Chapter 7. Analysis of Variance

Transcription

1 Deprtment of Appled Mthemtcs Chpter 7. Anlss of Vrnce Defnton The vrle mesured n the experment s clled the response vrle. (In ths chpter, ll response vrles re quntttve vrles.) The response wll e denoted wth the smol. Defnton The oject upon whch the response vrle s mesured s clled n expermentl (or smplng) unt. Defnton 3 The vrles, quntttve or qulttve, tht re relted to response vrle re clled fctors. Defnton 4 The ntenst settng of fctor (.e., the vlue ssumed fctor n n experment) s clled fctor level. Defnton 5 A tretment s prtculr comnton of levels of the fctors nvolved n n experment. Defnton 6 A sngle replcton of two-fctor fctorl experment s one n whch the response vrle s oserved once for ever possle fctor level comnton. Defnton 7 A complete two-fctor fctorl experment s one tht conssts of one or more replctons (Note: In prctce, the numer of replctons r for fctorl experment s lmost lws chosen to e or lrger.) Mrjore Chu, 9 7-

2 Defnton 8 Deprtment of Appled Mthemtcs A completel rndomzed desgn to compre k tretment (populton) mens s one n whch the tretments re rndoml ssgned to the expermentl unts, or n whch ndependent rndom smples re drwn from ech of the k trget popultons. Defnton 9 A rndomzed lock desgn s desgn n whch k tretments re compred wthn ech of locks. Ech lock contns k mtched expermentl unts nd the k tretments re rndoml ssgned, one to ech of the unts wthn ech lock. Defnton In fctorl experment, when the dfference etween the men levels of fctor A depends on the dfferent levels of fctor B, we s tht the fctors A nd B nterct. If the dfference s ndependent of the levels of B, then there s no ntercton etween fctors A nd B. Defnton A two-stge nested smplng desgn nvolves the rndom selecton of frst-stge (prmr) unts from populton. Su-smples of second-stge unts re then rndoml selected from wthn ech prmr unt. 7.. Completel rndomzed desgn Model: j j j,,, k, j,, n where j denotes the jth oservton of the tretment ; s the overll men; s the effect of tretment ; s the th tretment men; j s the error term. Assumptons: k. n Mrjore Chu, 9 7-

3 . ~ N, j ndependentl. Deprtment of Appled Mthemtcs 3. The smples from ech populton re rndom. pothess: : k (.e., there s no dfference n the tretment mens) : At lest two tretment mens dffer Formuls: SST k n j j k n j j k n (Totl sum of squres), SSA k n k n j j k n (Sum of squres for tretments), where n j, the totl of ll oservtons for tretment. j SSE = SST SSA (Sum of squres for error) MSA = SSA / (k-) (Men squre for tretments) MSE = SSE / (N-k) (Men squre error ), where N k n Test sttstc: F MSA MSE The results re summrzed n n ANOVA tle: Source SS df MS F Tretments SSA k- MSA = SSA/(k-) F=MSA/MSE Error SSE N-k MSE = SSE/(N-k) Totl SST N- Crtcl regon: F F ; k, N k Exmple : (dvertse) A stud reported n the Journl of Advertsng reserch (Oct./Nov. 985) exmned the effect of prce dvertsng on sles of eer n Mchgn. The stte prohted retlers from dvertsng the prce of eer products snce 975, except for ref perod (3/98-5/983) when the n ws temporrl lfted. The dt n the tle re Mrjore Chu, 9 7-3

4 Deprtment of Appled Mthemtcs the monthl totl sles of rewed everges (n thousnds of 3-gllon rrels). The dt llow us to compre totl sles of eer n three perods, efore (perod ), durng (perod ), nd fter (perod 3) the lftng of the prce dvertsng restrctons. Prce dvertsng restrcted (5/98-/98) No restrctons (5/98-5/983) Prce dvertsng restrcted (7/983-4/984) Is there suffcent evdence to ndcte dfferences n the verge totl sles of eer n the three perods? Fnd 95% confdence ntervl for the men sles of ech perod. Estmte the overll men sles f there s no sgnfcnt dfference n the three perods. Comment on our fndngs. : 3 : At lest two of the vrnces re dfferent : 3 : At lest one pr s not equl Input the monthl totl sles of the three perods n one vrle (sles) onl. Use nother vrle (perod) to specf the tpe of perod for ech oservton. Test for the homogenet of vrnce ssumpton nd test the effect of dvertsng perod Mrjore Chu, 9 7-4

5 Anlze Compre mens One-w ANOVA Dependent lst: prce Fctor: perod Post oc: LSD (Equl vrnce ssumed) Optons: Descrptve omogenet-of-vrnce Deprtment of Appled Mthemtcs Onew Descrptves PRICE 5/8-/8 5/8-5/83 7/83-4/84 Totl N 95% Confdence Intervl for Men Std. Lower Upper Men Devton Std. Error Bound Bound Mnmum Mxmum The ove tle summrzes the descrptve sttstcs of the three dvertsng perods. Test of omogenet of Vrnces PRICE Levene Sttstc df df Sg The ssumpton of equl vrnces s tested wth.635 sgnfcnce. There s no evdence to gnst tht the ssumpton of equl vrnces hold t 5% level of sgnfcnce. Mrjore Chu, 9 7-5

6 Deprtment of Appled Mthemtcs ANOVA PRICE Between Groups Wthn Groups Totl Sum of Men Squres df Squre F Sg F test s used to test for the eqult of mens nd elds.476 sgnfcnce. cn not e rejected tht there s no sgnfcnt dfference n the men monthl totl sles n the three dvertsng perods. Post oc Tests Dependent Vrle: PRICE LSD Multple Comprsons (I) d. perod 5/8-/8 5/8-5/83 7/83-4/84 (J) d. perod 5/8-5/83 7/83-4/84 5/8-/8 7/83-4/84 5/8-/8 5/8-5/83 Men 95% Confdence Intervl Dfference Lower Upper (I-J) Std. Error Sg. Bound Bound The pred comprson lso shows no sgnfcnt dfference mong the men monthl totl sles n the three perods. If the normlt ssumpton s not fulflled, non-prmetrc test wll e used nsted. Anlze Nonprmetrc Test Legc Dlogs K ndependent smples Mrjore Chu, 9 7-6

7 Test vrle lst: prce Groupng vrle: perod Mnmum = Mxmum = 3 Test tpe: Kruskl Wlls Deprtment of Appled Mthemtcs Kruskl-Wlls Test Rnks d. perod N Men Rnk prce 5/8-/8 5. 5/8-5/ /83-4/ Totl 8 Test Sttstcs, prce Ch-Squre.579 Df Asmp. Sg Kruskl Wlls Test. Groupng Vrle: d. Perod The Kruskl Wlls test hs the sgnfcnce of.454 whch ndctes tht the medn of monthl totl sles of the three dvertsng perods do not hve sgnfcnt dfference. 7.. Completel rndomzed lock desgn Model: j,,, k, j,, j j where j denotes the oservton of the tretment nd lock j; Mrjore Chu, 9 7-7

8 Deprtment of Appled Mthemtcs s the overll men; s the effect of tretment ; j s the effect of lock j; j s the error term. Assumptons: k.. j 3. ~ N, j j ndependentl. 4. The tretments re rndoml ssgned to the expermentl unts wthn ech lock. pothess: : (.e., there s no dfference n the tretment mens) k : At lest two tretment mens dffer : (.e., there s no dfference n the lock mens) : At lest two lock mens dffer Formuls: SST k j j where n = k. k j j n (Totl sum of squres), SSA k k j n (Sum of squres for tretments), j where j j, the totl of ll oservtons for tretment. SSB j k k j j n (Sum of squres for locks), j Mrjore Chu, 9 7-8

9 where k j j, the totl of ll oservtons for lock j. SSE = SST SSA - SSB (Sum of squres for error) MSA = SSA / (k-) (Men squre for tretments) MSB = SSB / (-) (Men squre for locks) MSE = SSE / (n-k) (Men squre error ) Test sttstc for the tretment mens: F = MSA/MSE Test sttstc for the lock mens: F = MSB/MSE Deprtment of Appled Mthemtcs The results re summrzed n n ANOVA tle: Source SS df MS F Tretments SSA k- MSA = SSA/(k-) F=MSA/MSE Block SSB - MSB = SSB/(-) F=MSB/MSE Error SSE (k-)(-) MSE = SSE/((k-)(-)) Totl SST n- Crtcl regon for the tretment mens: F F ; k, k Crtcl regon for the lock mens: F F ;, k Exmple : (supermrket) Perodcll, supermrkets conduct surve of prces of ther compettors nd nclude the results n newspper dvertsement. Alertson s supermrket, compre the prces of 49 grocer tems t three competng supermrkets wth the prces t ts store on recent d. The surve results for seven tems selected from 49 tems re shown n the tle. ow could we determne whether the men prces of grocer dffer mong the four supermrkets? Grocer tem Alertson s Ksh N Krr Pulx Food 4 Less Cerel $. $.8 $.39 $.8 Jell-o geltn Sop Ol Kleenex Tun Pes Enter the dt of prce n one vrle (prce) onl. Mrjore Chu, 9 7-9

10 Deprtment of Appled Mthemtcs Add two more vrles to ndcte the tem (tem) nd supermrket (mrket). Mrjore Chu, 9 7-

11 Anlze Generl Lner Model Unvrte Dependent vrle: prce Fxed fctor(s): mrket, tem Model Custom Mn effects: mrket, tem Sum of squres: Tpe III Post oc Post oc test for: mrket Equl vrnces ssumed: LSD Optons Dspl mens for: overll, mrket Dspl: omogenet tests Deprtment of Appled Mthemtcs : 3 4 : At lest two of the vrnces re dfferent : 3 4 (the men prces of the 4 supermrkets re the sme) : At lest one pr s not equl Unvrte Anlss of Vrnce Mrjore Chu, 9 7-

12 Deprtment of Appled Mthemtcs Between-Sujects Fctors supermrket Grocer tem Vlue Lel N Alertson's 7 Ksh 'N Krr 7 Pulx 7 Food 4 Less 7 Cerel 4 Jell-o geltn 4 Sop 4 Ol 4 Kleenex 4 Tun 4 Pes 4 Levene's Test of Eqult of Error Vrnces Dependent Vrle: PRICE F df df Sg.. 7. Tests the null hpothess tht the error vrnce of the dependent vrle s equl cross groups.. Desgn: Intercept+MARKET+ITEM Dependent Vrle: PRICE Source Corrected Model Intercept MARKET ITEM Error Totl Corrected Totl Tests of Between-Sujects Effects Tpe III Sum of Men Squres df Squre F Sg E R Squred =.96 (Adjusted R Squred =.889) The sgnfcnce of the supermrket nd grocer tem re equl to.5 nd. respectvel. The null hpothess of equl mens s rejected t 5% level. There s Mrjore Chu, 9 7-

13 Deprtment of Appled Mthemtcs sgnfcnt dfference of the men prces of grocer mong the four supermrkets. ence, onl the estmton of men prces of ndvdul supermrket s menngful. Estmted Mrgnl Mens. Grnd Men Dependent Vrle: PRICE 95% Confdence Intervl Men Std. Error Lower Bound Upper Bound The estmton of overll populton men s not relevnt when the hpothess of eqult of mens s rejected. We re nterested n the pont estmtes nd ntervl estmtes of the men prces of grocer tems of ech of the four supermrkets.. supermrket Dependent Vrle: PRICE 95% Confdence Intervl supermrket Men Std. Error Lower Bound Upper Bound Alertson's Ksh 'N Krr Pulx Food 4 Less From the smple dt, the 95% confdence ntervl of the men prce of grocer tems of Alertson s supermrket s.568 < <.87. Smlrl, the confdence ntervls of the men prce of the Ksh N Krr, Pulx nd Food 4 Less re.668 < <.97,.85 < <. nd.6 < <.85 respectvel. Mrjore Chu, 9 7-3

14 Deprtment of Appled Mthemtcs Post oc Tests supermrket Multple Comprsons Dependent Vrle: PRICE LSD (I) supermrket Alertson's Ksh 'N Krr Pulx Food 4 Less (J) supermrket Ksh 'N Krr Pulx Food 4 Less Alertson's Pulx Food 4 Less Alertson's Ksh 'N Krr Food 4 Less Alertson's Ksh 'N Krr Pulx Bsed on oserved mens. *. The men dfference s sgnfcnt t the.5 level. Men 95% Confdence Intervl Dfference Lower Upper (I-J) Std. Error Sg. Bound Bound -.E E- -.89* E E E * E E * * E * E E E * E- The F-test onl gves the result tht the men prces of the grocer tems of the four supermrkets re sgnfcnt dfferent, ut there s no ndcton of whch supermrket dffers from the others. Multple comprson s ppled to gve pred comprson of the men prces mong two supermrkets. The ove tle shows the results of (I) supermrket compre wth (J) supermrket. Onl the men prce of the Pulx supermrket sgnfcnt dffers from the other supermrkets. The men prces of the Alertson s, Ksh N Krr nd Food 4 Less do not hve sgnfcnt dfferent. ence, the men prce of the grocer tems of the Pulx supermrket s dfferent from the others. Mrjore Chu, 9 7-4

15 7.3. Two-fctor fctorl desgn Model: jk j jk,,,, j,,, k,, r j Deprtment of Appled Mthemtcs where jk denotes the oservton of the k th replcton from the th level of fctor A nd j th level of fctor B; s the overll men; s the effect of th level of fctor A; j s the effect of j th level of fctor B; j s the ntercton effect of th level of fctor A nd j th level of fctor B; jk s the error term. Assumptons:.. j j j 5. ~ N, j j jk ndependentl. 6. The tretments re rndoml ssgned to the expermentl unts. 7. The oservtons for ech fctor level comnton represent ndependent rndom smples. pothess: : There s no ntercton etween fctor A nd fctor B : Intercton effect exsts : There s no dfference mong the mens for mn effect of fctor A : At lest two of the mens of fctor A dffer Mrjore Chu, 9 7-5

16 Deprtment of Appled Mthemtcs ''' : There s no dfference mong the mens for mn effect of fctor B ''' : At lest two of the mens of fctor B dffer Formuls: SST r j k jk r j k jk n (Totl sum of squres), where n = r. SSA r r jk n (Sum of squres for fctor A), j k where r jk, the totl of ll oservtons for th level of fctor A. j k SSB j r r j jk n (Sum of squres for fctor B), j k where j jk k r, the totl of ll oservtons for jth level of fctor B. SSAB j j r r j k ntercton etween fctor A nd fctor B), jk n SSA SSB (Sum of squres for where r k j jk, the totl of ll replctons for th level of fctor A nd jth level of fctor B. SSE = SST SSA SSB - SSAB (Sum of squres for error) MSA = SSA / (-) (Men squre for fctor A) MSB = SSB / (-) (Men squre for fctor B) MSAB = SSAB/((-)(-) (Men squre for the ntercton) MSE = SSE / ((r-)) (Men squre error) Test sttstc for the fctor A mn effect: F = MSA/MSE Test sttstc for the fctor B mn effect: F = MSB/MSE Test sttstc for the ntercton effect: F = MSAB/MSE Mrjore Chu, 9 7-6

17 Deprtment of Appled Mthemtcs The results re summrzed n n ANOVA tle: Source SS df MS F Fctor A SSA - MSA = SSA/(-) F=MSA/MSE Fctor B SSB - MSB = SSB/(-) F=MSB/MSE Intercton SSAB (-)(-) MSAB = SSAB/((-)(-) F=MSAB/MSE Error SSE (r-) MSE = SSE/((r-)) Totl SST n- Crtcl regon for fctor A mn effect: F F ;, r Crtcl regon for fctor B mn effect: F F ;, r Crtcl regon for the ntercton effect: F F ;, r Exmple 3: (sles) A deprtment store conducted n experment to stud the sles of product for dfferent pckge desgns nd dfferent sellng prces. There were three dfferent pckge desgns t two dfferent sellng prces. Pckge desgn Sellng prce (gher) Sellng prce (Lower) The wnt to test:. The hpothess tht there re no nterctons for n comnton of pckge desgn nd sellng prce.. The hpothess tht the men populton mount of sles s the sme for three pckge desgns. 3. The hpothess tht the men populton sles mount s the sme for two sellng prces. 4. Locte where the dfference re f the hpothess s sgnfcnt. Mrjore Chu, 9 7-7

18 Anlze Generl Lner Model Unvrte Dependent vrle: sles Fctor(s): prce, desgn Model: Full fctorl Post oc Post oc tests for: desgn Equl vrnces ssumed: LSD Plots orzontl Axs: desgn Seprte Lnes: prce Add : 3 : Dfferent desgn effects exst : : Dfferent prce effects exst ''' : j, =,,3, j=, ''' : Intercton effects exst Deprtment of Appled Mthemtcs Unvrte Anlss of Vrnce Between-Sujects Fctors sellng prce DESIGN 3 Vlue Lel N hgher 6 lower Mrjore Chu, 9 7-8

19 Deprtment of Appled Mthemtcs Tests of Between-Sujects Effects Dependent Vrle: SALES Tpe III Source Sum of Squres df Men Squre F Sg. Corrected Model Intercept PRICE DESIGN PRICE * DESIGN Error Totl Corrected Totl 89.. R Squred =.975 (Adjusted R Squred =.953) The sgnfcnce of the prce effect, desgn effect nd ntercton effect re.555,. nd.37 respectvel. Therefore, t 5% level of sgnfcnce, the desgn effect on the men sles s sgnfcnt, wheres the prce effect nd ntercton effect of prce nd desgn re not sgnfcnt. Furthermore, LSD test s ppled to detect whch desgn s sgnfcnt dfferent from the others. Mrjore Chu, 9 7-9

20 Deprtment of Appled Mthemtcs Post oc Tests DESIGN Multple Comprsons Dependent Vrle: SALES LSD (I) DESIGN 3 (J) DESIGN 3 3 Bsed on oserved mens. *. The men dfference s sgnfcnt t the.5 level. Men 95% Confdence Intervl Dfference Lower Upper (I-J) Std. Error Sg. Bound Bound -4.5* * * * The effect of desgn s sgnfcnt dfferent from the effect of desgn nd desgn 3. Profle Plots 5 Estmted Mrgnl Mens of SALES 4 Estmted Mrgnl Mens 3 sellng prce hgher lower 3 DESIGN Mrjore Chu, 9 7-

21 Deprtment of Appled Mthemtcs Grphcl pproch s n lterntve method to dentf whether the ntercton effect etween two mn fctors exsts Two-stge nested desgn (fxed effect) Model: j,,,, j,, j j where j denotes the oservton of the j th level of fctor B nested under th level of fctor A; s the overll men; s the effect of th level of fctor A; s the effect of j th level of fctor B nested under th level of fctor A; j j s the error term. Assumptons:.. j j 3. ~ N, j ndependentl. pothess: : There s no dfference mong the mens for mn effect of fctor A ( =) : At lest two of the mens of fctor A dffer : There s no effect of the fctor B nested under fctor A ( j =) : Effects of fctor B nested under fctor A exst Formuls: SST r j k jk r j k jk n (Totl sum of squres), Mrjore Chu, 9 7-

22 Deprtment of Appled Mthemtcs where jk denotes the oservton of the k th replcton from the th level of fctor A nd j th level of fctor B nested under th level of fctor A, n = r. SSA r r jk n (Sum of squres for fctor A), j k where r jk j k, the totl of ll oservtons for th level of fctor A. j j SSB( A) r r (Sum of squres for fctor B), where r k j jk, the totl of ll replctons for th level of fctor A nd j th level of fctor B nested under th level of fctor A. SSE = SST SSA SSB(A) (Sum of squres for error) MSA = SSA / (-) (Men squre for fctor A) MSB(A) = SSB(A) / ((-)) (Men squre for fctor B nested under fctor A) MSE = SSE / ((r-)) (Men squre error) Test sttstc for the fctor A mn effect: F = MSA/MSE Test sttstc for the fctor B nested under fctor A: F = MSB(A)/MSE The results re summrzed n n ANOVA tle: Source SS df MS F Fctor A SSA - MSA = SSA/(-) F=MSA/MSE Fctor B wthn Fctor A SSB(A) (-) MSB = SSB/((-)) F=MSB/MSE Error SSE (r-) MSE = SSE/((r-)) Totl SST n- Crtcl regon for fctor A mn effect: F F ;, r Crtcl regon for fctor B nested under fctor A: F F ;, r Exmple 4: (lm) In lm-feedng experment, four lms were ssgned t rndom to ech of sx lots. Three rtons contnng dfferent percentges of corn were rndoml ssgned to the lots such tht ech rton ws used to feed two lots. The gns n weght over threemonth perod re gven elow. Mrjore Chu, 9 7-

23 Deprtment of Appled Mthemtcs Rton 3% corn 4% corn 5% corn Lot Weght Test to see f the hpothess tht the effects on the men weght gns due to three dfferent percentge of corn re the sme. Test to see f the lot wthn rton effect exsts. Three rton effect on men weght gn : 3 or = for ll : At lest two of the mens re dfferent The lots wthn ech rton effect on men weght gn : for ll j wthn rton j : j Anlze Generl Lner Model Unvrte Dependent vrle: weght Fxed fctor(s): rton, lot Model Custom Mn effect: rton Intercton effect: rton, lot Sum of squres: Tpe I Mrjore Chu, 9 7-3

24 Deprtment of Appled Mthemtcs Unvrte Anlss of Vrnce Between-Sujects Fctors rton LOT 3 Vlue Lel N 3% corn 8 4% corn 8 5% corn 8 Tests of Between-Sujects Effects Dependent Vrle: WEIGT Source Tpe I Sum of Squres df Men Squre F Sg. Corrected Model Intercept RATION RATION * LOT Error Totl Corrected Totl R Squred =.384 (Adjusted R Squred =.3) The sgnfcnce of the mn effect rton s., t mens tht there s evdence to gnst the null hpothess. On the other hnd, the sgnfcnce of the lots wthn ech rton effect s.646, s not rejected. ence, the rton effect on men weght gn exsts t 5% level of sgnfcnce, ut the lot wthn rton effect s not sgnfcnt. Mrjore Chu, 9 7-4

25 Deprtment of Appled Mthemtcs Exercse 7 Queston. (trnng) The personnel mnger of lrge nsurnce compn wshed to evlute the effectveness of four dfferent sles-trnng progrms desgned for new emploees. A group of 3 recentl hred college grdutes were rndoml ssgned to four progrms so tht there were eght sujects n ech progrm. At the end of the month-long trnng perod stndrd exm ws dmnstered to the 3 sujects; the scores re gven elow: Progrms A B C D To determne whether there s evdence of dfference n the four sles-trnng progrms t the.5 level of sgnfcnce. Queston. (cholesterol) The followng re the cholesterol contents, n mllgrms per pckge, whch four lortores otned for 6-ounce pckges of three ver smlr det foods: Lortor Det food A Det food B Det food C Test whether the cholesterol contents dffer mong the 3 det foods t the.5 level of sgnfcnce. Mrjore Chu, 9 7-5

26 Queston 3. (effluent) Deprtment of Appled Mthemtcs Ozonzton s secondr tretment for effluent, followng sorpton ferrous chlorde, ws studed for three recton tmes nd three p levels. The stud elded the followng results for effluent declne. Recton tme (mn) p level(7.) p level(9.) p level(.5) ) Test for sgnfcnt ntercton. ) Test for sgnfcnt dfferences mong recton tmes. c) Test for sgnfcnt dfferences mong p levels. Queston 4. (wter) Brcks for furnce lnngs re mde n two plnts; the re stored n one of two storge loctons n plnt I, nd n one of three storge loctons n plnt II. Rndom smples of rcks from ech locton re tested for wter content fter one week s storge. plnt I plnt II locton 3 wter content Test hpotheses tht loctons nd plnts hve no effect on wter content. Mrjore Chu, 9 7-6

27 Queston 5. (solvent) Deprtment of Appled Mthemtcs To estmte the degree of suspenson of suspenson polethlene, we extrct polethlene usng solvent, nd compre the gel contents (gel proporton). Ths method s clled the gel proporton estmton method. In the ook Desgn of Experments for the Qult Improvement pulshed the Jpnese Stndrds Assocton (989), stud ws conducted on the reltonshp etween the mount of gel generted nd three fctors (solvent, extrcton temperture, nd extrcton tme) tht nfluence mount of gel. The dt re s follows: Tme Solvent Temperture Ethnol Toluene Perform n nlss of vrnce to test for sgnfcnt mn nd ntercton effects. Mrjore Chu, 9 7-7