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# Class Note for PUBHLTH 744 at UMass(3)

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Date Created: 02/06/15

Linear models With applications in R PUBHLTH 744 Handout 8Disfn39huh39o77 Theory Instructor Andrea S Foulkes Division of Biostatistics and Epidemiolog UMass School of Public Health and Health Sciences Fall 2007 Sampling distribution of estimators Recall in the usual linear model when 00116 02 and VB is estimable where A P X the BLUE and UMVUE of P X is P MY We have the following results EP MY P MEY P MXB P X VB and COUPMY PMCO UYPM P Ma21MP UZP MP 02A X X A HandouLS 1 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 193 EX X 1X Y X XVlX X and 0mg CovX X 1X Y X X 1X 021XX X 1 02XX71 Therefore 3 N MVNW7 02X X 1 HandouLS 2 Theory of distributions Central and non central Chi square distributions Definition A random variable X has a central chi square distribution with 71 degrees of freedom written X N xg if the density of X is given by nZ mg 1 Wager2 HandouLS 3 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 m u P2in2 5 9MB where is a central chi square density with 2i i 71 degrees of freedom 1 2in 2 We 7 1 lt1gt V 2in27157z2 magma HandouLS 4 gt That is the non central chi square distribution is an infinite Poisson mixture of central chi square densities Equivalently the non central chi square distribution is a central chi square distribution with 2P l 71 degrees of freedom where P N P02380717 gt Non central X2 distributions are important for characterizing the distribution of tests statistics under alternative hypotheses gt Using moment generating functions ICBST if X N xi then EX n 2y and VarX 2n By HandouLS 5 gt Normal and multivariate normal distributions Definition A random variable X is said to have a normal distribution with mean u and variance 02 written X N N01 02 if the density of X is given by M lt2wgt12a1ezpltz 7 m2 lflu0and 02 1 then XN01 and we sayX hasa standard normal distribution HandouLS 6 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 HandouLS 7 Definition 84 Suppose X X17 Xn then X is said to have an n dimensional multivariate normal distribution with mean u and covariance matrix 2 if the density of X is given by fX 2w 2E 1Zezp x a WWW a W Note this requires 2 to be positive definite because a non positive definite matrix has 0 determinant HandouLS 8 gt Properties of MVN gt The moment generating function of X N MVan7 E is given by XO Ee X ezptM git2t gt A linear combination of MVNs is MVN That is suppose X N MVan7 E and define Y AX b where Arm and brxl are constants Then Y N MVNJAM bAEA HandoutS 9 gt Properties of MVN continued gt Proof of the last result is straightforward using moment generating functions gtyt Ee Y Ee AXI etbEetAX etbEeAtX e bgtXA t etbezp Atyu AtEAt exp iiAu b tAEAt But we know this is the mgf for a random variable with a MVN distribution mean Aub and Covariance AEA D HandouLS 10 gt Properties of MVN continued gt A linear combination of multivariate normals is multivariate normal That is if Xi N MVNnMEZ 239 11 7 k and Xi L Xj for 239 y j and Y a1X1 1 H aka for ai scalars then Y N MNVn 221 anti 21 am 1 gt Marginal distributions of MVNs are MVN Consider Y N MVNth 2 Partition Y into Y lt 1 gt where Y1 2 is 7 X 1 and Y2 is nir X 1 Further write it lt Z1 gt and 2 2 211 212 Then we have X N MVN 2 221 222 1 7 M17 11 and we say the marginal distribution of X1 is MVN gt Conditional distributions of MVNs are MVN Again suppose Y N MVNnuE ICBST YllYg yg MVNTM1E12E 2192M27 E112122321221 HandouLS 11 X1 Example SupposeX lt X2 gt MVN3ME where 1 1 0 1 0 The marginal 2 distribution of X gt is given by X3 X1 1 2 1 X3 gt MVN2ltlt 3 gt7lt1 2 The conditional distribution of lt igtiX2x2MVN2ltlt332gtlti Note that the dependency on X2 only enters into the mean HandouLS 12 gt T and non central t distributions Definition Suppose X N N01 Y N xi and X L Y independent Define X MYn We say T has a t distribution with 71 degrees of freedom written T N tn HandouLS 13 Definition Suppose X N Nu1 Y N xg and X L Y independent Define X MYn We say W has a non central t distribution with 71 degrees of freedom and noncentrality parameter In written W N tn HandouLS 14 gt F distribution Definition 8 Suppose X1 N xii The random variable 71 X2 N Xiz z and X1 L X2 Xlnl XZng is said to have a doubly noncentral F distribution with 711712 degrees of freedom and noncentrality parameters 39yl7 39yg written F N Fltn1n27172 If 39yg 0 then F is said to have a noncentral F distribution If m 0 and 39y2 0 then F is said to have a central F distribution F HandouLS 15 MMitMMitMMMMMMMMMMMH Code for geneaatmg ahetnhutmhe MMitMMitMMMMMMMMMMMH Generatlng 100 11d standard normals gt y lt7 ahoam1ooo1 gt soaty 1 7291292437 7193249953 7 7157199949 7159794143 13 7124213179 7120009597 19 7099590201 7094432424 25 7090444339 7090119079 31 7074057999 7099905114 37 7049194700 7044992399 43 7021722945 7017554997 49 7001991900 001939520 55 014759295 019442993 91 030993409 044997797 97 049909419 050179279 73 072237934 072530239 79 094045309 094999725 95 100990093 102494443 91 127399517 129533139 97 159459201 191202979 71 71 71 4 4 4 4 4 o HHHOOOOO 91947985 48920480 12994109 93197011 84471359 99893188 44574891 17419975 04992701 18587799 45041819 54509238 73179589 89989049 03877227 32427580 92907528 Plottmg a h1etogaam of the data h1sty gt postscrpt hlstyps gt gt devoff 71 71 71 4 4 4 4 4 o wHHooooo 90535191 43521924 05213599 92907528 84443919 53799744 40259803 13131978 07707743 20088112 47092395 54559739 75901810 93475729 04428198 38077299 18099509 Plottmg the kernel dehe1ty eetmate of the data d lt7 dehs1tyy gt postscrpt kdens ps gt gt plotd 71 71 71 a0 a0 a0 4 4 o HHOOOOO 87379999 40972999 02903979 92952735 80799821 49253338 35271715 08058519 10450990 20847479 48159957 59798922 81251957 99138390 20802999 47040024 85172423 30731359 02133102 91377382 75108915 49098706 34973077 08028638 13701799 22740782 49345912 95041410 81729919 99377979 24500979 50033783 Handout8 16 gt devoff t Determlng the 1 and 2aelded pavaluee for a ggven quantlle of the normal denslty gt pnom190mean0ed1 1 09750021 gt pnom1045 1 0950015 gt pnorm196lowertallFALSE 1 002499790 17 Determlnlng pavalue based on our slmulatlon gt sumsortygt196100 1 001 gt sumsortabsygt196100 1 002 t Determlnlng quantlle based on alpha level gt qnom005 1 71044954 gt qnom0025 1 71959904 17 Generatlng data from a central ehrequaae dlstrlbutlon wlth 1 df gt z lt7 rchlsq100df1ncp0 gt sortz 1 0002049204 0009072549 0004940019 0005047911 0005720099 0009924729 7 0011199900 0019720429 0017179920 0019099249 0029177999 0029799501 19 0095791405 0041095009 0041922199 0042209909 0049490599 0059171919 19 0050079050 0057090002 0000990919 0001050140 0009447509 0070110520 25 0090500111 0092925149 0094770029 0100120149 0107490921 0120592015 91 0190970445 0144991901 0170195942 0177759095 0194747995 0199150140 97 0210052179 0259571400 0272744919 0299779540 0904050799 0911129494 49 0997959995 0999975991 0407009212 0419449179 0417909405 0450492925 49 0509759145 0575915079 0579977299 0591094925 0599095922 0019210770 55 0025979017 0091099015 0711999921 0740797290 0749440205 0704227909 01 0900950042 0909591755 0922795010 0929210909 0995517751 0949999452 07 0957240214 1059019552 1079140470 1092240919 1120419971 1144025591 HandouLS 17 73 1166079491 1186530011 1193458213 1281009129 1296192680 1321850292 79 1382650102 1391320240 1428253314 1497552383 1532693308 1607659159 85 1610683975 1837818853 2263741748 2296067607 2361217204 2498717055 91 2539244569 2609111708 2921222928 3072399920 3105024880 3259695121 97 4410998643 4585548827 6961171162 9475800952 Plomng denslty of z gt postscrpt kdenszp5 gt plotdenstyz gt devoff Plomng denslty from a nonacencaal chrsquare dlstrlbutlon gt postscrpt kdenszp5 gt parmfrowc22 gt plotdenstyrchlsqSOOd gt plotdenstyrchlsqSOOd gt plotdenstyrchlsqSOOd gt plotdenstyrchlsqSOOd gt devoff 2Hxmco3oy1mco6ma1 p 1Hx1mco3oyhmco6ma1 ncp2 xlmc030 yhmco6 ma ncp4 xlmlt2030 yhmco6 ma Deternung pavalue based for a glven quantlle gt pchlsq384df1 1 09499565 gt 1apchasq384df1 1 005004352 Deternung pavalue based on smulaced data gt n lt7 1000 gt sumrchlsqndf1gt384n 1 0047 Deternunlng quantlle for a glven alpha W note need to enter 1aa1pha gt qchlsq006df1 1 000393214 gt qchlsq096df1 1 3841459 HandouLS 18 Smulatmg data from a central tad1stnbunon gt sortltrtlt100df2 1 6 11 16 21 26 31 36 41 46 61 66 61 66 71 76 81 86 91 96 Smulanng data from a central P gt sortrf10023 1 6 11 16 21 26 31 36 41 46 61 66 a6 a3 a2 a1 a1 a0 a0 a0 a0 a0 a0 a0 70 o o HHooo HHooooooooo 246461762 204597085 771532911 862491568 136329168 923179040 772015830 609144916 420285445 381616441 238610579 172425342 009044007 181307907 530907512 649849399 995273843 224393157 625088758 407508973 001228302 125144249 161863880 245444030 295895599 328043574 396380418 548205023 698160990 903560076 143767528 393710810 75 a3 a2 a1 a1 a0 a0 a0 a0 a0 a0 a0 0 HHHooo HHoooooooooo 317224454 179384553 645335360 649647139 122737125 836226167 686251317 599406425 412296917 360612612 222041939 160739788 044135917 248851536 556292116 696181966 006501414 505457601 834835946 490672457 009621800 132488251 178158533 259742733 314610248 334318782 422209832 622559862 726124658 938705110 230344837 400970431 74 a3 a2 a1 a1 a0 a0 a0 a0 a0 a0 a0 0 WHHHooo HHoooooooooo 531164926 74 092508553 72 264994127 72 429138728 71 117262515 71 834977129 70 659417862 70 584614940 70 405819852 70 339735203 70 180468479 70 125710363 70 052781865 307217103 558363987 829130428 114137911 552928897 933023032 234760869 d1str1but1on 010747406 134378779 188767036 279520462 314613674 347996478 442343556 649776788 849576487 980633582 266528526 430892842 0 mHHHooo HHHooooooooo ulth 2 df 013382296 73593475750 928580481 72878289046 166634696 72011764516 275194502 71181035099 114338428 71018336300 829707392 70787124604 642090510 70634601172 539935050 70531136182 396164353 70683440893 332170546 70251955796 179519030 70173642595 092072712 70082759760 085668389 0 151307672 456290735 0463244432 569792133 0576083043 975534175 0987076682 145065576 1162006371 568869650 1600674147 968315919 2 130377958 218534943 8982601125 ulth df23 022579176 0124469369 135415599 0156903413 190652890 0214303023 285467527 0286858659 323596480 0 327502219 360714019 0388461927 496892053 0 524252601 664546045 0667500328 860215345 0873134497 056599252 1074504111 343770554 1381751093 449967085 1496058941 HandouLS 19 61 66 71 76 81 86 91 96 mmwwwwHH 505332087 681348426 050614758 365206342 852878867 855603258 768086253 387461008 1 H48wwwHH 515132140 754867864 166247803 432308343 858168702 283713033 138818288 868124187 12 752708337 1 1625205893 1808022764 2290221120 2475553358 3077979040 4401970656 7177774652 1 1 2 2 3 5 7 5 634019336 920208365 303282595 765348959 119200066 738844767 416241410 149284430 3 mmwwwHH 643285823 875483875 321105885 862668262 277362654 817317852 656145543 254368858 HandouLS 20 Quadratic forms Motivation more in section on hypothesis testing Consider the linear model Y X3 l e where e N MVNn07UZI We are generally interested in testing a linear model against a reduced model We call our starting model the quotfull model and consider whether a simpler more parsimonious model is reasonable That is we consider the reduced model given by Y XO39yO l e where e N MVNn07UZI and CXO E CX For example in the regression setting we may have the full model given by EY X151 X252 and the reduced model given by EY X262 HandouLS 21 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 HandouLS 22 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 1 2 gar MY x if and only if M is an orthogonal projection operator of rank r HandouLS 23 Theorem Suppose Y N MVNMMUZI We have 1 gown X if and only if M is an orthogonal projection operator of rank r and M MM 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 2 Y AY and Y BY are independent if and only if AB 0 where A and B are symmetric HandouLS 24

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