Inference for Multivariate Means

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1 Inference for Multivariate Means Statistics 407, ISU Inference for the Population Mean Inference for the Population Mean Inference for the Population Mean his section focuses on the question: This section focuses on the question: Inference This section focuses for on the the question: Population Mean Whether a pre-determined mean is a plausible value for the normal Whether population a pre-determined mean, iven mean theisissample a plausible thatvalue we for for This section focuses on the question: have collected. the normal population mean, iven the sample that we we have collected. Whether a pre-determined mean is a value for the normal population hypothesis mean, testin iven the sample lanuae, that we this have means collected. we want to test In hypothesis testin lanuae, this means we want to totest In hypothesis testin lanuae, this means we want to test H o : µ = µ H o : 0 vs H µ = µ 0 vs A : µ = µ H A : : 0 µ = = µ 00 here µ where where =(µ,µ µ =(µ =(µ 2,...,µ is true mean is the,µ,µ p 2 ),...,µ is the p ) true is is thepopulation the true true population mean population mean and mean and µ and 0 = µ µ 0 = hypothesized mean. 0 = µ 0,µ 02 (µ,...,µ (µ 0,µ 0,µ 02 0p,...,µ ) is the 02,...,µ 0p ) hypothesized is hypothesized mean. mean. 0p is the hypothesized mean. 2

2 Univariate Testin of a Mean Univariate of Recall, in the univariate situation, when we want to test Recall, in the univariate situation, when we want to test H o : µ = µ 0 vs H A : µ = µ 0 we we would calculate calculate the t-statistic, the t-statistic, as follows: as follows: Then we would compare t t with = X µ 0 s/ values n. from a t-distribution with (n-) derees of freedom, and reject Ho only if t is larer than the critical values. Then we would compare t with values from a t-distribution with (n ) derees of freedom, and reject H o only if t is larer than the critical values. 2 3 Confidence Interval for a Mean ould also We could compute also compute a 00( a 00(- α)% confidence interval interval for the for ould also compute a 00( α)% confidence interval fo opulation mean, µ, usin population mean, µ,, usin X ± t n (α/2) s n X ± t n (α/2) s n 3 4

3 Multivariate Testin of a Mean Multivariate Testin Testin of of a Mean a Mean Multivariate Testin of a Mean Mean The multivariate analo of the t-statistic is: Multivariate Testin of a Mean Multivariate Testin of a Mean he The multivariate multivariate analo analo of of the the t-statistic t-statistic is: is: The multivariate The multivariate analo analo of the t-statistic is: te analo The multivariate of the t-statistic analo T 2 of the t-statistic = of is: n( the X t-statistic µ 0 ) is: Sis: T 2 T= 2 n( = X n( T 2 X µ = µ n( X 0 ) S µ 0 n ) S ( X n ( µ X 0 ) 0 ) S n ( X µ 0 ) µ 0 ) T 2 = n( X µ 0 ) S It T 2 is = called n( X µ Hotellins 0 ) S n ( X µ 0 ) It is called It is called It ishotellins called Hotellins T 2 T 2 n ( X µ 0 ). t is called T 2.. T 2. It is called Hotellins T 2. n ( X µ 0 ) tellins follows an (Remember? In T follows 2 2. T 2 follows an (n )p F p,n p,α distribution. in Tfollows 2 follows an an an (n )p n p F p,n p,α distribution. (Remember that, in the univariate n p (n )p case, when t t n, then t 2 that, in n p F p,n p,α distribution. (Remember that, in F,n. Note in the he (n )p the univariate multivariate case, when t t analo, that the n, then t 2 F derees of freedom,n.). Note in the incorporate the n p F case, p,n p,α distribution. when t t (Remember t 2 n, then t 2 F that,,n F,n. Note in. Note in the in the multivariate dimensionality analo, that that of the the the data.) derees freedom incorporate the case, when t t n of, the then t 2 derees of offreedom incorporate the the F,n. Note in the dimensionality of of the the of data.) the data.) nalo, that the derees of freedom incorporate the 4 of the data.) T 2 follows an (n )p n p F p,n p,α distribution. (Remember that, in the univariate case, when t t n, then t 2 F,n. Note in the multivariate analo, that the derees of freedom incorporate the BRIEF ARTICLE 4 5 THE AUTHOR Multivariate Testin of a Mean Null hypothesis: H 0 : µ = µ 0 Alternative hypothesis: H A : µ = µ 0 Why aren t there one-sided alternatives? 6

4 Example: Sweat data Example: Sweat data Perspiration Perspirationofof20 20 healthy females for 3 components: Sweat Sweat Rate, Perspiration Rate, Sodium Sodiumof Content Content 20 healthy and and females Potassium Potassium for 3 components: Content. Sweat Rate, Sodium Content and Potassium Content SweatRate SweatRate Na Na K K X = 45.6 X S n = S n S S n = n = We want to test if We want to test H o : µ = (4 50 0) vs H A : µ = (4 50 0) We want to test if H o : µ = ( ) vs H A : µ = (4 50 0) Calculate: T 2 = T 2 = = =

5 Takin α =0.0, (n )p Takin Usin =0.0, (n )p n p F n p p,n p(0.0) = p,n p(0.0) = 8.8. Then Then we then would we would reject H we would reject o at the atho the at the 0% 0% 0% level level level of of of sinificance. sinificance. Takin Takin Usin α =0.05, =0.05, (n )p (n )p n p F n p p,n p(0.05) p,n p(0.05) = = Then Then we then we would Ho we would would NOT NOT reject reject H o at at the theat 5% 5% the 5% level level level of of of sinificance. sinificance Confidence Reions and Simultaneous Confidence Reions and Simultaneous Comparisons of Comparisons Confidence Component Reions and of Component Simultaneous Means Means Comparisons Means of fidence Reions and Simultaneous Comparisons of Confidence Reions Component and Simultaneous Means Comparisons of Component Means A 00( α)% confidence reion for the mean of a p-dimensional A 00( α)% normal distribution confidence is the reion ellipsoid for the determined mean of a p-dimensional by all µ such that A 00( α)% confidence reion for the mean of a p-dimensional normal distribution is the ellipsoid determined by all µ such that that normal distribution is the ellipsoid determined by all µ such that n( X µ) S n( X µ) n n ( p(n ) X µ) p(n ) X µ) n p F p,n p(α) p,n p(α) n( X n ( µ) p(n ) X S n µ) ( p(n ) X µ) n p n p F F p,n p(α) where X,...,X n are sample observations from a N p (µ, Σ) population.,...,x are sample observations from p (µ, Σ) pop- where where X,...,X n are sample observations from a N p (µ, Σ) population. population. This equation defines an ellipse in p-space. This This equation defines an ellipse in in p-space. p-space. in p-space. α)% confidence reion for the mean of a p-dimensional l distribution is the ellipsoid determined by all µ such that n( X µ) S X,...,X n are sample observations from a N p (µ, Σ) pop-. quation defines an ellipse in p-space

6 Lenths of the major axes. Lenths of the major axes. The shape of the confidence ellipse is dictated by the variancecovariance matrix, S. The orientation is iven by the eienvectors of S, and the lenths of the major axes of the confidence The shape of the confidence ellipse is dictated by the variancecovariance matrix,. The is iven by the eienvectors, ellipse ellipse are are iven iven by by and the of the major axes of the confidence ellipse are iven by p(n ) λ i p(n ) λ n(n p) F i p,n p(α) where λ i is the i th eienvalue of S. where is the i'th eienvalue. Lenths of the major axes. The shape of the confidence ellipse is dictated by the varia covariance matrix, S. The orientation is iven by the eien tors of S, and the lenths of the major axes of the confide n(n p) F p,n p(α) where λ i is the i th eienvalue of S. Inference for the Populati 9 9 This section focuses on the question: Inference for the Population Mea Example: Sweat data This section Whether focuses on a pre-determined the question: mean is a the normal population mean, iven t Multivariate Testin of a Mea have collected. Whether a pre-determined mean is a plausibl thethe normal multivariate population 90% confidence analomean, of reion theiven t-statistic the is: samp have collected. In hypothesis testin lanuae, this mean x data points + sample T 2 mean, = n( X µ 0 ) S n ( X µ 0 ) In hypothesis testin hypothesized lanuae, H o : mean, µ this = µ 0 means vs we H A wa : µ confidence reion (3D) It is called Hotellins T 2. where µ =(µ,µ 2,...,µ p ) is the true popu Notice H that is the (µ 0,µ 02,...,µ o : µ = µ confidence 0p ) is 0 vs H the hypothesized A : µ = µ 0 T 2 follows an (n )p mea n p reion, F p,n p,α which distribution. (Re the univariate corresponds case, when to the t t where µ =(µ,µ result 2,...,µ for p the ) ishypothesis true n, then t test. population 2 F m multivariate analo, that derees of freedom (µ 0,µ 02,...,µ 0p ) is the hypothesized mean. dimensionality of the data.) 2

7 Relationship between Interval and Hypothesis Test Hypothesis testin is equivalent to examinin the hypothesized value in relation to the simultaneous confidence ellipse. If the hypothesized value lies the ellipse, then the null hypothesis is. 3 Simultaneous Confidence Intervals Bonferroni Method vs T 2 intervals method for computin simultaneous confidence intervals Scheffe s: (n )p X i ± n p F p,n p,α s i n called Scheffe s method. These intervals are the confilipse projected into the coordinate axes. 4

8 0% t-interval Simultaneous Confidence Intervals s i X i ± t n, α 2 n o contain the true mean with probability ( α). in multiple (p) confidence intervals the joint probey contain the true means needs to be at least hieve this each interval will need to have a hiher t it contains the true mean. One method for enhe Bonferroni correction: Bonferroni: X i ± t n, α 2p s i n Inference for the Populati This section focuses on the question: 5 Intervals vs Reion Whether a pre-determined mean is a the normal population mean, iven t Multivariate Testin of a Mea have collected. The multivariate analo of the t-statistic is: In hypothesis 90% confidence testinreion lanuae, this mean x data points + sample T 2 mean, = n( X µ 0 ) S n ( X µ 0 ) hypothesized H o : mean, µ = µ 0 vs H A : µ confidence reion (3D) It is called Hotellins T where µ =(µ Bonferroni confidence 2.,µ 2,...,µ p ) is the true popu interval, reion T 2 follows (µ 0,µ 02 an,...,µ (n )p 0p ) is the hypothesized mea n p F p,n p,α distribution. (Re the univariate Ellipse case, extends whenoutside t t n of box,, then in t 2 F multivariatesome analo, directions. that Hypothesized derees of freedom dimensionality mean of is the inside data.) box always. 6

9 Paired Samples! the second value from the first and treat as a one sample problem.! Example: Municipal wastewater treatment plants are required by law to monitor their dischares into rivers and streams on a reular basis. Concern about the reliability of data from one of these self-monitorin prorams led to a study in which samples of effluent were divided and sent to two laboratories were divided and sent to two laboratories for testin. One half of each sample was sent to the Wisconsin state lab and the other half sent to a private commercial lab routinely used in the monitorin proram. Measurements were made on biochemical oxyen demand (BOD) and suspended solids (SS) were obtained, on n= samples.! Are the analyses producin equivalent results? 7 Data Sample BOD SS BOD SS BOD SS

10 Plots BOD SS Lab D 2= SS SS Lab Differences x SS S d =! Plot vs! Are the values for BOD the same from labs & 2?! Are the values for SS the same from labs & 2?! Plot in multivariate plots. +! Is there a systematic shift 222 from x one rep to another ! Plot BOD, alon BOD with sample mean and hypothesized mean.! How similar are the 99.3 sample mean 88.3 and hyp mean? SS Differences Lab BOD T = [ ] S d = 88.3Two 48.6 Sample Hotellins T 2 o Sample Two Sample Hotellins T 2 BOD F Two Sample Hotellins T 2 For two 2,9(0.05) samples Two = collected 9.47 Sample independently Reject Hotellins H o T. 2 from multivariate normal populations, For two N(µ samples 2 Two Sample Hotellins T, Σ),N(µ collected 2, Σ), independently the test statistic from multivariate for H 0 : n For two For samples two mal collected populations, collected independently 9.36 = from 3.6 multivariate normal normal populations, µ N(µ, Σ),N(µ , Σ),, the test statistic for for H 0 : µ = For µ 2 two vs Hsamples A : µ collected = µ 2 is: N(µ, Σ),N(µ 2, Σ), the test statistic for H independently from multivariate normal populations, = µ 2 vs H A : µ N(µ µ = µ 2 vs H A : µ, Σ),N(µ = µ 2 is: 2, Σ), the test statistic for H 0 : vs = µ 2 is: is ected independently from multivaria µ = µ 2 vs H A : µ = µ 2 is: µ Reject, Σ),N(µ T 2 =( X H o. 2 X, ) Σ), (( + the T test statistic f 2 T µ 2 is: 2 =( X X 2 ) (( =( X )S pooled ) T 2 =( X + )S X 2 ) (( n X + n 2 ) (( + )S( pooled X ) X ( 2 X ) X 2 ) n )S pooled ) n 2 ( X X X 22 )) n n n 2 n 2 2 is distributed is as distributed (n +n 2 2)p as is distributed is distributed as (n as n +n (n +n 2 2)p 2 p F (n +n 2 2)p 2)p n n +n +n 2 p 2 p F p,n F np,n +n +n 2 p 2 p,α F p,n +n and 2 p,α and p,n +n +n 2 p,α 2 p,α and and S S pooled S pooled = = n n +n n +n 2 2 S n pooled = n S pooled n n n +n 2 2 S 2. S +n 2 2 S = n + n n +n2 2 2 S n +n 2 2. S + n 2 n 2. +n 2 2 S 2. X 2 ) (( + )S pooled ) 22 ( X X 22 2 ) n n = 22

11 THE AUTHOR BRIEFn i ARTICLE # K! wij yij yi yij yi ni ni j Testin 3 " 83 " " > = for 8 6 if Equality of Variance-Covariance S i = % & K THE 0 if n AUTHOR i $ ' Test for Homoeneity of Variancecovariances H 0 : µ = µ 2 Bartlett s test for homoeneity of variances, for univariate data is: Covariance Matrix H A : µ = µ 2 H 0 : µ which To test eneralizes = µ % 2 the hypothesis to " S " > H 0 : Σ = Σ 2 =... = Σ aainst the alternative S = &K! that they n i 6 0 are n not 5 i if n i= all equal, SAS PROC DISCRIM uses a LRT test 'K 0H A : µ = µ 2 if n$ Λ = i= S i (n i )/2 H 0 : Σ = Σ 2 =... = Σ S pooled (n p)/2 M Statistic (extends Bartlett s univariate test.) No exact distribution, but for lare M % & ' χ 2 data can be used to obtain p-values. Also Box s M test, has no exact distribution. K0n" 5lo S " n =! i " 6lo S i if S > 0, there s R code for this. BUT, both tend to be K i= sensitive, SYSMIS to!! if S $ Comparin several Multivariate Means - MANOVA Comparin Comparin several several Multivariate Means - MANOVA Comparin several Multivariate Means - MANOVA Data: Data:! Data: samples samples samples of of sizeof of size nsize,...n n n,...n,...n ; p variables. ; ; p variables. ; p variables. Population:! Population: populations; populations; different different means means (µ, Population: populations;, different means,...,µ (µ (µ,...,µ,...,µ ), same ), same ), same covariance covariance (Σ), (Σ), multivariate multivariate normal. normal. covariance (Σ), multivariate normal.! Arises Arises Arises from from () from desined () () desined desined experimental experimental experimental studies, studies, studies, where where where there there are are there are multiple responses measured for one or more treatments, (2) in Arises multiple multiple from responses () responses desined measured measured experimental for one foror one more studies, treatments, more treatments, where (2) there in (2) are in classification problems, where we want to classify a new observation multiple classification classification into responses problems, a class, based problems, measured where we on two where or more for want we oneto measured want orclassify more to a variables. classify treatments, newaobser- observation into ainto class, (2) in classification vation problems, a class, based based on where twonor we two more want measured more to classify measured variables. a variables. new observation into a class, based on two or more measured variables

12 Recall Univariate ANOVA Recall Univariate ANOVA We have random samples X l,x l2,...,x lnl from N(µ l, σ 2 ), l =,...,. We are interested to know if the population means of Recall Recall Univariate ANOVA the roups are different, that is, if µ = µ 2 =...= µ. e We have The have We random problem have random issamples parametrized XX l,x l2 l, l,x into l2,...,x the followin lnl from model N(µ formulation: l=, We..., We are. We are interested are to toknow if if ifthe population population means of means the of of l, σ 2 ), ), l l =,...,.,...,. e the roups roups are are different, that that is, if is, is, if if µ = µ 2 =...=. µ. The problem is Xparametrized lj = µ + into the τ l followin + emodel lj formulation: he The problem problem is parametrized is parametrized into into the the followin model formulation: formulaon: with with the the constraint l= n lτ l = 0,, which leadsto tothethe null hypothesis null hypothesis notation: notation: X lj = µ + τ l lj X lj = µ + τ l + e lj H with the constraint o : τ = τ 2 =...= τ =0 l= n lτ l = 0, which leads to the null hy- constraint notation: ithpothesis l= n 29 lτ l = 0, which leads to the null hyothesis notation: H o : τ = τ 2 =...= τ =0 H o : τ = τ 2 =...= τ =0 The model is fit to the data by splittin each observation into The model is fit to the data by splittin each each observation into into components: components: X lj = X + ( X l X) + (X lj X l ), j =,...,n l ; l =,..., X lj = X + ( X l X) + (X lj X l ), j =,...,n l ; l =,..., and computin the sums of squares for each component: and computin the sums of squares for each component: and computin the sums of squares for each component: n l (X lj X) 2 = n l ( X l X) 2 n l n l (X lj X l ) (X 2 l= j= lj X) 2 = n l= l ( X l X) 2 n l + (X l= j= lj X l ) 2 l= j= l= l= j=

13 ANOVA Table ANOVA Table SourceTreatment (Between) SS SS Tr df Treatment (Between) SS Tr Error (Within) SS Res l= n Error (Within) SS Res l Total SS CorTot l= n l Total SS CorTot l= n l Then Then we we test test the hypothesisusin H o : τ = τ 2 =... = τ = 0 vs H A : not all equal 0, usin: F = ANOVA Table Source SS df Then we test the hypothesis H o : τ = τ 2 =... = τ = 0 vs H A : not all equal 0, usin: F = SS Tr /( ) SS Res /( l= n l ) F, n l (α) SS Tr /( ) SS Res /( l= n l ) F, n l (α) Multivariate ANOVA Multivariate ANOVA The correspondin model formulation is The correspondin model formulation is X lj = µ + τ l + e lj And the correspondin data decomposition where X lj, e l j are n l p matrices, µ, τ l are p-d vectors, ivin the correspondin data representation: X lj = X + ( X l X) + (X lj X l ), j =,...,n l ; l =,..., 32 26

14 H A : µ = µ 2 H 0 : Σ = Σ 2 =... = Σ The correspondin sums of squares χ The correspondin sums of squares 2 is iven by: is iven by: n l l= j= n l (X lj X)(X lj X) = j= n l (X lj X l )(X lj X l ) l= l= (X lj X)(X lj X) = l= j= n l ( X l X)( X l X) + l= = B + W n l ( X l X)( X l X) + n l l= j= (X lj X l )(X lj MANOVA Table Source SS df Treatment (Between) B Error (Within) W l= n l Total B + W l= n l 33 Testin the hypothesis, H o : τ =...= τ l 27 = 0 he bi complication Testinhere the hypothesis, Hthat 0 : µ = we µ are H o comparin : τ 2 =...= τmatrices l = 0 now nstead The bi Testin of sinle complication the numbers. here is hypothesis, The thatmatrices we are comparin H o : correspond matrices τ =...= toτ variance now l = 0 llipses instead The p-d of sinle bi space. complication numbers. So there The H A : isµ are matrices that = µ 2 numerous we are correspond comparin test statistics. tomatrices variance now The ellipses most instead p-d popular of space. sinleis So numbers. Wilks there are The lambda: numerous matrices test correspond statistics. to variance The ost popular The bi complication is Wilks lambda: here H 0 : is Σ that = Σwe 2 =... are = comparin Σ matrices now most instead popular ellipses in of sinle is p-d Wilks space. numbers. lambda: So there are numerous test statistics. The The matrices correspond to variance ellipses most popular is Wilks lambda: χ in p-d space. So there are numerous 2 test statistics. The most popular is Wilks' lambda: Λ = W W n l Λ = W (X lj X)(X lj B + W B X) = n + W l ( X l X)( X l X) + l= j= Λ = W l= B + W e bi complication here is that we are comparin matrices now tead of sinle numbers. The matrices correspond to variance ipses in p-d space. So there are numerous test statistics. The st popular is Wilks lambda: f ΛIf is Λ If too is too is too small small then then reject H. If o. If l= j= n l n= l = n is n lare, is lare, (n (n If Λ is too W p + )/2) (p ln W W small then reject H o is Λ distributed = W. If (p + )/2) ln is distributed as asχ B+W 2 χ as as χ 2 B+W 2 n l = n is lare, (n (n (p + )/2) ln W W (α). p( ) For p( ) p( ) smaller n, (p + )/2) ln B+W is distributed as χ n, B+W 2 (α). For. For smaller n, p( ) some combinations of p, the test statistic has an F distribution. ome combinations n, some combinations of p, of of p, the p, the test the test statistic has an has F distribution. an F distribution. some combinations of p, B the + W test statistic has an F distribu Λ is too small then reject H o. If n l = n is lare, (n + )/2) ln Testin the hypothesis, THE AUTHOR H o : τ =...= τ l = 0 Testin the hypothesis, H o : τ =...= τ l = 0 The bi complication here is that we are comparin matrice instead Testin of sinle the numbers. hypothesis The matrices correspond to var ellipses in p-d space. So there are numerous test statistics. If Λ is too small then reject H o. If n l = n is lare, (n W B+W BRIEF ARTICLE n l (X lj X l )(X lj X l ) (α). For smal is distributed as χ 2 28 (α). For smaller n p( ) 34

15 Note that we can also express Λ as Note that we can also express Note that Note we that can we can also also express Λ Λ as as as Λ min(p, ) = Λmin(p, ) min(p, ) = i= +λ i i= +λ i Λ = +λ i i= where where λ i ; i λ=,...,min(p, ) are the eienvalues of W i ; i ) are the eienvalues of of WB. B. values of W B.. Also note, Also note, bein bein based based on on determinants of the of the matrices matrices that that e Wilks matrices lambda Wilks Also that note, lambda isbein essentially isbased essentially on comparin determinants comparin volumes of volumes the matrices of the of the incorrespond- ellipses. lambda ellipses. Another is essentially Another way of calculatin of calculatin volumes of volumes the correspondin of ellipses of ellipses is the is the that correspond- Wilks f the products products ellipses. of the Another oflenths way lenths of the calculatin ofleadin the leadin volumes axes axes of ellipses (eienvalues). is the of ellipses is the products of the lenths of the leadin axes (eienvalues). envalues). where λ i ; i =,...,min(p, ) are the eienvalues of W B. Also note, bein based on determinants of the matrices that Wilks lambda is essentially comparin volumes of the correspondin ellipses. Another way of calculatin volumes of ellipses is the products of the lenths of the leadin axes (eienvalues) Other Test Statistics Other Test Statistics Other Test Statistics Other Test Statistics Roy s! Roy's larest eienvalue of ofbw BW Roy s larest eienvalue of BW Lawley! Lawley and and Hotellins: T trace(bw ) ) Lawley and Hotellins: T = trace(bw ) Pillai s trace: trace(b(b + W) ) Pillai s! Pillai s Pillai's trace: trace: V V = trace(b(b + W) W) ) )

16 Example: Dominican Republic Student Testin 82 Students, enrolled in one of Technoloy (23), Architecture (38), Medical Technoloy (2) prorams. The students were iven 4 tests: Aptitude, Math, Lanuae, General Knowlede. Is there is a BRIEF difference ARTICLE in the averae test scores for each student type? THE AUTHOR H 0 : µ = µ 2 H A : µ = µ 2 H 0 : Σ = Σ 2 =... = Σ χ 2 3 n l (X lj X)(X lj X) = l= j= l= n Summary l (X lj Statistics X l )(X lj X l ) l= j= n l ( X l X)( X l X) + n=82, n =23, n 2 =38, n 3 =2, p=4, =3 2 (n (p + )/2) ln W THE AUTHOR B+W THE AUTHOR X = X = X 2 = X = THE AUTHOR S = S = S = S 2 = S 2 = S 2 = S 3 = S 3 = Can we consider that S 3 = the population variance-covariances miht be equal? 32

17 Descriptive raphics Can we consider that the population variances miht be equal? Are the population means different? Apt Math Lan GenKnow 5 count Tech Arch MedTech StudType Tech Arch MedTech value Descriptive raphics Apt Corr: Corr: 0.52 Corr: 0.39 Can we consider that the population variance-covariances miht be equal? Are the means different? Lan Math Corr: 0.43 Corr: Corr: GenKnow Apt Math Lan GenKnow 34

18 Descriptive raphics Can we consider that the population variance-covariances miht be equal? Are the means different? 35 Test OR B = W =

19 S 2 = S 3 = Test µ = µ 2 = µ 3 τ = τ 2 = τ 3 = 0 H o : µ = µ 2 = µ 3 H o : τ = τ 2 = τ 3 = S 3 = OR B = THE AUTHOR W = S = > summary(manova(cbind(apt, Math, Lan, GenKnow)~StudType, S 2 = domrep), test="wilks") Df Wilks approx 67.5 F num 28.3 Df den 0.57 Df 68.2 Pr(>F) StudType e-07 *** Residuals 79 S 3 = H o : µ = µ 2 = µ 3 H o : τ = τ 2 = τ 3 = 0 B = Which variables? Analysis of variance on each variable W = Df Sum Sq Mean Sq F value Pr(>F) Apt e-07 *** Math ** Lan e-05 *** GenKnow * All variables have sinificant difference between means! In order of importance: 38

20 B = W = Which roups? Df Sum Sq Mean Sq F value Pr(>F) e-07 *** h ** e-05 *** Know * BRIEF ARTICLE 3 BRIEF ARTICLE Tukey s multiple comparisons Apt diff lwr lwr upr padj Arch-Tech MedTech-Tech MedTech-Arch Lan diff lwr upr p adj Arch-Tech MedTech-Tech MedTech-Arch Apt: diff lwr upr p adj Arch-Tech Math: MedTech-Tech Lan: MedTech-Arch GenKnow: BRIEF ARTICLE diff lwr upr p adj Arch-Tech MedTech-Tech MedTech-Arch diff lwr upr p adj Math diff lwr upr adj Arch-Tech Arch-Tech MedTech-Tech MedTech-Arch GenKnow diff lwr lwr upr p adj Arch-Tech MedTech-Tech MedTech-Arch diff lwr upr p adj Arch-Tech MedTech-Tech MedTech-Arch Summary In eneral, usin a multivariate test, such as, is preferable to multiple univariate tests, such as. This controls for the correlation structure in the data. Its possible to et non-sinificant results in the univariate tests, but sinificant results in the multivariate tests. That is, the univariate tests may not detect the differences, in the presence of correlation between the responses. 40

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