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1、试验设计简介,PioneerRonald. A. Fisher,Introduce three basic principles, analysis of variance,Frank Yates,theory ofanalysis of varianceYatess algorithmbalanced incompleteblock design,David John Finney,became assistant toDr Frank Yates atRothamsted Experimental Stationin 1939,R.C. Bose,Combinatorial Designs
2、,George E.P. Box,Industrial Era: process modelling, optimization, response surfacemethodology.,Essentially, all models are wrong, but some are useful. - G.E.P. Box,G. Taguchi,Quality improvement, Robust parameter designs,Introduction to Experimental Design Statistical Models Review of Regression Ana
3、lysis,Some examples,How to know the weights by balance?,Method 1：Take the mean of the 4 measures for each object.,Method 2：Combinatorial method,16 times,4 times,Sheldon (1960),Goal: Is there any difference between the operators?,One-way Layout,化工试验,在某化工产品的合成工艺中，考虑反应温度(A)、压力(B) 和催化剂用量(C)，并选择了试验范围分别为：
4、,温度(A): 80oC120oC；压力(B): 46 大气压；催化剂用量(C): 0.5%1.5%；,我们需要选择这三个因素的最佳组合，以达到高产的目的。,许多产品都是混合多种成分在一起形成的。,面粉,水,糖,蔬菜汁,椰子汁,盐,发酵粉,乳酸,钙,咖啡粉,香料,色素,面包,怎样确定各种成分的比例呢？,经验,试验,混料试验,加工面包试验,环保试验,在水及食物中的某些化学元素，吃多了对人体是有害的，为了研究这些元素对人体健康的影响。,24,137,569,The Usefulness of Experimental Design,Experiments are performed by inve
5、stigators in virtually all fields of inquiry, usually to discover something about a particular process or system.,A well designed experiment is an efficient method of learning about the world (Atkinson and Donev, 1992). 一个精心设计的试验是认识世界的有效方法.,Make it your motto day and night.,Experiment!,And it will l
6、ead you to the light.,Experiment!,Two Problems:Design: How to choose a set of experimental points on the domain Modeling: How to find an approximate model to fit the experimental data.,Design Objectives,Treatment ComparisonScreeningModel BuildingParameter EstimationOptimizationPredictionConfirmation
7、etc.,Design Methodology,Treatment ComparisonFull & Fractional Factorial DesignCombinatorics DesignCoding TheoryResponse Surface MethodologyANOVA Type DesignOptimal DesignBayesian (Optimal) Design,Design Methodology (Continued),Saturated (Minimal Point) DesignTaguchi Product (Robust) DesignMixture Ex
8、perimentComputer ExperimentSupersaturated DesignUniform DesignMicroArray Design,The design of experiments is an important part of scientific research. A good experimental design should minimize the number of experiments to acquire as much information as possible.,Some Concepts in Experimental Design
9、,Factor：controllable variable that is of interest in the experiment. Quantitative factor：whose values can be measured on a numerical scale and that fall in an interval, e.g., temperature, pressure, ratio of two raw materials, etc.Qualitative factor：(categorical factor or indicator factor) whose valu
10、es are categories such as different equipment of the same type, different operators,etc.,Experimental domain： the space where the factors take values.Level：a few selected values of a factor in the experimental domain. Level-combination：(treatment combination) one of the possible combinations of leve
11、ls of the factors.Run：(trial) the implementation of a level-combination in the experimental environment.,Response：the result of a run based on the purpose of the experiment. The response can be numerical value or qualitative or categorical, and can be a function that is called functional response.Ra
12、ndom error：exists in any industrial or laboratory experiments. It can often be assumed to be distributed as a normal distribution N(0, ) in most experiments.,More Concepts,BlockRandomization RepeatThe type of designsThe organization and management of experiment,The type of design,(1) Operating envir
13、onment physical experiment：an experiment is implemented in a laboratory, a factory, or agricultural filed. exist random error so that we might obtain different outputs under the identical experimental setting. the underlying model is known or unknown. computer experiment：model the physical processes
14、 by computer code. no random error the underlying model is known.,Different considerations have different methods to classify,(2) Constraint condition,No constraint experiment：the choice of the level of every factor is independent. For example, an experiment has s factors and the interval of ith fac
15、tor is ai,bi, i=1,s, then the experimental domain is a1,b1 as,bs.Experimental design with mixture：the choice of the level of every factor is dependent with other factors. For example, the ratio of every factor is xi (i=1,s), which has the following constrain:,Single factor experimentMultiple factors
16、 experiment,(3) Number of factors,(4) Number of responses,Single response Multiple responseFunctional response,(5) Number of turns(轮),(6) Number of blocks,One turn, Sequential experiment,One block, block experiment,Example: free-fall motion,If physical law for the free-fall motion is unknown, i.e.,
17、the relationship between time (t) and free-fall distance (y) is unknown. We want to estimate its relationship by some experiments.,We choose different free-fall distances y1, , yn , and record their falling time t1, tn . Assume the model is,where the random error follows,Statistical Models,From this
18、 example, we know that the advantages of statistical model are as follows:,Base on the statistical model, we can estimate the response where the time has not been measured yet.Using the data, we can guess/discover a physical law. We can estimate the magnitudeof random error without repeated experime
19、nt.,The type of statistical Models,There are many kinds of experimental designs. Each design is based on its own statistical model. The following are some of statistical models.,ANOVA ModelRegression ModelNonparametric Regression ModelRobust Regression Design,Example. Growth Curve Model,Rapid growth
20、,Assume the experimenter does not know the relationship between x and y before the experiment.,To estimate the response function, we could perform an experiment by taking several sample responses(ys) to approximate the real model.,A. ANOVA Model,Levels: x1, x2, x3, x4,Response (yij): jth response of
21、 at ith level,Error,Testing:,Model:,or,B. Regression Model,Optimal design: A design; based on the underlying model, which satisfy one of the following criterion,We estimate the model by,and others.,C. Non-parametric Regression Model,g(x): unknown function,By the experiment one wishes to find an appr
22、oximation,In this case, the uniform design is useful.,: random error,D. Robust Design,A robust design is used when we know partially of a model,We can use some robust designs or uniform designs.,where f(x) is some known function and h(x) denotes the departure of the model , from the true model,Desig
23、n: the way that the experiment is performedAnalysis: including model fitting, assessment of the model assumption, drawing the conclusionRelationship between design and analysisThe choice of the design is often linked to a particular modelImpact of the design on the analysis criterion,Regression anal
24、ysis (review),Regression Model:yi, xi1, xi2, , xi,p-1, i = 1, , ny = b0 + b1x1 + b2x2 + + b p-1 x p-1 + eE(e) = 0,Var(e) = s2 unknownor yi = b0 + b1xi1 + b2xi2 + + b p-1 xi, p-1 + eiei,en are iid. E(ei) = 0, Var(ei) = s2.Moreover, yi= b1 g1(xi) + b2 g2(xi) + + b p gp(xi) + eiei,en are iid. E(ei) = 0
25、, Var(ei) = s2.xi=( xi1, xi2, , xi,p-1), i=1, , n,42,General Regression Model in terms of Matrix:y = Gb + eE(e) = 0,Cov(e) = s2Inwhere y : n1, G : np, b : p1, e : n1 with iid components.,(1),The linear model (1) includes many useful model:,Linear modelLinear model through origin Quadratic modelCente
26、red quadratic model,y = b0 + b1x1 + b2x2 + + b p-1 x p-1 + e,y = b1x1 + b2x2 + + b p-1 x p-1 + e,(2),For the linear model (1),The least square estimation of model (1) is Properties:,where M=GG is denoted as the information matrix,or some time call M=GG/n as the information matrix.,(a) Estimation,For
27、 the linear model (2), the following hypotheses are often needed in the practice: (k=p-1),A. Testing whether the model makes the sense. H0: b1 = = bk = 0 VS H1: Some bj 0, 1 j k,B. Testing whether the variable xj gives a significant contribution to the model. H0: bj = 0 VS H1: bj 0,(b) Hypothesis Te
28、sting,C. General Hypothesis H0: Cb = f VS H1: Cb f,Case A and B are just the special cases of Case C,y = Xb + e, e Nn (0,s2 In)Ab = c, A: q p (q p), c: q 1. Rank(A) = q,under the constraints Ab = c;estimates without constraints.,Theorem 1.1 Under the constraint model, we have(i)(ii)where,Theory of h
29、ypothesis testing:,Theorem 1.2 Denote are given in the theorem 1.1. Under the above notation for case C, we have:(i)(ii)(iii) When H0 is true, the statistics,The proof can refer to Seber (1977).,Theory of hypothesis testing (cont.),48,Case A H0: b1 = = bk = 0 VS H1: Some bj 0, 1 j k,Taking A = (0, I
30、k), c = 0, we have:,From theorem 3.3, the testing statistic becomes:,Case A (cont.) H0: b1 = = bk = 0 VS H1: Some bj 0, 1 j k This test is usually expressed as an ANOVA table:,Case A (cont.) ANOVA Table: ( k = p - 1),Case B H0: bj = 0 VS H1: bj 0Take A = (0,0,1,0,0) ej+1, where 1 is in the j +1 th position, c = 0. Then,Denote(GG)-1 = C = (cij)A (GG) -1A -1 = cj+1,j+1-1,