The 2 k Factorial Design. Dr. Mohammad Abuhaiba 1
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1 The 2 k Factorial Design Dr. Mohammad Abuhaiba 1
2 HoweWork Assignment Due Tuesday 1/6/ , 6.2, 6.17, 6.18, 6.19 Dr. Mohammad Abuhaiba 2
3 Design of Engineering Experiments The 2 k Factorial Design Special case of the general factorial design; k factors, all at two levels The two levels are usually called low and high (they could be either quantitative or qualitative) It provides the smallest number of runs with which k factors can be studied in a complete factorial design. Assumptions: Factors are fixed Completely randomized designs Usual normality assumptions are satisfied Response is nearly linear over the range of the factors levels chosen Dr. Mohammad Abuhaiba 3
4 The 2 2 Design Chemical Process Example Study the effect of concentration of reactant and amount of catalyst on conversion in a chemical process. - and + denote low and high levels of a factor Low and high are arbitrary terms Geometrically, the four runs form the corners of a square Factors can be quantitative or qualitative Dr. Mohammad Abuhaiba 4
5 Chemical Process Example A = reactant concentration, B = catalyst amount, y = recovery Dr. Mohammad Abuhaiba 5
6 Analysis Procedure for a Factorial Design Estimate factor effects Formulate model Statistical testing (ANOVA) Refine the model Analyze residuals (graphical) Interpret results Dr. Mohammad Abuhaiba 6
7 Estimation of Factor Effects ab a b (1) 1 A y y 2n [ ab a b (1)] A A 2n 2n ab b a (1) 1 B y y 2n [ ab b a (1)] B B 2n 2n ab (1) a b 1 AB 2n [ ab (1) a b] 2n 2n Dr. Mohammad Abuhaiba 7
8 Sum of Squares Sum of squares for any contrast can be computed from Eq SS T is given by Eq. 6-9 SS T has 4n-1 DOF Contrast A ab a b (1) Contrast ab b a (1) B Contrast ab (1) a b SS SS SS AB A B AB ab a b (1) 4n ab b a (1) 4n ab (1) a b 4n Dr. Mohammad Abuhaiba 8
9 Statistical Testing - ANOVA Dr. Mohammad Abuhaiba 9
10 Standards Order (Yates) Effests (1) a b ab A B AB Treatment combination Factorial effect I A B AB (1) a b ab Dr. Mohammad Abuhaiba 10
11 The Regression Model y x x o x 1 is a coded variable that represents reactant concentration x 2 is a coded variable that represents amount of catlyst Relationship between natural variables and cosed variables is given by: x con x con x /2 high conlow x 1 grand average x x /2 o 1 2 A B /2 /2 x 2 con high high high con x x x low cat cat cat x cat x cat low low /2 /2 Dr. Mohammad Abuhaiba 11
12 The Response Surface Dr. Mohammad Abuhaiba 12
13 Residuals and Diagnostic Checking Dr. Mohammad Abuhaiba 13
14 The 2 3 Factorial Design Dr. Mohammad Abuhaiba 14
15 Effects in The 2 3 Factorial Design A y y A B y y B C y y C See Eqs 6-11 to 6.17 for the factors' effects A B C Dr. Mohammad Abuhaiba 15
16 Table of and + Signs for the 2 3 Factorial Design Dr. Mohammad Abuhaiba 16
17 Properties of the Table Except for column I, every column has an equal number of + and signs The sum of the product of signs in any two columns is zero Multiplying any column by I leaves that column unchanged (identity element) The product of any two columns yields a column in the table: AB AB 2 ABBC AB C AC Orthogonal design Orthogonality is an important property shared by all factorial designs Each effect has a single DOF Sum of squares for any effect is: Dr. Mohammad Abuhaiba SS Contrast 2 8n 17
18 Example of a 2 3 Factorial Design Example 6-1: The Fill Height Experiment Run Coded Factors Fill Height Deviation Factor Levels A B C Replicate 1 Replicate 2 Low (-1) High (+1) A (%) B (psi) C (bpm) Dr. Mohammad Abuhaiba 18
19 Example of a 2 3 Factorial Design Estimation of Factor Effects ANOVA Model Coefficients Full Model Remove non-significant factors Model Coefficients Reduced Model The AB interaction is significant at about 10%. Thus, there is some mild interaction between carbonation and pressure. Run the process at low pressure and high line speed. Reduce variablity in carbonation by controlling temperature more precisely Dr. Mohammad Abuhaiba 19
20 Model Summary Statistics for Reduced Model R 2 and adjusted R 2 R R 2 SS SS Model T 2 SS E / Adj 1 SST / DOF DOF R 2 for prediction (based on PRESS) R 2 Pred 1 E T PRESS SS T Dr. Mohammad Abuhaiba 20
21 Model Summary Statistics (pg. 222) Standard error of model coefficients (full model) 2 ˆ ˆ MS E se( ) V ( ) k k n2 n2 Confidence interval on model coefficients ˆ t se( ˆ) ˆ t se( ˆ) / 2, df / 2, df E E Dr. Mohammad Abuhaiba 21
22 The General 2 k Factorial Design There will be k main effects, and k two-factor interactions 2 k three-factor interactions 3 1 k factor interaction Dr. Mohammad Abuhaiba 22
23 The General 2 k Factorial Design Analysis Procedure for a 2 k Design 1. Estimate factor effects 2. Form initial model 3. ANOVA 4. Refine model 5. Analyze residuals 6. Interpret results Dr. Mohammad Abuhaiba 23
24 The General 2 k Factorial Design ANOVA Contrast AB... K ( a 1)( b 1)...( k 1) The sign in each set of parentheses is negative if the factor is included in the effect and positive if the factor is not included. Examples: 2 3 and 2 5 designs Effects and sum of squares are estimated as 2 AB... K Contrast k AB n2 1 SS k Contrast n2... K AB... K AB... K Dr. Mohammad Abuhaiba 24 2
25 Unreplicated 2 k Factorial Designs 2 k factorial designs with one observation at each treatment combination An unreplicated 2 k factorial design is also sometimes called a single replicate Risks: If there is only one observation at each corner, is there a chance of unusual response observations spoiling the results? Modeling noise? Dr. Mohammad Abuhaiba 25
26 Unreplicated 2 k Factorial Designs Spacing of Factor Levels in the If the factors are spaced too closely, it increases the chances that the noise will overwhelm the signal in the data More aggressive spacing is usually best Dr. Mohammad Abuhaiba 26
27 Unreplicated 2 k Factorial Designs Lack of replication causes potential problems in statistical testing Replication admits an internal estimate of error With no replication, fitting the full model results in zero degrees of freedom for error Potential solutions to this problem Pooling high-order interactions to estimate error Normal probability plotting of effects (Daniels, 1959) Dr. Mohammad Abuhaiba 27
28 Example of an Unreplicated 2 k Design A 2 4 factorial was used to investigate the effects of four factors on the filtration rate of a resin The factors are A = temperature, B = pressure, C = mole ratio, D= stirring rate Experiment was performed in a pilot plant Process engineer is interested in maximizing filteration rate. Factor C currently at the high level Would reduce formaldehyde concentration as much as possible. Dr. Mohammad Abuhaiba 28
29 The Resin Plant Experiment Dr. Mohammad Abuhaiba 29
30 The Resin Plant Experiment Dr. Mohammad Abuhaiba 30
31 Contrast Constants for the 2 k Design A B AB C AC BC ABC D AD BD ABD CD ACD BCD ABCD {1} a b ab c ac bc abc d ad bd abd cd acd bcd abcd Dr. Mohammad Abuhaiba 31
32 Estimates of the Effects Dr. Mohammad Abuhaiba 32
33 The Normal Probability Plot of Effects Dr. Mohammad Abuhaiba 33
34 The Half Normal Plot of Effects A plot of the absolute value of the effect estimates against their cummulative normal probabilities. Figure 6-15 The straight line always passes through the origin and should also pass close to the fiftieth percentile data value. Dr. Mohammad Abuhaiba 34
35 Main Effects and Interactions Dr. Mohammad Abuhaiba 35
36 Design Projection Example 6-2 Because B is not significant and all interactions involving B are negligible, we may discard B from the experiment so that the design becomes a 2 3 factorial in A, C, and D with two replicates. ANOVA for the 2 3 design is shown in Table 6.13 By projecting the single replicate of the 2 4 into a replicated 2 3, we now have both an estimate of the ACD interaction and an estimate of error based on what is sometimes called hidden replication. Dr. Mohammad Abuhaiba 36
37 Design Projection General Case If we have a single replicate of 2 k design, and if h (h<k) factors are negligible and can be dropped, then the original data correspond to a full two-level factorial in the remaining k h factors with 2 h replicates Dr. Mohammad Abuhaiba 37
38 ANOVA Summary for the Model Dr. Mohammad Abuhaiba 38
39 The Regression Model Dr. Mohammad Abuhaiba 39
40 Model Residuals Dr. Mohammad Abuhaiba 40
41 Model Interpretation Response Surface Plots With concentration at either the low or high level, high temperature and high stirring rate results in high filtration rates Dr. Mohammad Abuhaiba 41
42 Example 6-3: The Drilling Experiment A = drill load, B = flow, C = speed, D = type of mud, y = advance rate of the drill Dr. Mohammad Abuhaiba 42
43 The Drilling Experiment Normal Probability Plot of Effects Dr. Mohammad Abuhaiba 43
44 DESIGN-EXPERT Plot adv._rate Residuals Residuals vs. Predicted Predicted Residual Plots Dr. Mohammad Abuhaiba 44
45 Residual Plots The residual plots indicate that there are problems with the equality of variance assumption Employ a transformation on the response Power family transformations are widely used * y y Transformations are typically performed to Stabilize variance Induce normality Simplify the model Dr. Mohammad Abuhaiba 45
46 Selecting a Transformation Empirical selection of lambda Prior (theoretical) knowledge or experience can often suggest the form of a transformation Dr. Mohammad Abuhaiba 46
47 Effect Estimates Following Log Transformation Three main effects are large No indication of large interaction effects Dr. Mohammad Abuhaiba 47
48 ANOVA Following Log Transformation Dr. Mohammad Abuhaiba 48
49 Following Log Transformation Dr. Mohammad Abuhaiba 49
50 Addition of Center Points to a 2 k Design Based on the idea of replicating some of the runs in a factorial design Runs at the center provide an estimate of error and allow the experimenter to distinguish between two possible models: First-order model (interaction) Second-order model 0 0 k k k y x x x i i ij i j i1 i1 ji k k k k 2 i i ij i j ii i i1 i1 ji i1 y x x x x Dr. Mohammad Abuhaiba 50
51 y F y C The hypotheses are: SS Pure Quad H H 0 1 no "curvature" k : 0 i1 k : 0 i1 ii ii nfnc ( yf yc ) n n This sum of squares has a single degree of freedom F C 2 Dr. Mohammad Abuhaiba 51
52 Example 6-6 Refer to the original experiment shown in Table Suppose that four center points are added to this experiment, and at the points x1=x2 =x3=x4=0 the four observed filtration rates were 73, 75, 66, and 69. The average of these four center points is 70.75, and the average of the 16 factorial runs is Since are very similar, we suspect that there is no strong curvature present. nc 4 Usually between 3 and 6 center points will work well Design-Expert provides the analysis, including the F-test for pure quadratic curvature Dr. Mohammad Abuhaiba 52
53 ANOVA for Example 6-6 Dr. Mohammad Abuhaiba 53
54 If curvature is significant, augment the design with axial runs to create a central composite design. The CCD is a very effective design for fitting a second-order response surface model Dr. Mohammad Abuhaiba 54
55 Practical Use of Center Points (pg. 250) Use current operating conditions as the center point Check for abnormal conditions during the time the experiment was conducted Check for time trends Use center points as the first few runs when there is little or no information available about the magnitude of error Center points and qualitative factors? Dr. Mohammad Abuhaiba 55
56 Center Points and Qualitative Factors Dr. Mohammad Abuhaiba 56
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