ࡱ> /1.5@ u!bjbj22 JXXuRRRRRRRf\DfH&::%%%%%%%$'R)%R  %RR&X#X#X# ~RR%X# %X#X##?$RRg$. 0x"*W$$<&0H&_$*,#"*g$ffRRRR*Rg$$>,X#B$f%%ff  N# ff AMS588. One-way Repeated Measures ANOVA: The Univariate and the Multivariate Analysis Approaches Suppose there are k regions of interest (ROIs) and n subjects. Each subject was scanned on baseline (soda) as well as after drinking alcohol. Our main hypothesis is whether the change between baseline and alcohol is homogeneous among the ROIs. That is  EMBED Equation.DSMT4 , where  EMBED Equation.DSMT4  is the effect of alcohol on the jth ROI, EMBED Equation.DSMT4  The Univariate Analysis Approach For subject i, let  EMBED Equation.DSMT4  denote the paired difference between baseline and alcohol for the jth ROI, then the (univariate) repeated measures ANOVA model is:  EMBED Equation.DSMT4 , where  EMBED Equation.DSMT4  is the (fixed) effect of ROI j, EMBED Equation.DSMT4  is the (random) effect of subject i, EMBED Equation.DSMT4 is the random error independent of  EMBED Equation.DSMT4 . With normality assumptions, we have  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4 . Let  EMBED Equation.DSMT4 , we have  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 , where  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4  with  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4 . This particular structure of the variance covariance matrix is called compound symmetry. For each subject, it assumes that the variances of the k ROIs are equal  EMBED Equation.DSMT4 and the correlation between each ROI pair is constant  EMBED Equation.DSMT4 , which may not be realistic. The univariate approach to one-way repeated measures ANOVA is equivalent to a two-way mixed effect ANOVA for a randomized block design with subject as the blocks and ROIs as the treatments. The degrees of freedom for the ANOVA F-test of equal treatment effect is  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4  respectively. That is, EMBED Equation.DSMT4 . We will reject the null hypothesis at the significance level ( if  EMBED Equation.DSMT4 . The Multivariate Analysis Approach Alternatively, we can use the multivariate approach where no structure, other than the usual symmetry and non-negative definite properties, is imposed on the variance covariance matrix  EMBED Equation.DSMT4  in  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 . Certainly we have more parameters In this model than the univariate repeated measures ANOVA model, and thus we must have at least  EMBED Equation.DSMT4  subjects. The test statistic is  EMBED Equation.DSMT4  where  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 , and  EMBED Equation.DSMT4 . Under the null hypothesis,  EMBED Equation.DSMT4  EMBED Equation.DSMT4 . The Hotellings  EMBED Equation.DSMT4 statistic has the following relationship with the F statistics:  EMBED Equation.DSMT4 . We will reject the null hypothesis at the significance level ( if  EMBED Equation.DSMT4 . When to use what approach? There are more parameters to be estimated in the multivariate approach than in the univariate approach. Thus, if the assumption for univariate analysis is satisfied, one should use the univariate approach because it is more powerful. Huynh and Feldt (1970) give a weaker requirement for the validity of the univariate ANOVA F-test. It is referred to as the Type H Condition. A test for this condition is called the Machlys sphericity test. In SAS, this test is requested by the PrintE option in the repeated statement. Example. One-way Repeated Measures ANOVA (n=4, k=4) SubjectROI 1ROI 2ROI 3ROI 415961127128931112101443858 SAS Program: One-way Repeated Measures Analysis of Variance data repeatM; input ROI1-ROI4; datalines; 5 9 6 11 7 12 8 9 11 12 10 14 3 8 5 8 ; proc anova data=repeatM; title 'one-way repeated measures ANOVA'; model ROI1-ROI4 = /nouni; repeated ROI 4 (1 2 3 4)/printe; run; SAS Output: One-way Repeated Measures Analysis of Variance Estimated Error Variance-Covariance Matrix ROI_1 ROI_2 ROI_3 ROI_1 10.00 8.00 7.00 ROI_2 8.00 16.75 11.75 ROI_3 7.00 11.75 8.75 2. Test for Type H Condition --- Mauchly's Sphericity Tests (Note: p-value for the test is big, so we can use the univariate approach) Variables DF Criterion Chi-Square Pr > ChiSq Orthogonal Components 5 0.0587599 4.8812865 0.4305 3. Multivariate Analysis Approach --- Manova Test Criteria and Exact F Statistics for the Hypothesis of no drug Effect Statistic Value F Value Num DF Den DF Pr > F Wilks' Lambda 0.00909295 36.33 3 1 0.1212 Pillai's Trace 0.99090705 36.33 3 1 0.1212 Hotelling-Lawley Trace 108.97530864 36.33 3 1 0.1212 Roy's Greatest Root 108.97530864 36.33 3 1 0.1212 4. Univariate Analysis Approach --- Univariate Tests of Hypotheses for Within Subject Effects Adj Pr > F Source DF Anova SS Mean Square F Value Pr > F G - G H - F ROI 3 50.25000000 16.75000000 11.38 0.0020 0.0123 0.0020 Error(ROI) 9 13.25000000 1.47222222 Greenhouse-Geisser Epsilon 0.5998 Huynh-Feldt Epsilon 1.4433 Interpretation Note that the multivariate F-test has value of 36.33, degrees of freedom of 3 and 1, and the p-value is 0.1212. While the univariate F-test has value of 11.38, degrees of freedom of 3 and 9, and the p-value is 0.0020. In this case, since the assumption for the univariate approach is satisfied, we use the univariate approach which is more powerful (smaller p-value).  *-acdwxc d { | } ~       2 ضؙ|plldljh6Uh6jh6CJEHU j^2G@ h6UVmHnHujh6CJEHU j'G@ h6UVmHnHujh6CJEHU j1G@ h6UVmHnHujh6CJU h66CJ h6CJhtBh6CJ hM\CJhM\h6CJhtB htBCJ',de  *%Zsu %+$If$a$gdM\u!2 3 4 5 6 x {     / 0 1 2 U V W X o p ӱӔwkZ j'G@ h6UVmHnHujh6CJEHU j,G@ h6UVmHnHujh6CJEHU j'G@ h6UVmHnHuj h6CJEHU j,G@ h6UVmHnHujh6CJU h66CJ h6CJh6jh6Ujeh6EHU jG@ h6UVmHnHu p q r       0 1 2 3 = > U V W X 㵩㘌{o^Rj|"h6CJEHU j .G@ h6UVmHnHujh6CJEHU jS,G@ h6UVmHnHujh6CJEHU j+G@ h6UVmHnHujh6CJEHU j~T@ h6UVmHnHujh6CJEHU j+,G@ h6UVmHnHu h6CJjh6CJUj h6CJEHU X Z [ r s t u } ~ ¶|k_X h66CJj3h6CJEHU j/G@ h6UVmHnHuj0h6CJEHU j/G@ h6UVmHnHuj ,h6CJEHU j>.G@ h6UVmHnHuj(h6CJEHU j-G@ h6UVmHnHuj%h6CJEHU j1G@ h6UVmHnHujh6CJU h6CJ"    45LMNOTUlmno¶|k_V jah6CJjBBh6CJEHU jqh> h6UVmHnHuj?h6CJEHU jrqh> h6UVmHnHuj<h6CJEHU jdqh> h6UVmHnHuj9h6CJEHU j0G@ h6UVmHnHujh6CJUj6h6CJEHU j0G@ h6UVmHnHu h6CJ!%2345Ӿӡӄxg[jDRh6CJEHU jzAG@ h6UVmHnHujOh6CJEHU j1G@ h6UVmHnHujLh6CJEHU j .G@ h6UVmHnHujIh6CJEHU j|1G@ h6UVmHnHuh6 h6CJjh6CJUj Fh6CJEHU jqh> h6UVmHnHu 012356MNOPVWnopq¶ӥӈ|k_N jb;G@ h6UVmHnHujTch6CJEHU j8;G@ h6UVmHnHuj_h6CJEHU jX5G@ h6UVmHnHuj\h6CJEHU j9G@ h6UVmHnHujXh6CJEHU j8G@ h6UVmHnHu h6CJjh6CJUjUh6CJEHU j4G@ h6UVmHnHu34KLMN  `㵩㏃zvzkR1hM\h656CJOJQJ^JaJhnH tH h6CJhnH tH h6h6hnH tH jph6CJEHU j>@G@ h6UVmHnHu jah6CJjkh6CJEHU j?G@ h6UVmHnHujih6CJEHU j;G@ h6UVmHnHu h6CJjh6CJUjHfh6CJEHU+,.0247YSSSSS$Ifkds$$Iflrj U@+"04 la78:<?ACYSSSSS$Ifkds$$Iflrj U@+"04 laCDFILORYSSSSS$IfkdMt$$Iflrj U@+"04 laRSUWY[]YSSSSS$Ifkdt$$Iflrj U@+"04 la]^_`YWWWWWWWWWkdu$$Iflrj U@+"04 la  027LTYZ\]^_`abceԺԺԺlPPPPP6h65B* OJQJfHhnH phq tH 3h6B*OJQJfHhnH phq tH 3h6B* OJQJfHhnH phq tH 3h6B*OJQJfHhnH phq tH 3h6B*OJQJfHhnH phq tH 6h65B* OJQJfHhnH phq tH h6CJOJQJhnH tH  2LmrtD789 l & Fekmprs9xz/ʮʟ{k{kYk{L{k{?{k{?{ *h6CJhnH tH  *h6CJhnH tH " *hM\h6CJaJhnH tH hM\h6CJaJhnH tH h6CJhnH tH 1hM\h656CJOJQJ^JaJhnH tH hM\CJOJQJhnH tH 6h65B* OJQJfHhnH phq tH 3h6B*OJQJfHhnH phq tH 6 *h6B*OJQJfHhnH phq tH lQ*/FG  u!   u!̷%hM\h6B*CJOJQJaJphhM\B*CJOJQJaJph)hM\hM\B*CJOJQJ^JaJph/hM\h656B*CJOJQJ^JaJphhM\CJOJQJaJh6CJOJQJhnH tH / =!"#$%Dd HhB  S A? 2􌲦N6J(}Ö]D `!U􌲦N6J(}Ör@ @|#xTkQfvc%kh `dmVWhJ5)cn#i`E? ăWE`&ͦ}dw73oޛBn1؇g,9N۟qE9ā^,NTRucwzqZ.fckN8_BczG2M2yD+AmlܤQ,*7@&Dhj[*>RpW^q\wݪ% q^u{2=]nľbrόHːJIJH [=j& rkΡpaupىգS6%Unaf}@JH;Լy^Dd ,|B  S A? 2,'w"٭t6x&,[ `!,'w"٭t6x&,\`0xeQJA>gV-`̋(ʄRM R$W RZ5ҫG誧7"Nh;gVDj|sf7|! d@@xh DRDydqCf](*`bJ|b~H-Q+* QEt~+umd 3`Nl&& DړHl:v?USѦ$;c E6kl}1楖, f-s}rȣ\ByA.)1`+3+B״{|@LAk3&Ŝ^u5ҫ;ۄ8L 7c܉)>h.O]!IGA5ŷ*B.&w-r-Yw\廧Me:]S<~ynDd 8@B  S A? 22ઁw|VvQ\ `!ઁw|VvQ\"v xRKA&ZҦ$Dz 5KUf1#SD<NJ݃P0&"vx߾y~;ȥD%'Cϕ'+8T %E|\ %dKqNQIʭn{e\ m(ǷJ"MRSaC^e'.&tnhE'e%Rhvy]2U+$o]i|El$\4 n9j5PCufu3/|MmUv*hʶbˍfoCX@S^p{.^zxڰ8l}12eiN0]YbYa`{y gIb$xqĩi$8sп,Ϡ~FZgg#|ZCqܟ29$+«#OO5Lh4wKDd hhB  S A? 2@(jO, `!@(jO,@@||xR;kPl];G ִ${PS rp YBg1[ }`s-5-Gs$ h-H%JK  !gBÀ{!CmLqMOƂǘ A-o|&uZ*=reIFi ~˓?bP(u[g{=Ͼ5#BfȘjKس56}%t.P/L/P? (;&Dj홝NܽRߩmC2VwwRH 6x)_x2|eٌ֦YW ~_'EP(6>OuDkW 񏬲U5f IB *T*VgPĩ58T&xyjg(Z} .F#.qz@Swӳ@uDd @|B  S A? 2V\v!E2ҽ:2^  `!*\v!E2ҽ:~ `p0xSAkQԦ{Ec@ (6L)n iwqDlN,^Ox@d"{iT̼oͼ%P`D0YHD[霶\т@ b[ _bt5g {r,Fg9]2BggEF?D%E5W"fuvبnF^EX0 p[SʽO{r+sr[Gr[Bzq)hRiF3lh Ͳlk+IZ⍀~O9Q4Ͳ7up.ݶEƋ؍ģ;mFȍH[n~Yjn*7/>-e:gV-`̋(ʄRM R$W RZ5ҫG誧7"Nh;gVDj|sf7|! d@@xh DRDydqCf](*`bJ|b~H-Q+* QEt~+umd 3`Nl&& DړHl:v?USѦ$;c E6kl}1楖, f-s}rȣ\ByA.)1`+3+  !"#$%'()*+,-03S456789:;=<>?@ABCDEFGHIJKLMNOPQRTUtVXWYZ[\]^`_abcdefghijklmnoqprsvuwxzy|{}~Root Entry Fx2XData &!vWordDocumentJObjectPoolJ0xx_1078407635YF0x0xOle CompObjiObjInfo  !"%()*+,/2347:;<?BCDGJKLMNQTUVWX[^_`abcfijklmpstux{|}~ FMathType 4.0 Equation MathType EFEquation.DSMT49q8@6M6GlxDSMT4WinANSITimes New RomanSymbolCourier NewMT Extra!/ED/APG_APAPAE%B_AC_AE*_HA@AHA*_D_E_E_A  HH 00 ::m Equation Native T_1078404997h" F0x0xOle  CompObj i11 ==m 22 =="L==m kk FMathType 4.0 Equation MathType EFEquation.DSMT49q@6M6GlxDSMT4WinANSITimes New RomanSymbolCourier NewMT Extra!/ED/APG_APAPAObjInfo Equation Native _1078407774F0x0xOle E%B_AC_AE*_HA@AHA*_D_E_E_A  m jj FMathType 4.0 Equation MathType EFEquation.DSMT49q@6M6GlxDSMT4WinANSITimes New RomanSymbolCourier NewMT Extra!/ED/APG_APAPACompObjiObjInfoEquation Native _1078395026F0x0xE%B_AC_AE*_HA@AHA*_D_E_E_A  jj==11,,"L,,kk.. FMathType 4.0 Equation MathType EFEquation.DSMT49q@6M6GHDSMT4WinAllBasicCodePagesOle CompObjiObjInfoEquation Native Times New RomanSymbolCourier NewMT Extra!/'_!!/G_APAPAE%B_AC_A %!AHA_D_E_E_A  ijY FMathType 4.0 Equation MathType EFEquation.DSMT49q8@6M6GlxDSMT4WinANSITimes New Rom_1078406159 'F0x0xOle #CompObj$iObjInfo&Equation Native 'T_1078406171F0x0xOle -CompObj .ianSymbolCourier NewMT Extra!/ED/APG_APAPAE%B_AC_AE*_HA@AHA*_D_E_E_A  YY iijj ==m jj ++SS ii ++e iijj FMathType 4.0 Equation MathType EFEquation.DSMT49qObjInfo!0Equation Native 1_10784051051$F0x0xOle 5@6M6GlxDSMT4WinANSITimes New RomanSymbolCourier NewMT Extra!/ED/APG_APAPAE%B_AC_AE*_HA@AHA*_D_E_E_A  SS ii FMathType 4.0 Equation MathType EFEquation.DSMT49qCompObj#%6iObjInfo&8Equation Native 9_1078406187E)F0x0x@6M6GlxDSMT4WinANSITimes New RomanSymbolCourier NewMT Extra!/ED/APG_APAPAE%B_AC_AE*_HA@AHA*_D_E_E_A  e iijj FMathType 4.0 Equation MathType EFEquation.DSMT49qOle =CompObj(*>iObjInfo+@Equation Native A@6M6GlxDSMT4WinANSITimes New RomanSymbolCourier NewMT Extra!/ED/APG_APAPAE%B_AC_AE*_HA@AHA*_D_E_E_A  SS ii FMathType 4.0 Equation MathType EFEquation.DSMT49q_1079279308.F0x0xOle ECompObj-/FiObjInfo0H@@6M6GHDSMT4WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT Extra!/ED/APG_APAPAE%B_AC_AE*_HA@AHA*_D_E_E_A  S i  <": iid N0,s s2 ()Equation Native I\_10784061053F0x0xOle OCompObj24Pi FMathType 4.0 Equation MathType EFEquation.DSMT49qD@6M6GlxDSMT4WinANSITimes New RomanSymbolCourier NewMT Extra!/ED/APG_APAPAE%B_AC_AE*_HA@AHA*_D_E_E_A  e iijj  <": iiiidd NN00,,s ObjInfo5REquation Native S`_10784062278F0x0xOle Ye22 () FMathType 4.0 Equation MathType EFEquation.DSMT49q|@6M6GlxDSMT4WinANSITimes New RomanSymbolCourier NewMT Extra!/ED/APG_APAPAE%B_AC_AE*_HA@AHA*_D_E_E_A CompObj79ZiObjInfo:\Equation Native ]_1078406667=F0x0x  YY ii ==YY ii11 ,,YY ii22 ,,"L,,YY iikk () '' FMathType 4.0 Equation MathType EFEquation.DSMT49qOle dCompObj<>eiObjInfo?gEquation Native hdH@6M6GlxDSMT4WinANSITimes New RomanSymbolCourier NewMT Extra!/ED/APG_APAPAE%B_AC_AE*_HA@AHA*_D_E_E_A   YY ii  <": iiiidd NN kk  m,,S() FMathType 4.0 Equation MathTy_1078407570|BF0x0xOle nCompObjACoiObjInfoDqpe EFEquation.DSMT49q@6M6GlxDSMT4WinANSITimes New RomanSymbolCourier NewMT Extra!/ED/APG_APAPAE%B_AC_AE*_HA@AHA*_D_E_E_A  ii==11,,"L,,nn`8 FMathType 4.0 Equation MathTyEquation Native r_10784065376;GF0x0xOle vCompObjFHwipe EFEquation.DSMT49qT@6M6GlxDSMT4WinANSITimes New RomanSymbolCourier NewMT Extra!/ED/APG_APAPAE%B_AC_AE*_HA@AHA*_D_E_E_A   m==m 11 ,,m 22 ,,"L,,m kk () ''ObjInfoIyEquation Native zp_1078406718LF0x0xOle  FMathType 4.0 Equation MathType EFEquation.DSMT49q֜@6M6GlxDSMT4WinANSITimes New RomanSymbolCourier NewMT Extra!/ED/APG_APAPAE%B_AC_AE*_HA@AHA*_D_E_E_A  S==s CompObjKMiObjInfoNEquation Native _1078407072TQF0x0xss22 ++s e22 s ss22 "Ls ss22 s ss22 s ss22 ++s e22 "Ls ss22 "M"M"O"Ms ss22 s ss22 "Ls ss22 ++s e22 []==s 22 11r"Lrr11"Lr"M"M"O"Mrr"L11[] FMathType 4.0 Equation MathType EFEquation.DSMT49q4@6M6GlxDSMT4WinANSITimes New RomOle CompObjPRiObjInfoSEquation Native PanSymbolCourier NewMT Extra!/ED/APG_APAPAE%B_AC_AE*_HA@AHA*_D_E_E_A  r== s ss22 s ss22 ++s e22 FMathType 4.0 Equation MathType EFEquation.DSMT49q_1078407110VF0x0xOle CompObjUWiObjInfoX@6M6GlxDSMT4WinANSITimes New RomanSymbolCourier NewMT Extra!/ED/APG_APAPAE%B_AC_AE*_HA@AHA*_D_E_E_A  s 22 ==s ss22 ++s e22 FMathType 4.0 Equation MathTyEquation Native 8_1078407319Ow[F0x0xOle CompObjZ\ipe EFEquation.DSMT49q@6M6GlxDSMT4WinANSITimes New RomanSymbolCourier NewMT Extra!/ED/APG_APAPAE%B_AC_AE*_HA@AHA*_D_E_E_A  s 22 () FMathType 4.0 Equation MathTyObjInfo]Equation Native _1078407357`F0x0xOle CompObj_aiObjInfobEquation Native _1047032164eF0x0xpe EFEquation.DSMT49q@6M6GlxDSMT4WinANSITimes New RomanSymbolCourier NewMT Extra!/ED/APG_APAPAE%B_AC_AE*_HA@AHA*_D_E_E_A  r() FMathType 4.0 Equation MathTyOle CompObjdfiObjInfogEquation Native pe EFEquation.DSMT49qQ@6M6GHDSMT4WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT Extra!/'_!!/G_APAPAE%B_AC_A %!AHA_D_E_E_A  k"-1()_1047032178crjF0x0xOle CompObjikiObjInfol FMathType 4.0 Equation MathType EFEquation.DSMT49qQ @6M6GHDSMT4WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT Extra!/'_!!/G_APAPAE%B_AC_A %!AHA_D_E_E_A  n"-1()k"-1()Equation Native %_1047032247oF0x0xOle CompObjnpi FMathType 4.0 Equation MathType EFEquation.DSMT49qQs@6M6GHDSMT4WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT Extra!/ED/APG_APAPAE%B_AC_AE*_HA@AHA*_D_E_E_A  F 0  <ObjInfoqEquation Native _1047032276mtF0x0xOle ": H 0 F k"-1,n"-1()k"-1() FMathType 4.0 Equation MathType EFEquation.DSMT49qQT@6M6GHDSMT4WinAllBasicCodePagesCompObjsuiObjInfovEquation Native p_1078407548^@yF0x0xTimes New RomanSymbolCourier NewMT Extra!/ED/APG_APAPAE%B_AC_AE*_HA@AHA*_D_E_E_A  F 0 >>F k"-1,n"-1()k"-1()a FMathType 4.0 Equation MathType EFEquation.DSMT49qOle CompObjxziObjInfo{Equation Native @6M6GlxDSMT4WinANSITimes New RomanSymbolCourier NewMT Extra!/ED/APG_APAPAE%B_AC_AE*_HA@AHA*_D_E_E_A  S FMathType 4.0 Equation MathType EFEquation.DSMT49q_1078407591~F0x0xOle CompObj}iObjInfo@6M6GlxDSMT4WinANSITimes New RomanSymbolCourier NewMT Extra!/ED/APG_APAPAE%B_AC_AE*_HA@AHA*_D_E_E_A  ii==11,,"L,,nn`8 FMathType 4.0 Equation MathType EFEquation.DSMT49qEquation Native _1078411642,F0x0xOle CompObjiObjInfoEquation Native _1078408408F0x0xOle @6M6GlxDSMT4WinANSITimes New RomanSymbolCourier NewMT Extra!/ED/APG_APAPAE%B_AC_AE*_HA@AHA*_D_E_E_A  kk++11() FMathType 4.0 Equation MathType EFEquation.DSMT49qCompObjiObjInfoEquation Native _1078409383F0x0x@6M6GlxDSMT4WinANSITimes New RomanSymbolCourier NewMT Extra!/ED/APG_APAPAE%B_AC_AE*_HA@AHA*_D_E_E_A  TT 0022 ==nnnn"-11()CC  YY() '' CCQQCC '' () "-11    !$'()*-012369:;<?BCDEFGHKNOPQRTUVWXYZ\]^_`bCC  YY() FMathType 4.0 Equation MathType EFEquation.DSMT49q$@6M6GlxDSMT4WinANSITimes New RomanSymbolCourier NewMT Extra!/ED/APG_APAPAE%B_AC_AE*_HA@AHA*_D_E_E_A Ole CompObjiObjInfoEquation Native @   YY==p YY iiii==11nn " FMathType 4.0 Equation MathType EFEquation.DSMT49q֬@6M6GlxDSMT4WinANSITimes New RomanSymbolCourier NewMT Extra!/ED/APG_APAPA_1078409655F0x0xOle  CompObj iObjInfo Equation Native _1078408536F0x0xOle CompObjiE%B_AC_AE*_HA@AHA*_D_E_E_A  QQ==p YY ii "-  YY() ii==11nn "  YY ii "-  YY() '' FMathType 4.0 Equation MathType EFEquation.DSMT49q@6M6GlxDSMT4WinANSITimes New RomanSymbolCourier NewMT Extra!/ED/APG_APAPAE%B_AC_AE*_HA@AHA*_D_E_E_A  CC==11"-1100"L00000011"-11"L0000"M"M"ObjInfoEquation Native _1078410040F0x0xOle "M"M"M000000"L11"-11[]  FMathType 4.0 Equation MathType EFEquation.DSMT49q @6M6GlxDSMT4WinANSITimes New RomanSymbolCourier NewMT Extra!/ED/APG_APAPACompObj#iObjInfo%Equation Native &<_1078410082F0x0xE%B_AC_AE*_HA@AHA*_D_E_E_A  TT 0022 <":HHootteelllliinngg''ss FMathType 4.0 Equation MathType EFEquation.DSMT49q@6M6GlxDSMT4WinANSITimes New RomOle +CompObj,iObjInfo.Equation Native /anSymbolCourier NewMT Extra!/ED/APG_APAPAE%B_AC_AE*_HA@AHA*_D_E_E_A  TT kk"-11,,nn"-1122 FMathType 4.0 Equation MathType EFEquation.DSMT49q_1078410207F0x0xOle 4CompObj5iObjInfo7Equation Native 8_1078411229F0x0xOle =CompObj>i@6M6GlxDSMT4WinANSITimes New RomanSymbolCourier NewMT Extra!/ED/APG_APAPAE%B_AC_AE*_HA@AHA*_D_E_E_A  TT kk"-11,,nn"-1122 FMathType 4.0 Equation MathType EFEquation.DSMT49qObjInfo@Equation Native A_1078411326F0x0xOle I@6M6GlxDSMT4WinANSITimes New RomanSymbolCourier NewMT Extra!/ED/APG_APAPAE%B_AC_AE*_HA@AHA*_D_E_E_A  FF 00 ==TT 0022   nn"-kk++11kk"-11()nn"-11()  <": HH 00 FF kk"-11,,nn"-kk++11 FMathType 4.0 Equation MathType EFEquation.DSMT49q(@6M6GlxDSMT4WinANSITimes New RomanSymbolCourier NewMT Extra!/ED/APG_APAPAE%B_AC_AE*_HA@AHA*_D_E_E_A CompObjJiObjInfoLEquation Native MD1Table* FF 00 >>FF kk"-11,,nn"-kk++11aOh+'0 4@ \ h t ;Repeated Measures ANOVA versus Multivariate ANOVA (MANOVA)epeAMSMS Normal.dB״{|@LAk3&Ŝ^u5ҫ;ۄ8L 7c܉)>h.O]!IGA5ŷ*B.&w-r-Yw\廧Me:]S<~yn~Dd hB  S A? 2f =~ `!f =~f@R|xRJQU+`EeB-~I RQZN *)\=@V#f4{G$ja.s3('@ia^$|WZV]W LQBĈR4—I@G^N{J_ևpXOH$^)BiF"뫢H^fN;fm>Ne.&m_aX޴Y3"2dUԙ61$Cְuѹb:]ydRϘ~E,lq)o<];f3Mo|h3ҵtceo"weF.٦'?CKN;&n-cVx2jnR$(tyְwZ"PO>%!MpDd |B  S A? 2E(7]ޅN `!E(7]ޅd`H0x]RMK@}ij!Q TbA+b[< SZ?z+z'" ƙM)!vvۙN-b<Ia#+4#kj4EIu:4hq4BI=,thrgtO%+ +, S;}bя(<" ~ŻNzwjtAѷePM)kҒV󣚿}|JYDl_ɋYq^,!.)|;/>R2SFT|zSl"}V(VVv> ,s׵ggO6'XZ.#rtd;f]'6I;fD:ƞ^ wDV>|(C '&@W-Aj?/7^ND5 8p~Dd hB   S A? 2f =~ `!f =~f@R|xRJQU+`EeB-~I RQZN *)\=@V#f4{G$ja.s3('@ia^$|WZV]W LQBĈR4—I@G^N{J_ևpXOH$^)BiF"뫢H^fN;fm>Ne.&m_aX޴Y3"2dUԙ61$Cְuѹb:]ydRϘ~E,lq)o<];f3Mo|h3ҵtceo"weF.٦'?CKN;&n-cVx2jnR$(tyְwZ"PO>%!MpDd OB   S A?  2_LȽ#ˤ[ `!_LȽ#ˤ`0, :xToA~3 ]b"6V&lؘBKPֺaQ '=5^==CM'ѽ7+ u̼7C`@L(l{QBlvͥ3=w:cyEPơ`8Kb>+BFָ4a2inq~W>6,@QZ)Eѥ旺߇uHq7r `°be?|^L&|d 9ecD`fj=|R.BO{0 #j #O9j+0:o/,yNL`&O1wP@ӊ<̃gFB>wCݬ=i7~Aowzl}p.kҧ# 'j҆l 9ضqrxYX %=d*J ]LaE!ug1$،CxvUSlBN#;>l%f \əǎ| 4OJ@ 9[4[|&E\"4+y& ?x>% &⊆ y]_!f,Ow.ǥnhkIuo?vGXu8ƫU>zM-C"^#SCN/XDd xB   S A ?  2mzS"&Y `!mzS"&Y,` :dxoQg[ Ev1B m"Ԙ5l#`%R `,'z0@O1{1*0 {@.+G5`<\H%['xVHd?@֕07m'=V^TBQy NQָonR |U]%/dc8$ib@)҈ EŊ`BfՌZy \+s8AMԒΟYc.Dr%~I8ЈK]5Dvu@F踞(H|] u@a i]k5}Ce%#f0[d[qV)]oWL% 79TYö m$.dswϋ^ dNv,=⪛qNWynQb s.f~pI: (S;A\{QZ^W6TVJp:㣽]<+U$2qM =>$ikqKHUoIS +@S'ͷ-S+S˹vDd lB   S A ?  2vVE#~J `!vVE#~n XhxtxTMOSA=wޫӖB%Z_T>R%mܘRJ-14)1qQWltCK&?L ދis{̹B0>)6!k}j"*VktYbNWêXI똺i9I? *wE[su1L"=JM0_Н;-lUnd^|Wʋ 9nE)oOa>!/&s+Q2l۝{{0;۠(AՅt-zRIMOA/N^5'F3v,zy<},}n)åI=c{s;`Ci<+%<<  QE1p %؁{l' ~ˉ AyX>μRP9/?F(ʊum߈>7_ZR٣^Л__N|GnAU+A#CͺKAJ BK JVlַv^Yd̦ 5Ѧ&.P'ZC Y|zriOrG+\<~nUX&LXC,=6X}R|1鴫2mEtF}R@zQBF#;[k=i)8 +~fE_"e}}t4Kau* ɮW4OslrO#3EVH-%'VL0A UīUEwkkP86 PB4! @b =KA.F~jS<J{%h,(@dFwSo'!T%nDd @B  S A ?  2."Wv"S:UjilT *& `!"Wv"S:UjilT ~ xR]KA=wv5vTh$U_ݚ FB6‚ ͦĈER'G"PZ1;$m@vg9sa 0Y&oE[RNG{hJ{j4ʎL0z=b9:r {%xWǓ V%JeLg$N™BɥC_SpkvgIg5Oo nFPS P“nS~H Dd B  S A? 2"#7XCT# O, `!#7XCT# Z %@ \9PVxVkW?cvf7 [7bLb 4d i&+E,)l1`>؉>Yh, +h5@QŇ? "/B*`zLLffg{~w=a@FǏ8uc0F –JphJCQ肺0ÑUo V[Vȍ^-TfJI*X‚ϐ\(mTvW%d01Yғ ӓS8(;6946U11Ys Sa #AR'BI2Y4#f#7܏+cd-tjG:)OɎy4!F:ټğ(^=HĚ!%GE|7׀v<_ִn?2w\+8;3KmܥȌFzFpκYjV; ?R~^J%{ѷՃ}JmEknְĬ%*NvFgVnaÖSj`nW:n:σxx/(}N bOدAsjmEc /jAczӶ]&Ǔqo'Lޜ-{3s`騟%;ӭ3áGΊr3F?+^L"_V.^)bӌs ʌ激>lkŕRY(!έ~lu/s)fοt ԲC"JZOlj a, H1[L`b#L̼"/,O0Y E@87'L3^^Ii@!F=dm(Q oNhH\wI)I9G>y ,|pO M!(W|?P ?XK^LSZ<8Ьl7,y@5=6޽MR>[+TL'Dd B  S A? 2_R/vSy\m1 `!e_R/vSy\4 3xSkQ4c1HCIBSIa`49Yxxփ='*{RptC+>vvf7YQ' vA_Bg *Rkh B9Ҿe .p6d[Ư\b(*^gؗ9O+ JJ 'FHf:^DWgJAH\&@`vz}+:CfT,@F![Q|'e*8ݙh4DU2cN+Tlw=:qXBGs3kVR>7}z+_`: U'>j}eFUVøחPtvX۰V7!-/$z}9tB^ku͡ۅpbsX|Ӹa[k?ΝeKUB'נEV`}e:uzQ5CωTsDHYclji>+mӽﺛTsُYճWUs÷~8"Vƣ.bƚ7m[]y[_c]f<}+^lSy w5h^,[)ZR|"ӑywiަ(IEY8\R֧ZU)sc>D3۵]8o <3t6AlsşDp3N8Dd B  S A? 2; ㉐eK7 `!; ㉐eK@ @CxRJP=sj! ZE`A+b+.Ckh W.]W~"ىƙIs'3瞙;N8#@b)"ْ @{_{fjPPRԼhI,꜑;ƾjNSz3MWuU]ʱ+(oWvAj.Hp\qW/=P EU(Zz;LoT=l,dQڢOVqѧnn>".)|0*k?ŦT|rP-yUT,WH;Gjq{ݷ%^dgo.,sQL'e=lOؓnu|ym3I,/4#lYFQ{f$!!*vq8g]Mn}ϒLaU{ÎZh׻;=|HQ{eC}VDd B  S A? 2֧mPR^x9 `!֧mPR^xhxROKAoV+Y ʊ„ЦKiJ`!!)Ykxo:t@{o\Dٝyy7f|1h7u^(áqAzJࢠ舞ژRa Z3^p)5/ P,>Dr ޤ;>VЧە% "/?pH?~s˅q.ps^,>O3]igZבZT&kWvٲyBL!bv faf%atS =hꁩ>ϫ:8tMBd5vҳG&M@)p t(OK}RlZa~G^i0-g}vz5n4ʨj7ȸouGDd B  S A? 2n9@l{JR< `!B9@l{@hxSAkA&M`"ؤmܦA#ISRx'^ēw<ɋ=y֛X o&&B7o}fȖ."`0P5b879&M\@a|`G]iTQw}eFҚhD䇪셄%.4S[Mm>իq'!P kTkڴ[Idn9&5r񊾊! RLrn/wr0Qw!0>g1h 4w9xs(0Atӻ YFRjYm<(Ug"jQ5Dp;rYRϕxoR`՛U .[i!,WV,,^`y+<ްˡ,3ML7ւ%,MItB6X8*sy,+R;pG;z^iW/Ak~s[}ӷ9I}6穩R0Dd (B  S A? 2; AL^,vV? `!n; AL^,@ h<xTkAfv&&(b=%EiPH0)k]%` KEփ'/^FÔR|eC{h-ɤ@:;{;{m-j!W6y˪Rú5uX18|bLA2b15Wg6%:[\HtKWt7s.4}dE }z3=| rWj;?cY[i=h1@.&-E.ғ*tb6ɱ_̋FA8#0]Wre~f]ج|C3.mǂ:~8U42m8` ƑTs_)%p|0CDhQ;Q4G@ c}@8w5XM`.]Z_!b3i@^<nOȗeEicw/Ķz,~=cJ;ZɂEvxvc SnD=ݖ̺DR᫹RPe )]4)tpN%'UX#8M$g>ڌ~!L~@=rT Z{}Qy6<~hg3LTfFP|p0r YE68Hcx~BÏDFlLypϸixb]-[HDG Eܑ>˘,3>rta곆9|9E9852M ͳNz7O*$6\8K28[ r%*eTm3-ѸmRUL n1Dղw߹KNx)W]{`/8Fkjg5bC'._5 p:_ ȃP|0 *䴑\»Mv2oЕue;>/#]4pI_&L-$,;} <*;+pțhVJꍱqx} u=CA\Dd B  S A? 2HCHZ-' aNAI `!HCHZ-' aNA`!RhxMQMKQ=V~d-)`Rt EDJjrz*{Jxoywǹ@ b!Ζt:U=ZS}A(%0Gi1s0]бL{^8OjF97`* YOH"TPbHEw.z *^揃!.OӞxh Nb5HHDIs_7.D2=Bx0 :̑*GSq"``jDd OB  S A ? 2&-ZL `!&-4`0, :vxߋQ9wt]l(ʄbk)"hpZ%q Ph0p]~<`BAH?2Osu[g>{s;`Ci<+%<<  QE1p %؁{l' ~ˉ AyX>μRP9/?F(ʊum߈>7_ZR٣^Л__N|GnAU+A#CͺKAJ BK JVlַv^Yd̦ 5Ѧ&.P'ZC Y|zriOrG+\<~nUX&LXC,=6X}R|1鴫2mEtF}R@zQBF#;[k=i)8 +~fE_"e}}t4Kau* ɮW4OslrO#3EVH-%'VL0A UīUEwkkP86 PB4! @b =KA.F~jS<J{%h,(@dFwSo'!T%nDd @B  S A ? 2."Wv"S:UjilT O `!"Wv"S:UjilT ~ xR]KA=wv5vTh$U_ݚ FB6‚ ͦĈER'G"PZ1;$m@vg9sa 0Y&oE[RNG{hJ{j4ʎL0z=b9:r {%xWǓ V%JeLg$N™i}'R|\mz+7ȦfdljӪ-]vQܟ;yo5vׁ#㎇؏ŝW(Yԡ3R1?.fb ubd$[z%6J_TGyRF0u2Mzt1dN̙ev@W,woDd B  S A? 2M9ǖz E+I4:)HU `!!9ǖz E+I4:@p xUkQMRͦIjJ& ^,4nlIV H +&6bz(ғG?C{^ xY%'?KÈd} xGuoB7?*ar+-AjaAN *`*+|~q_5Ʀ 7;Y%x$s.'y6*VK,H/ne[$1amX`fA]ש < HHl:氹{o6Te̦h9X)zs7;eԺ7EgӳjKo0TX lbIJfrx^ln:{3tP=fljAo-&SA]/ azp]hn x7ѨhDŦ8TCϛs43$5[]z:EخȞjCWQnn79ݒHz: 6O/A(J@TKݹ B5f4t@[@42H E;Z?~Nr͖7壔,٣ (>{N?oG/i\-=\=89\`^B :"Q΋ x+^(I&@-:t(zxUw 0sPu!UsJ熾"u+ZE|!jQ* 4Dd B  S A? 2-Je}d`z+Y `!r-Je}d`@@xSAkQ6dXQ",CAk՛6L-&ErZ"0II#6[x({+(ۍ5*>y~3;͛ySr"#axF1 /q%` 4#)8 6bf?}v$+ Zw[H Jr^B$BiFQs \N<54]S)uO3De1VѬodOZI P\?lGD4K4R牨Q:ejkl{#fi5#5QTy\ D&όdrϵh|֘lp5rrEhdszwp2NVaCVs<.UN*إ|jŤU*.ENj7K208a/l1%\섈ˣz["Mmc+p4iz&gh(` ܉>ͣ{5glb{R\_\aOl {|߱ۇ5!{/'3$frrr<{Bm zgtU\JL.^Y^,@'xӕ`r6kH2` vj$S%mBFh?FB?ʧmMFҠ<$Lx8 PJ* >3"/I iUg,!i|.ՑPp7i{PD"+4A)!b@Ӕ4bB P!ēR<؃g^U"{Rnaw7y3 >|)! L`00%)IR_(B!Dt- Ep'%;ؗkh|ڨ(9hT4# ^e f|,~^+Ǘ@͐U2i2!)0i;s\tNCKhx 'RkfAg^ly%*g󧜵Ъ=+p#4x>8gkW폫 ɏwGy(!=S!|<1N.(~ߏ\QQ٭flWo;a^I8'deGDR/2y/BTX #*OJ+!*8@2feC_/!u0U{H|f҂5%83L)kG#C'V`!" i*tRbB&BJ IHS'~w5LC]<-|HHʈ0x~?>/W5QC%-f$.ls ] RAi[- vP)SڣϪ0R>` g6qsK}zql!,H6uIJr+5=;U.N4-oo].r'&O^-0v}8L#DL$L}L3 `"dCOoh0{-o?ᡚabAéۣ885yM#Dd B " S A? !2AQ *yH.gf `!Q *yH.gX@2hxRMKQ=$хЏ4`$ Ј4#.L$I1Y)Jvq#;Y}/c*>fwy~f Cȥ/`K""yIKSМ@T@2z4>L\3l8*Jw|HJ>!тF~|D\>K,t-soۭZNsT #\ј 'g̢P;OwLGxolo8,5)4 gǼ1/M;u2VGHUo]e]a܋$bRZqAكF`V_#2NCdsN yʫ^h<{SƋ і49኷g{+{N(?H/N89m\S'Lh |SP`_BxXp6ިڎ+m]g[x5M\3Dd B # S A? "2AQ *yH.gci `!Q *yH.gX@2hxRMKQ=$хЏ4`$ Ј4#.L$I1Y)Jvq#;Y}/c*>fwy~f Cȥ/`K""yIKSМ@T@2z4>L\3l8*Jw|HJ>!тF~|D\>K,t-soۭZNsT #\ј 'g̢P;OwLGxolo8,5)4 gǼ1/M;u2VGHUo]e]a܋$bRZqAكF`V_#2NCdsN yʫ^h<{SƋ і49኷g{+{N(?H/N89m\S'Lh |SP`_BxXp6ިڎ+m]g[x5M\3 Dd 0B $ S A? #2u>Z5Aut(OqOQ:l `!I>Z5Aut(OqOjkxUOQymii-DJK<" Zk"/&bN$Ch<=x '/^@7/ĐD3otwQltNg}}yP+*T"x艢lJy<&=)V UeSuΫ h Xy6{U*FV&JK +wx#b[x=W>PL3F875 NϦzDϦKi"kS!ûĽT6+M奅AmN!BϮm4Ϯa\-=a{9ŠA;0W^ UAtSv9}n- Ll>Z$+isAt_qi7"IRM Z =IRG 0 SN:l RB@aՠFcb +P!;qJLQʊhd x?`KdգQRlʣVf G7 zg C l^1}oZX~eq\#O"~z=Dd (B % S A ? $2~6oל'jn9ZEp `!R6oל'jn9 @ h xSMo@}v$ +D*&1 QMP$RU$HD!Iќ(!nĩ/p_B!0k;VYB p9t~blIJ` WNK5~6ޑN{ > 1܄J勀z ;zAJc6i.&?‰\eV](SU7u%&> {¢'vvV(&k5VުA:6i-Q:{o52^gA"֜QyLvveÆx+VilH3rƦ7Ru6ثrc)b$xpĥi& *r",W%0|F#Z|yE799S{r#ULL̫ٱ,8[ ~؟$$If!vh55555#v:V l054$$If!vh55555#v:V l054$$If!vh55555#v:V l054$$If!vh55555#v:V l054$$If!vh55555#v:V l054SummaryInformation(SDocumentSummaryInformation8[XCompObjajotaApplied Mathematics VA 26lMicrosoft Word 10.0@v#@9@<L@Rx՜.+,0( hp  SUNYSB. hA ;Repeated Measures ANOVA versus Multivariate ANOVA (MANOVA) Title  FMicrosoft Word Document MSWordDocWord.Document.89q<@< Normal_HmH nHsH tHP@P Heading 1$<@&5CJKHOJQJN@N Heading 2$<@&56CJOJQJDA@D Default Paragraph FontVi@V  Table Normal :V 44 la (k@(No List 2B@2 Body TextCJLP@L Body Text 2B*CJOJQJhtH uuJ,de*%Z s u %+,.02478:<?ACDFILORSUWY[]^_` 2LmrtD789 lQ*/FGw0000000000-0-0-0-00 0000000000000000000000000000000000000000000000000000 00000000000000000 0 0 000 00 0 000 00 0 00 0 de*%Z s u %+,.02478:<?ACDFILORSUWY[]^_` 2LmrtD789 lQ*/FGw 000000000-0-0-0-00 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 000000000000000000000 0000000000000@0@00 0 000 00 0 00 00 0 00 00(0 02 p X eu! "$+7CR]lu!!#u!c{}24/1Woq02=UWZrt} 4LNTln   2 4    0 2 5 M O V n p 3 K M u:::::::::::::::::::::::::::::::::::::HLDHL< HL!HL̦BBwddw:*urn:schemas-microsoft-com:office:smarttagsStreet;*urn:schemas-microsoft-com:office:smarttagsaddress9*urn:schemas-microsoft-com:office:smarttagsplace8*urn:schemas-microsoft-com:office:smarttagsCity 0:x{UV l t u  EJek[dfpeuLQ|~ 6@w`cBJ  1 3  27LTmpVdw333333333333333333333333 `9wwAMSAMSAMSAMSAMSAMSAMSAMSAMSApplied Mathematics Yo hh^h`o(.YotBM\6Z s %+,.02478:<?ACDFILORSUWY[]^w@+}up@UnknownG: Times New Roman5Symbol3& : ArialC5  SAS Monospace?5 z Courier New"qhcӻrYdF . .!24hh3H(?tB:Repeated Measures ANOVA versus Multivariate ANOVA (MANOVA)AMSApplied Mathematics