Chapter 1 Calculating Sample Size in Anthropometry

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1 Chpter Clcultig Smple Sie i Athropometry Crie A. Beller, Bethy J. Foster, d Jmes A. Hley Abstrct Smple sie estimtio is fudmetl step whe desigig cliicl trils d epidemiologicl studies for which the primry objective is the estimtio or the compriso of prmeters. Oe my be iterested i the prevlece of overweight childre i give popultio; however, the true prevlece will remi ukow d cot be observed uless the whole popultio is studied. Sttisticl iferece is the use of sttistics d rdom smplig to mke ifereces cocerig the true prmeters of popultio. By choosig represettive smple, iferece bsed o the observed prevlece leds to estimtio of the true prmeter. But how my subjects should be smpled to obti ccurte estimte of the prevlece? Similrly, how my subjects should we smple to show tht this prmeter is differet from some fixed vlue? We first review bsic sttisticl cocepts icludig rdom vribles, popultio d smple sttistics, s well s probbility distributios such s the biomil d orml distributios. Priciples of poit d itervl estimtio, s well s hypothesis testig, re preseted. We cosider severl commoly used sttistics: sigle proportios, differeces betwee two proportios, sigle mes, differeces betwee two mes, d referece limits. For ech prmeter, poit estimtors re preseted s well s methods for costructig cofidece itervls. We the review geerl methods for clcultig smple sies. We first cosider precisio-bsed estimtio procedures, where the smple sie is estimted s fuctio of the desired degree of precisio. Next, lthough there is greter emphsis o precisio-drive estimtio procedures, we lso briefly describe powerbsed estimtio methods. This pproch requires defiig priori the differece oe wishes to detect, the desired sigificce level, d the desired power of the test. Smple sie estimtio procedures re preseted for ech prmeter, d exmples re systemticlly provided. Abbrevitios d Nottios N ME e Popultio sie Smple sie Mrgi of error Precisio C. A. Beller ( ) Deprtmet of Cliicl Epidemiology d Cliicl Reserch, Istitut Bergoié, Regiol Comprehesive Ccer Ceter, 9 Cours de l Argoe, Bordeux, Frce e-mil: beller@bergoie.org V.R. Preedy (ed.), Hdbook of Athropometry: Physicl Mesures of Hum Form i Helth d Disese, DOI 0.007/ _, Spriger Sciece+Busiess Medi, LLC 0 3

2 4 C.A. Beller et l. m m s s p p H 0 H A b p BMI DBP Popultio me Smple me Popultio vrice Smple vrice Popultio proportio Smple proportio Null hypothesis Altertive hypothesis Type I error rte Type II error rte 00p% stdrd orml devite Body mss idex Distolic blood pressure. Itroductio Smple sie estimtio is fudmetl step whe desigig cliicl trils d epidemiologicl studies for which the primry objective is the estimtio or the compriso of prmeters. Oe my be iterested i the prevlece of give helth coditio, e.g. obesity, i specific popultio; however, the true prevlece will lwys remi ukow d cot be determied uless the whole popultio is observed. Sttisticl iferece is the use of sttistics d rdom smplig to mke ifereces cocerig the true prmeters of popultio. By selectig represettive smple, iferece bsed o the observed prevlece leds to estimtio of the true prmeter. But how my subjects should be smpled to obti ccurte estimte of the prevlece? Smple sie estimtio c be either precisio-bsed or power-bsed. I the first scerio, oe is iterested i estimtig prmeter, such s proportio, or differece betwee two mes, with specific level of precisio. O the other hd, oe might oly be iterested i testig whether two prmeters differ. The smple sie will be estimted s fuctio of the sie of the differece oe wishes to detect s well s the degree of certity oe wishes to obti. To uderstd the process of smple sie estimtio, it is importt to be fmilir with bsic sttisticl cocepts. We first review sttisticl priciples, s well s geerl cocepts of sttisticl iferece, icludig estimtio d hypothesis testig. Methods for smple sie estimtio re preseted for vrious prmeters usig precisio-bsed d power-bsed pproches, lthough there is greter emphsis o precisio-drive estimtio procedures (Grder d Altm 986, 988 ). Most cocepts preseted i this chpter re vilble i itroductory sttisticl textbooks (Altm et l. 000 ; Armitge et l. 00 ) d texts focusig o the methodology of cliicl trils (Mchi et l. 997 ; Friedm et l. 998 ; Sckett 00 ; Pitdosi 005 ). We refer the iterested reders to these works.. Bsic Sttisticl Cocepts.. Rdom Vrible A rdom vrible ssigs vlue to ech subject of popultio, such s weight, hir colour, etc. By rdom, it is implied tht the true vlue of the vrible cot be kow util it is observed. A vrible

3 Clcultig Smple Sie i Athropometry 5 (for simplicity, we will ofte discrd the term rdom throughout the rest of this chpter) is either qutittive or qulittive. A qutittive vrible is oe tht c be mesured d c tke rge of vlues, for exmple, wist circumferece, sie, ge, or the umber of childre i household. Qutittive vribles iclude discrete d cotiuous vribles. A discrete vrible is oe tht c tke oly limited rge of vlues, or similrly, the possible vlues re distict d seprted, such s the umber of childre i household. O the other hd, cotiuous vrible c tke ifiite rge of vlues, or similrly, c ssume cotiuous uiterrupted rge of vlues, such s height or ge. A qulittive or ctegoricl vrible is oe tht cot be umericlly mesured, such s the presece or bsece of disese, geder, or the colour of hir. A dichotomous or biry vrible is oe tht c tke oe of two vlues, such s the presece or bsece of trit or stte, or whether or ot oe is overweight... Popultio Versus Smple Sttistics Suppose we re iterested i describig the sie of 0-yer old girls ttedig Eglish schools. Height i this popultio c be summried by vrious qutities, such s the me, the medi or the vrice. These qutities re clled popultio sttistics d re usully represeted with Greek letters. Uless ll 0-yer old girls ttedig Eglish schools re mesured, the true vlue of popultio sttistics, such s the me height i our exmple, cot be observed d is ukow. It is however possible to estimte the true vlue with some degree of certity. This ivolves rdomly smplig from the whole popultio of iterest. Bsed o rdom smple of 0-yer old girls ttedig Eglish schools, oe observes the distributio of heights i this smple d clcultes the observed me. Qutities derived from observed smple re clled smple sttistics, d re usully deoted usig Rom letters. Two rdom smples of equl sie will usully ot yield the sme vlue of the smple sttistic. The possible differeces betwee the estimtes from ll possible smples (coceptul), or betwee ech possible estimte d the true vlue re referred to s smplig vritio. As result, it is ot possible to coclude tht the observed smple me correspods to the true popultio me. By usig pproprite sttisticl methods, smple sttistics c be used to mke ifereces bout popultio sttistics. I the ext sectio, we preset commoly used sttistics...3 Summriig Dt..3. Ctegoricl Vribles Summriig ctegoricl vribles ivolves coutig the umber of observtios for ech ctegory of the vrible. These couts re usully referred to s frequecies. The proportio of such couts mog the totl c lso be represeted...3. Qutittive Vribles Cotiuous vribles c be summried usig mesures of loctio d dispersio. Mesures of loctio, such s the me or the medi, represet the cetrl tedecy of distributios. Dispersio mesures, such s the vrice, represet the reprtitio of vrible roud the cetrl tedecy.

4 6 C.A. Beller et l. Give popultio of sie N d vrible X with observed vlues x,, x N, the popultio me N xi is give by: m = å. If oly rdom smple of sie is vilble, the smple me m is clculted similrly d give by m = å. i= N xi i= If observtios re ordered i icresig order, the medi is the middle observtio of the smple. th If the umber of observtios i smple is odd, the medi is the vlue of the ( ) + observtio of the ordered smple, while it is the me of the vlues of the two middle observtios if the umber of observtios is eve. The most commo mesure of dispersio is the vrice, or its squre root, the stdrd devitio. Give popultio of sie N, the vrice s is give by s popultio me. The popultio stdrd devitio is s = å å i = å i ( x m) ( x m) i N i N, where m is the. If smple of sie is ( xi m) vilble, the smple vrice s i is provided by s =, where m is the smple me. The deomitor is slightly differet from tht of popultio vrice. This correctio esures tht the prmeter s is ubised estimtor of the popultio vrice s. Similrly, the smple stdrd ( xi m) i devitio, usully deoted s SD, is clculted s SD =. Other mesures of dispersio iclude the rge, the iter-qurtile rge d referece limits. The rge correspods to the differece betwee the mximum d the miimum vlues. Whe i icresig order, the first or lower qurtile correspods to the vlue below which 5% of the dt fll. The third or upper qurtile correspods to the vlue below which 75% of the dt fll. There re severl methods to compute qurtiles: oe ivolves clcultig the first d third qurtiles s the rk of the ( ) 4 + th d 3 ( + ) th observed vlues. Other methods re vilble, but will usully 4 led to reltively close results (Armitge et l. 00 ). The iter-qurtile rge is the differece betwee the upper d lower qurtiles. Note tht the secod or middle qurtile correspods to the vlue below which 50% of the dt fll, d s such, is equivlet to the medi. More geerlly, the 00p% referece limit, where 0 < p <, is the vlue below which 00p% of the vlues fll. For exmple, the medi is equivlet to the 50% referece limit. Referece limits re lso clled referece vlues, percetiles, or qutiles. Exmple : A rdom smple of te 0-yer-old wome leds to the followig observed weights (i kg): 50, 55, 60, 6, 45, 5, 6, 54, 48, 53. The vrible of iterest X is the weight, which is cotiuous vrible. We first reorder this rdom sequece of = 0 observtios: 45, 48, 50, 5, 53, 54, 55, 60, 6, 6. Bsed o previous formule, we hve the followig results: 0 xi The smple me is clculted s m = å = = 54 kg i= The smple medi is clculted s med = = 53.5 kg å

5 Clcultig Smple Sie i Athropometry 7 Tble. Smple probbility distributio for the ctegoricl vrible X defied s the umber of copies of llele A x 0 P(X = x) 3/0 5/0 /0 The smple vrice is clculted s The rge is give by 6 45 = 7 kg s å ( x 54) (45 54) + + (6 54) = = = 3 kg 0 9 i i..4 Probbility Distributios A distributio is defied s the set of frequecies of the vlues or ctegories of mesuremet mde o group of persos. The distributio tells us either how my or wht proportio of the group ws foud to hve ech vlue (or ech rge of vlues) out of ll of the possible vlues tht the qutittive mesure c hve (Lst 00 ). Cosider the vrible X correspodig to the umber of copies of certi llele A. The vrible X c tke three distict vlues: 0,,. Te subjects re rdomly selected from the geerl popultio d the followig series of outcomes is observed: 0, 0,,,,,, 0,,. This series is clled rdom series, sice previous vlues cot be used to predict future observtio. Oe c the clculte the proportio of subjects crryig two copies of the llele. This observed proportio equls /0 i our exmple. If subjects re rdomly smpled idefiitely, this proportio will ted towrds limitig vlue, clled the probbility of crryig two lleles, deoted by P(X = ) = p. The smple probbility distributio of the vrible X is represeted i Tble.. We hve the followig result: å P( X = x) =. This leds us to fudmetl property of probbility distributios: for give vrible, the sum of the probbilities of ech possible evet equls...4. Beroulli d Biomil Distributios Cosider dichotomous rdom vrible X whose vlues c be either oe stte or the other, or the bsece or presece of specific trit. The true (ukow) proportio of the popultio tht is i the idex ctegory of the stte or trit (the probbility tht rdomly selected idividul would be i this ctegory (X = )) is deoted s p d the probbility of beig i the other (referece) ctegory (X = 0) is thus p. The prmeter p defies the probbility distributio of X, d is clled the Beroulli distributio (fter J Beroulli, Swiss mthemtici). The vrible X is sid to follow Beroulli distributio with prmeter p. If, s here, the vrible X is coded s 0 (referece ctegory) or (idex ctegory), its me d vrice re give respectively by me(x) = p d vr(x) = p ( p). Defie the rdom vrible Y s the observed umber of subjects with prticulr stte of iterest out of rdomly selected subjects. If the probbility tht this stte is preset is p, the vrible Y, tht is, the sum of Beroulli rdom vribles with prmeter p, is sid to follow biomil distributio with me d vrice give respectively by me(y) = p d vr(y) = p ( p). The probbility tht this k k k! k k stte is preset for k out of subjects is give by: P( Y = k) = Cp ( p) = p ( p). ( k)! k!

6 8 C.A. Beller et l. Tble. Probbility distributio for the ctegoricl vrible Y defied s the umber of subjects with stte S out of 3 rdomly selected subjects y 0 3 P(Y = y) /8 3/8 3/8 /8 Whe is lrge, the distributio of Y will coverge to orml distributio with me p d vrice p ( p). Exmple : The presece of helth stte S is ssessed i = 3 rdomly selected subjects. The vrible Y correspods to the umber of subjects with stte S out of the = 3 subjects, d c thus hve 4 possible vlues: 0,,, or 3. Assume tht the stte S is preset i p = 50% of the popultio. The probbility of observig k = subjects with stte S out of = 3 rdomly selected subjects is 3 3! 3 3 give by: P( Y = ) = C3p ( p) = ( 0.50) ( 0.50) =. (3 )!! 8 The complete probbility distributio of Y c be computed d is represeted i Tble The Norml Distributio Cosider cotiuous rdom vrible with vlues tht vry betwee d +. The most importt cotiuous probbility distributio is the Gussi distributio (fter Krl Guss, Germ mthemtici), lso clled the orml distributio, which is idexed by two prmeters: the me d the vrice. If X follows orml distributio with me m d vrice s, we use the followig ottio: X ~ N (m,s ).The curve represetig the orml distributio is symmetric, so tht the me d the medi fll t the cetre of the symmetry. It is described by the fuctio æ x m ç è s ø f ( x) = e. The probbility tht rdom vrible flls betwee two vlues, A d B, is s p represeted grphiclly by the re uder the curve of the probbility distributio d betwee the two verticl lies with coordites x = A d x = B, s illustrted i Fig... B Numericlly, this probbility is obtied by computig the itegrl P( A < x < B) = f ( x) dx = æ x m ç è s ø e dx. Clcultio of this itegrl is ot fesible usig simple clcultio tools; however, s p the properties of the orml distributio simplify the probbility estimtio i some cses. For exmple, the probbility tht y rdom orml vrible X ~ N (m,s ) flls withi oe stdrd devitio of the me is kow to be bout 68%, tht is P( m s < x < m + s) = 68%, s preseted i Fig... Similrly, bout 95% of the distributio flls withi two stdrd devitios of the me (P( m s < x < m + s) = 95%), d 99.7% withi three stdrd devitios (P(m 3s < x < m + 3s) = 99.7%). For other vlues of the orml distributio, oe hs to either clculte the itegrl B A P( A < x < B) = ò f ( x) dx (by hd or usig mthemticl or sttisticl softwre) or rely o sttisticl tbles for which these probbilities re tbulted for vlues of m d s. There is however ifiite umber of vlues for these prmeters, d it is therefore ot possible to hve tbles for ll of them. Iterestigly, every orml distributio with prmeters m d s c be expressed i terms of orml distributio with me 0 d vrice, clled the stdrd orml distributio. Ideed, X m if X ~ N (m,s ), the oe c defie the vrible Z such tht Z =. It c be show tht Z ~ N (0,). s ò A

7 Clcultig Smple Sie i Athropometry 9 Fig.. Norml probbility distributio fuctio for vrible X. The shded re represets the probbility tht the vrible X flls betwee A d B. Numericlly, this probbility is obtied by computig the itegrl P( A < x < B) = B ò A æ x m ç è s ø f ( x) dx = e dx s p Fig.. Probbility distributio fuctio for orml rdom vrible X with me m d vrice s. The shded re represets the probbility tht the orml vrible X flls withi the itervl from the me mius oe stdrd devitio to the me plus oe stdrd devitio, tht is P(m s < x < m + s), which is bout 68% Thus, computig the probbility tht vrible X ~ N (m,s ) belogs to the itervl [A;B], is equivlet to computig the probbility tht vrible Z ~ N (0,) belogs to the itervl éa m B m ù ;. ê s s ú ë û Tbles of the stdrd orml distributio re vilble i most sttisticl textbooks. Importt tbles re ssocited with the stdrd orml distributio, icludig the tble P ( ), which provides for ech

8 0 C.A. Beller et l. X Probbility Vlues.98σ 95% of vlues.98σ Probbility of Cses i portios of the curve Stdrd Devitios 4σ From The Me Cumultive % σ.58σ 99% of vlues σ σ σ 0 +σ +σ +3σ +4σ 0.%.3% 5.9% 50% 84.% 97.7% 99.9% Z Scores Fig..3 The orml distributio. This plot illustrtes stdrd results of the orml distributio icludig stdrd devitios from the mes d the Z -scores vlue of, the probbility tht Z flls outside the itervl [ ; ]: P( ) = P (Z < or Z > ) = P( < Z < ). Some stdrd orml vlues re commoly used: P(.64) = 0.0 d P(.96) = 0.05, tht is, P(.96 < Z <.96) = 95%. Thus, if stdrd orml vrible is rdomly selected, there is 95% chce tht its vlue will fll withi the itervl [.96;.96]. Equivletly, the probbility tht stdrd orml rdom vrible Z ~ N (0,) flls outside the itervl [.96;.96] is 5%. Give the symmetry of the orml distributio, P(Z >.96) = P(Z <.96)=.5%. These stdrd results re illustrted i Fig..3. The Z -vlues re usully referred to s the Z -scores d re commoly used i thropometry. They re discussed i greter detil i subsequet chpter. The orml distributio is commoly used i thropometry, i prticulr to costruct referece rges or itervls. This llows oe to detect mesuremets which re extreme d possibly borml. A typicl exmple is the costructio of growth curves (WHO Child growth stdrds 006 ). I prctice, however, dt c be skewed (i.e. ot symmetric) d thus observtios do ot follow orml distributio (Elvebck et l. 970 ). I such cses, trsformtio of the observtios, such s logrithmic, c remove or t lest reduce the skewess of the dt (Hrris d Boyd 995 ; Wright d Roysto 999 )..3 Priciples of Sttisticl Estimtio There re two estimtio procedures: poit estimtio d itervl estimtio. Poit estimtio provides vlue tht we hope to be s close s possible to the true ukow prmeter vlue. Itervl estimtio provides itervl tht hs fixed priori probbility of cotiig the true prmeter vlue.

9 Clcultig Smple Sie i Athropometry.3. Poit Estimtio.3.. Poit Estimtor d Poit Estimte Poit estimtio is the process of ssigig vlue to popultio prmeter bsed o the observtio of smple drw from this popultio. The resultig umericl vlue is clled the poit estimte ; the mthemticl formul/fuctio used to obti this vlue is clled the poit estimtor. While the poit estimtor is ideticl whtever the smple, the poit estimte vries cross smples..3.. Poit Estimtio for Proportio The vrible of iterest is biry, such s the presece or bsece of specific helth coditio. We re iterested i p, the true prevlece of this coditio. Assume we rdomly drw smple of subjects d deote the observed umber of subjects with the coditio of iterest by k. The poit k estimtor of p is give by p =, while poit estimtes correspod to the umericl vlues p, p, ¼, obtied from distict smples of observtios Poit Estimtio for Me d Vrice X is cotiuous vrible with popultio me d vrice deoted respectively by m d s. If subjects re rdomly selected with subjects i= to hvig respectively observed vlues x to x, the x me m, the vrice s i d the stdrd devitio SD re estimted respectively by m = å, i= ( xi m) ( xi m) s = å d SD = å. i= i=.3..4 Poit Estimtio for Referece Limit Let X, X,, X be the mesuremets for rdom smple of idividuls. The 00p% referece limit, where 0 < p <, is the vlue below which 00p% of the vlues fll. Referece limits my be estimted usig oprmetric or prmetric (i.e. distributio-bsed) pproches (Wright d Roysto 999 ). The simplest pproch is to fid the empiricl referece limit, bsed o the order sttistics. The k th order sttistic, deoted s X (k), of sttisticl smple is equl to its k th -smllest vlue. Thus, give our iitil smple X, X,, X, the order sttistics re X (), X (),, X (), d represet the observtios with first, secod,, th smllest vlue, respectively. A empiricl estimtio of the 00p% referece limit is give by the vlue which hs rk [p( + )], where [.] deotes the erest iteger. For exmple, for smple of sie = 99 d p = 0.05, [p( + )]=5, d thus the vlue of the fifth order sttistic, X (5), provides poit estimte of the.5% referece limit. Referece limits c lso be estimted bsed o prmetric methods. Assume tht the observtios follow orml distributio with me m d vrice s. For rdom smple of sie, the smple me d smple stdrd devitio re give by m d SD, respectively. The 00p% referece limit is the estimted s m + p SD, where p is the 00p% stdrd orml devite.

10 C.A. Beller et l..3. Itervl Estimtio.3.. Priciples A sigle-vlue poit estimte, without y idictio of its vribility, is of limited vlue. Becuse of smple vritio, the vlue of the poit estimte will vry cross smples, d it will ot provide y iformtio regrdig precisio. Oe could provide estimtio of the vrice of the poit estimtor. However, it is more commo to provide rge of possible vlues. A cofidece itervl hs specified probbility of cotiig the prmeter vlue (Armitge et l. 00 ). By defiitio, this probbility, clled the coverge probbility, is ( ), where 0 < <. If oe selects smple of sie t rdom, usig sy set of rdom umbers, the the smple tht oe gets to observe is just oe smple from mog lrge umber of possible smples oe might hve observed, hd the ply of chce bee otherwise. A differet set of rdom umbers would led to differet possible smple of the sme sie d differet poit estimte of the prmeter of iterest, log with its cofidece itervl. Of the my possible itervl estimtes, some 00( )% of these itervls will cpture the true vlue. This implies tht we cot be sure tht the 00( )% cofidece itervl clculted from the ctul smple will cpture the true vlue of the prmeter of iterest. The computed itervl might be oe of the possible itervls (with proportio 00 %) tht do ot coti the true vlue. The most commoly used coverge probbility is 0.95, tht is = 0.05 = 5%, i which cse the itervl is clled 95% cofidece itervl..3.. Itervl Estimtio for Proportio I this situtio, the vrible of iterest is biry, such s the presece or bsece of specific helth coditio, d we re iterested i p, the prevlece rte of this coditio. Give rdom smple k of subjects, poit estimtor for the proportio p is give by p =, where k is the observed umber of subjects with the coditio of iterest i the smple. Whe is lrge, the smplig distributio of p is pproximtely orml, with me p d vrice p( p). Thus, 00( )% cofidece é p( p) ù itervl for p is give by ê p ± ú. I smller smples ( p < 5 or ( p ) < 5), exct ê ë úû methods bsed o the biomil distributio should be pplied rther th orml pproximtio to it (Mchi et l. 997 ). Exmple : Assume oe is iterested i estimtig p, the proportio of overweight childre. I selected rdom smple of = 00 childre, the observed proportio of overweight childre is p = 40%.A 95% cofidece itervl (.96 ) is give by é ê0.40 ±.96 ë = for the true proportio p of overweight childre 0.40(0.60) ù ú, tht is [30%; 50%]. 00 û.3..3 Itervl Estimtio for the Differece Betwee Two Proportios The vribles of iterest re biry, d we re iterested i p d p, the prevlece rtes of specific coditio i two idepedet smples of sie d. Let k d k deote the umber of

11 Clcultig Smple Sie i Athropometry 3 observed subjects with the coditio of iterest i these two smples. A poit estimtor for the k k differece p p is give by p p =, where p d p re the two smple proportios. Whe d re lrge, the smplig distributio of p p is pproximtely orml with me p( p) p ( p ) p p d vrice +. Thus, 00( )% cofidece itervl for the true é p ( p ) p ( p ) ù differece i proportios (p p ) is give by ê( p p ) ± + ú. I cse of ê ë úû smller smples ( p < 5 or ( p ) < 5 or p < 5 or ( p ) < 5), exct methods bsed o the biomil distributio should be pplied rther th orml pproximtio to it (Mchi et l. 997 ). Exmple : Assume rdomied tril is coducted to compre two physicl ctivity itervetios for reducig wist circumferece. A totl of 0 d 00 subjects re ssiged to receive itervetio A or B, respectively. The observed success rte is 40% i group A d 0% i group B. A 95% cofidece itervl for the true differece i success rtes is give by é ù ê( ) ±.96 + ú, tht is, [8%; 3%]. ë 0 00 û.3..4 Itervl Estimtio for Me The vrible of iterest is cotiuous with true me m d true vrice σ. We re iterested i estimtig the true popultio me m. A totl of subjects hve bee rdomly selected from this popultio of iterest. Whe is lrge ( > 30), the distributio of the smple me m is pproximtely orml with me m d vrice s. A 00( ) cofidece itervl for m is thus give by é s ù êm ± ú. Oe c use s s estimte of the popultio vrice, uless the vrice σ is êë úû kow. If the smple sie is smll, d if the vrible of iterest is kow to be orml, the é s ù 00( ) cofidece itervl for m is give by êm ± t, ú, where t êë, correspods to the % úû tbulted poit of the t, distributio. Exmple : Oe is iterested i estimtig the verge weight of 5-yer-old girls. After selectig rdom smple of 00 girls, the observed verge weight is 55 kg, d the stdrd devitio is 5 kg. A 95% cofidece itervl for the me weight of 5-yer-old girls is thus give by: é 5 ù ê55 ±.96 ú, tht is [5; 58]. êë 00 úû.3..5 Itervl Estimtio for the Differece Betwee Two Mes Give two idepedet smples of sie d with respective mes m d m, d vrices d s, poit estimtor for the differece m m is give m m, where m d m correspod s

12 4 C.A. Beller et l. to the two smple mes. Whe d re lrge, the distributio of m m is pproximtely orml with me m m d vrice s s +. A 00( )% cofidece itervl for the true dif- ferece i mes is the give by é s s ê( m m ) ± + ê ë ù ú. Uless the vrices úû s d s re kow, oe c use the smple vrices é s s ù ê( m m ) ± + ú. ê ë úû s d s s their respective estimtes: Exmple : Oe wishes to compre the distolic blood pressure (DBP) betwee two popultios of subjects A d B. I smple of 50 rdomly selected subjects from popultio A, the observed DBP me is 0 mmhg, while the me DBP i 60 ptiets rdomly selected from popultio B is bout 00 mmhg. The observed stdrd devitios re, respectively, 0 mmhg d 8 mmhg. A 95% cofidece itervl for the true differece i DBP mes is thus tht is, [6.6; 3.4]. é 0 8 ù ê(0 00) ±.96 + ú, êë úû.3..6 Itervl Estimtio for Referece Limit Usig oprmetric pproch, cofidece itervl for referece limit c be expressed i terms of order sttistics (Hrris d Boyd 995 ). A pproximte 00( )% cofidece itervl for the 00p% referece limit is give by the order sttistics x ( r ) d x ( s ), where r is the lrgest iteger less th or equl to p + p( p), d s is the smllest iteger greter th or equl to p + + p( p). Exmple : I their exmple, Hrris d Boyd re iterested i the 90% cofidece itervl (.645 ) = for the.5% ( p = 0.05) referece limit bsed o rdom smple of sie = 40 (Hrris d Boyd 995 ). The lower boud of this itervl correspods to the vlue of the r th order sttistic, where r is the lrgest iteger less th or equl to p + p( p) = , tht is,. Similrly, the upper boud correspods to the vlue of the s th order sttistic, where s is the smllest iteger greter th or equl to p + + p( p), tht is,. Referrig to the ordered observtios, bouds of the 90% cofidece itervl for the.5% referece limit re thus provided by the vlues of the secod d eleveth order sttistics.

13 Clcultig Smple Sie i Athropometry 5 Cofidece itervls c lso be built usig prmetric methods. Assume tht the observtios follow orml distributio with me m d vrice σ. For rdom smple of sie, the smple me d smple stdrd devitio re give by m d SD, respectively. The prmetric estimtor of the 00 p % referece limit is m + p SD, where p is the 00p% stdrd orml devite. The vrice of this estimtor is give by s æ p ç +. A 00( )% cofidece itervl for the 00p% è ø é SD æ ù p referece limit is thus êm + psd ± ç + ú. See Hrris d Boyd for worked exmples ê è ø ú ë û (Hrris d Boyd 995 )..4 Precisio-Bsed Smple Sie Estimtio Smple sie c be computed usig either precisio-bsed or power-bsed pproch. I thropometry, however, there is greter emphsis o precisio-drive estimtio procedures (Grder d Altm 986, 988 ). For y give prmeter, the legth of the cofidece itervl is fuctio of the smple sie. More specificlly, the lrger the smple, the rrower the cofidece itervl. Thus, the mi purpose of cofidece itervls is to idicte the (im)precisio of the smple study estimtes s popultio vlues (Grder d Altm 988 ). Coversely, oe c fix the desired legth of the itervl d estimte the umber of subjects eeded ccordigly. Formule re vilble to estimte the smple sie s fuctio of the precisio, which c be expressed i two wys. Oe c express the legth of the itervl i bsolute terms bsed o the bsolute mrgi of error, ME, which represets hlf the width of the cofidece itervl, d is the qutity ofte quoted s the plus or mius i ly reports of surveys. I such cse, the cofidece itervl is expressed s estimte ± ΜΕ. It is lso possible to express the legth of the itervl i reltive terms bsed o the precisio usully deoted by e. I such cses, the cofidece itervl is expressed s estimte ± e estimte. There is direct reltioship betwee the mrgi of error d the precisio sice ME = e estimte; smple sie c thus be estimted s fuctio of either prmeter. Note tht the term error is ofte source of cofusio whe delig with smple sie d power clcultios. Therefore, it is lwys very importt to provide precise defiitio of this term, tht is, to clrify whether we re referrig to bsolute or reltive error. Throughout this sectio, we cosider tht observtios from the sme smple re idepedet. I the cse of two-smple problems, we cosider tht the two smples re idepedet..4. Dichotomous Vribles.4.. Smple Sie for Estimtig Sigle Proportio with Give Precisio A 00( )% cofidece itervl for the true proportio p give smple sie d ticipted é p( p) ù vlue p is give by: ê p ± ú. The width of the cofidece itervl is thus give by ë û

14 6 C.A. Beller et l. p( p) d depeds o the umber of subjects i the smple. Coversely, the umber of subjects eeded to estimte the proportio will deped o the degree of precisio (idicted by the mximum width of the cofidece itervl) oe is willig to ccept. The mrgi of error, ME, is hlf the width of the cofidece itervl. If oe wts the bsolute mrgi of error to be t most ME, tht p( p) p( p) is, oe wts ME, the the umber of subjects should be t lest =. ME This problem c lso be expressed i terms of precisio (or reltive error). The cofidece itervl é p( p) ù ê p ± ú c be rewritte s [p ± e p], implyig tht the estimtio of p is provided to ë û withi 00e % of its ticipted vlue, where e is the precisio of the estimtio. Sice ME = pe,the ( p) required smple sie c thus be expressed s =. Note tht the smple sie is mxi- pe mied for p = If either p or ( p) re smll (i.e. below 5), exct pproximtios bsed o the biomil distributio should be pplied rther th orml pproximtio to it (Mchi et l. 997 ). Exmple : Oe is iterested i estimtig the prevlece of overweight childre with bsolute mrgi of error smller th 0.05 (ME = 0.05). If the prevlece, p, is expected to be roud æ 50% ( p = 0.50), d iterest is i estimtig 95% cofidece itervl ç =.96, the miimum è ø umber of subjects eeded should be ( 0.5), tht is, t lest 384 subjects Smple Sie for Estimtig the Differece Betwee Two Proportios with Give Precisio Give two proportios p d p for two idepedet smples of sie d, 00( )% cofidece itervl for the true differece i proportios (p p ) is estimted by é p ( p ) p ( p ) ù ê( p p ) ± + ú, where p ê ë d p re the observed smple proportios. úû The width of the cofidece itervl is give by p ( p ) p ( p ) +. Assumig smples re of equl sie = = d bsolute mrgi of error ME, the umber of subjects i ech smple should be t lest ( ( ) + ( )) p p p p =. I terms of precisio ME e or reltive error, where ME= e ( p p ), the required smple sie is expressed s = ( ( ) + ( )) p p p p e ( p p ) smple sies (Mchi et l. 997 ).. Similr formule hve bee derived for the cse of uequl

15 Clcultig Smple Sie i Athropometry 7 Exmple: A study is set up to compre ew physicl ctivity itervetio to stdrd oe for overweight subjects. The im is to reduce the body mss idex (BMI) dow to 30 cm/kg or lower. The ticipted success rtes re expected to be pproximtely p = 0% d p = 5% for the ew d stdrd itervetios respectively. The ivestigtor would like to recruit two groups of ptiets with equl smple sies = =, d provide 95% cofidece itervl for differece i success rtes with bsolute mrgi of error ME = 0%. The miimum umber of subjects per group ( ) should be t lest =, tht is, 07 subjects per group Cotiuous Vribles.4.. Smple Sie for Estimtig Sigle Me with Give Precisio A 00( )% cofidece itervl for me m give ticipted me m d ssumig lrge é s ù s smple is give by êm ± ú. The width of the cofidece itervl is. If we wt ë û the bsolute mrgi of error to be t most ME, the the umber of subjects hs to be greter th s =. If oe wishes to express the smple sie i terms of precisio e, where ME = em, the ME required smple sie is expressed s s =. Uless the vrice s is kow, oe c use e m literture-bsed or experiece-bsed s s estimte of the popultio vrice. Exmple : A ivestigtor is iterested i estimtig distolic blood pressure (DBP) i specific popultio. The me DBP d stdrd devitio re ticipted to be bout 05 mmhg (m = 05) d 0 mmhg (SD = 0). If the desired reltive precisio is 5% (e = 5%), the required smple sie.96 0 for estimtig 95% cofidece itervl for the me DBP should be t lest =, tht is subjects..4.. Smple Sie for Estimtig the Differece Betwee Two Mes with Give Precisio Assumig idepedet smples of sie d, with respective smple mes m d m d commo vrice s, 00( )% cofidece itervl for the true differece i mes is give é ù ± + ú. The width of the cofidece itervl is thus give by û by ê( m m ) s ë s +. Assumig smples of equl sie = = d mrgi of error ME, the umber of subjects i ech smple should be t lest s =. If oe wishes to express the ME

16 8 C.A. Beller et l. smple sie i terms of precisio e, where ME = e( m m ), the required smple sie is expressed s s =. Uless the vrice s is kow, oe c use literture-bsed or experiece- e ( m m ) bsed s s estimte of the popultio vrice. Similr formule hve bee derived for the cse of uequl smple sies (Mchi et l. 997 ). Exmple : A ivestigtor is iterested i evlutig two tretmets (A d B) imed t decresig cholesterol level. The ticipted me cholesterol levels followig tretmets A d B re, respectively, 00 d 50 mg/dl, with 0 mg/dl commo stdrd devitio. To obti 95% cofidece itervl for the differece i cholesterol levels fter tretmet with (reltive) precisio e = 0%, the.96 0 smple sie should be t lest =, tht is 3 subjects per group. 0.0 (00 50).4..3 Smple Sie for Estimtig Regressio-Bsed Referece Limit with Give Precisio I some cses, the vrible of iterest might be idexed by secodry vrible. Suppose we hve cotiuum of distributios idexed by covrite. For exmple, ssume tht we re studyig BMI i group of childre of differet ges. Isted of the me BMI, we might be iterested i other prmeters of the BMI distributio, more prticulrly i the 95% referece limit of the BMI distributio for vrious ges. How my childre should we smple i order to hve precise estimte of this referece limit d for every possible vlue of ge? This questio c be swered by pplyig lier regressio techiques to estimte the referece limit s fuctio of ge. Methods hve bee developed to estimte smple sies for regressio-bsed referece limits uder vrious situtios (Beller d Hley 007 ). We provide overview of these pproches, d refer the iterested reder to this literture for dditiol detils. It is ssumed tht the me vlue of the respose vrible of iterest (e.g., BMI) vries lierly with the covrite (e.g., ge), d tht the respose vlues re pproximtely ormlly distributed roud this me. The respose vrible d the covrite of iterest re deoted by Y d X, respectively. Assume tht t y give vlue x 0 of ge, the me vlue of iterest, such s BMI, is pproximte lier fuctio of X d tht idividul BMI vlues re ormlly distributed roud this me (the Y x ~ N b + b x, s. ltter evetully fter suitble trsformtio) with costt vrice: 0 ( 0 0 ) The 00p% referece limit for Y t this specific ge poit x 0 is give by: Q0 = b0 + bx0 + ps, where p is the stdrd orml devite correspodig to the 00p% referece limit of iterest. Give selected idividuls with dt poits ( xi, yi), i,, ) Ù Ù Ù = ¼, poit estimtor for the 00p% referece limit is give by: Q0 = b + b x0 + s, where b Ù 0 d re obtied by lest-squres 0 p Y X estimtio of the regressio coefficiets b 0 d b, d s is the observed root me squre error. Y X Smple sie estimtio for regressio-bsed referece limits requires defiig the followig prmeters: The 00 p% referece limit of iterest, where 0 < p <, d the correspodig oe-sided stdrd orml devite, p. For exmple, if we re iterested i the 95% referece limit, the oe-sided stdrd orml devite is 0.95 =.64. b Ù

17 Clcultig Smple Sie i Athropometry 9 The 00( )% cofidece itervl for the referece limit of iterest, d its correspodig twosided stdrd orml devite, where 0 < <. For exmple, if we wt the 95% cofidece itervl, the = 0.05 d =.96. The 00( b )% referece rge, which ecompsses 00( b )% of the vlues (e.g., BMI) s well s its correspodig two-sided stdrd orml devite b, where 0 < b <. For exmple, if we wt the 95% referece rge, the b = 0.05 d b =.96. The reltive mrgi of error, defied s the rtio of the width of the 00(-)% cofidece itervl for the referece limit to the width of the 00( b)% referece rge. This mes tht we wt smple sie lrge eough so tht the width of the 00( )% cofidece itervl for our referece limit is smll whe compred to the width of the 00( b)% referece rge (we usully tke = b). The desig of the study, tht is, the distributio of the covrite (e.g., ge) i the smple ivestigted, which will ifluece the computtio of the smple sie. Oce the bove prmeters hve bee specified, we c estimte the smple sie. Assume, first tht we choose our smple so tht the covrite (e.g., ge) follows uiform distributio. I such cses, the vrice of the estimtor Q Ù of the referece limit is pproximtely equl to Ù s æ p 0 = + vr( Q ) ç 4. The width of the 00( )% cofidece itervl for the 00p% referece è ø limit t the extreme vlue of ge is therefore s 4 p +. Assume we wt reltive error of, defied s the rtio of the width of the 00( )% cofidece itervl for the referece limit to the width of the 00( b)% referece rge. The width of the 00( )% referece rge is give by b s. Thus, if we wt the rtio of the width of the 00( )% cofidece itervl for the referece limit to the width of the 00( b)% referece rge to be smller th the reltive error, we require b p 4 +. Tht is, the miimum smple sie,, required to estimte the 00( )% cofidece itervl for the 00p% referece limit, with reltive mrgi of error of, whe compred to the 00( b)% referece rge, should be t lest = æ p ç 4 + è ø. Similr formule b hve bee derived ssumig other smplig strtegies (Beller d Hley 007 ). For exmple, isted of uiform ge distributio, oe might tke oe-third of the smple t oe ge extreme, oe-third t the midpoit, d oe-third t the other ge extreme. I this study desig, the miimum

18 0 C.A. Beller et l. smple sie requiremet is the æ 5 p ç + è ø b. Similrly, we c lso expect tht the ge distributio i the smple will follow orml distributio. If we ssume tht the rge of X is pproximtely 4 times the stdrd devitio of X, the we show tht the smple sie requiremet becomes: æ p ç 5 + è ø. b Notice tht the previous formule were derived uder the worst-cse scerio, tht is, ssumig tht we re iterested i estimtig the referece limit t the extreme ed of ge, where the vribility is highest, d thus the lrgest smple sie is eeded. If oe is iterested i the 00p% referece limit t the verge ge vlue (where the vribility is miimied), the the smple sie is reduced to æ p ç + è ø, for y give ge distributio (or similrly, ssumig homogeeous popultio ot b idexed by covrite). Put simply, iformtio from either side of the verge ge dds stregth to the iformtio t the verge ge. I cotrst, iformtio t the extremes of the ge distributio c oly gther stregth from oe side of the ge distributio; o the other side, there is ifiite ucertity. Severl other fctors c impct the smple sie (Cole 006 ), such s for exmple the rge of the covrite of iterest. If oe is iterested i the height of childre, lrger smple will be eeded whe cosiderig birth to 8 yers th whe cosiderig 5 yers. No-costt vribility c lso ffect the smple sie, sice the vrice of the estimtor, vr( Q Ù 0 ), is proportiol to s, where s vries with ge. This rtio c be mde costt cross ge groups by esurig tht the smple sie is proportiol to s. Thus, t the ges t which the vribility is icresed, for exmple durig puberty, the smple sie eeds to be icresed ppropritely to compeste (Goldstei 986 ; Cole 006 ). Similrly, i cse of heteroscedsticity of the vrible of iterest cross the covrite, regressio techiques c be used to model the stdrd devitio s fuctio of the me, d previous formule c still be used s rough guide for smple sie plig. Notice tht the vritio s to be used i plig icludes both the true iter-idividul vribility d the vribility of the mesurig istrumets used: mesuremet tools with differig precisios will provide differet smple sie estimtes. Filly, the ture of the reltio betwee the covrite d the vrible of iterest c lso ffect the smple sie, s more subjects will be eeded to cpture wiggles i the reltio compred to simple lier reltio. If there is some olierity i the covrite, such s for exmple qudrtic reltioship, the formule c lso be ccommodted by djustig the poit estimtor of the referece limit of iterest. Exmple : We re iterested i estimtig specific BMI referece limit s lier fuctio of ge. æ Specificlly, we wish to produce 95% cofidece itervl ç =.96 for the 95% BMI referece è ø

19 Clcultig Smple Sie i Athropometry limit ( 0.95 =.64), with reltive mrgi of error = 0%, whe compred with the 95% referece æ rge ç b =.96. If ge is uiformly distributed i the smple, the the miimum required smple è ø ( ) sie is =, i.e., we would eed t lest 536 observtios to obti this precise estimte of the 95% BMI referece limit t y plce i the ge rge. If, o the other hd, oe is iterested i the 95% referece limit oly t the verge ge vlue, or i homogeeous popultio ( ) ot idexed by covrite, the we would eed t lest =, tht is, observtios..5 Priciples of Hypothesis Testig We hve preseted formule for the estimtio of the smple sie required to estimte vrious prmeters with desired degree of precisio. Similrly, oe my wt to esure tht sufficiet umber of subjects re vilble to show differece betwee two prmeters. We discuss here some bsic priciples of hypothesis testig which c be foud i itroductory sttisticl textbooks, s well s works devoted to cliicl trils or epidemiology (Lchi 98 ; Friedm et l. 998 ; Armitge et l. 00 ). Oe wishes to compre two groups, A d B, with true prevlece rtes, p A d p B. From sttisticl poit of view, this compriso is expressed i terms of ull hypothesis, clled H 0, which sttes tht o differece exists betwee the two groups: H 0 : p A p B = 0. Hypothesis testig cosist i testig whether or ot H 0 is true, more specificlly, whether or ot it should be rejected. Thus, util otherwise prove, H 0 is cosidered to be true. The true prevlece rtes, p A d p B re ukow. If two groups of subjects re properly smpled, oe c obti pproprite estimtes P A d P B. Although p A d p B might ot differ, it is possible tht by chce loe, the observed proportios P A d P B re differet. I such cses, oe might flsely coclude tht the two groups hve differet prevlece rtes. Such flse-positive error is clled type error, d the probbility of mkig such error correspods to the sigificce level d is deoted by. The probbility of mkig type error should be miimied. However, decresig the sigificce level icreses the smple sie. The probbility of observig differece s extreme s or more extreme th the differece ctully observed, give tht the ull hypothesis is true, is clled the p-vlue d is deoted by P. The ull hypothesis H 0 will be rejected if p <. If the ull hypothesis is ot correct, the ltertive hypothesis, deoted by H A must be true, tht is H A :p A p B =d, where d ¹ 0. It is possible tht by chce loe, the observed proportios P A d P B differ oly by smll mout. As result, the ivestigtor my fil to reject the ull hypothesis. This flse-egtive error is clled type error, d the probbility of mkig such error is deoted by b. The probbility of correctly cceptig H 0 is thus b d is referred s the power. It defies the cpbility of sttisticl test to revel give differece betwee two prmeters, if this differece relly exists. The power depeds o the sie of the differece we wish to detect, the type error, d the umber of subjects. Tht is, if the type error d the smple sie re held costt, study will hve lrger power if oe wishes to detect lrge differece compred to smll oe.

20 C.A. Beller et l..6 Power-Bsed Smple Sie Estimtio The umber of subjects eeded should be pled crefully i order to hve sufficiet power to detect sigifict differeces betwee the groups cosidered. To provide power-bsed (or testbsed) estimtio of smple sie requires defiig priori the differece oe wishes to detect, the desired sigificce level d the desired power of the test. As will be discussed below, smple sie formule ivolve the rtio of the vrice of the observtios over the differece oe wishes to detect, or more geericlly, oise sigl rtio, where the sigl correspods to the differece oe wishes to detect, d the oise (or ucertity) is the sum of ll the fctors (sources of vritio) tht c ffect the sigl (Sckett 00 ). Note tht we oly discuss the cse of idepedet observtios. I cliicl trils, for exmple, it my ot lwys be possible to rdomie idividuls. For exmple, physicl ctivity itervetio might be implemeted by rdomiig schools. Idividuls re the grouped or clustered withi schools. They cot be cosidered s sttisticlly idepedet, d the smple sie eeds to be dpted sice stdrd formule uderestimte the totl umber of subjects (Doer et l. 98 ; Friedm et l. 998 )..6. Dichotomous Vribles.6.. Smple Sie for Comprig Proportio to Theoreticl Vlue The vrible of iterest is dichotomous. For exmple, oe is iterested i the prevlece of obese childre p. Bsed o rdom smple, the objective is to compre this proportio to trget (fixed) vlue p 0. The ull d ltertive hypotheses re give respectively by H 0 : p = p 0 d H A : p ¹ p 0. The miimum umber of subjects eeded to perform this compriso ssumig observed prevlece æ ç p0 ( p0 ) + b p ( p ) è ø rte p, sigificce level d power b is = p p ( ) 0 Exmple: Oe is iterested i evlutig specific diet imed t reducig weight i obese subjects. A success is defied s reducig BMI dow to 5 or lower. Oe wishes to test whether this ew diet hs better success rte th the stdrd diet for which the efficcy rte is kow to be p 0 = 0%. The ticipted efficcy rte of the ew diet is p = 40%. The smple sie eeded to show differece betwee the efficcy rte of the ew diet d the trget efficcy rte p 0 ssumig æ sigifi cce level = 0.05 ç =.96, d power b = 0.90 ( b =.8) is t lest è ø = ( ( 0.0) ( 0.40) ) ( ), i.e., 36 ptiets. If the ticipted efficcy rte is p = 30%, tht is, oe wishes to detect smller differece, the the smple sie must be icresed to t lest = ( ( 0.0) ( 0.30) ) ( ), tht is, 37 subjects..

21 Clcultig Smple Sie i Athropometry 3 Note tht the clculted smple sies re quite low. This is becuse we re comprig oe smple to oe historicl (or literture-bsed) smple. I prctice, oe will usully be comprig two smples (ext sectio), s i cliicl trils. I this cse, the resultig smple sie is much higher s vribility of the secod smple hs to be ccouted for..6.. Smple Sie for Comprig Two Proportios The outcome of iterest is dichotomous d two idepedet groups of equl sie re beig smpled d compred. The objective is to detect differece betwee two proportios p 0 d p, tht is, the ull d ltertive hypotheses re give respectively by H 0 : p p 0 = 0 d H A : p p 0 = d, where d ¹ 0. The sie of ech smple required to detect ticipted differece p p 0, ssumig sigificce ( ) 0 æ ç p( p) + b p0 ( p0 ) + p ( p ) è ø level, d power b is =, where p is the ver- p p ge of the two ticipted proportios p 0 d p. Similr formule re vilble for the cse of uequl smple sies (Pitdosi 005 ). Exmple : A tril is set up to compre two physicl ctivity itervetios with ticipted success rtes of 50% d 30%. The smple sie per group eeded to show differece betwee the two itervetios ssumig differece i success rtes d = 50% 30% = 0%, æ sigificce level = 0.05 ç =.96, d power b = 0.90 ( b =.8) is è ø ( ( 0.40) ( 0.30) ( 0.50) ) = tht is, 84 subjects per group Cotiuous Vribles.6.. Smple Sie for Comprig Me to Theoreticl Vlue The vrible of iterest is cotiuous. For exmple, oe is iterested i the me wist circumferece, m, followig physicl ctivity itervetio. Bsed o rdom smple of subjects, the objective is to compre this me circumferece to trget me vlue m 0. The ull d ltertive hypotheses re thus give respectively by H 0 : m = m 0 d H A : m ¹ m 0. For this oe smple problem, the umber of subjects eeded to perform this compriso ssumig ticipted me vlue m, stdrd devitio s, sigificce level d power b is = s æ + ç b è ø ( m m ) 0. Uless the vrice s is kow, oe c use literture-bsed or experiece-bsed s s estimte of the popultio vrice.

22 4 C.A. Beller et l. Exmple : Oe wishes to test whether me wist circumferece i give popultio followig ew physicl ctivity itervetio is reduced compred to stdrd itervetio. Followig the stdrd itervetio, the me wist circumferece is kow to be bout m 0 = 40 cm. It is ticipted tht the ew itervetio will reduce this circumferece to m = 30 cm. The smple sie eeded to show differece betwee the me wist circumferece with the ew itervetio d æ trget vlue m 0, ssumig sigificce level = 0.05 ç =.96, 90% power ( b =.8) è ø ( ) ( ) d stdrd devitio SD = 0cm, is = Smple Sie for Comprig Two Mes, tht is, 4 ptiets. s æ + ç b è ø The vrible of iterest is cotiuous d two idepedet groups of equl sie re beig smpled d compred. The objective is to detect differece betwee two mes, tht is, the ull d ltertive hypotheses re give respectively by H 0 : m m 0 = 0 d H A : m m 0 = d, where d ¹ 0. The smple sie of ech group required to detect this differece ssumig ticipted differece m m 0, commo vrice s, sigificce level d power b is =. Uless the vrice s is kow, ( m m0 ) oe c use literture-bsed or experiece-bsed s s estimte of the popultio vrice. Exmple : I their exmple, Armitge et l. re iterested i comprig two groups of me usig the forced expirtory volume (FEV) (Armitge et l. 00 ). From previous work, the stdrd devitio of FEV is 0.5 L. A two-sided sigificce level of 0.05 æ ç =.96 is to be used with 80% è ø power ( b = 0.84). I order to show me differece of 0.5 L betwee the groups, d ssumig 0.5 ( ) smples of equl sies, the totl umber of me should be t lest =, tht 0.5 is, 63 me per group..7 Other Prmeters, Other Settigs Smple sies c be estimted for vrious prmeters d uder vrious settigs. As such, it is ot possible to cover ll possible situtios ito sigle book chpter! We hve reviewed formule for the estimtio of smple sies for commoly used prmeters such s mes, proportios d referece limits. Other prmeters such s time-to-evet outcomes (Freedm 98 ; Schoefeld 983 ; Dixo d Simo 988 ), correltio coefficiets (Boett 00 ), cocordce coefficiets (Doer 998 ), or eve multiple edpoits c be cosidered (Gog et l. 000 ). Similrly, methods for clcultig smple sie ssumig other desigs hve bee ivestigted. Isted of detectig specific differece, oe might be iterested i showig equivlece or oiferiority (Flemig 008 ) ; observtios my be clustered (Doer et l. 98 ; Hsieh 988 ), etc. Smple sie estimtio procedures hve bee developed for these settigs d we refer the iterested reder to specilied literture or geerl works o smple sie (Friedm et l. 998 ; Mchi et l. 997 ; Altm et l. 000 ; Pitdosi 005 ).