Factorial Experiment PDF

Factorial Experiment  What is a Factorial Experiment? Factorial Experiment- – Experimental plans used to study the effects of two or more factors on process/product attributes, where each level of each factor is varied simultaneously with the other factors in the experime...

Factorial Experiment  What is a Factorial Experiment? Factorial Experiment- – Experimental plans used to study the effects of two or more factors on process/product attributes, where each level of each factor is varied simultaneously with the other factors in the experiment.  Advantages of Factorial Designs  Effects of varying all factors simultaneously can be evaluated over a wide range of experimental condition.  More sensitive to detecting differences than will the single-step approach.  All possible interactions among the factors can be estimated and hence it is efficient in locating the true optimum.  Estimates of main and interaction effects are obtained by optimal use of experimental materials.  2k Factorial Design An experiment that consists of k factors A, B, C, …, each at two levels, is called a 2k factorial experiment. In a 2k factorial design, there are 2k combinations of factor levels. Each combination can be viewed as a treatment. For two factor study, k = 2, the total number of combinations is 22 = 4.  2k Factorial Design The two levels of each factor are denoted by -1 and +1. Experimental conditions represented by means of a special notation and listed in a form called "standard order". – Each experimental condition represented by the product of lowercase letters corresponding to the factors which are taken at level +1, called the "highest level". – If a lowercase letter corresponding to a factor is missing, this means that the factor is taken at level -1, called the "lower level".  Screening Experimental Designs- 2k Factorial Design  Thus in a three-factor (A, B, C) experiment, – ac, for example, denotes the experimental condition where factors A and C are taken at the higher levels and factor B is taken at the lower level, – c represents the experimental condition where factor C is taken at the higher level and factors A and B are taken at the lower level, and so forth. – The symbol "1" is used to denote the experimental condition in which all factors are taken at the lower level.  The experimental conditions are applied in a random order during the experiment itself.  For purpose of analysing the results it is convenient to arrange them in standard order. For k = 2, this order is 1, a, b, ab.  Procedure for obtaining treatment combination in standard order  On the first column, label the first row (1), then the product of a with (1). Then all products of b with the terms which are already there; b  (1) = b; b  a = ab.  Then all products of c with the terms which are already there; c  (1) = c; c  a = ac; c  b = bc; c  ab = abc. This is the 'standard' order.  Now put in the signs following the rule that each letter in the symbol corresponds to its factor at the high level (+1) in the combination.  The experiment can then be carried out using the treatment combinations in randomised order and the data analysed using ANOVA  Factor A 23 factorial experiment treatment combinations Level Level of factor Treatment A B C combination 1 -1 -1 -1 a +1 -1 -1 b -1 +1 -1 ab +1 +1 -1 c -1 -1 +1 ac +1 -1 +1 bc -1 +1 +1 abc +1 +1 +1 Number of Runs for a 2k Full Factorial Number of Factors Number of Runs 2 4 3 8 4 16 5 32 6 64 7 128 Fractional Factorial Designs  Fractional Factorial Designs Example, Suppose we want to study four factors, A, B, C, D, but we want to use 8 runs rather than the 16 runs required by a full 24 design. To do this, we first write down the extended design matrix for the full 23 design (i.e. the 2k design with 8 runs, required):  Fractional Factorial Designs A B C AB AC BC ABC -1 -1 -1 +1 +1 +1 -1 +1 -1 -1 -1 -1 +1 +1 -1 +1 -1 -1 +1 -1 +1 +1 +1 -1 +1 -1 -1 -1 -1 -1 +1 +1 -1 -1 +1 +1 -1 +1 -1 +1 -1 -1 -1 +1 +1 -1 -1 +1 -1 +1 +1 +1 +1 +1 +1 +1  Fractional Factorial Designs Next, since the highest-order interaction is least likely to be important, replace the ABC column by the letter D. – Abbreviated by writing D = ABC. Then erase all remaining interaction columns to obtain the design matrix  Fractional Factorial Designs A B C D -1 -1 -1 -1 +1 -1 -1 +1 -1 +1 -1 +1 +1 +1 -1 -1 -1 -1 +1 +1 +1 -1 +1 -1 -1 +1 +1 -1 +1 +1 +1 +1  Fractional Factorial Designs This 4-column matrix is the design matrix of a fractional factorial design based on four factors. – Because the 8 test runs comprise only a fraction of the 16 runs required in a full 24 design, we say that the 8-run experiment is a fractional factorial experiment. – Since this design uses only half of the 16 runs, we say that it is a half fraction of the full factorial design based on four factors. The particular fractional factorial design we have created is denoted as a 24-1 design.  Fractional Factorial Designs This notation (24-1) carries the following information: – The design has 8 runs because 24-1 = 23 = 8. – Four factors are studied in the experiment. – Each factor has two levels. – One factor (factor D) has been added to a full design based on 8 runs. – The design uses a fraction, 1/21, of the runs of a full 2k design.  Fractional Factorial Designs Generally, any fractional factorial design can be described by the notation 2k-p, which is intended to convey that: – The design has a total of 2k-p test runs. – k factors are studied in the experiment. – Each factor has two levels. – p factors have been added to a full design based on 2k-p runs. – The design uses a fraction, 1/2p, of the runs of a full 2k design.  Confounding in a 2k factorial experiments When the number of factors is 5 or greater, a full factorial design requires a large number of runs and the standard error per unit increases. This standard error is therefore likely to be higher in a large factorial experiment than for a comparable single- factor experiment. This increase in standard errors can usually be kept small by the device known as confounding.  Alias structure  Reward for using fractional factorial design is a substantial reduction in the required number of test runs.  However, ability to clearly distinguish some of the effects from one another is lost.  Consider the 24-1 design created earlier by the assignment D = ABC.  The D effect and the ABC effect cannot be distinguished from one another because the same column of +1's and -1"s in the design matrix is used to compute both the ABC and D effects.  Consequently, D and ABC are said to be aliases of one another or – D effect is confounded with the ABC effect.  The reason that we chose to alias D with the ABC column was that we hoped that ABC effect would be negligible.  If that is true, then we will have obtained a main effect estimate for D using only 8 runs.  Method for writing Alias Structure  Depends on some simple observations about multiplying columns of +1's and -1's:  First, the letter I denotes the column consisting entirely of +1's.  Any column multiplied by itself yields column I. For example, A*A = A2 = I, B*B = B2 =I, and so forth.  Multiplying column I by any other column does not change the column. For example, A*I = I*A = A.  Using these facts, we can obtain the alias structure of any fractional factorial as follows:  Method for writing Alias Structure  First write the p assignments of additional factors in equation form. These p equations are called the design generators.  Multiply each generator from step 1 by its left side to put each generator into the form I = w, where w is a "word" composed of several letters representing particular experimental factors (e.g. D = ABC becomes I = ABCD). It is also possible to create words with "-" signs, such as D = -ABC. If this is done, the resulting design will use a different fraction of the runs from the full 2k design.  Letting I = w1, I = w2, …, I = wp denote the p design generators from step 2, form all possible products of the words wi (one at a time, two at a time, three at a time, etc.). Use the fact that squares of factors can be eliminated (e.g., A 2 = I and multiplying by I does not change anything). There will be a total of 2p words formed. This collection is called the defining relation of the design.  Multiply each word in the defining relation by all 2k-1 effects based on k factors. Use the fact that squares of factors cancel out to simplify the products. The result is called the alias structure of the design.  Design Resolution The length of the shortest word in the defining relation (excluding I) is termed the design resolution  What does Design Resolution mean?  Design Resolution = III: Main effects are confounded with two factor interactions.  Design Resolution = IV: Main effects are confounded with three factor interactions. It can also be two factor interactions are confounded with one another.  Design Resolution = V: Main effects confounded with 4 factor or, two-factor interactions are confounded with three factor interactions.  Design Resolution = II: Main effects confounded with one another. This would be very, very bad.  Exercise 1 Design a five factor fractional factorial using 8 test runs (25-2). a. Determine the alias structure of this design. b. What is the design resolution? Explain.  Exercise 1 Six factors are thought to influence the taste of a beverage namely, type of sweetener (A), ratio of syrup to water (B), carbonation level (C), pasteurisation temperature (D), pasteurisation time (E) and pH (F). The objective is to reduce the number of factors for future R&D work; hence an experiment was designed with each of the factors at two levels designated as low (-1) and high (+1). 1 What type of experimental design is appropriate for this objective? 2 Should the researcher conduct the experiment using the 6 factors, how many experimental runs will be required assuming no replication? 3. The researcher wishes to conduct a fraction of the runs in 2. Construct a 26-3 fractional factorial design for the researcher setting D = AB, E = AC and F = BC. Write down the aliases for the design.