Needed for random error estimate

Partitioning for Two-Factor ANOVA(with Replication)

Total variation

Factor A variation

Factor B variation

SST (X ijk X)2

k1

n'

j1

c

i1

r

SSAcn' (X i X)2

i1

r

X 1

rcn 'X ijk

k1

n '

j1

c

i1

r

X i 1

cn'X ijk

k1

n'

j1

c

(Grand mean)

(Mean for i-th level of factor A)

SSBrn' (X j X)2

j1

c

X j 1

rn'X ijk

k1

n'

i1

r

(Mean for j-th level of factor B)

Partitioning, cont’d Variation due to interaction of A and B

Random error

SSE SST SSA SSB SSAB (X ijk X ij )2

k

n '

j1

c

i1

r

r = number of levels for factor A c = number of levels for factor Bn’ = number of replications for eachn = total number of observations (rcn’)Xijk = k-th observation for i-th level of factor A and j-th level of factor B

SSABn' (X ij X i X j X)2

j1

c

i1

r

'

1'

1 n

kijkij X

nX

(Mean for replications of i-j combination)

Mean Squares (Variances) Total mean square

MST = SST / (rcn’-1) Factor A mean square

MSA = SSA / (r-1) Factor B mean square

MSB = SSB / (c-1) A-B interaction mean square

MSAB = SSAB / (r-1)(c-1) Mean square error

MSE = SSE / rc(n’-1)

F Tests for Effects Factor A effect

F = MSA / MSE Reject H0 if F > FU(,r-1,rc(n’-1))

Factor B effect F = MSB / MSE Reject H0 if F > FU(,c-1,rc(n’-1))

A-B interaction effect F = MSAB / MSE Reject H0 if F > FU(,(r-1)(c-1),rc(n’-1))

Two-Way ANOVA (with Repetition) Summary Table

Source Degrees of Freedom (df)

Sum of Squares (SS)

Mean Square (MS) (Variance)

F p-value

A r-1 SSA MSA = SSA/(r-1) MSA/MSE

B c-1 SSB MSB = SSB/(c-1) MSB/MSE

AB (r-1)(c-1) SSAB MSAB = SSAB/(r-1)(c-1) MSAB/MSE

Error rc(n’-1) SSE MSE = SSE/rc(n’-1)

Total rcn’-1 SST

Tukey-Kramer Comparisons Critical range (Q) test for levels of factor A

qU(,r,rc(n’-1)) from Table A.9 Perform on each of the r(r-1)/2 pairs of levels

Critical range (Q) test for levels of factor B

qU(,c,rc(n’-1)) from Table A.9 Perform on each of the c(c-1)/2 pairs of levels

X i X i' qUMSE

rn'

X j X j ' qUMSE

cn'

Main Effects and Interaction Effects

No interaction Interaction Crossing Effect

Three-Way ANOVA (with Repetition) Summary Table

Source Degrees of Freedom (df)

Sum of Squares (SS)

Mean Square (MS) (Variance) F p-value

A i-1 SSA MSA = SSA/(i-1) MSA/MSE

B j-1 SSB MSB = SSB/(j-1) MSB/MSE

C k-1 SSC MSC = SSC/(k-1) MSC/MSE

AB (i-1)(j-1) SSAB MSAB = SSAB/(i-1)(j-1) MSAB/MSE

BC (j-1)(k-1) SSBC MSBC = SSBC/(j-1)(k-1) MSBC/MSE

AC (i-1)(k-1) SSAC MSAC = SSAC/(i-1)(k-1) MSAC/MSE

ABC (i-1)(j-1)(k-1) SSABC MSABC = SSABC/(i-1)(j-1)(k-1) MSABC/MSE

Error ijk(n’-1) SSE MSE = SSE/ijk(n’-1)

Total Ijkn’-1 SST

Main and Interaction Effects For a k-factor design

Number of main effects

Number of 2-way interaction effects

Number of 3-way interaction effects

See text (p 529) for sample plots

k

1

k!

1!(k 1)!k

k

2

k!

2!(k 2)!k(k 1) /2

k

3

k!

3!(k 3)!k(k 1)(k 2) /6

3-Factor 2-Level Design Notation

ABC(1) = a-lo, b-lo, c-lo - - -a = a-hi, b-lo, c-lo + - -b = a-lo, b-hi, c-lo - + -c = a-lo, b-lo, c-hi - - +ab = a-hi, b-hi, c-lo + + -bc = a-lo, b-hi, c-hi - + +ac = a-hi, b-lo, c-hi + - +abc = a-hi, b-hi, c-hi + + +

Contrasts and Estimated Effects

A = (1/4n’)[a + ab + ac + abc - (1) - b - c - bc]B = (1/4n’)[b + ab + bc + abc - (1) - a - c - ac]C = (1/4n’)[c + ac + bc + abc - (1) - a - b - ab]AB = (1/4n’)[abc - bc + ab - b - ac + c - a + (1)]BC = (1/4n’)[(1) - a + b - ab - c + ac - bc + abc]AC = (1/4n’)[(1) + a - b - ab - c - ac + bc + abc]ABC = (1/4n’)[abc - bc - ac + c - ab + b + a - (1)]

Effect = (1/n’2k-1)ContrastSS = (1/n’2k)(Contrast)2

Sum over replications

k = number of factorsn’ = number of replicates

3-Factor 2-Level Contrast Table

Notation A B C AB AC BC ABC

(1) - - - + + + -

a + - - - - + +

b - + - - + - +

c - - + + - - +

ab + + - + - - -

ac + - + - + - -

bc - + + - - + -

abc + + + + + + +

Using Normal Probability Plots Cumulative percentage for i-th ordered effect

pi = (Ri - 0.5)/(2k - 1)

Ri = ordered rank of I-th effect k = number of factors

Plot on normal probability paper, or use PHStat

Note deviations from zero and from the nearly straight vertical line for normal random variation

See example in text (p 535)

Fractional Factorial Design Choose a defining contrast

Typically highest interaction term Halves the number of combinations But introduces confounding interactions

Aliasing

Resolution III, IV, V designs based on types of confounding interactions

remaining in design

==> http://www.itl.nist.gov/div898/handbook/pri/section3/pri3347.htm==> http://www.statsoft.com/textbook/stexdes.html

Taguchi Approach Parameter design Quadratic Loss Function

Loss = k(Yi-T)2

Partition into design parameters (inner array) and noise factors

Use Signal-to-Noise (S/N) ratios to meet target or minimize/maximize response

Use of orthogonal arrays

Homework Work through Appendix 11.1 Work through Problems

11.36-38 Review for Exam #2

Chapters 8-11 Take-home “Given out” end of class Oct 25 Due beginning of class Nov 1