BIOSTATISTICAL METHODS FOR TRANSLATIONAL & CLINICAL RESEARCH

Transcription

1 BIOSAISICAL MEHODS FOR RANSLAIONAL & CLINICAL RESEARCH Direct Bioassays: REGRESSION APPLICAIONS

2 COMPONENS OF A BIOASSAY he subject is usually a aimal, a huma tissue, or a bacteria culture, he aget is usually a drug, a chemical he resose is usually a chage i a articular characteristic or eve the death of a subject; resoses ca be biary or measured o cotiuous scale.

3 BASIC PROCESS For stochastic assays, our oly targets, we refer to the relatioshi betwee stimulus level ad the resose it roduces as a regressio model. A test rearatio of the stimulus - havig a ukow otecy - is assayed to fid the resose. We fid the dose of the stadard rearatio which roduces the same resose (as that by test rearatio.

4 here are two tyes of bioassays: ( direct assays ad ( idirect assays. hey are both stochastic

5 DIREC ASSAYS I direct assays, the doses of the stadard ad test rearatios are directly measured for (or util a evet of iterest. Resose is fixed (biary, dose is radom. 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 resose/evet for each subject. he value is called idividual effect dose (IED. For examle, we ca icrease the dose util the heart beat (of a aimal ceases to get IED.

6 yical Exerimet: A grou of subjects (e.g. aimals are radomly divided ito two subgrous ad the IED of a stadard rearatio is measured i each subject of grou ; the IED of the test or ukow rearatio is measured i each subject of grou. he aim is to estimate the relative otecy, that is the ratio of cocetratios of the test relative to stadard to roduce the same biological effect/evet.

7 Kee i mid that 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 the ratio of cocetratios of the test to stadard as the ratio of doses of the stadard to test: ρ Dose Dose S

8 Whe the relative otecy ρ >, the est Prearatio is stroger (we eed a larger dose of the est i order to roduce the same resose ad vice versa. Pairs of doses that give the same resose are termed equiotet, meaig same stregth.

9 Data are very simle: two ( ideedet samles, the tye you usually have for twosamle t-test or Wilcoxo test; but we will ot comare them usig a test of sigificace. We wat to estimate Relative Potecy.

10 he questio is: How to estimate the ratio of two arameters? he oit estimate ca be easily obtaied; the more difficult arts is the Stadard Error ad Cofidet Itervals here are a few more examles

11 RELAIVE RISK 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 revalece; 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 or robabilities.

12 ODDS 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 Perhas it s the oly ratio that we have hadled roerly.

13 Ad the aalysis of Idirect Bioassays ivolves Ratio of Regressio Coefficiets. Other examles iclude Ratio of Sesitivities, Ratio of Secificities, ad Stadardized Mortality Ratio.

14 HE GAP Most teachig ad learig rograms i Statistics 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? Statistics uts more emhasis o additive models ; most lausible biological ad biomedical models are multilicative.

15 Eve the Likelihood Fuctio L. he model is multilicative; the likelihood fuctio is a roduct. But we take the log to tur it ito a sum because it is easier to take derivative; however, we have o roblem i that case: we oly wat to maximize L ad logarithm is a mootoic fuctio L ad log of L are maximize at the same time.

16 he method we cover here is called Fieller s theorem. Fieller s theorem is ot a theorem; it s a historical ame. It should be called Fieller s method, a statistical method for the estimatio of a ratio of two arameters.

17 COMMON FORM r A B Both statistics, A ad B, are asymtotic ally distribute d as " ormal " with "estimable variaces "

18 r A B A ad B could be samle meas (direct assays, roortio s (relative risk, or regressio coefficiets (quatitat ive assays

19 If we do the usual way by takig logs: log r log A log B he, i formig cofidece itervals for ρ (r is a estimate of ρ, 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 S HEOREM If r A/B is a estimate of ρ, we cosider the statistic (A- ρb which is distributed as ormal because both A ad B are ormally distributed ad is ρ a costat. We derive mea ad variace of that statistic which lead to cofidece limits for ρ. Let C A- ρb, distributed as ormal We first fid the mea & variace of C

22 Examle: I Direct Bioassays, r is the ratio of two samle meas; ad ρ - the (ukow Relative Potecy is the ratio of the two oulatio meas. I Eidemiology, r the Relative Risk is the ratio of two samle roortios.

23 Recall: C A- ρb is distributed as ormal We first fid the mea & variace of C E(C 0 Var(C V; V is estimated by v C/ v is distribute d as "t" Pr(-t Pr( C Pr( C / v vt C t / v t ;

24 Pr( C Pr{( Solve A the vt ρ B (A ρb.95 "quadratic to obtai lower ad vt vt.95; equatio" uer limits : for ρ

25 he two solutios (or roots of that equatio are the lower ad the uer edoit of the 95% Cofidece Iterval for the (ukow ratio ρ.

26 DIREC ASSAYS.95 ] ( } Pr[{ ( ( 0 ( + + S S S S S r s t X X X X Var X X E ρ ρ σ ρ ρ where t. 975 is the 97.5 th ercetile of the t distributio with ( S + - 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.

27 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

28 0 } ( { 4 } { exist because roots two for ρ : Solve 0 } ( { ( } { > S S S S S S S S r s t X X X X r s t X X X X r s t X X ρ ρ ρ

29 RESULS 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. S S r x s t r x s t g r g x s t r g } ( { ( + ± + ± Exact Aroximatio

30 EXAC RESUL Stadard est otal Mea Variace { r ± ( g t g s g t (df x s {.8 ±.09 (0.95,.48 t.79 s. 975 ( g + } x S (6(.36 + (6(.65 (.79(.3464 { } ( r (.8 7 }

31 APPROXIMAE RESUL Stadard est otal Mea Variace t s g r (df ± t t x s s x.79 (6(.36 + (6( ± ( (0.9,.44 + S 7 + r ( } vs. (.95,.48

32 RAIO OF PROPORIONS ρ r C π π Var( C ρ π ( π + ρ π ( π

33 APPROACH # i last ste variace estimated use we : meas of o aroach i ratio (Similar t for ρ α00% C.I. form ( roots wo ( ( ( ( ( ( ( ( ( / ^ ^ + + α ρ π π ρ π π ρ z C Var C Var C Var C

34 APPROACH # 006 rials Cotemorary Cliical (Liu, for ρ α00% C.I. form ( roots wo ( ( ( ( ( / + α ρ π π ρ π π ρ z C Var C Var C

35 HE CHOICES 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.

36 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 direct bioassays as well as quatitative bioassays e.g. ratio of regressio coefficiets.

37 ODDS 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 heorembased 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.