Department of Epidemiology and Biostatistics, McGill University. EPI (Inferential Statistics) Midterm Examination, May 31, 1994

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1 INSTRUCTIONS Dprtmnt of Epimiology n Biosttistis, MGill Univrsity B rif n W R I T E C L E A R L Y. EPI (Infrntil Sttistis) Mitrm Exmintion, My 31, 1994 Unlss spifilly sk for, omplt lultions [or vn omplt sntns] r not n. Answr in point form whn possil. Answrs [intifi vi nom--plum known only to you] to hn in t/for th ginning of lss on Tusy Jun 7. W will olltivly orrt/gr som of th work in lss. 1 Tru or fls, n xplin rifly. f g h i j k If you 7 to h vlu on list, you 7 to th SD. If you oul h vlu on list, you oul th SD. If you hng th sign of h vlu on list, you hng th sign of th SD. If you uplit h vlu in list, you lv th SD pproximtly unhng. If ll th vlus in list r positiv, you nnot hv SD whih is lrgr thn th mn. Hlf th vlus on list r lwys low th mn. In lrg list, th istriution of msurmnts follows th norml urv quit losly. If two lrg popultions hv xtly th sm vrg vlu of 50 n th sm SD of 10, thn th prntg of vlus twn 40 n 60 must xtly th sm for oth popultions. Th vrin of th sum of two rnom vrils is lwys iggr thn th vrin of h on. An rsrhr hs omputr fil of pr-trtmnt WBCs for ptints. Thy rng from 2,800 to 38,600. By int, th highst WBC gts hng to 386,000. This ffts th mn ut not th min n th IQR. For th mn in lrg U.S. smpl survy ( th HANES stuy), mn inom in th iffrnt g groups inrs with g until 50 or so n thn grully lin. Thus, th inom of typil mn inrss s h gs until 50 or so n thn strts rsing. l Suppos ll stunts in lss of 20 got th sm wrong nswr to multipl hoi xm qustion with 4 hois. To tst whthr th stunts ollu [ont trihé] whil th monitor ws out of th room for 2 minuts, th shool prinipl lult th proility tht rnom vril Y with Binomil(20,0.25) 0.25 x 0.75 istriution woul 20. H i this y first lulting µ=20(.25) = 5 n SD=. H 20 thn lult Pro[ Z 20 µ ] n, fining tht th P-vlu ws vry smll, h onlu tht th SD stunts h "lmost rtinly" ollu. [Hint: thr r svrl prolms; onntrt on th min lultion rror n lso on th iggr prolm of possil logil rror in infrn; ignor th issu of ontinuity orrtions n th ury of th Gussin pproximtion] Cours MiTrm pg 1 / 6

2 2 Rfr to th lttr to th BMJ from lft-hn mil sttistiin onrning srious is in th omprison of gs t th of lft-hn n right-hn prsons. Th sm point woul pply, vn mor rmtilly, if w wr to ompr g t th of prsons who wnt through 'th nw mth' urriulum in lmntry shool with g t th of thos who h th 'ol mth' urriulum (th 'nw mth' urriulum ws introu into wstrn ountris t vrious stgs in th 1960's n 1970's). It woul lso pply to omprison, vi th oitury olumns of th mil journls, of g-t-th of riologists (thirs is long stlish spilty] n mrgny-miin spilists (n mrging spilty). To monstrt tht you unrstn Pto's point,... onstrut rlisti 2-wy tl sriing th g istriutions in ths two typs of prsons {l/r or, if you prfr, nwr/olr} in 1994 popultion [or s Pto looks t it, th prvlns of ths two typs of prson s funtion of g]. To kp it simpl, limit yourslf to on gnr. Apply th sm g-spifi th rts to th two typs of prsons [if you wish, you n us th th rts riv from 1990 Qu mortlity t givn in pg 3 of th mtril on M&M $4.1; you n still mk th sm point if you us fwr g tgoris to ru th mount of rithmti]. Thn, for h of th two typs, lult th vrg g-t th of thos tht i in th nxt svrl yrs. How ig iffrn o you gt in th "vrg g t th" of th two typs of prsons? Commnt. 3 In th ightnth ntury, yllow fvr ws trt y ling th ptint. On minnt physiin of th tim, Dr. Bnjmin Rush, wrot: I gn y rwing smll quntity t tim. Th pprn of th loo n its ffts upon th systm stisfi m of its sfty n ffiy. Nvr for i I xprin suh sulim joy s I now flt in ontmplting th suss of my rmis... Th rr will not wonr whn I short xtrt from my notook, t 10th Sptmr, 1793]. ''Thnk Go, of th on hunr ptints, whom I visit, or prsri for, this y, I hv lost non.'' Explin som of th sign prolms in Rush's stuy. 4 A snil strts out to lim wll. During th y it movs upwrs n vrg of 22 m (SD 4 m); uring th night, inpnntly of how wll it os uring th y, it slips k own n vrg of 12 m (SD 3 m). Th forwr n kwr movmnts on on y/night r lso inpnnt of thos on nothr y/night. Aftr 16 ys n 16 nights, how muh vrtil progrss will it hv m? Wht is th hn tht, ftr 16 ys n 16 nights, it will hv progrss t lst 150 m? Ovr n ov th ssumption of inpnn, whih ws 'givn', i you hv to mk strong [n possily unwrrnt] istriutionl ssumptions in orr to nswr prt? Explin rfully. Cours MiTrm pg 2 / 6

3 5 A ovrviw of rnomiz linil trils of ntipltlt thrpy s prophylxis ginst p vnous thromosis [BMJ on 22 Jn. 1994] foun th following: Ctgory of tril % os rution (SD) gnrl surgry 37% ( 8) trumti orthopi surgry 31% (13) ltiv orthopi surgry 49% (11) Is 37% sttisti or prmtr? Why? Dos h SD rfr to (i) vrition of iniviuls or (ii) smpling vrition ssoit with th stimt? Explin your rsoning. Us th SD of h stimt to rgu tht th pprnt htrognity in th prnt rutions, i.. th spr from 31% to 49%, oul simply rflt rnom vrition lon [iffrns mong thr stimts r mor iffiult thn w hv lrn to lt with, so for simpliity, onntrt on th iffrn of two stimts] Sin w hv nithr sttistil nor iologi sis for ssuming iffrnt siz ffts for iffrnt typs of ptints, in th spirit of Om's rzor, w n onstrut on ovrll stimt from th thr. On wy to o this is tk simpl vrg of th thr rutions, giving h stimt wight of 1/3 i.. ( )/3 = 39%. If w rt this qul-wight vrg, th unrtinty {SD} ssoit with it shoul smllr thn th SD of th omponnts. Clult th SD for 1 3 stimt stimt stimt 3 [M&M 4.3 & xriss shoul hlp] Sin w hv thr stimts with iffrnt grs of unrtinty, it mks sns to lult n vrg of thm whih givs mor wight to th iniviul stimts with smllr SD's. It n shown mthmtilly tht th wight vrg with th lowst SD is th on with wights tht r invrsly proportionl to th iniviul vrins. In our xmpl hr, this woul l to wights proportionl to 1 64, n rsptivly, or n ovrll stimt of 0.52stimt stimt stimt 3. This givs wight vrg of just ovr 39%. [th ft tht th two mthos giv lmost th sm nswr is oinin in this xmpl; it osn't hppn gnrlly] As stt, this informtion-wight vrg hs lowr unrtinty {SD} ssoit with it thn simpl qully-wight vrg. Clult th SD for 0.52stimt stimt stimt 3 ; ompr it with th SD in ov. Th ovrviw rport n stimt of 42% (SD 17) for high risk mil ptints. f Comin th singl stimt for surgil ptints n th 42% for mil ptints. Your ovrll stimt shoul losr to 39 thn to 42%. Why? Clult its SD. Why os th 'vrging' of th two stimts not iminish th SD vry muh? Commnts: 1. It might prfrl (in trms of mor urt CI's) to omin th stimts on log sl, whr th smpling vrition shoul mor Gussin; 2. th ov mtho of omining stimts (n prtiulrly of lulting SD's, or SE's if you prfr) is not pproprit if thr is finit htrognity in th stimts (lso, th vrg hs muh lss mning). Cours MiTrm pg 3 / 6

4 6 A hlth prtmnt srvs 50,000 houshols. As prt of survy, simpl rnom smpl of 400 of ths houshols r survy. Th vrg numr of ults in th smpl houshols is 2.35, n th SD is 1.1. Skth possil frquny istriution showing th vriility in th numr of ults pr houshol [on't spn lot of tim on tril n rror gtting th istriution to mth th mn n SD xtly; if you n show on whih oms within 0.1 of th mn n 0.2 of th SD, tht's goo nough] If possil, fin n pproximt 95%-onfin intrvl for th vrg numr of ults in ll 50,000 houshols, n from it n pproximt 95%-onfin intrvl for th totl numr of ults in ll 50,000 houshols. If this isn't possil, xplin why not. All ults in th 400 smpl houshols r intrviw. This mks 940 popl. On th vrg, th smpl popl wth 4.2 hours of tlvision th Suny for th survy, n th SD ws 2.1 hours. If possil, fin n pproximt 95%-onfin intrvl for th vrg numr of hours spnt wthing tlvision y ll ults in th 50,000 houshols on tht Suny. If this isn't possil, xplin why not. 7 Nw lsr ltimtrs n msur lvtion to within fw inhs, without is, n with no trn or pttrn to th msurmnts. As prt of n xprimnt, 25 rings wr m on th lvtion of mountin pk. Thir mn ws 81,411 inhs, n thir SD ws 30 inhs. Fill in th lnks in prt (), thn sy whthr h of () is tru or fls; xplin your nswrs rifly. (You my ssum Gussin vrition of th msurmnts, with no is.) Th lvtion of th mountin pk is stimt s. Thr is pproximtly % hn tht w r ovr-stimting or unr-stimting th tru lvtion y mor thn 6 inhs. 81,411 ± 12 inhs is 95%-onfin intrvl for th vrg of th 25 rings. 81,411 ± 12 inhs is 95%-onfin intrvl for th lvtion of th mountin pk. A lrg mjority of th 25 rings wr in th rng 81,411 ± 12 inhs. Th lvtion of th mountin pk is th sttisti hr; th 81,411 is prmtr. 8 An invstigtor t lrg univrsity is intrst in th fft of xris in mintining mntl ility. H is to stuy th fulty mmrs g 40 to 50, looking sprtly t two groups: th ons who xris rgulrly n th ons who on't. Thr r lrg numrs in h group, so h tks simpl rnom smpl of 32 from h group, for til stuy. On of th things h os is to ministr n IQ tst to th smpl popl, with th following rsults: rgulr xris no rgulr xris smpl siz vrg sor SD of sors Th iffrn twn th vrgs is "highly sttistilly signifint". Th invstigtor onlus tht xris os in hlp to mintin mntl ility mong th fulty mmrs g 40 to 50 t his univrsity. Stt th null n ltrntiv hypothss, lult th p-vlu n vrify th sttmnt out th iffrn ing "highly sttistilly signifint". Is th uthor's onlusion justifi? Why/why not? Cours MiTrm pg 4 / 6

5 9 An invstigtor wnts to show tht first-orn hilrn sor highr on voulry tsts thn sonorns. Sh will us th WISC voulry tst (ftr stnrizing for g, hilrn in gnrl hv mn of 30 n SD of 10 on this tst). Sh onsirs two stuy signs: i ii In shool istrit fin numr of 2-hil fmilis with oth 1st-orn n 2n-orn nroll in lmntry shool. From shools in th istrit, tk smpl of 1st-orn n smpl of 2n orn hilrn nroll in lmntry shool. List 1 sttistil n 1 prtil vntg of h pproh. For th sign you prfr, wht woul you rommn s sttistil tst of th hypothsis? For th sign you prfr, n ssuming sh tlls you tht iffrn of 3 points on th stnriz tst woul importnt, trmin n pproprit smpl siz. If you on't hv suffiint informtion to mk th trmintion, xplin to hr xtly wht sh ns to provi you for you n trmin th smpl siz. 10 Consir RCT tht l to th rommntion of lumptomy n rition s n qully fftiv ut lss isfiguring ltrntiv to msttomy in trting rst nr. In th originl stuy thr wr thr trtmnt groups: totl msttomy (n = 590), lumptomy (n = 636), n lumptomy n irrition (n =629). At th n of th follow-up prio (vrg 81 months), th numrs liv with no vin of iss wr: totl msttomy 373 (63.2%), lumptomy 371 (58.3%) n lumptomy n irrition 412 (65.5%). [I hvn't hk ths numrs; thy, n qustions - tht follow r tkn from n rtil in Chn Nws 1 ] Clult mrgin of rror ssoit with h of th prntgs liv with no vin of iss. Likwis, lult mrgin of rror ssoit with th iffrn of th first n thir prntgs. Stt your lvl of onfin tht th rrors in th stimts r no mor thn wht you hv lult. Wht r th most importnt ssumptions r you mking in lulting ths limits of rror? Wht woul th fft on ths mrgins of rror if th t on rnom 19% of th stuy sujts wr rmov? Crry out th lultions. Suppos tht som womn nroll wr thnilly inligil for th stuy, lthough th rnomiz ssignmnt n follow-up wr proprly rri out in n unis wy. Th rsrh group si tht nw nlysis, with th t on 19% of th ptints rmov, shows tht th stuy's originl pulish onlusions rmin vli. But govrnmnt spoksmn rmrk tht rmoving 19% of th smpl iminish th sttistil powr of th stuy. Wht os this lttr sttmnt mn? You wr sk to prtiipt in iing th smpl siz for nw two-rm stuy to rvisit th qustion of totl msttomy vrsus lumptomy n irrition. Givn th intns puli intrst in th nw tril, th onologists in th rsrh group sk you, s th most sttistilly rtiult, to provi thnilly urt intrprttions of th 3 Grk symols (α, β, ) in th smpl siz formul tht woul unrstnl to journlists n ut non-xprts in sttistis (or for tht mttr in linil trils). You might lso intrviw y lol tlvision sttion. Prpr suh n intrprttion, limiting yourslf to 200 wors [or 150 sons of 'soun its'] in totl. If you r fml, wht vlu of o you think shoul us? If you r ml, sk som (sttistilly?) signifint fml in your lif (who hsn't tkn ours in sttistis) wht vlu of shoul us. How woul you wor your qustion to hr? 1 Prpr y J. Luri Snll s prt of th CHANCE Cours Projt support y th Ntionl Sin Fountion n th Nw Engln Consortium for Unrgrut Sin Eution. Currnt n prvious issus of CHANCE Nws n foun on th intrnt vi gophr to: hn.rtmouth.u. Cours MiTrm pg 5 / 6

6 11 Rfr to th rtil "Hir onntrtions of niotin n otinin in womn n thir nworn infnts y Eliopoulos t l (JAMA 1994; 271: ). f g h i j Th uthors stt tht th smpl siz ws hosn to tt twofol mor otinin in infnts of pssiv smokrs thn infnts of non-smokrs [lst prgrph of Sujts n Mthos]. This "twofol mor otinin", roughly spking, orrspons to iffrn of 0.3 twn mns in th log 10 (onntrtion) sl, sl on whih th osrvtions r mor nrly [ut still not quit] Gussin thn in th onntrtion sl. Suppos tht thir pr-stuy informtion ws tht th twn-infnt SDs on this log 10 otinin sl woul pproximtly 0.4 for h of th two groups ing ompr. Assuming thy wr going to rruit qul numrs of pssiv smoking n nonsmoking mothrs, n with th lph n powr thy mntion, how mny of h woul rquir? If otinin msurmnts wr Gussin on th log 10 sl, woul thy Gussin on th ln i.. log sl? Not tht log 10 (otinin ) = log (otinin ) = ln(otinin ). For th 36 tiv smoking womn, th mn numr of igrtts us ily ws Wht ws th SD? Why woul this twn-womn SD of littl us in sriing th pttrn of twnwomn vrition in rport onsumption [stt to hv vri from 1 to 40]? In th lst sntn of th first prgrph of Rsults, wht o (i) th sttmnt tht "r=.75" n (ii) th wor "signifint" mn? Why o you think "thr ws no orrltion twn th ily numr of igrtts rport y th mothrs n ithr mtrnl or nontl onntrtions of niotin or otinin"? Put th sttmnt "P<0.001" [ftr th r=.49 t top of thir olumn] into wors tht ths prnts woul unrstn. Don't us th irulr xplntion tht us P<0.001, it is "signifint". "Mtrnl onntrtions of niotin wr invrily highr thn nontl lvls (P<0.001)" [nxt sntn]. Sin this rtinly isn't th s for ll 94 mothr pirs in Figur 1, th uthors must rfrring only to th n=36 pirs whr th mothr smok. Th uthors on't sy in thir sttistil mthos stion wht tst thy us to lult this p-vlu [thy only rfr to tsts for 'twn groups']. Wht 2 tsts of hypothss tht r ovr in M&M Ch 7 wr vill to thm? Extly wht hypothsis os h on tst? In plin wors, wht is mnt y th phrs "onntrtions of otinin i not iffr signifintly twn mothrs n infnts"? Th primry npoint of intrst ws stt to infnts' hir onntrtion of otinin, n th smpl siz lultion onntrt on th pssiv smoking vrsus non smoking mothrs. Mn {SEM] onntrtions of otinin in infnts of pssiv smoking n nonsmoking mothrs wr 0.60[0.15] n 0.26[0.04] rsptivly. Th uthors sy tht ths onntrtions wr signifintly iffrnt. Just from th numril summris {mn[sem]} thy provi, n you n prform sttistil tst to vrify this? Do you fl omfortl rrying out this tst? Why/Why not? If not, n if you h ss to th til t, wht othr options woul you propos? Th Figur lgn osn't sy, ut wht o th rror rs in Fig 3 rprsnt? Woul you hv us somthing ls? Why/Why not? Cours MiTrm pg 6 / 6