/ department of mathematics and computer science BV04 Screening Designs

/ department of mathematics and computer science Contents regression analysis and effects 2 p -experiments blocks 2 p-k -experiments (fractional factorial experiments) software literature

/ department of mathematics and computer science Three factors: example Response:deviation filling height bottles Factors:carbon dioxide level (%)A pressure (psi)B speed (bottles/min)C

/ department of mathematics and computer science Effects How do we determine whether an individual factor is of importance? Measure the outcome at 2 different settings of that factor. Scale the settings such that they become the values +1 and -1.

/ department of mathematics and computer science setting factor A measurement +1

/ department of mathematics and computer science setting factor A measurement +1

/ department of mathematics and computer science setting factor A measurement +1 effect

/ department of mathematics and computer science setting factor A measurement +1 effect N.B. effect = 2 * slope slope

/ department of mathematics and computer science setting factor A measurement Effect factor A = 50 – 35 = 15

/ department of mathematics and computer science More factors We denote factors with capitals: A, B,… Each factor only attains two settings: -1 and +1 The joint settings of all factors in one measurement is called a level combination.

/ department of mathematics and computer science More factors AB Level Combination

/ department of mathematics and computer science Notation A level combination consists of small letters. The small letters denote which factors are set at +1; the letters that do not appear are set at -1. Example: ac means: A and C at 1, the remaining factors at -1 N.B. (1) means that all factors are set at -1.

/ department of mathematics and computer science An experiment consists of performing measurements at different level combinations. A run is a measurement at one level combination. Suppose that there are 2 factors, A and B. We perform 4 measurements with the following settings: A -1 and B -1 (short: (1) ) A +1 and B -1 (short: a ) A -1 and B +1 (short: b ) A +1 and B +1 (short: ab )

/ department of mathematics and computer science A 2 2 Experiment with 4 runs AByield (1) b 1 a1 ab11

/ department of mathematics and computer science Note: CAPITALS for factors and effects small letters for level combinations ( = settings of the experiments) (A, BC, CDEF) (a, bc, cde, (1))

/ department of mathematics and computer science Graphical display A B a ab (1) b

/ department of mathematics and computer science B A

/ department of mathematics and computer science B A estimates for effect A:

/ department of mathematics and computer science B A estimates for effect A: = 15

/ department of mathematics and computer science B A estimates for effect A: = = 15

/ department of mathematics and computer science B A estimates for effect A: = = 15 Which estimate is superior?

/ department of mathematics and computer science B A estimates for effect A: = = 15 Combine both estimates: ½(50-35) + ½(60-40) = 17.5

/ department of mathematics and computer science B A In the same way we estimate the effect B (note that all 4 measurements are used!): ½(40-35) ½(60-50) + = 7.5

/ department of mathematics and computer science B A The interaction effect AB is the difference between the estimates for the effect A: ½(60-40)½(50-35)-= 2.5

/ department of mathematics and computer science Interaction effects Cross terms in linear regression models cause interaction effects: Y = x A + 4 x B + 7 x A x B x A  x A +1  Y  Y x B, so increase depends on x B. Likewise for x B  x B +1 This explains the notation AB.

/ department of mathematics and computer science No interaction Factor A Output lowhigh B low B high

/ department of mathematics and computer science Interaction I Factor A Output lowhigh B low B high

/ department of mathematics and computer science Interaction II Factor A Output lowhigh B low B high 20 45

/ department of mathematics and computer science Interaction III Factor A Output lowhigh B low B high 20 45

/ department of mathematics and computer science Trick to Compute Effects AByield (1) 35 b140 a150 ab1160 (coded) measurement settings

/ department of mathematics and computer science AByield (1) 35 b140 a150 ab1160 Effect estimates Trick to Compute Effects

/ department of mathematics and computer science AByield (1) 35 b140 a150 ab1160 Effect estimates Effect A = ½( ) = 17.5 Effect B = ½( – ) = 7.5 Trick to Compute Effects

/ department of mathematics and computer science ABAByield (1) ?35 b1?40 a1?50 ab11?60 Trick to Compute Effects Effect AB = ½(60-40) - ½(50-35) = 2.5

/ department of mathematics and computer science ABAByield (1) 135 b1 40 a1 50 ab11160 Trick to Compute Effects Effect AB = ½(60-40) - ½(50-35) = 2.5 × = AB equals the product of the columns A and B

/ department of mathematics and computer science IABAByield (1) b a ab Trick to Compute Effects Computational rules: I×A = A, I×B = B, A×B=AB etc. This holds true in general (i.e., also for more factors).

/ department of mathematics and computer science Factors: a 2 3 Design

/ department of mathematics and computer science Factors: a 2 3 Design ABCYield (1)---5 a+--2 b-+-7 ab++-1 c--+7 ac+-+6 bc-++9 abc+++7

/ department of mathematics and computer science (1)=5 a=2 ab=1b=7 ac=6 abc=7bc=9 c=7      effect A = ¼(+16-28)=-3 A B C scheme 2 3 design

/ department of mathematics and computer science effect AB = ¼(+20-24)=-1 scheme 2 3 design (1)=5 a=2 ab=1b=7 ac= 6 abc=7bc=9 c=7      A B C

/ department of mathematics and computer science Back to 2 factors – Blocking IABAB (1)+--+ b+-+- a++-- ab++++ Suppose that we cannot perform all measurements at the same day. We are not interested in the difference between 2 days, but we must take the effect of this into account. How do we accomplish that? day 1 day 2

/ department of mathematics and computer science Back to 2 factors – Blocking IABABday (1)+--+1 b+-+-1 a++--2 ab++++2 Suppose that we cannot perform all measurements at the same day. We are not interested in the difference between 2 days, but we must take the effect of this into account. How do we accomplish that? “hidden” block effect

/ department of mathematics and computer science Back to 2 factors – Blocking IABABday (1)+--+- b+-+-- a++--+ ab+++++ We note that the columns A and day are the same. Consequence: the effect of A and the day effect cannot be distinguished. This is called confounding or aliasing).

/ department of mathematics and computer science Back to 2 factors – Blocking IABABday (1)+--+? b+-+-? a++--? ab++++? A general guide-line is to confound the day effect with an interaction of highest possible order. How can we accomplish that here?

/ department of mathematics and computer science Back to 2 factors – Blocking Solution: day 1: a, bday 2: (1), ab or interchange the days! IABABday (1)+--++ b+-+-- a++--- ab+++++

/ department of mathematics and computer science Back to 2 factors – Blocking Solution: day 1: a, bday 2: (1), ab or interchange the days! IABABday (1)+--++ b+-+-- a++--- ab+++++ Choose within the days by drawing lots which experiment must be performed first. In general, the order of experiments must be determined by drawing lots. This is called randomisation.

/ department of mathematics and computer science Here is a scheme for 3 factors. Interactions of order 3 or higher can be neglected in practice. How should we divide the experiments over 2 days? day 1 day 2

/ department of mathematics and computer science Fractional experiments Often the number of parameters is too large to allow a complete 2 p design (i.e, all 2 p possible settings -1 and 1 of the p factors). By performing only a subset of the 2 p experiments in a smart way, we can arrange that by performing relatively few, it is possible to estimate the main effects and (possibly) 2nd order interactions.

/ department of mathematics and computer science Fractional experiments IABABCACBC AB C (1) a b ab c ac bc abc

/ department of mathematics and computer science Fractional experiments IABABCACBC AB C (1) a b ab c ac bc abc

/ department of mathematics and computer science Fractional experiments IABABCACBC AB C (1) a b ab

/ department of mathematics and computer science Fractional experiments IABABCACBC AB C (1) a b ab With this half fraction (only 4 = ½×8 experiments) we see that a number of columns are the same (apart from a minus sign): I = -C, A = -AC, B = -BC, AB = -ABC

/ department of mathematics and computer science Fractional experiments IABABCACBC AB C (1) a b ab We say that these factors are confounded or aliased. In this particular case we have an ill-chosen fraction, because I and C are confounded. I = -C, A = -AC, B = -BC, AB = -ABC

/ department of mathematics and computer science Fractional experiments – Better Choice: I = ABC IABABCACBC AB C (1) a b ab c ac bc abc

/ department of mathematics and computer science Fractional experiments – Better Choice: I = ABC IABABCACBC AB C a b c abc The other “best choice” would be: I = -ABC Aliasing structure: I = ABC, A = BC, B = AC, C = AB

/ department of mathematics and computer science IABABCACBC AB C a b c abc In the case of 3 factors further reducing the number of experiments is not possible in practice, because this leads to undesired confounding, e.g. : I = A = BC = ABC, B = C = AB = AC,

/ department of mathematics and computer science IABABCACBC AB C a abc Other quarter fractions also have confounded main effects, which is unacceptable.

/ department of mathematics and computer science Further remarks on fractions there exist computational rules for aliases. E.g., it follows from A=C that AB = BC. Note that I = A 2 = B 2 = C 2 etc. always holds (see the next lecture) tables and software are available for choosing a suitable fraction. The extent of confounding is indicated by the resolution. Resolution III is a minimal ; designs with a higher resolution are very much preferred.

/ department of mathematics and computer science Plackett-Burman designs So far we discussed fractional designs for screening. This is sensible if one cannot exclude the possibility of interactions. If one knows based on foreknowledge that there are no interactions or if one is for some reason is only interested in main effects, than Plackett-Burman designs are preferred. They are able to detect significant main effects using only very few runs. A disadvantage of these designs is their complicated aliasing structure.

/ department of mathematics and computer science Number of measurements For every main or interaction effect that has to estimated separately, at least one measurement is necessary. If there are k blocks, then this requires additional k - 1 measurements. The remaining measurements are used for estimation of the variance. It is important to have sufficient measurements for the variance.

/ department of mathematics and computer science Choice of design After a design has been chosen, the factors A, B, … must be assigned to the factors of the experiment. It is recommended to combine any foreknowledge on the factors with the alias structure. The individual measurements must be performed in a random order. never confound two effects that might both be significant if you know that a certain effect will not be significant, you can confound it with an effect that might be significant.

/ department of mathematics and computer science Centre points and Replications If there are not enough measurements to obtain a good estimate of the variance, then one can perform replications. Another possibility is to add centre points. B A a ab (1) b Adding centre points serves two purposes: better variance estimate allow to test curvature using a lack-of-fit test Centre point

/ department of mathematics and computer science Curvature A design in which each factor is only allowed to attain the levels -1 and 1, is implicitly assuming a linear model. This is because knowing only the functions values at -1 and +1, then 1 and x 2 cannot be distinguished. We can distinguish them by adding the level 0. This is the idea behind adding centre points.

/ department of mathematics and computer science Analysis of a Design ABCYield (1)---5 a+--2 b-+-7 ab++-1 c--+7 ac+-+6 bc-++9 abc+++7

/ department of mathematics and computer science Analysis of a Design – With 2-way Interactions Analysis Summary File name: Estimated effects for Yield average = 5.5 +/ A:A = /- 0.5 B:B = 1.0 +/- 0.5 C:C = 3.5 +/- 0.5 AB = /- 0.5 AC = 1.5 +/- 0.5 BC = 0.5 +/ Standard errors are based on total error with 1 d.f.

/ department of mathematics and computer science Analysis of a Design – With 2-way Interactions Analysis of Variance for Yield Source Sum of Squares Df Mean Square F-Ratio P-Value A:A B:B C:C AB AC BC Total error Total (corr.) R-squared = percent R-squared (adjusted for d.f.) = percent Standard Error of Est. = Mean absolute error = 0.25 Durbin-Watson statistic = 2.5 Lag 1 residual autocorrelation =

/ department of mathematics and computer science Analysis of a Design – Only Main Effects Analysis Summary File name: Estimated effects for Yield average = 5.5 +/ A:A = / B:B = 1.0 +/ C:C = 3.5 +/ Standard errors are based on total error with 4 d.f. Effect estimates remain the same!

/ department of mathematics and computer science Analysis of a Design – Only Main Effects Analysis of Variance for Yield Source Sum of Squares Df Mean Square F-Ratio P-Value A:A B:B C:C Total error Total (corr.) R-squared = percent R-squared (adjusted for d.f.) = percent Standard Error of Est. = Mean absolute error = Durbin-Watson statistic = (P=0.3180) Lag 1 residual autocorrelation =

/ department of mathematics and computer science Analysis of a Design with Blocks BlockABCYield (1)1---5 ab1++-1 ac1+-+6 bc1-++9 a2+--2 b2-+-7 c2--+7 abc2+++7

/ department of mathematics and computer science Analysis of a Design with Blocks – With 2-way Interactions Analysis of Variance for Yield Source Sum of Squares Df Mean Square F-Ratio P-Value A:A B:B C:C AB AC BC blocks Total error Total (corr.) R-squared = percent R-squared (adjusted for d.f.) = percent Saturated design: 0 df for the error term → no testing possible

/ department of mathematics and computer science Analysis of a Design with Blocks – Only Main Effects Analysis of Variance for Yield Source Sum of Squares Df Mean Square F-Ratio P-Value A:A B:B C:C blocks Total error Total (corr.) R-squared = percent R-squared (adjusted for d.f.) = percent Standard Error of Est. = Mean absolute error = 0.75 Durbin-Watson statistic = (P=0.0478) Lag 1 residual autocorrelation =

/ department of mathematics and computer science Analysis of a Fractional Design (I = -ABC) ABCYield (1)---5 ac+-+6 bc-++9 ab++-1

/ department of mathematics and computer science Analysis of a Fractional Design (I = -ABC) Analysis of Variance for Yield Source Sum of Squares Df Mean Square F-Ratio P-Value A:A-BC B:B-AC C:C-AB Total error Total (corr.) R-squared = percent R-squared (adjusted for d.f.) = 0.0 percent Estimated effects for Yield average = 5.25 A:A-BC = -3.5 B:B-AC = -0.5 C:C-AB = No degrees of freedom left to estimate standard errors.

/ department of mathematics and computer science ABYield (1)--5 a+-6 b-+9 ab Pure Error = Analysis of a Design with Centre Points

/ department of mathematics and computer science Analysis of a Design with Centre Points Analysis of Variance for Yield Source Sum of Squares Df Mean Square F-Ratio P-Value A:A B:B AB Lack-of-fit Pure error Total (corr.) R-squared = percent R-squared (adjusted for d.f.) = percent Standard Error of Est. = Mean absolute error = Durbin-Watson statistic = (P=0.1157) Lag 1 residual autocorrelation = P-Value < 0.05 → Lack-of-fit!

/ department of mathematics and computer science Software Statgraphics: menu Special -> Experimental Design StatLab: Design Wizard (illustrates blocks and fractions): Box (simple optimization illustration):

/ department of mathematics and computer science Literature J. Trygg and S. Wold, Introduction to Experimental Design – What is it? Why and Where is it Useful?, homepage of chemometrics, editorial August 2002: Introduction from moresteam.com: V. Czitrom, One-Factor-at-a-Time Versus Designed Experiments, American Statistician 53 (1999), Thumbnail Handbook for Factorial DOE, StatEase