Design of Experiments. Study Support. Josef Tošenovský

Transcription

1 VYSOKÁ ŠKOLA BÁŇSKÁ TECHNICKÁ UNIVERZITA OSTRAVA FAKULTA METALURGIE A MATERIÁLOVÉHO INŽENÝRSTVÍ Design of Experiments Study Support Josef Tošenovský Ostrava 15 1

2 Title: Design of Experiments Code: Author: Josef Tošenovský Edition: first, 15 Number of pages: 55 Academic materials for the Economics and Management of Industrial Systems study programme at the Faculty of Metallurgy and Materials Engineering. Proofreading has not been performed. Execution: VŠB - Technical University of Ostrava Z českého originálu přeložil: Mark Landry

3 CONTENTS 1. FULL FACTORIAL DESIGN AT TWO LEVELS (4) 1.1 Coded variables (5) 1. Effect of factor (7) 1.3 Effect of factor graphs (8) 1.4 Dispersion estimate of the effect of factor (9) 1.5 Effect significance test (1) 1.6 Effect of factor graphic evaluation (1) 1.7 Graph interaction (13) 1.8 Regression model of experiment (14). FRACTIONAL FACTORIAL DESIGN AT TWO LEVELS (15).1 Factor operations and their properties (16). Determing comfounding patterns (17).3 Stating the number of comfounding factors (3).4 Generator selection for medium designs (4).5 Design projection (4).6 Supplementary fractions: creation and properties (6).7 Design of experiments for eight factors (8) 3. SIGNIFICANT POINTS OF DESIGN (9) 4. SECOND ORDER RESPONSE SURFACES MODELS (3) 4.1 Model determining conditions (3) 4. Metods of determining second order response surfaces models (3) 4..1 Composite design (33) 4.. Calculating the optimal level of significant factors ( Box - Behnken design (BBD) (33) 4..4 Designs for factors at three levels (35) 5. BLOCKING IN EXPERIMENTAL DESIGN (36) 5.1 Block construction (37) 5. Evaluating block effects (38) 6. Lack-of-fit TEST (41) 7. STEEPEST ASCENT (45) 3

4 STUDY INSTRUCTIONS For the subject Design Planning in the second semester of the follow-up masters study branch you will obtain study material for part-time study. PREREQUISITES Theory of probability, mathematical statistics, econometry SUBJECT OBJECTIVE AND LEARNING RESULTS The objective of the subject is to present the basics of design planning theory AFTER STUDYING THROUGH THE SUBJECTS A STUDENT SHOULD BE ABLE TO: Skill results, for example: A student should basically understand design matrix planning A student should be able to be oriented in connected regressive analysis, econometry and design matrix planning A student will gain a useful overview of the effective procedures in forming regressive models Skill results, for example. A student will know how to plan an industrial design matrix A student will be able to evaluate the effects of input parameters A student will discover a design matrix model. IN STUDYING EVERY CHAPTER WE RECOMMEND THE FOLLOWING PROCEDURES: To repeat gone-through chapters, to clarify the objectives of new chapters and connections to previous ones. METHOD OF COMMUNICATION WITH TEACHERS: For example.. At the beginning of a semester, a teacher requests a semestral project on an assigned topic from the area of regressional analysis. The project will be checked by a teacher up to 14 days from its submission and the results will be sent to students by means of IS EDISON CONSULTATION WILL TAKE PLACE WITH A SUBJECT S GUARANTOR OR A LECTURER at lectures individual consultation after a previously agreed to or telephone call. 4

5 1. Full factorial design at Two Levels Time for study: 5 hours Objective: Managing these operations a) putting a design matrix together b) calculation of the effect of factor c) evaluating the effect s significance Lecture Under the concept to design matrix it is further understood as to change usual working conditions with the objective of finding working procedures and at the same time to gain deeper knowledge about a product s properties and the production process. It is possible to divide design matrix procedures into: a) unplanned, b) planned. Planned designed matrices comply with an design matrix and a plan. A design of experiment consists of 3 events: a) number of experiments, b) conditions, under which an individual experiment is done, c) order of experiments. It is obvious from the above mentioned that it is important to distinguish between concepts experiment = determining quality indicator values for certain, previously planned, production conditions; design of experiments = the system (matrix) of all experiments; If we indicate the following quality indicator Y (resp. quality indicators Y 1,...,Y k ), the factors which influence it A,B,C,D..., then the objective of the design of experiment is 1) to quantify the rate of significance of each factor A,B,C,D, and to decide which factors are a decisive method of influencing quality indicators, ) to determine the level of significant factors in order for Z to be optimal and stable. Example 1 (Data: lit. [4], s.137, ex.4.8, corrected) It studied the influence of content C, Mn and Ni on the temperature of martenized transformations. The result of the design matrix is given in the table. Convert the variables into coded values and evaluate the influential factors and interaction. Solution: 5

6 We will set up a table of factors and their considered levels (Tab.1): Tab.1 Factor and level list Factor Sign Lower level Upper level % C A,1,9 % Mn B, 1,4 % Ni C,1, Now let s build a design matrix plan (Tab ) There exists many methods how to make a plan, according to which individual experiments will be carried out. Among the most used plans are full factorial designs, which look like this in a given case: Tab. Design matrix plan A B C Y1 Y Average Y,1,, ,9,, ,1 1,4, ,9 1,4, ,1,, ,9,, 7 9 8,1 1,4, ,9 1,4, 19 1 Yi = Starting temperature ( o F) The design matrix contains all variants of inputs (the number of experiments, from which 3, is commonly k ) 1.1 Coded variables It is more suitable to write the design matrix plan using these symbols: if each of the factors is considered at two levels, then the lower level will be designated -1 (resp.only "-" and the upper level +1 (resp. "+"). Table will then have the shape: Tab.3 Design matrix plan in coded variables Experimen A B C Y i 1 t

7 The recalculation of real variables into coded variables can be carried out not only from extreme values x max (= +1) and x min (= -1) in such a way that: x c x o x x max max x x min min (1) where x o = variable x in original units, x c = coded variable, x max = upper level x, x min = lower level x. Of course, formula (1) is possible to use for calculating real variables for coded value knowledge. Factor A value recalculation: a) for lower level,9,1,1,4 x c A,9,1,4, b) for upper level,9,1,9,4 x c A,9,1,4, 1 1 For qualitative variables formula (1) is not used: this variable gains only two values, so that it doesn t make sense to calculate (x max + x min )/. The number of experiments, from which the full design matrix is made, is calculated using the k relation n, where k denotes the numer of factors. Thus, in this case, where there are k = 3 factors, the number of experimental runs equals n = 3 = 8. Therefore the table has 8 rows. A graphic illustration of the design matrix plan is called CPLOT. For two factors there is a square, for three factors a cube with the centre at the beginning of the coordinate system. Coordinate peaks correspond to individual design matrices. Task 1 Draw an design matrix plan and write appropriate coordinates to the peaks. If the design plan is stated, individual experiments are carried out under some condition, it is possible to carry out the full design and to record the values of the observed indicator Y. In our case each experiment will be repeated twice. The results are in Table 4: 7

8 Tab.4 Experiment results with repetition Test A B C Y1 Y Average Y Creating Table 4 is the end of the preparation and design work. Further on, only calculations follow. Their objectives will state which of the factors influence Y in a important way. In regards to this, several factors cannot be independently significant, while in co-operation with another factor (which is called interaction) they are already significant. They complete Table 4 with all possible interactions. Only then can the significance of factors and the interaction be calculated. The signs in columns AB, AC, BC, ABC increase as the product of signs in corresponding columns (Table 5). Tab.5 Factors and their interactions A B C AB AC BC ABC Average Y There will further be the calculation of the effect (influence) of individual factors. 1. Effect of factor The effect of factor is understood as a change of the quality Y indicator, which causes a changeover of this factor from the lower level (-1) to the upper level (+ 1), without regard to the level of other factors. The calculation of the effect of factor can be carried out by various methods, for example through the sign method, Yates method, using regression coefficients. Sign method We gain the calculation of effect A through a sign shift from column A to column Y and their sum, which we divide by the number n/, n is the number of experiments. For example, for Ef(A), we get 8

9 Ef(A) = 4 1 ( ) = =,5( ) =,5(-3) = Tab.6 The effect of factors using the sign method A B C AB AC BC ABC Average Y effect factor ,5 Average Y A B AB C AC BC ABC A calculation record for other factors and interactions using the sign method: Ef(B) =,5 ( ) = -9 Ef(C) =,5 ( ) = -65 Ef(AB ) =,5 ( ) = 1 Ef(AC) =,5 ( ) = 15 Ef(BC) =,5 ( ) = 1 Ef(ABC ) =,5 ( ) = Effect of factor A, B, C graph At first we calculate the average Y for the lower ( Y ) and upper ( Y ) levels, then we construct the graphs. Tab.8 Average Y for upper and lower factor levels A B C Y Y ,5 585 Y 65 57,5 5 Example for A: Y =,5( ) = 84; Y =,5( ) = 65. It is possible also to calculate effect A as the difference of averages Y - Y : ef(a) = = Y a Y are used to construct a sectional graph, for example, for factor A:

10 Řada ,5-1 -,5,5 1 1,5 Fig 1. Sectional Graph (Excel) for factor A The difference Y in the final points of the graph show the effect size: a zero effect corresponds to a section parallel with axis X. A growing direction (a positive effect) means that by the changeover the effect from the lower to the upper level factor level increases, a decreasing direction (a negative effect) conversely means a drop in effect. 1.4 Dispersion estimate of the effect of factor The results of individual experiments Y 1,,Y n are random variable with normal distribution. Vector Y = (Y 1,,Y n ) is the same as in v regression analysis ( Y X. ). For vector Y it was proven: Y ~ N( EY,var Y) ; it is assumed that var Y I which means, that all random variables of Yi have normal distribution and the same dispersion. The normality assumption Yi is significant: it is used further a) when deriving the distribution of dispersion s of the estimate of effect, b) in the graphic evaluation of the effect s significance. Using the calculation of effect as the average difference we can characteristically obtain the effect. If each random variables Yi has the normal distribution, that is. ~ N(, ), then for their average it is valid that Y ~ N(, ) N and also the effect of factor distribution ef ~ N(, ). e e e ef Y Y ( for it is linear combination ) has a normal Y i Y a Y Mean value E( Y Y ). e The estimate of the dispersion effect e (the same for all factors) will be: 1

11 4 () N. s e where N is the total number of experiments (including repetition). 4 We have then ef ~ N(, ) N The dispersion of the estimate of effect is used for testing the significance of the effect of factors. In the case of repeated experiments, σ it is estimated using dispersion s, which is calculated s s s k k k (3) n 1, i i i n is the number of repeated i-th experiment, s is a weighted dispersion. s i is the dispersion Y in the i-t experiment. It is possible to calculate dispersion easier in two repetitions (than using the dispersion definition) using the relation 1 s i ( Y1 Y ) (4) If individual experiments are not repeated, the estimate σ is carried out using what are called central points (see further). After calculating the factor effect it is necessary to carry out an evaluation of the importance of the obtained values so we can objectively decide if ef(a) is a sufficient enough value and that is why A is an influential factor. We can use various methods of evaluating the significance of effects, especially a) Tests, b) graphic methods, for example a normal probability plot, c) ANOVA. Furthermore, we will deal with a test and graph for evaluating the significance of effects. 1.5 Significance of effect test The importance of repeated experiments is the possibility of testing the significance of effects or of carrying out a residue anaysis. The disadvantage of repetition is increasing the number of tests and thus costs. Other possibilities, also enabling the same without testing and therefore for lower costs, are for example: 11

12 a) two pairs of measurements, b) placing central points into the design matrix, c) a projection plan. Two pairs of measuring mean that during each test many measurements are carried out. For example, an idential product is measured in many places. Two pairs of measurements is not identical with repeated measuring. Central points and plan projection are dealt with in other chapters. In this chapter we deal with all points of the significant importance effect test. Testing procedure 1. Zero hypothesis H o : the factor effect is unsignificant Alternative hypothesis H 1 : the effect of factor is important.testing criterium efekt t (5) s e 3. Critical value t n n n n 1... ( ) k (6) where n 1,..., n k are the number of repeated experients, here n i = ; n is the number of experiments without repetition, here n = Test conclusion: for t ( ) rejects a zero hypothesis which means that the t n 1n... n n k effect (and thus the factor) is significant. In example 1 t ( n... ) 16 (,5),36 1 n n n t k 8 Dispersion s k i1 k i1 ( n 1) s i ( n 1) i i s ,5 1

13 4. s 4.531,5 s ( ef ) 13,81 ; s(ef) = 11,5 N 16 N = 16 is the number of tests including repetition. The calculations are arranged in Table 9. Tab.9 Testing the significance of effect Y1 Y Aver.Y si si effect factor T=ef/s(ef) ,847 55, , A , B -7, , AB, , C -5, , AC 1, ,847 1 BC, , ABC -,86957 The critical value exceeds testing criterium factors A,B and C in an absolute value. 1.6 Graphic evaluation of the effect of factor If repeated individual experiments are not carried out, a graphic method is used to determine significant factors, specifically the normal probability graph. In the graph the effect is carried out on the horizontal access and on the vertical axis is the relative cumulative frequency in percent 1( i,5) Pi, m (7) where i = 1,,..., m is the number of factors and interactions. As to significance we consider those factors, which are found considerably outside the main lines (illustrated somewhere by a straight line, which represents a distribution function on a normal probability paper). In using graphic methods it is useful to place these data into auxiliary tables: Tab.1 Table for the graphic evaluation of effects i Effect Factor o A B C ABC AB BC AC es P i 7,14 1,43 35,71 5, 64,9 78,57 9,86 The effects in the second row are arranged in ascending order. The graph, (the normal probability plot), is built from data in the second row (the horizontal axis) and in the fourth row (the vertical axis) is in Fig. 3. Values of P i (percentage) are transferred to normal probability paper (the scale is not even). In the graph it is seen that outside the main lines there are factors, for which the testing criterion exceeds the critical value. The factors are with the greatest effects: A (the most significant) B and C. 13

14 percentage 99, Normal Probability Plot for Y 1, Standardized effects Fig 3 Normal probability plot 1.7 Interaction graphs For significant interaction it is necessary to make graphs, enabling the discussion about the optimal level of individual factors in dependence of another factor. The resultant optimal level can be distinguished from those which are stated individually from the graph of levels. This graph is then decisive. As an example let s make a graph for AB interaction of influence A on quality indicator Y depending on factor B. In the design matrix plan we can find the necessary values. Tab.11 Design matrix plan A B C Average Y Let s make two tables, necessary for making two line graphs Tab.1 Determining the extreme points of graph interaction A B average A B average 14

15 Manual construction of graph interaction AB: effect A on Y depending on B. From the graph it is seen that for example for maximalizing Y an A and B lower level is optimal Y 89 -B 35 +B Fig. 5 Effect A on Y depending on B A 1.8 Regression model of experiment After determing the significant factors it is possible to make a model of experiment. Furthermore, we will distinguish these models (for two factors): a) linear model: Y = b + b 1 x 1 + b x This model is sufficient for modelling small areas, b) full second order model: Y = b + b 1 x 1 + b x + b 11 x 1 + b x + b 1 x 1 x is often enough even for describing complicated forms of dependence, the coefficients are easily determined, c) non-full second order model Y = b + b 1 x 1 + b x + b 1 x 1 x. Calculation of coefficients for individual models: For model a) and c) the coefficients can be calculated using various methods, for example: 1) using effects of model: b o Y for example.,5.ef(x 1 ) = b 1, ) least square method., b 1, b,, b k is half the effect of an appropriate factor, For determining model b) it is not possible to use the above mentioned design matrix plan. They use special plans, for example: 1) combined plan, ) three-level plan, 15

16 3) Box - Behnken plan. The linear model in example 1 will be Y = 55,5 + (-575/).A + (-9/).B + (-65/).C Y = 55,5-87,5.A - 45.B - 3,5.C Interaction factors are not important therefore they are not in the model. It is possible to use the model for calculating theoretical values. For example for A = B = C -1 it will be Y ˆ1 55,5 87,5( 1) 45( 1) 3,5( 1) 917,5. The difference between the empirical value Y 1 = 94 and theoretical value Y ˆ 1 = 917,5 is the residual deviation e 1 = ,5 =,5. We substitute coded variables into the equation. It is possible to even substitute non-coded variables. The problem is for qualitative variables: those not possible to replace. For example: how do you substitute a new machine, an afternoon shift, material of type A, etc. into the? The problem is solved by introducing auxilary variables. Concept summary: Effect of factor, effect of factor graphs, distribution estimate of the effect of factors, effect significance test, graphic evaluation of the effect of factors, interaction graphs Questions: a) Can you describe the procedures for making a design maxtrix plan? b) Which calculation method of the effect of factor do you know? Describe it. c) How do you evaluate the effect through a test or graphically? d) Can you make tables necessary for constructing a factor interaction graph?. Fractional Factorial Design at Two Levels Time for study: 5 hours 16

17 Objective a) Making a fractional design plan b) Determining a group of factors with the same effect c) Methods of optimally generating factors Lecture In full factorial designs a design plan is made for every factor. For fractional factorial designs a plan is made only for several factors. We call these main factors. Other factors we call secondary factors, which are created (generated) using main factors. Through this we achieve a reduction in the number of experiments. Designating main and secondary factors is in no way connected to the size of their effect on observed indicator Y. It is not even possible at the beginning of an experiment to limit it, for what comes from it that we do not know anything about the effect. If it concerns the method of stating auxilary factors using main ones, it is significant how it is done. We can devote ourselves to this problem later. If k is the designation of a full factorial design at two levels, = the number of the level of factors, k = the number of factors k p then is the designation for the fractional factorial design, p = reduction degree. For example, we want in plan 7, which is represented by n = 18 experiments, to reduce the number of experiments to half, tj. we get the fractional factorial design, which is 71 represented by n 64 experiments, that is half. It is at least possible to reduce the number of experiments. Plans with the number of experiments reduced to one half are called half plans. The degree of reduction p can be even higher than 1, for example 7 4, where there will only be n = 8 experiments. It is for k = 7 factors that there is the highest possible reduction. For so called linear plans (plans of the first order), used for determining models, which contain only factors without their conbination or power, the greatest possible reduction comes from the rule according to which the number of experiments can not be smaller than the number of factors. Then n k must be valid. For second order plans, which are determined for determining second order models, there are more conditions, as we can see in the fourth chapter. The plans with the greatest possible reduction are saturated plans. In the example mentioned above k = 7 and n = 7-4 = 8, so that p = 4 is the greatest possible degree of reduction. For example for k = 15 factors the greatest possible reduction is 15-11, for the number of experiments it is when p = 11 equalling n = 16 and the number of factors is k = 15. If we still carry out a higher reduction, for example. 15-1, then k = 15, but n = 8, so that n < 17

18 k. Between two-level and saturated plans there can still be a series of possible reductions. Plans with degrees of reduction between maximum and minimum are called central plans. For example between 7-1 and 7-4 there are central plans 7- and 7-3. It is possible to then divide fractional factorial plans in three groups: a) plans with the lowest reduction (half plans) b) plans with the highest reduction (saturated plans) c) central plans.1 Operations with factors and their properties Let s mark factor I, containing only the level + 1. Such a factor is called per unit. For operations with factors these obvious relations are valid: A.A = I (8) A.I = I.A = A (9) (A.B).C = A(B.C) (1) A.B = B.A (11) In evaluating fractional plans there are several important concepts used, which we will explain in the following general examples. Example 1 Create a half plan for factors A,B,C,D and E. Solution: We procede in a way that we determine 4 main factors (for example A,B,C,D) for which we build a full plan and the remaining (secondary) factor E is expressed as their combination, for example E = ABCD. (1) Every combination of factors makes a word. A word consists of the letters (of factors) The number of letters in a word is th word length. Relation (1) is called the generator. In plan k p there is p secondary factors, from which each has to be created (generated) using main factors, that is why p is also a number of generators. By multiplying the generator (1) by the left side, that is by factor E and by using the relation (8) (11) we get E.E = E. ABCD I = ABCDE. (13) The words, which are equal to factor I, we call th defining relation, for example (13). As we can further see, defining relations can also be more than one. The length of the shortest word 18

19 in defining relations is called the resolution of the design and is recorded to the type of plan in Roman numerals as an index. For example here it is V. We read from this the recording: the factors are at two levels, the number of factors is 5, the auxilary factor is one, it deals with a two-level plan and the resolution of the design is V. The resolution of the design is V because the word in defining relation (13) has the length ( the number of letters) Determining confounding patterns Using the defining relation, it is possible to find pairs of factors (resp. interactions) which create the same sequence of signs and which are calle confounding factors. For example, the confounding interaction for DE: I = ABCDE / DE According to (8) and (9) we get DE.I = DE.ABCDE DE = ABC. The product of signs in the columns under D and E are in agreement with the product of the signs for factors A, B and C. Try to calculate this. Example 7 (Data: [4], p.18/5.7, modified) The effect of 7 factors is observed for the time (Y in min), needed to wrap 1 standard packages. Factors, which can effect performance, we designate by numbers 1 to 7. They are Factor Lower level (-1) Upper level (+1) 1 Management Present Absent Sex Man Woman 3 Time of Day Morning Afternoon 4 Temperature Air Conditioning No Air Conditioning 5 Music There is There is not 6 Age Up to 5 Over 5 7 Place of Work Countryside Town/City 7 4 Create a factorial plan with generators 4 = -13, 5 = 13, 6 = -1, 7 = -3. Determine confounding factors and evaluate the importance of the factors. Is it an optimal place of work? Solution: Design plan (saturated) factors 1 3 4=-13 5=13 6=-1 7=-3 Y(min.) 19

20 tests , , , , , , , ,4 p 4 The number of defining relations will be I = - 16 = 135 = - 37 = (the product of two) = = 1367 = 346 = = - 45 = 147 (the product of three) = 567 = = 1456 = 3457 (the product of four) = There are then 15 defined equations in total. Determining confounding factors using defining relations: 1 = -6 = 35 = -137 = -34 = = 367 = 1346 = -57 = -145 = 47 = =1567= = 456 = = = -16 = 135 = -37 = -134 = -356 = 1367 = 346 = -157 = -45 = 147 = = 567 = -467 = 1456 = 3457 = It makes sense to leave only the interaction of two of them, the higher is considered insignificant: 1 = - 6 = 35 = 47; = - 16 = - 37 = - 45 Calculation of the effect of factors No.1 and using the sign method: Ef(1) = 1/4 ( - 46,1 + 55,4 44,1 + 58,7 56,3 + 18,9 46,4 + 16,4 ) Ef(1) = - 1, Ef() = 1/4 ( - 46,1 55,4 + 44,1 + 58,7 56,3 18,9 + 46,4 + 16,4 ) Ef() = -, Graphic evaluation of the significance of effects. In manually constructing graphs it is necessary to make a table Number Effect -,85-16,575-1,875-3,175 -,775 -,55 3,45 Factor Pi (%) 7,14 1,43 35, ,9 78,57 9,86

21 percentage On the graph on the horizontal axis are the effects, on the vertical axis Pi. Significant are those factors which are found outside the main line, illustrated with a straight line. Factors 5,3 and 1 are outside the main lines. 99, Normal Probability Plot for Y 1, Standardized effects Interaction graphs Determining the and points of factor interactions 1 and 3: 1 foreman: absent + present 3 time of day: morning + afternoon In the design plan we determine (for factor levels 1 and 3 according to the tables) the relevant Y. 1 3 average ,1 44,1 45, ,4 58,7 57,5 1 3 average ,3 46,4 51, ,9 16,4 17,65 Y 57,5 51,35-3 morning 1

22 45,1 +3 afternoon 17,65-1 absent +1 present 1 foreman Fig.1 Factor interaction of "foreman" and "time of day" Determining the and points in factor interaction 1 and 5: 1 foreman: absent + present 5 music: none + playing 1 5 average ,3 46,4 51, ,4 58,7 57,5 1 5 average ,1 44,1 45, ,9 16,4 17,65 Y - no music 57,5 51,35 45,1 + with music 17,65-1 absent +1 present Fig Factor interaction of "foreman" and "music" foreman Determining the and points of interaction factors 3 and 5 3 time of day - morning + afternoon 5 music - none + playing 3 5 average

23 - - 55,4 58,7 57, ,3 46,4 51, average ,1 44,1 45, ,9 16,4 17,65 Y 57,5-5 no music 45,1 +5 with music 51,35 17,65-3 morning +3 afternoon day part Fig 3 Factor interaction of "time of day" and "music" Conclusion From the given graph it is possible to evaluate that the shortest time for wrapping 1 standard items (with the greatest work productivity) will take place in the afternoon in the presence of a foreman, also playing music has a positive effect on work productivity. Interaction 3 and more factors were not examined, for it is generally valid that the longer the word, the lesser the effect. Generator selection As was already said, in creating a half plan, one of the factors is expressed as an interaction of the rest. When we say "the rest" it does not mean necessarily all the rest. So for example, for five factors A,B,C,D,E it is possible to express E by a number of methods for the half plan. Let s compare the two: a) E = AB, b) E = ABCD The relevant defined equations are a) I = ABE b) I = ABCDE 5 1 In case a) we have plan 5 1 III, in case b) plan V. This second design is better, for it has design resolution V which means in looking for confounding patterns there will be short and long words in the sum; for example for A, we have a) A + BE b) A + BCDE 3

24 In case b) it creates a confounding pattern with the A interaction of more factors, and that is why it has a smaller share on the total effect, for more than two factors, even so small, it is negligible and it works only with short words, here with factor A. Thus, it is easier to discuss it towards the calculated effect. The optimal selection of generators for the various number of factors (k) and different degrees of resolutions are given in the tables. One of the shortest is designed by L.W.Robinson (Tab.1) In the table there are 4 arrows: for example, for k=3 and mid-resolution IV: if resolution IV has to be solved, a complete design has to be done. The number of experiments are given in bold letters for given k and the level of resolution. For k = 1 to 15 generators are not given. It is always better to do a halfing design and then, on the basis on analysis, to fill it in if need be (by other fractions), than to immediately do a full design. The same principles as for two-level designs are valid for medium and saturated designs. Saturated designs are those where the number of parameters p = the number of experiments n. In Table No.1 are given the above-mentioned optimal selection of generators (L.W. Robinson, QE 15, No 3, 3, p ). Tab.1 Optimal selection of generators Resolution level k III IV V 3 3 = 1. 4 Full = Full = = = =.3 5 = = = = = = = = = = = = = = = = = = = = = 1. 1 = = = = = = = = = = = = =.3 13= =.4 14 = = = Determining the number of confounding factors In design k-p there is p of secondary factors. That is why p generators are necessary from which it is possible to create 4

25 p p p... 1 p (14) of defined equations (the given defined equation + the product of pairs, of which p is over two, the product of triads, from which p is over three, etc.) It is possible to write statement (14) in the form k1 ( p k p p p resp. 1 resp. k k k p pk k p p 1.1 ) 1 (1 1) 1 k p The number of defined equations in plan k-p je p -1. From each defined equation it is k p possible to create for a given factor a confounding factor, so for plan is the number of confounding factors equal to the number of defining relations + a given factor (plus one) 1 ( p -1) + 1 = p Number of confounding factors in plan k-p is p. Example 3 (determining a defined equation) Let s consider the situtation when 7 factors A,B,C,D,E,F,G are observed, effecting quality 7 4 indicator Y. If we make saturated design, the main factors A,B,C are selected, from which we make a full design. The remaining four factors (D,E,F,G) are auxiliary. For example, generators are selected so that (their number is p = 4): D = AB, E = AC, F = BC, G = ABC. p The defined equation will be (their number is 1 = ): ABD = ACE = BCF = ABCG (the product of two)=bcde=acdf=cdg=abef=beg = AFG (the product of three) = DEF = ADEG = BDFG = CEFG (the product of four) = ABCDEFG Because instead of 7 18 of experiments there will only be 4 8, every design is a 4 fraction, which is of the whole design, so that 16 fractions exist and in each there are 16 confounding factors. For example for factor A we have this group of confounding factors. A+BD+CE+ABCF+BCG+ABCDE+CDF+ACDG+BEF+ABEG +FG+ADEF+DEG+ABDFG+ACEFG+BDCEFG 5

26 p (their number is 4 16 ) It makes sense to leave only the interaction of two of them, so that for A it will be A+BD+CE+FG..4 Generator selection for central designs Central plans should be used to connect the good properties of half and saturated plans. Among central plans such a plan can be sought, which has fewer experiments than half plan and if possible the most suitable groups of confounding factors. We should be reminded that the best confounding interaction is considered that which is created by the greatest number of factors. The composition of confounding factors is connected to the selection of generators. Example 4 To compare the results of the various selections of generators for half plan A,B,C,D,E,F,G. 7 3 with factors a) Generators: E = ABCD, F = ABC, G = BCD The defined equations will be I =ABCDE = ABCF = BCDG = DEF = AEG = ADFG = BCEFG 7 3 The level of resolution is III: III b) Generators: E = ABC, F = BCD, G = ACD, Defining relation I = ABCE = BCDF = ACDG = ADEF = BDEG = CEFG = = ABFG, 7 3 where the level of resolution is IV: IV..5 Design projection If some of the factors have an insignificant effect, it is possible to leave them out. Such a design can be obtained with repetition. For example for three factors A,B,C it has the full design of a 3 shape A B C By leaving factor C out a design with repetition is established. It is generally valid that if we have full design k without repetition and if we leave h (h < k) factors out of it, we establish a full design for k-h factors with the number of repetition h. 6

27 If we leave out factor 1 from the halfing design, we establish a full design. Example 5 Let s take design 3-1 with factors A, B and C = AB. A B C = AB By leaving out C, we establish a full design. Similarly, if we leave out B we establish a full design A B C = AB If we generally leave out d of factors in design k-p without repetition, we establish either a full design or fractional design with repetition. The established design has the identical defining relation which also doesn t contain the left-out factors. When analyzing fractional designs complications arise as a result of the sum of factors, by which a calculated effect is found. In designing fractional designs there is an attempt to summarize because the sum a) was short and long words, b) was the smallest word in the sum It is possible to solve the first objective by the correct selection of generators; in the final result this causes a higher level of resolution. As an example we can make a comparison of 5 1 generators E = AB resp. E = ABCD, which leads to the design 5 1 III resp. V, that is a reolution level III resp. V..6 Supplementary fractions: creation and properties One possibility how to achieve objective b) is to supplement basic fractions of the plan by other supplementary fractions, that is combining fractions (sequence). Supplementary fractions for two-level designs Example Let s take factors 1,, 3, 4, 5 and half design. 7

28 The generator has the form 5 = 134. Using this generator the first half (the first fraction) of the full design is created. Together with all interactions of the two factors there is the design in the table: Y The effect of the individual factors: 1 = - 14 = -,75 =,5 15 = 1,5 3 = 3 = 1,5 4 = 1,5 4 = 1,75 5 = -6,5 5 = 1,5 1 = 1,5 34 =,5 13 =,5 35 =,5 45 = -9,5 In this design, several interactions have the same sequence of signs and therefore they also result in the same effect, for example 13 and 45. Interaction effect comes from ef(45) = 9,5. Because the joint effect with interaction is 13, we write ef(45) = -9, The second half (the second fraction) of the design 5-1 has the generator 5 = so that I = Confounding factor, for example to is 134 in the first fraction and in the second fraction Factor effect is in the first fraction 8

29 and in the second fraction ef() =, ef() = 18, The "pure" effect of factors and 1345 should be calculated only when there is knowledge of two fractions. Since only one fraction is available, it is not possible to calculate the "pure" effect. It is therefore required that there is the greatest interaction in the confounding patterns with factors. This is achieved if there is a high level of resolution. For half designs, if there is observed k factors 1,, 3,... k, this procedure is the best: a) to create a full design for factors 1,..., k-1 b) to express the k-th factor using the factors 1,,3..., (k-1): k = (k-1) Supplementary fractions for half and saturated designs In design 7-4 these generators will be available for forming all 16 fractions (each fraction is made up of 8 experiments): D AB, E AC, F BC, G ABC. Design 7-4 and experiment results (for generators with positive signs) are in the table: A B C D = AB E = AC F = BC G=ABC Y For example, we create another fraction design of 7-4 in this way: so that only for factor D the sign is changed. D = -AB, E = AC, F = BC, G = ABC, The design of this fraction and experiment results are in the following table: A B C D = -AB E = AC F = BC G = ABC Y Exper

30 For the first fraction there is for example for A the effect and confounding patterns (ex.3) ef ( A) A BD CE FG In the second case it comes that ef ( A).75. The confounding patterns for the second fractions are obtained from the first so that signs can change in defined equations where there is factor D. ef ( A). 75 A BD CE FG. It is possible to obtain the effect of factor A and confounding patterns in united plan as an average of both fractions. 1 1 ( ef 1( A) ef ( A)) (3.5.75) 1 [( A BD CE FG) ( A BD CE FG)] A CE FG,15.7 Design of experiment for eight factors Another way to reduction a design with a simple analysis of results is to construct special designs. For example, for 8 factors it is possible to make special designs in these steps: a) to plan the first 8 experiments in the same way as for design 7 4 with which the eighth column is created only with plus signs, b) the other eight experiments (that is experiments 9 to 16) are created by supplementary fractions with contrary signs. The advantages of this procedure are: a) the fewest experiments, b) it calculates the pure effect (without summation) Y , , , , , , , , , ,1 3

31 , , , , , ,3 At the conclusion of this chapter we make recommendations for creating fractional designs: a) the design should begin with a limited number of factors, which are enough to control the process, b) to select generators according to the recommendations in the tables, c) after calculating the effects to leave non-significant factors of the design out, d) according to need to add supplementary fractions, e) not to make a full design or a design with repetitions at the beginning of planning, f) for a greater number of factors look for special designs. Concept summary Operations with factors and their properties, determining confounding patterns, determining the number of confounding factors, selecting generators for half designs, projection designs, design for eight factors. Questions: a) Can you describe the procedures of fractional design plans? b) What problems are brought about by shortening designs? c) How do you evaluate and solve these problems? d) How are confounding patterns determined? 3. Significant points of design Time for study: 1 hour Objective: a) Types of significant points b) Coordinates and numbers of significant points c) Significant point use Lecture When looking for second-order regressive models (Chapter Four) it is necessary to fill in a design plan about the points with which we have in the meantime not worked with. That is why it is necessary to go through the points in this chapter. 31

32 For each of the mentioned points there is mentioned what coordinates it has, its recommended number and significance. In the design plan there is used according to need three types of points: 1) factorial points ) central points 3) axial points resp. star points Factorial points are always in the design. k p The number of factorial points is n. The coordinates for three factors: ( 1, 1, 1). Significance: it is used to calculate the effect of factor. Central points The recommended number of central points is 3-5 or a number determined from a table. The coordinates for three factors: (,,). Significance: a) from Yi in central points there is calculated the average and it is compared with the average Yi in factorial points (test of curvature), b) from Yi in the central points there is calculated the estimate of dispersion σ as a substitute for the estimate σ calculated from repeated experiments (Example 1), c) from residue e i in the central points there is calculated pure errors in measuring, d) the number of central points effects the shape of graph var(y ˆ). Central points can be added to design k-p, if all factors are quantitative. For qualitative factors level doesn t exist. Central points are not used for calculating the effect of factor. Axial points For two factors, x 1,x the design points lie on a circle with a centre (,), which goes through the top of a square. For three factors they are those points, where a round surface with a centre at the beginning and factorial points go through the design intersecting coordinate axes. Number of axial points: n = k Coordinates for three factors: (,,),(,,),(,, ), where recommended 4, n k is the number of factorial points. Significance: a) it enables the calculation of the coefficients in the full second-order model, b) it makes the calculation of regression coefficents more precise, c) it ensures a suitable selection of design rotatability. n k E (, ) B (-1, 1) A (1, 1) F (-, ) S (, ) H (, ) 3

33 Fig 1 Significant design points of F Example 1 ( (calculation of s e in the design without repetition) Calculate the dispersion of the effect estimate using central points for the following design. Factors and levels ,1,15, 3,49,7,1 Design matrix Solution: The average Y in central points is Test 1 3 Y Y C The dispersion Y in central points is , 5 s 66 75, , ,5 8 75, ,6 4 s 4 69,6 Dispersion effect estimate s e 34, 83 n 8 and standard deviation s e = 5,9. k 33

34 Concept summary Operations with factors and their properties, determining confounding patterns, determining the number of confounding patterns, generator selection for medium designs, design projection, supplementary fractions, design plan for eight factors. Questions a) What is the number of axial points? Coordinates of axial points? b) What is the numer of factorial points? Coordinates of factorial points? c) What is recommended number of central points? Coordinates of central points? 4. Second-Order Response Surface Model Time for study: 4 hours Objectives: a) Combined design b) Tri-level design c) Box Behnken design Lecture Up until now (in Chapter One) we have been dealing with the method of determining a linear regression model. Since a linear model is sometimes not enough, it is necessary to look for more of a complex second-order response surface model. The first question is when this model is necessary and then to find a method of determining it and evenutaully to use it in calculating an optimal setting process. Briefly, it is possible to say that in comparison with the procedure for linear models the second-order response surface model is completely different. In this chapter we show how to solve the above-mentioned problem. 4.1 Model determining conditions For determining the full second-order response surface model it is necessary to fullfil at least two conditions: 1) the number of experiments should not be less than the number of coefficients in the sought-for model; for linear models we then get the condition more simplified: n k ; n = number of experiments, k = number of linerar model factors. ) every variable in the design has to be at least 3 levels (for this it is possible to meet either by using a supplementary design with axial points = a combined design, or to directly to select 3 levels of factors = tri-level plan). 34

35 Before we begin to look for a second-order response surface model we have to verify if such a model is necessary: this is possible to determine using various methods, for example: a) by a curvature test, b) by a lack-of-fit test (Chap.6), c) using ANOVA (Chap.6). 4. Methods of determining a second-order response surface model The coefficients of full second-order response surface models are not one-half of half plan, as it was for linear and non-full second-order response surface models. That is why other methods are used to determine them. We present a further three: a) Composite design b) Three-level design c) Box-Behnken design 4..1 Composite design The composite design is made up of: factorial points + axial points + central points For central designs, when the center is point (; ), the axial design is easily created, only using two new levels. If is calculated using the relation 4 n k, where n k is the number of factorial points in a design, we speak about the rotatability of the design. For rotatable designs, dispersion Y is the same in points at the same distance from the centre. For example, dispersion Y in points ( 1; 1), ( ; ) a (; ) is the same. 4.. Calculating the optimal level of significant factors Let s take full second-order response surface model k i1 k i1 y ixi iixi ijxi x j (1) i j which is also possible to write using a matrix T T y x b x Bx, () o where B is the symmetrical matrix of the coefficients b ii and b ij /; b is the vector of coefficients b i. Determining stationary points We calculate the vector of stationary points from the equation 35

36 x 1 s 1 B b (3) Example 3 They should find the stationary points for the second-order response surface model with two variables x 1,x. The design matrix has the form and is completed by axial points and four central points. x 1 (min) =, x 1 (max) = 5,x (min) = 15, x (max) = 5. Design matrix Solution: There comes a model x 1 x Y , , , , From which Y = 79,75 + 9,83x 1 + 4,x 8,88x 1-5,13x -7,75x 1 x 8,88 3,875 9,83 B a b 3,875 5,13 4, x s 1 B 1 b 1,16798,1689,1689 9,83,5579,978 4,,11 The optimal coded values (stationary points) of variables x 1, x are a.11. For this optimal entry selection is Y = 8,47. In addition to the optimal level of parameters it is possible to also find an optimal area, in which the best level of the observed Y output and also minimal variability are attained. This area is discovered as the intersection of contour graphs for optimal Y and graph for minimal variability Box-Behnken design (BBD) The setup of a design plan is carried out for three factors in such a way: one factor is at the level (the first here) and for the remaining two factors a full design (in blue) is made Exp x y z 36

37 Another factor is at level (the second) and for the remaining two factors a full design is set up, etc. The last row is the vector of central points. For four factors : a full design is created gradually for all pairs of factors, and at the same time the others are at level. BBD design for five factors: This entry is a shortened form of the design: for example in the first row 1, 1,,, of the entry means that the first and the second variable appears in the full design, the rest. The recording 1, 1,,, then represents 4 rows of the design. The last row is the vector of the central points. k The number of tests for k = 3,4,5: n = 4.. For two factors with BBD it does not appear Designs for factors at three levels Full factorial designs for three factors at three levels: The design is created similarly as for two-level factors: a) we set the number of experiments: n = 3 k, 37

38 b) with creation we begin for example at factor x 3 : we alternate level -1,, 1; we continue at x : the level change is three times the previous one ; finally at x 3 -the change is three times the previous one, thus after nine of the same levels: Test x 1 x x 3 Y Example 5 Make a full design for two factors A, B, which have three levels. With the creation, we begin at factor A. Solution: We begin the formation from the first factor, at the second we triple the period of change: A B Y

39 Note: As opposed to a two-level design, the length of this design is not very distinctive. For three factors there is n = 3 3 = 7, whereas for two levels the number of experiments is n = 3 = 8. That is why if it is possible, we make the two-level design or composite design more precise. Concept summary Determining model conditions, methods of determining second-order response surface models, composite designs, calculation of the optimal level of important factors, Box-Behnken design (BBD), designs for factors at three levels Questions a) What is the basic problem in creating designs for second-order response surface models? b) How to solve the problem? c) What are the disadvantages of a tri-level design? 5. Blocking in Experimental Designs Time for study: hours Objectives: a) Forming blocks according to input b) Evaluating block effects Lecture While doing experiment, only what is stated in the design is changes. If it is not possible to keep this basic condition (for example, it is not possible to carry out an entire design matrix during one day or from one batch of raw materials), a distribution of experiments are carried out in what are called blocks. To create blocks practically means to divide experiments into groups, in which the conditions are the same for their working (for example, experiments carried out the first day make the first block, experiments from the second day then the second block). Blocks are created so that 39

40 a) during effect calculations there is eliminated the appropriate effect that during a design matrix it is not possible to ensure the same conditions for all experiments, b) in evaluating the design matrix results there was the possibility to state if the blocks have an effect on the result. Blocks are created using generator blocks, which are given in the tables. Thus, besides generators of secondary factors and generators of fractions, we also have generators of blocks. Tab.1 A selection of generator blocks from the chart: Number of variables Size of block Block generator k 3 4 B1=13 B1=1, B= B1=134 4 B1=14, B=134 B1=1,B=3,B3= B1= B1=13,B=345 4 B1=15,B=35,B3=345 B1=1,B=13,B3=34, B4= B1= B1=136,B= B1=135,B=156, B3=134 4 B1=16,B=136,B3=346,B4=456 B1=1,B=3,B3=34, B4=45,B5= Block construction Example 1 (three factors, four blocks) Divide the experiments into 4 blocks in design 3. Solution: It is necessary to divide 3 = 8 experiments into 4 blocks, which make 8/4 = experiments. In Table 1 for k = 3 and block size the block generators are B1=1 and B = 13. There exists 4 variants of signs, which make B1, B. We number the appropriate blocks: We then have these blocks: 1 3 B1 = 1 B = 13 block IV I III II II III I IV Block I II III IV (B1,B) (-,-) (+,-) (-,+) (+,+) 4

41 The final matrix distribution into the blocks will be Factors Block variables Test No. 1 3 B1=1 B=13 Block No IV I III II II III I IV 5. Evaluating block effects If the design matrix is divided into blocks, it is necessary to state if the differences in the design matrix results are significant in the individual blocks. For this purpose the S R part is set aside from the residual sum of squares, occuring on blocks and ANOVA is carried out. If dividing the design matrix into blocks affects the experiment results, introduce block variable x B, so that the model has the form Y = f(x) + d.x B + e For two blocks the coefficient block variable d is calculated in this way: the coded block value x B is indicated as -1 in the first block and as +1 in the second block. The coefficient of the block variable will be (n = the number of all experiments) d n j1 n x j1 Bj x Y Bj j Example 3 (evaluating block effect) The full design for three factors is supplemented by two central points and axial points with the selection. Experiments are divided into two blocks. The first 8 experiments make up the first block (-1), the second eight experiments the second block (+ 1) The design and its distribution into blocks X1 X X3 Y Block , , , , , , ,6-1 41

42 ,7-1 57,5 1 59, ,5 1 44, ,58 1 8, ,5 1 4, 1 The coefficient block variable should be calculated and ANOVA carried out to evaluate the block influence. The considered model is full second-order. Solution: Manual calculation The regression coefficient of the block variable is d 5, , 57,5... 4,,64 ( 1)... ( 1) 16 1,9 For two blocks with a sum of square deviations, occuring on the blocks, it determines SS(blocks) = n.d SS 16.1,9 6,656 Generally for m block it is m Bi SS( blocks) n i1 i G n 318, , ,4 16 6,656 m = number of blocks Bi = sum of experimental results (Y) in the i-th block ni = number of experiments in i blok G = sum of all experimental results (Y) n = total number of experiments Residual sum of squares "purified" by the block effect s R e SS( blocks) i In evaluating SS B (block) by the ANOVA test a degree of freedom of m-1 is used. Testing criterium SS( blocks)/( m 1) T SR / dfr ( m 1) m = number of blocks 4

43 Calculation with Statgraphics Basic ANOVA for the full second-order response surface model (calculated without the Block column) SS Df dispersion F p-value Model S M = 1451,6 k-1 = 9 161,9 3,9,5 Residual S R = 41,6 n-k = 16-1 = 6 6,93 Summary S T = 1493,17 15 k = number of parameters: 1 absolute component + 3 linear + 3 mixed + 3 quadratic = 1 n = number of points =16 The expanded ANOVA for every component of the model including blocks: variable SS xb 6,6 x1 36 x 7,59 x3 5,41 x11 17,88 x 86,55 x33 434,3 x1 46,69 x13 85,54 x3 59,6 ANOVA for evaluating block effect: SS df SS/df T Critic. value blocks 6,6 m-1 = 1 6,6/1 = 6,6 residual without 41,6-6,6 = 15 df R -(m-1) = /5 = 3 blocks 6,6/3 = 8,9 F 1,5 (,5) = 6,6 T exceeds F 1,5 (,5) which means that the blocks affects the experiment results. Calculation of coefficients (Statgraphics) parameter estimate C 41,14 Blok 1,9 X1 1,5 X -,13 X3 1,81 X11-4,69 X -6,7 X33-5,1 X1-7,13 X13-3,7 X3 -,73 Concept summary Block generators, evaluating of blocks effect, blocks formation significance 43

44 Questions a) Why are blocks created? b) How to create blocks? c) How to evaluate blocks effect? 6. Lack-of-fit Test Time for study: hours Objective Mastering the Lack-of-fit test procedure Lecture In basic dispersion analysis (ANOVA) in an regression analysis S T and S R are calculated S e ; S ( Y ˆ Y R Y ˆ i Yi ) ( i ˆ i ) Y It is also possible to calculate their sum according to the formula: S T ( Yi Y ) S Yˆ S R SY ˆ S T represents the entire sum of the square deviation of empirical values from the average. For the needs of testing the model S R is further analysed in the Lack-of-fit test. The test is founded on the decomposition S R = S P + S L significant difference between S L and S P. and the evaluation if there is a If the difference is significant, we consider the model as insufficient. The test can be carried out a) for repeated experiments, b) for central points and it is for one variable (x) and generally for k of variables (x 1,,x k ). It then requires to observe Y for the same selection of regressors (x 1,,x k ), that is the same rows in matrix X. Basic test points 44

45 1. Ho: Ho : SL SP; H1 : SL SP. The agreement of S P and S L evaluates by F test with the testing criterion F S df S df L L P P s s L P 3. Critical value for level of significance α: F m,n (α), where m = df L, n = df P. df L resp. df P are degrees of freedom for S L resp. S P. 4. H o is rejected in the extension of critical values. The calculation of the degrees of freedom df R and df L df P m i1 df R = n-p ( n 1) n m n i df R = df L + df P C 1 df L = n-p-(n-m) = m-p n i = number of repetitions of i-th experiments p = number of parameters in the regression function n c = number of central points m = number of various experiments n = number of points We can show the calculation of S L and S P in the following cases for two situations: a) repeated experiments, b) a design with central points. Example 1 (the lack-of-fit test for a design with repetition ). For (x,y) should be found a regression line and ANOVA should be done, then decompose S R on S P and S L and to carry out a lack-of-fit test. x Y 1 1,3,3 1,3 1,8 3,,8 4, 1,5 5,7, 6 3,3 3,8 7 3,3 1,8 8 3,7 3,7 9 3,7 1,7 1 4,,8 11 4,,8 45

46 1 4,, 13 4,7 3, 14 4,7 1,9 15 5, 1,8 16 5,3 3,5 17 5,3,8 18 5,3,1 19 5,7 3,4 6, 3, 1 6, 3, 6,3 3, 3 6,7 5,9 Solution: The regression line has the equation Yˆ 1,46, 316x. ANOVA, which is part of the regression analysis is in Tab.1. Tab.1 ANOVA SS df Dispersion Testing criterion Critical value SY ˆ 5,499 p-1 = - 1 SY ˆ /1=5,499 T = 5,499/,78 F 1,1 (,5) = 4,35 T = 7,56 S R 15,78 n-p = 3- S R /1=,78 S T,777 Calculation of S P : In the first column the values are x, in which the repeated experiment was carried out. Tab. Calculation of S P and df P Repeated tests in point x j Test results of Y ju Y j nj u1 ( n j -1 Y ju Y j ) 1,3.3, 1,8,5,15 1,.8, 1.5,15, ,3 3.8, 1.8,8, 1 3,7 3.7, 1.7,7, 1 4,.8,.8,.,6,4 4,7 3., 1.9,55, ,3 3.5,.8,.1,8,98 6, 3., 3 3,1, 1 sum S p = 7,55 df P = 1 Tab.3 ANOVA with S R breakdown SS df Dispersion Testing criterion F S ˆ 5, ,499 Y S R 15,78 df R = 3- = 1,78 S L S L = S R -S P = 8,3 df L = 1-1 = 11 8,3/11 =,748 F =,748/,76 = 1,6 S P 7,55 df P = 1 7,55/1 =,76 S T,777 df R = n-p = 3- = 1 m df P ( n j 1) 1 (tab.) j1 df L = df R - df P = 1-1 = 11 46

47 Critical value: F 11,1 (,5) =,94 Since F = 1,61 < F 11,1 (,5) =,94, item S L is not significant; that is why the model is convennient (it agrees well with the theoretic). If S L is significant, the found model would be insufficiently in agreement with the theoretic. It is then necessary to try to determine where and what caused this disagreement. The cause can be an autocorelation (A) and heteroscedasticity (H). A and H are determined using residue analysis. Example ( lack-of-fit test for a design with central points) To find S R for a full second-order response surface model and to carry out the lack-of-fit test. Solution Yˆ 79,7 9,7x 1 4,1 x 8,7x1 5,1 x 7, 8 x 1 x Y , , , , x x S R = 36,5; SY ˆ = 173,46; Y C 79, 75 S P = (76-79,75) + (79-79,75) + (83-79,75) + (81-79,75) = 6,75 ~6,8 S L = S R S P = 36,5 6,8 = 9,7 1 df rozptyl T S R 36,5 df R = n-p = 1-6 = 6 F 3,3 (,5) = 9,7 S L 9,7 df L = df R -df P = 3 9,7/3 = 3, 3,/8,9 =,36 S P 6,8 df P = n c -1 = 3 6,8/3 = 8,9 T < 9,7; Let s leave Ho; n = 1 the number of points; p = 6 the number of parameters of the regression function, n c = 4 the number of central points. Concept summary Lack-of-fit test : zero hypothesis, testing criterion, critical value, test result, repeated experiments, experiments with central points. Questions a) What is the importance of the lack-of-fit test? b) How to proceed with experiments with repetition? c) How to procede with tests with central points? 47

48 7. Steepest Ascent Time for study: hours Objective Managing the procedure of steepest acsent Lecture 7. Steepest Ascent (SA) In planning the design matrix in the way they have been described so far an design of a certain type was created, the effect of individual factors was calculated and a model established. This enables the determination of the optimal values of effecting factors. It however only deals with local optimum. For determining the absolute optimum of factors it is necessary to continue in constructing other designs, created in wider designing areas. The main stages in the steepest ascent method are: 1. Create a first design of a linear model, find the first equation of the linear model and determine from it the shift of the first design.. In a given direction determine points of "exploratory" experiments and determine the centre of the second design. Creating a second design for a linear model, determining the equation of the second linear model, determining the new direction of the design shift, carrying out exploratory experiments in the new direction, etc. The procedure is repeated until Y output is improved. 48

49 3. The last design is supplemented by axial points for determining the full second-order response surface model 4. The absolutely optimal setting of factors is calculated. We can see the above-mentioned procedure in Example No.1. Example 1 (data: [4], s. 514, corrected) The observed output Y depends on two factors: x 1 a x. The full design, supplemented by three central points, is in Table 1. Tab.1 First SA design method Operational values Coded values x 1 x ξ 1 ξ Y , ,3 8 17, , , , , , , 6, , 64, , 6,3 Find a linear model, the procedure direction and using the steepest acsent method find the new centre for another design. The objective is the maximalization of output Y. Solution: Coding and decoding equation for x 1 and for x x x x 13 x,5 13.,5 Calculation of the effect of factors: 1 ef ( x 1 ) ( 54,3 6,3 64,6 68) 4,7 1 ef ( x ) ( 54,3 6,3 64,6 68) 9, Average values of Y in the first design is (it is calculated from Y in the central points) 1 Y 1 (54,3 6,3 64,6 68 6,3 64,3 6,3) 6,1. 7 Dispersion estimate in the central points of the first design: 49

50 3 1 The linear model has the equation 6,3 6,3 6,3 6,3 64,3 6, s Y 4,7 9 6,1 x1 x 6,1,35x1 4, 5x The direction of design movement is determined by a vector whose coordinates are the coefficents of factors x 1 a x in the linear model: (,35; 4,5). We can modify the coordinates for simplifying further calculations:,35 4,5 ( 1; 1,91),35,35 Multiples of these coordinates are coded factor values. Exploratory experiments will be carried out at these points. Multiples of the basic directive vector, that is coordinates for exploratory experiments with the result Yi:,35 4,5 ( 1; 1,91) Y 1 = 73,3,35,35 ( ;.1,91 3,8) ( 3;3.1,91 5,73) Y 3 = 86,8 ( 4;4.1,91 7,64) ( 5;5.1,91 9,55) Y 5 = 58, Since the objective is to maximalize Y, the new centre point (with the coded values of coordinates) is ( 3;5,73). The new centre in real values then is: For x 1 : = 9, for x :,5.5, = 144,3; we round off to x = 145. Coding and decoding equations for x 1 x1 9 1 x and for x x 145 x In the second design there are two central points. Tab. Second SA method design Operational values Coded values x 1 x ξ 1 ξ Y , , , 5

51 , , ,8 The average value in central points is Y C 88, 5. The dispersion estimate in central points in the second design: 89,7 88,5 86,8 88,5 4, 1 1 s. 1 By connecting the estimates in the central points we make the estimate (3 1).4 ( 1).4,1 e 4,7. 3 Effect calculation in the second design: 1 ef ( x 1 ) ( 78,8 84,5 91, 77,4) 4,5 1 ef ( x ) ( 78,8 84,5 91, 77,4),65 Average value Y in the second design is Y 84, 73. The model in the second design has the equation Y = 84,73-,5x 1 + 1,35x. e more precise: Using the same method we can proceed in setting out farther in a new direction. The procedure is repeated so long as the new direction brings improvement to output Y, resp. if we do not reach the edge of the technological possibilities of setting factors. Let s consider the procedure in this case as finished. We now supplement the second design with axial points (and two central points for further improving the estimate e ) in such a way it will be possible to look for a second-order response surface model: Tab.3 Third SA method design Operational values Coded values Output x 1 x ξ 1 ξ Y , , , , , ,8 7 76* 145-1,41 83,3 8 14* 145 1,41 81, * -1,41 81, * 1,41 79, , , 51

52 *) Calculation of real values: For x 1 1 1,41: 1. x1 9 1( 1,41) 9 75,9 1,41: 1.1, ,1 1 For x 1,41: 5. x 145 5( 1,41) ,95 1,41: 5.1, ,5 Average value Y in central points in the third design Y C 1 ( ) 86,5 Dispersion estimate in central points e from the third design: s , ,5, By connecting estimates s, s we make more precise estimate 1 s, 3 e :.4 1.4,11.,5 e 3,18 4 The full second-order response surface model has the equation Y 87,36 1,38x x x. Coefficient determination r =, ,36x,14x1 3,1x 4, 87 We calculate the optimal input values x 1 and x. From the coefficent of the second-order reponse surface model we create matrix b, B and calculate the vector of the optimal input x s: 1,38,14,44 1 3,78 b, B,36, x 1 s B b,44 3,1 3,4 In real values it is x 1 = 1(-3,78) + 9 = 5, and x = 5.3, = 16, If we do not carry out steepest ascent, optimal x 1 and x come out from the first design: 1 x 1 x Y , , ,6 5

53 + + 68, Y- 59,45 57,3 Y+ 64,15 66,3 When maximalizing Y then the optimal selection is for the upper limit x 1 a x ; in real values it is x 1 = 8 and x = 13,5. By attaining the coded values in the determined regression function it comes out a) for optimal selection x 1 = -3,78 a x = 3,4: Y = 9,41 b) for the selection stated from the graph of levels where x 1 = x = +1: Y = 76,3. Concept summary Procedure direction of the design shift, exploratory experiments, new design centre Questions a) Why is the steepest acsent carried out? b) How to determine the direction of the design shift? c) How long does the design shift? Literature used [1] Myers,R.H., Montgomery,D.C. Response Surface Methodology, nd Ed., Wiley,NY. [] Software: Design Expert. [3] Raszka, J.: Plánování experimentů a jeho aplikace v Třineckých železárnách. Design projection (M s. Diplomová práce, VŠB-TU Ostrava 1. [4] Box,G.E.P.-Hunter,W.-Hunter,S.:Statistics for experimenters. Wiley [5] Box,GE.P..- Draper,N.R.: Empirical Model Building and Response Surface. John Wiley, NY Anglická terminologie některých výrazů (v textu červených): V české verzi nejsou vždy anglické překlady originálních termínů, ale upravená terminologie Kódované proměnné = coded variables Pokus = experiment Experiment = design matrix kvalitativní proměnné = qualitative variables Efektem faktoru = effect of factor Úsečkový graf úrovní = graph of levels normální pravděpodobnostní graf = normal probability plot normálním rozdělením = normal distribution definiční rovnice = defining relation řešení plánu = resolution (of the design) is the lengh of the shortest word in the defining relation 53

54 délka slova = lenght of the word generátor = generator zaměnitelné dvojice = confounding patterns Dynamické plánování experimentů = Steepest Ascent Projekce plánu = Design projection (MyNo s.116) 54