1 2 3 4 5 6 7 8 Heading (x1) Generic(-) Detailed(+) Generic(-) Detailed(+) Generic (-) Detailed(+) Generic(-) Detailed(+) sum+ sumave + aveeffect 163 147.5 40.75 36.875 3.875 Run b0 = b1= b2= b3= b4= b5 = b6= b7= Email Open(x2) No(-) No(-) Yes(+) Yes(+) No(-) No(-) Yes(+) Yes(+) 179.5 131 44.875 32.75 12.125 Body(x3) Text(-) Text(-) Text(-) Text(-) HTML(+) HTML(+) HTML(+) HTML(+) x1 x2 + + + + x1x3 + + + + 101 168 152.5 209.5 143 158 25.25 42 38.13 52.375 35.6 39.5 -27.125 6.38 -1.38 x2x3 + + + + R Rate R Rate x1x2x3 1 2 Ave RR SD RR 46 38 42 5.656854 + 34 38 36 2.828427 + 56 59 57.5 2.12132 68 80 74 8.485281 + 25 27 26 1.414214 22 32 27 7.071068 21 23 22 1.414214 + 19 33 26 9.899495 126 185 31.5 46.1 -15 145.5 165 36.375 41.25 -4.875 77.625 1.9375 6.0625 -13.5625 3.1875 -0.6875 -7.3125 -2.4375 Regression Model is y = b0 + b1x1 + b2x2 + b3x3 + b4x1x2 + b5x1x3 + b6x2x3 + b7 x1 x2 x3 y = 77.625 + 1.9375 x1 + 6.0625 X2 -13.5625 x3 + 3.1875 x1 x2 -0.6875x1x3 -7.3125 x2x3 - 2.4375 x1 x3 x3 Graphical Analysis Most highest response is detailed, open email, and text. WEEK 8 HOMEWORK Table: Invoice Experiment Error Customer Size + + + + Customer Location + + + + Product Type + + + + Number of Errors 15 18 6 2 19 23 16 21 Customer Size: Small (-), Large (+) Customer Location: Foreign (-), Domestic (+) Product Type: Commodity (-), Specialty (+) Customer Size effect 2 Customer Location effect 7,5 Product Type 9,5 Size x Location Effect average errors (large) 16 average errors (small) 14 average errors (Foreign) 18,75 average errors (domestic) 11,25 average error (Commodity) average error (specialty) 10,25 19,75 + + + + 1,5 Interactions Size x Product Effect + + + + 2,5 Product x Location Effect + + + + 5 WEEK 8 - LECTURE NOTES USING PROCESS EXPERIMENTATION TO BUILD MODELS I. STATISTICAL DESIGN OF EXPERIMENTS A) ONE-FACTOR-AT-A-TIME EXPERIMENTATION 1) In this approach each process variable is studied individually by varying that factor and measuring the response, while holding all other variables constant. This is repeated for each variable. 2) A key assumption is that the process variables function independently of each other. In reality, significant interactions are often found and are what make some process effective. As a result, this method performs poorly in the presence of interacting variables. 3) Another limitation is that it takes a long time to test all the variables when you test them one at a time. B) THE STATISTICAL APPROACH 1) Statistical models are developed by quantitatively measuring the relationship between one or more process variables and the process responses. This is more effective than testing one factor at a time and it minimizes the number of tests required. 2) Some of the reasons for the success of this approach: a) Increases the chance of identifying key drivers b) Provides a systematic approach to testing c) Enables collection of quality data d) Evaluates a large number of variables e) Controls nuisance variables f) Produces quantitative estimates of effects g) Identifies interaction effects h) Measures experimental uncertainty i) Enables effective and efficient use of data C) EXAMPLES OF TEST STUDIES 1) Chocolate Milk Study a) Test Design To find the optimum combinations of chocolate cream and thickness, a 2X2X2 factorial design was use (See page 275) b) A total of three experiments were run, with middle levels being the starting point. c) The results are indicated in the table on page 275. 2) Plastic Parts Case Study a) The defect rate of molded plastic parts was running around 1.7% and needed to be reduced. b) Four experiments were run. Each run consisted of a 3X3 factorial design for temperature and pressure. Each of the four experiments involved a different combination Raw material (A or B) and Process (Old or New). c) The results are indicated in the tables on pages 278 and 279. d) Three key points resulted from these experiments: i) The iterative nature of experimentation is evident. ii) It illustrates the power of the factorial design iii) The factorial design does an effective job of of sampling the experimental region. II. REVISED STATISTICAL APPROACH TO EXPERIMENTATION A) NEW STATISTICAL APPROACH - an iterative approach Hypothesis A Design A Data A Analysis A Hypothesis B B) THREE MAJOR EXPERIMENTAL ENVIRONMENTS 1) Screening - This is the early stage of the design, with many potential variables to study. a) Which variables have the largest effects? 2) Characterization - We are working with fewer variables which had the largest effects from the screening stage. 3) Optimization - We are working with 2 to 6 variables to create s prediction equation to help identify the optimum operating conditions. (See pages 280 and 281) C) PLANNING TEST PROGRAMS 1) The first step is to obtain a clear statement of the problem. This includes the problem, questions needing answers, available resources and the time frame 2) Design the program based on the information provided 3) Plan and conduct the experiment 4) Analysis of the results of the experiment 5) Reporting and discussing results D) DESIGNING THE EXPERIMENT 1) The first step is to agree on overall objective 2) Identify the responses (y values) or measures of performance 3) Identify the process variables (x values) that appear to have the greatest effects on the results. Usually 2 or 3 levels of the variables are sufficient. 4) The size of the experiment depends on amount of desired replication and the time and resources available for experimentation. 5) The degree of randomization is important (See page 286) 6) Determine the method of data collection. What software to be used? III. CASE STUDIES A) TWO-FACTOR EXPERIMENT (Page 287) 1) This is a two-level factorial design with 5 replicates per group. 2) The concept of interaction effects is very important (See page 290) 3) Regression analysis (See Table 7.8 page 290 and 292) The analysis of factor effects and the regression analysis give identical results. Regression analysis is more useful when the factors are continuous quantitive measures such as time and temperature. B) THREE FACTOR EXPERIMENT (Page 293) 1) This is a 3 factor factorial design with 2 levels for each factor and 2 replicates for each group. (See page 297 for details of the process) 2) As the number of factors or number of levels increases, the total size of the experiment increases exponentially. As a result, when the number of factor levels is large the number of factors studied tends to be small, and vice-versa. IV. BLOCKING, RANDOMIZATION AND CENTER POINTS A) BLOCKING 1) Experiments are conducted in blocks to eliminate the effect of extraneous variables such as time effects (hour, day, week or season), and location effects (different sites, different machines) 2) The effect of the blocking variable is of little interest, but it must be taken into account for the results of the experiment to be valid. (See discussion on page 301) B) RANDOMIZATION 1) If we know the factors that might cause extraneous variation, we can remove their effects by blocking. To protect ourselves against other extraneous factors that are unknown, we use randomization. 2) There are two main types of randomization: complete and restricted. Complete randomization is the preferred method when practical. (See page 302) C) CENTER POINTS 1) Center points are used in response surface experiments when it is of interest to detect and quantify curved response functions, by using 3 or more levels. 1 2 3 4 5 6 7 8 effect Avg 1 2 no = yes = + Gen = 1 NO = 1 text = 1 head open body A B C 1 1 1 2 1 1 1 2 1 2 2 1 1 1 2 2 1 2 1 2 2 2 2 2 -37 -48.5 108.5 -18.5 -24.25 54.25 det yes text 131 209.5 179.5 101 AxB 1 2 2 1 1 2 2 1 25.5 12.75 1 2 3 4 effect Avg. regress. script train A B NO NO Yes NO NO Yes Yes Yes 25.6 36.4 12.8 18.2 6.4 9.1 A x B int. yes no no yes Results 10.8 15.2 20.6 41.8 16.8 8.4 4.2 AxC 1 2 1 2 2 1 2 1 -5.5 -2.75 BxC 1 1 2 2 2 2 1 1 -58.5 -29.25 126 184.5 AxBxC 1 2 2 1 2 1 1 2 19.5 9.75 Avg. 42 36 57.5 74 26 27 22 26 rep. 1 46 34 56 68 25 22 21 19 rep. 2 38 38 59 80 27 32 23 33
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