Contents. Introduction to Bayesian methods. Introduction to Bayesian methods Meta Analysis. Models and Methods. Mantel-Haenzel methods for 2x2 tables

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1 Cotet Meta-aalye: Modeller og metoder Itroducto to Bayea method Meta Aaly. Model ad Method. Matel-Haezel method for x table Sta Lydere KLMED8006 Avedt med tatt St303 med tatt 8 Aprl Frequett v Bayea tattc. Itroducto to Bayea method Partly baed o Evertt, B. S.: Moder Medcal Stattc. Arold Publher, 003. Secto 8. The probablty dtrbuto P(D θ of our data D deped o ome parameter( θ. Example: X,..., ~ (, X N μσ Data D Paramter θ The frequett regard θ a a uow cotat The Bayea regard θ a a (uoberved radom varable from a probablty dtrbuto, pror dtrbuto, P(θ. 5 6 Baye rule (oer, eq 3.0 B B A B Followg Baye theorem, the probablty dtrbuto of the parameter gve the data, poteror dtrbuto, P( θ P( D θ P( θ D P( θ P( D θ dθ P( B A j P( A BP( B P( A B P( B The deomator a cotat (doe ot deped o θ, o P( θ D P( θ P( D θ poteror dtrbuto pror dtrbuto lelhood

2 7 8 Bayea etmate: ˆ θb E( θ D (-α Bayea cofdece terval (credblty terval: The α/ quatle (θ l, θ h of the poteror dtrbuto Iterpretato (!: P( θ < θ < θ α l h 9 0 Example: Normal dtrbuto. X,..., ~ (, X N μσ Pror dtrbuto: μ ~ N ( ν, τ Poteror dtrbuto: μ ( X,..., X ~ N( Bν + ( B X,( B σ / σ / where B σ / + τ Baye etmate (poteror mea ˆ μb Bν + ( B X A weghted average betwee the pror mea ν ad X If the pror varace τ large, we have a vague or uformatve pror, ad ˆ μ B X Example: Bomal dtrbuto: umber of evet tral: X ~ b(, θ Pror dtrbuto: θ ~ Beta( a, b ( a b a b f ( θ Γ + ( for 0< < ( a ( b θ θ Γ Γ θ a>0, b>0 a E( θ a + b f(theta The Beta dtrbuto , 0,4 0,6 0,8 Theta ab ab/ a, b4 ab6,5 a8, b

3 3 4 Poteror dtrbuto: θ X ~ Beta( a+ x, b+ ( x Poteror mea: a+ x E( θ x a + b + a peudo umber of evet a+b peudo umber of tral 5 6 Meta Aaly. Model ad Method. Maly baed o: Normad, Sharo-Le T: Tutoral Botattc. Meta-Aaly: Formulatg, Evaluatg, Combg, ad eportg. Stattc Medce, 8, ( Def. Meta-Aaly may be broadly defed a the quattatve revew ad ythe of the reult of related but depedet tude. Meta-aalye er gjeomgag og ammefatg av relaterte, uavhegge tuder. Example: adomzed cotrolled tral of ldocae v placebo for patet wth myocardal farcto LOS (legth of tay hoptal for troe patet, pecalt patet troe care v opecalt troe care 3

9 0 3 4 Fxed effect v radom effect model Fxed effect: The tude have detcal charactertc ad tudy effect.

4 Fxed effect v radom effect model Fxed effect: The tude have detcal charactertc ad tudy effect. adom effect: The tude may have dfferet effect ad dfferet charactertc Debate a to the choce of approprate model Alway reaoable to aume ome betwee-tudy varato ad few reao to beleve t zero. Combg tude: Dfferet clae of outcome (tudy ummary Dcrete outcome uch a dfferece proporto Cotuou outcome uch a mea Tet tattc (ot exhautve lt 4

5 5 6 Bary outcome: dfferece, relatve r, odd rato Etmator ad cofdece terval a decrbed oer (005, Secto 3.3 outcome Etmator Stadard devato Group Ye No T a b a+b T C c d c+d C dfferece elatve r ( rato d pˆ pˆ T C r pˆ / pˆ T C Log ( r d pt ( pt pc ( pc + T pt p + p p C T T C C C Study umber : pˆ T a c, pˆc T C Odd rato pˆ /( ˆ T pt ω pˆ /( pˆ C C ω a b c d Log ( 7 8 Cotuou outcome: Dfferece mea from tudy umber : Y xt xc wth tadard devato calculated a p + T C where ( + ( p + T T C C T C Poble effect ze: The tadardzed mea dfferece T μ μ δ σ whe C Y N j T T j ~ ( μ, σ ;,,..., T Y N j C C j ~ ( μ, σ ;,,..., C (Normad, 999, tate the above wthout ubcrpt 9 30 Etmator for δ (deoted Hedge g: T C Y Y h, ˆ ˆ δ Var( h + + T C ( T + C p-value a outcome May method ext for combg p, p,, p. (Cooper ad Hedge (994 lt 6 method Much ued (Darlgto & Haye, 000: The Stouffer ad the Fher method Uder H 0, uder qute geeral codto, p (approxmately uformly dtrbuted o (0, where ˆ δ the ample etmate of δ. 5

6 3 3 The Stouffer (949 method: Compute the z-value correpodg to the p-value: z Φ ( p ~ N(0, uder H 0 z+ z z Combed z-value: z ~ N(0, uder H 0 Combed p-value: pφ ( z Example (Darlgto & Haye, 000: p-value.59,.33,.,.09 z-value -.999, , -.39 combed z-.330, combed p0.00 The Fher (93 method: A low value of pp... p or equvaletly, log( pp... p log( p tae a evdece agat H 0 Uder H 0,, log( p~ χ Example: p-value.59,.33,.,.09 χ 6.88, df *48 Fher p Publcato ba: Fle-Drawer (Fal-Safe umber eearcher may have upublhed, ot gfcat reult ther fle-drawer How may uow tude (N FS,α wth average z0 eed to be added to the ow N to mae the outcome of Stouffer tet ot gfcat at level α? (oethal, 979 N Sum of N FS,α term z N N N N z + 0 z / or α FS, α z FS, α + N α / Example (cot d N4, combed p0.00, combed zz , z N FS, dv N FS, Source of varato meta-aaly but: Combg p-value gve lttle ght effect ze ad t drecto. Samplg error may vary betwee tude. Varyg ample ze. Study-level charactertc may vary (for example forproft v ot-for-proft hotptal Iter-tudy varato (radom effect model 6

7 37 38 Y ummary tattc from tudy o (for example treatmet effect Approxmately ormally dtrbuted Fxed effect model: Y N ~ ( θ, for,,..., where θ the mea treatmet effect (ame all tude Fg. 3. Fxed effect model adom-effect model Smlar to: - mult level model - herarchcal model - mxed effect model θ ~ N( θτ, Y N θ, ~ ( θ, whereθ the tudy pecfc mea draw from a uperpopulato wth hyperparameter θ ad τ, Y N θ ~ ( θ, θ θ τ θ τ, ~ N (, Fg. 4. adom effect model. 4 4 Ucodtoal dtrbuto of Y : Y N ~ ( θ, + τ where ad τ are wth-tudy ad betwee tudy varato. The dtrbuto of the tudy-pecfc effect θ, codtoal o the oberved data ad the hyperparameter, θ θ τ θ + ( Y,..., Y,, ~ N( B ( B Y, ( B where B the hrage factor for the th tudy + τ 7

8 43 44 Fxed-effect model: Maxmum lelhood etmator for commo mea f ow: WY ˆ θ MLE wth W W, ˆ θmle ~ N θ, ( W Tet for homogeety of tudy mea: H 0 : θ θ... θ H : At leat two are dfferet Uder H 0, for large ample ze, ˆ QW W( Y θmle ~ χ adom-effect model: Maxmum lelhood etmator for commo mea f τ wa ow: W τ Y ˆ( MLE wth W ( τ θτ ( W ( τ + τ EML (etrcted Maxmum Lelhood Apply lear fucto K y uch that K y cota oe of the fxed effect. Etmate the radom effect (varace parameter by applyg ML to K y. For fxed effect, EML ML. Uually τ uow. Two commo etmato method are - EML (etrcted Maxmum Lelhood - Bayea Example: Whe X, X,..., X depedet N( μσ, ˆ σ (, EML X X ( ˆ σ ( ML X X EML or ML? (Dggle & al, 00, p 69. EML etmator hould be le baed (McCulloch ad Searle, 00, p A growg preferece for EML mxed model For balaced ad ormal data, EML oluto are mmal varace ubaed. EML for ubalaced data et yeld o clea algebrac reult EML etmator eem to be le etve to outler the data 8

9 49 50 EML: Etmate of θ ad τ may be foud a oluto to ˆ τ ˆ ( ˆ τ w ( ˆ ( τ Y θ w Etmator for θ (emprcal Baye: ˆ ˆ ˆ ( ˆ θ B θ + B Y where Bˆ ˆ + τ w( ˆ ˆ τ Y θ wth ( ˆ w τ w ( ˆ τ + ˆ τ 5 5 Full Bayea approach: θτ, are regarded a radom varable wth oe realzato For example θ ~ N(0, a ad τ ~ gamma( c, d The hyperparameter (a,c,d are pecfed a pror, ot etmated from data Matel-Haezel method for x table oer, Secto 3.5 9

10 55 56 K table: outcome Ye No T a b Group C c d Matel-Haezel tet for codtoal depedece. Uder H 0, table o, codtoal o the row ad colum um, a hypergeometrc dtrbuted (a Fher exact tet, wth mea ad varace ( a + b( a + c E ( a + b( c + d( a + c( b + d V ( The (Cochra-Matel-Haezel tet for codtoal depedece Null hypothee: The commo O: ( O E χmh ~ χ uder H 0 V where O O a ( a + b( a + c E a + b c + d a + c b + d V ( E V ( ( ( ( oer Eq 3.4 p 653 ue a cotuty correcto, a propoed by Matel & Haezel (959. ( O E 0.5 χmh V Th approxmate a exact codtoal tet, but ted to be coervatve (Agret The Matel-Haezel etmator for the commo odd rato: oer, eq 3.5 p 655 O ˆ MH ad bc Cofdece terval for commo O (oer eq 3.6 exp l O ˆ ˆ MH ± z α / Var(l OMH where Var(l O ˆ P ( PS + Q QS MH + + ( ( ( S ( S ad a + d b + c ad bc P, Q,, S 0

11 6 6 Tet for homogeety of odd rato: Strata: Woolf method, oer eq 3.7, Brelow-Day method (SPSS, StatXact Meta-aaly: oer eq 3.40 Example Doll ad Hll (950, oer exerce Me: moe omoe total lug cacer cotrol total Wome moe omoe total lug cacer cotrol total Me ad wome eparately: etmate 95% c.. p-value Me to 59.7E-6 Wome.47. to exact StatXact etmate 95% c.. p-value Me to.3e-6 Wome.47. to Logtc regreo a alteratve to Matel Haezel method. Strata (or tudy a categorcal covarate (or coded wth - dcator varable Approxmately ame reult a Matel- Haezel method. Tet for homogeety of O: Brelow & Day tattc 5., df, p0.0 MH etmate for commo O: % c..:.4 to eferece Cooper, H, Hedge, L. V. The Hadboo of eearch Sythe. uell Sage Foudato, 994. Darlgto,. B., Haye, A. F.: Combg Idepedet p Value: Exteo of the Stouffer ad the Bomal Model. Pychologcal Method, 000, Vol 5, No 4, Dggle, P. J., Heagerty, P., Lag, K-Y, Zeger, S. L.: Aaly of Logtudal Data d ed, Oxford Uverty pre, 00. Evertt, B. S.: Moder Medcal Stattc. Arold Publher, 003. eferece McCulloch, C. E., Searle, S. : Geeralzed, Lear ad Mxed Model. Wley, 00. Normad, Sharo-Le T: Tutoral Botattc. Meta- Aaly: Formulatg, Evaluatg, Combg, ad eportg. Stattc Medce, 8, (999 oethal,. The fle-drawer problem ad tolerace for ull reult. Pychologcal Bullet, 86, (979 oer, B. Fudametal of Botattc. 6th Ed. Thomo, 005.

8 The independence problem

8 The independence problem Noparam Stat 46/55 Jame Kwo 8 The depedece problem 8.. Example (Tua qualty) ## Hollader & Wolfe (973), p. 87f. ## Aemet of tua qualty. We compare the Huter L meaure of ## lghte to the average of coumer