Jon Deeks and Julian Higgins. on Behalf of the Statistical Methods Group of The Cochrane Collaboration. April 2005

Transcription

1 Standard statstcal algorthms n Cochrane revews Verson 5 Jon Deeks and Julan Hggns on Behalf of the Statstcal Methods Group of The Cochrane Collaboraton Aprl 005 Data structure Consder a meta-analyss of k studes When the studes have a bnary outcome the results of each study can be presented n a x table (Table ) gvng the numbers of subjects who do or do not experence the event n each of the two groups (here called nterventon and control) Table Bnary data Study Event o event Total Interventon a b Control c d n n If the outcome s a contnuous measure the number of subjects n each of the two 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 Interventon n m sd Control n m sd If the outcome s analysed by comparng observed wth 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 are also entered by the revew 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) (nterventon) V Z n Group sze (control) n For other outcomes a generc approach can be used the user drectly specfyng the values of the treatment effect and ts standard error for each tral (the standard error may be calculable from a confdence nterval) Rato treatment effects (eg odds rato rsk rato hazard rato rato of means) wll normally be expressed on a log-scale dfference treatment effects (eg rsk

2 dfference dfferences n means) wll normally be expressed on ther natural scale Group szes can optonally be entered by the revew author but are not nvolved n the analyss Study Estmate of effect Table 4 Generc data Standard error of Group sze estmate (nterventon) SE θ { } θ n Group sze (control) n Formulae for ndvdual studes Indvdual study responses: bnary outcomes Peto odds rato For study denote the cell counts as n Table and let n = a + For the Peto method the ndvdual odds ratos are gven by Z ORPeto = exp V The logarthm of the odds rato has standard error SE{ ln ( ORPeto )} = V where Z s the O E statstc: E[ ] Z = a a wth n ( a + c) E[ a ] = (the expected number of events n the nterventon group) and nn ( a + c)( b + d) V = (the hypergeometrc varance of a ) ( ) b n = c + d and = n + n 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 wth standard error a c = RD n n = ab cd n + n { RD } 3 3 SE Empty cells Where zeros cause problems wth computaton of effects or standard errors 05 s added to all cells ( a b c d ) for that study except when a c = 0 or b d = 0 when the relatve effect measures OR and RR are undefned = Indvdual study responses: contnuous outcomes Denote the number of subjects mean and standard devaton as n Table and let = n + n and s = ( ) + ( ) n sd n sd be the pooled standard devaton across the two groups = Dfference n means (mean dfference) The dfference n means (referred to as mean dfference) s gven by MD = m m wth standard error = sd + sd SE{ MD } n n Standardsed dfference n means (standardsed mean dfference) There are several popular formulatons of the standardsed mean dfference The one mplemented n Cochrane revews s Hedges adjusted g whch s very smlar to Cohen's d but ncludes an adjustment for small sample bas m m 3 SMD = s 4 9 wth standard error SE { SMD } SMD = + nn 394 ( ) 3

4 Indvdual study responses: O E and varance For study the effect estmate s gven by wth 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 how the observed and expected values were derved Indvdual study responses: Generc method As the user drectly enters the treatment effects and ther standard errors no further processng s needed All types of treatment effects are elgble for ths method but t mght be most useful when treatment effects have been calculated n a way whch makes specal consderaton of desgn (eg cluster randomsed and cross-over trals) are adjusted for other effects (adjusted effects from nonrandomsed studes) or are not covered by exstng methods (eg ratos of means relatve event rates) Poolng methods Mantel-Haenszel methods for combnng trals Odds rato The Mantel-Haenszel pooled odds rato s gven by w OR = w OR where each study s odds rato s gven weght The logarthm of where w OR has standard error gven by bc = { ( OR )} = + + SE ln R ad E F + G H R RS S = ; S ( + ) a d ad E = ; ( + ) b c ad G = ; bc = ; ( + ) a d bc F = ; ( b + ) c bc H = 4

5 Rsk rato The Mantel-Haenszel pooled rsk rato s gven by RR w = w RR where each study s rsk rato s gven weght The logarthm of where w RR has standard error gven by ( + ) c a b = { ( RR )} SE ln ( ) P = RS nn a+ c ac an cn P = ; R = ; S = Rsk dfference The Mantel-Haenszel pooled rsk dfference s gven by w RD RD = w where each study s rsk dfference s gven weght nn w = RD has standard error gven by where SE { RD } 3 3 nn J = K abn + cdn nn J = ; K = Test for heterogenety The heterogenety statstc s gven by ( ) Q = w θ θ where θ represents a log odds rato log rsk rato or rsk dfference and the w are the weghts calculated as SE { } θ rather than the weghts used for the Mantel-Haenszel meta-analyses Under the null hypothess that there are no dfferences n treatment effect among trals ths follows a ch-squared dstrbuton wth k degrees of freedom (where k s the number of studes contrbutng to the meta-analyss) The statstc I s calculated as 5

6 ( 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 trals 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 standardsed mean dfferences for contnuous data and also for combnng treatment effects n the generc method In the general formula the treatment effect s denoted by θ whch s the tral s log odds rato log rsk rato rsk dfference mean dfference or standardsed mean dfference or the estmate of treatment effect n the generc method The ndvdual effect szes are weghted accordng to the recprocal of ther varance (calculated as the square of the standard error gven n the ndvdual study secton above) gvng w = These are combned to gve a pooled estmate wth ( SE{ θ }) w θ IV w θ = { IV } SE θ = The heterogenety statstc s gven by a smlar formula as for the Mantel-Haenszel method usng the nverse varance form of the weghts w w ( ) QIV = w θ θiv Under the null hypothess that there are no dfferences n treatment effect among trals ths follows a ch-squared dstrbuton wth k degrees of freedom (where 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 trals Here the overall odds rato s gven by where the odds rato Peto tral secton and V ( ORPeto ) V Vln ORPeto = exp OR s calculated usng the approxmate method descrbed n the ndvdual are the hypergeometrc varances The logarthm of the odds rato has standard error { ( ORPeto )} SE ln = V 6

7 The heterogenety statstc s gven by Q ( ln ) ( ) Peto = V ORPeto ln ORPeto { } Under the null hypothess that there are no dfferences n treatment effect among trals ths follows a ch-squared dstrbuton wth k degrees of freedom (where 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 trals Ths s an mplementaton of the Peto method whch allows ts applcaton to tme-to-event data as well as bnary data The overall effect estmate s gven by Vθ θ= exp V where the estmate θ from study s calculated from Z and V as for ndvdual studes The overall effect s ether a log odds rato or a log hazard rato (the user should specfy whch) The logarthm of the effect estmate 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 treatment effect among trals ths follows a ch-squared dstrbuton wth k degrees of freedom (where k s the number of studes contrbutng to the meta-analyss) I s calculated as QPeto ( k ) I = max 00% 0 QPeto DerSmonan and Lard random-effects models Under the random-effects model the assumpton of a common treatment effect s relaxed and the effect szes are assumed to have a dstrbuton θ θ τ τ V ( ) The estmate of s gven by Q ( k ) τ = max 0 w ( w ) w where the are the nverse-varance weghts calculated as w w = SE { θ } for log odds rato log rsk rato rsk dfference mean dfference standardsed mean dfference or for the treatment effect n the generc method as approprate 7

8 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 between 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 weght w = SE { } θ +τ The pooled effect sze s gven by w θ θ DL = w and SE{ θ DL } = w ote that n the case where the heterogenety statstc Q s less than or equal to ts degrees of freedom ( k ) the estmate of the between tral varaton τ s zero and the weghts concde wth those gven by the nverse-varance method Confdence ntervals The 00( α)% confdence nterval for θ s gven by SE θ+ SE θ Φ α θ { θ} Φ( α ) to { } ( ) where θ s the log odds rato log rsk rato rsk dfference mean dfference standardsed mean dfference or generc treatment effect and Φ s the standard normal devate For log odds ratos log rsk ratos and generc treatment effects entered on the log scale (and dentfed as such by the revew author) the pont estmate and confdence nterval lmts are exponentated for presentaton Test statstcs Test for presence of an overall treatment effect In all cases the test statstc s gven by θ Z = SE θ where the odds rato rsk rato and other rato treatment effects are agan consdered on the log scale Under the null hypothess that there s no overall effect of treatment effect ths follows a standard normal dstrbuton ( ) Test for comparson of subgroups The test s vald for all methods except the Mantel-Haenszel methods for bnary data The Q statstc defned by ether Q or Q s calculated separately for each of the S subgroups and for IV Peto the totalty of studes yeldng statstcs Q Q and Q The test statstc s gven by Q Q Q nt tot S S j= j tot = 8

9 Under the null hypothess that there are no dfferences n treatment effect among subgroups ths follows a ch-squared dstrbuton wth S degrees of freedom (where S s the number of subgroups) The statstc I s calculated as ( S ) Qnt I = max 00% 0 Qnt Ths measures the extent of nconsstency among 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 Bblography Breslow E Day E 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 o3 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 Revewes and Healthcare: meta-analyss n context BMJ Publcatons (n press) DerSmonan R Lard Meta-analyss n clncal trals Controlled Clncal Trals 986; 7: Greenland S Robns J Estmaton of a common effect parameter from sparse follow-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 Haenszel W Statstcal aspects of the analyss of data from retrospectve studes of dsease Journal of the atonal Cancer Insttute 959;: Robns J Greenland S Breslow E 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 ew York: Russell Sage Foundaton 994 Snclar JC Bracken MB Effectve Care of the ewborn nfantoxford: Oxford Unversty Press 99Chapter Yusuf S Peto R Lews J Collns R Sleght P Beta blockade durng and after myocardal nfarcton: an overvew of the randomzed trals Progress n Cardovascular Dseases 985;7: