Multifactor Experiments

November 26, 2013Gui Citovsky, Julie Heymann, Jessica Sopp, Jin Lee, Qi Fan, Hyunhwan Lee,

Jinzhu Yu, Lenny Horowitz, Shuvro Biswas

Outline

• Two-Factor Experiments with Fixed Crossed Factors

• 2k Factorial Experiments

• Other Selected Types of Two-Factor Experiments

Two-Factor Experiments with

Fixed Crossed Factors

First, single factor

• Comparison of two or more treatments (groups)• Single treatment factor• Example: A study to compare the average flight

distances for three types of golf balls differing in the shape of dimples on them: circular, fat elliptical, thin elliptical• Treatments circular, fat elliptical, thin elliptical• Treatment factor type of ball

Single factor continued

Two-Factor Experiments With Fixed Crossed Factors

• Two fixed factors, A with a ≥ 2 levels and B with b ≥ 2 levels

• ab treatment combinations• If there are n observations obtained under

each treatment combination (n replicates), then there is a total of abn experimental units

Two-Factor Experiments With Fixed Crossed Factors

• Example: Heat treatment experiment to evaluate the effects of a quenching medium (two levels: oil and water) and quenching temperature (three levels: low, medium, high) on the surface hardness of steel

• 2 x 3 = 6 treatment combinations• If 3 steel samples are treated for each

combination, we have N = 18 observations

Model and Estimates of its Parameters

Let yijk=kth observation on the (i,j)th treatment combination, i=1,2,…,a , j=1,2,…,b, and k=1,2,…,n.

Let random variable Yijk correspond to observed outcome yijk.

Basic Model: and independent

where

Table format

Parameters

Grand Mean: ith Row Average:

jth Column Average:

(i,j)th Row Column Interaction

ith Row Main Effect:jth Column Main Effect:

Least Squares Estimates

Variance

• Sample variance for (i, j)th cell is:

• Pooled estimate for σ2:

Example• Experiment to study how mechanical bonding strength of

capacitors depends on the type of substrate (factor A) and bonding material (factor B).

• 3 substrates: Al2O3 with bracket, Al2O3 no bracket, BeO no bracket

• 4 types of bonding material: Epoxy I, Epoxy II, Solder I and Solder II

• Four capacitors were tested at each factor level combination

Example continued

Pooled sample variance:

Example continued: Sample Means

Example continued: Other Model Parameters

Two- Way Analysis of VarianceWe define the following sum of squares:

Analysis of Variance

• Degrees of Freedom:• SST: N – 1• SSA: a – 1 • SSB: b – 1 • SSAB: (a – 1)(b – 1)• SSE: N – ab

• SST = SSA + SSB + SSAB + SSE.• Similarly, the degrees of freedom also follow

this identity, i.e.

Analysis of Variance

• Mean squares =

Hypothesis Test

We test three hypotheses:

Not all

Not all

Not all

If all interaction terms are equal to zero, then the effect of one factor on the mean response does not depend on the level of the other factors.

When do we reject H0?

• Use F-statistics to test our hypotheses by taking the ratio of the mean squares to the MSE.

Reject

Reject

Reject

• We test the interaction hypothesis H0AB first.

Summary (Table 13.5)

Source of Variation (Source)

Sum of Squares

(SS)

Degrees of

Freedom (d.f.)

Mean Square (MS)

F

Main Effects A

a – 1

Main Effects B

b – 1

Interaction AB

(a – 1)(b – 1)

Error N – ab

Total N – 1

Example: Bonding Strength of Capacitors

Data Capacitors;input Bonding $ Substrate $ Strength @@;Datalines;Epoxy1 Al203 1.51 Epoxy1 Al203 1.96 Epoxy1 Al203 1.83 Epoxy1 Al203 1.98…;

proc GLM plots=diagnostics data=Capacitors;TITLE "Analysis of Bonding Strength of Capacitors";CLASS Bonding Substrate;Model Strength = Bonding | Substrate; run;

Bonding Strength of Capacitors ANOVA Table

• At α=0.05, we can reject H0B and H0AB but fail to reject H0A.

• The main effect of bonding material and the interaction between the bonding material and the substrate are both significant.

• The main effect of substrate is not significant at our α.

Main Effects Plot

• Definition: A main effects plot is a line plot of the row means of factor and A and the column means of factor B.

Al2O3_x000d_ Al2O3 + Brckt Be00

1

2

3

4

Factor A Effects Plot

Mea

n Re

sons

e

Epoxy I Epoxy II Solder I Solder II0

1

2

3

4

Factor B Main Effects Plot

Mea

n Re

sons

e

Interaction Plot

Model Diagnostics with Residual Plots

• Why do we look at residual plots? • Is our constant variance assumption true?• Is our normality assumption true?

2k Factorial Experiments

2k Factorial Experiments

• 2k factorial experiments is a class of multifactor experiments consists of design in which each factor is studied at 2 levels. • If there are k factors, then we have 2k

treatment combinations

• 2-factor and 3-factor experiments can be generalized to >3-factor experiments

22 experiment• 22 Experiment: experiment with factors A and B,

each at two levels.

ab = (A high, B high) a = (A high, B low)b = (A low, B high) (1) = (A low, B low)

22 experiment cont’d

Yij ~ N(µi, σ2) i = (1), a, b, ab j = 1, 2, … , n

Assume a balanced design with n observations for each treatment combinations, denote these

observations by yij

22 experiment cont’d

• Main effect of factor A (): difference in the mean response between the high level of A and the low level of A, averaged over the levels of B

• Main effect of factor B (: difference in the mean response between the high level of B and the low level of B, averaged over the levels of A

• Interaction effect of AB (): difference between the mean effect of A at the high level of B and at the low level of B

= = ( =

22 experiment cont’d

Est. Main Effect A = = Est. Main Effect B = = Est. Interaction AB = =

The least square estimates of the main effects and the interaction effects are obtained by

replacing the treatment means by the corresponding cell sample means.

Contrast Coefficients for Effects in a 22 Experiment

Treatment

combination

EffectI A B AB

(1) + - - +a + + - -b + - + -ab + + + +*Notice that the term-by-term products of

any two contrast vectors equal the third one

22 experiment cont’d

23 experiment

• 23 Experiment: experiment with factors A, B, and C with n observations.

Yij ~ N(µi, σ2),

i = (1), a, b, ab, c, ac, bc, abc j = 1, 2, … , n.

Est. Main Effect A = Est. Main Effect B = Est. Main Effect C =

23 experiment cont’dEst. Interaction Effect AB =

Est. Interaction Effect BC =

Est. Interaction Effect AC =

Est. Interaction Effect ABC =

Contrast coefficients for Effects in a 23 ExperimentTreatment Combinati

on

EffectI A B AB C AC BC ABC

(1) + - - + - + + -a + + - - - - + +b + - + - - + - +ab + + + + - - - -c + - - + + - - +ac + + - - + + - -bc + - + - + - + -abc + + + + + + + +

23 experiment cont’d

23 experiment example

Factors affecting bicycle performance:

Seat height (Factor A): 26" (-), 30" (+)Generator (Factor B): Off (-), On(+) Tire Pressure (Factor C): 40 psi (-), 55

psi (+)

23 experiment example cont’d

Travel times from Bicycle ExperimentFactor Time (Secs.)

A B C Run 1 Run 2 Mean- - - 51 54 52.5+ - - 41 43 42.0- + - 54 60 57.0+ + - 44 43 43.5- - + 50 48 49.0+ - + 39 39 39.0- + + 53 51 52.0+ + + 41 44 42.5

23 experiment example cont’d

A = = -10.875B = = 3.125C = = -3.125AB = = -0.625AC = = 1.125BC = = 0.125ABC = = 0.875

significant

2k experiment

• 2k experiments, where k>3.

• n iid observations yij (j = 1,2,…n) at the ith treatment combination and its sample mean yi (i = 1,2,…, 2k) has the following estimated effect.

Est. Effect =

Statistical Inference for 2k Experiments Basic Notations and Derivations

•

d.f.

CI and Hypotheses Test with t Test

• Therefore a CI for any population effect is given by

• The t-statistic for testing the significance of any estimated effect is

Hypotheses Test with F Test

• Equivalently, we can use F test to do it

• The estimated effect is significant at level if

Sums of Squares for Effects

The effects are mutually orthogonal contrasts.

Regression Approach to 2k Experiments

• a 22 experiment

• Multiple regression model

Regression Approach to 2k Experiments

• 23 experiment

• If all interactions are dropped from the model, the new fitted model is

Regression Approach to 2k Experiments

• The interpolation formula

Bicycle Example: Main Effects Model

• main effects model

• minimum travel time

A(seat height)= -10.875 B(generator) = 3.125 C(tire pressure) = -3.125 = 47.1875

Bicycle Example: Main Effects Model

= 1.56+5.0625+0.0625+4.1875 = 10.875

d.f. = 4

Pure SSE = 33.5 d.f = 8

pooled SSE = 33.5 + 10.875 d.f. = 12 (total)

MSE = = 3.698

Sums of squares for omitted interactions effects

Bicycle Example: Residual

DiagnosticsTo check model assumptions

proc glm plots=diagnostics data = biker;class A B C;model travel= A|B|C;run;

Residuals

• Normality • Equal error variance

Single Replicated Case

• Unreplicated case: n =1• Problems in statistical testing• 0 degrees of freedom for error, • cannot use formal tests and C.I. to estimate of error and

assess effects

• Potential solutions• Pooling high-order interactions to estimate error• Graphical approach: normal plot against effects

• Estimated effects• Independent, orthogonal, normally distributed, common

variance (

Unusual response?

Noise? Spoiling the

results?

Single Replicated Case

• Effect Sparsity principle• If number of effects is large (e.g. k= 4, 15 effects), a majority

of them are small ~N (0,σ2), few a large and more influential ~ (u≠0, σ2)

• Reduced model• retaining only significant effects, omitting non-significant

ones• Obtain sums of squares for omitted effects => pooled error

sum of squares (SSE) (Error due to ignoring negligible effects)

• Error d.f. = # pooled omitted effects• MSE = SSE/error d.f. • Perform formal statistical inferences

Other Types of Two-Factor Experiments

Section 13.3

Two-Factor Experiments with (Crossed and) Mixed Factors

• A is fixed factor with a levels• B is random factor with b levels• Assume a balanced design with n ≥ 2 obs’s

at each of (a x b) treatment combinations

Example:

• Compare three testing laboratories• Material tested comes in batches• Several samples from each batch tested in each

laboratory• Laboratories represent a fixed factor• Batches represent a random factor• Two factors are crossed, since samples are tested from

each batch in each laboratory• Model?

Mixed Effects Model

• Yijk = µ + τi + ßj + (τß)ij + Єijk

µ,τi are fixed parameters

ßj, (τß)ij are random parameters

Єijk i.i.d. N(0,σ2) random errors

The (Probability) Distribution of the Random Effects

The random βj are the main effects of B, which are assumed to be i.i.d. N(0, σβ

2) where σβ2 is called

the variance component of the B (random factor) main effect. The distribution of βj would therefore be

f βj (x) = exp(-x2/2σβ2 )

• +• Variance Components Model• SST = SSA + SSB +SSAB +SSE

(same as fixed-effects model)

Variance Components Model

Expected Mean Squares

• E(MSA) = σ2 + nσ2AB + nΣi

a τi2

/(a-1)

• E(MSB) = σ2 + nσ2AB + anσ2

B

• E(MSAB) = σ2 + nσ2AB

• E(MSE) = σ2

Unbiased estimators of variance components

• 2 = MSE

• 2AB

= (MSAB - 2 )/n

• 2B

= (MSB - 2 - n 2AB) /an

Common tests

• H0A: τ1 = … =τa = 0 vs. H1A: At least one τi ≠ 0

• H0B: σ2B = 0 vs. H1B: σ2

B > 0

• H0AB: σ2AB = 0 vs. H1AB: σ2

AB > 0

Common tests: results

• Reject H0A if FA = MSA/MSAB > fa-1,(a-1)(b-1),α

• Reject H0B if FB = MSB/MSAB > fb-1,(a-1)(b-1),α

• Reject H0AB if FAB = MSAB/MSE > f(a-1)(b-1),v,α

Two-Factor Experiments w. Nested and Mixed Factors

• Model:• Where,

Two-Factor Experiments w. Nested and Mixed Factors

• Orthogonal Decomposition of Sum of Squares

Two-Factor Experiments w. Nested and Mixed Factors

• ANOVA Table

Illustrative Example

• Consider the Following Experiment:~ A Concentration of Reactant~ B Concentration of Catalyst

Analysis with SAS

• Code

Analysis with SAS• Selected Output

Summary• Two factor experiments with multiple levels

• Model:

• We can decompose the Sum of Squares as:

• And compute test statistics under Ho, as:

Summary• 2^k Factorial Experiments

• k factors, 2 levels each

• Calculate the Sum of Squares due to an effect as

Acknowledgements• Tamhane, Ajit C., and Dorothy D. Dunlop. "Analysis of

Multifactor Experiments." Statistics and Data Analysis: From Elementary to Intermediate. Upper Saddle River, NJ: Prentice Hall, 2000.

• Cody, Ronald P., and Jeffrey K. Smith. "Analysis of Variances: Two Independent Variables." Applied Statistics and the SAS Programming Language. 5th ed. Upper Saddle River, NJ: Prentice Hall, 2006.

• Prof. Wei Zhu

• Previous Presentations