CHAPT II : Prob-stats, estimation

Transcription

1 CHAPT II : Prob-stats, estaton Randoness, probablty Probablty densty functons and cuulatve densty functons. Jont, argnal and condtonal dstrbutons. The Bayes forula. Saplng and statstcs Descrptve and nferental statstcs. Varablty: precson vs. dsperson, uncertanty vs. nforaton. Estators, optal estaton Desgnng an estator. Mau a posteror, au lkelhood. Propertes of estators: consstency, accuracy, effcency, suffcency. Sulaton analyss The fundaental atr, replcatons n assayng. Analyss of the error varance odel. Absolute and relatve norally dstrbuted easureent errors. Bo-Co transforaton. MLE crteron n PKs Condtonal dstrbuton and lkelhood functon. The clearance estaton eaple. General for of MLE and MAP crtera. Model valdaton Analyss of resduals, posteror check of workng hypotheses. Akake Inforaton Crteron. Etended least-squares. Mult-output systes. MLE crteron for Posson easureent error.

2 Functonal schee - Chapt II Measureent nose nose PK PK process Observaton Adnstraton protocol Pror Pror nforaton Equvalence crteron PK PK odel Predcton Nonlnear prograng

3 Randoness Rando eperent events arsng out of an eperent cannot be predcted wth certanty ; rando eperent repeated several tes leads to dfferent events. Intal condtons Rando Eperent Rando outputs Rando event, true or false Logcal assessent on rando outputs. Certan events occur ore often than others descrbe such phenoena by attachng precse easures or ndces called probabltes. 3

4 Dscrete events Events and operatons ( Venn's dagras ) A A : Copleent of Countable set of dsjont events A A A A j A k A A j A A j A Unon or : Intersecton and : A j A A j A A j A A j 4

5 Defne probablty The probablty achne A j j = : n Let be the set of dsjont events. The probablty achne,, converts Venn's dagra surfaces n nubers n a noralzed scale :. [, ] Pr A A j A n Pr Pr{ A } s propertes j Margnal probablty Pr Pr = 5 { A A } = Pr{ A } + Pr{ A } j { B } = Pr{ B } j Pr A j j Pr{ A } Pr Pr { A } n { A } j

6 Jont rando events E : A B se : grl, boy. studes :, and for the scences, econocs and pharacy. What s the probablty to have a grl dong econocs? What s the probablty that ths grl do econocs? Jont and condtonal probabltes Questons a a b b b 3 { A = a, B = b }? Pr = { B = b / A = a }? Pr = { A B } Pr{ B / A } Before the rando eperent,? A After the event has been observed,? Pr 6

7 Knds of probabltes Condtonal probabltes Defnton n the causal way : predct event after has been observed n the opposte way : Margnal and jont probabltes Pr{ A } Pr{ B } Pr{ A B } Pr{ A, B } Pr A/ B Pr B / A sooth event after has been observed. and : argnal probabltes, or : jont probablty, { } and : condtonal probabltes. The Bayes forula B A { } Pr Pr Pr { A } = Pr{ A B } Pr{ A } B / A A/ { B } = Pr{ A B } Pr{ B } B { B / A } Pr = Pr { B } { A } Pr { A / B } 7

8 Matheatc tools : PDF, CDF Contnuous atheatc functons Probablty Densty Functon ( PDF ) to copute an nterval probablty Pr { } X < = f d Cuulatve Densty Functon ( CDF ) to copute an nterval probablty { < } = Pr X F F f f () F F () Note : df d = f and area under the PDF curve epresses probablty. 8

9 Operatng wth PDFs Propertes of f PDF : F CDF : F f f d = nondecreas ng F( ) = F( + ) = Multvarate PDF The entre nforaton s contaned n the jont argnals condtonals The Bayes forula y f (, y) (, ) = f y dy f f ( y) f ( y) f ( y) f (, y) and. One has successvely (, ) = f y d f y, = and f (, y) f = f ( y ) [ ] f y = f f y f y 9

10 Do statstcs Statstcs conssts n perforng eperents to obtan data, analyzng, suarzng, nterpretng data, desgn new eperents. Statstcal database { T, j : n } j = k p k p j j jk jp n n nk np Rectangular array crossng slar ndvduals and recorded varables Key factor s varablty aong slar ndvduals. Populaton of ndvduals : fnte ( countng ) or nfnte ( probng / saplng technques ). Contnuous and dscrete varables. Array reducton : classfcaton for ndvduals, factorzaton for varables.

11 Statstcal analyses Estaton Populaton Rando draw adnstraton Drug saplng dosage Indvdual odelng Sulaton Modelng PDF Structural and nuerc knowledge Varablty ˆ j Descrptve statstcs graphcs :: hstogra, nuerc :: average, Inferental statstcs PDF PDF ath ath functons epectaton,

12 Saple vs. odel characterstcs Rando varable X ( nae ), ( realzaton ) Saple Rando drawn fro a populaton j j =: n Model Probablty densty functon f Central tendency Average Epectaton Varablty Varance ( arth ) Varance s var ave n ( ) = j j n j= n ( ) = ( ) j j n j= E[ ] X = f d [ ] ( [ ] ) V X = E X f d

13 Modelng rando eperents Analyss Saple Model Identfcaton Nuerc Central tendency Varablty s E [ X ] V [ X ] Paraetrc For γ,γ Γ [ X ], Γ [ X ] 3 hstogra PDF, CDF.4 Structural Graphc

14 Skewness and kurtoss.5 Γ =.57.5 Skewness Ref Γ ( N ) = f () Γ =.57.5 Kurtoss Ref Γ ( N ) = 3 f () 4 3 Γ 3 Γ 4 = 5. =

15 The Gaussan densty Dstrbuton odel f λ = ep Paraeters πσ λ σ Dstrbuton characterstcs E [ X] λ V[ X] Saple characterstcs s = = σ + + σ f ( ) Paraeter estates X [ X] [ X] λ = E = σ = V = s ( λ, ) ~ N σ 5

16 Observed data n n the saple [ X ] j =: 9 - ng L - L h { } j = 5.97 s = = nbr. of cases

17 Dstrbuton «estaton» Saple = 5.97 s = Dstrbuton odel X Selected odel λ = σ = X ( λ, ) ~ N σ s (, ) ~ N s.73 Probablty densty

18 Precson vs. dsperson.3 dsperson Intra- vs. nter-nd. varablty 7 ndvduals overlap Dsperson ( standard devaton, std ) s condtoned by precson ( standard error, se ) k [ h - ].. precson V [L] 8

19 Uncertanty vs. nforaton The confdence nterval quantfes the uncertanty for a rando varable. It ay be assocated wth precson or dsperson..35 The narrower the confdence nterval, the hgher the nforaton.3.5 nforaton low hgh precson precse accurate dsperson wde narrow f () CI CI = 7.44 =

20 Inforatve use of condtonal PDFs As causalty s not needed for Bayesan nference, we propose to sharng out rando varables n two sub-sets : drectly avalable rando varables,, and estated rando varables wth large varablty,. Y X Wth ths organzaton, t s ore nforatve to use condtonal PDFs for estaton X = CL E. :, or X = CL E. :, Y = BSA Y = drug concentraton CL f ( CL/ BSA) f ( CL) More precse estaton for fro than fro [ f ( y) ] > f nfo nfo [ ]

21 Condtonal densty ( CL ) The densty of CL s nfluenced by PC & AG covarates f ( CL) PC=.5 kg & AG=34 weeks 8% PC=.5 kg 35% Margnal 55% CL [ L/h]

22 Optal estaton Estaton s the operaton assgnng nuercal values to unknown paraeters, based on nose-corrupted observatons. Organzaton of the varables The observed drug concentratons over te, ( The rando paraeters to be estated, ( f, y f f y Consder the jont PDF and then : y p densonal vector). densonal vector). the argnal s called pror PDF [ the argnal s not of nterest ]. the condtonal s called posteror PDF : f y arg a{ f y } L = the condtonal leads to the lkelhood functon : arg a{ f y } B = f y MAP MLE

23 Desgnng an estator Rule : Cross the speces of avalable PDF wth the rando or y characterstcs..4 Lkelhood Posteror Pror f () or f (/y) or f (y/).3.. Mode Epectaton Medan Mode Epectaton Medan Rando varable or y 3

24 Mau a posteror (MAP) Desgn : A reasonable estate of would be the ode B of the posteror densty for the gven observaton y y = arg a{ } = f y ˆ B.35.3 E :.5 f f y = y B = y = y B = 3 6 f (/y)..5 y y The role of the dsperson of at the sae saplng tes for several ndvduals : Dsperson of y y 3

25 Mau lkelhood (MLE) Desgn : After the observaton y = y has been obtaned, a reasonable estate of would be L, the value whch gves to the partcular observaton the hghest probablty of occurrence arg a{ } = f y ˆ L.35.3 y E : f f y = 5. = L y = 6. = L 3 f (y/).5..5 = = 3. 3 = 8 The role of the precson n y assay : screen slar ndvduals y

26 MLE crteron for sngle - output Intal for Hypotheses H : Addtve error H : Noral error H : The odel s an eact descrpton of the real process. Error ˆ L = M (, ) e = y y t E ~ N, σ arg a { ln f ( y ) } State (, ) y = y t + e M Y ~ N ym t,, σ σ σ j σ σ j H3 : Independence e (, ) j = j f y, y = f ( y ) f ( y ) j j f e e f e f e e j y M( t ) y y j y ( t, ) M j 6

27 General for of the MLE crteron For avalable observed data and under the H3 hypothess the estator becoes ˆ L where J = arg a = arg a ln = arg n{ J } { ln f ( y / ) } = arg a ln f ( y, = : / ) = s the crteron functon to be nzed. The st ter s known as the etended SE ter. f { } ( y / ) = arg a ln f ( y / ) ( ln π σ ) J The nd ter s called the weghted SE ter. It s the only one nvolvng observed data and t s weghted by the uncertanty of eperent. y y = (, ) y ym t = + = = σ f y ~ N 7

28 The fundaental atr n n a real case j The nde : denotes the row and the n of saplng pont ( a ) ; : denotes the colun and the n of replcaton n assayng ( a ). n y... y... y j n y... y... y j n y... y... y j n ave ave ave ( y ) = y var( y ) j... j ( y ) = y var( y ) j y = y var y = s j j j = = s s 8

29 Varance odel of eperental error Two basc stuatons The absolute or hooscedastc error s α= y The relatve or heteroscedastc error s α = y y y t t Model the varance as s = K α y 9

30 Quantfcaton assay curve LOQ s [ng/l] Absolute error Relatve error y [ng/l] 3

31 Crteron and error varance odel After ntroducng the error varance odel : J s nu along the drecton when : J K = Then : We recognze : or K K SE the etended SE ter as nd ter, the weghted SE ter as 3rd ter. = ( π) J σ s = K y α wth the second ter s needed when s unknown. Otherwse : SE = [ + ] + α ln ln y + ln = SE = = y y y M α ( t, ) α J J = SE 3

32 Least Squares vs. MLE Fro an eprcal pont of vew : The eperental error s accounted fro replcated assays n real saples : The varance of the eperental error nvolvng observatons substtutes the odel nvolvng predctons. The crteron to be nzed becoes : s = K y σ ( π) J α α K ym ( t ) =, y α SE = [ ln + ] + ln ym( t, ) + ln = ( t ) M, y wth SE = = (, ) ( t, ) y ym t α ym 3

33 Relevant ponts n n MLE In a practcal way, reove H : K L ˆ L = argn{ J } Jontly, : s the posteror estaton of n the error varance odel ; evaluates the resdual error (RMSE) (eperent and structural errors, and respectvely) ; K e K s quantfes the goodness of ft. K K L = ( L) SE Copare K L K e wth avalable fro the eperent error analyss (reproducblty). K = K + K E : Let L e s K L = % Ke % Ks %

34 General for of the MAP crteron Intal for : ˆ B = arg a + { ln f ln f ( y ) } Under the hypotheses H, H, H and H3 for the MLE, the estator becoes : Where : { ln[ f ] + ln[ L( y, = : / ) ]} J y ym( t, ) = ln f + ln π σ + ˆ B = arg a = J J [ ] s the crteron functon to be nzed. The st ter epresses the pror nforaton. = = The nd and 3rd ters epress the lkelhood functon. K Use, the pror knowledge about, n the error varance ter. f J arg n{ } K [ ] [ K ] α SE = ln + ln π + ln + ln ym( t, ) + K σ = 34