Statistical algorithms in Review Manager 5

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1 Statstcal algorthms n Reve Manager 5 Jonathan J Deeks and Julan PT Hggns on behalf of the Statstcal Methods Group of The Cochrane Collaboraton August 00 Data structure Consder a meta-analyss of k studes When the studes have a dchotomous (bnary) outcome the results of each study can be presented n a table (Table ) gvng the numbers of partcpant ho do or do not experence the event n each of the to groups (here called expermental (or ) and control (or )) Table : Bnary data Study Event No event Total Expermental a b n Control c d n If the outcome s a contnuous measure the number of partcpants n each of the to groups ther mean response and the standard devaton of ther responses are requred to perform meta-analyss (Table ) Table : Contnuous data Group Mean Standard Study sze response devaton Expermental n m sd Control n m sd If the outcome s analysed by comparng observed th expected values (for example usng the Peto method or a log-rank approach for tme-to-event data) then O E statstcs and ther varances are requred to perform the meta-analyss Group szes may also be entered by the reve author but are not nvolved n the analyss Study Table 3: O mnus E and varance Varance of Group sze O mnus E (O mnus E) (expermental) V Z n Group sze (control) n For other outcomes a generc approach can be used the user drectly specfyng the values of the nterventon effect estmate and ts standard error for each study (the standard error may be calculable from a confdence nterval) Rato measures of effect effects (eg odds rato rsk rato hazard rato rato of means) ll normally be expressed on a log-scale dfference measures of

2 effect (eg rsk dfference dfferences n means) ll normally be expressed on ther natural scale Group szes can optonally be entered by the reve author but are not nvolved n the analyss Study Estmate of effect Table 4: Generc data Standard error of Group sze estmate (expermental) SE θ { } θ n Group sze (control) n Formulae for ndvdual studes Indvdual study estmates: dchotomous outcomes Peto odds rato For study denote the cell counts as n Table th n a + b n c + d and let n n For the Peto method the ndvdual odds ratos are gven by N + Z ORPeto exp V The logarthm of the odds rato has standard error SE{ ln ( ORPeto )} V here Z s the O E statstc: E[ ] Z a a th n ( a + c) E[ a ] N (the expected number of events n the expermental nterventon group) and nn ( a + c)( b + d) V N N (the hypergeometrc varance of a ) ( ) Odds rato For methods other than the Peto method the odds rato for each study s gven by ad OR bc the standard error of the log odds rato beng SE{ ln ( OR )} a b c d Rsk rato The rsk rato for each study s gven by RR a / n c / n

3 the standard error of the log rsk rato beng { ( RR )} SE ln + a c n n Rsk dfference The rsk dfference for each study s gven by th standard error a c RD n n ab cd n + n { RD} 3 3 SE Empty cells Where zeros cause problems th computaton of effects or standard errors 05 s added to all cells ( a b c d ) for that study except hen a c 0 or b d 0 hen the relatve effect measures OR and RR are undefned Indvdual study estmates: contnuous outcomes Denote the number of partcpants mean and standard devaton as n Table and let N n + n and s ( ) + ( ) n sd n sd N be the pooled standard devaton across the to groups Dfference n means (mean dfference) The dfference n means (referred to as mean dfference) s gven by MD m m th standard error SE{ MD } n n sd + sd Standardzed dfference n means (standardzed mean dfference) There are several popular formulatons of the standardzed mean dfference The one mplemented n RevMan s Hedges adjusted g hch s very smlar to Cohen's d but ncludes an adjustment for small sample bas m m 3 SMD s 4 N 9 th standard error SE { SMD } N SMD + nn 394 ( N ) 3

4 Indvdual study estmates: O E and varance For study the effect estmate s gven by th standard error Z θ V { } SE θ V The effect estmate s ether of a log odds rato or a log hazard rato dependng on ho the observed and expected values ere derved Indvdual study estmates: Generc method As the user drectly enters the nterventon effect estmates and ther standard errors no further processng s needed All types of nterventon effects are elgble for ths method but t mght be most useful hen nterventon effects have been calculated n a ay hch makes specal consderaton of desgn (eg cluster randomzed and cross-over trals) are adjusted for other effects (adjusted effects from non-randomzed studes) or are not covered by exstng methods (eg ratos of means relatve event rates) Meta-analyss methods All summatons are over from to the number of studes unless otherse specfed Mantel-Haenszel methods for combnng results across studes Odds rato The Mantel-Haenszel summary log odds rato s gven by ln ( ) ln OR OR () and the Mantel-Haenszel summary odds rato by OR here each study s odds rato s gven eght bc N The summary log odds rato has standard error gven by E F + G H SE{ ln ( OR )} + + R RS S () here ad bc R ; S ; N N OR 4

5 ( + ) a d ad E ; N ( + ) b c ad G ; N F ( + ) a d bc N ( b + ) c bc H N ; Rsk rato The Mantel-Haenszel summary log rsk rato s gven by ln ( ) ln RR RR (3) and the Mantel-Haenszel summary rsk rato by RR here each study s rsk rato s gven eght c( a + b) N The summary log rsk rato has standard error gven by P SE{ ln ( RR )} (4) RS here nn ( a + c) acn an cn P ; R ; S N N N RR Rsk dfference The Mantel-Haenszel summary rsk dfference s gven by RD RD (5) here each study s rsk dfference s gven eght nn N The summary rsk dfference has standard error gven by J SE{ RD } K (6) here 3 3 abn + cdn nn J ; K nn N N Test for heterogenety The heterogenety test statstc s gven by Q ( ) θ θ 5

6 here θ represents the log odds rato log rsk rato or rsk dfference and the are the eghts calculated as SE { } θ rather than the eghts used for the Mantel-Haenszel meta-analyses Under the null hypothess that there are no dfferences n nterventon effect among studes ths follos a ch-squared dstrbuton th k degrees of freedom (here k s the number of studes contrbutng to the meta-analyss) The statstc I s calculated as ( k ) Q I max 00% 0 Q Ths measures the extent of nconsstency among the studes results and s nterpreted as approxmately the proporton of total varaton n study estmates that s due to heterogenety rather than samplng error Inverse-varance methods for combnng results across studes Inverse-varance methods are used to pool log odds ratos log rsk ratos and rsk dfferences as one of the analyss optons for bnary data to pool all mean dfferences and standardzed mean dfferences for contnuous data and also for combnng nterventon effect estmates n the generc method In the general formula the nterventon effect estmate s denoted by θ hch s the study s log odds rato log rsk rato rsk dfference mean dfference or standardzed mean dfference or the estmate of nterventon effect n the generc method The ndvdual effect szes are eghted accordng to the recprocal of ther varance (calculated as the square of the standard error gven n the ndvdual study secton above) gvng SE θ These are combned to gve a summary estmate th ( { }) θ IV θ (7) { IV } SE θ (8) The heterogenety statstc s gven by a smlar formula as for the Mantel-Haenszel method: Q ( ) IV θ θiv Under the null hypothess that there are no dfferences n nterventon effect among studes ths follos a ch-squared dstrbuton th k degrees of freedom (here k s the number of studes contrbutng to the meta-analyss) I s calculated as QIV ( k ) I max 00% 0 QIV Peto's method for combnng results across studes The Peto summary log odds rato s gven by 6

7 ln ( ) ln ( ) V ORPeto ORPeto V (9) and the summary odds rato by Vln ( ORPeto ) ORPeto exp V here the odds rato OR Peto s calculated usng the approxmate method descrbed n the ndvdual study secton and The log odds rato has standard error V are the hypergeometrc varances { ( ORPeto )} SE ln (0) V The heterogenety statstc s gven by Q {( ln ) ( ) } Peto V ORPeto ln ORPeto Under the null hypothess that there are no dfferences n nterventon effect among studes ths follos a ch-squared dstrbuton th k degrees of freedom (here k s the number of studes contrbutng to the meta-analyss) I s calculated as QPeto ( k ) I max 00% 0 QPeto O E and varance method for combnng studes Ths s an mplementaton of the Peto method hch allos ts applcaton to tme-to-event data as ell as bnary data The summary effect estmate s gven by V θ θ V () here the estmate θ from study s calculated from Z and V as for ndvdual studes The summary effect s ether a log odds rato or a log hazard rato (the user should specfy hch) The effect estmate (on a non-log scale) s gven by effect estmate exp V θ V and s ether an odds rato or a hazard rato The effect estmate (on the log scale) has standard error SE θ () { } The heterogenety statstc s gven by Q ( ) Peto V θ θ Under the null hypothess that there are no dfferences n nterventon effect among studes ths follos a ch-squared dstrbuton th k degrees of freedom (here k s the number of studes contrbutng to the meta-analyss) I s calculated as V 7

8 I ( k ) Q max 00% 0 QPeto Peto DerSmonan and Lard random-effects models Under the random-effects model the assumpton of a common nterventon effect s relaxed and the effect szes are assumed to have a dstrbuton ( N ) θ θ τ The estmate of τ s gven by Q ( k ) τ max 0 ( ) here the are the nverse-varance eghts calculated as SE { θ } for log odds rato log rsk rato rsk dfference mean dfference standardzed mean dfference or for the nterventon effect n the generc method as approprate For contnuous data and for the generc method Q s Q For bnary data ether Q or IV IV Q may be taken Both are mplemented n RevMan 5 (and ths s the only dfference beteen randomeffects methods under Mantel-Haenszel and nverse-varance optons) Agan for odds ratos rsk ratos and other rato effects the effect sze s taken on the natural logarthmc scale Each study s effect sze s gven eght { } SE θ +τ The summary effect sze s gven by θ θ DL (3) and SE{ θ DL } (4) Note that n the case here the heterogenety statstc Q s less than or equal to ts degrees of freedom ( k ) the estmate of the beteen study varaton τ s zero and the eghts concde th those gven by the nverse-varance method Confdence ntervals The 00( α)% confdence nterval for θ s gven by θ SE θ Φ α θ+ SE θ Φ α { } ( ) to { } ( ) here θ s the log odds rato log rsk rato rsk dfference mean dfference standardzed mean dfference or generc nterventon effect estmate and Φ s the standard normal devate For log odds ratos log rsk ratos and generc nterventon effects entered on the log scale (and dentfed 8

9 as such by the reve author) the pont estmate and confdence nterval lmts are exponentated for presentaton Test statstcs Test for presence of an overall nterventon effect In all cases the test statstc s gven by θ Z SE θ here the odds rato rsk rato and other rato measures are agan consdered on the log scale Under the null hypothess that there s no overall effect of nterventon effect ths follos a standard normal dstrbuton ( ) Test for comparson of subgroups The test s vald for all methods It s based on the noton of performng a test for heterogenety across subgroups rather than across studes Let θ j be the summary effect sze for subgroup j th standard error SE { θ j } The summary effect sze may be based on ether a fxed-effect or a random-effects meta-analyss For fxed-effect meta-analyses these numbers correspond to above equatons () and (); (3) and (4); (5) and (6); (7) and (8); (9) and (0); or () and () each appled thn each subgroup For random-effects meta-analyses these numbers correspond to equatons (3) and (4) each appled thn each subgroup Note that for rato measures all computatons here are performed on the log scale Frst e compute a eght for each subgroup: j SE { θ } j then e perform a (fxed-effect) meta-analyss of the summary effect szes across subgroups: jθ j θ tot j The test statstc for dfferences across subgroups s gven by Q θ θ ( ) nt j j tot Under the null hypothess that there are no dfferences n nterventon effect across subgroups ths follos a ch-squared dstrbuton th S degrees of freedom (here S s the number of subgroups th summary effect szes) I for dfferences across subgroups s calculated as Qnt ( S ) I max 00% 0 Qnt Ths measures the extent of nconsstency across the subgroups results and s nterpreted as approxmately the proporton of total varaton n subgroup estmates that s due to genune varaton across subgroups rather than samplng error 9

10 Note An alternatve formulaton for fxed-effect meta-analyses (nverse varance and Peto methods only) s as follos The Q statstc defned by ether Q or Q s calculated separately for each of the S subgroups and for the totalty of studes yeldng statstcs Q QS and Qtot The test statstc s gven by S IV Q Q Q nt tot j j Ths s dentcal to the test statstc gven above n these specfc stuatons Peto 0

11 Bblography Borensten M Hedges LV Hggns JPT Rothsten HR Introducton to Meta-analyss John Wley & Sons 009 Breslo NE Day NE Combnaton of results from a seres of x tables; control of confoundng In: Statstcal Methods n Cancer Research Volume : The analyss of case-control data IARC Scentfc Publcatons No3 Lyon: Internatonal Agency for Health Research on Cancer 980 Deeks JJ Altman DG Bradburn MJ Statstcal methods for examnng heterogenety and combnng results from several studes n a meta-analyss In: Egger M Davey Smth G Altman DG Systematc Revees and Healthcare: meta-analyss n context BMJ Publcatons (n press) DerSmonan R Lard N Meta-analyss n clncal trals Controlled Clncal Trals 986; 7: Greenland S Robns J Estmaton of a common effect parameter from sparse follo-up data Bometrcs 985;4: Greenland S Salvan A Bas n the one-step method for poolng study results Statstcs n Medcne 990; 9:47-5 Hedges LV Olkn I Statstcal Methods for Meta-analyss San Dego: Academc Press 985 Chapter 5 Hggns JPT Thompson SG Deeks JJ Altman DG Measurng nconsstency n meta-analyss BMJ 003; 37: Mantel N Haenszel W Statstcal aspects of the analyss of data from retrospectve studes of dsease Journal of the Natonal Cancer Insttute 959;: Robns J Greenland S Breslo NE A general estmator for the varance of the Mantel-Haenszel odds rato Amercan Journal of Epdemolgy 986; 4:79-73 Rosenthal R Parametrc measures of effect sze In: Cooper H Hedges LV (eds) The Handbook of Research Synthess Ne York: Russell Sage Foundaton 994 Snclar JC Bracken MB Effectve Care of the Neborn nfantoxford: Oxford Unversty Press 99Chapter Yusuf S Peto R Les J Collns R Sleght P Beta blockade durng and after myocardal nfarcton: an overve of the randomzed trals Progress n Cardovascular Dseases 985;7:335-37

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