PubH 7470: STATISTICS FOR TRANSLATIONAL & CLINICAL RESEARCH

Transcription

1 PubH 7470: AIIC FOR RANLAIONAL & CLINICAL REEARCH ulemet for Aalysis: Use of FIELLER HEOREM for HE EIMAION OF RAIO

2 HE GAP Most teachig ad learig rograms i tatistics ad Biostatistics ours icluded - focus o the differeces & the sums of arameters, statistics, or radom variables However, i may alicatios we have to deal with ratios of arameters, statistics, or radom variables Reaso? tatistics uts more emhasis o additive models ; most lausible biological ad biomedical models are multilicative.

3 RELAIVE RIK Relative Risk has bee a oular arameter i eidemiology studies; a cocet used for the comariso of two grous or oulatios with resect to a uwated evet. It is the ratio of icidece rates or disease revaleces; usually, oe grou is uder stadard coditio agaist which the other grou exosed is measured. Relative Risk is a ratio: Risk Ratio, a ratio of two roortios.

4 ODD RAIO Whe icidece ad revalece are low rare diseases, the Relative Risk ad the Odds Ratio are aroximately equal. Odds Ratio is more oular because it is estimable i retrosective desigs; i ractice, we calculate Odds Ratio ad iterret it like Relative Risk. But Odds Ratio is still a ratio of arameters; maybe it s a differet kid of ratios a ratio of ratios

5 DIAGNOIC E ome of the idices of diagostic accuracy are the Likelihood Ratios, each is the ratio of two robabilities Both are exressible as fuctios of sesitivity ad secificity. LR Pr Pr D D LR Pr Pr D D

6 COMPARION OF CREENING E WIH BINARY ENDPOIN We ca erform two searate Chi-square tests or McNemar Chi-square tests deedig o the desig, oe for cases ad oe for cotrols; for a overall level of, each test is erformed at /. hat is, we comare sesitivities ad we comare secificities searately: No Problem here.

7 MEAURING DIFFERENCE If the differece betwee two diagostic tests are foud to be sigificat; the level of differece should be summarized ad reseted. he two commoly used arameters are the ratio of two sesitivities R + ad the ratio of two secificities R - ; ratios of two roortios.

8 here are may other examles: Etiologic Fractio Causal Iferece, tadardized Mortality Ratio Evirometal & Occuatioal Health, Effect ize Cliical rials, etc

9 DIREC AAY I direct assays, the doses of the stadard ad test rearatios are directly measured for a evet of iterest with itra-subject dose escalatio. Whe a evet of iterest occurs, e.g.. the death of the subject, ad the variable of iterest is the dose required to roduce that evet for each subject. he value is called idividual effect dose IED.

10 ice the cocetratio ad the dose are iversely roortioal - whe cocetratio is high, we eed a smaller dose to reach the same resose. I other words, we defie the relative otecy or ratio of cocetratios of the test to stadard as the ratio of doses of the stadard to test: Dose Dose hat is a "Ratio of Meas"

11 PARALLEL-LINE AAY Parallel-lie assays are those i which the resose is liearly related to the log dose. From the same defiitio of relative otecy, the two doses are related by D = D. he model: he above assumtio leads to: E[Y X =logd ] = +X, E[Y X =logd = D ] = + log + X We have arallel lies with a commo sloe ad differet itercet.

12 MULIPLE REGREION A commo aroach is oolig data from both rearatios ad usig Multile Regressio. Deedet Variable: Y = Resose; wo Ideedet Variables are: X = logdose & P = Prearatio a dummy variable coded as P = for est ad P = 0 for tadard

13 Multile Regressio Model : E Y β β is the commo sloe ad is the "differece of itercets"; M log 0 X P hat is " Ratio of Regressio Coefficiets"

14 LOPE RAIO AAY loe-ratio assays are those i which the resose is liearly related to the dose itself. From the same defiitio of relative otecy, the two doses are related by D = D. he model: he above assumtio leads to: E[Y X =D ] = +X, E[Y X =D = D ] = + X. We have straight lies with a commo itercet ad differet sloes.

15 MULIPLE REGREION ame regressio setu, differet models; Deedet Variable: Y = Resose; wo Ideedet Variables are: X = Dose & P = Prearatio a dummy variable coded as P = for est Prearatio ad P = 0 for tadard Prearatio

16 " hat ivolves a " the commo itercet ad is β 0 0 Regressio Coefficiets Ratio of β β # : Regressio Model Multile PX X Y E

17 MULIPLE REGREION # Let Y be the resose, X ad X the doses; defied for use with the combied samle as follows: for ay observatio o, set X =0, for ay observatio o, set X =0: E Y β 0 0 Commo Itercet β β X ; aother " Ratio" X ;

18 COMMON FORM A r B Both statistics,a ad B,are asymtotically distributed as " ormal" with "estimable variaces"

19 If we do the usual way by takig logs: log r log A log B he, i formig cofidece itervals for r is a estimate of, we assume that loga ad logb are asymtotically/aroximately ormally distributed which cotradict the fact that A ad B themselves are ormally distributed. he result is based o iflated variaces variace of logormal distributio is larger tha variace of ormal distributio which is iefficiet because cofidece itervals are too log uecessarily.

20 Examle: Focusig o Risk Ratio ratio of roortios, Lui Cotemorary Cliical rials, 006 foud that the log trasformatio method could lead to itervals which are may times loger tha those by cometig methods - as much as 40 times i some cofiguratios a obvious loss of efficiecy.

21 FIELLER HEOREM If r = A/B is a estimate of, we cosider the statistic A- B which is distributed as ormal because both A ad B are ormally distributed ad is a costat. We derive mea ad variace of that statistics which lead to cofidece limits for. Let C = A- B, distributed as ormal We first fid the mea & variace of C

22 Recall: C = A- B is distributed as ormal We first fid the mea & variace of C EC 0 VarC V;V is estimated by v C/ v is distributed as "t" Pr-t Pr C Pr C / v vt C t / v t ;

23 Pr C Pr{ A vt B.95.95; olve the "quadratic equatio": A ρb vt vt to obtai lower ad uer limits for ρ

24 DIREC AAY.95 ] } Pr[{ 0 r s t X X X X Var X X E where t. 975 is the 97.5 th ercetile of the t distributio with + - degrees of freedom. he two roots for obtaied by solvig the quadratic equatio i withi the robability statemet will yield the 95% cofidece limits r L ad r U.

25 Recall: Whe you have a quadratic equatio ax + bx + c = 0; first ste is checkig b -4ac. If it s ositive, roots exist: x b b a 4ac

26 0 } { 4 } { two roots exist because olve for ρ : 0 } { } { r s t X X X X r s t X X X X r s t X X

27 REUL he first oe is the 95% CI directly from the Fieller s theorem, the secod oe is a aroximatio because the term g is ofte rather small. r x s t r x s t g r g x s t r g } { Exact Aroximatio

28 EXAC REUL tadard est otal Mea Variace t { r g g s g t df x s t { ,.48 s. 975 g } x { } r.8 7 }

29 APPROXIMAE REUL tadard est otal Mea Variace t s g df r t t x s s x ,.44 7 r } vs..95,.48

30 PARALLEL-LINE AAY.95 ] } Pr[{ } { 0 } { D M s t Mb y y D M b Var M y y Var Mb y y Var Mb y y E where t. 975 is the 97.5 th ercetile of the t distributio with df E degrees of freedom.

31 PROCE FOR 95% C.I. he two roots for obtaied by solvig the quadratic equatio i withi the robability statemet will yield the 95% cofidece limits M L ad M U..95 ] } Pr[{ D M s t Mb y y D x x x x X X

32 REUL 95% Cofidece limits from the Fieller s theorem g is ofte very small; sometimes ca treat -g as 975. } { Db s t g D m g b s t m g D x x x x X X

33 EXAMPLE Prearatio tadard Prearatio est Prearatio Dose D; mmgcc X = log0dose Resose Y; mm

34 NUMERICAL REUL I our umerical examle, we have m=.454, t. 975 df=35=.03, commo sloe is b=., = =4, D=.8998, ad s =.583 leadig to: g = % cofidece limits for M is.4,.67 95% cofidece iterval for relative otecy is.33,.47 which icludes oit estimate of.4

35 LOPE-RAIO AAY Let Y be the resose, X ad X the doses; defied for use with the combied samle as follows: for ay observatio o, set X =0, for ay observatio o, set X =0: aother "Ratio" ; Commo Itercet β ; 0 0 X X Y E

36 UE OF FIELLER HEOREM.95 ] Pr[ 0 X R X s t b b X X b Var b Var b b Var b b E where t. 975 is the 97.5 th ercetile of the t distributio with df E degrees of freedom.

37 PROCE FOR 95% C.I. Pr[ b b t s X R X ].95 he two roots for obtaied by solvig the quadratic equatio i withi the robability statemet will yield the 95% cofidece limits R L ad R U.

38 REUL he first oe is the 95% CI directly from the Fieller s theorem, the secod oe is for the secial case of the 5-oit sloe ratio assays. { r g P X t b s X [ P ; ad g g X t b s X r X ]}

39 RAIO OF PROPORION r C Var C

40 APPROACH # we use estimated variace i last ste meas : imilar to aroach i ratio of I.for ρ α00% C. wo roots form / ^ ^ z C Var C Var C Var C

41 APPROACH # C Var C Var C / wo roots form z α00% C. I.for ρ Liu, Cotemorary Cliical rials 006

42 HE CHOICE It s ot clear if it s better to use the variace or the estimated variace as i Biological Assays; Liu CC, 006 used variace but gave o exlaatio/justificatio. But he got ito a ew roblem: the resultig quadratic equatio may have o real roots i some simulatio cofiguratios.

43 Lui Cotemorary Cliical rials, 006 alied Fieller s heorem to study Risk Ratio ; showed that the use of Fieller s heorem/method would lead to more efficiecy i.e. shorter itervals but, more imortat, it imroves coverage robability. I believe that the results aly to quatitative bioassays e.g. ratio of regressio coefficiets.

44 ODD RAIO Does Fieller s heorem work for Odds Ratio? Odds Ratio is a ratio of ratios ; its estimated umerator ad deomiator are ot ormally distributed more like log ormal; is Fieller s heorem-based method robust i this case? Maybe ot, I do ot kow; at least I m ot sure. Perhas the log trasformatio method works well for Odds Ratio; ad it has bee oe of a few ratios that we hadle roerly.

45 #8. IUE OF HE DAY: Read ad reset the article by Lui i Cotemorary Cliical rials, 006.