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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

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## Statistical Analysis

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**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**Extra Sum of Squares F-Testfor 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 Change in Average Response Due to Changes in the Levels of Factor A Main Effects for Factor B Change in Average Response Due to Changes in the Levels of Factor B Interaction Effects for Factors A & B Change in Average Response Due Joint Changes in Factors A & B in Excess of Changes in the Main Effects**Ho: q = q0 vs Ha: qq0**Reject Ho if | t | > t a/2 Individual Prespecified Comparisons(Single-factor Model Used for Illustration) Very general result MGH Sec 6.2.2**Ho: ai = ai’ vs Ha: ai ai’**Reject Ho if | t | > t a/2 Ordinary t-Test Pairwise Comparisons Specific application**Algebraic Main Effects Representation**Two Factors, Two Levels (No Repeats for Simplicity) y = (y11 y12 y21 y22) mA= ( -1 -1 +1 +1) mB= ( -1 +1 -1 +1) M(B) = mB’y/2**Algebraic Interaction Effects Representation**Two Factors, Two Levels (No Repeats for Simplicity) y = (y11 y12 y21 y22) mAB= ( +1 -1 -1 +1)**Effects Representation :Two-Level, Two-Factor Factorial**Factor Levels: Lower = - 1 Upper = +1 Factor LevelsEffects Representation Factor A Factor B A B AB Lower Lower Upper Upper Lower Upper Lower Upper -1 -1 1 1 -1 1 -1 1 1 -1 -1 1 Mutually Orthogonal Contrasts Note: AB = A x B MGH Table 5.6**Calculated Effects**y = Vector of Averages Across Repeats (Response Vector if r = 1) m = Effects RepresentationVector Effect = m’ y / 2k-1**Pilot Plant Chemical-Yield Study**Conclusion ? MGH Table 6.4**Multiple Comparisons**Several comparisons of factor means or of factor effects using procedures that control the overall significance or confidence level Comparisonwise Error Rate aC = Pr(Type 1 Error) for one statistical test Experimentwise Error Rate aE = Pr(One or More Type 1 Errors) for two or more rests**Experimentwise Error Rate :k Independent Statistical Tests**Experimentwise & Comparisonwise Error Rates Assumes Independence aC = .05 Dependent Tests Common MSE Lack of Orthogonality**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 ni : Use MGH Exhibit 6.13**Studentized Range Statistic**Assume Studentized Range unequal ni**Tukey’s “Honest” Significant Difference (HSD or TSD)**MGH Exhibit 6.11**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 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)**Simultaneous**Test Single Degree-of- Freedom Contrasts Sums of Squares and Contrasts a-1 Mutually Orthonormal Contrast Vectors Orthonormal Basis Set ANY Set of Orthonormal Contrast Vectors**Model and Assumptions**yijk = m + ai + bj+ (ab)ij+eijk where yij = warping measurement for the kth repeat at the ith temperature using a plate having the jth amount of copper m = overall mean warping measurement ai = fixed effect of the ith temperature on the mean warping bi = fixed effect of the jth copper content on the mean warping (ab)ij = fixed effect of the interaction between the ith temperature and the jth copper content on the mean warping eij = random experimental error, NID(0,s2)**Warping of Copper Plates**Quantitative Factor Levels HOW Does Mean Warping Change with the Factor Levels ? MGH Table 6.7**Warping of Copper Plates**Are There Contrast Vectors That Quantify Curvature ? 35 30 25 Average Warping 20 15 10 0 50 75 150 100 125 Temperature (deg F)**Warping of Copper Plates**Are There Contrast Vectors That Quantify Curvature ? 35 30 25 Average Warping 20 15 10 0 20 40 60 80 100 Copper Content (%)**Main Effects Contrasts :Equally Spaced Quantitative Factor**Levels n=4 q1 = Linear q2 = Quadratic q3 = Cubic**Linear Combinations of Parameters**Estimable Functions of Parameters Estimator Same for Contrasts Standard Error t Statistic**Scaled Contrasts**Note: Need Scaling to Make Polynomial Contrasts Comparable**Resolution III (R = 3)**Main Effects (s = 1) are unconfounded with other main effects (R - s = 2) Example : Half-Fraction of 23 (23-1) Design Resolution Resolution R Effects involving s Factors are unconfounded with effects involving fewer than R-s factors**Designing a 1/2 Fraction of a 2k Complete Factorial**Resolution = k • 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**Designing Higher-Order Fractions**• Total number of factor-level combinations = 2k • Experiment size desired = 2k/2p = 2k-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 selectedrepeat test runs • Randomize**Design Resolution for Fractional Factorials**• Determine the p defining equations • Determine the 2p - p - 1 implicit defining equations: symbolically multiply all of the defining equationsResolution = Smallest “word’ length in the defining & implicit equations • Each effect has 2p aliases**26-2 Fractional Factorials :Confounding Pattern**Build From 1/4 Fraction RIII I = ABCDEF = ABC = DEF A = BCDEF = BC = BDEF B = ACDEF = AC = ADEF . . . (I + ABCDEF)(I + ABC) = I + ABCDEF + ABC + DEF Defining Contrasts Implicit Contrast**Optimal 1/4 Fraction**RIV I = ABCD = CDEF = ABEF A = BCD = ACDEF = BEF B = ACD = BCDEF = AEF . . . 26-2 Fractional Factorials :Confounding Pattern Build From 1/2 Fraction RIII I = ABCDEF = ABC = DEF A = BCDEF = BC = BDEF B = ACDEF = AC = ADEF . . .**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**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

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