Experimental design. Matti Hotokka Department of Physical Chemistry Åbo Akademi University

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1 Experimental design Matti Hotokka Department of Physical Chemistry Åbo Akademi University

2 Contents Elementary concepts Regression Validation Design of Experiments Definitions Random sampling Factorial designs Response surface designs Robust parameter design [1] Otto, Chemometrics, Wiley, [2] Wu, Hamada, Experiments, Wiley, [3] Snedecor & Cochran: Statistical Methods, Iowa State Univ. Press [4] Cochran, Experimental designs, Wiley, 1966.

3 Definitions Analytical function Signal y: to be modeled or optimized. E.g., yield, measuring time, figure-of-merit, deviation from a model etc. y x

4 Definitions Factors Factor, or feature: ph, concentration, temperature,... A huge number of factors govern every measurement. The chemist must know which are important and must be tested. The others are kept as constant as possible.

5 Definitions Replications Every measurement is repeated from start a number of times so that a mean, a standard error and a confidence limit can be determined. Observation: mean of a set of parallel measurements. Blank: reference observation with default value of all the important factors, y B.

6 Definitions Replications vs. Repetitions Repetition: Repeated reading of the meter. Replication: New measurement from start. Repetitions test your ability to read a digital meter. Replications test the experimental errors in the measuring procedure.

7 Definitions Calibration parameters Sensitivity Detection limit Precision and trueness Specificity and selectivity

8 Definitions Calibration curve Sensitivity = slope Signal, y y = b0 + b1 x x y y b1 = x b 0 Intercept b 0 can be ignored if the sample is obtained against a blank (reference). Concentration, x

9 Definitions Analytical range Dynamic range: The valid range of x where the signal y depends functionally on x. Analytical range: the interval of x where the signal y can be determined accurately.

10 Definitions Signal, y DL Dynamic range Analytical range LoD Concentration, x

11 Definitions Detection limit Detection limit: lowest value of x where the signal can still be separated from noise. Noise is measured as the variance of the blank. y = y + 3 s x DL B B DL = y DL b b 0 1

12 Definitions Limit of determination Limit of determination: lowest value of x (concentration) where y can be determined with a useful accuracy.

13 Definitions Bias Error e in variable x (say, concentration) e = x x true x

14 Definitions Bias Error e in variable x (say, concentration) ( ) ( ) e = x x = x x + x x true Random error true Bias Random error x true x x Systematic error

15 Definitions Precision and trueness Precision = repeatability s = ( x x) n i 1 2 Trueness = deviation from true value x RR(%) = 100 x true

16 Definitions Selectivity Selectivity: possibility to measure in presence of interfering components. Specificity: sensitivity for a given analyte. Analytical resolution: N = x/ x. x x

18 Random sampling Why randomization All experiments must be made in random order. Response Response Systematic True slope Drift Random True slope Concentration Concentration

19 Random sampling Random number lists The random sequences are obtained from tables of random numbers. Nowadays random number generators of pocket calculators may be used. Normally, you get the same sequence every time. This is OK. If you want truly random numbers you should use a random seed.

20 Random sampling Always randomize Assume that four different concentrations are to be tested. Name them A, B, C, D. Make four parallel measurements for each: A 1, A 2, A 3, A 4 etc.

21 Random sampling Always randomize First run: Measure A 1, B 1, C 1, D 1. Second run: Start from scratch and do A 2, B 2, etc. A 1, B 1, C 1, D 1 A 2, B 2, C 2, D 2 A 3, B 3, C 3, D 3 Wrong! Systematic errors will not be found. A 4, B 4, C 4, D 4

22 Random sampling Always randomize Randomize the order of concentrations in the runs. A 1, B 1, C 1, D 1 C 2, D 2, A 2, B 2 D 3, A 3, B 3, C 3 B 4, C 4, D 4, A 4

23 Random sampling Always randomize Use linear (or non-linear) regression to analyse the results. Specifically, plot the residues to see whether some effects were not captured.

24 Random sampling Analysis y A B C D Conc.

25 Random sampling Residues Residues A 1, B 1, C 1, D 1 C 2, D 2, A 2, B 2 D 3, A 3, B 3, C 3 B 4, C 4, D 4, A 4 Drift! A B C D Conc.

26 Random sampling Types of factors Controlled factors Varied systematically or kept constant Known factors that cannot be controlled E.g., drift of instrument Unknown factors that can be anticipated E.g., impurities of the chemicals Truly unknown effects

27 Random sampling Blocking Some constant factors cannot be kept fixed but vary from batch to batch, day to day,... Make a series of measurements varying one factor and keeping the other conditions as constant as possible => A 1, B 1, C 1, D 1. This is a block. Then measure A 2, B 2, C 2, D 2 keeping the conditions constant but not necessarily the same as in block 1 if this is not possible.

28 Random sampling Latin square designs Randomize the blocking experiment. Run Sample A B D C 2 D C A B 3 B D C A 4 C A B D Observe the good balance.

30 Factorial designs What? Typically two-level experiments A low level and a high level for each factor. Typically for screening Study which of the presumed factors really show a significant effect.

31 Factorial designs Two levels Each factor is tested at a low and a high level. Designate the levels symbolically -1 and +1. Rate of p-phenylenediamine (PPD) oxidation at constant enzyme level of 13.6 mg L -1 is studied using spectrophotometry: Factor Level T, C ph [PPD], mm

32 Factorial designs Experiment plan Run Factors T PPD ph y 1 - y 4 y s but like this, randomized.

33 Factorial designs 2 k design Run Coded factor levels Main effects Interaction effects T PPD ph TxPPD TxpH PPDxpH y Experimental accuracy? Four parallel determinations => s = D.f.=3. Compute the differences high level - low level: D T =(y 1 +y 2 +y 3 +y 4 )/4 - (y 5 +y 6 +y 7 +y 8 )/4

34 Factorial designs 2 k design D T = 0.53 D PPD = 5.73 D ph = 2.19 D TxPPD = D TxpH = D PPDxpH = Statistically significant effects at 95 % confidence: D >Student t s = 0.18, 3 degrees of freedom => t = 3.18 D > = D PPD, D ph and D PPDxpH are significant. s

35 Factorial designs Another analysis method Consider the normal distribution. P y 1 y 1 0 x 0 0 Most statistical quantities are normally distributed. x y

36 Factorial designs Half-normal quantiles Given the integrated normal distribution in a y vs y plot, calculate what is the x that gives your y value. i Φ N In the example there are six effects, N = 6. What is x if y is *(1-0.5)/6 = 0.542? Answer: This you obtain from the tables of normal distribution.

37 Factorial designs Half-normal quantiles Six effects: N = 6. Numbered 1,2,3,...,6 in ascending order. Therefore i y = *(i-0.5)/8 x

38 Factorial designs Half-normal quantiles Sort the D values in ascending order x y TxpH TxPPD T PPDxpH ph PPD TxpH PPD ph T PPDxpH TxPPD 1 2

39 Factorial designs Orthogonal arrays The 2 k design gives 64 combinations for k = 8. Too many degrees of freedom! Choose half of the combinations, 2 k-1. However, you cannot choose any set of combinations. The arrays must be orthogonal.

40 Factorial designs Orthogonal arrays There are many ways of choosing orthogonal arrays. Plackett and Burmann, and Hall, and Taguchi, have published large selections based on Hadamard matrices.

41 Taguchi table L4 (2 3 ) Taguchi table L4 (2 3 ) Full set of experiments Eight experiments

42 Taguchi table L4 (2 3 ) Taguchi table L4 (2 3 ) Taguchi design Four experiments

43 Factorial designs Orthogonal arrays What you loose when using orthogonal arrays is (some of) the interaction effects.

44 Factorial designs More reduction Designs of size 2 k-p, p>1, also have been proposed.

45 Factorial designs Three-level designs +1 x x 1

46 Factorial designs Three-level designs Response -1 0 Factor +1

47 Factorial designs Central composite design y y y y y y y y 8 9 -a 0 0 y a 0 0 y a 0 y a 0 y a y a y 14 15, 16, y 15, y 16, y 17

48 Factorial designs Box-Behnken design

49 Factorial designs Lattice design

50 Factorial designs Analysis Use multivariate regression.

52 Response surfaces Optimization tasks Biggest is best Find a set of factor values that give maximal response (e.g., yield) Smallest is best Find minimum Nominal is best Minimize the difference (measured - nominal)

53 Response surfaces The response ph PPD

54 Response surfaces Optimization techniques Any optimization strategy can be used Single factor at a time (the engineering method) may miss the optimum Fixed-size simplex algorithm may work better

55 Response surfaces Engineering method ph Max Measure at the indicated points. PPD

56 Response surfaces Simplex method Code the factor values to the range (0,1). Generate the initial simplex. Measure at the indicated points. ph If there are N factors (here N=2) the simplex has N+1 points. Here the points are 0,0;1,0; 0.5, PPD Unknown surface

57 Response surfaces Simplex method ph 1 0 w 0 p 1 Remove the worst point. Calculate the centroid of the remaining points. PPD p = 1 N N + 1 v j + 1 j = 1 j w

58 Response surfaces Simplex method Measure at the indicated point. ph 1 0 w 0 p r 1 Generate a new point. PPD r = p + ( p w)

59 Response surfaces Simplex method Measure at the indicated points. ph 1 0 w 0 1 PPD

61 Robust parameters Factor categories Control factors Can be kept fixed once chosen Noise factors Cannot be controlled Create the variations in the quality of the product

62 Robust parameters Typical procedure Usually, the quality of the product is improved by reducing the noise. Unfortunately the noise factors are difficult (=expensive) to reduce.

63 Robust parameters 2 k design, a reminder Run Coded factor levels Main effects Interaction effects T PPD ph TxPPD TxpH PPDxpH y

64 Robust parameters Control of noise Main effects: All control and noise factors. Interaction effects between control factors and noise factors may be quite large. If this is the case, then variations in the product quality may be reduced by adjusting the control factors so that the effect of noise is reduced.

65 Robust parameters An example Consider the dependence of y (signal level) on x (voltage over detector = control factor). y Width of noise A B x

Experimental design. Matti Hotokka Department of Physical Chemistry Åbo Akademi University

Experimental design. Matti Hotokka Department of Physical Chemistry Åbo Akademi University Experimental design Matti Hotokka Department of Physical Chemistry Åbo Akademi University Contents Elementary concepts Regression Validation Hypotesis testing ANOVA PCA, PCR, PLS Clusters, SIMCA Design