Statistical Analysis Professor Lynne Stokes Department of Statistical Science Lecture 15 Review

Brief Review Complete Factorial Experiments Completely Randomized Designs Main Effects Interactions Analysis of Variance Sums of Squares Degrees of Freedom Tests of Factor Effects Multiple Comparisons of Means Orthogonal polynomials Fractional Factorial Experiments Half Fractions, Higher-Order Fractions Aliases (Confounding), Design Resolution Screening experiments

Extra Sum of Squares Hierarchy Principle An interaction is included only if ALL main effects and lower-order interactions involving the interaction factors are included in the model and analysis Full Model All hierarchical model terms Reduced Model One or more model terms deleted All remaining model terms are hierarchical

Hierarchical Models

Extra Sum of Squares F-Test for Hierarchical Models For Balanced Designs, F Tests for Fixed Effects Derived from QF, Mean Squares, and ESS are Identical

General Method for Quantifying Factor Main Effects and Interactions Main Effects for Factor A Main Effects for Factor B Interaction Effects for Factors A & B Change in Average Response Due to Changes in the Levels of Factor A Change in Average Response Due to Changes in the Levels of Factor B Change in Average Response Due Joint Changes in Factors A & B in Excess of Changes in the Main Effects

Individual Prespecified Comparisons (Single-factor Model Used for Illustration) Ho:  =  0 vs Ha:   0 Reject Ho if | t | > t  /2 Very general result MGH Sec 6.2.2

Pairwise Comparisons Specific application Specific application Ho:  i  i’ vs Ha:  i  i’ Reject Ho if | t | > t  /2 Ordinary t-Test

Algebraic Main Effects Representation y = (y 11 y 12 y 21 y 22 ) m A = ( ) M(B) = m B ’y /2 m B = ( ) Two Factors, Two Levels (No Repeats for Simplicity)

Algebraic Interaction Effects Representation y = (y 11 y 12 y 21 y 22 ) m AB = ( ) Two Factors, Two Levels (No Repeats for Simplicity)

Effects Representation : Two-Level, Two-Factor Factorial Factor Levels: Lower = - 1 Upper = +1 Factor Levels Effects Representation Factor AFactor B A B AB Lower Upper Lower Upper Lower Upper Note: AB = A x B MGH Table 5.6 Mutually Orthogonal Contrasts Mutually Orthogonal Contrasts

Calculated Effects y = Vector of Averages Across Repeats (Response Vector if r = 1) m = Effects RepresentationVector Effect = m’ y / 2 k-1

Pilot Plant Chemical-Yield Study MGH Table 6.4 Conclusion ?

Multiple Comparisons Several comparisons of factor means or of factor effects using procedures that control the overall significance or confidence level Several comparisons of factor means or of factor effects using procedures that control the overall significance or confidence level Comparisonwise Error Rate  C = Pr(Type 1 Error) for one statistical test Experimentwise Error Rate  E = Pr(One or More Type 1 Errors) for two or more rests

Experimentwise Error Rate : k Independent Statistical Tests Experimentwise & Comparisonwise Error Rates Dependent Tests Common MS E Lack of Orthogonality Common MS E Lack of Orthogonality  C  Assumes Independence

Fisher’s Least Significant Difference (LSD) Protected: Preceded by an F Test for Overall Significance Unprotected: Not Preceded by an F Test – Individual t Tests MGH Exhibit 6.9

Least Significant Interval (LSI) Plot LSI Plot Plot the averages, with bars extending LSD/2 above & below each average. Bars that do NOT overlap indicate significantly different averages. If Unequal n i : Use MGH Exhibit 6.13

Studentized Range Statistic Assume Studentized Range unequal n i

Tukey’s “Honest” Significant Difference (HSD or TSD) MGH Exhibit 6.11

Bonferroni Method

Bonferroni Multiple Comparisons (BSD) Number of Pairwise Comparisons

Contrasts of Effects Estimable Factor Effects Contrasts Elimination of the Overall Mean Requires Contrasts of Main Effect Averages. (Note: Want to Compare Factor Effects.) Elimination of Main Effects from Interaction Comparisons Requires Contrasts of the Interaction Averages. (Note: Want Interaction Effects to Measure Variability that is Unaccounted for by or in Addition to the Main Effects.)

Statistical Independence Orthogonal Linear Combinations are Statistically Independent Orthogonal Linear Combinations are Statistically Independent Orthogonal Contrasts are Statistically Independent Orthogonal Contrasts are Statistically Independent

Main Effects Contrasts : Qualitative Factor Levels Three statistically independent contrasts of the response averages A partitioning of the main effects degrees of freedom into single degree-of-freedom contrasts (a = 4: df = 3)

Sums of Squares and Contrasts Single Degree-of- Freedom Contrasts Single Degree-of- Freedom Contrasts Simultaneous Test Simultaneous Test a-1 Mutually Orthonormal Contrast Vectors a-1 Mutually Orthonormal Contrast Vectors Orthonormal Basis Set Orthonormal Basis Set ANY Set of Orthonormal Contrast Vectors ANY Set of Orthonormal Contrast Vectors

y ijk =  +  i +  j  +  ij  +  e ijk where y ij = warping measurement for the kth repeat at the ith temperature using a plate having the jth amount of copper  = overall mean warping measurement  i = fixed effect of the ith temperature on the mean warping  i = fixed effect of the jth copper content on the mean warping (  ij = fixed effect of the interaction between the ith temperature and the jth copper content on the mean warping e ij = random experimental error, NID(0,  2 ) Model and Assumptions

Warping of Copper Plates MGH Table 6.7 Quantitative Factor Levels HOW Does Mean Warping Change with the Factor Levels ? Quantitative Factor Levels HOW Does Mean Warping Change with the Factor Levels ?

Warping of Copper Plates Temperature (deg F) Average Warping Are There Contrast Vectors That Quantify Curvature ? Are There Contrast Vectors That Quantify Curvature ?

Warping of Copper Plates Copper Content (%) Average Warping Are There Contrast Vectors That Quantify Curvature ? Are There Contrast Vectors That Quantify Curvature ?

Main Effects Contrasts : Equally Spaced Quantitative Factor Levels   = Linear   = Quadratic   = Cubic n=4

Linear Combinations of Parameters Estimable Functions of Parameters Estimator Standard Error t Statistic Same for Contrasts

Warping of Copper Plates

Scaled Contrasts Note: Need Scaling to Make Polynomial Contrasts Comparable

Design Resolution Resolution R Effects involving s Factors are unconfounded with effects involving fewer than R-s factors Resolution III (R = 3) Main Effects (s = 1) are unconfounded with other main effects (R - s = 2) Example : Half-Fraction of 2 3 (2 3-1 )

Designing a 1/2 Fraction of a 2 k Complete Factorial Write the effects representation for the main effects and the highest-order interaction for a complete factorial in k factors Randomly choose the +1 or -1 level for the highest-order interaction (defining contrast, defining equation) Eliminate all rows except those of the chosen level (+1 or -1) in the highest-order interaction Add randomly chosen repeat tests, if possible Randomize the test order or assignment to experimental units Resolution = k

Designing Higher-Order Fractions Total number of factor-level combinations = 2 k Experiment size desired = 2 k /2 p = 2 k-p Choose p defining contrasts (equations) For each defining contrast randomly decide which level will be included in the design Select those combinations which simultaneously satisfy all the selected levels Add randomly selected repeat test runs Randomize

Design Resolution for Fractional Factorials Determine the p defining equations Determine the 2 p - p - 1 implicit defining equations: symbolically multiply all of the defining equations Resolution = Smallest “word’ length in the defining & implicit equations Each effect has 2 p aliases

2 6-2 Fractional Factorials : Confounding Pattern Build From 1/4 Fraction I = ABCDEF = ABC = DEF A = BCDEF = BC = BDEF B = ACDEF = AC = ADEF... R III (I + ABCDEF)(I + ABC) = I + ABCDEF + ABC + DEF Defining ContrastsImplicit Contrast

2 6-2 Fractional Factorials : Confounding Pattern Build From 1/2 Fraction I = ABCDEF = ABC = DEF A = BCDEF = BC = BDEF B = ACDEF = AC = ADEF... R III Optimal 1/4 Fraction I = ABCD = CDEF = ABEF A = BCD = ACDEF = BEF B = ACD = BCDEF = AEF... R IV

Screening Experiments Very few test runs Ability to assess main effects only Generally leads to a comprehensive evaluation of a few dominant factors Potential for bias Highly effective for isolating vital few strong effects should be used ONLY under the proper circumstances Highly effective for isolating vital few strong effects should be used ONLY under the proper circumstances

Plackett-Burman Screening Designs Any number of factors, each having 2 levels Interactions nonexistent or negligible Relative to main effects Number of test runs is a multiple of 4 At least 6 more test runs than factors should be used

Fold-Over Designs Reverse the signs on one or more factors Run a second fraction with the sign reversals Use the confounding pattern of the original and the fold-over design to determine the alias structure Averages Half-Differences