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Design and Analysis Af Experiments with K Factors Having P Levels Design and Analysis af Experiments with k Factors having p Levels Henrik Spliid Lecture notes in the Design and Analysis of Experiments 1st English edition 2002 Informatics and Mathematical Modelling Technical University of Denmark, DK–2800 Lyngby, Denmark 0 c hs. Design of Experiments, Course 02411, IMM, DTU 1 Foreword These notes have been prepared for use in the course 02411, Statistical Design of Ex- periments, at the Technical University of Denmark. The notes are concerned solely with experiments that have k factors, which all occur on p levels and are balanced. Such ex- periments are generally called pk factorial experiments, and they are often used in the laboratory, where it is wanted to investigate many factors in a limited - perhaps as few as possible - number of single experiments. Readers are expected to have a basic knowledge of the theory and practice of the design and analysis of factorial experiments, or, in other words, to be familiar with concepts and methods that are used in statistical experimental planning in general, including for example, analysis of variance technique, factorial experiments, block experiments, square experiments, confounding, balancing and randomisation as well as techniques for the cal- culation of the sums of squares and estimates on the basis of average values and contrasts. The present version is a revised English edition, which in relation to the Danish has been improved as regards contents, layout, notation and, in part, organisation. Substantial parts of the text have been rewritten to improve readability and to make the various methods easier to apply. Finally, the examples on which the notes are largely based have been drawn up with a greater degree of detailing, and new examples have been added. Since the present version is the first in English, errors in formulation an spelling may occur. Henrik Spliid IMM, March 2002 April 2002: Since the version of March 2002 a few corrections have been made on the pages 21, 25, 26, 40, 68 and 82. Lecture notes for course 02411. IMM - DTU. c hs. Design of Experiments, Course 02411, IMM, DTU 2 c hs. Design of Experiments, Course 02411, IMM, DTU 3 Contents 1 4 1.1Introduction................................... 4 1.2 Literature suggestions concerning the drawing up and analysis of factorial experiments................................... 5 22k{factorial experiment 7 2.1 Complete 2k factorialexperiments....................... 7 2.1.1 Factors.................................. 7 2.1.2 Design.................................. 7 2.1.3 Modelforresponse,parametrisation.................. 8 2.1.4 Effects in 2k–factor experiments .................... 9 2.1.5 Standardnotationforsingleexperiments.............. 9 2.1.6 Parameterestimates.......................... 10 2.1.7 Sumsofsquares............................. 11 2.1.8 Calculationmethodsforcontrasts.................. 11 2.1.9 Yates’algorithm............................ 12 2.1.10Replicationsorrepetitions....................... 13 2.1.11 23 factorialdesign............................ 14 2.1.12 2k factorialexperiment ........................ 17 2.2 Block confounded 2k factorialexperiment.................. 18 2.2.1 Construction of a confounded block experiment . ......... 23 2.2.2 Aone-factor-at-a-timeexperiment.................. 25 2.3 Partially confounded 2k factorialexperiment................. 26 2.3.1 Somegeneralisations.......................... 29 2.4 Fractional 2k factorialdesign......................... 32 chs. Design of Experiments, Course 02411, IMM, DTU 2 2.5Factorson2and4levels............................ 41 3 General methods for pk-factorial designs 46 3.1 Complete pk factorialexperiments....................... 46 3.2 CalculationsbasedonKempthorne’smethod................ 55 3.3 Generalformulationofinteractionsandartificialeffects.......... 58 3.4 Standardisationofgeneraleffects....................... 60 3.5 Block-confounded pk factorialexperiment.................. 63 3.6 Generalisation of the division into blocks with several defining relations . 68 3.6.1 Constructionofblocksingeneral................... 72 3.7 Partial confounding ............................... 76 3.8 Constructionofafractionalfactorialdesign................. 84 3.8.1 Resolutionforfractionalfactorialdesigns.............. 88 3.8.2 Practicalandgeneralprocedure.................... 89 3.8.3 Alias relations with 1/pq × pk experiments.............. 93 3.8.4 Estimation and testing in 1/pq × pk factorialexperiments...... 99 3.8.5 Fractionalfactorialdesignlaidoutinblocks.............103 Index ..... 114 My own notes ..... 116 chs. Design of Experiments, Course 02411, IMM, DTU 3 Tabels 2.1Asimpleweighingexperimentwith3items.................. 32 2.2 A 1/4×25 factorialexperiment......................... 38 2.3 A 2×4experimentin2blocks......................... 42 2.4 A fractional 2×2×4factorialdesign...................... 43 3.1 Making a Graeco-Latin square in a 32 factorialexperiment......... 48 3.2 Latin cubes in 33 experiments......................... 51 3.3 Estimation and SSQ in the 32-factorialexperiment.............. 56 3.4 Index variation with inversion of the factor order .............. 59 3.5Generalisedinteractionsandstandardisation................. 60 3.6 Latin squares in 23 factorialexperimentsandYates’algorithm....... 61 3.7 23 factorialexperimentin2blocksof4singleexperiments......... 63 3.8 32 factorialexperimentin3blocks....................... 64 3.9 Division of a 23 factorial experiment into 22 blocks.............. 67 3.10 Dividing a 33 factorialexperimentinto9blocks............... 69 3.11 Division of a 25 experiment into 23 blocks.................. 70 3.12 Division of 3k experiments into 33 blocks................... 71 3.13 Dividing a 34 factorial experiment into 32 blocks............... 73 3.14 Dividing a 53 factorialexperimentinto5blocks............... 75 3.15 Partially confounded 23 factorialexperiment................. 76 3.16 Partially confounded 32 factorialexperiment................. 80 3.17FactorexperimentdoneasaLatinsquareexperiment............ 84 3.18 Confoundings in a 3−1 × 33 factorialexperiment,aliasrelations...... 86 3.19 A 2−2 × 25 factorialexperiment........................ 90 3.20 Construction of 3−2 × 35 factorialexperiment................ 94 chs. Design of Experiments, Course 02411, IMM, DTU 4 3.21 Estimation in a 3−1 × 33-factorialexperiment................ 99 3.22 Two SAS examples...............................102 3.23 A 3−2 × 35 factorialexperimentin3blocksof9singleexperiments....104 3.24 A 2−4 × 28 factorialin2blocks........................108 3.25 A 2−3 × 27 factorialexperimentin4blocks..................112 chs. Design of Experiments, Course 02411, IMM, DTU 5 1 1.1 Introduction These lecture notes are concerned with the construction of experimental designs which are particularly suitable when it is wanted to examine a large number of factors and often under laboratory conditions. The complexity of the problem can be illustrated with the fact that the number of possible factor combinations in a multi-factor experiment is the product of the levels of the single factors. If, for example, one considers 10 factors, each on only 2 levels, the number of possible different experimentsis2×2×::: × 2=2k = 1024. If it is wanted to investigate the factors on 3 levels, this number increases to 310 = 59049 single experiments. As can be seen, the number of single experiments rapidly increases with the number of factors and factor levels. For practical experimental work, this implies two main problems. First, it quickly becomes impossible to perform all experiments in what is called a complete factor structure, and second, it is difficult to keep the experimental conditions unchanged during a large number of experiments. Doing the experiments, for example, necessarily takes a long time, uses large amounts of test material, uses a large number of experimental animals, or involves many people, all of which tend to increase the experimental uncertainty. These notes will introduce general models for such multi-factor experiments where all factors are on p levels, and we will consider fundamental methods to reduce the experimental work very considerably in relation to the complete factorial experiment, and to group such experiments in small blocks. In this way, both savings in the experimental work and more accurate estimates are achieved. An effort has been made to keep the notes as "non-mathematical" as possible, for example by showing the various techniques in typical examples and generalising on the basis of these. On the other hand, this has the disadvantage that the text is perhaps somewhat longer than a purely mathematical statistical run-through would need. Generally, extensive numerical examples are not given nor examples of the design of experiments for specific problem complexes, but the whole discussion is kept on such a general level that experimental designers from different disciplines should have reasonable possibilities to benefit from the methods described. As mentioned in the foreword, it is assumed that the reader has a certain fundamental knowledge of experimental work and statistical experimental design. Finally, I think that, on the basis of these notes, a person would be able to understand the idea in the experimental designs shown, and would also be able to draw up and analyse experimen- tal designs that are suitable in given problem complexes. However, this must not prevent the designer of experiments from consulting the relevant specialist literature on the subject. Here can be found many numerical examples,
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