Meta Analysis. LJ Wei Harvard University. Outline

Transcription

1 Meta Analyss LJ We Harvard Unversty Outlne Introducton to meta-analyss Prncples of meta-analyss Formulatng hypothess and effect measures Methods for combnng results across studes Fxed effects poolng of prmary study results Between study heterogenety Random effects analyss Meta Regresson Meta analyss of rare events Exact nference for combnng 2x2 tables

2 Introducton What s Meta Analyss What s meant by the word Meta-analyss Meta s Greek for later n tme Meta s now used to denote somethng that goes to a hgher level or s more comprehensve. How s an analyss made more comprehensve? In emprcal research, there are often multple studes addressng the same research queston A standard analyss attempts to reach a concluson based on a sngle study wthout reference to any other studes. A meta-analyss attempts to reach a concluson based on a set of studes that address the same hypothess. Introducton What s Meta Analyss The Natonal Lbrary of Medcne (989, pp. -40) defnes metaanalyss as A quanttatve method of combnng the results of ndependent studes (usually drawn from the publshed lterature) and syntheszng summares and conclusons whch may be used to evaluate therapeutc effectveness, plan new studes, etc., wth applcaton chefly n the areas of research and medcne Meta-analyss began to be used as an ndex term that year. However, Gene V. Glass had begun usng the term n 976 (p. 3). Meta-analyss refers to the analyss of analyses. I use t to refer to the statstcal analyss of a large collecton of results from ndvdual studes for the purpose of ntegratng the fndngs. It connotes a rgorous alternatve to the casual, narratve dscussons of research studes whch typfy our attempts to make sense of the rapdly expandng research lterature. 2

3 Introducton Hstory of Meta Analyss In 805, Legendre developed least squares to combne data on the orbts of comets from dfferent observatores. In 930 s, statstcans workng n agrcultural research developed methods for combnng the results of studes. Most notable are Fsher and Cochrane. In 960 s, Cohen popularzed the noton of effect sze for use n sample sze determnaton n the socal and behavoral scences Effect sze measures the dfferences between null hypothess and the truth Effect sze + sample sze determnes the power. In 976, Grass publshed an artcle Prmary, secondary and meta-analyss of research. Ths s when the term meta-analyss was frst used. Introducton Hstory of Meta Analyss In 980 s, meta-analyss became a wdely used tool n socal scences and also receved ncreasng attenton from clncal research: tatstcal methods for meta-analyss by Hedges & Olkn publshed n 980 Peto s method for bnary data publshed n 985; Dermonan and Lard s random effects models for meta-analyss publshed n 986. By 990 s, Meta-analyss starts beng done n large scale n medcne wthn a new emphass on evdence based medcne: The Englsh Cochrane center for systematc revew was establshed n 992 The Internatonal Cochrane Collaboraton was formed n 993 3

4 Introducton Why Meta Analyss There are several reasons for conductng a meta-analyss of the results of prevous studes: The ncreasngly large # of research studes 40,000 journals for the scences 5000 artcle every 30 seconds Unsystematc expert revews of an area of research are often based or years behnd the current research. ystematc and quanttatve revews are needed to summarze fndngs n a tmely manner wthout bas. Introducton Why Meta Analyss centfc research s supposed to be replcable, and t s common to have several studes addressng the same hypothess n dfferent settngs. Meta-analyss provdes a way to consder replcaton and consstency of results across a set of studes wthout requrng that each study necessarly have large enough power to reach sgnfcance. Meta analyss can be used to ncrease power. Meta analyss can be used to examne whether studes do not replcate each other and reach dfferent conclusons. Ths may lead to scentfc advances n several ways: Determne varatons of the treatment effect that produce a greater effect Identfy sub-populatons that respond to treatment better 4

5 Prncples of Meta-analyss Meta-analyss s typcally a two stage process ) ummary statstc calculated from each study. For controlled trals, these values descrbe the treatment effects observed n each tral. 2) Pooled effect s calculated by combnng treatment effect estmates from ndvdual studes. Typcally a weghted average of ndvdual effects The combnaton of treatment effect estmates across studes may assume Fxed effects: treatment effects the same across studes. Random effects: treatment effects ~ a dstrbuton across studes. Prncples of Meta-analyss The standard error or samplng dstrbuton of the pooled effect can be used to derve A confdence nterval A p-value for testng whether there s a treatment dfference. In addton to provdng a measure of overall average treatment effect, meta-analyss methods can provde an assessment of Whether the varaton among the results of the separate studes s compatble wth random varaton Whether t s large enough to ndcate nconsstency of treatment effects across studes heterogenety 5

6 Formulatng Hypothess and Effect Measures Before conductng a meta analyss, t s mportant to decde the hypothess or am of the analyss. When formulatng a hypothess for meta-analyss, t s mportant to determne the precse queston the meta-analyss ams to address whether the meta analyss s exploratory or hypothess testng Hypothess testng: s the ntent of the study to provde a defntve test (usually a test of average effect = 0) Exploratory: are there varatons n the treatment or characterstcs of the studes that lead to better outcomes? One also needs to select an approprate effect measure Type of Data and Effect Measures Dchotomous or bnary outcome Interventon Control Event m m 0 No Event n -m n 0 -m 0 Total n n 0 Event Rates p =m /n p 0 =m 0 /n 0 Relatve Rsk (RR) = Odds Rato (OR) = Rsk Dfference (RD) = m m 0 m m 0 / n / n 0 /( n /( n p 0 p 0 p = p 0 m ) = m ) 0 p p 0 /( p) /( p ) 0 6

7 Type of Data and Effect Measures Dchotomous or bnary outcome when events of nterest are rare p and p 0 RR OR essental to assess the absolute rsk n addton to relatve rsk RR and OR = f there are no events n the control group RR and OR not defned f both groups have zero events tandard procedures ether exclude studes wth 0/0 events or add 0.5 to empty cells. Type of Data and Effect Measures Contnuous outcome Mean dfference (MD) tandardzed mean dfference (MD) = MD td Devaton of Outcome All trals assess the same outcome, but measure t n a varety of ways (e.g depresson measured wth dfferent psychometrc scales.) Inherently assumes that the dfferences n the standard devaton among trals reflect dfferences n the measurement scales and not real dfferences n varablty among tral populatons. Overall treatment effect dffcult to nterpret (unts n the standard devaton), but related to the term effect sze whch s frequently used n the socal scences. 7

8 Type of Data and Effect Measures Ordnal outcome Takes values y < y 2 < < y K Example: mld, moderate and severe. When the number of categores s large, such data are often analyzed as contnuous data. One may transform ordnal data nto bnary data by combnng adjacent categores. Proportonal odds rato under a proportonal odds model logt P( Y y k Trt) = α + β Trt k β, the log-odds rato, summarzes the treatment effect Type of Data and Effect Measures Tme to event (survval) outcome Examples: tme to death, tme to the onset of dsease Event tme outcomes are subject to censorng due to loss to follow up or end of study Tme to event data sometmes are analyzed as dchotomous data by consderng the probablty of t-year survval. The most common approach s to express the treatment effect as a hazard rato under a proportonal hazard assumpton logt P( Y y k Trt) = α + β Trt k 8

9 tatstcal Methods for Combnng Results Across tudes What s the true effect? Depends on the underlyng assumpton about the study specfc effect Fxed Effects Assumpton Assumes that all studes have the same true effect Varablty only wthn each study Precson depends manly on study sze Random Effects Assumpton tudes allowed to have dfferent underlyng or true effects Allows varaton between studes as well as wthn studes Basc assumpton: study results are ndependent. Fxed Effects Poolng of Prmary tudy Results Under the fxed effects framework, varous procedures have been proposed Vote count p-value methods Mnmum p-value um of -2log p-value Probt of the p-value Poolng usng effect szes Inverse varance Bnary outcomes Cochrane-Mantel-Hansel Peto method 9

10 Fxed Effects Vote Count Consder # of studes n favor of the concluson (say, reach the 0.05 level of sgnfcance ) and examne f they are the majorty Ths approach has been used a lot due to ts smplcty, but has several drawbacks gnfcance depends on study sample sze and effect sze. Even f the null hypothess s wrong and studes are not small, the percentage of trals reachng sgnfcance could stll be less than 50% low power of detectng a treatment effect Vote counts do not provde an estmate of effect sze. Fxed Effects P-value Methods Consder studes wth the same null hypothess H 0. Each study has a test statstc T and p-value p. We are nterested n testng usng all studes at α level. Under H 0, p ~ Unform(0,) and Φ (p ) ~ N(0, ). Mnmum p-value (Tppett): mn{p,, p } < ( α) / um of -2log p-value (Fsher) 2 2log( p = ) > χ2 ( α) Probt of the p-value / 2 = Φ ( p ) < Φ ( α) 0

11 Fxed Effects Poolng Usng Effect zes nce Glass s work n 976, combnng effect szes has become the man form of meta-analyss. uppose the estmated effect szes are { ˆ β, =,..., } To ascertan the true underlyng effect, a common approach s to consder a weghted average of the effect estmates from ndvdual studes: ˆ β = = w ˆ β w = Fxed Effects Poolng Usng Effect zes Inverse Varance Method: Under fxed effects framework: ˆ β = β0 + ε, =,..., β 0 s the true value of the common effect ε represents the samplng varablty of 2 E(ε ) 0, var(ε ) = σ = var(βˆ ) ε Pooled estmate of β 0 : ˆ β = = Weghts chosen to mnmze the varance Optmal mnmum varance weghts: w w ˆ β βˆ wˆ = w mn var = ˆ σ 2 ε

12 Bnary Outcomes Fxed Effects Poolng Usng Effect zes When the event rates are low or tral szes are small, the standard error estmates used n the nverse varance method may be poor. Cochrane-Mantel-Hansel uses a dfferent weghtng scheme that depends upon the effect measure (eg RR, OR, RD). Cochrane-Mantel-Hansel pooled OR: OR MH = = MH wˆ OR wˆ = MH MH m0 ( n m ) wˆ = n + n0 th tudy Interventon Control Event m m 0 No Event n -m n 0 -m 0 Total n n 0 OR m ( n0 m = m ( n m 0 0 ) ) Bnary Outcomes Peto s Method: Fxed Effects Poolng Usng Effect zes based on an approxmaton to the data lkelhood, expressed as a dfference between observed and expected counts to estmate the pooled log OR and to test heterogenety. th tudy Interventon Control Event m m 0 No Event n -m n 0 -m 0 Total n n 0 O = m V = E, n ( n m + n0 m0 ) ( n + n )( n + n ) E 0 = n n + n 0 0 ( m + m 0 ) log OR Peto = = (O E ) = V var(log OR Peto ) = = V 2

13 Between tudy Heterogenety The key assumpton of fxed effects meta analyss methods s that all prmary studes are estmatng the same underlyng true effect The underlyng effects across studes may be heterogeneous Each study effect sze βˆ s estmatng an ndvdual populaton effect β. As study sample sze N, ˆ β β ome of the β may be the same, but not all of them. Between tudy Heterogenety ources of heterogenety Patent selecton ncluson/excluson crtera, dsease severty/ type, patent characterstcs, geographc dfferences Treatment admnstraton duraton of treatment, dose, blndng of treatment, complance tudy type clncal tral, case control study, cohort study Types of controls hosptal controls, populaton controls, dfferent dsease controls Analyss performed ntent to treat vs completer analyss, outcome measure used 3

14 Between tudy Heterogenety Testng for heterogenety 2 2 Q = wˆ (βˆ βˆ ) ~ χ Cochran s Q-test: = pooled Provdes a measure of between study varaton. Other descrptve measures of heterogenety H statstc: H = Q /( ) has mean under H 0. under H Hoggns and Thompson (2002) suggested: H >.5 cauton regardng heterogenety; H <.2 lttle heterogenety I 2 statstc = (H 2 )/H 2 % of total varablty n effect sze due to between study varaton I 2 ~ 0 lttle heterogenety; I 2 ~ hgh heterogenety termed the nconsstency of the trals ncluded n meta-analyss and has become a preferred measure of heterogenety. 0 Random Effects Under random effects model, βˆ estmates an underlyng study specfc effect sze β. ˆ 2 2 β = β + ε, ε ~ N(0, σ ), β ~ N( β0, τ ), =,..., Bayesan model wth a Gaussan pror dstrbuton for β The varance τ 2 represents between study varaton. 2 The study specfc varance σ represents wthn study varaton whch goes to 0 as study sample sze Pooled estmate: ˆ RE RE RE = w ˆ RE β ˆ β wˆ ˆ = 2 2 = = w ˆ τ +σˆ ε 4

15 Meta Regresson The pooled effect sze estmates the average effect across all studes In the presence of heterogenety the valdty of such an average measure? not a sngle populaton effect sze that apples to all studes random effects poolng addresses heterogenety to some extent Meta regresson provde an alternatve approach that allows exploraton of why studes have vared effect szes. one uses characterstcs of the studes to explan the excess varaton n effect szes Thompson and Hggns (2002) revewed several meta-regresson methods Meta Regresson tram (996) proposed a lnear mxed effects model Y ˆ = X α + Z β + e + ζˆ Related Effect Estmates (k x ) (e.g. Mean Outcome or Dfference) Fxed Effects Random Effects Remanng Between tudy Heterogenety amplng Error β ~ N(0, D) e ~ N(0, σ 2 I) ςˆ ~ N(0, V ) V 0 as n 5

16 Meta Regresson pecal Cases The Dermonan and Lard (986) Model: Ŷ = α + β + ζˆ The Begg and Plote (99) Model: Y ˆ = X α + β + ζˆ Ths model allows the ncluson of sngle treatment hstorcal controls as well as comparatve trals n a treatment effect assessment. Yˆ α = X δ chemo + β + ˆ ζ 20 estmates of the probablty of 2-year dsease free survval 4 trals wth both BMT and Chemotherapy, Y s a 2x vector, 2 sngle arm studes: = 4 BMT only Y s a scalar, X = (, ) 8 Chemotherapy only Y s a scalar, X = (, 0) X 0 6

17 Exact Procedure for Combnng 2x2 Tables for Rare Events Exact Procedure for Combnng 2x2 Tables for Rare Events tandard nference procedures for meta-analyss rely on large sample approxmatons to the dstrbutons of the combned pont estmators. However, such approxmaton may be naccurate when the ndvdual study sample szes are small, or total number of studes s not large, or event rates are low When the events of nterest are very rare, many studes may have 0 events n one or both groups. tandard procedures ether excludng studes wth 0 events, or add 0.5 to empty cells 7

18 Example Effect of Rosgltazone on MI or CVD Deaths Nssen and Wolsk (2007) performed a meta analyss to examne whether Rosgltazone (Avanda, GK), a drug for treatng type 2 dabetes melltus, sgnfcantly ncreases the rsk of MI or CVD related death. Avanda was ntroduced n 999 and s wdely used as monotherapy or n fxed-dose combnatons wth ether Avandamet or Avandaryl. The orgnal approval of Avanda was based on the ablty of the drug to reduce blood glucose and glycated hemoglobn levels. Intal studes were not adequately powered to determne the effects of ths agent on mcrovascular or macrovascular complcatons of dabetes, ncludng cardovascular morbdty and mortalty. Example Effect of Rosgltazone on MI or CVD Deaths However, the effect of any ant-dabetc therapy on cardovascular outcomes s partcularly mportant because more than 65% of deaths n patents wth dabetes are from cardovascular causes. Of 6 screened studes, 48 satsfed the ncluson crtera for the analyss proposed n Nssen and Wolsk (2007). 42 studes were reported n Nssen and Wolsk (2007), the remanng 6 studes have zero MI or CVD death 0 studes wth zero MI events 25 studes wth zero CVD related deaths 8

19 Event Rates from 0% to 2.70% for MI Event Rates from 0% to.75% for CVD Death MI CVD Death?????? Log Odds Rato 95% CI: (.03,.98); p-value = 0.03 (n favor of the control) Log Odds Rato 95% CI: (0.98, 2.74); p-value =

20 Exact Meta-Analyss Procedure Combnng Exact Intervals Questons: Could we combne nformaton across studes wthout excludng studes wth 0 events or artfcal mputaton? Could we make exact nference wthout relyng on possbly naccurate large sample approxmatons when the total number of studes s small, or the sample szes of ndvdual studes are small, or when the event rates are low. Exact Meta-Analyss Procedure Combnng Exact Intervals uppose we are nterested n makng nferences about a effect measure Δ whose true value s Δ 0 For example, the rsk dfference between two groups for the dabetes studes wth respect to MI ncdences. pecfcally, suppose we are nterested n constructng a 00( α)% one-sded confdence nterval (a, ) for Δ 0 based on all data from ndependent studes. 20

21 Exact Meta-Analyss Procedure Combnng Exact Intervals For a gven confdence level η, one may obtan study specfc one-sded η-level confdence ntervals for the rsk dfference. each nterval s constructed based on the data only from ts correspondng study. Iˆ ( η) ˆ ( η) I ˆ ( η) I tudy tudy tudy For any gven value of Δ, we examne whether Δ s the true value. If Δ = Δ 0, then by the defnton of η-level confdence ntervals any gven such nterval should contan Δ wth probablty η on average Δ should belong to at least 00η% of the above ndependent ntervals. Exact Meta-Analyss Procedure Combnng Exact Intervals To determne whether a gven value Δ should be ncluded n the one-sded confdence nterval (a, ) we examne whether the probablty that η-level study specfc confdence ntervals contan Δ s ndeed at least η Iˆ ( η ) ˆ ( η ) I ˆ ( η ) I tudy tudy tudy Does Δ belong to ths nterval? y (Δ, η) = f yes, 0 f not Under the null that Δ = Δ 0, P{ y ( Δ, η) = } η 2

22 Exact Meta-Analyss Procedure Combnng Exact Intervals Thus, we propose to nclude Δ n the 00( α)% level confdence nterval (a, ) f t( Δ, η) = w { y ( Δ, η) η } c where w s a study specfc postve weght (e.g. sample sze) c s chosen such that P(T(η) < c) α, = T ( η) = w ( B η ) s the null counterpart of t(δ,η) = {B, =,, } are n ndependent Bernoull random varable wth success probablty of η. We repeat ths process wll all other possble values for Δ and obtan the fnal nterval. Exact Meta-Analyss Procedure Combnng Exact Intervals One may further mprove the nterval estmate by usng multple ntervals wth a range of η levels from each study. Let 0 < η < < η K < and j= Iˆ ( η ) v j s a postve weght for the η j level ntervals ˆ ( η I K For any gven Δ, we nclude Δ n the fnal nterval (a, ) f K K tcomb( Δ) = v jt( Δ, η j ) d where P v jtj ( η j ) < d α k = { T ( η),..., T K ( ηk )} s constructed based on correlated Bernoull K random vectors such that v represents the null j = jt j (η j ) counterpart of t comb (Δ) ) 22

23 Exact Meta-Analyss Procedure Combnng Exact Intervals mlarly, we obtan combned ( α)% one sded nterval (, b) based on the correspondng one-sded study specfc ntervals. Thus, (a, b) would be a ( 2α)% two-sded nterval for the rsk dfference. A pont estmator for Δ 0 may be obtaned as, the md-pont of the ntersecton of all two-sded ntervals for Δ 0 across all possble values of α Δˆ s the value of Δ wth the least evdence of beng rejected as the truth. Δˆ Event Rates from 0% to 2.70% for MI Event Rates from 0% to.75% for CVD Death 23

24 Example Effect of Rosgltazone on MI or CVD Deaths For RR or OR effect measures unless pror nformaton about the underlyng event rates s avalable, t s not clear how to utlze studes wth zero events wthout contnuty correcton. RD may be used as an alternatve effect measure appealng nterpretaton exact nference may be used We examne the effect of Rosgltazone on MI or CVD deaths based on Δ = RD (Rosgltazone Control). Example Effect of Rosgltazone on MI or CVD Deaths Each study, we construct 20 exact confdence ntervals at levels {η, η 20 } whch are equally spaced from 0. to Based on these ndvdual ntervals, we then construct the fnal combned nterval based on the hypothess testng procedure. 24

25 Exact Inference Asymptotc Inference CVD Death Δ ˆ = 0.063% Δ ˆ = 0.% 95% CI: (-0.3, 0.23)% 95% CI: (0.00, 0.3)% P-value = 0.83 P-value = 0.05 Exact Inference Asymptotc Inference MI Δ ˆ = 0.8% Δ ˆ = 0.9% 95% CI: (-0.08, 0.38)% 95% CI: (0.02, 0.42)% P-value = 0.27 P-value =

26 Non-parametrc Inference for the Random Effects Dstrbuton n Meta Analyss Random Effects Meta Analyss Fxed effects meta analyss methods assume that the true effects of nterest are the same across all prmary studes The estmated study specfc effect βˆ converges to the same quantty as study sample sze n The underlyng effects across studes may be heterogeneous Each study effect sze s estmatng an ndvdual βˆ populaton effect β wth ˆ β β as n ome of the β may be the same, but not all of them. 26

27 Random Effects Meta Analyss uper Populaton β ~ F( ) β β... 2 β - β... tudy Data tudy 2 Data tudy - Data tudy Data 2 n (βˆ β ) ~ N(0, ˆ σ ) n (βˆ β ) ~ N(0, ˆ ) n (βˆ β 2 ) ~ N(0, ˆ σ ) n (βˆ 2 β ) ~ N(0, ˆ σ ) 2 2 σ Random Effects Meta Analyss Under random effects model wth normal pror, Dermonan and Lard (DL) (986) proposed procedures for estmatng the mean of the random effects β 0 : 2 n ˆ 2 β = β + τ, ( β β ) ~ N(0, ˆ σ ), =,..., 0 E(τ ) = 0 and var(τ ) represents between study varaton. The study specfc varance σˆ 2 / n 0 as n May not work well wth when the number of studes s small! 27

28 Random Effects Meta Analyss Non-parametrc Estmaton of the Medan Wang et al (2008) proposed nterval estmaton procedures for the quantles of β wthout requrng the number of studes to be large. uppose we are nterested n estmatng the medan of β denoted by μ 0. If β known, exact confdence nterval for μ 0 can be obtaned by nvertng a sgn test: T ( μ) = { I( β < μ) 0.5} = The null dstrbuton of T(μ 0 ) + /2 s a Bnomal(,0.5). Random Effects Meta Analyss Non-parametrc Estmaton of the Medan ˆβ ' If sconsstently estmate β, then one may consder the test statstc ~ T ( μ) = { ( ˆ I β < μ) 0.5} = ~ Uncondtonal null dstrbuton of T ( μ 0 ) + /2 s approxmately Bnomal(, 0.5). However, the Bernoull varable I( ˆ β < μ) may not be a good surrogate for I( β < μ) If the varance of s not small relatve to the dstance between βˆ β and μ. 28

29 Random Effects Meta Analyss Non-parametrc Estmaton of the Medan Alternatvely, one may replace I( ˆ β < μ) wth a measure of lkelhood for the event β < μ Example: the observed coverage level of the nterval (-,μ) for β whch s Φ{( μ ˆ β ) / ˆ σ } The test statstc based on the coverage level s Tˆ( μ) = [ {( ˆ Φ μ β ) / ˆ σ } 0.5] = tudes wth data that are more nformatve for the event β < μ would yeld coverage level closer to ether 0 or and thus carry more weght n the test statstc. Random Effects Meta Analyss Non-parametrc Estmaton of the Medan nce Φ{( μ ˆ β ) / ˆ σ } I( β < μ) 0 n probablty Φ{( μ ˆ β ) / ˆ σ }symmetrc around 0.5 one may approxmate the null dstrbuton of based on Tˆ( μ) = [ {( ˆ Φ μ β ) / ˆ σ } 0.5] T * = ( μ) = ˆ Φ{( μ β ) / ˆ σ } 0.5 (2Δ = {Δ, =,,} ~ Bernoull(0.5) ndependent of data ) 29

30 Random Effects Meta Analyss Non-parametrc Estmaton of the 00p th Percentle For the 00p th percentle, the test statstc s Uncondtonally Tˆ ( μ) = [ {( ˆ Φ μ β ) / ˆ σ } 0.5] = The null dstrbuton of T * p p {ε, =,,} ~ Bernoull(p) ndependent of data ( μ) = ˆ Φ{( μ β ) / ˆ σ } 0.5 (2ε ) = ˆ ( μ) s asymptotcally Bnomal(,p) T p ˆ ( μ) can be approxmated by T p Example Effect of EA on the Rsk Mortalty Bennett et al (2008): meta analyss to examne whether the erythropoetn-stmulatng agents (EA) for treatng cancer chemotherapy-assocated anema would ncrease the rsk of mortalty 5 phase III comparatve trals (EA vs placebo or standard of care) Effect measure: hazard rato From each study, s a consstent estmator of the underlyng βˆ study specfc hazard rato β. Confdence ntervals of β are also avalable to nfer the wthn study varaton. 30

31 Example Effect of EA on the Rsk Mortalty 95% Confdence Intervals for the Hazard Rato Medan 25 th Percentle 75 th Percentle DL (.0,.20) ~ T ( ) ) wth I( ) (0.90,.26) (0.49, 0.93) (.25,.72) Tˆ ( wth Φ( ) (0.94,.2) (0.70, 0.99) (.8,.48) 3