A Bluffer Guide to Sphericity Andy Field Univerity of Suex The ue of repeated meaure, where the ame ubject are teted under a number of condition, ha numerou practical and tatitical benefit. For one thing it reduce the error variance caued by between-group individual difference, however, thi reduction of error come at a price becaue repeated meaure deign potentially introduce covariation between experimental condition (thi i becaue the ame people are ued in each condition and o there i likely to be ome conitency in their behaviour acro condition). In between-group ANOVA we have to aume that the group we tet are independent for the tet to be accurate (Scariano & Davenport, 1987, have documented ome of the conequence of violating thi aumption). A uch, the relationhip between treatment in a repeated meaure deign create problem with the accuracy of the tet tatitic. The purpoe of thi article i to explain, a imply a poible, the iue that arie in analying repeated meaure data with ANOVA: pecifically, what i phericity and why i it important? What i Sphericity? Mot of u are taught during our degree that it i crucial to have homogeneity of variance between condition when analying data from different ubject, but often we are left to aume that thi problem goe away in repeated meaure deign. Thi i not o, and the aumption of phericity can be likened to the aumption of homogeneity of variance in between-group ANOVA. Sphericity (denoted by ε and ometime referred to a circularity) i a more general condition of compound ymmetry. Imagine you had a population covariance matrix Σ, where: BPS-MSC Newletter 6 (1) Page 13
11 1 Σ = 31 n1 1 3 n 13 3 33 n3 1n n 3n nn Equation 1 Thi matrix repreent two thing: (1) the off-diagonal element repreent the covariance between the treatment 1 n (you can think of thi a the untandardied correlation between each of the repeated meaure condition); and () the diagonal element ignify the variance within each treatment. A uch, the aumption of homogeneity of variance between treatment will hold when: 11 33 nn Equation (i.e.. when the diagonal component of the matrix are approximately equal). Thi i comparable to the ituation we would expect in a between-group deign. However, in repeated meaure deign there i the added complication that the experimental condition covary with each other. The end reult i that we have to conider the effect of thee covariance when we analye the data, and pecifically we need to aume that all of the covariance are approximately equal (i.e. all of the condition are related to each other to the ame degree and o the effect of participating in one treatment level after another i alo equal). Compound Symmetry hold when there i a pattern of contant variance along the diagonal (i.e. homogeneity of variance ee Equation ) and contant covariance off of the diagonal (i.e. the covariance between treatment are equal ee Equation 3). While compound ymmetry ha been hown to be a ufficient condition for conduction ANOVA on repeated meaure data, it i not a neceary condition. 1 13 3 1 3 Equation 3 Sphericity i a le retrictive form of compound ymmetry (in fact much of the early reearch into repeated meaure ANOVA confued compound ymmetry with phericity). Sphericity n n n BPS-MSC Newletter 6 (1) Page 14
refer to the equality of variance of the difference between treatment level. Wherea compound ymmetry concern the covariation between treatment, phericity i related to the variance of the difference between treatment. So, if you were to take each pair of treatment level, and calculate the difference between each pair of core, then it i neceary that thee difference have equal variance. Imagine a ituation where there are 4 level of a repeated meaure treatment (A, B, C, D). For phericity to hold, one condition mut be atified: A B A C A D B C B D C D Equation 4 Sphericity i violated when the condition in Equation 4 i not met (i.e. the difference between pair of condition have unequal variance). How i Sphericity Meaured? The implet way to ee whether or not the aumption of phericity ha been met i to calculate the difference between pair of core in all combination of the treatment level. Once thi ha been done, you can imply calculate the variance of thee difference. E.g. Table 1 how data from an experiment with 3 condition (for implicity there are only 5 core per condition). The difference between pair of condition can then be calculated for each ubject. The variance for each et of difference can then be calculated. We aw above that phericity i met when thee variance are roughly equal. For thi data, phericity will hold when: A B A C B C Where: A B A C B C = 15.7 = 10.3 = 10.3 A uch, A B A C = B C BPS-MSC Newletter 6 (1) Page 15
Condition A Condition B Condition C A-B A-C B-C 10 1 8-5 15 15 1 0 3 3 5 30 0-5 5 10 35 30 8 5 7 30 7 0 3 10 7 Variance: 15.7 10.3 10.3 Table 1: Hypothetical data to illutrate the calculation of the variance of the difference between condition. So there i, at leat ome deviation from phericity becaue the variance of the difference between condition A and B i greater than the variance of the difference between condition A and C, and between B and C. However, we can ay that thi data ha local circularity (or local phericity) becaue two of the variance are identical (). Thi mean that for any multiple comparion involving thee difference, the phericity aumption ha been met (for a dicuion of local circularity ee Rouanet and Lépine, 1970). The deviation from phericity in the data in Table 1 doe not eem too evere (all variance are roughly equal). Thi raie the iue of how we ae whether violation from phericity are evere enough to warrant action. Aeing the Severity of Departure from Sphericity Luckily the advancement of computer package make it needle to ponder the detail of how to ae departure from phericity. SPSS produce a tet known a Mauchley tet, which tet the hypothei that the variance of the difference between condition are equal. Therefore, if Mauchley tet tatitic i ignificant (i.e. ha a probability value le than 0.05) we mut conclude that there are ignificant difference between the variance of difference, ergo the condition of phericity ha not been met. If, however, Mauchley tet tatitic i nonignificant (i.e. ) then it i reaonable to conclude that the variance of difference are not ignificantly different (i.e. they are roughly equal). So, in hort, if Mauchley tet i ignificant then we mut be wary of the F-ratio produced by the computer. BPS-MSC Newletter 6 (1) Page 16
Figure 1: Output of Mauchley tet from SPSS verion 7.0 Figure 1 how the reult of Mauchley tet on ome fictitiou data with three condition (A, B and C). The reult of the tet i highly ignificant indicating that the variance between the difference were ignificantly different. The table alo diplay the degree of freedom (the df are imply N 1, where N i the number of variance compared) and three etimate of phericity (ee ection on correcting for phericity). What i the Effect of Violating the Aumption of Sphericity? Rouanet and Lépine (1970) provided a detailed account of the validity of the F-ratio when the phericity aumption doe not hold. They argued that there are two different F-ratio that can be ued to ae treatment comparion. The two type of F-ratio were labelled F and F repectively. F refer to an F-ratio derived from the mean quare of the comparion in quetion and the interaction of the ubject with that comparion (i.e. the pecific error term for each comparion i ued thi i the F-ratio normally ued). F i derived not from the pecific error mean quare but from the total error mean quare for all of the repeated meaure comparion. Rouanet and Lépine (1970) argued that F i le powerful than F and o it may be the cae that thi tet tatitic mie genuine effect. In addition, they howed that for F to be valid the covariation matrix, Σ, mut obey local circularity (i.e. phericity mut hold for the pecific comparion in quetion) and Mendoza, Toothaker & Crain (1976) have upported thi by demontrating that the F ratio of an L J K factorial deign with two repeated meaure are valid only if local circularity hold. F" require only overall circularity (i.e. the whole data et mut be circular) but becaue of the non-reciprocal nature of circularity and compound ymmetry, F doe not require compound ymmetry whilt F' doe. So, given that F i the BPS-MSC Newletter 6 (1) Page 17
tatitic generally ued, the effect of violating phericity i a lo of power (compared to when F i ued) and an tet tatitic (F-ratio) which imply cannot be validity compared to tabulated value of the F-ditribution. Correcting for Violation of Sphericity If data violate the phericity aumption there are a number of correction that can be applied to produce a valid F-ratio. SPSS produce three correction baed upon the etimate of phericity advocated by Greenhoue and Geier (1958) and Huynh and Feldt (1976). Both of thee etimate give rie to a correction factor that i applied to the degree of freedom ued to ae the oberved value of F. How each etimate i calculated i beyond the cope of thi article, for our purpoe all we need know i that each etimate differ lightly from the other. The Greenhoue-Geier etimate (uually denoted a εˆ ) varie between (where k i the number of repeated meaure condition) and 1. The cloer that εˆ i to 1.00, the more homogeneou are the variance of difference, and hence the cloer the data are to being pherical. Figure 1 how a ituation with three condition and hence the lower limit of εˆ i 0.5, it i clear that the calculated value of εˆ i 0.503 which i very cloe to 0.5 and o repreent a ubtantial deviation from phericity. Huynh and Feldt (1976) reported that when εˆ > 0.75 too many fale null hypothee fail to be rejected (i.e. the tet i too conervative) and Collier, Baker, Mandeville & Haye (1967) howed that thi wa alo true with εˆ a high a 0.90. Huynh and Feldt, therefore, propoed a correction to εˆ to make it le conervative (uually denoted a ε ~ ). However, Maxwell and Delaney (1990) report that ε ~ actually overetimate phericity. Steven (199) therefore recommend taking an average of the two and adjuting the df by thi averaged value. Girden (199) recommend that when εˆ > 0.75 then the df hould be corrected uing ε ~ ; If εˆ < 0.75, or nothing i known about phericity at all, then the conervative εˆ hould be ued to adjut the df. BPS-MSC Newletter 6 (1) Page 18
Figure : Output of epilon corrected F value from SPSS verion 7.0. Figure how a typical ANOVA table for a et of data that violated phericity (the ame data ued to generate Figure 1). The table in Figure how the F ratio and aociated degree of freedom when phericity i aumed, a can be een, thi reult in a ignificant F tatitic indicating ome difference() between the mean of the three condition. Underneath are the corrected value (for each of the three etimate of phericity). Notice that in all cae the F ratio remain the ame, it i the degree of freedom that change (and hence the critical value of F). The degree of freedom are corrected by the etimate of phericity. How thi i done can be een in Table. The new degree of freedom are then ued to acertain the critical value of F. For thi data thi reult in the oberved F being nonignificant at p < 0.05. Thi particular data et illutrate how important it i to ue a valid critical value of F, it can mean the difference between a tatitically ignificant reult and a nonignificant reult. More importantly, it can mean the difference between making a Type I error and not. Etimate of Sphericity Ued Value of Etimate Term df Correction New df None Effect Error 8 0.503 Effect 0.503 1.006 Error 8 0.503 8 4.04 0.506 Effect 0.506 1.01 Error 8 0.506 8 4.048 Table : Show how the phericity correction are applied to the degree of freedom. BPS-MSC Newletter 6 (1) Page 19
Multivariate v. Univariate Tet A final option, when you have data that violate phericity, i to ue multivariate tet tatitic (MANOVA) becaue they are not dependent upon the aumption of phericity (ee O Brien & Kaier, 1985). There i a trade off of tet power between univariate and multivariate approache although ome author argue that thi can be overcome with uitable matery of the technique (O Brien and Kaier, 1985). MANOVA avoid the aumption of phericity (and all the correponding conideration about appropriate F ratio and correction) by uing a pecific error term for contrat with 1 df and hence, each contrat i only ever aociated with it pecific error term (rather than the pooled error term ued in ANOVA). Davidon (197) compared the power of adjuted univariate technique with thoe of Hotelling T (a MANOVA tet tatitic) and found that the univariate technique wa relatively powerle to detect mall reliable change between highly correlated condition when other le correlated condition were alo preent. Mendoza, Toothaker and Nicewander (1974) conducted a Monte Carlo tudy comparing univariate and multivariate technique under violation of compound ymmetry and normality and found that a the degree of violation of compound ymmetry increaed, the empirical power for the multivariate tet alo increaed. In contrat, the power for the univariate tet generally decreaed (p 174). Maxwell and Delaney (1990) noted that the univariate tet i relatively more powerful than the multivariate tet a n decreae and propoed that the multivariate approach hould probably not be ued if n i le than a + 10 (a i the number of level for repeated meaure) (p 60). A a general rule it eem that when you have a large violation of phericity (ε < 0.7) and your ample ize i greater than (a + 10) then multivariate procedure are more powerful whilt with mall ample ize or when phericity hold (ε > 0.7) the univariate approach i preferred (Steven, 199). It i alo worth noting that the power of MANOVA increae and decreae a a function of the correlation between dependent variable (Cole et al, 1994) and o the relationhip between treatment condition mut be conidered alo. Multiple Comparion So far, I have dicued the effect of phericity on the omnibu ANOVA. A a final flurry ome dicuion of the effect on multiple comparion procedure i warranted. Boik (1981) provided an etimable account of the effect of nonphericity on a priori tet in repeated meaure deign, and concluded that even very mall departure from phericity produce large biae in the F-tet and recommend againt uing thee tet for repeated meaure contrat. BPS-MSC Newletter 6 (1) Page 0
When experimental error term are mall, the power to detect relatively trong effect can be a low a.05 (when phericity =.80). He argue that the ituation for a priori comparion cannot be improved and conclude by recommending a multivariate analogue. Mitzel and Game (1981) found that when phericity doe not hold (ε < 1) the pooled error term conventionally employed in pairwie comparion reulted in nonignificant difference between two mean declared ignificant (i.e. a lenient Type 1 error rate) or undetected difference (a conervative Type 1 error rate). They therefore recommended the ue of eparate error term for each comparion. Maxwell (1980) ytematically teted the power and alpha level for 5 a priori tet under repeated meaure condition. The tet aeed were Tukey Wholly Significant Difference (WSD) tet which ue a pooled error term, Tukey procedure but with a eparate error term with either ( 1 n ) df [labelled SEP1] or ( 1)( k 1) n df [labelled SEP], Bonferroni procedure (BON), and a multivariate approach the Roy-Boe Simultaneou Confidence Interval (SCI). Maxwell teted thee a priori procedure varying the ample ize, number of level of the repeated factor and departure from phericity. He found that the multivariate approach wa alway "too conervative for practical ue" (p 77) and thi wa mot extreme when n (the number of ubject) i mall relative to k (the number of condition). Tukey tet inflated the alpha rate a the covariance matrix depart from phericity and even when a eparate error term wa ued (SEP1) alpha wa lightly inflated a k increaed whilt SEP alo lead to unacceptably high alpha level. The Bonferroni method, however, wa extremely robut (although lightly conervative) and controlled alpha level regardle of the manipulation. Therefore, in term of Type I error rate the Bonferroni method wa bet. In term of tet power (the Type II error rate) for a mall ample (n = 8) WSD wa the mot powerful under condition of nonphericity. Thi advantage wa everely reduced when n = 15. Keelman and Keelman (1988) extended Maxwell work and alo invetigated unbalanced deign. They too ued Tukey WSD, a modified WSD (with non-pooled error variance), Bonferroni t- tatitic, and a multivariate approach, and looked at the ame factor a Maxwell (with the addition of unequal ample). They found that when unweighted mean were ued (with unbalanced deign) none of the four tet could control the Type 1 error rate. When weighted mean were ued only the multivariate tet could limit alpha rate although Bonferroni t tatitic were coniderably better than the two Tukey method. In term of power they concluded that a the number of repeated treatment level increae, BON i ubtantially more powerful than SCI (p 3). BPS-MSC Newletter 6 (1) Page 1
So, in term of thee tudie, the Bonferroni method eem to be generally the mot robut of the univariate technique, epecially in term of power and control of the Type I error rate. Concluion It i more often the rule than the exception that phericity i violated in repeated meaure deign. For thi reaon, all repeated meaure deign hould be expoed to tet of violation of phericity. If phericity i violated then the reearcher mut decide whether a multivariate or univariate analyi i preferred (with due conideration to the trade off between tet validity on one hand and power on the other). If univariate method are choen then the omnibu ANOVA mut be corrected appropriately depending on the level of departure from phericity. Finally, if pairwie comparion are required the Bonferroni method hould probably be ued to control the Type 1 error rate. Finally, to enure that the group ize are equal otherwie even the Bonferroni technique i ubject to inflation of alpha level. BPS-MSC Newletter 6 (1) Page
Reference Boik, R. J. (1981). A Priori tet in repeated meaure deign: effect of nonphericity, Pychometrika, 46 (3), 41-55. Cole, D. A., Maxwell, S. E., Arvey, R., & Sala, E. (1994). How he power of MANOVA can both increae and decreae a a function of the intercorrelation among the dependent variable, Pychological Bulletin, 115 (3), 465-474. Girden, E. R. (199). ANOVA: Repeated Meaure (Sage univerity paper erie on qualitative application in the ocial cience, 84), Newbury Park, CA: Sage. Greenhoue, S. W., & Geier, S. (1959). On method in the analyi of profile data. Pychometrika, 4, 95 11. Huynh, H., and Feldt, L. S. (1976). Etimation of the Box correction for degree of freedom from ample data in randomied block and plit-plot deign. Journal of Educational Statitic, 1 (1), 69-8. Keelman, H. J. & Keelman, J. C. (1988). Repeated meaure multiple comparion procedure: Effect of violating multiample phericity in unbalanced deign. Journal of educational Statitic, 13 (3), 15-6. Maxwell, S. E. (1980). Pairwie multiple comparion in repeated meaure deign. Journal of Educational Statitic, 5 (3), 69-87. Maxwell, S. E. & Delaney (1990). Deigning experiment and analyzing data. Belmont, CA: Wadworth. Mendoza, J. L., Toothaker, L. E. & Nicewander, W. A. (1974). A Monte Carlo comparion of the univariate and multivariate method for the group by trial repeated meaure deign. Multivariate Behavioural Reearch, 9, 165-177. Mendoza, J. L., Toothaker, L. E. & Crain, B. R. (1976). Neceary and ufficient condition for F Ratio in the L x J x K Factorial deign with two repeated factor. Journal of the American Statitical Aociation, 71, 99-993. Mitzel, H. C., & Game, P. A. (1981). Circularity and multiple comparion in repeated meaure deign, Britih Journal of Mathematical and Statitical Pychology, 34, 53-59. O Brien, M. G., & Kaier, M. K. (1985). MANOVA method for analyzing repeated meaure deign: An extenive primer, Pychological Bulletin, 97 (), 316-333. BPS-MSC Newletter 6 (1) Page 3
Rouanet, H. & Lépine, D. (1970). Comparion between treatment in a repeated-meaurement deign: Anova and multivariate method. The Britih Journal of mathematical and Statitical Pychology, 3, 147-163. Scariano, S. M. & Davenport, J. M. (1987). The effect of violation of independence in the oneway ANOVA. The American Statitician, 41 (), 13 19. Steven, J. (199). Applied multivariate tatitic for the ocial cience (nd edition). Hilldale, NJ: LEA. BPS-MSC Newletter 6 (1) Page 4