# Lecture 8 Topic 5: Multiple Comparisons (means separation)

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1 Lectue 8 Topic 5: Multiple Compaisons (means sepaation) ANOVA: H 0 : µ 1 = µ =... = µ t H 1 : The mean of at least one teatment goup is diffeent If thee ae moe than two teatments in the expeiment, futhe analysis is equied to detemine which means ae significantly diffeent. Thee ae two stategies: 1. Planned, single d.f. F tests (othogonal contasts) Motivated by the teatment stuctue Independent Poweful and pecise Limited to (t 1) compaisons Well-defined method. Multiple compaisons (means sepaation) Motivated by the data Useful when no paticula elationship exists among teatments Up to unlimited compaisons Many, many methods/philosophies to choose fom Eo ates Selection of the most appopiate multiple compaison test is heavily influenced by the eo ate. Recall that a Type I eo occus when one incoectly ejects a tue null hypothesis H 0. The Type I eo ate is the faction of times a Type I eo is made. In a single compaison, this is α. When compaing thee o moe teatment means: 1. Compaison-wise Type I eo ate (CER) The numbe of Type I eos, divided by the total numbe of compaisons.. Expeiment-wise Type I eo ate (EER) The numbe of expeiments in which at least one Type I eo occus, divided by the total numbe of expeiments. 1

2 Example: An expeimente conducts an expeiment with 5 teatments. In such an expeiment, thee ae 10 possible paiwise compaisons that can be made: Total possible paiwise compaisons: p = t( t 1) Suppose that thee ae no tue diffeences among the teatments (i.e. H 0 is tue), but that one Type I eo is made. CER = (1 Type I eo) / (10 compaisons) = 0.1 o 10% EER = (1 expeiments with a Type I eo) / (1 expeiment) = 1 o 100% Things to conside: 1. EER is the pobability of thee being a Type I eo somewhee in the expeiment. As the numbe of teatments inceases, the EER à 100%.. To maintain a low EER, the CER has to be kept vey low. Convesely, a easonable CER will inflate the EER to a potentially unacceptable level. 3. The elative impotance of contolling these two Type I eo ates depends on the objectives of the study: When incoectly ejecting one compaison jeopadizes the entie expeiment o when the consequence of incoectly ejecting one compaison is as seious as incoectly ejecting a numbe of compaisons, EER contol is moe impotant. When one eoneous conclusion will not affect othe infeences in an expeiment, CER contol is moe impotant. 4. Diffeent multiple compaison pocedues have been developed based on diffeent philosophies egading contol of these two kinds of eo.

3 Computing EER So you set CER = α what is EER? The EER is difficult to compute because, fo a given set of data, Type I eos ae not independent. But it is possible to compute an uppe bound fo the EER by assuming that the pobability of a Type I eo fo any single compaison is α and is independent of all othe compaisons. In that case: Uppe bound EER = 1 - (1 - α) p whee p = t( t 1), as befoe Example: Fo 10 teatments and α = 0.05: t( t 1) 10(10 1) p = = = 45 Uppe bound EER = 1 (1 0.05) 45 = 0.90 This fomula may also be used to detemine a value fo α fo some fixed maximum EER. 0.1 = 1 (1 α) 45 (1 α) 45 = 0.9 (1 α) = 0.9 (1/45) α =

4 Patial null hypothesis Suppose thee ae 10 teatments, one of which shows a significant effect while the othe 9 ae appoximately equal: x Y i. x x x x x x x x x Y Teatment numbe ANOVA will pobably eject H 0. Even though one mean is tuly diffeent, thee is still a chance of making a Type I eo in each paiwise compaison among the 9 simila teatments. An uppe bound the EER is computed by setting t = 9 in the above fomula: t( t 1) 9(9 1) p = = = 36 Uppe bound EER = 1 (1 0.05) 36 = 0.84 Intepetation: The expeimente will incoectly conclude that two tuly simila effects ae diffeent 84% of the time. This is called the expeiment-wise eo ate unde a patial null hypothesis. Some teminology: CER = compaison-wise eo ate EERC = expeiment-wise eo ate unde a complete null hypothesis (standad EER) EERP = expeiment-wise eo ate unde a patial null hypothesis MEER = maximum expeiment-wise eo ate unde any complete o patial null hypothesis. 4

5 Multiple compaisons tests Statistical methods fo making two o moe infeences while contolling cumulative Type I eo ates ae called simultaneous infeence methods: 1. Fixed-ange tests: Those which povide confidence intevals and tests of hypotheses. Multiple-ange tests: Those which povide only tests of hypotheses Equal eplications. Results (mg shoot dy weight) of an expeiment (CRD) to detemine the effect of seed teatment by diffeent acids on the ealy gowth of ice seedlings. Teatment Replications Mean Contol HCl Popionic Butyic t = 4, = 5, oveall mean = 3.86 Souce df SS MS F Total Teatment Eo Unequal eplications. Results (lbs/animal day) of an expeiment (CRD) to detemine the effect of diffeent foage genotypes on animal weight gain. Teatment Replications (Animals) Mean Contol Foage-A Foage-B Foage-C t = 4, = vaiable, oveall mean = Souce df SS MS F Total Teatment Eo

6 Fixed-ange tests These tests povide a single ange fo testing all diffeences in balanced designs and can povide confidence intevals. LSD à Dunnett à Tukey à Scheffe Less consevative à Moe consevative Moe likely to declae diffeences à Less likely to declae diffeences Highe Type I eo ates à Lowe Type I eo ates Highe powe à Lowe powe Least significant diffeence (LSD), the epeated t test One of the oldest, simplest, and most widely misused multiple paiwise compaison tests. The LSD test declaes the diffeence between means significant when: Y i Y i and Y j > LSD, whee 1 1 LSD = t + α fo unequal, df 1 LSD = tα fo equal, df Y j of teatments T i and T j to be Seed teatment data: = and df = 16. LSD = t α, df = = So, if Y i Y j > 0.143, they ae declaed significantly diffeent. Contol 4.19 HCl 3.87 Popionic 3.73 Butyic

7 Teatment Mean LSD Contol 4.19 a HCl 3.87 b Popionic 3.73 c Butyic 3.64 c All acids educed shoot gowth. The eduction was moe sevee with butyic and popionic acid than with HC1. We do not have evidence to conclude that popionic acid is diffeent in its effect than butyic acid. When teatments ae equally eplicated, only one LSD value is equied to test all possible compaisons. Foage data: = and df =. In cases of unequal eplication, diffeent LSD values must be calculated fo each compaison involving diffeent numbes of eplications. The 5% LSD fo compaing the contol with Feed B: LSD = t α, df 1 Cont 1 + =.074 B = A vs. Contol = A vs. B = A vs. C = B vs. C = C vs. Contol = Teatment Mean LSD Feed B 1.45 a Feed A 1.36 b Feed C 1.33 b Contol 1.0 c At the 5% level, we conclude all feeds cause significantly geate weight gain than the contol. Feed B causes the highest weight gain; Feeds A and C ae equally effective. 7

8 Confidence intevals The (1 α) confidence limits of the quantity (µ A - µ B ) ae given by: (1 α) CI fo (µ A - µ B ) = ( YA Y ) ± LSD B Geneal consideations fo LSD The LSD test is much safe when the means to be compaed ae selected in advance of the expeiment (i.e. befoe looking at the data). The LSD test is the only test fo which CER equals α. This is often egaded as too libeal. It has been suggested that the EEER can be maintained at α by pefoming the oveall ANOVA test at the α level and making futhe compaisons if and only if the F test is significant (Fishe's Potected LSD test). Howeve, it was then demonstated that this assetion is false if thee ae moe than thee means: A peliminay F test contols only the EERC, not the EERP. Bonfeoni to the escue... Again conside the case of 5 teatments and thus 5*4/ = 10 paiwise compaisons (i.e. hypotheses): α = 0.05 α Bon = 0.05/10 = Uppe bound EER = 1 ( ) 10 =

9 Dunnett's Test Paiwise compaison of a contol to all othe teatment means, while holding MEER α. This test uses the t* statistic (Table A-9b), a modified t statistic based on the numbe of compaisons to be made (p = numbe of teatment means, excluding the contol). DLSD = t * 1 α, p, df fo unequal ( 0 i ) * DLSD = t fo equal ( α 0 = i ) p, df, Seed teatment data: = , df = 16, and p = 3. DLSD = t * α, p, df = = (DLSD = > LSD = 0.143) The smallest diffeence between the contol and any acid teatment is: Contol - HC1 = = 0.3 > All othe diffeences, being lage, ae also significant. The 95% simultaneous confidence intevals fo all thee diffeences take the fom: (1 α) CI fo (µ 0 - µ i ) = ( Y0 Yi ) ± DLSD Contol Butyic = 0.3 ± 0.15 Contol HC1 = 0.46 ± 0.15 Contol Popionic = 0.55 ± 0.15 We have 95% confidence that the 3 tue diffeences fall simultaneously within the above anges. 9

10 Animal foage data: = 0.004, df =, and p = 3. When teatments ae not equally eplicated, thee ae diffeent DLSD values fo each of the compaisons. The 5% DSLS to compae the contol with Feed-C: DLSD = t * α, p, df = = Since Y0 YC = 0.15 > , the diffeence is significant. All othe diffeences with the contol, being lage than this, ae also significant. 10

11 Tukey's w pocedue All possible paiwise compaisons, while holding MEER α. Sometimes called the "honestly significant diffeence" (HSD) test, Tukey's contols the MEER when the sample sizes ae equal. Instead of t o t*, it uses the statistic q α, p, df that is obtained fom Table A-8: YMAX YMIN qα, p, df = s The citical diffeence in this method is labeled w: 1 1 w = q + α, p, df fo unequal 1 w = q fo equal α,p,df We do not multiply by a facto of because Table A-8 (class website) aleady includes the facto in its values: Fo p =, df =, and α= 5%, the citical value is.77 = 1.96 * Y Tukey citical values ae lage than those of Dunnett because the Tukey family of contasts is lage (all pais of means). Seed teatment data: = , df = 16, and p = 4. w = q α , p, df = 4.05 = (w = > DLSD = > LSD = 0.143) Teatment Mean w Contol 4.19 a HCl 3.87 b Popionic 3.73 b c Butyic 3.64 c 11

12 Animal foage data: = 0.004, df =, and p = 4. The 5% w fo the contast between the Contol and Feed-C: w = q α ,, 3.93 = p df + = + Cont C 6 7 Since Y Y = 0.15 > , it is significant. As in the LSD, the only paiwise Cont C compaison that is not significant is that between Feed C ( Y C = ) and Feed A ( Y =1.361). A Scheffe's F test Compatible with the oveall ANOVA F test: Scheffe's neve declaes a contast significant if the oveall F test is nonsignificant. Scheffe's test contols the MEER fo ANY set of contasts. This includes all possible paiwise and goup compaisons. Since this pocedue allows a lage numbe of compaisons, it is less sensitive than othe multiple compaison pocedues. Fo paiwise compaisons, the Scheffe citical diffeence (SCD) has a simila stuctue as that descibed fo pevious tests: SCD = 1 1 df F +,, fo unequal Tt α df Tt df 1 SCD = dftt F df Tt df fo equal α,, Seed teatment data: = , df Tt = 3, df = 16: SCD = df 3(3.4) Tt Fα, df Tt, df = = (SCD = > w = > DLSD = > LSD = 0.143) 1

13 The table of means sepaations: Teatment Mean F s Contol 4.19 a HCl 3.87 b Popionic 3.73 b c Butyic 3.64 c Animal foage data: = 0.004, df Tt = 3, df =. The 5% SCD fo the contast between the Contol and Feed-C: SCD = df = (3.05) F,, = Tt α df Tt df Since Y Y = 0.15 > , it is significant. Cont C Scheffe's pocedue is also eadily used fo inteval estimation: (1 α) CI fo (µ 0 - µ i ) = ( Y0 Yi ) ± SCD The esulting intevals ae simultaneous in that the pobability is at least (1 α) that all of them ae tue simultaneously. Scheffe's F test fo goup compaisons The most impotant use of Scheffe's test is fo abitay compaisons among goups of means. To make compaisons among goups of means, you fist define a contast, as in Topic 4: Q = t i= 1 t c i Y i, with the constaint that c i = 0 (o ic i = 0fo unequal ) i= 1 We eject the null hypothesis (H 0 ) that the contast Q = 0 if the absolute value of Q is lage than some citical value F S : t i= 1 13

14 Citical value F S = df Tt F α, df Tt, df t i= 1 c i i (The pevious paiwise expessions ae fo the paticula contast 1 vs. -1.) Example: If we wish to compae the contol to the aveage of the thee acid teatments, the contast coefficients ae (+3, 1, 1, 1). In this case: Q = c i Y i t i= 1 = 4.190(3) ( 1) ( 1) ( 1) = The citical F s value fo this contast is: F S c = dftt F 3 3(3.4) ( 1) + ( 1) 5 + ( 1) t i α, df, df = = Tt i= 1 i Since Q = > = F s, we eject H 0. The aveage of the contol (4.190 mg) is significantly lage than the aveage of the thee acid teatments (3.745 mg). 14

15 Multiple-stage tests (MSTs) / Multiple-ange tests Allow simultaneous hypothesis tests of geate powe by fofeiting the ability to constuct simultaneous confidence intevals. Duncan à Student-Newman-Keuls (SNK) à REGWQ All thee use the Studentized ange statistic (qα), and all thee ae esult-guided. With means aanged in ode, an MST povides citical distances o anges that become smalle as the paiwise means to be compaed become close togethe in the aay. Such a stategy allows the eseache to allocate test sensitivity whee it is most needed, in disciminating neighboing means. The geneal stategy: µ 1 α 4 µ α 4 α 3 α α 1 α α 3 α 3 µ 3 µ 4 µ 5 α 4 α 4 α 1 < α < α 3 < < α t-1 "Confidence" is eplaced by the concept of "potection levels" So if a diffeence is detected at one level of the test, the eseache is justified in sepaating means at a fine esolution with less potection (i.e. with a highe α). 15

16 Duncan's multiple ange test As the test pogesses, Duncan's method uses a vaiable significance level (α p-1 ) depending on the numbe of means involved: α p-1 = 1 - (1 - α) p-1 Despite the level of potection offeed at each stage, MEER is uncontolled. The highe powe of Duncan's method compaed to Tukey's is due to its highe Type I eo ate. Duncan citical anges (R p ): R = q α p p, df p 1, Fo the seed teatment data: p 3 4 α p 1, p, R p q Identical to LSD fo adjacent means (LSD = 0.14). Duncan's used to be the most popula method of means sepaation, but many jounals no longe accept it. It is not ecommended. The Student-Newman-Keuls (SNK) test As the test pogesses, SNK uses a fixed significance level (α), which is always less than o equal to Duncan's vaiable significance level: α SNK = α 1 - (1 - α) p-1 Moe consevative than Duncan's, holding EERC α. Accepted by some jounals that eject Duncan's. Poo behavio in tems of EERP and MEER. Not ecommended. 16

17 Assume the following patial null hypothesis: µ 1 µ µ 3 µ 4 µ 5 µ 6 µ 7 µ 8 µ 9 µ 10 The SNK method educes to five independent tests, one fo each pai, by LSD. The pobability of at least one false ejection is: 1 (1 α) 5 = 0.3 As the numbe of means inceases, MEER à 1. To find the SNK citical ange (W p ) at each level of the analysis: W = q p α, p, df Fo the seed teatment data: p 3 4 q 0.05, p, R p Again, identical to LSD fo adjacent means (LSD = 0.14). The Ryan, Einot, Gabiel, and Welsh (REGWQ) method Not as well known as the othes, REGWQ method appeas to be among the most poweful step-down multiple ange tests and is ecommended by some softwae packages (e.g. SAS) fo equal eplication (i.e. balanced designs). Contols MEER by setting: α p-1 = 1 - (1 - α) p/t fo p < (t 1) and α p-1 = α fo p (t 1) 17

18 Assuming the sample means have been aanged in descending ode fom Y 1 to Y t, the homogeneity of means Y,..., Y, with i < j, is ejected by REGWQ if: i j Y Yj > qα p i 1, p, df Fo the seed teatment data: p 3 4 α p q 0.05, p, R p >SNK =SNK =SNK <Tukey =Tukey Tukey w = The diffeence between the HCl and Popionic teatments is declaed significant with SNK but not with REGWQ ( < 0.145). Teatment Mean REGWQ Contol 4.19 a HCl 3.87 b Popionic 3.73 b c Butyic 3.64 c Some suggested ules of thumb: 1. When in doubt, use Tukey.. Use Dunnett's (moe poweful than Tukey's) if you only wish to compae each teatment level to a contol. 3. Use Scheffe's if you wish to "mine" you data. 18

19 One final point to note is that seveely unbalanced designs can yield vey stange esults: Teatment Data Mean A B C D * NS Data fom ST&D page

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