Presentation on theme: "Design and Analysis of Experiments"— Presentation transcript:
1Design and Analysis of Experiments Dr. Tai-Yue WangDepartment of Industrial and Information ManagementNational Cheng Kung UniversityTainan, TAIWAN, ROCThis is a basic course blah, blah, blah…
2Experiments with Blocking Factors Dr. Tai-Yue WangDepartment of Industrial and Information ManagementNational Cheng Kung UniversityTainan, TAIWAN, ROCThis is a basic course blah, blah, blah…
3Outline The Randomized Complete Block Design The Latin Square Design The Graeco-Latin Square DesignBalanced Incomplete Block Design
4The Randomized Complete Block Design In some experiment, the variability may arise from factors that we are not interested in.A nuisance factor (擾亂因子)is a factor that probably has some effect on the response, but it’s of no interest to the experimenter … however, the variability it transmits to the response needs to be minimizedThese nuisance factor could be unknown and uncontrolled use randomization
5The Randomized Complete Block Design If the nuisance factor are known but uncontrollable use the analysis of covariance.If the nuisance factor are known but controllable use the blocking techniqueTypical nuisance factors include batches of raw material, operators, pieces of test equipment, time (shifts, days, etc.), different experimental units
6The Randomized Complete Block Design Many industrial experiments involve blocking (or should)Failure to block is a common flaw in designing an experiment (consequences?)
7The Randomized Complete Block Design-example We wish determine whether or not four different tips produce different readings on a hardness testing machine.One factor to be consider tip typeCompletely Randomized Design could be used with one potential problem the testing block could be differentThe experiment error could include both the random and coupon errors.
8The Randomized Complete Block Design-example To reduce the error from testing coupon, randomize complete block design(RCBD) is used
9The Randomized Complete Block Design-example Each coupon is called a “block”; that is, it’s a more homogenous experimental unit on which to test the tips“complete” indicates each testing coupon (BLOCK) contains all treatmentsVariability between blocks can be large, variability within a block should be relatively smallIn general, a block is a specific level of the nuisance factor
10The Randomized Complete Block Design-example A complete replicate of the basic experiment is conducted in each blockA block represents a restriction on randomizationAll runs within a block are randomizedOnce again, we are interested in testing the equality of treatment means, but now we have to remove the variability associated with the nuisance factor (the blocks)
11The Randomized Complete Block Design– Extension from ANOVA Suppose that there are a treatments (factor levels) and b blocks
12The Randomized Complete Block Design– Extension from ANOVA Suppose that there are a treatments (factor levels) and b blocks
13The Randomized Complete Block Design– Extension from ANOVA A statistical model (effects model) for the RCBD isThe relevant (fixed effects) hypotheses are
14The Randomized Complete Block Design– Extension from ANOVA Or
15The Randomized Complete Block Design– Extension from ANOVA Partitioning the total variability
16The Randomized Complete Block Design– Extension from ANOVA The degrees of freedom for the sums of squares inare as follows:Therefore, ratios of sums of squares to their degrees of freedom result in mean squares and the ratio of the mean square for treatments to the error mean square is an F statistic that can be used to test the hypothesis of equal treatment means
17The Randomized Complete Block Design– Extension from ANOVA Mean squares
18The Randomized Complete Block Design– Extension from ANOVA F-test with (a-1), (a-1)(b-1) degree of freedomReject the null hypothesis ifF0>F α,a-1,(a-1)(b-1)
19The Randomized Complete Block Design– Extension from ANOVA ANOVA Table
20The Randomized Complete Block Design– Extension from ANOVA Manual computing:
21The Randomized Complete Block Design– Extension from ANOVA Meaning of F0=MSBlocks/MSE?The randomization in RBCD is applied only to treatment within blocksThe Block represents a restriction on randomizationTwo kinds of controversial theories
22The Randomized Complete Block Design– Extension from ANOVA Meaning of F0=MSBlocks/MSE?General practice, the block factor has a large effect and the noise reduction obtained by blocking was probably helpful in improving the precision of the comparison of treatment means if the ration is large
24The Randomized Complete Block Design– Example To conduct this experiment as a RCBD, assign all 4 pressures to each of the 6 batches of resinEach batch of resin is called a “block”; that is, it’s a more homogenous experimental unit on which to test the extrusion pressures
25The Randomized Complete Block Design– Example—Minitab Vascular-Graft.MTWThe Randomized Complete Block Design– Example—MinitabStatANOVATwo-wayTwo-way ANOVA: Yield versus Pressure, BatchSource DF SS MS F PPressureBatchErrorTotalS = R-Sq = 77.12% R-Sq(adj) = 64.92%
27The Randomized Complete Block Design– Example —Residual Analysis Basic residual plots indicate that normality, constant variance assumptions are satisfiedNo obvious problems with randomizationNo patterns in the residuals vs. blockCan also plot residuals versus the pressure (residuals by factor)These plots provide more information about the constant variance assumption, possible outliers
31The Randomized Complete Block Design– Other Example
32The Randomized Complete Block Design– Other Example
33The Randomized Complete Block Design– Other Example Blocking effectWithout blocking effectTwo-way ANOVA: 濃度 versus 化學品類別, 樣品Source DF SS MS F P化學品類別樣品ErrorTotalS = R-Sq = 96.30% R-Sq(adj) = 94.14%One-way ANOVA: 濃度 versus 化學品類別Source DF SS MS F P化學品類別ErrorTotalS = R-Sq = 70.24% R-Sq(adj) = 64.66%
34The Randomized Complete Block Design– Other Example Blocking effectWithout blocking effect
35The Randomized Complete Block Design– Other Aspects The RCBD utilizes an additive model – no interaction between treatments and blocksTreatments and/or blocks as random effectsMissing valuesWhat are the consequences of not blocking if we should have?
36The Randomized Complete Block Design– Other Aspects Sample sizing in the RCBD? The OC curve approach can be used to determine the number of blocks to run..see page 133
37The Latin Square Design These designs are used to simultaneously control (or eliminate) two sources of nuisance variabilityThose two sources of nuisance factors have exactly same levels of factor to be consideredA significant assumption is that the three factors (treatments, nuisance factors) do not interactIf this assumption is violated, the Latin square design will not produce valid resultsLatin squares are not used as much as the RCBD in industrial experimentation
38The Latin Square Design The Latin square design systematically allows blocking in two directionsIn general, a Latin square for p factors is a square containing p rows and p columns.Each cell contain one and only one of p letters that represent the treatments.
39A Latin Square Design – The Rocket Propellant This is a
40Statistical Analysis of the Latin Square Design The statistical (effects) model isThe statistical analysis (ANOVA) is much like the analysis for the RCBD.
43The Standard Latin Square Design A square with first row and column in alphabetical order.
44Other Topics Missing values in blocked designs RCBD Latin square Estimated by
45Other Topics Replication of Latin Squares To increase the error degrees of freedomThree methods1. Use the same batches and operators in each replicate2. Use the same batches but different operators in each replicate3. Use different batches and different operator
46Other TopicsReplication of Latin SquaresANOVA in Case 1
47Other TopicsReplication of Latin SquaresANOVA n Case 2
48Other TopicsReplication of Latin SquaresANOVA n Case 3
49Other Topics Crossover design p treatments to be tested in p time periods using np experiment units.Ex : 20 subjects to be assigned to two periodsFirst half of the subjects are assigned to period 1 (in random) and the other half are assigned to period 2.Take turn after experiments are done.
51Graeco-Latin SquareFor a pxp Latin square, one can superimpose a second pxp Latin square that treatments are denoted by Greek letters.
52Graeco-Latin SquareIf the two squares have the property that each Greek letter appears once and only once with each Latin letter, the two Latin squares are to be orthogonal and this design is named as Graeco-Latin Square.It can control three sources of extraneous variability.