Engineering Statistics Handbook Episode 6 Part 12 potx

5. Process Improvement 5.3. Choosing an experimental design 5.3.3.How do you select an experimental design? A design is selected based on the experimental objective and the number of factors The choice of an experimental design depends on the objectives of the experiment and the number of factors to be investigated. Experimental Design Objectives Types of designs are listed here according to the experimental objective they meet Types of designs are listed here according to the experimental objective they meet. Comparative objective: If you have one or several factors under investigation, but the primary goal of your experiment is to make a conclusion about one a-priori important factor, (in the presence of, and/or in spite of the existence of the other factors), and the question of interest is whether or not that factor is "significant", (i.e., whether or not there is a significant change in the response for different levels of that factor), then you have a comparative problem and you need a comparative design solution. ● Screening objective: The primary purpose of the experiment is to select or screen out the few important main effects from the many less important ones. These screening designs are also termed main effects designs. ● Response Surface (method) objective: The experiment is designed to allow us to estimate interaction and even quadratic effects, and therefore give us an idea of the (local) shape of the response surface we are investigating. For this reason, they are termed response surface method (RSM) designs. RSM designs are used to: Find improved or optimal process settings ❍ ● 5.3.3. How do you select an experimental design? http://www.itl.nist.gov/div898/handbook/pri/section3/pri33.htm (1 of 3) [5/1/2006 10:30:23 AM] Troubleshoot process problems and weak points❍ Make a product or process more robust against external and non-controllable influences. "Robust" means relatively insensitive to these influences. ❍ Optimizing responses when factors are proportions of a mixture objective: If you have factors that are proportions of a mixture and you want to know what the "best" proportions of the factors are so as to maximize (or minimize) a response, then you need a mixture design. ● Optimal fitting of a regression model objective: If you want to model a response as a mathematical function (either known or empirical) of a few continuous factors and you desire "good" model parameter estimates (i.e., unbiased and minimum variance), then you need a regression design. ● Mixture and regression designs Mixture designs are discussed briefly in section 5 (Advanced Topics) and regression designs for a single factor are discussed in chapter 4. Selection of designs for the remaining 3 objectives is summarized in the following table. Summary table for choosing an experimental design for comparative, screening, and response surface designs TABLE 3.1 Design Selection Guideline Number of Factors Comparative Objective Screening Objective Response Surface Objective 1 1-factor completely randomized design _ _ 2 - 4 Randomized block design Full or fractional factorial Central composite or Box-Behnken 5 or more Randomized block design Fractional factorial or Plackett-Burman Screen first to reduce number of factors Resources and degree of control over wrong decisions Choice of a design from within these various types depends on the amount of resources available and the degree of control over making wrong decisions (Type I and Type II errors for testing hypotheses) that the experimenter desires. 5.3.3. How do you select an experimental design? http://www.itl.nist.gov/div898/handbook/pri/section3/pri33.htm (2 of 3) [5/1/2006 10:30:23 AM] Save some runs for center points and "redos" that might be needed It is a good idea to choose a design that requires somewhat fewer runs than the budget permits, so that center point runs can be added to check for curvature in a 2-level screening design and backup resources are available to redo runs that have processing mishaps. 5.3.3. How do you select an experimental design? http://www.itl.nist.gov/div898/handbook/pri/section3/pri33.htm (3 of 3) [5/1/2006 10:30:23 AM] Balance Balance dictates that the number of replications be the same at each level of the factor (this will maximize the sensitivity of subsequent statistical t (or F) tests). Typical example of a completely randomized design A typical example of a completely randomized design is the following: k = 1 factor (X1) L = 4 levels of that single factor (called "1", "2", "3", and "4") n = 3 replications per level N = 4 levels * 3 replications per level = 12 runs A sample randomized sequence of trials The randomized sequence of trials might look like: X1 3 1 4 2 2 1 3 4 1 2 4 3 Note that in this example there are 12!/(3!*3!*3!*3!) = 369,600 ways to run the experiment, all equally likely to be picked by a randomization procedure. Model for a completely randomized design The model for the response is Y i,j = + T i + random error with Y i,j being any observation for which X1 = i (or mu) is the general location parameter T i is the effect of having treatment level i Estimates and Statistical Tests 5.3.3.1. Completely randomized designs http://www.itl.nist.gov/div898/handbook/pri/section3/pri331.htm (2 of 3) [5/1/2006 10:30:23 AM] Estimating and testing model factor levels Estimate for : = the average of all the data Estimate for T i : - with = average of all Y for which X1 = i. Statistical tests for levels of X1 are shown in the section on one-way ANOVA in Chapter 7. 5.3.3.1. Completely randomized designs http://www.itl.nist.gov/div898/handbook/pri/section3/pri331.htm (3 of 3) [5/1/2006 10:30:23 AM] Table of randomized block designs One useful way to look at a randomized block experiment is to consider it as a collection of completely randomized experiments, each run within one of the blocks of the total experiment. Randomized Block Designs (RBD) Name of Design Number of Factors k Number of Runs n 2-factor RBD 2 L 1 * L 2 3-factor RBD 3 L 1 * L 2 * L 3 4-factor RBD 4 L 1 * L 2 * L 3 * L 4 . . . k-factor RBD k L 1 * L 2 * * L k with L 1 = number of levels (settings) of factor 1 L 2 = number of levels (settings) of factor 2 L 3 = number of levels (settings) of factor 3 L 4 = number of levels (settings) of factor 4 . . . L k = number of levels (settings) of factor k Example of a Randomized Block Design Example of a randomized block design Suppose engineers at a semiconductor manufacturing facility want to test whether different wafer implant material dosages have a significant effect on resistivity measurements after a diffusion process taking place in a furnace. They have four different dosages they want to try and enough experimental wafers from the same lot to run three wafers at each of the dosages. Furnace run is a nuisance factor The nuisance factor they are concerned with is "furnace run" since it is known that each furnace run differs from the last and impacts many process parameters. 5.3.3.2. Randomized block designs http://www.itl.nist.gov/div898/handbook/pri/section3/pri332.htm (2 of 4) [5/1/2006 10:30:24 AM] Ideal would be to eliminate nuisance furnace factor An ideal way to run this experiment would be to run all the 4x3=12 wafers in the same furnace run. That would eliminate the nuisance furnace factor completely. However, regular production wafers have furnace priority, and only a few experimental wafers are allowed into any furnace run at the same time. Non-Blocked method A non-blocked way to run this experiment would be to run each of the twelve experimental wafers, in random order, one per furnace run. That would increase the experimental error of each resistivity measurement by the run-to-run furnace variability and make it more difficult to study the effects of the different dosages. The blocked way to run this experiment, assuming you can convince manufacturing to let you put four experimental wafers in a furnace run, would be to put four wafers with different dosages in each of three furnace runs. The only randomization would be choosing which of the three wafers with dosage 1 would go into furnace run 1, and similarly for the wafers with dosages 2, 3 and 4. Description of the experiment Let X1 be dosage "level" and X2 be the blocking factor furnace run. Then the experiment can be described as follows: k = 2 factors (1 primary factor X1 and 1 blocking factor X2) L 1 = 4 levels of factor X1 L 2 = 3 levels of factor X2 n = 1 replication per cell N =L 1 * L 2 = 4 * 3 = 12 runs Design trial before randomization Before randomization, the design trials look like: X1 X2 1 1 1 2 1 3 2 1 2 2 2 3 3 1 3 2 3 3 4 1 4 2 4 3 5.3.3.2. Randomized block designs http://www.itl.nist.gov/div898/handbook/pri/section3/pri332.htm (3 of 4) [5/1/2006 10:30:24 AM] Matrix representation An alternate way of summarizing the design trials would be to use a 4x3 matrix whose 4 rows are the levels of the treatment X1 and whose columns are the 3 levels of the blocking variable X2. The cells in the matrix have indices that match the X1, X2 combinations above. By extension, note that the trials for any K-factor randomized block design are simply the cell indices of a K dimensional matrix. Model for a Randomized Block Design Model for a randomized block design The model for a randomized block design with one nuisance variable is Y i,j = + T i + B j + random error where Y i,j is any observation for which X1 = i and X2 = j X1 is the primary factor X2 is the blocking factor is the general location parameter (i.e., the mean) T i is the effect for being in treatment i (of factor X1) B j is the effect for being in block j (of factor X2) Estimates for a Randomized Block Design Estimating factor effects for a randomized block design Estimate for : = the average of all the data Estimate for T i : - with = average of all Y for which X1 = i. Estimate for B j : - with = average of all Y for which X2 = j. 5.3.3.2. Randomized block designs http://www.itl.nist.gov/div898/handbook/pri/section3/pri332.htm (4 of 4) [5/1/2006 10:30:24 AM] Advantages and disadvantages of Latin square designs The advantages of Latin square designs are: They handle the case when we have several nuisance factors and we either cannot combine them into a single factor or we wish to keep them separate. 1. They allow experiments with a relatively small number of runs.2. The disadvantages are: The number of levels of each blocking variable must equal the number of levels of the treatment factor. 1. The Latin square model assumes that there are no interactions between the blocking variables or between the treatment variable and the blocking variable. 2. Note that Latin square designs are equivalent to specific fractional factorial designs (e.g., the 4x4 Latin square design is equivalent to a 4 3-1 fractional factorial design). Summary of designs Several useful designs are described in the table below. Some Useful Latin Square, Graeco-Latin Square and Hyper-Graeco-Latin Square Designs Name of Design Number of Factors k Number of Runs N 3-by-3 Latin Square 3 9 4-by-4 Latin Square 3 16 5-by-5 Latin Square 3 25 3-by-3 Graeco-Latin Square 4 9 4-by-4 Graeco-Latin Square 4 16 5-by-5 Graeco-Latin Square 4 25 4-by-4 Hyper-Graeco-Latin Square 5 16 5-by-5 Hyper-Graeco-Latin Square 5 25 Model for Latin Square and Related Designs 5.3.3.2.1. Latin square and related designs http://www.itl.nist.gov/div898/handbook/pri/section3/pri3321.htm (2 of 6) [5/1/2006 10:30:24 AM] [...]... http://www.itl.nist.gov/div898 /handbook/ pri/section3/pri3321.htm (4 of 6) [5/1/20 06 10:30:24 AM] 1 2 4 3 4 3 1 2 2 5.3.3.2.1 Latin square and related designs 3 3 3 4 4 4 4 2 3 4 1 2 3 4 4 3 1 3 1 2 4 with k = 3 factors (2 blocking factors and 1 primary factor) L1 = 4 levels of factor X1 (block) L2 = 4 levels of factor X2 (block) L3 = 4 levels of factor X3 (primary) N = L1 * L2 = 16 runs This can alternatively... Further information B D A C E C E B D A D A C E B E B D A C More details on Latin square designs can be found in Box, Hunter, and Hunter (1978) http://www.itl.nist.gov/div898 /handbook/ pri/section3/pri3321.htm (6 of 6) [5/1/20 06 10:30:24 AM] 5.3.3.2.2 Graeco-Latin square designs with k = 4 factors (3 blocking factors and 1 primary factor) L1 = 3 levels of factor X1 (block) L2 = 3 levels of factor X2... 5-Level Factors X1 X2 X3 row column treatment blocking blocking factor factor factor 1 1 1 1 1 2 2 2 2 2 3 3 3 3 3 4 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 http://www.itl.nist.gov/div898 /handbook/ pri/section3/pri3321.htm (5 of 6) [5/1/20 06 10:30:24 AM] 1 2 3 4 5 3 4 5 1 2 5 1 2 3 4 2 5.3.3.2.1 Latin square and related designs 4 4 4 4 5 5 5 5 5 2 3 4 5 1 2 3 4 5 3 4 5 1 4 5 1 2 3 with k = 3 factors (2 blocking... Latin square design be randomly selected from those available, then randomize the run order Latin Square Designs for 3-, 4-, and 5-Level Factors http://www.itl.nist.gov/div898 /handbook/ pri/section3/pri3321.htm (3 of 6) [5/1/20 06 10:30:24 AM] 5.3.3.2.1 Latin square and related designs Designs for 3-level factors (and 2 nuisance or blocking factors) 3-Level Factors X1 X2 X3 row column treatment blocking... (block) L2 = 3 levels of factor X2 (block) L3 = 3 levels of factor X3 (primary) L4 = 3 levels of factor X4 (primary) http://www.itl.nist.gov/div898 /handbook/ pri/section3/pri3322.htm (2 of 4) [5/1/20 06 10:30:25 AM] 5.3.3.2.2 Graeco-Latin square designs N = L1 * L2 = 16 runs This can alternatively be represented as (A, B, C, and D represent the treatment factor and 1, 2, 3, and 4 represent the blocking factor):... 2 3 4 5 3 4 5 1 2 5 1 2 3 4 2 3 4 5 1 4 5 1 2 3 with k = 4 factors (3 blocking factors and 1 primary factor) L1 = 3 levels of factor X1 (block) http://www.itl.nist.gov/div898 /handbook/ pri/section3/pri3322.htm (3 of 4) [5/1/20 06 10:30:25 AM] . level = 12 runs A sample randomized sequence of trials The randomized sequence of trials might look like: X1 3 1 4 2 2 1 3 4 1 2 4 3 Note that in this example there are 12! /(3!*3!*3!*3!) = 369 ,60 0. (1978). 5.3.3.2.1. Latin square and related designs http://www.itl.nist.gov/div898 /handbook/ pri/section3/pri3321.htm (6 of 6) [5/1/20 06 10:30:24 AM] with k = 4 factors (3 blocking factors and 1 primary factor) L 1 . Latin Square 3 16 5-by-5 Latin Square 3 25 3-by-3 Graeco-Latin Square 4 9 4-by-4 Graeco-Latin Square 4 16 5-by-5 Graeco-Latin Square 4 25 4-by-4 Hyper-Graeco-Latin Square 5 16 5-by-5 Hyper-Graeco-Latin