IE 361 Module 20Design and Analysis of Experiments: Part 1Prof.s Stephen B. Vardeman and Max D. MorrisReading: Section 6.1, Statistical Quality Assurance Methods for Engineers1After one brings a process to physical stability and qu anti fies what it is capableof doing, it’s reasonable to consider fundamental changes to its configura-tion/how it is run. Intelligent/efficient d ata collection and analysis aimed atfinding fundamental process improvements is the subject of the final set ofmodules of this course. The topic is the "design and analysis of experiments,"with the goal of eventually addressing complex situations where there are many"process knobs" (factors), each with multiple settings (levels) and thus manymany potential ways that things could be done, and the object is to find goodcombinations of levels of important factors.Figure 1 illustrates the problem addressed in these modules. The noisy processoutput is affected by variables x1,x2,x3and potentially other v ariables (bothrecognized and unrecognized). The question is how to set up the "controlpanel" (the settings of the "knobs" or values of some variables x1,x2,x3)to collect data and efficiently learn how to optimize the process to get desiredvalues of y.2Figure 1: A Process With Many Inputs x or Factors Affecting a Response y3Samples from r Different Sets of Process Condi-tions and A One-Way Normal Model for Experi-mental ResultsWe begin with a most basic experimental scenario, where one has data consist-ing of observed responses, y, for some number, r,different processes conditions.We’ll writeyij= the jth response in sample i (madeunder the ith set of process conditions)wheresamplesizesaren1,n2,...,nr.Example 20-1 A classic data set from Devore’s Probability and Statistics forEngineering and the Sciences concerns the current required to achieve a target4brightness on a type of television tube. All combinations of 2 types of glassand 3 types of phosphor created r =6types of tubes. Tests on 3 tubes ofeach type produced the measured current requirements yij(in μA) recordedin the following table with corresponding summary statistics.5Typ e 1 Tub es(Glass 1, Phosphor 1)Typ e 2 Tub es(Glass 1, Phosphor 2)Typ e 3 Tub es(Glass 1, Phosphor 3)y11= 280y12= 290y13= 285y21=300y22=310y23=295y31=270y32=285y33=290¯y1=285s21=25¯y2= 301.67s22=58.33¯y3=281.67s23= 108.33Typ e 4 Tub es(Glass 2, Phosphor 1)Typ e 5 Tub es(Glass 2, Phosphor 2)Typ e 6 Tub es(Glass 2, Phosphor 3)y41= 230y42= 235y43= 240y51=260y52=240y53=235y61=220y62=225y63=230¯y4=235s24=25¯y5=245s25=175¯y6= 225s26=25It is often useful to model observations from r samples of respective sizesn1,n2,...,nras independent random samples from normal distributions with6possibly different means μ1,μ2,...,μrbut a common standard deviations σ.Figure 2 illustrates these distributional assumptions.Figure 2: Distributions of Responses Under r Different Sets of Process Condi-tions7This basic "one-way normal model" is sometimes expressed in symbolic formasyij= μi+ ijwhere the ijare independent normal random variables with mean 0 and s tan-dard deviation σ.Section 6.1 of SQAME discusses ways (based on examination of residuals muchas in the regression analysis of Stat 231) for investigating the reasonablenessof the one way normal model in a particular application. In this module andthe ones that follow, we will take for granted that such work has been tak encare of, and consider what then can be done in the way of statistical inferenceand planning.8Estimation of σ and Linear Combinations of MeansWhere the one-way normal model is appropriate, it makes sense to pool togetherthe r standard deviations s1,s2,...,srto make a single pooled estimate ofthe common group standard deviation, σ. The way we will do this it to uses2pooled=(n1− 1) s21+(n2− 1) s22+ ···+(nr− 1) s2r(n1− 1) + (n2− 1) + ···+(nr− 1)=(n1− 1) s21+(n2− 1) s22+ ···+(nr− 1) s2rn − ras an estimate of σ2,wheren = n1+ n2+ ···+ nris the total number ofobservations in the study. This estimate of σ2is a weighted average of the r9sample variances. Corresponding to it is the estimatespooled=rs2pooledof σ, the standard deviation of responses for any fixed one of the conditions1, 2,...,r. Thispooledsamplestandarddeviationcanbeusedtomakeconfidence limits for σ.Thesearespooledvuutn − rχ2upperand spooledvuutn − rχ2low erwhere the appropriate degrees of freedom are ν = n − r.Example 20-1 continued The r =6sample standard deviations for thedifferent tube types in the glass-phosphor study are pooled to makespooled=s2(25)+2(58.33) + 2 (108.33) + 2 (25) + 2 (175) + 2 (25)18 − 6=8.3 μA10This intends to measure the variation in current required to produce the stan-dard brightness in tubes of any single type. Appropriate degrees of freedomforthisestimateareν =18− 6=12and 95% confidence limits are8.3s1223.337and 8.3s124.404that is6.0 μA and 13.8 μAσ (and its estimate, spooled) is a measure of basic background noise in an ex-periment, a baseline against which any apparent differences in average responsefor different conditions are to be measured. One specific way in which thesecomparisons can be made, is to make confidence limits for interesting linearcombinations of means μ1,μ2,...,μr.Thatis,we’llletL = c1μ1+ c2μ2+ ···+ crμr11stand for an arbitrary linear combination of group mean responses. The cor-responding linear combination of sample means isˆL = c1¯y1+ c2¯y2+ ···+ cr¯yrandisanobviousestimateofL. Useful special cases of this formalism are:ci=1and all others 0 L = μiˆL =¯yici=1,ci0= −1 and all others 0 L = μi− μi0ˆL =¯yi− ¯yi0The first of these is the mean for condition i and the second is the differencebetween the condition i and condition i0response means.Confidence limits for L can be based onˆL and spooledasˆL ± tspooledvuutc21n1+c22n2+ ···+c2rnr12The degrees of freedom for t are those associated with spooled,namelyν =n−r. The quantity spooledrc21n1+c22n2+ ···+c2rnris an estimate of the standarddeviation ofˆL and t times this is a kind of "margin of error" for estimating L.Example 20-1 continued In the glass-phosphor study, 95% confidence limitsfor the mean current requirement for tube type i are¯yi± tspooledvuut(1)2nithat is¯yi± 2.179 (8.3)s13or ¯yi± 10.44 μAThat is, each of the 6 sample means is in some sense "good to within 10.44 μA"as representing the corresponding tube mean current requirement.13As