Design and Analysis of Single-Factor Experiments: The Analysis of Variance

Transcription

1 13 CHAPTER OUTLINE Design nd Anlysis of Single-Fctor Experiments: The Anlysis of Vrince 13-1 DESIGNING ENGINEERING EXPERIMENTS 13-2 THE COMPLETELY RANDOMIZED SINGLE-FACTOR EXPERIMENT An Exmple The Anlysis of Vrince Multiple Comprisons Following the ANOVA More About Multiple Comprisons (CD Only) Residul Anlysis nd Model Checking Determining Smple Size Technicl Detils About the Anlysis of Vrince (CD Only) 13-3 THE RANDOM EFFECTS MODEL Fixed Versus Rndom Fctors ANOVA nd Vrince Components Determining Smple Size in the Rndom Model (CD Only) 13-4 RANDOMIZED COMPLETE BLOCK DESIGN Design nd Sttisticl Anlysis Multiple Comprisons Residul Anlysis nd Model Checking Rndomized Complete Block Design with Rndom Fctors (CD Only) LEARNING OBJECTIVES After creful study of this chpter, you should be ble to do the following: 1. Design nd conduct engineering experiments involving single fctor with n rbitrry number of levels 2. Understnd how the nlysis of vrince is used to nlyze the dt from these experiments 3. Assess model dequcy with residul plots 4. Use multiple comprison procedures to identify specific differences between mens 5. Mke decisions bout smple size in single-fctor experiments 6. Understnd the difference between fixed nd rndom fctors 7. Estimte vrince components in n experiment involving rndom fctors 468

2 13-1 DESIGNIING ENGINEERING EXPERIMENTS Understnd the blocking principle nd how it is used to isolte the effect of nuisnce fctors 9. Design nd conduct experiments involving the rndomized complete block design CD MATERIAL 10. Use operting chrcteristic curves to mke smple size decisions in single-fctor rndom effects experiment 11. Use Tukey s test, orthogonl contrsts nd grphicl methods to identify specific differences between mens. Answers for most odd numbered exercises re t the end of the book. Answers to exercises whose numbers re surrounded by box cn be ccessed in the e-text by clicking on the box. Complete worked solutions to certin exercises re lso vilble in the e-text. These re indicted in the Answers to Selected Exercises section by box round the exercise number. Exercises re lso vilble for some of the text sections tht pper on CD only. These exercises my be found within the e-text immeditely following the section they ccompny DESIGNING ENGINEERING EXPERIMENTS Experiments re nturl prt of the engineering nd scientific decision-mking process. Suppose, for exmple, tht civil engineer is investigting the effects of different curing methods on the men compressive strength of concrete. The experiment would consist of mking up severl test specimens of concrete using ech of the proposed curing methods nd then testing the compressive strength of ech specimen. The dt from this experiment could be used to determine which curing method should be used to provide mximum men compressive strength. If there re only two curing methods of interest, this experiment could be designed nd nlyzed using the sttisticl hypothesis methods for two smples introduced in Chpter 10. Tht is, the experimenter hs single fctor of interest curing methods nd there re only two levels of the fctor. If the experimenter is interested in determining which curing method produces the mximum compressive strength, the number of specimens to test cn be determined from the operting chrcteristic curves in Appendix Chrt VI, nd the t-test cn be used to decide if the two mens differ. Mny single-fctor experiments require tht more thn two levels of the fctor be considered. For exmple, the civil engineer my wnt to investigte five different curing methods. In this chpter we show how the nlysis of vrince (frequently bbrevited ANOVA) cn be used for compring mens when there re more thn two levels of single fctor. We will lso discuss rndomiztion of the experimentl runs nd the importnt role this concept plys in the overll experimenttion strtegy. In the next chpter, we will show how to design nd nlyze experiments with severl fctors. Sttisticlly bsed experimentl design techniques re prticulrly useful in the engineering world for improving the performnce of mnufcturing process. They lso hve extensive ppliction in the development of new processes. Most processes cn be described in terms of severl controllble vribles, such s temperture, pressure, nd feed rte. By using designed experiments, engineers cn determine which subset of the process vribles hs the gretest influence on process performnce. The results of such n experiment cn led to 1. Improved process yield 2. Reduced vribility in the process nd closer conformnce to nominl or trget requirements 3. Reduced design nd development time 4. Reduced cost of opertion

3 470 CHAPTER 13 DESIGN AND ANALYSIS OF SINGLE-FACTOR EXPERIMENTS: THE ANALYSIS OF VARIANCE Experimentl design methods re lso useful in engineering design ctivities, where new products re developed nd existing ones re improved. Some typicl pplictions of sttisticlly designed experiments in engineering design include 1. Evlution nd comprison of bsic design configurtions 2. Evlution of different mterils 3. Selection of design prmeters so tht the product will work well under wide vriety of field conditions (or so tht the design will be robust) 4. Determintion of key product design prmeters tht ffect product performnce The use of experimentl design in the engineering design process cn result in products tht re esier to mnufcture, products tht hve better field performnce nd relibility thn their competitors, nd products tht cn be designed, developed, nd produced in less time. Designed experiments re usully employed sequentilly. Tht is, the first experiment with complex system (perhps mnufcturing process) tht hs mny controllble vribles is often screening experiment designed to determine which vribles re most importnt. Subsequent experiments re used to refine this informtion nd determine which djustments to these criticl vribles re required to improve the process. Finlly, the objective of the experimenter is optimiztion, tht is, to determine which levels of the criticl vribles result in the best process performnce. Every experiment involves sequence of ctivities: 1. Conjecture the originl hypothesis tht motivtes the experiment. 2. Experiment the test performed to investigte the conjecture. 3. Anlysis the sttisticl nlysis of the dt from the experiment. 4. Conclusion wht hs been lerned bout the originl conjecture from the experiment. Often the experiment will led to revised conjecture, nd new experiment, nd so forth. The sttisticl methods introduced in this chpter nd Chpter 14 re essentil to good experimenttion. All experiments re designed experiments; unfortuntely, some of them re poorly designed, nd s result, vluble resources re used ineffectively. Sttisticlly designed experiments permit efficiency nd economy in the experimentl process, nd the use of sttisticl methods in exmining the dt results in scientific objectivity when drwing conclusions THE COMPLETELY RANDOMIZED SINGLE-FACTOR EXPERIMENT An Exmple A mnufcturer of pper used for mking grocery bgs is interested in improving the tensile strength of the product. Product engineering thinks tht tensile strength is function of the hrdwood concentrtion in the pulp nd tht the rnge of hrdwood concentrtions of prcticl interest is between 5 nd 20%. A tem of engineers responsible for the study decides to investigte four levels of hrdwood concentrtion: 5%, 10%, 15%, nd 20%. They decide to mke up six test specimens t ech concentrtion level, using pilot plnt. All 24 specimens re tested on lbortory tensile tester, in rndom order. The dt from this experiment re shown in Tble 13-1.

4 13-2 THE COMPLETELY RANDOMIZED SINGLE-FACTOR EXPERIMENT 471 Tble 13-1 Tensile Strength of Pper (psi) Hrdwood Observtions Concentrtion (%) Totls Averges This is n exmple of completely rndomized single-fctor experiment with four levels of the fctor. The levels of the fctor re sometimes clled tretments, nd ech tretment hs six observtions or replictes. The role of rndomiztion in this experiment is extremely importnt. By rndomizing the order of the 24 runs, the effect of ny nuisnce vrible tht my influence the observed tensile strength is pproximtely blnced out. For exmple, suppose tht there is wrm-up effect on the tensile testing mchine; tht is, the longer the mchine is on, the greter the observed tensile strength. If ll 24 runs re mde in order of incresing hrdwood concentrtion (tht is, ll six 5% concentrtion specimens re tested first, followed by ll six 10% concentrtion specimens, etc.), ny observed differences in tensile strength could lso be due to the wrm-up effect. It is importnt to grphiclly nlyze the dt from designed experiment. Figure 13-1() presents box plots of tensile strength t the four hrdwood concentrtion levels. This figure indictes tht chnging the hrdwood concentrtion hs n effect on tensile strength; specificlly, higher hrdwood concentrtions produce higher observed tensile strength. Furthermore, the distribution of tensile strength t prticulr hrdwood level is resonbly symmetric, nd the vribility in tensile strength does not chnge drmticlly s the hrdwood concentrtion chnges Tensile strength (psi) σ2 σ2 σ2 σ Hrdwood concentrtion (%) () µ + τ 1 µ + τ 2 µ µ + τ 3 µ + τ 4 µ 1 µ 2 µ 3 µ 4 Figure 13-1 () Box plots of hrdwood concentrtion dt. (b) Disply of the model in Eqution 13-1 for the completely rndomized single-fctor experiment. (b)

5 472 CHAPTER 13 DESIGN AND ANALYSIS OF SINGLE-FACTOR EXPERIMENTS: THE ANALYSIS OF VARIANCE Grphicl interprettion of the dt is lwys useful. Box plots show the vribility of the observtions within tretment (fctor level) nd the vribility between tretments. We now discuss how the dt from single-fctor rndomized experiment cn be nlyzed sttisticlly The Anlysis of Vrince Suppose we hve different levels of single fctor tht we wish to compre. Sometimes, ech fctor level is clled tretment, very generl term tht cn be trced to the erly pplictions of experimentl design methodology in the griculturl sciences. The response for ech of the tretments is rndom vrible. The observed dt would pper s shown in Tble An entry in Tble 13-2, sy y ij, represents the jth observtion tken under tretment i. We initilly consider the cse in which there re n equl number of observtions, n, on ech tretment. We my describe the observtions in Tble 13-2 by the liner sttisticl model Y ij i ij i 1, 2, p, e j 1, 2, p, n (13-1) where Y ij is rndom vrible denoting the (ij)th observtion, is prmeter common to ll tretments clled the overll men, i is prmeter ssocited with the ith tretment clled the ith tretment effect, nd ij is rndom error component. Notice tht the model could hve been written s Y ij i ij i 1, 2, p, e j 1, 2, p, n where i i is the men of the ith tretment. In this form of the model, we see tht ech tretment defines popultion tht hs men i, consisting of the overll men plus n effect i tht is due to tht prticulr tretment. We will ssume tht the errors ij re normlly nd independently distributed with men zero nd vrince 2. Therefore, ech tretment cn be thought of s norml popultion with men i nd vrince 2. See Fig. 13-1(b). Eqution 13-1 is the underlying model for single-fctor experiment. Furthermore, since we require tht the observtions re tken in rndom order nd tht the environment (often clled the experimentl units) in which the tretments re used is s uniform s possible, this experimentl design is clled completely rndomized design. Tble 13-2 Typicl Dt for Single-Fctor Experiment Tretment Observtions Totls Averges 1 y 11 y 12 p y 1n y 1. 2 y 21 y 22 p y 2n y 2. o o o o o o o y 1 y 2 p y n y. y.. y 1. y 2. o y. y..

6 13-2 THE COMPLETELY RANDOMIZED SINGLE-FACTOR EXPERIMENT 473 The fctor levels in the experiment could hve been chosen in two different wys. First, the experimenter could hve specificlly chosen the tretments. In this sitution, we wish to test hypotheses bout the tretment mens, nd conclusions cnnot be extended to similr tretments tht were not considered. In ddition, we my wish to estimte the tretment effects. This is clled the fixed-effects model. Alterntively, the tretments could be rndom smple from lrger popultion of tretments. In this sitution, we would like to be ble to extend the conclusions (which re bsed on the smple of tretments) to ll tretments in the popultion, whether or not they were explicitly considered in the experiment. Here the tretment effects i re rndom vribles, nd knowledge bout the prticulr ones investigted is reltively unimportnt. Insted, we test hypotheses bout the vribility of the i nd try to estimte this vribility. This is clled the rndom effects, or components of vrince, model. In this section we develop the nlysis of vrince for the fixed-effects model. The nlysis of vrince is not new to us; it ws used previously in the presenttion of regression nlysis. However, in this section we show how it cn be used to test for equlity of tretment effects. In the fixed-effects model, the tretment effects i re usully defined s devitions from the overll men, so tht (13-2) Let y i. represent the totl of the observtions under the ith tretment nd y i. represent the verge of the observtions under the ith tretment. Similrly, let y.. represent the grnd totl of ll observtions nd y.. represent the grnd men of ll observtions. Expressed mthemticlly, y i. n j 1 y.. n j 1 y ij i 0 y ij y i. y i. n i 1, 2,..., y.. y.. N (13-3) where N n is the totl number of observtions. Thus, the dot subscript nottion implies summtion over the subscript tht it replces. We re interested in testing the equlity of the tretment mens 1, 2,...,. Using Eqution 13-2, we find tht this is equivlent to testing the hypotheses H 0 : 1 2 p 0 H 1 : i 0 for t lest one i (13-4) Thus, if the null hypothesis is true, ech observtion consists of the overll men plus reliztion of the rndom error component ij. This is equivlent to sying tht ll N observtions re tken from norml distribution with men nd vrince 2. Therefore, if the null hypothesis is true, chnging the levels of the fctor hs no effect on the men response. The ANOVA prtitions the totl vribility in the smple dt into two component prts. Then, the test of the hypothesis in Eqution 13-4 is bsed on comprison of two independent estimtes of the popultion vrince. The totl vribility in the dt is described by the totl sum of squres SS T 1 y ij y..2 2 n j 1

7 474 CHAPTER 13 DESIGN AND ANALYSIS OF SINGLE-FACTOR EXPERIMENTS: THE ANALYSIS OF VARIANCE The prtition of the totl sum of squres is given in the following definition. Definition The sum of squres identity is or symboliclly n j 1 1 y ij y..2 2 n 1 y i. y..2 2 SS T SS Tretments SS E n j 1 1 y ij y i.2 2 (13-5) (13-6) The identity in Eqution 13-5 (which is developed in Section on the CD) shows tht the totl vribility in the dt, mesured by the totl corrected sum of squres SS T, cn be prtitioned into sum of squres of differences between tretment mens nd the grnd men denoted SS Tretments nd sum of squres of differences of observtions within tretment from the tretment men denoted SS E. Differences between observed tretment mens nd the grnd men mesure the differences between tretments, while differences of observtions within tretment from the tretment men cn be due only to rndom error. We cn gin considerble insight into how the nlysis of vrince works by exmining the expected vlues of SS Tretments nd SS E. This will led us to n pproprite sttistic for testing the hypothesis of no differences mong tretment mens (or ll i 0). The expected vlue of the tretment sum of squres is E1SS Tretments n nd the expected vlue of the error sum of squres is E1SS E 2 1n i There is lso prtition of the number of degrees of freedom tht corresponds to the sum of squres identity in Eqution Tht is, there re n N observtions; thus, SS T hs n 1 degrees of freedom. There re levels of the fctor, so SS Tretments hs 1 degrees of freedom. Finlly, within ny tretment there re n replictes providing n 1 degrees of freedom with which to estimte the experimentl error. Since there re tretments, we hve (n 1) degrees of freedom for error. Therefore, the degrees of freedom prtition is The rtio n 1 1 1n 12 MS Tretments SS Tretments 1 12 is clled the men squre for tretments. Now if the null hypothesis H 0 : 1 2 p is true, MS Tretments is n unbised estimtor of 2 becuse g 0. However, if H 1 is true, MS Tretments estimtes 2 i 0 plus positive term tht incorportes vrition due to the systemtic difference in tretment mens.

8 13-2 THE COMPLETELY RANDOMIZED SINGLE-FACTOR EXPERIMENT 475 Note tht the error men squre MS E SS E 31n 124 is n unbised estimtor of 2 regrdless of whether or not H 0 is true. We cn lso show tht MS Tretments nd MS E re independent. Consequently, we cn show tht if the null hypothesis H 0 is true, the rtio F 0 SS Tretments 1 12 SS E 31n 124 MS Tretments MS E (13-7) hs n F-distribution with 1 nd (n 1) degrees of freedom. Furthermore, from the expected men squres, we know tht MS E is n unbised estimtor of 2. Also, under the null hypothesis, MS Tretments is n unbised estimtor of 2. However, if the null hypothesis is flse, the expected vlue of MS Tretments is greter thn 2. Therefore, under the lterntive hypothesis, the expected vlue of the numertor of the test sttistic (Eqution 13-7) is greter thn the expected vlue of the denomintor. Consequently, we should reject H 0 if the sttistic is lrge. This implies n upper-til, one-til criticl region. Therefore, we would reject H 0 if f 0 f, 1, 1n 12 where f 0 is the computed vlue of F 0 from Eqution Efficient computtionl formuls for the sums of squres my be obtined by expnding nd simplifying the definitions of SS Tretments nd SS T. This yields the following results. Definition The sums of squres computing formuls for the ANOVA with equl smple sizes in ech tretment re nd SS T n j 1 SS Tretments y 2 ij y..2 N The error sum of squres is obtined by subtrction s y 2 i n y..2 N SS E SS T SS Tretments (13-8) (13-9) (13-10) The computtions for this test procedure re usully summrized in tbulr form s shown in Tble This is clled n nlysis of vrince (or ANOVA) tble. Tble 13-3 The Anlysis of Vrince for Single-Fctor Experiment, Fixed-Effects Model Source of Degrees of Vrition Sum of Squres Freedom Men Squre F 0 Tretments SS Tretments 1 MS Tretments MS Tretments MS E Error SS E (n 1) MS E Totl SS T n 1

9 476 CHAPTER 13 DESIGN AND ANALYSIS OF SINGLE-FACTOR EXPERIMENTS: THE ANALYSIS OF VARIANCE EXAMPLE 13-1 Consider the pper tensile strength experiment described in Section We cn use the nlysis of vrince to test the hypothesis tht different hrdwood concentrtions do not ffect the men tensile strength of the pper. The hypotheses re H 0 : H 1 : i 0 for t lest one i. We will use The sums of squres for the nlysis of vrince re computed from Equtions 13-8, 13-9, nd s follows: SS T 4 SS Tretments 4 6 j 1 y 2 ij y.. 2 N p y 2 i. n y2.. N SS E SS T SS Tretments The ANOVA is summrized in Tble Since f 0.01,3, , we reject H 0 nd conclude tht hrdwood concentrtion in the pulp significntly ffects the men strength of the pper. We cn lso find P-vlue for this test sttistic s follows: P P1F 3, Since P is considerbly smller thn 0.01, we hve strong evidence to conclude tht H 0 is not true. Minitb Output Mny softwre pckges hve the cpbility to nlyze dt from designed experiments using the nlysis of vrince. Tble 13-5 presents the output from the Minitb one-wy nlysis of vrince routine for the pper tensile strength experiment in Exmple The results gree closely with the mnul clcultions reported previously in Tble Tble 13-4 ANOVA for the Tensile Strength Dt Source of Degrees of Vrition Sum of Squres Freedom Men Squre f 0 P-vlue Hrdwood concentrtion E-6 Error Totl

10 13-2 THE COMPLETELY RANDOMIZED SINGLE-FACTOR EXPERIMENT 477 Tble 13-5 Minitb Anlysis of Vrince Output for Exmple 13-1 One-Wy ANOVA: Strength versus CONC Anlysis of Vrince for Strength Source DF SS MS F P Conc Error Totl Individul 95% CIs For Men Bsed on Pooled StDev Level N Men StDev ( * ) ( * ) ( * ) ( * ) Pooled StDev Fisher s pirwise comprisons Fmily error rte Individul error rte Criticl vlue Intervls for (column level men) (row level men) The Minitb output lso presents 95% confidence intervls on ech individul tretment men. The men of the ith tretment is defined s i i i 1, 2, p, A point estimtor of i is ˆ i Y i.. Now, if we ssume tht the errors re normlly distributed, ech tretment verge is normlly distributed with men i nd vrince 2 n. Thus, if 2 were known, we could use the norml distribution to construct CI. Using MS E s n estimtor of 2 (The squre root of MS E is the Pooled StDev referred to in the Minitb output), we would bse the CI on the t-distribution, since T Y i. i 1MS E n hs t-distribution with (n 1) degrees of freedom. This leds to the following definition of the confidence intervl.

11 478 CHAPTER 13 DESIGN AND ANALYSIS OF SINGLE-FACTOR EXPERIMENTS: THE ANALYSIS OF VARIANCE Definition A 100(1 ) percent confidence intervl on the men of the ith tretment i is y i. t 2,1n 12 B MS E n i y i. t 2,1n 12 B MS E n (13-11) Eqution is used to clculte the 95% CIs shown grphiclly in the Minitb output of Tble For exmple, t 20% hrdwood the point estimte of the men is y , MS E 6.51, nd t 0.025, , so the 95% CI is or 3y 4. t 0.025,20 1MS E n psi psi It cn lso be interesting to find confidence intervls on the difference in two tretment mens, sy, i j. The point estimtor of i j is Y i. Y j., nd the vrince of this estimtor is Now if we use MS E to estimte 2, V1Y i. Y j.2 2 n 2 n 2 2 n T Y i. Y j. 1 i j 2 12MS E n hs t-distribution with (n 1) degrees of freedom. Therefore, CI on i j my be bsed on the t-distribution. Definition A 100(1 ) percent confidence intervl on the difference in two tretment mens i j is y i. y j. t 2,1n 12 B 2MS E n i j y i. y j. t 2,1n 12 B 2MS E n (13-12) A 95% CI on the difference in mens 3 2 is computed from Eqution s follows: 3y 3. y 2. t 0.025,20 12MS E n4 or

12 13-2 THE COMPLETELY RANDOMIZED SINGLE-FACTOR EXPERIMENT 479 Since the CI includes zero, we would conclude tht there is no difference in men tensile strength t these two prticulr hrdwood levels. The bottom portion of the computer output in Tble 13-5 provides dditionl informtion concerning which specific mens re different. We will discuss this in more detil in Section An Unblnced Experiment In some single-fctor experiments, the number of observtions tken under ech tretment my be different. We then sy tht the design is unblnced. In this sitution, slight modifictions must be mde in the sums of squres formuls. Let n i observtions be tken under tretment i (i 1, 2,..., ), nd let the totl number of observtions N g n i. The computtionl formuls for SS T nd SS Tretments re s shown in the following definition. Definition The sums of squres computing formuls for the ANOVA with unequl smple sizes n i in ech tretment re nd SS T SS Tretments ni yij 2 y2.. j 1 N y 2 i. n i y2.. N SS E SS T SS Tretments (13-13) (13-14) (13-15) Choosing blnced design hs two importnt dvntges. First, the ANOVA is reltively insensitive to smll deprtures from the ssumption of equlity of vrinces if the smple sizes re equl. This is not the cse for unequl smple sizes. Second, the power of the test is mximized if the smples re of equl size Multiple Comprisons Following the ANOVA When the null hypothesis H 0 : 1 2 p 0 is rejected in the ANOVA, we know tht some of the tretment or fctor level mens re different. However, the ANOVA doesn t identify which mens re different. Methods for investigting this issue re clled multiple comprisons methods. Mny of these procedures re vilble. Here we describe very simple one, Fisher s lest significnt difference (LSD) method. In Section on the CD, we describe three other procedures. Montgomery (2001) presents these nd other methods nd provides comprtive discussion. The Fisher LSD method compres ll pirs of mens with the null hypotheses H 0 : i j (for ll i j) using the t-sttistic t 0 y i. y j. 2MS E B n Assuming two-sided lterntive hypothesis, the pir of mens i nd j would be declred significntly different if 0 y i. y j. 0 LSD

13 480 CHAPTER 13 DESIGN AND ANALYSIS OF SINGLE-FACTOR EXPERIMENTS: THE ANALYSIS OF VARIANCE where LSD, the lest significnt difference, is LSD t 2, 1n 12 B 2MS E n (13-16) If the smple sizes re different in ech tretment, the LSD is defined s LSD t 2,N MS B E 1 n 1 i n b j EXAMPLE 13-2 We will pply the Fisher LSD method to the hrdwood concentrtion experiment. There re 4 mens, n 6, MS E 6.51, nd t 0.025, The tretment mens re y psi y psi y psi y psi The vlue of LSD is LSD t 0.025,20 12MS E n Therefore, ny pir of tretment verges tht differs by more thn 3.07 implies tht the corresponding pir of tretment mens re different. The comprisons mong the observed tretment verges re s follows: 4 vs vs vs vs vs vs From this nlysis, we see tht there re significnt differences between ll pirs of mens except 2 nd 3. This implies tht 10 nd 15% hrdwood concentrtion produce pproximtely the sme tensile strength nd tht ll other concentrtion levels tested produce different tensile strengths. It is often helpful to drw grph of the tretment mens, such s in Fig. 13-2, with the mens tht re not different underlined. This grph clerly revels the results of the experiment nd shows tht 20% hrdwood produces the mximum tensile strength. The Minitb output in Tble 13-5 shows the Fisher LSD method under the heding Fisher s pirwise comprisons. The criticl vlue reported is ctully the vlue of t 0.025,20 5% 10% 15% 20% psi Figure 13-2 Results of Fisher s LSD method in Exmple 13-2.

14 13-2 THE COMPLETELY RANDOMIZED SINGLE-FACTOR EXPERIMENT Minitb implements Fisher s LSD method by computing confidence intervls on ll pirs of tretment mens using Eqution The lower nd upper 95% confidence limits re shown t the bottom of the tble. Notice tht the only pir of mens for which the confidence intervl includes zero is for 10 nd 15. This implies tht 10 nd 15 re not significntly different, the sme result found in Exmple Tble 13-5 lso provides fmily error rte, equl to in this exmple. When ll possible pirs of mens re tested, the probbility of t lest one type I error cn be much greter thn for single test. We cn interpret the fmily error rte s follows. The probbility is tht there re no type I errors in the six comprisons. The fmily error rte in Tble 13-5 is bsed on the distribution of the rnge of the smple mens. See Montgomery (2001) for detils. Alterntively, Minitb permits you to specify fmily error rte nd will then clculte n individul error rte for ech comprison More About Multiple Comprisons (CD Only) Residul Anlysis nd Model Checking The nlysis of vrince ssumes tht the observtions re normlly nd independently distributed with the sme vrince for ech tretment or fctor level. These ssumptions should be checked by exmining the residuls. A residul is the difference between n observtion y ij nd its estimted (or fitted) vlue from the sttisticl model being studied, denoted s ŷ ij. For the completely rndomized design ŷ ij y i. nd ech residul is e ij y ij y i., tht is, the difference between n observtion nd the corresponding observed tretment men. The residuls for the pper tensile strength experiment re shown in Tble Using y i. to clculte ech residul essentilly removes the effect of hrdwood concentrtion from the dt; consequently, the residuls contin informtion bout unexplined vribility. The normlity ssumption cn be checked by constructing norml probbility plot of the residuls. To check the ssumption of equl vrinces t ech fctor level, plot the residuls ginst the fctor levels nd compre the spred in the residuls. It is lso useful to plot the residuls ginst y i. (sometimes clled the fitted vlue); the vribility in the residuls should not depend in ny wy on the vlue of y i. Most sttistics softwre pckges will construct these plots on request. When pttern ppers in these plots, it usully suggests the need for trnsformtion, tht is, nlyzing the dt in different metric. For exmple, if the vribility in the residuls increses with y i., trnsformtion such s log y or 1y should be considered. In some problems, the dependency of residul sctter on the observed men y i. is very importnt informtion. It my be desirble to select the fctor level tht results in mximum response; however, this level my lso cuse more vrition in response from run to run. The independence ssumption cn be checked by plotting the residuls ginst the time or run order in which the experiment ws performed. A pttern in this plot, such s sequences of positive nd negtive residuls, my indicte tht the observtions re not independent. Tble 13-6 Residuls for the Tensile Strength Experiment Hrdwood Concentrtion (%) Residuls

15 482 CHAPTER 13 DESIGN AND ANALYSIS OF SINGLE-FACTOR EXPERIMENTS: THE ANALYSIS OF VARIANCE Residul vlue 0 5% 10% 15% 20% 1 2 Norml score z j Figure 13-4 Plot of residuls versus fctor levels (hrdwood concentrtion) Residul vlue Figure 13-3 Norml probbility plot of residuls from the hrdwood concentrtion experiment. Residul vlue y i 2 4 Figure 13-5 Plot of residuls versus y i. This suggests tht time or run order is importnt or tht vribles tht chnge over time re importnt nd hve not been included in the experimentl design. A norml probbility plot of the residuls from the pper tensile strength experiment is shown in Fig Figures 13-4 nd 13-5 present the residuls plotted ginst the fctor levels nd the fitted vlue y i. respectively. These plots do not revel ny model indequcy or unusul problem with the ssumptions Determining Smple Size In ny experimentl design problem, the choice of the smple size or number of replictes to use is importnt. Operting chrcteristic curves cn be used to provide guidnce in mking this selection. Recll tht the operting chrcteristic curve is plot of the probbility of

16 13-2 THE COMPLETELY RANDOMIZED SINGLE-FACTOR EXPERIMENT 483 type II error ( ) for vrious smple sizes ginst mesure of the difference in mens tht it is importnt to detect. Thus, if the experimenter knows the mgnitude of the difference in mens tht is of potentil importnce, the operting chrcteristic curves cn be used to determine how mny replictes re required to chieve dequte sensitivity. The power of the ANOVA test is 1 P5Reject H 0 0 H 0 is flse6 P5F 0 f, 1, 1n 12 0 H 0 is flse6 (13-17) To evlute this probbility sttement, we need to know the distribution of the test sttistic F 0 if the null hypothesis is flse. It cn be shown tht, if H 0 is flse, the sttistic F 0 MS Tretments MS E is distributed s noncentrl F rndom vrible, with 1 nd (n 1) degrees of freedom nd noncentrlity prmeter. If 0, the noncentrl F-distribution becomes the usul or centrl F-distribution. Operting chrcteristic curves re used to evlute defined in Eqution These curves plot ginst prmeter, where 2 n 2 i 2 (13-18) The prmeter 2 is (prt from n) the noncentrlity prmeter. Curves re vilble for 0.05 nd 0.01 nd for severl vlues of the number of degrees of freedom for numertor (denoted v 1 ) nd denomintor (denoted v 2 ). Figure 13-6 gives representtive O.C. curves, one for 4 (v 1 3) nd one for 5 (v 1 4) tretments. Notice tht for ech vlue of there re curves for 0.05 nd O.C. curves for other vlues of re in Section on the CD. In using the operting curves, we must define the difference in mens tht we wish to detect in terms of g 2 i. Also, the error vrince 2 is usully unknown. In such cses, we must choose rtios of g 2 i 2 tht we wish to detect. Alterntively, if n estimte of 2 is vilble, one my replce 2 with this estimte. For exmple, if we were interested in the sensitivity of n experiment tht hs lredy been performed, we might use MS E s the estimte of 2. EXAMPLE 13-3 Suppose tht five mens re being compred in completely rndomized experiment with The experimenter would like to know how mny replictes to run if it is importnt to reject H 0 with probbility t lest 0.90 if g 5 2 i The prmeter 2 is, in this cse, 2 n 2 i 2 n 152 n 5 nd for the operting chrcteristic curve with v , nd v 2 (n 1) 5(n 1) error degrees of freedom refer to the lower curve in Figure As first guess, try n 4 replictes. This yields 2 4, 2, nd v 2 5(3) 15 error degrees of freedom. Consequently, from Figure 13-6, we find tht Therefore, the power of the test is pproximtely , which is less thn the required 0.90, nd so we

17 484 CHAPTER 13 DESIGN AND ANALYSIS OF SINGLE-FACTOR EXPERIMENTS: THE ANALYSIS OF VARIANCE Probbility of ccepting the hypothesis ν1 = 3 α = α = ν 2 = φ (for α = 0.01) 2 3 φ (for α = 0.05) Probbility of ccepting the hypothesis ν 1 = 4 α = α = 0.01 ν 2 = φ (for α = 0.01) Figure φ (for α = 0.05) Two Operting chrcteristic curves for the fixed-effects model nlysis of vrince.

18 13-2 THE COMPLETELY RANDOMIZED SINGLE-FACTOR EXPERIMENT 485 conclude tht n 4 replictes is not sufficient. Proceeding in similr mnner, we cn construct the following tble: n 2 (n 1) Power (1 ) Thus, t lest n 6 replictes must be run in order to obtin test with the required power Technicl Detils bout the Anlysis of Vrince (CD Only) EXERCISES FOR SECTION In Design nd Anlysis of Experiments, 5th edition (John Wiley & Sons, 2001) D. C. Montgomery describes n experiment in which the tensile strength of synthetic fiber is of interest to the mnufcturer. It is suspected tht strength is relted to the percentge of cotton in the fiber. Five levels of cotton percentge re used, nd five replictes re run in rndom order, resulting in the dt below. Cotton Observtions Percentge () Does cotton percentge ffect breking strength? Drw comprtive box plots nd perform n nlysis of vrince. Use (b) Plot verge tensile strength ginst cotton percentge nd interpret the results. (c) Anlyze the residuls nd comment on model dequcy In Orthogonl Design for Process Optimiztion nd Its Appliction to Plsm Etching (Solid Stte Technology, My 1987), G. Z. Yin nd D. W. Jillie describe n experiment to determine the effect of C 2 F 6 flow rte on the uniformity of the etch on silicon wfer used in integrted circuit mnufcturing. Three flow rtes re used in the experiment, nd the resulting uniformity (in percent) for six replictes is shown below. C 2 F 6 Flow (SCCM) Observtions () Does C 2 F 6 flow rte ffect etch uniformity? Construct box plots to compre the fctor levels nd perform the nlysis of vrince. Use (b) Do the residuls indicte ny problems with the underlying ssumptions? The compressive strength of concrete is being studied, nd four different mixing techniques re being investigted. The following dt hve been collected. Mixing Technique Compressive Strength (psi) () Test the hypothesis tht mixing techniques ffect the strength of the concrete. Use (b) Find the P-vlue for the F-sttistic computed in prt (). (c) Anlyze the residuls from this experiment An experiment ws run to determine whether four specific firing tempertures ffect the density of certin type of brick. The experiment led to the following dt. Temperture ( F) Density () Does the firing temperture ffect the density of the bricks? Use (b) Find the P-vlue for the F-sttistic computed in prt (). (c) Anlyze the residuls from the experiment.

19 486 CHAPTER 13 DESIGN AND ANALYSIS OF SINGLE-FACTOR EXPERIMENTS: THE ANALYSIS OF VARIANCE An electronics engineer is interested in the effect on tube conductivity of five different types of coting for cthode ry tubes in telecommunictions system disply device. The following conductivity dt re obtined. Coting Type Conductivity () Is there ny difference in conductivity due to coting type? Use (b) Anlyze the residuls from this experiment. (c) Construct 95% intervl estimte of the coting type 1 men. Construct 99% intervl estimte of the men difference between coting types 1 nd The response time in milliseconds ws determined for three different types of circuits in n electronic clcultor. The results re recorded here. Circuit Type Response () Using 0.01, test the hypothesis tht the three circuit types hve the sme response time. (b) Anlyze the residuls from this experiment. (c) Find 95% confidence intervl on the response time for circuit three An rticle in the ACI Mterils Journl (Vol. 84, 1987, pp ) describes severl experiments investigting the rodding of concrete to remove entrpped ir. A 3-inch 6-inch cylinder ws used, nd the number of times this rod ws used is the design vrible. The resulting compressive strength of the concrete specimen is the response. The dt re shown in the following tble. Rodding Level Compressive Strength () Is there ny difference in compressive strength due to the rodding level? (b) Find the P-vlue for the F-sttistic in prt (). (c) Anlyze the residuls from this experiment. Wht conclusions cn you drw bout the underlying model ssumptions? An rticle in Environment Interntionl (Vol. 18, No. 4, 1992) describes n experiment in which the mount of rdon relesed in showers ws investigted. Rdon-enriched wter ws used in the experiment, nd six different orifice dimeters were tested in shower heds. The dt from the experiment re shown in the following tble. Orifice Dimeter Rdon Relesed (%) () Does the size of the orifice ffect the men percentge of rdon relesed? Use (b) Find the P-vlue for the F-sttistic in prt (). (c) Anlyze the residuls from this experiment. (d) Find 95% confidence intervl on the men percent of rdon relesed when the orifice dimeter is A pper in the Journl of the Assocition of Asphlt Pving Technologists (Vol. 59, 1990) describes n experiment to determine the effect of ir voids on percentge retined strength of sphlt. For purposes of the experiment, ir voids re controlled t three levels; low (2 4%), medium (4 6%), nd high (6 8%). The dt re shown in the following tble. Air Voids Retined Strength (%) Low Medium High () Do the different levels of ir voids significntly ffect men retined strength? Use (b) Find the P-vlue for the F-sttistic in prt (). (c) Anlyze the residuls from this experiment. (d) Find 95% confidence intervl on men retined strength where there is high level of ir voids. (e) Find 95% confidence intervl on the difference in men retined strength t the low nd high levels of ir voids An rticle in the Mterils Reserch Bulletin (Vol. 26, No. 11, 1991) investigted four different methods of prepring the superconducting compound PbMo 6 S 8. The uthors contend

20 13-3 THE RANDOM-EFFECTS MODEL 487 tht the presence of oxygen during the preprtion process ffects the mteril s superconducting trnsition temperture T c. Preprtion methods 1 nd 2 use techniques tht re designed to eliminte the presence of oxygen, while methods 3 nd 4 llow oxygen to be present. Five observtions on T c (in K) were mde for ech method, nd the results re s follows: Preprtion Method Trnsition Temperture T c ( K) () Is there evidence to support the clim tht the presence of oxygen during preprtion ffects the men trnsition temperture? Use (b) Wht is the P-vlue for the F-test in prt ()? (c) Anlyze the residuls from this experiment. (d) Find 95% confidence intervl on men T c when method 1 is used to prepre the mteril Use Fisher s LSD method with 0.05 to nlyze the mens of the five different levels of cotton content in Exercise Use Fisher s LSD method with 0.05 test to nlyze the mens of the three flow rtes in Exercise Use Fisher s LSD method with 0.05 to nlyze the men compressive strength of the four mixing techniques in Exercise Use Fisher s LSD method to nlyze the five mens for the coting types described in Exercise Use Use Fisher s LSD method to nlyze the men response times for the three circuits described in Exercise Use Use Fisher s LSD method to nlyze the men mounts of rdon relesed in the experiment described in Exercise Use Apply Fisher s LSD method to the ir void experiment described in Exercise Using 0.05, which tretment mens re different? Apply Fisher s LSD method to the superconducting mteril experiment described in Exercise Which preprtion methods differ, if 0.05? Suppose tht four norml popultions hve common vrince 2 25 nd mens 1 50, 2 60, 3 50, nd How mny observtions should be tken on ech popultion so tht the probbility of rejecting the hypothesis of equlity of mens is t lest 0.90? Use Suppose tht five norml popultions hve common vrince nd mens 1 175, 2 190, 3 160, 4 200, nd How mny observtions per popultion must be tken so tht the probbility of rejecting the hypothesis of equlity of mens is t lest 0.95? Use THE RANDOM-EFFECTS MODEL Fixed versus Rndom Fctors In mny situtions, the fctor of interest hs lrge number of possible levels. The nlyst is interested in drwing conclusions bout the entire popultion of fctor levels. If the experimenter rndomly selects of these levels from the popultion of fctor levels, we sy tht the fctor is rndom fctor. Becuse the levels of the fctor ctully used in the experiment were chosen rndomly, the conclusions reched will be vlid for the entire popultion of fctor levels. We will ssume tht the popultion of fctor levels is either of infinite size or is lrge enough to be considered infinite. Notice tht this is very different sitution thn we encountered in the fixed effects cse, where the conclusions pply only for the fctor levels used in the experiment ANOVA nd Vrince Components The liner sttisticl model is i 1, 2, p, Y ij i ij e (13-19) j 1, 2, p, n where the tretment effects i nd the errors ij re independent rndom vribles. Note tht the model is identicl in structure to the fixed-effects cse, but the prmeters hve different

21 488 CHAPTER 13 DESIGN AND ANALYSIS OF SINGLE-FACTOR EXPERIMENTS: THE ANALYSIS OF VARIANCE interprettion. If the vrince of the tretment effects i is 2, by independence the vrince of the response is V1Y ij The vrinces 2 nd 2 re clled vrince components, nd the model, Eqution 13-19, is clled the components of vrince model or the rndom-effects model. To test hypotheses in this model, we ssume tht the errors ij re normlly nd independently distributed with men 0 nd vrince 2 nd tht the tretment effects i re normlly nd independently distributed with men zero nd vrince 2.* For the rndom-effects model, testing the hypothesis tht the individul tretment effects re zero is meningless. It is more pproprite to test hypotheses bout. Specificlly, H 0 : 2 0 H 1 : 2 0 If 2 0, ll tretments re identicl; but if 2 0, there is vribility between tretments. The ANOVA decomposition of totl vribility is still vlid; tht is, 2 SS T SS Tretments SS E (13-20) However, the expected vlues of the men squres for tretments nd error re somewht different thn in the fixed-effect cse. In the rndom-effects model for single-fctor, completely rndomized experiment, the expected men squre for tretments is E1MS Tretments 2 E SS Tretments b 1 2 n 2 (13-21) nd the expected men squre for error is SS E E1MS E 2 E c 1n 12 d 2 (13-22) From exmining the expected men squres, it is cler tht both MS E nd MS Tretments estimte 2 when H 0 : 2 0 is true. Furthermore, MS E nd MS Tretments re independent. Consequently, the rtio F 0 MS Tretments MS E (13-23) *The ssumption tht the { i } re independent rndom vribles implies tht the usul ssumption of from the fixed-effects model does not pply to the rndom-effects model. g i 0

22 13-3 THE RANDOM-EFFECTS MODEL 489 is n F rndom vrible with 1 nd (n 1) degrees of freedom when H 0 is true. The null hypothesis would be rejected t the -level of significnce if the computed vlue of the test sttistic f 0 f, 1,(n 1). The computtionl procedure nd construction of the ANOVA tble for the rndomeffects model re identicl to the fixed-effects cse. The conclusions, however, re quite different becuse they pply to the entire popultion of tretments. Usully, we lso wnt to estimte the vrince components ( 2 nd 2 ) in the model. The procedure tht we will use to estimte 2 nd 2 is clled the nlysis of vrince method becuse it uses the informtion in the nlysis of vrince tble. It does not require the normlity ssumption on the observtions. The procedure consists of equting the expected men squres to their observed vlues in the ANOVA tble nd solving for the vrince components. When equting observed nd expected men squres in the one-wy clssifiction rndomeffects model, we obtin MS Tretments 2 n 2 nd MS E 2 Therefore, the estimtors of the vrince components re nd ˆ 2 MS E ˆ 2 MS Tretments MS E n (13-24) (13-25) Sometimes the nlysis of vrince method produces negtive estimte of vrince component. Since vrince components re by definition nonnegtive, negtive estimte of vrince component is disturbing. One course of ction is to ccept the estimte nd use it s evidence tht the true vlue of the vrince component is zero, ssuming tht smpling vrition led to the negtive estimte. While this pproch hs intuitive ppel, it will disturb the sttisticl properties of other estimtes. Another lterntive is to reestimte the negtive vrince component with method tht lwys yields nonnegtive estimtes. Still nother possibility is to consider the negtive estimte s evidence tht the ssumed liner model is incorrect, requiring tht study of the model nd its ssumptions be mde to find more pproprite model. EXAMPLE 13-4 In Design nd Anlysis of Experiments, 5th edition (John Wiley, 2001), D. C. Montgomery describes single-fctor experiment involving the rndom-effects model in which textile mnufcturing compny weves fbric on lrge number of looms. The compny is interested in loom-to-loom vribility in tensile strength. To investigte this vribility, mnufcturing engineer selects four looms t rndom nd mkes four strength determintions on fbric smples chosen t rndom from ech loom. The dt re shown in Tble 13-7 nd the ANOVA is summrized in Tble From the nlysis of vrince, we conclude tht the looms in the plnt differ significntly in their bility to produce fbric of uniform strength. The vrince components re estimted by ˆ nd ˆ

23 490 CHAPTER 13 DESIGN AND ANALYSIS OF SINGLE-FACTOR EXPERIMENTS: THE ANALYSIS OF VARIANCE Tble 13-7 Strength Dt for Exmple 13-4 Observtions Loom Totl Averge Tble 13-8 Anlysis of Vrince for the Strength Dt Source of Sum of Degrees of Men Vrition Squres Freedom Squre f 0 P-vlue Looms E-4 Error Totl Therefore, the vrince of strength in the mnufcturing process is estimted by I V1Y ij 2 ˆ 2 ˆ Most of this vribility is ttributble to differences between looms. This exmple illustrtes n importnt ppliction of the nlysis of vrince the isoltion of different sources of vribility in mnufcturing process. Problems of excessive vribility in criticl functionl prmeters or properties frequently rise in qulityimprovement progrms. For exmple, in the previous fbric strength exmple, the process men is estimted by y psi, nd the process stndrd devition is estimted by ˆ y 2Vˆ1Y ij psi. If strength is pproximtely normlly distributed, the distribution of strength in the outgoing product would look like the norml distribution shown in Fig. 13-7(). If the lower specifiction limit (LSL) on strength is t 90 psi, substntil proportion of the process output is fllout tht is, scrp or defective mteril tht must be sold s second qulity, nd so on. This fllout is directly relted to the excess vribility resulting from differences between looms. Vribility in loom performnce could be cused by fulty setup, poor mintennce, indequte supervision, poorly trined opertors, nd so forth. The engineer or mnger responsible for qulity improvement must identify nd remove these sources of vribility from the process. If this cn be done, strength vribility will be gretly reduced, perhps s low s ˆ Y 2 ˆ psi, s shown in Fig. 13-7(b). In this improved process, reducing the vribility in strength hs gretly reduced the fllout, resulting in lower cost, higher qulity, more stisfied customer, nd enhnced competitive position for the compny Determining Smple Size in the Rndom Model (CD Only) Process fllout psi LSL () Figure psi LSL (b) The distribution of fbric strength. () Current process, (b) improved process.