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Multifactor Experiments November 26, 2013 Gui Citovsky, Julie Heymann, Jessica Sopp, Jin Lee, Qi Fan, Hyunhwan Lee, Jinzhu Yu, Lenny Horowitz, Shuvro Biswas

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Outline Two-Factor Experiments with Fixed Crossed Factors 2 k Factorial Experiments Other Selected Types of Two-Factor Experiments

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Two-Factor Experiments with Fixed Crossed Factors

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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

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Single factor continued

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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

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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

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Model and Estimates of its Parameters Let y ijk =k th observation on the (i,j) th treatment combination, i=1,2,…,a, j=1,2,…,b, and k=1,2,…,n. Let random variable Y ijk correspond to observed outcome y ijk. Basic Model: and independent where

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Table format

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Parameters Grand Mean: i th Row Average: j th Column Average: ( i,j ) th Row Column Interaction i th Row Main Effect: j th Column Main Effect:

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Least Squares Estimates

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Variance Sample variance for ( i, j ) th cell is: Pooled estimate for σ 2 :

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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: Al 2 O 3 with bracket, Al 2 O 3 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

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Example continued Pooled sample variance:

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Example continued: Sample Means

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Example continued: Other Model Parameters

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Two- Way Analysis of Variance We define the following sum of squares:

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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.

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Analysis of Variance

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Hypothesis Test We test three hypotheses: 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.

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When do we reject H 0 ? Use F -statistics to test our hypotheses by taking the ratio of the mean squares to the MSE. Reject We test the interaction hypothesis H 0AB first.

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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

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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;

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Bonding Strength of Capacitors ANOVA Table At α =0.05, we can reject H 0B and H 0AB but fail to reject H 0A. 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.

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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.

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Interaction Plot

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Model Diagnostics with Residual Plots Why do we look at residual plots? Is our constant variance assumption true? Is our normality assumption true?

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2 k Factorial Experiments

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2 k 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 2 k treatment combinations 2-factor and 3-factor experiments can be generalized to >3-factor experiments

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2 2 experiment 2 2 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)

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2 2 experiment cont’d Y ij ~ N(µ i, σ 2 ) i = (1), a, b, ab j = 1, 2, …, n Assume a balanced design with n observations for each treatment combinations, denote these observations by y ij

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2 2 experiment cont’d

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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.

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Contrast Coefficients for Effects in a 2 2 Experiment Treatment combinati on Effect IABAB (1)+--+ a++-- b+-+- ab++++ *Notice that the term-by-term products of any two contrast vectors equal the third one 2 2 experiment cont’d

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2 3 experiment 2 3 Experiment: experiment with factors A, B, and C with n observations. Y ij ~ N(µ i, σ 2 ), i = (1), a, b, ab, c, ac, bc, abc j = 1, 2, …, n.

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2 3 experiment cont’d

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Contrast coefficients for Effects in a 2 3 Experiment Treatment Combination Effect IABABCACBCABC (1)+--+-++- a++----++ b+-+--+-+ ab++++---- c+--++--+ ac++--++-- bc+-+-+-+- abc++++++++ 2 3 experiment cont’d

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2 3 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 (+)

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2 3 experiment example cont’d Travel times from Bicycle Experiment FactorTime (Secs.) ABCRun 1Run 2Mean ---515452.5 +--414342.0 -+-546057.0 ++-444343.5 --+504849.0 +-+39 39.0 -++535152.0 +++414442.5

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2 3 experiment example cont’d significant

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2 k experiment 2 k experiments, where k>3. n iid observations y ij ( j = 1,2,… n ) at the i th treatment combination and its sample mean y i ( i = 1,2,…, 2 k ) has the following estimated effect.

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Statistical Inference for 2 k Experiments Basic Notations and Derivations

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CI and Hypotheses Test with t Test

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Hypotheses Test with F Test

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Sums of Squares for Effects The effects are mutually orthogonal contrasts.

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Regression Approach to 2 k Experiments

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Bicycle Example: Main Effects Model

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Sums of squares for omitted interactions effects

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Bicycle Example: Residual Diagnostics To check model assumptions proc glm plots=diagnostics data = biker; class A B C; model travel= A|B|C; run; Normality Equal error variance

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Single Replicated Case Unusual response? Noise? Spoiling the results?

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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, σ 2 ), few a large and more influential ~ (u≠0, σ 2 ) 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

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Other Types of Two- Factor Experiments Section 13.3

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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

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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?

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Mixed Effects Model Y ijk = µ + τ i + ß j + ( τ ß) ij + Є ijk µ, τ i are fixed parameters ß j, ( τ ß) ij are random parameters Є ijk i.i.d. N(0, σ 2 ) random errors

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The (Probability) Distribution of the Random Effects

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Variance Components Model

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Expected Mean Squares E(MSA) = σ 2 + n σ 2 AB + n Σ i a τ i 2 /(a-1) E(MSB) = σ 2 + n σ 2 AB + an σ 2 B E(MSAB) = σ 2 + n σ 2 AB E(MSE) = σ 2

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Unbiased estimators of variance components

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Common tests H 0A : τ 1 = … = τ a = 0 vs. H 1A : At least one τ i ≠ 0 H 0B : σ 2 B = 0 vs. H 1B : σ 2 B > 0 H 0AB : σ 2 AB = 0 vs. H 1AB : σ 2 AB > 0

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Common tests: results Reject H 0A if F A = MSA/MSAB > f a-1,(a-1)(b-1), α Reject H 0B if F B = MSB/MSAB > f b-1,(a-1)(b-1), α Reject H 0AB if F AB = MSAB/MSE > f (a-1)(b-1),v, α

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Two-Factor Experiments w. Nested and Mixed Factors Model: Where,

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Two-Factor Experiments w. Nested and Mixed Factors Orthogonal Decomposition of Sum of Squares

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Two-Factor Experiments w. Nested and Mixed Factors ANOVA Table

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

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Analysis with SAS Code

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Analysis with SAS Selected Output

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Summary Two factor experiments with multiple levels Model: We can decompose the Sum of Squares as: And compute test statistics under Ho, as:

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Summary 2^k Factorial Experiments k factors, 2 levels each Calculate the Sum of Squares due to an effect as

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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

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