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# Comp Analys Hlth Sci PUBHLTH 744

UMass

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This 14 page Class Notes was uploaded by Agustin Bechtelar on Friday October 30, 2015. The Class Notes belongs to PUBHLTH 744 at University of Massachusetts taught by Staff in Fall. Since its upload, it has received 21 views. For similar materials see /class/232292/pubhlth-744-university-of-massachusetts in Public Health at University of Massachusetts.

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Date Created: 10/30/15

If we additionally assume 6 N MVNOUZI then we have P MY N MVNA B02A X X A For X full rank 3 is estimable and the unique least squares estimate is 3 X X 1X Y We then have the following results EltBgt EX X 1X Y X XVlX X 0mg CovX X 1X Y X X 1X 021XX X 1 02X X 1 Therefore 3 N MVNW7 02X X 1 Definition A random variable X is said to have a non central chi square distribution with 71 degrees of freedom and non centrality parameter 39y written X N xi if the density of X is given by l 2in2 Y 5 7 1 lt1 m2in27leim2 u F2i n2 5 view 9MB 2 where is a central chi square density with 2i i 71 degrees of freedom Theorem Suppose Z17 7Zn are independent and identically distributed iid N01 random variables Further suppose X ELI Z12 then X N xi Theorem Suppose Y17 Yn are independent random variables and K N NM702 Further suppose X i Egg12 then X N xi where 39y i 21 12 Definition 84 Suppose X X17 Xn then X is said to have an i 439 39 39 multivariate normal distribution with mean u and covariance matrix 2 if the density of X is given by fX 2w 2E 1Zezp212x a WWW a W Note this requires 2 to be positive definite because a non positive definite matrix has 0 determinant distribution of X1 is given by X3 X1 1 2 1 X3 gt MVN2ltlt 3 gt7lt1 2 The conditional distribution of X1 7 3 7 x2 1 1 X3 iX272MVN2 3 7 1 2 Note that the dependency on X2 only enters into the mean Definition Suppose X N Nu1 Y N xg and X L Y independent Define X MYn W We say W has a non central t distribution with 71 degrees of freedom and noncentrality parameter In written W N tn MMMMMMMMMMMMMMH Code for generatlng dlstrlbutlons MMMMMMMMMMMMMMH Generatlng 7100 11 standard normals 7 rnorm10001 rt 87379699 85172423 40972669 7 Y 61282437 7 57188648 93248953 91647985 7190535191 71 71 71 71 7148920480 7143521924 71 7 46194700 70 7044 4891 7 40259803 7 35271715 21722945 13131978 008058516 7 701 0 104 20802666 24500979 1 7 47040024 50033783 62907528 2 18069509 00860093 27398517 58458201 02494443 29533138 61202978 0 0 0 0 0 1 1 1 0 0 0 0 1 1 1 Plottlng a hlstogram of the data gt postscrpt hlstyps gt h1sty gt devoff Plottlng the kernel denslty estlmate of the gt postscrpt kdensps gt a lt7 dehs1tyy gt plotd gt devoff Determlng the 1 and 2aelded pavaluee for a ggven quantlle of the normal denslty mlt196mean0ed1 gt pnom1645 1 0950015 gt pnorm196lowertallFALSE 1 002499790 lug pava ue based on our slmulatlon t Determln l gt sumsortygt196100 1 001 gt sumsortabsygt1961OO 1 002 Determlnlng quantlle based on alpha level 1 i t Generatlng data from a central ehrequaae dlstrlbutlon wlth 1 df gt z lt7 rchlsq100df1ncp0 002048204 0003072549 0004346618 0005647811 005720099 009924723 056679650 0057686002 0060986913 0061056146 063447509 070116520 080560111 0092825148 0094776029 0100120143 107496821 120582615 130370445 0144391901 0176185342 0177759035 184747385 193150146 67 0957246214 1053018552 1079146476 1092246818 126413971 144025531 73 1166079491 1186530011 1193458213 1281009129 1296192680 1821850292 2361217204 2498717055 9105024880 3259695121 97 4410998643 4585548827 6961171162 9475800952 otnng dens1ty of z gt postscrpt kdenszp5 gt plotdenstyz gt devoff Plomng denslty from a nonacencaal chrsquare dlstrlbutlon scrpt kdenszps gt pamme d 2Hthco3o ylmc06ma 6 ma 7 so 11 10 clto gt devoff Deternung pavalue based for a glven quannle gt pchlsq384df1 09499565 gt 1apchasq384df1 1 005004352 Deternung pavalue based on smulaced data gt n lt7 1 o o o gtSum hlsqndf1gt384n 7 re 1 004 Deternunlng quannle for a glven alpha W note need to enter 1aa1pha gt 1 3841459 Smula 11 16 21 61 66 61 66 71 76 81 86 91 96 Smul 66 mag data a2771532811 a1 862481568 71136329168 7 70238610579 7 70172425342 70 008044007 0 181307807 0 530807512 0648848388 0 885273843 1224383157 1625088758 2407508873 anng data 0245444030 0285885588 0 328043574 1383710810 75 317224454 73479384553 72645335360 648647138 160738788 0044135817 1834835846 2480672457 from a central 0 008621800 0 258742733 0 314610248 0 334318782 1400870431 74 082508553 531164826 72 264884127 428138728 122737125 7 222041838 7 833023032 234760868 austnbut 010747406 278520462 314613674 347886478 430882842 from a central tad1stnbunon 74 828580481 72166634696 7 275184502 117262515 7 114338428 7 180468478 7 125710363 052781865 968315 218534843 022579176 ulth 2 d 013382286 73593475750 878288046 011764516 181035099 018336300 8030 a 1 092072712 085668 388 56280735 568782133 875534175 145065576 568868650 818 2 130377858 8882601125 ulth df23 0 124468368 135415588 190652890 28546 2 32358 7 6480 360714018 496892053 664546045 860215345 343770554 449967085 486058841 606332087 962878867 866603269 769096263 387461008 515132140 959168702 293713o33 139919289 969124197 1626206893 3o7797904o 4401970666 7177774662 12752708337 634019336 119200066 738844767 416241410 149284430 643285923 277362654 917317852 656145543 254368858 Our null hypothesis might be that EY E CXo and the disjoint alternative is EY E CX and CX CXo If we let M and M0 be the orthogonal projection operators onto CX and CXo respectively then under the full model the UMVUE of EY is MY and under the reduced model the UMVUE of EY is MOY Suppose the reduced model is true This implies that MY and MOY are estimating the same quantity On the other hand if MY and MOY are different then they are not estimating the same quantity and the models must differ and so the reduced model is not correct That is the correctness of the reduced model depends on the quantity M7 M0Y We measure this by the squared length given by Y M7 M0Y Definition Suppose Ynxl is a random vector and 14an is a matrix of constants A quadratic form is a random variable given by Y AY We assume A is symmetric Theorem Suppose Y N MVNnOUZI We have WY x i 02 if and only if M is an orthogonal projection operator of rank r Theorem Suppose Y N MVNMMUZI We have 1 02 mm X if andonly if M is an orthogonal projection operator of rank r and 7 202 39 Theorem Suppose Y N MVNMMUZI then 1 Y AY and BY are independent if and only if AB 0 where A is symmetric Y AY and Y BY are independent if and only if AB 0 where A and B are symmetric

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