Showing papers on "Fractional factorial design published in 1999"

Designing Experiments and Analyzing Data: A Model Comparison Perspective, Third Edition

Abstract: Contents: Preface. Part I: Conceptual Bases of Experimental Design and Analysis. The Logic of Experimental Design. Introduction to the Fisher Tradition. Part II: Model Comparisons for Between-Subjects Designs. Introduction to Model Comparisons: One-Way Between-Subjects Designs. Individual Comparisons of Means. Testing Several Contrasts: The Multiple-Comparison Problem. Trend Analysis. Two-Way Between-Subjects Factorial Designs. Higher Order Between-Subjects Factorial Designs. Designs With Covariates: ANCOVA and Blocking. Designs With Random or Nested Factors. Part III: Model Comparisons for Designs Involving Within-Subjects Factors. One-Way Within-Subjects Designs: Univariate Approach. Higher-Order Designs With Within-Subjects Factors: Univariate Approach. One-Way Within-Subjects Designs: Multivariate Approach. Higher Order Designs With Within-Subjects Factors: Multivariate Approach. Part IV: Alternative Analysis Strategies. An Introduction to Multilevel Models for Within-Subjects Designs. An Introduction to Multilevel Hierarchical Mixed Models: Nested Designs. Appendices: Statistical Tables. Part 1. Linear Models: The Relation Between ANOVA and Regression. Part 2. A Brief Primer of Principles of Formulating and Comparing Models. Notes. Solutions to Selected Exercises. References.

Design of Experiments: Statistical Principles of Research Design and Analysis

Abstract: 1 RESEARCH DESIGN PRINCIPLES The Legacy of Sir Ronald A Fisher / Planning for Research / Experiments, Treatments, and Experimental Units / Research Hypotheses Generate Treatment Designs / Local Control of Experimental Errors / Replication for Valid Experiments / How Many Replications? / Randomization for Valid Inferences / Relative Efficiency of Experiment Designs / From Principles to Practice: A Case Study 2 GETTING STARTED WITH COMPLETELY RANDOMIZED DESIGNS Assembling the Research Design / How to Randomize / Preparation of Data Files for the Analysis / A Statistical Model for the Experiment / Estimation of the Model Parameters with Least Squares / Sums of Squares to Identify Important Sources of Variation / A Treatment Effects Model / Degrees of Freedom / Summaries in the Analysis of Variance Table / Tests of Hypotheses About Linear Models / Significance Testing and Tests of Hypotheses / Standard Errors and Confidence Intervals for Treatment Means / Unequal Replication of the Treatments / How Many Replications of the F Test? / Appendix: Expected Values / Appendix: Expected Mean Squares 3 TREATMENT COMPARISONS Treatment Comparisons Answer Research Questions / Planning Comparisons Among Treatments / Response Curves for Quantitative Treatment Factors / Multiple Comparisons Affect Error Rates / Simultaneous Statistical Inference / Multiple Comparisons with the Best Treatment / Comparison of All Treatments with a Control / Pairwise Comparisons of All Treatments / Summary Comments on Multiple Comparisons / Appendix: Linear Functions of Random Variables 4 DIAGNOSING AGREEMENT BETWEEN THE DATA AND THE MODEL Valid Analysis Depends on Valid Assumptions / Effects of Departures from Assumptions / Residuals Are the Basis of Diagnostic Tools / Looking for Outliers with the Residuals / Variance-Stabilizing Transformations for Data with Known Distributions / Power Transformations to Stabilize Variances / Generalizing the Linear Model / Model Evaluation with Residual-Fitted Spread Plots / Appendix: Data for Example 41 5 EXPERIMENTS TO STUDY VARIANCES Random Effects Models for Variances / A Statistical Model for Variance Components / Point Estimates of Variance Components / Interval Estimates for Variance Components / Courses of Action with Negative Variance Estimates / Intraclass Correlation Measures Similarity in a Group / Unequal Numbers of Observations in the Groups / How Many Observations to Study Variances? / Random Subsamples to Procure Data for the Experiment / Using Variance Estimates to Allocate Sampling Efforts / Unequal Numbers of Replications and Subsamples / Appendix: Coefficient Calculations for Expected Mean Squares in Table 59 6 FACTORIAL TREATMENT DESIGNS Efficient Experiments with Factorial Treatment Designs / Three Types of Treatment Factor Effects / The Statistical Model for Two Treatment Factors / The Analysis for Two Factors / Using Response Curves for Quantitative Treatment Factors / Three Treatment Factors / Estimation of Error Variance with One Replication / How Many Replications to Test Factor Effects? / Unequal Replication of Treatments / Appendix: Least Squares for Factorial Treatment Designs 7 FACTORIAL TREATMENT DESIGNS: RANDOM AND MIXED MODELS Random Effects for Factorial Treatment Designs / Mixed Models / Nested Factor Designs: A Variation on the Theme / Nested and Crossed Factors Designs / How Many Replications? / Expected Mean Square Rules 8 COMPLETE BLOCK DESIGNS Blocking to Increase Precision / Randomized Complete Block Designs Use One Blocking Criterion / Latin Square Designs Use Two Blocking Criteria / Factorial Experiments in Complete Block Designs / Missing Data in Blocked Designs / Experiments Performed Several Times / Appendix: Selected Latin Squares 9 INCOMPLETE BLOCK DESIGNS: AN INTRODUCTION Incomplete Blocks of Treatments to Reduce Block Size / Balanced Incomplete Block (BIB) Designs / How to Randomize Incomplete Block Designs / Analysis of BIB Designs / Row-Column Designs for Two Blocking Criteria / Reduce Experiment Size with Partially Balanced (PBIB) Designs / Efficiency of Incomplete Block Designs / Appendix: Selected Balanced Incomplete Block Designs / Appendix: Selected Incomplete Latin Square Designs / Appendix: Least Squares Estimates for BIB Designs 10 INCOMPLETE BLOCK DESIGNS: RESOLVABLE AND CYCLIC DESIGNS Resolvable Designs to Help Manage the Experiment / Resolvable Row-Column Designs for Two Blocking Criteria / Cyclic Designs Simplify Design Construction / Choosing Incomplete Block Designs / Appendix: Plans for Cyclic Designs / Appendix: Generating Arrays for a Designs 11 INCOMPLETE BLOCK DESIGNS: FACTORIAL TREATMENT DESIGNS Taking Greater Advantage of Factorial Treatment Designs / 2 to the nth Power Factorials to Evaluate Many Factors / Incomplete Block Designs for 2 to the nth Power Factorials / A General Method to Create Incomplete Blocks / Incomplete Blocks for 3 to the nth Power Factorials / Concluding Remarks / Appendix: Incomplete Block Design Plans for 2 to the nth Power Factorials 12 FRACTIONAL FACTORIAL DESIGNS Reduce Experiment Size with Fractional Treatment Designs / The Half Fraction of the 2 to the nth Power Factorial / Design Resolution Related to Aliases / Analysis of Half Replicate 2^n - 1 Designs / The Quarter Fractions of 2 to the nth Power Factorials / Construction of 2^(n - p) Designs with Resolution III and IV / Genichi Taguchi and Quality Improvement / Concluding Remarks / Appendix: Fractional Factorial Design Plans 13 RESPONSE SURFACE DESIGNS Describe Responses with Equations and Graphs / Identify Important Factors with 2 to the nth Power Factorials / Designs to Estimate Second-Order Response Surfaces / Quadratic Responses Surface Estimation / Response Surface Exploration / Designs for Mixtures of Ingredients / Analysis of Mixture Experiments / Appendix: Least Squares Estimation of Regression Models / Appendix: Location of Coordinates for the Stationary Point / Appendix: Canonical Form of the Quadratic Equation 14 SPLIT-PLOT DESIGNS Plots of Different Size in the Same Experiment / Two Experimental Errors for Two Plot Sizes / The Analysis for Split-Plot Designs / Standard Errors for Treatment Factor Means / Features of the Split-Plot Design / Relative Efficiency of Subplot and Whole-Plot Comparisons / The Split-Split-Plot Design for Three Treatment Factors / The Split-Block Design / Additional Information About Split-Plot Designs 15 REPEATED MEASURES DESIGNS Studies of Time Trends / Relationships Among Repeated Measurements / A Test for the Huynh-Feldt Assumption / A Univariate Analysis of Variance for Repeated Measures / Analysis When Univariate Analysis Assumptions Do Not Hold / Other Experiments with Repeated Measures Properties / Other Models for Correlation Among Repeated Measures / Appendix: The Mauchly Test for Sphericity / Appendix: Degrees of Freedom Adjustments for Repeated Measures Analysis of Variance 16 CROSSOVER DESIGNS Administer All Treatments to Each Experimental Unit / Analysis of Crossover Designs / Balanced Designs for Crossover Studies / Crossover Designs for Two Treatments / Appendix: Coding Data Files for Crossover Studies / Appendix: Treatment Sum of Squares for Balanced Designs 17 ANALYSIS OF COVARIANCE Local Control with a Measured Covariate / Analysis of Covariance for Completely Randomized Block Designs / The Analysis of Covariance for Blocked Experiment Designs / Practical Consequences of Covariance Analysis / REFERENCES / APPENDIX TABLES / ANSWERS TO SELECTED EXERCISES / INDEX

Fractional Factorial Plans

Abstract: Fractional Plans and Orthogonal Arrays. Symmetric Orthogonal Arrays. Asymmetric Orthogonal Arrays. Some Results on Nonexistence. More on Optimal Fractional Plans and Related Topics. Trend-Free Plans and Blocking. Some Further Developments. Appendix. References. Index.

Minimum $G_2$-aberration for nonregular fractional factorial designs

Abstract: Deng and Tang proposed generalized resolution and minimum aberration criteria for comparing and assessing nonregular fractional factorials, of which Plackett–Burman designs are special cases.A relaxed variant of generalized aberration is proposed and studied in this paper.We show that a best design according to this criterion minimizes the contamination of nonnegligible interactions on the estimation of main effects in the order of importance given by the hierarchical assumption.The new criterion is defined through a set of $B$ values, a generalization of word length pattern. We derive some theoretical results that relate the $B$ values of a nonregular fractional factorial and those of its complementary design. Application of this theory to the construction of the best designs according to the new aberration criterion is discussed. The results in this paper generalize those in Tang and Wu, which characterize a minimum aberration (regular) $2^{m-k}$ design through its complementary design.

Minimum-aberration two-level fractional factorial split-plot designs

Abstract: It is often impractical to perform the experimental runs of a fractional factorial in a completely random order, In these cases, restrictions on the randomization of the experimental trials are imposed and the design is said to have a split-plot structure. We rank these fractional factorial split-plot designs similarly to fractional factorials using the aberration criterion to find the minimum-aberration design. We introduce an algorithm that constructs the set of all nonisomorphic two-level fractional factorial split-plot designs more efficiently than existing methods. The algorithm can be easily modified to efficiently produce sets of all nonisomorphic fractional factorial designs, fractional factorial designs in which the number of levels is a power of a prime, and fractional factorial split-plot designs in which the number of levels is a power of a prime.

Minimum aberration and model robustness for two-level fractional factorial designs

Abstract: Summary. The performance of minimum aberration two-level fractional factorial designs is studied under two criteria of model robustness. Simple sufficient conditions for a design to dominate another design with respect to each of these two criteria are derived. It is also shown that a minimum aberration design of resolution IlIl or higher maximizes the number of two-factor interactions which are not aliases of main effects and, subject to that condition, minimizes the sum of squares of the sizes of alias sets of two-factor interactions. This roughly says that minimum aberration designs tend to make the sizes of the alias sets very uniform. It follows that minimum aberration is a good surrogate for the two criteria of model robustness that are studied here. Examples are given to show that minimum aberration designs are indeed highly efficient.

Theory of optimal blocking of $2^{n-m}$ designs

Abstract: In this paper, we define the blocking wordlength pattern of a blocked fractional factorial design by combining the wordlength patterns of treatment-defining words and block-defining words. The concept of minimum aberration can be defined in terms of the blocking wordlength pattern and provides a good measure of the estimation capacity of a blocked fractional factorial design. By blending techniques of coding theory and finite projective geometry, we obtain combinatorial identities that govern the relationship between the blocking wordlength pattern of a blocked $2^{n-m}$ design and the split wordlength pattern of its blocked residual design. Based on these identities, we establish general rules for identifying minimum aberration blocked $2^{n-m}$ designs in terms of their blocked residual designs. Using these rules, we study the structures of some blocked $2^{n-m}$ designs with minimum aberration.

Medium optimization for spore production of coniothyrium minitans using statistically-based experimental designs

Abstract: Statistically-based experimental designs were used to optimize a chemically defined solid medium for the spore production of Coniothyrium minitans. In the first optimization step the influence of starch, urea, phosphate, magnesium, calcium, thiamin and trace elements on spore production was evaluated using a fractional factorial design. Starch and trace elements influenced spore production positively while urea affected spore production negatively. The other components had no significant influence on spore production. In the second and third steps the concentrations of starch, urea and trace elements were further optimized using central composite designs and response surface analysis. This optimization strategy allowed the spore production to be increased by a factor 7 from 4 x 10(9) to almost 3 x 10(10) spores per Petri dish of 9 cm diameter. Copyright 1999 John Wiley & Sons, Inc.

Some theoretical results for fractional factorial split-plot designs

Abstract: Fractional factorial (FF) designs are commonly used in industrial ex-periments to identify factors affecting a process. When it is expensive or difficult to change the levels of some of the factors, fractional factorial split-plot (FFSP) designs represent a practical design option. Though FFSP de-sign matrices correspond to FF design matrices, the randomization struc-ture of the FFSP design is different. In this paper, we discuss the impact of randomization restrictions on the choice of FFSP designs and develop theoretical results. Some of these results are very closely related to those available for FF designs while others are more specific to FFSP designs and are more useful in practice. We pay particular attention to the min-imum aberration criterion (MA) and emphasize the differences between FFSP and FF designs.

Experimental approach to selection of pulsing parameters in pulsed GMAW

Abstract: An efficient method of iden- tifying power supply pulsing parameters for pulsed gas metal arc welding based on statistical experimental design is pre- sented. Fractional factorial screening ex- periments are combined with D-optimal experimental designs to allow the user to develop an accurate wire feed rate model for varying pulsing conditions and to characterize the desirable one droplet per pulse (ODPP) operating region for a given wire type and diameter. Equations defining the wire feed rates and time at a given peak current required for ODPP transfer are presented. Compared to con- ventional techniques, a very small num- ber of experiments is required. A power- supply-dependent approach and a more generic method employing measured rather than nominal current values are presented. Joints produced using this ap- proach are evaluated and found to meet applicable bead geometry standards.

A top-down in-house validation based approach for the investigation of the measurement uncertainty using fractional factorial experiments

Abstract: The recently presented in-house validation concept developed on the basis of a variance component model is extended and optimised by basing it on a fractional factorial sampling scheme. The presented approach relies on a comprehensive mathematical model to assess simultaneously the in-house reproducibility and its components in addition to uncertainty-related performance characteristics like power functions and critical concentrations. The influence of individual factors and the corresponding uncertainties can be determined separately in a single experiment with optimum utilization of the generated data, in this way achieving a favourable ratio between costs and result.

A comparison of factorial and random experimental design methods for the development of regression and neural network simulation metamodels

Abstract: This paper compares two forms of experimental design methods that may be used for the development of regression and neural network simulation metamodels. The experimental designs considered are full factorial designs and random designs. The paper shows that, for two example problems, neural network metamodels using a randomised experimental design produce more accurate and efficient metamodels than those produced by similar sized factorial designs with either regression or neural networks. The metamodelling techniques are compared by their ability to predict the results from two manufacturing systems that have different levels of complexity. The results of the comparison suggest that neural network metamodels outperform conventional regression metamodels, especially when data sets based on randomised simulation experimental designs are used to produce the metamodels rather than data sets from similar sized full factorial experimental designs.

Blocking in regular fractional factorials: a projective geometric approach

Abstract: A projective geometric characterization is given of the existence of any regular main effect $s^{n-k}$ design in $s^{\gamma}$ blocks It leads to a constructive method for finding a maximal blocking scheme for any given fractional factorial design A useful sufficient condition for admissible block designs is given in terms of the minimum aberration property of a certain unblocked design

An optimisation procedure to determine the coefficients of the extended Taylor's equation in machining

Abstract: An optimum experimental design to determine the coefficients of the Extended Taylor's Equation in machining is proposed. The technique is based on the minimisation of the ratio between maximum and minimum singular values of the matrix of sensitivity of the tool life related to the machining parameter variations. This procedure generates the best set of cutting conditions to be used in tool life tests which results in a fast convergence of the coefficients and their confidence intervals. This technique was compared to the commonly used fractional factorial design when face milling AISI 1045 steel with cemented carbide cutting tools. The results showed a considerable reduction in the number of tests required to obtain a reliable equation when the optimum experimental procedure was used when compared to the factorial design.

Design of Experiments with Two- and Four-Level Factors

Abstract: [This abstract is based on the author's abstract.] A guide is provided for the inclusion of four-level factors into standard two-level factorial designs. Practitioners familiar with two-level fractional factorials often are frustrated when confronted ..

Statistical Application of Orthogonal Arrays

Abstract: Rao (1947) introduced orthogonal arrays because of their desirable statistical properties when used in “fractional factorial” experiments. Nowadays the main statistical application of orthogonal arrays, with mixed levels or otherwise, is still as fractional factorials, although other applications have been discovered. We will present the main application in considerable detail, while only giving key references for the other applications. Unless stated otherwise, throughout this chapter the term orthogonal array is to be interpreted as including mixed level arrays.

Saturated main-effect designs for factorial experiments

Abstract: The results of a computer search for saturated designs for 2^n factorial experiments with n runs is reported, (where n = 2 mod 4). A complete search of the design space is avoided by focussing on designs constructed from cyclic generators. A method of searching quickly for the best generators is given. The resulting designs are as good as, and sometimes better than, designs obtained via search algorithms reported in the literature. The addition of a further factor having three levels is also considered. Here, too, a complete search is avoided by restricting attention to the most efficient part of the design space under p-efficiency.

An algorithm for the design of factorial experiments when the data are correlated

Abstract: Since the development of methods for the analysis of experiments with dependent data, see for example Gleeson and Cullis (1987), the design of such experiments has been an area of active research. We investigate the design of factorial experiments, complete and fractional, for various dependency structures. An algorithm for generating optimal or near optimal designs is presented and shown to be useful across a wide range of dependency structures.