to a 2 k Designs Section 6-6 page 271 Based on the idea of replicating some of the runs in a factorial design 2 level designs assume linearity. If interaction terms are added to model some curvature results from the twisting of the planes! Runs at the center provide an estimate of error and allow the experimenter to distinguish between two possible models: i.e Provides a check for curvature! First-order model (interaction) Second-order model 0 0 k k k y = β + β x + β x x + ε i i ij i j i= 1 i= 1 j> i k k k k 2 i i ij i j ii i i= 1 i= 1 j> i i= 1 y = β + β x + β x x + β x + ε 1
to a 2 k Designs Section 6-6 page 271 First-order model (interaction) Second-order model 0 0 k k k y = β + β x + β x x + ε i i ij i j i= 1 i= 1 j> i k k k k 2 i i ij i j ii i i= 1 i= 1 j> i i= 1 y = β + β x + β x x + β x + ε There is the possibility that the 2 nd order model is a better fit, So how do we ensure we test for this? The addition of replicated center points adds protection against curvature from second order effects. These replicated points at the design center do not affect the estimates of the effects. When we add centerpoints we assume the Factors are quantitative. 2
to a 2 k Designs Adding centerpoints to a factorial design. Add a zero level to the FACTORS levels tested Define Centerpoint replicates: n c 3
to a 2 k Designs Adding centerpoints to a factorial design. This is a wise thing to do as there are significant benefits: 1. Improved model and determining if curvature is significant. Factorial designs without centerpoints assume linear relationships. Centerpoints validate quadratic models such as: curvature 2. Independent estimate of error 4
BASIC IDEA: Average of the four corners minus the average of the center : If large significant Curvature exists! y F = y C The hypotheses are: SS Pure Quad H H 0 1 = no "curvature" k : β = 0 i= 1 k : β 0 i= 1 ii ii nn F C( yf yc) n + n F C 2 This sum of squares has a single degree of freedom 5
Example 6-6, Pg. 273: Chemical Engineer adds centerpoints to experiment because they are not sure about assumption of linearity n C = 5 Usually between 3 and 6 center points will work well Design-Expert provides the analysis, including the F-test for pure quadratic curvature 6
to a 2 k Designs ANOVA for Example 6-6 page 274 Response: yield ANOVA for Selected Factorial Model Analysis of variance table [Partial sum of squares] Sum of Mean F Source Squares DF Square Value Prob > F Model 2.83 3 0.94 21.92 0.0060 A 2.40 1 2.40 55.87 0.0017 B 0.42 1 0.42 9.83 0.0350 AB 2.500E-003 1 2.500E-003 0.058 0.8213 Curvature 0.0027 1 0.0027 0.063 0.8137 Pure Error 0.17 4 0.043 Cor Total 3.00 8 Std. Dev. 0.21 R-Squared 0.9427 Mean 40.44 Adj R-Squared 0.8996 C.V. 0.51 Pred R-Squared N/A PRESS N/A Adeq Precision 14.234 So what can we say about the 2nd order curvature for this model? 7
to a 2 k Designs ANOVA for Example 6-6 page 274 Hypothesis Test for Curvature: Test statistic: n f = 4; n c = 5 ; d.o.f. = 7 s e = s c =0.21 t curv = 0.072 t 0.25,7 = 2.36 Thus we conclude: Curvature is not significant! 8
to a 2 k Designs ANOVA for Example 6-6 page 274 Hypothesis Test for Curvature: EXCEL statistics: For this Example: Curvature is not significant! Corners Centerpoints Yf Yc 40 40.3 41.6 40.5 40.9 40.7 39.3 40.2 40.6 AVERAGE 40.45 40.46 SD 0.21 t curv 0.07188852 t 0.025,7 2.36462256 TINV(.05,7) 9
: CCD design 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 ( Chap 11) Four axial runs 10
Practical Use of Center Points (pg. 275) Like the control group! 1. Use current operating conditions as the center point. 2. Check for abnormal conditions during the time the experiment was conducted. 3. Check for time trends ( process stability) Run one at start, middle and end or experimental runs. 4. Use center points as the first few runs when there is little or no information available about the magnitude of error. See if variation is large before proceeding with experiment. If too large STOP and figure out why before running experiment! 5. Center points and qualitative factors? >>>>> 11
Center Points and Qualitative Factors Placed at center for quantitative factors Temperature and time 12