CHAPTE 7 Multvaate effect szes ndces Seldom does one fnd that thee s only a sngle dependent vaable nvolved n a study. In Chapte 3 s Example A we have the vaables BDI, POMS_S and POMS_B, n Example E thee ae 5 cholesteol measuements as well as systolc and dastolc blood pessue measuements. Many tmes one wll analyse and detemne effect szes ndces fo the vaables sepaately. Howeve, t s also sensble to consde the vaables as a sngle unt whee not only the dffeent means and SDs play a ole, but also the nte-coelatons between the vaables. Based on these analyses, called vaate analyses, one can also obtan effect szes ndces. In Fgue 7. we epoduce Olejnk & Algna (000) s Fgue and use t to explan how a vaate standadzed dffeence n means can be ntepeted. Ths epesentaton s a scatte-plot of the two vaables x and x n samples dawn fom populatons A, B, C and D. The ponts shown on the dagam each
epesent one of the goup mean s ( x,x ), whle the ellpses ae dawn n such a way that most of that goup s measuements (say 90%) fall wthn t. In goups A and B we fnd a postve coelaton between x and x, whle t s negatve n goups C and D. Note that the goups A and B ovelap, whle thee s almost no ovelappng n goups C and D. The fact that the ellpses ae almost the same sze suggests that the SDs ae equal fo all the goups fo x and also fo x. If one wants to compaes the means x and x of goups A and B, then the Mahalanobs-dstance D can be used, snce t s an analogue of the unvaate standadzed dffeence. It s defned as: p ( δˆ δˆ δδ ˆ ˆ p ) D = + (7.) whee ˆδ and ˆδ ae the standadzed dffeences accodng to (4.3), and ae defned as: ˆδ x x x x =, A B A B = and sp s p ˆδ whee x A and x B ae the means of x fo goups A and B and s p s the common SD fo x ove goups A and B, etc. Futhe, p s the pooled wthn-goup coelaton : p sp =, s s p p whee S p s the pooled covaance between x and x. Suppose that, n the example of the data n Fgue 7., the pooled SD fo x s 8 and fo x t s 3. Futhe, let the p fo goups A and B be 0,5 and fo C and D let t be equal to -0,5. Fom the fgue t s clea that f we want to compae goups A and B then we calculate the followng: ( ) ˆδ = 30-0 / 8 =, 5 and
ˆδ = ( 30-0) / 3 =,67, whle ( ) D =,5 +,67-0,5, 5,67 =, 64/0,75-0,5 Theefoe, D =, 74. = 3,09, If we want to compae goups C and D, then ˆδ and ˆδ ae the same as befoe but, because p = 0, 5, we fnd that D =,93. The followng thee ponts ae mpotant to note: Fst, the value of D s lage than the values of ethe of ˆδ and ˆδ because t epesents the Eucldean dstance between the -dmensonal means on a scale whch s detemned by both the SDs of, and the coelaton between, the vaables x and x. The ˆδ value, on the othe hand, s only the dffeence between the one-dmensonal means n the same unts as the SD. Second, thee s a lage sepaaton between the ponts n goups C and D than n goups A and B, even though the mean ponts ae equally fa fom one anothe. Ths sepaaton can be ascbed to the fact that A and B have a postve coelaton between x and x, whle the coelaton between the vaables fo goups C and D s negatve. The values of D eflect ths occuence, because, fo A and B, the value δδ ˆ ˆ p s subtacted fom the postve value δ ˆ + δˆ, whle fo C and D t s added (because p < 0 ). Thus whle D =, 74 fo A and B, t s substantally hghe,.e.,,93, fo C and D. Thd, n addton to the sgn, the magntude of p also plays a ole. 3
In the followng paagaph the Mahalanobs-dstance D s genealzed to moe than vaables. Late n the chapte we look at detemnng omnbus effects as well as contast effects fo the case wth moe than goups. 7. Compang two goups wth m vaables In the ntoducton we llustated the case whee one compaes the two-vaable means of two populatons. Suppose now that two m -vaable populatons ae completely obsevable and that δ ( δ, δ,..., δ m ) = (7.) epesents the vecto of effect sze ndces fo the vaables x, x,..., x m fo both populatons. Futhe, let ρ ρ3 ρ m ρ = (7.3) ρm denote the common coelaton matx consstng of the nte-coelatons between x, x,..., x m fo both populatons. Fo compang the m -vaable means between two populatons the genealzed Mahalanobs-dstance fo populatons can be used, namely D = δ δ '. (7.4) Ths nvolves matx calculatons whch ae dffcult to do by hand. An ease fomula fo calculatng ths quantty s: D = ( N + N ) ( Λ) A B N N A B Λ, (7.5) 4
whee N A and N B ae the populaton szes. The quantty Λ s known as Wlks lambda, t s calculated n a vaate analyss of vaance (MANOVA) whee two mean vectos ae compaed and s povded n the output of any statstcal compute package capable of MANOVA such as SPSS, SAS o STATISTICA. In the case whee andom samples of sze n A and n B ae dawn fom two populatons, D can be estmated by D ɶ, defned as ( n )( ˆ A nb ) ɶ A B. (7.6) D = + Λ + n n Λˆ Hee ˆΛ s based on the sample data. Unfotunately, ths estmato s actually based. The estmato ˆD s consdeed pefeable snce t s appoxmately unbased. It s defned as ˆ Dˆ Λ = ( na + nb m 3) m ˆ +. (7.7) Λ n A n B Note that f defnton. ˆD s negatve, then the value s changed to 0, seeng as D 0 pe Note also that the value of Uˆ ˆ ( ) Λ = (whch s called the Hotellng-statstc) s Λ ˆ usually epoted along wth Wlk s lambda ( ˆΛ ) n most of the above mentoned compute packages. Futhe, the so called Hotellng s expesson ( ) T na nb = + Λˆ Λˆ so that D, D ɶ and ˆD can also be calculated usng, ( ) U, T s obtaned usng the ˆ ( ) U o T. 5
An appoxmate ( α ) 00% CI fo D can also be calculated. Ths follows snce ˆ c. Λ has an appoxmate non-cental- F dstbuton wth non-centalty Λˆ paamete ncp D = n n A B ( n + n ) topc can be found n Appendx B. A B D and c s a constant. Moe detals egadng ths A SAS-pogam named CI_D s avalable on the web page of ths manual and make use of the followng nputs: ˆΛ, n A, n B, m and α. Example 7. Consde Example A fom Chapte 3. Suppose that we want to compae the expemental and contol goups wth espect to the mean vecto of the BDI befoe tests, afte tests and follow-up tests. The MANOVA output obtaned fom STATISTICA s: Λ ˆ = 0,53 ; F ( 3;46) = 86,0 ( p < 0,000). Wth n = n = 5 and m = 3 ; t follows that A B ˆ Λ Λˆ ˆ ( ) = = 5,6 U and Dɶ = 48 5,6 + = 4,64 5 5 and D ˆ = (44 5,6 3) ( 0,04 + 0,04) = 4, 4, whch s smalle than D ɶ because t s appoxmately unbased. Applyng the pogam CI_D poduces the 95% CI fo D : ( 3,44;5,60 ). Ths means that the Mahalonobs dstance between the two goups 3- dmensonal mean vectos can be as small as 3, 44 and as lage as 5,60 wth a 95% pobablty 6
7.. Gudelne values fo D: Snce the value D s a dstance between mean vectos whch s weghted by the nte-coelatons of x,..., x m, t s measued on a dffeent scale as δ, δ,..., δ m, the ndvdual effect sze ndces fo the m vaables. Only n vey specal ccumstances ae thee elatonshps between D and δ,..., δ m. Fo smplcty, we wll estct ou dscusson to the case nvolvng only vaables whee D s gven by equaton (7.). In the exteme case, whee p = 0, then D smplfes to ˆ δ + ˆ δ, but snce thee s no coelaton between x and x, each one can be consdeed sepaately and a vaate analyss would be edundant. When p tends to, then D tends to nfnty, but n these cases x descbes almost all of the vaance of x so that one only needs to analyse x o x, and not both. In the case whee ˆ δ ˆ = δ and whee p tends to, then D tends to ˆ δ ˆ = δ. Table 7. dsplays, fo selected values of ˆ δ ˆ = δ and p, the values of D. It s clea that: D > ˆ δ = ˆ δ D becomes lage when p becomes smalle The value of D s at a maxmum value when p s close to. Smla to the case whee p s close to, f p s close to -, then only one of x o x should be analysed because almost all of the vaance of the one vaable s descbed by the othe. Table 7. povdes some ndcatons of the possble gudelne values when we know the p values and when ˆ δ and ˆ δ ae almost equal to one anothe, t s not possble n the moe geneal case wth m vaables. 7
δ Table 7.: Values of Mahalanobs- δ = δ dstance f k=, m=, -0.9-0.8-0.7-0.6-0.5-0.4-0.3-0. -0. 0. 0.45 0.3 0.6 0. 0.0 0.8 0.7 0.6 0.5 0. 0.89 0.63 0.5 0.45 0.40 0.37 0.34 0.3 0.30 0.3.34 0.95 0.77 0.67 0.60 0.55 0.5 0.47 0.45 0.4.79.6.03 0.89 0.80 0.73 0.68 0.63 0.60 0.5.4.58.9..00 0.9 0.85 0.79 0.75 0.6.68.90.55.34.0.0.0 0.95 0.89 0.7 3.3..8.57.40.8.8..04 0.8 3.58.53.07.79.60.46.35.6.9 0.9 4.0.85.3.0.80.64.5.4.34 4.47 3.6.58.4.00.83.69.58.49 δ 0.0 0. 0. 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0. 0.4 0.3 0.3 0. 0. 0. 0. 0. 0. 0.0 0. 0.8 0.7 0.6 0.5 0.4 0.3 0. 0. 0. 0. 0.3 0.4 0.40 0.39 0.37 0.36 0.35 0.34 0.33 0.3 0.3 0.4 0.57 0.54 0.5 0.50 0.48 0.46 0.45 0.43 0.4 0.4 0.5 0.7 0.67 0.65 0.6 0.60 0.58 0.56 0.54 0.53 0.5 0.6 0.85 0.8 0.77 0.74 0.7 0.69 0.67 0.65 0.63 0.6 0.7 0.99 0.94 0.90 0.87 0.84 0.8 0.78 0.76 0.74 0.7 0.8.3.08.03 0.99 0.96 0.9 0.89 0.87 0.84 0.8 0.9.7..6..08.04.0 0.98 0.95 0.9.4.35.9.4.0.5..08.05.03 8
7. Effect szes of contast effects fo m vaables and k goups As n Chapte 6, one can compae moe than goups usng contasts. contast now compaes populaton mean vectos wth one anothe and s defned as: whee and ψ = cµ + cµ +... + ckµ k, (7.8) µ = ( µ, µ,..., µ ) m ( ψ ψ ψ ) ψ =,,..., m, whle c, c,, c k ae the contast weghts of the k populatons. A Example 7. In Example E of Chapte 3 the mean vectos of cholesteol-values (chol_0- chol_4) fo the heat patents whch dd lttle o no execse can be compaed to those patents who execsed modeately to fequently wth the contast: whee ψ = 0,5µ + 0,5µ 0,5µ 0,5µ, 3 4 µ s the mean vecto wth the 5 cholesteol-values as ts components, whle ψ s the contast of each cholesteol-value as ts components. The Mahalanobs dstance D ψ can be used as an effect sze ndex fo ψ, whee: D ψ = N k c Λ = N Λ ( ψ ) ψ, (7.9) and whee, as befoe, N s the populaton sze and N = N +... + N, whle Λ ψ s k Wlks lambda fo the contast ψ. 9
When k andom samples ae dawn fom the populatons, an estmato (Klne, 004b : 0) Dɶ ψ s defned as k c Λ n ( n k )( ˆ ψ ) ɶ = ψ =, (7.0) Λˆ ψ D whee n = n +... + nk and Λ ˆ ψ s now based on the sample data. The esults of the MANOVA ae used to test ψ = 0. An appoxmate ( α ) 00% CI fo D ψ s obtaned usng the same method as n paagaph 7.. The SAS-pogam CI_D-contast s avalable on the manual s web page and uses the followng nputs: Λ ˆ, m, n,..., n, c,..., c and α. ψ k k Example 7.3: Contnung wth Example 7. the esults of the MANOVA to test ψ = 0 ae: ˆ 0, 765; ψ Λ = F ( 5; 4),58( p 0, 0404) = =. Futhe, n = 50, k = 4, n = 9, n = 0, n 3 = 9, n 4 = and c = 0,5, c = 0, 5, c 3 = 0,5 and c 4 = 0,5, so that 0,5 0,5 0,5 0,5 ( )( )( 9 0 9 ) 50 4 0, 765 + + + Dɶ ψ = 0,765 thus Dɶ ψ =,588,99 = =,5, 0,765 The 95% CI fo D ψ s then: ( 0;, 48 ) (the lowe bound could not be calculated, and was smply assumed to be equal to zeo). 0
7.3 Multvaate omnbus effect In the unvaate case n Chapte 5, the effect sze ndex η s employed to measue the omnbus effect of dffeences n k populaton means. Ths was the popoton of the dependent vaable s vaance whch could be attbuted to populaton membeshp. Howeve, snce Wlks lambda Λ can be consdeed as the popoton of the genealzed vaance of a vecto of m vaables not attbuted to populaton membeshp (see Klne, 004b: 4), an obvous choce fo effect sze ndex s Wlks genealzed coelaton ato: η = Λ. (7.) v Cohen(988) suggests a genealzaton of the effect sze f used n ple lnea egesson, whch s based on Λ,.e / f = Λ (7.) whee m (k -) - 4 = m + (k -) - 5. (7.3) Notes:. Wth ple lnea egesson, m =, theefoe = and also Λ =, consequently f = = (see Chapte 5, paagaph 5..3).. Table 7. can be used fo eadng off values of fo selected k and m values, whle Table 7.3 dsplays values of f fo gven Λ and values. 3. In contast to ts unvaate countepat, f, f depends upon k and m whch complcates the fxng of gudelnes whethe lage. Table 7.: Values of fo gven k and m f s small, medum o
m 3 4 5 6 7 8 9 0 5 0 3 4.43.65.76.83.87.90.9.94.97.98 5.65 3.06 3.3 3.49 3.6 3.69 3.75 3.79 3.90 3.94 6.76 3.3 3.7 4.00 4. 4.36 4.47 4.56 4.79 4.88 7.83 3.49 4.00 4.39 4.69 4.9 5.0 5.4 5.6 5.78 8.87 3.6 4. 4.69 5.08 5.39 5.63 5.83 6.40 6.64 k 9.90 3.69 4.36 4.9 5.39 5.77 6.08 6.34 7. 7.47 0.9 3.75 4.47 5.0 5.63 6.08 6.46 6.78 7.78 8.5.94 3.79 4.56 5.4 5.83 6.34 6.78 7.6 8.38 8.99.95 3.83 4.63 5.35 5.99 6.56 7.05 7.48 8.93 9.68 3.95 3.85 4.68 5.44 6. 6.74 7.8 7.76 9.43 0.34 4.96 3.87 4.73 5.5 6.3 6.89 7.47 8.00 9.89 0.95 5.97 3.89 4.76 5.57 6.3 7.0 7.64 8. 0.30.5 0.98 3.94 4.87 5.76 6.6 7.4 8.8 8.90.8 3.8 30.99 3.97 4.94 5.89 6.8 7.73 8.6 9.48 3.35 6.50 Table 7.3 : Values of f fo gven Λ and Wlks se Λ 0.99 0.96 0.9 0.9 0.88 0.85 0.8 0.75 0.7 0.6 0.5 0.4 0.3 0. 0. 0.0 0.04 0.08 0. 0. 0.5 0. 0.5 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.0 0.04 0.09 0. 0.4 0.8 0.5 0.33 0.43 0.67.00.50.33 4.00 9.00 0.0 0.0 0.04 0.05 0.07 0.08 0. 0.5 0.0 0.9 0.4 0.58 0.83.4.6.5 0.00 0.0 0.03 0.04 0.05 0.07 0.09 0. 0.5 0.3 0.3 0.44 0.6 0.90.5 3 0.00 0.0 0.03 0.04 0.04 0.06 0.08 0.0 0.3 0.9 0.6 0.36 0.49 0.7.5 3.5 0.00 0.0 0.0 0.03 0.04 0.05 0.07 0.09 0. 0.6 0. 0.30 0.4 0.58 0.93 4 0.00 0.0 0.0 0.03 0.03 0.04 0.06 0.07 0.09 0.4 0.9 0.6 0.35 0.50 0.78 4.5 0.00 0.0 0.0 0.0 0.03 0.04 0.05 0.07 0.08 0. 0.7 0.3 0.3 0.43 0.67 5 0.00 0.0 0.0 0.0 0.03 0.03 0.05 0.06 0.07 0. 0.5 0.0 0.7 0.38 0.58 5.5 0.00 0.0 0.0 0.0 0.0 0.03 0.04 0.05 0.07 0.0 0.3 0.8 0.4 0.34 0.5 6 0.00 0.0 0.0 0.0 0.0 0.03 0.04 0.05 0.06 0.09 0. 0.6 0. 0.3 0.47 6.5 0.00 0.0 0.0 0.0 0.0 0.03 0.03 0.05 0.06 0.08 0. 0.5 0.0 0.8 0.43 7 0.00 0.0 0.0 0.0 0.0 0.0 0.03 0.04 0.05 0.08 0.0 0.4 0.9 0.6 0.39 7.5 0.00 0.0 0.0 0.0 0.0 0.0 0.03 0.04 0.05 0.07 0.0 0.3 0.7 0.4 0.36 8 0.00 0.0 0.0 0.0 0.0 0.0 0.03 0.04 0.05 0.07 0.09 0. 0.6 0. 0.33 9 0.00 0.00 0.0 0.0 0.0 0.0 0.03 0.03 0.04 0.06 0.08 0. 0.4 0.0 0.9 0 0.00 0.00 0.0 0.0 0.0 0.0 0.0 0.03 0.04 0.05 0.07 0.0 0.3 0.7 0.6 0.00 0.00 0.0 0.0 0.0 0.0 0.0 0.0 0.03 0.04 0.06 0.08 0. 0.4 0. 5 0.00 0.00 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.03 0.05 0.06 0.08 0. 0.7 Effek: Klen Medum Goot
In the case of k samples, η = Λ can be estmated by: ˆ η u v ˆ + = Λ v, (7.4) by usng a esult of Cohen & Nee (984). Hee u = m (k ), v = w + m (k ) /, wth w = n (m + k) /. (7.5) Substtuton of (7.4) n (7.) gves the estmato: v u + v ˆ ˆ / f = Λ. (7.6) Smla to the F-statstc n (5.) n ple lnea egesson, the statstc F u v ˆ v = Λ, u / (, ) ( ) (7.7) has unde the null hypothess of no dffeence between mean vectos, appoxmately a F-dstbuton wth u and v degees of feedom (Cohen, 988). When ths hypothess s not tue t follows also that F(u,v) has appoxmately a non-cental F-dstbuton wth u and v degees of feedom and non-centalty paamete (7.8) ncp u v / = ( Λ )( + + ). As n paagaph 5..5, the ( α) 00% CI can be detemned fo follows: Detemne as n Appendx B a ( α) 00% CI η and f as fo ncp: ( ncp, ncp ). ncp Fom (7.8) follows: Λ = + u + v + (7.9) By means of substtuton of ncpo and ncpb en O f B = ncp u + v +. n (7.9), the CI s esult n η ( L), η ( U ) en f ( L), f ( U ) fo η and f espectvely. 3
These ( α ) 00% CI's fo η meev and f can be calculated by means of the SAS-pogam VI_eta_meev, avalable on the web page of the manual. Othe effect sze ndces based on Wlks Λ ( τ ), the Hotellng-Lawley statstc ( ζ ) and on Plla s statstc ( ) ξ, ae gven by Hubety (994 : 94). An appoxmate ( α) 00% CI can be calculated fo dscuss ths ndex (see Steyn & Ells, 009). ζ, but we wll fst befly It s defned as whee U ( s) ζ =, (7.0) + s U ( s) ( s) U s the Hotellng-Lawley quantty (see Appendx C) fo m -vaable populatons and s mn( m,k ) =. An estmato whch s appoxmately unbased fo ζ s gven by (see Appendx C.): ˆζ ( ) ˆ ( ) ( ) ( ) ( s) n-k-m- U m k = ( s) ns + n-k-m- Uˆ m k, (7.) whee ( s) Û s the Hotellng-Lawley statstc based on k samples fom the m - vaable nomal populatons. Ths statstc s usually epoted togethe wth Wlks lambda n the output of a MANOVA n compute packages such as SAS, SPSS o STATISTICA. An appoxmate ( α ) 00% CI fo ζ can be calculated usng the SAS-pogam VI_zeta_kwad on the manual s web page. Let ( L, U ) (see Appendx C.). ζ ζ denote ths nteval 4
Both the estmato and the CI ae sutable fo use wth smalle samples. When the sample sze, n, s lage, an asymptotcally unbased estmato (see Appendx C.) s gven by: ( ) ˆ ( ) ( s) ( ) ( ) ˆ n k U m k ζ = ˆ ( s) ns + n k U m k. (7.) Both estmatos fo ζ can be negatve f ( s) Û s vey small. The quantty ζ s theoetcally non-negatve, and so, f the estmato tuns out to be negatve we set the estmato to zeo. An asymptotc ( α ) 00% CI fo ζ can also be obtaned usng a SAS-pogam (VI_zeta_kwad) whch s avalable on the manual s web page. Let ( ζ L, ζ U ) be used to denote ths nteval (see Appendx C.). Example 7.4: Consde Example F of Chapte 3. The esults of a MANOVA on the vaables S_Cho, S_T, HDL_C and LDL_C fo compang the 3 actvty goups, was: = ˆΛ = 0,88, F ( 8; 7), 8 ( p< 0,000) n 36, =. Also, ˆ () U =0,36, F ( 8; 70), 95 ( p< 0,000) =. 4 (3 -) - 4 = = 60 =, u = 4 (3 ) = 8, w = 36 (4 + 3 ) / = 357,5, 4 + (3 -) - 5 5 v = 357,5 () + 4 () / = 7. ˆ = 0,88 =,066 (0,997) = 0,063. 8 + 7 / 7 f 95% CI fo VI_eta_meev. f : (0,045; 0,084) obtaned wth the SAS-pogam 5
We estmate η by: 8 7 ˆ + η = 0, 88 = 0,88 (,006) = 0,5. 7 Estmaton of ζ : Snce n s vey lage, we fnd that 359 0, 36 8 76, 8 ζ = = 36 359 0, 36 8 74 76, 8 = 0, 0609, whee the appoxmate 95% CI s: ( 0, 045 ; 0, 079) whch means that ζ can be as small as 0, 045 and as lage as 0, 079 wth 95% pobablty 7.4 Effect szes ndces fo canoncal coelaton In paagaph 5. ρ y.a and y.a ae consdeed to be the effect szes fo the popoton of y s vaance whch s explaned by the ple lnea egesson ŷ = a + b x + b x +... + bu xu, n a populaton and andom sample espectvely. The notaton A efes to the set of pedctos x, x,, x u, whle b, b,, b u ae the egesson coeffcent,.e., weghts whch ae chosen so that the coelaton, ρ y.a o y.a, between y and ŷ s a maxmum. The constant a does not play a ole n detemnng ρ y.a and y.a and can be assumed to be zeo, whch mples that y - y, x - x,, xu - x u ae used nstead of y, x,, x u. 6
Suppose that, nstead of usng only one ctea vaable y, we use a set B whch conssts of y, y,, y v. The queston s now: How can we detemne the elatonshp between the two sets of vaables? The fst step s to detemne the canoncal vaables o pncpal components of A and B. In othe wods, we must detemne a lnea combnaton of the elements of A, x, x,, x u,.e., A = ax + a x +... + au xu, such that t explans the maxmum popoton of the total vaance n A. Next a second lnea combnaton A s found whch attempts to explan the maxmum popoton of the emanng vaance n A. We poceed n ths way untl the lnea combnaton A u s obtaned. These lnea combnatons of x,, x u ae known as the canoncal vaables. Thee ae as many canoncal vaables as thee ae ognal vaables. The same technque can be appled to B; the fst canoncal vaable s B = b y + b y +... + bv yv, and the emanng vaables ae defned smlaly and denoted by B,...,B v. Each of the coelatons of x,, x u wth A, ae known as the loadngs (o stuctue coeffcents) of x,, x u on the fst canoncal vaable. The mean of the sum of squaes of the loadngs descbes the popoton vaance of the set A whch s explaned by A and s denoted by VA. Smlaly VA,, VA u and also VB,...,VB v can be obtaned. The second step s to otate the fst pa of canoncal vaables A and B so that the coelaton between them s a maxmum. otaton means that the set of weghts a,...,a u and b,...,b v ae manpulated to obtan canoncal weghts whch ae defned n such a way that the popotons of vaance VA and VB eman unchanged. The maxmum coelaton s known as the fst canoncal coelaton and s denoted by λ. In a smla fashon we can obtan λ,, λ s, called the canoncal coelatons between the pas ( A, B ),,( s the mnmum of u and v. A s, B s ), whee s 7
The fst canoncal coelaton λ s based on A and B (both of whch descbe the maxmum popotons vaances fom the espectve sets) and t s a good ndcaton of the elatonshp between the sets A and B. The squaed value, λ, s then the popoton vaance whch A o B explan, and s called the egenvalue. The lnea combnatons A and B only explan a poton of the total vaance n A and B (the popotons VA and VB ); as a esult, λ can not actually be used to measue the elatonshp between the sets of vaables x,...,x u and y, y v. athe, t epesents a sot of patal coelaton the coelaton between A and B wthout consdeng A,, A s and B,, B (the so called edundancy ). s A bette measue of the edundancy of B gven A whch s defned by Stewat & Lowe (968) as: s s v B.A = λ VB = λ j / v = = j= whee l j s the loadng of l, (7.3) y j on B. In ths fomulaton, the set B s consdeed to be the cteon values and A as the pedctos so that If A and B s oles ae evesed, then B.A s the edundancy of the cteon set gven the pedctos. s s u A.B = λ VA = λ j / u = = j= l, (7.4) whee l j s the loadng of x j on A. Anothe method to obtan B.A s: 8
= / v (7.5) v B.A = whee s the ple coelaton coeffcent of x,, x u wth y. It s fo ths eason that we use the notaton B.A. Lkewse, t follows that = / u, (7.6) u A.B = whee s the ple coelaton coeffcent of y,, y v wth x. The edundances B.A and A.B ae defned fo populatons as well as samples and because the only dffeence between them s the type of data used (complete populaton data o sample data) we wll not dstngush between the two. Ths notaton wll be mantaned thoughout the text. In ode to llustate all of the concepts and the ntepetatons, we wll dscuss them by efeng to the example that follows ths paagaph. Canoncal coelatons and edundancy can be vey dffcult to calculate by hand and we usually make use of compute packages to pefom canoncal coelaton analyss. As a esult, the output of one of these compute packages, e.g., STATISTICA, wll be used n the example. Example 7.5: In Example A of Chapte 3, we want to detemne the elatonshp between the BDI s befoe tests, afte tests and follow-up tests (BECKPE, BECKPO and BECKPO) on the one sde (B) and the POMS_A (TENSPE, TENSPOS and TENSPOS) and POMS_D (DEPE, DEPPOS and DEPPOS) on the othe (A). The output of a canoncal analyss fom STATISTICA s the followng: (a) 9
N=50 No. of vaables Vaance extacted Total edundancy Vaables: 3 4 5 6 Canoncal Analyss Summay (kdklek_total scales) Canoncal :.8938 Ch²(8)=70.097 p=0.0000 Include condton: goup=0 o goup=03 Left ght Set Set 3 6 00.000% 77.46% 48.4597% 38.5355% BECKPE TENSPE BECKPO DEPPE BECKPO TENSPOS DEPPOS TENSPOS DEPPOS (b) Egenvalues (kdklek_total scales) Include condton: goup=0 o goup=03 oot oot oot oot 3 Value 0.6738 0.4453 0.8338 c) Facto Stuctue, left set (kdklek_total scales) Include condton: goup=0 o goup=03 Vaable oot oot oot 3 BECKPE BECKPO BECKPO 0.34803 0.7394-0.576496 0.85877 0.3369 0.45633 0.98646-0.58999-0.09555 (d) Facto oot oot oot 3 Vaance Extacted (Popotons), left set (kdklek_total Include condton: goup=0 o goup=03 Vaance eddncy. extactd 0.58965 0.395856 0.869 0.055345 0.86 0.033397 (e) 0
Facto Stuctue, ght set (kdklek_total scales) Include condton: goup=0 o goup=03 Vaable oot oot oot 3 TENSPE DEPPE TENSPOS DEPPOS TENSPOS DEPPOS 0.48037 0.6737-0.444477 0.3308 0.73679-0.5475 0.77995-0.0487 0.9677 0.875580 0.369798 0.306 0.776935-0.8875-0.86 0.8777-0.006656-0.3357 (f) Vaable oot oot oot 3 Vaance Extacted (Popotons), ght set (kdklek_tota Include condton: goup=0 o goup=03 Vaance eddncy. extactd 0.47666 0.39688 0.850 0.044883 0.3335 0.00784 (g) Canoncal Weghts, left set (kdklek_total scales) Include condton: goup=0 o goup=03 Vaable oot oot oot 3 BECKPE BECKPO BECKPO 0.05973 0.7849-0.69047 0.4337 0.79963.7768 0.79069-0.949809-0.739669 (h) Canoncal Weghts, ght set (kdklek_total scales) Include condton: goup=0 o goup=03 Vaable oot oot oot 3 TENSPE DEPPE TENSPOS DEPPOS TENSPOS DEPPOS -0.48088 0.46440-0.587 0.70837 0.35866-0.637 0.378857-0.85508-0.34040 0.865.3057.4870 0.8008 0.349648 0.884 0.346035-0.8896 -.04977
The table n (a) shows the fst canoncal coelaton to be λ = 0, 89. The second and thd canoncal coelatons ae λ = 0, 45 = 0, 49 and λ 3 = 0, 834 = 0, 48, whch wee obtaned fom λ, λ and λ 3 n table (b). Note that the mnmum of the numbe of vaables n each goup s s = 3, theefoe thee ae only 3 canoncal coelatons. The BDI goup s fst canoncal vaable, B, explans 0, 59 of the vaance of the 3 vaables (see table (d)) whch s obtaned fom the sum of squaes of the fst column of table (c) dvded by 3: VB = (0,348 + 0,86 + 0,983 ) / 3 = 0,590. Table (c) shows the loadngs of each vaable on the canoncal vaables. In the same way VA = 0, 476 fom table (f), whch s the popoton vaance of the 6-POMS-vaables whch s explaned by A. The canoncal coelaton λ = 0, 89 s thus the coelaton between A and B, but whch each explan the popotons of 0, 476 and 0, 590 fo the espectve goups. The edundancy of B gven A: = 0, 4846 whch s obtaned fom Table (a) and s calculated usng equaton B.A (7.8): = 0, 674 0, 5896 + 0, 45 0, 83 + 0, 834 0, 8 B.A = 0, 3959 + 0, 0554 + 0, 0334 = 0, 4847 (Note that the last column of (d) povdes the ndvdual poducts calculated above). If we un a ple egesson of all of the POMS vaables on each of the BDI vaables, then the ple coelatons ae: 0, 54 ; 0, 73 ; 0,80, and the = 0, 54 + 0, 73 + 0, 80 / 3 = 0, 4844. edundancy s B.A ( )
If the POMS ae used as the cteon, then = 0, 3854 (see table (a)). A.B 7.5 Gudelnes fo vaate omnbus-effects As n the case of the unvaate omnbus effect η and ts estmatos dscussed n Chapte 6, the queston now s: When should these effects be consdeed lage o small? Klne (004b) attempts to solve ths poblem by fst lookng at the case nvolvng two vaate populatons, whee η s the popoton vaance of the dscmnant functon (DF) explaned by populaton membeshp. The DF s nothng othe than the lnea combnaton of the esponse vaables y,..., y m whch epesents the maxmum coelaton of the dchotomous goupng vaable. Theefoe η = λ, wth λ the canoncal coelaton between DF and the goupng vaable. The edundancy of the m esponse vaables, gven the dchotomous goupng vaable, s then λ V, whee V s the popoton vaance of the DF. Klne(004b) supples two easons why η can be substantally lage than any ndvdual y s η : (a) η epesents the popoton of genealzed vaance based on m vaables saneously; (b) Populaton membeshp explans only a poton of the vaance n the DF, whle the DF, n tun, only explans a poton of the vaance of the set of esponse vaables (see the pevous paagaph: the DF s a canoncal vaable, whch s otated, wth a vaance smalle than the total vaance of the esponse vaables). Thus η s the popoton of a poton of the vaance and must thus be lage than the tue popoton (e.g., f the DF s vaance s 3,5 and the total vaance s 5, whle f η = 0, 4, then t s 0, 4 of the 3, 5 whch s lage when t s 0, 4 of 5 ). 3
Klne s aguments can be extended to k populatons. In ths case thee s a set of k dchotomous goupng vaables (also called dummy vaables) whch ndcate populaton membeshp. Consequently the elatonshp between the m esponse vaables and the k goupng vaables can be expessed as the λ λ =. Wlks lambda can be wtten n canoncal coelaton,...,,s mn( k,m) s tems of canoncal coelatons as: ( λ )( ) ( λ... λs ) Λ =. (7.7) The values λ,, λ ae each a popoton vaance of a canoncal vaable s based on the esponse vaables, then, wth espect to a canoncal vaable whch s based on goupng vaables, t does not captue the ente popoton vaance wthn the esponse vaables. Consequently, the popoton η s lage than when t s expessed n tems of the esponse vaable s vaance. Accodng to the pevous paagaph the edundancy of the esponse vaables, gven the goupng vaables (set A), epesents the tue popoton. It can thus seve as an effect sze ndex and s gven by: =, (7.8) s y.a λ V whee V s the popoton vaance explaned by the th canoncal vaable of y,..., y m. In pactce t s ease to calculate m y.a η = y.a usng: = / m, (7.9) whee η s the popoton vaance attbuted to populaton membeshp fo the usual effect sze ndces as defned n Chapte 6. Note that η s nothng y, othe than the ple coelaton coeffcent of goupng vaables. y on the k dchotomous 4
If one s wokng wth samples, the edundancy s obtaned by eplacng η wth ts estmato ˆη. Example 7.6 Once agan, consde Example 7.4. A MANOVA s conducted on the vaables S_CHOL, S_TI, HDL_C and LDL_C to compae the thee actvty goups of men. Wth the ad of STATISTICA, the followng esults ae obtaned: n = 359, k = 3, m = 4, Λ ˆ = 0, 8804. Afte that an ANOVA s conducted fo each vaable and the followng F-values ae obtaned: 60, 49; 53, 43; 4, 59 and 40, 70. The estmato fo η was 0, 8. We use equaton (6.) n the estmaton of S_CHOL s popoton vaance whch s explaned by populaton membeshp: ( ) ( ) ( ) ( ) ( ) ( ) n k F / n k 354 60, 49 / 356 ˆ η = = n k F / n k + n k / k 354 60, 49 / 356 + 356 / = 0, 0804 Smlaly, t follows that ˆ η = 0, 076, ˆ, η 3 = 0 0335 and ˆ, η 4 = 0 055. The edundancy s thus ( ) y.a = 0, 0804 + 0, 076 + 0, 0335 + 0, 055 4 = 0, 060. Whle 0,8 epesents the popoton vaance whch the two canoncal vaables contbute to populaton membeshp, the edundancy s the popoton vaance of the fou esponse vaables whch s explaned. It s almost half the sze of η. An altenatve ndex whch s also based on Λ s (Hubety, 997: 94): 5
wth estmato whee s mn( m,k ) τ = Λ s, (7.30) / s ˆ s u v ˆ + τ = Λ, (7.3) v =, gven by (7.3) and u and v by (7.5).. Fom expesson (7.7) t follows that Λ s s the geometc mean of the ( λ ) s. Then we have that Because τ < η. s ( s) λ λ U =, s the athmetc mean of the s: s s λ ζ = λ λ = + s λ s s / n = λ s = λ. (7.3) Howeve ζ also depends on λ,..., λ s, and t s thus a moe complex elatonshp than fo η and τ. The smplest functon of λ,..., λ s s based on Plla s statstc (Hubety, 994: 94): s λ s = ξ =, (7.33) of whch oy s statstc s a specal case: Note that whle weghted mean of the θ = λ. (7.34) ξ s the mean of the λ s, the edundancy, fom (7.8), s a explaned by the canoncal vaables. λ s wth weghts equal to the popoton vaances, canoncal vaable, V, s close to, then θ, ξ and and any of these can be used as an effect sze ndex. V, If the vaance explaned by the fst y.a ae almost the same, 6
Each of the ndces η, ζ, τ, ξ and θ ae functons of the canoncal coelatons n some way o anothe and they all le between 0 and. As a esult they can be consdeed as dffeenced popotons; the lage the value, the lage the effect. All of these quanttes expess popotons n tems of the s canoncal vaables whch, afte otaton, have the maxmum coelatons wth the goupng vaable. These canoncal vaables do not captue all of the vaance n the m esponse vaables, except when m k. In these cases s = m and λ,..., λ s explans all the vaance of y,..., y m. To ecap: It s dffcult to povde gudelnes fo detemnng when the omnbus effect s small o lage. easons fo ths nclude: The ndces η, τ, ζ, ξ and θ ae based on the MANOVA statstcs: Wlks, Hotellng-Lawley s, Plla s and oy s statstcs. Howeve, whle they ae all functons of the canoncal coelatons, λ,..., λ s, they ae all dffeent functons, and so all the values ae dffeent. The canoncal coelatons measue the elatonshp between the s dscmnant functons (based on the m esponse vaables) and the set of k dchotomous goupng vaables. They do not povde a dect elatonshp between m esponse vaables and thus the popotons η, τ, ζ, ξ and θ ae all too lage. The edundancy y.a pesents the popoton vaance, but compensates fo the potons of the vaances explaned by the dscmnant functons. Theefoe we ecommend that y.a be used as the pmay effect sze ndex fo jontly measung the popoton of the esponse vaables whch can be attbuted to populaton membeshp. Howeve, snce t s the mean of the unvaate η s, the same gudelnes can be used as those povded n Chapte 6. Unfotunately, ths mean can be geatly nfluenced by one vey lage η -value, and so a geat 7
amount of consdeaton and cae should be employed befoe makng use of these gudelnes. Example 7.7: Consde Example 7.4. STATISTICA: The followng tables show the esults obtaned fom Effect Intecept Akt_gp Multvaate Tests of Sgnfcance (dvdwesthuzen) Sgma-estcted paametezaton Effectve hypothess decomposton Include condton: geslag= Test Value F Effect Eo p df df Wlks 0.03053 0766.06 4 356 0.00 Plla's 0.96947 0766.06 4 356 0.00 Hotellng 3.7588 0766.06 4 356 0.00 oy's 3.7588 0766.06 4 356 0.00 Wlks 0.88044.8 8 7 0.00 Plla's 0.98.6 8 74 0.00 Hotellng 0.3550.95 8 70 0.00 oy's 0.3330 45. 4 357 0.00 Test of SS Whole Model vs. SS esdual (dvdwesthuzen) Include condton: geslag= Dependnt Vaable Multple Multple ² Adjusted ² SS Model df Model MS Model SS esdual df esdual MS esdual F p S_CHO 0.8597 0.08749 0.080397 87693 93846.6 059796 359 506.55 60.4934 0.000000 S_TI 0.69990 0.07895 0.07530 4633 730566.6 858335 359 3674.8 53.463 0.000000 HDL_C 0.86886 0.03496 0.033506 300 500.8 89543 359 60.4 4.5943 0.000000 LDL_C 0.3773 0.056508 0.0559 95893 476446.7 59035 359 707.3 40.6965 0.000000 Pevously we obtaned the followng estmates fo the dffeent effect szes: η : ζ : ˆω = 0, 86 ˆζ = 0 0609, ;., wth 95% CI : ( 0 045 0,079) When Wlks lambda s equal to 0, 8804 t follows that the estmato of τ s: ˆ, τ = 0 8804 = 0, 067. 8
When Plla s statstc s equal to 0, 98 t follows that the estmato of ξ s ˆ ξ = 98 = 0, 0599, and when oy s statstc s equal to 0, 333 the estmato s ˆ θ = 0, 333. Fom the second table the values of ˆη follow dectly as the adjusted -values,.e., 0, 0804 ; 0, 075 ; 0, 0335 and 0, 055, as calculated n Example 7.4. The mean of these values s then the edundancy: = 0, 060. y.a All these values epesent the popotons of vaances of S_CHOL, S_TIG, HDL_C and LDL_C whch can be attbuted to the actvty goups. The edundancy ndex s 0,06 whch s also oughly gven by the estmates ˆζ, ˆτ, ˆξ and thus, n tems of the gudelnes n Chapte 6, t ndcates a medum effect. 9