2-level factorials

Lecture 6: 2-level Factorials

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Treatments and BlocksTwo-Way Analysis of Variance - MANOVAThe Importance of TransformationTwo-Level Factorials

Two-Level Factorials

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What is a factorial?

• So far, we analyzed data based on the type of treatment they received.

• This analysis is known as "one-way" analysis of variance.

• Now we discuss data that can be classified in more than one "ways". Each of these ways is known as a "factor".

• In the beginning we will call the first classification a "treatment" and the second classification a "block".

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y ti = μ + β i + τ t + ε ti yti = y + (yi - y) + (yt - y) + (yti - y t - y i + y)

Randomized Block Designs

• In general we "block" the effect that we want to eliminate so that we can see the effects of the "treatment".

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tube 1

tube 2

Gas A Gas B Gas C

Example: Particle Contamination Study

• We have two LPCVD tubes and three gas suppliers. We are interested in finding out if the choice of the gas supplier makes any difference in terms of average particle counts on the wafers.

• Experiment: we run three full loads on each tube, one for each gas. We report the average particle adders for each load (excluding the first wafer in the boat…)

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A B C1 7 36 2 152 13 44 18 25

10 40 10 20

7 36 2 20 20 20 -5 -5 -5 -10 20 -10 2 1 -313 44 18 20 20 20 5 5 5 -10 20 -10 -2 -1 3

S = SA + SB + ST + SR

3,778 = 2,400 + 150 + 1,200 + 286 = 1 + 1 + 2 + 2

2,400 150 600 14

Treatment (Gas)

Block (Tube)

Decompose according to the equation: yti = μ + βi + τt + ε ti

yti = y + (yi - y) + (yt - y) + (yti - yt - yi + y)

A Simple Two-Way ANOVA for particles...

Sums of squaresDegrees of freedomMean squares

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yti = μ + βi + τt + εti

yti = y + (yi - y) + (yt - y) + (yti - yt - yi + y)

Decompose particle counts to global average, deviationsfrom that average due to machine choice and gas source choice. What remains is the residual.

The Two-Way ANOVA

moments

estimates

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Source of Sum DFs Mean sq Variance of sq

Average SA 1bet. blocks SB n-1 sB

2

bet. treatm ST k-1 sT2

residuals SR (n-1)(k-1) sR2

total S nk

Assumptions: - additivity- IIND of residuals

Expected values: s2B : σ2 + k Σ β2i / (n-1)s2T : σ2 + n Σ τ2t / (k-1)

Two-Way ANOVA Table

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Analysis of Variance

SourceModelErrorC Total

DF325

Sum of Squares1350.00

28.001378.00

Mean Square450.0014.00

F Ratio32.14Prob > F

0.03

Effect Test

SourceBlock Treatment

Nparm12

DF12

Sum of Squares150.00

1200.00

F Ratio10.7142.86

Prob > F0.080.02

MANOVA for our Example

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This model assumes additivity as well as IIND residuals:

yti = μ + βi +τt + εti

yti = μ + βi +τt

Note that this model “predicts” 20 values using 4+5+1=10 parameters.

The Model of the two-way ANOVA

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D = Y - A

D = B + T + R

Easy to prove that R ⊥ T and A ⊥ R.

D

B T

R

Geometric Interpretation of the two-way ANOVA

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ytij = μti + ε tij

y tij = yti + (ytij - yti)

μti = μ + τt + βi + ωti

yti = y + (yt - y) + (yi - y) + (yti - yt - yi + y)

A Two-Way Factorial Design

• When both effects are of interest, then the same arrangement is called a factorial design.

• The only addition to our model is the interaction term.

moments

estimates

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Source Sum DFs Mean sq of Var of sq

bet. bake SB n-1 sB2

bet. etch ST k-1 sT2

interaction SI (n-1)(k-1) sI2

residuals Se nk(m-1) sE2

total S nkm-1

Assumptions: residuals IIND

Expected values: sB2 : σ2 + mk Σ β2i / (n-1)

sT2 : σ2 + mn Σ τ2t / (k-1)

sI2 : σ2 + mΣΣ ω2ti / (n-1)(k-1)

ANOVA Table for Two-Way Factorials

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What are your conclusions?

Effect Test

SourceBlock TreatmentBlock *Treatment

DF122

Sum of Squares150.00

1200.0028.00

F-Ratio•••

Prob > F•••

Analysis of VarianceSourceModelErrorC Total

DF505

Sum of Squares1378.00

0.001378.00

Mean Square275.60

•

F Ratio•

Prob > F•

Factorial for our Example

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Etch Rec Bake DLeff1 1 0.312 1 0.823 1 0.434 1 0.451 1 0.452 1 1.103 1 0.454 1 0.711 1 0.462 1 0.883 1 0.634 1 0.661 1 0.432 1 0.723 1 0.764 1 0.621 2 0.362 2 0.923 2 0.444 2 0.561 2 0.292 2 0.613 2 0.354 2 1.02

Etch Rec Bake DLeff1 2 0.402 2 0.493 2 0.314 2 0.711 2 0.232 2 1.243 2 0.404 2 0.381 3 0.222 3 0.303 3 0.234 3 0.301 3 0.212 3 0.373 3 0.254 3 0.361 3 0.182 3 0.383 3 0.244 3 0.311 3 0.232 3 0.293 3 0.224 3 0.33

Two way factorial experiment for DLeff.

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Effect TestSourceEtch RecipeBake ProcEtch Rec*Bake Proc

DF326

Sum of Squares0.921.030.25

F Ratio13.8123.221.87

Prob>F0.000.000.11

Analysis of VarianceSourceModelErrorC Total

DF113647

Sum of Squares2.200.803.01

Mean Square0.200.02

F Ratio9.01

Prob>F0.00

Anova Table for 2-way DLeff Factorial

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ResidualDLeff

-0.4

-0.2

0

0.2

0.4

0.6

0.8

0.25 0.35 0.45 0.55 0.65 0.75

Predicted DLeff

ResidualDLeff

-0.4

-0.2

0

0.2

0.4

0.6

0.8

Bake ProcA B C

Diagnostics for DLeff

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-3

-2

-1

0

1

2

3

-0.4 -0.2 0.0 0.2 0.4 0.6 0.8

Residual Statistics for DLeff

MaxMin 75%50%25%

Est Mean and its 95% conf. interval

Normal Probability Plot (must be a straight line)

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σy prop μα

Y = yλ

λ = 1 - α

Variance Stabilization

If the variance is a function of μ, then the appropriate transformation is needed to correct it. In general:

Need for Transformation

where σ can be determined empirically.Non-additivityAn empirically selected transformation can be tested using a formal Anova-based test.

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Effect TestSourceEtch RecipeBake ProcEtch Rec*Bake Pro

DF326

Sum of Squares3.565.240.40

F Ratio21.9348.431.22

Prob>F0.000.000.32

Analysis of VarianceSourceModelErrorC Total

DF113647

Sum of Squares9.191.95

11.14

Mean Square0.840.05

F Ratio15.45Prob>F0.00

Anova Table for 2-way ln(DLeff) Factorial

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-0.6-0.5-0.4-0.3-0.2-0.10.00.10.20.30.40.5

-1.5 -1.0 -0.5 0.0

lnDLeff Predicted

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

1.0 1.5 2.0 2.5 3.0

Bake Proc

Residual ln(DLeff) Residual ln(DLeff)

Diagnostics for ln(DLeff)

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-3

-2

-1

0

1

2

3

-0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5

Residual Statistics for ln(DLeff)

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DLeff

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 0.2 0.4 0.6 0.8 1 1.2 1.4

DLeff Predicted

lnDLeff

DLeff

-1.5

-1.0

-0.5

0.0

-1.5 -1.0 -0.5 0.0lnDLeff Predicted

ln(DLeff)

Improvement due to Transformation

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235 x 5 Latin Square design.

<- treatmentsblocks

Other Blocked Arrangements• Sometimes we might have to deal with more than one

blocking variables.• Example: testing the effect of a develop recipe on wafers

and resists that come from different vendors.• Attach two types of labels, A,B.. and I, II... and combine

them to balance the blocking effects:

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y ti = μ+ τ t + β i + ω ti

y = f(x1,x2...,xn)

Measuring the Effect of Variables

• The first step is always to find whether any variable has an effect on the outcome.

• Nothing more can be done for qualitative (categorical) variables such as recipe type, vendor name etc.

• Some times we deal with quantitative variables (temperature, pressure).

• Once the effect has been confirmed, the next step is to build empirical quantitative models of the process:

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they can be used to visualize, control, design, diagnose etc.

How can such a model be "extracted" from the process?

Quantitative Models of the Process

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x

y

x1 x2

level (-1) level (+1)

y1

y2

Estimated model:y = Avgy +1/2 Effectx x'

Estimated model:y = Avgy +1/2 Effectx x'^

“True” model y = η + 1/2 Ε x’

Effectx = y2 - y1

y = (y2 + y1) / 2

x' = 2 (x - Avgx) / ( x2 - x1 )Avgx = (x2 + x1) / 2

A Simple 1-Factor, 2-Level Factorial

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(4)

(1) (2)

(3)

x1

x2

A Simple 2-Factor 2-Level Factorial

y

2y

3y4y

1y

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Calculations for the 2 Factor Factorial

• Average response• Effect for x1

• Effect for x2

• Interaction of x1 and x2

• Model

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Note on Economy in experimentation:

(1) (2)

(3) (4)

(5) (6)

(7) (8)

Simple, few runs, can be augmented.2k

l1 × l2 × ...lk

General Factorial Designs at Two Levels

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(1) (2)

(3) (4)

x1

x2

(1)

(2)

(3)

(4)

x1

x2

(5)

How many runs do we need for the same precision?

Factorial vs the "one-factor-at-a-time" experiment

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23 Pilot Plant Example

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Calculation of Main Effects and Interactions

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oPressure (P) 300-550 mtorrTemperature (T) 605-650 CSilane flow (F) 100-250 sccmDep. Rate R in Å/minP T F R Effects- - -+ - -- + -+ + -- - ++ - +- + ++ + +

94.8111.0214.1255.894.1

145.9286.7340.5

192.940.9

162.847.96.9

11.930.8-5.9

AVGPTFPTPFTFPTF

σ experiment (estimate from replicates) = 9.05 (8) σ mean (estimate from replicates) = 3.20σ effect (estimate from replicates) = 6.40

Two Level Factorial for Polysilicon Deposition

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s2 = ν1s1

2+ν2s22+...+ νgsg2

ν1 + ν2 +...+ νg

N ( 0, 4N

σ2 )

Sum of squares can be used to estimate σ2

Example from previous page:σ effect (estimate from insignificant effects) = 8.65 (3)

V(effect) = V ( y+ - y-) = ( 14

+ 14

)σ2 = 12

σ2

in general:V(effect) = 4

Nσ2

Variance Estimation via replicated runs:

Via insignificant high order effects:High order effects can be seen as samples from one distribution:

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Pi = 100(i - 12

)/m i = 1,2, ..., m

Used to identify insignificant effects.

Normal Probability Plots

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0.0 2.0 4.0 6.0 8.0 10.01.1

15

102030507080909599

99.999.99

Normal Probability Plot - Example

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Normal Probability Plot - 25 Factorial Example

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Normal Probability Plot - 25 Factorial Example

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y = 192.9+20.4P'+81.4T'+23.9F'+15.4T'F'^

86420-100

0

100

200

1086420-20

-10

0

10

20

30

Polysilicon Deposition Example (cont.)

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Natural log of Deposition Rate R in ln(Å/min)

P T F ln(R) Effects- - -+ - -- + -+ + -- - ++ - +- + ++ + +

5.150.240.900.21-0.060.070.08-0.08

AVGPTFPTPFTFPTF

σ experiment (estimated from replications) = .028 (8) σ mean (estimated from replications) = .010σ effect (estimated from replications) = .020σ effect (estimated from insignificant effects) = .073 (4)

4.554.715.375.544.544.985.665.83

Using a Transformation to improve the model

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Mean++++++++

8

PT+--++--+

4

PF+-+--+-+

4

TF++----++

4

PTF-++-+--+

4

Avg.4.554.715.375.544.544.985.665.83

T--++--++

4

P-+-+-+-+

4

F----++++

4

Quick Calculation of Effects - Contrasts

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12345678

Avg.4.554.715.375.544.544.985.665.83

(1)9.26

10.919.52

11.490.160.170.440.17

(2)20.1721.010.330.611.651.970.01

-0.27

(3)41.180.943.62

-0.260.840.280.32

-0.28

d84444444

est 5.150.240.90

-0.060.260.070.08

-0.08

effAVGPTPTFPFTFPTF

T--++--++

P-+-+-+-+

F----++++

Quick Calculation of Effects - Yate's

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y = 51.15+.12P'+.45T'+.11F'^

86420-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1086420-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

Effects, interactions and residuals must be checked...

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Conclusions

• From One way ANOVA to MANOVA.• From “table” models to continuous models.• The most important concepts are:

– There is a “true” model (with unknown “moments”)– There is an “estimated” model.– Each estimated model parameter is a “statistic”.– Critical underlying assumptions must be checked.– Residuals must be Independently, Identically,

Normally Distributed (IIND).

(See chapter 7 and chapter 10 in BHH, or chapter 12 in class textbook)