PASS Sample Size Software. Logistic Regression. Age vs Prob(Death From Disease) Age. P Log 1

Transcription

1 Chaptr 86 Introduction Logistic rgrssion xprsss th rlationship btwn a binary rspons variabl and on or mor indpndnt variabls calld covariats. A covariat can b discrt or continuous. Considr a study of dath from disas at various ags. This can b put in a logistic rgrssion format as follows. Lt a binary rspons variabl Y b on if dath has occurrd and zro if not. Lt X b th individual s ag. Suppos a larg group of various ags is followd for tn yars and thn both Y and X ar rcordd for ach prson. In ordr to study th pattrn of dath vrsus ag, th ag valus ar groupd into intrvals and th proportions that hav did in ach ag group ar calculatd. Th rsults ar displayd in th following plot.. Ag vs Prob(Dath From Disas) Prob(Dath From Disas) Ag As you would xpct, as ag incrass, th proportion dying of disas incrass. Howvr, sinc th proportion dying is boundd blow by zro and abov by on, th rlationship is approximatd by an S shapd curv. Although a straight-lin might b usd to summariz th rlationship btwn ags 4 and 6, it crtainly could not b usd for th young or th ldrly. Undr th logistic modl, th proportion dying, P, at a givn ag can b calculatd using th formula β + β X P = β + β X + This formula can b rarrangd so that it is linar in X as follows Log P = β X + β Not that th lft sid is th logarithm of th odds of dath vrsus non-dath and th right sid is a linar quation for X. This is somtims calld th logit transformation of P. Whn th scal of th vrtical axis of th plot is modifid using th logit transformation, th following straight-lin plot rsults. 86-

2 . Ag vs Log[Prob/(-Prob)] 6. Log[Prob/(-Prob)] In th logistic rgrssion modl, th influnc of X on Y is masurd by th valu of th slop of X which w hav calld β. Th hypothsis that β = vrsus th altrnativ that β = B is of intrst sinc if β =, X is not rlatd to Y. Undr th altrnativ hypothsis that β = B, th logistic modl bcoms Undr th null hypothsis, this rducs to Ag - P = BX β + - P = β To tst whthr th slop is zro at a givn valu of X, th diffrnc btwn ths to quantitis is formd giving which rducs to P β + BX β = log - P - P BX = - P - P = log = log P / ( - P ) P / ( - P ) ( OR) whr OR is odds ratio of P and P. This rlationship may b solvd for OR giving OR = This shows that th odds ratio of P and P is dirctly rlatd to th slop of th logistic rgrssion quation. It also shows that th valu of th odds ratio dpnds on th valu of X. For a givn valu of X, tsting that B is zro is quivalnt to tsting OR is on. Sinc OR is commonly usd and wll undrstood, it is usd as a masur of ffct siz in powr analysis and sampl siz calculations. BX 86-2

3 Powr Calculations Suppos you want to tst th null hypothsis that β = vrsus th altrnativ that β = B. Hsih, Block, and Larsn (998) hav prsntd formula rlating sampl siz, α, powr, and B for two situations: whn X is normally distributd and whn X is binomially distributd. Whn X is normally distributd, th sampl siz formula is N = ( z α / 2 + z β ) ( ) P * P * B whr P * is th vnt rat (probability that Y = ) at th man of X. Not that B is dfind in trms of an incras of on standard dviation of X abov th man. Whn X is binomially distributd and X = or, th sampl siz formula is N = z P ( P) α / 2 R 2 ( P P) ( R) 2 2 P ( P )( R) + z P ( P ) + β R whr P is th vnt rat at X = and P is th vnt rat at X =, R is th proportion of th sampl with X =, and P is th ovrall vnt rat givn by ( ) ( ) P = R P + R P. 2 Multipl Th multipl logistic rgrssion modl rlats th probability distribution of Y to two or mor covariats X, X 2,, X k by th formula - P = β X... X + β + + β k whr P is th probability that Y = givn th valus of th covariats. It is a simpl xtnsion of th simpl logistic rgrssion modl that was just prsntd. In powr analysis and sampl siz work, attntion is placd on a singl covariat whil th influnc of th othr covariats is statistically rmovd by placing thm at thir man valus. Whn thr ar multipl covariats, th following adjustmnt was givn by Hsih (998) to giv th total sampl siz, N m N = N - ρ m 2 whr ρ is th multipl corrlation cofficint btwn X (th variabl of intrst) and th rmaining covariats. Notic that th numbr of xtra covariats dos not mattr in this approximation. k 86-3

4 Procdur Options This sction dscribs th options that ar spcific to this procdur. Ths ar locatd on th Dsign tab. For mor information about th options of othr tabs, go to th Procdur Window chaptr. Dsign Tab Th Dsign tab contains most of th paramtrs and options that you will b concrnd with. Solv For Solv For This option spcifis th paramtr to b solvd for from th othr paramtrs. Th paramtrs that may b slctd ar P, Sampl Siz, Alpha, and Powr. Undr most situations, you will slct ithr Powr for a powr analysis or Sampl Siz for sampl siz dtrmination. Slct Sampl Siz whn you want to calculat th sampl siz ndd to achiv a givn powr and alpha lvl. Slct Powr whn you want to calculat th powr of an xprimnt. Tst Altrnativ Hypothsis Spcify whthr th tst is on-sidd or two-sidd. Whn a two-sidd hypothsis is slctd, th valu of alpha is halvd by PASS. Evrything ls rmains th sam. Commonly, accptd procdur is to us th Two-Sidd option unlss you can justify using a on-sidd tst. Powr and Alpha Powr This option spcifis on or mor valus for powr. Powr is th probability of rjcting a fals null hypothsis, and is qual to on minus Bta. Bta is th probability of a typ-ii rror, which occurs whn a fals null hypothsis is not rjctd. A typ-ii rror occurs whn you fail to rjct th null hypothsis of qual probabilitis of th vnt of intrst whn in fact thy ar diffrnt. Valus must b btwn zro and on. Historically, th valu of.8 (Bta =.2) was usd for powr. Now,.9 (Bta =.) is also commonly usd. A singl valu may b ntrd hr or a rang of valus such as.8 to.95 by.5 may b ntrd. Alpha This option spcifis on or mor valus for th probability of a typ-i rror (alpha). A typ-i rror occurs whn you rjct th null hypothsis of qual probabilitis whn in fact thy ar qual. Valus of alpha must b btwn zro and on. Historically, th valu of.5 has bn usd for alpha. This mans that about on tst in twnty will falsly rjct th null hypothsis. You should pick a valu for alpha that rprsnts th risk of a typ-i rror you ar willing to tak in your xprimntal situation. You may ntr a rang of valus such as..5. or. to. by

5 Sampl Siz N (Sampl Siz) This option spcifis th total numbr of obsrvations in th sampl. You may ntr a singl valu or a list of valus. Effct Siz Baslin Probability P (Baslin Probability that Y=) This option spcifis on or mor P valus. Th intrprtation of P dpnds on whthr X is binary or continuous. Binomial Covariat Whn X is binary, P is th probability that Y = whn X =. All othr covariats ar assumd to b qual to thir man valus. In this cas, th logistic quation rducs to so that - P P = = β + Normal Covariat Whn X is normally distributd, P is th probability that Y = whn X = µ X, whr µ X is th man of X. That is, P is th baslin probability that Y = whn X is ignord. All othr covariats ar assumd to b qual to thir man valus. In this cas, th logistic quation rducs to so that β β P = β + β µ X - P P = + β β µ + X β β µ + X Effct Siz Altrnativ Probability Us P or Odds Ratio This option spcifis th whthr to spcify P dirctly or to spcify it by spcifying th odds ratio. Sinc th rlationship btwn th odds ration, P, and P is givn by OR P / P / ( - P) ( - P ) = spcifying OR and P implicitly spcifis P. This options lts you spcify whthr you want to stat th altrnativ hypothsis in trms of P or th odds ratio. 86-5

6 P (Altrnativ Probability that Y=) This option spcifis th ffct siz to b dtctd by spcifying P. As was shown arlir, th slop of th logistic rgrssion can b xprssd in trms of P and P. Hnc, by spcifying P, you ar also spcifying th slop. This option is only usd whn th Usr P or Odds Ratio option is st to P. Its intrprtation dpnds on whthr X is binomial or normal. Binomial Covariat Whn X is binary, P is th probability that Y = whn X =. All othr covariats ar assumd to b qual to thir man valus. In this cas, th logistic quation rducs to sinc X =. - P = β + β Normal Covariat Whn X is normally distributd, P is th probability that Y = whn X = µ x + σ x. That is, whn x is on standard dviation abov th man. All othr covariats ar assumd to b qual to thir man valus. In this cas, th logistic quation rducs to so that - P = β + β x P = β + + βx β + βx Odds Ratio (Odds/Odds) This option spcifis th odds ratio to b dtctd by th study. As was shown arlir, th slop of th logistic rgrssion can b xprssd in trms of P and th odds ratio. Hnc, by spcifying OR, you ar also spcifying th slop. Using th formula spcifying OR and P implicitly spcifis P. OR P P = P + ( ) OR( P ) This option is only usd whn th Usr P or Odds Ratio option is st to Odds Ratio. Its intrprtation dpnds on whthr X is binomial or normal. Binomial Covariat Whn X is binary, this option givs th odds ratio of P and P. All othr covariats ar assumd to b qual to thir man valus. In this cas, th logistic quation rducs to sinc X =. - P = β + β 86-6

7 This odds ratio compars th odds of obtaining Y = whn X = to th odds of obtaining Y = whn X =. Normal Covariat Whn X is normally distributd, this option givs th odds ratio of P and P, whr P is th probability that Y = whn X = x, whr x is a valu othr than µ x. All othr covariats ar assumd to b qual to thir man valus. In this cas, th logistic quation rducs to so that - P = β + β x P = β + + βx β + βx This odds ratio compars th odds of obtaining Y = whn X = x to th odds of obtaining Y whn X = µ x. Effct Siz Covariats (X is th Variabl of Intrst) R-Squard of X with Othr X s This is th R-Squard that is obtaind whn X is rgrssd on th othr X s (covariats) in th modl. Us this to study th influnc on powr and sampl siz of adding othr covariats. Not that th numbr of additional variabls dos not mattr in this formulation. Only thir ovrall rlationship with X through this R-Squard valu is usd. Of cours, this valu is rstrictd to bing gratr than or qual to zro and lss than on. Us zro whn thr ar no othr covariats. X (Indpndnt Variabl of Intrst) This option spcifis whthr th covariat is binary (binomial) or continuous (normal). This is a vry important distinction sinc th sampl siz rquird for a particular powr lvl is much largr for a binary covariat than for a continuous covariat. This slction also changs th maning of P and P. Prcnt of N with X = Whn X is binary, this option spcifis th proportion, R, of th sampl in which X =. Not that th valu is spcifid as a prcntag. 86-7

8 Exampl Powr for a Continuous Covariat A study is to b undrtakn to study th rlationship btwn post-traumatic strss disordr and hart rat aftr viwing vido taps containing violnt squncs. Hart rat is assumd to b normally distributd. Th vnt rat is thought to b 7% among soldirs. Th rsarchrs want a sampl siz larg nough to dtct an odds ratios of.5 or 2. with 9% powr at th.5 significanc lvl with a two-sidd tst. Thy dcid to calculat th powr at lvl sampl sizs btwn 2 and 2. Stup This sction prsnts th valus of ach of th paramtrs ndd to run this xampl. First, from th PASS Hom window, load th procdur. You may thn mak th appropriat ntris as listd blow, or opn Exampl by going to th Fil mnu and choosing Opn Exampl Tmplat. Option Valu Dsign Tab Solv For... Powr Altrnativ Hypothsis... Two-Sidd Alpha....5 N (Sampl Siz) P (Baslin Probability that Y=)....7 Us P or Odds Ratio... Odds Ratio Odds Ratio (Odds/Odds) R-Squard of X with Othr X s... X (Indpndnt Variabl of Intrst)... Continuous (Normal) Annotatd Output Click th Calculat button to prform th calculations and gnrat th following output. Numric Rsults Odds R Powr N P P Ratio Squard Alpha Bta

9 Rport Dfinitions Powr is th probability of rjcting a fals null hypothsis. It should b clos to on. N is th siz of th sampl drawn from th population. P is th rspons probability at th man of th covariat, X. P is th rspons probability whn X is incrasd to on standard dviation abov th man. Odds Ratio is th odds ratio whn P is on top. That is, it is [P/(-P)]/[P/(-P)]. R-Squard is th R2 achivd whn X is rgrssd on th othr indpndnt variabls in th rgrssion. Alpha is th probability of rjcting a tru null hypothsis. Bta is th probability of accpting a fals null hypothsis. Summary Statmnts A logistic rgrssion of a binary rspons variabl (Y) on a continuous, normally distributd, indpndnt variabl (X) with a sampl siz of 2 obsrvations achivs 7% powr at a.5 significanc lvl to dtct a chang in Prob(Y=) from th valu of.7 at th man of X to. whn X is incrasd to on standard dviation abov th man.this chang corrsponds to an odds ratio of.5. This rport shows th powr for ach of th scnarios. Th rport shows that a powr of 9% is rachd at a sampl siz of about 3 for an odds ratio of 2. and for an odds ratio of.5. Plot Sction 86-9

10 Ths plots show th powr vrsus th sampl siz for th two valus of th odds ratio. 86-

11 Exampl 2 Sampl Siz for a Continuous Covariat Continuing with th prvious study, dtrmin th xact sampl siz ncssary to attain a powr of 9%. Stup This sction prsnts th valus of ach of th paramtrs ndd to run this xampl. First, from th PASS Hom window, load th procdur. You may thn mak th appropriat ntris as listd blow, or opn Exampl 2 by going to th Fil mnu and choosing Opn Exampl Tmplat. Option Valu Dsign Tab Solv For... Sampl Siz Altrnativ Hypothsis... Two-Sidd Powr....9 Alpha....5 P (Baslin Probability that Y=)....7 Us P or Odds Ratio... Odds Ratio Odds Ratio (Odds/Odds) R-Squard of X with Othr X s... X (Indpndnt Variabl of Intrst)... Continuous (Normal) Output Click th Calculat button to prform th calculations and gnrat th following output. Numric Rsults Odds R Powr N P P Ratio Squard Alpha Bta This rport shows th powr for ach of th scnarios. Th rport shows that a powr of 9% is achivd at a sampl siz of 335 for an odds ratio of 2. and 98 for an odds ratio of

12 Exampl 3 Effct Siz for a Continuous Covariat Continuing th prvious study, suppos th rsarchrs can only afford a sampl siz of 5 individuals. Thy want to dtrmin if a maningful odds ratio can b dtctd with this sampl siz. Stup This sction prsnts th valus of ach of th paramtrs ndd to run this xampl. First, from th PASS Hom window, load th procdur. You may thn mak th appropriat ntris as listd blow, or opn Exampl 3 by going to th Fil mnu and choosing Opn Exampl Tmplat. Option Valu Dsign Tab Solv For... P > P or Odds Ratio > Altrnativ Hypothsis... Two-Sidd Powr....9 Alpha....5 N (Sampl Siz)... 5 P (Baslin Probability that Y=)....7 R-Squard of X with Othr X s... X (Indpndnt Variabl of Intrst)... Continuous (Normal) Output Click th Calculat button to prform th calculations and gnrat th following output. Numric Rsults Odds R Powr N P P Ratio Squard Alpha Bta This rport shows that this xprimntal dsign can dtct an odds ratio of.765. That is, it can dtct a shift in th vnt rat from.7 to

13 Exampl 4 Sampl Siz for a Binary Covariat A study is to b undrtakn to study th rlationship btwn post-traumatic strss disordr and gndr. Th vnt rat is thought to b 7% among mals. Th rsarchrs want a sampl siz larg nough to dtct an odds ratio of.5 with 9% powr at th.5 significanc lvl with a two-sidd tst. Stup This sction prsnts th valus of ach of th paramtrs ndd to run this xampl. First, from th PASS Hom window, load th procdur. You may thn mak th appropriat ntris as listd blow, or opn Exampl 4 by going to th Fil mnu and choosing Opn Exampl Tmplat. Option Valu Dsign Tab Solv For... Sampl Siz Altrnativ Hypothsis... Two-Sidd Powr....9 Alpha....5 P (Baslin Probability that Y=)....7 Us P or Odds Ratio... Odds Ratio Odds Ratio (Odds/Odds)....5 R-Squard of X with Othr X s... X (Indpndnt Variabl of Intrst)... Binary (X= or ) Prcnt of N with X=... 5 Output Click th Calculat button to prform th calculations and gnrat th following output. Numric Rsults Pcnt N Odds R Powr N X= P P Ratio Squard Alpha Bta Th sampl siz is stimatd at This should b vnly dividd among mals and fmals. 86-3

14 Exampl 5 Validation for a Continuous Covariat Hsih (998) pag 628 givs th powr as 95% whn N = 37, alpha =.5 (two-sidd), P =.5, and th odds ratio is.5. Th covariat is assumd to b continuous. Stup This sction prsnts th valus of ach of th paramtrs ndd to run this xampl. First, from th PASS Hom window, load th procdur. You may thn mak th appropriat ntris as listd blow, or opn Exampl 5 by going to th Fil mnu and choosing Opn Exampl Tmplat. Option Valu Dsign Tab Solv For... Powr Altrnativ Hypothsis... Two-Sidd Alpha....5 N (Sampl Siz) P (Baslin Probability that Y=)....5 Us P or Odds Ratio... Odds Ratio Odds Ratio (Odds/Odds)....5 R-Squard of X with Othr X s... X (Indpndnt Variabl of Intrst)... Continuous (Normal) Output Click th Calculat button to prform th calculations and gnrat th following output. Numric Rsults Odds R Powr N P P Ratio Squard Alpha Bta PASS calculats a powr of.9549 which matchs Hsih. 86-4

15 Exampl 6 Validation for a Binary Covariat Hsih (998) pag 626 givs th powr as 95% whn N = 282 (qual sampl sizs for both groups), alpha =.5 (two-sidd), P =.4, and th P =.5. Th covariat is assumd to b binary. Stup This sction prsnts th valus of ach of th paramtrs ndd to run this xampl. First, from th PASS Hom window, load th procdur. You may thn mak th appropriat ntris as listd blow, or opn Exampl 6 by going to th Fil mnu and choosing Opn Exampl Tmplat. Option Valu Dsign Tab Solv For... Powr Altrnativ Hypothsis... Two-Sidd Alpha....5 N (Sampl Siz) P (Baslin Probability that Y=)....4 Us P or Odds Ratio... P P (Altrnativ Probability that Y=)....5 R-Squard of X with Othr X s... X (Indpndnt Variabl of Intrst)... Binary (X = or ) Prcnt of N with X=... 5 Output Click th Calculat button to prform th calculations and gnrat th following output. Numric Rsults Pcnt N Odds R Powr N X= P P Ratio Squard Alpha Bta PASS calculats a powr of.952 which matchs Hsih. 86-5