Meta-Analysis of Correlated Proportions

Transcription

1 NCSS Statstcal Softare Chapter 457 Meta-Analyss of Correlated Proportons Introducton Ths module performs a meta-analyss of a set of correlated, bnary-event studes. These studes usually come from a desgn n hch to dchotomous responses are made on each subject (or subject par). The results of each study can be summarzed as counts n a 2-by-2 table. For example, the bnary response s recorded after treatment A and agan after treatment B. The response s f the event of nterest occurs or 0 otherse. Ths analyss also apples to matched pars data n hch each case subject s matched th a smlar subject from a control group. The program provdes a complete set of numerc reports and plots to allo the nvestgaton and presentaton of the studes. The plots nclude the forest plot, radal plot, and L Abbe plot. Both fxed-, and random-, effects models are avalable for analyss. Meta-Analyss refers to methods for the systematc reve of a set of ndvdual studes th the am to combne ther results. Meta-analyss has become popular for a number of reasons:. The adopton of evdence-based medcne hch requres that all relable nformaton s consdered. 2. The desre to avod narratve reves hch are often msleadng. 3. The desre to nterpret the large number of studes that may have been conducted about a specfc treatment. 4. The desre to ncrease the statstcal poer of the results be combnng many small-sze studes. The goals of meta-analyss may be summarzed as follos. A meta-analyss sees to systematcally reve all pertnent evdence, provde quanttatve summares, ntegrate results across studes, and provde an overall nterpretaton of these studes. We have found many boos and artcles on meta-analyss. In ths chapter, e brefly summarze the nformaton n Sutton et al (2000) and Thompson (998). Refer to those sources for more detals about ho to conduct a metaanalyss. Treatment Effects Suppose you have obtaned the results for studes, labeled =,,. Each study conssts of to dchotomous measurements Y and Y 2 on each of n subjects (the subject may be a par of matched ndvduals). Measurement Y represents the treatment response and Y 2 represents the control response. The results of each study are summarzed by four counts: a the number of Y and Y. = 2 = b the number of Y and Y 0. = 2 = 457- NCSS, LLC. All Rghts Reserved.

2 NCSS Statstcal Softare Meta-Analyss of Correlated Proportons c the number of Y 0 and Y. = 2 = d the number of Y 0 and Y 0. = 2 = Occasonally, one of these counts ll be zero hch causes calculaton problems. To avod ths, the common procedure s to add a small value of 0.5 or 0.25 to all counts so that zero counts do not occur. Odds Rato When a pared desgn s used, Saha and Khurshd (995) ndcate that the odds rato s estmated usng the follong smple formula of McNemar hch s based on the Mantel-Haenszel estmator. OR For statstcal analyss, the logarthm of the odds rato s usually used because ts dstrbuton s more accurately approxmated by the normal dstrbuton for smaller sample szes. Saha and Khurshd (995) page 9 gve the varance of the sample log odds rato s estmated by b = c V ( ln ( OR ) ) = + b c Rs Rato or Relatve Rs Follong Saha and Khurshd (995) page 39, the rs rato s estmated as follos. RR a = a Le the odds rato, the logarthm of the rs rato s used because ts dstrbuton s more accurately approxmated by the normal dstrbuton for smaller sample szes. The varance of the sample log rs rato s estmated by ( ( ) ) + + b c ( + ) ( a + c )( a + b ) b c V ln RR = Rs Dfference Follong Saha and Khurshd (995) page 39, the rs dfference s calculated as follos. RD b c = n The estmated varance of the sample rs dfference s gven by ( ) V RD ( + ) ( ) n b c b c = 3 n NCSS, LLC. All Rghts Reserved.

3 NCSS Statstcal Softare Meta-Analyss of Correlated Proportons Defnng the Study Parameters Let θ represent the outcome measure created from the 2-by-2 table. That s, θ may be the odds rato, rs rato, or rs dfference. Let θ represent the estmate of θ from the study. Confdence ntervals based on the normal dstrbuton may be defned for θ n the usual manner. θ ± z α / 2 In the case of the odds rato and the rs rato, the nterval s created on the logarthmc scale and then transformed bac to the orgnal scale. ( θ) V It ll be useful n the sequel to mae the follong defnton of the eghts. v = V ( θ ) = / v Hypothess Tests Several hypothess tests have be developed to test the varous hypotheses that may be of nterest. These ll be defned next. Overall Null Hypothess To statstcal tests have been devsed to test the overall null hypothess that all treatment effects are zero. The null hypothess s rtten H0: θ = 0 =,, Nondrectonal Test The nondrectonal alternatve hypothess that at least one θ 0 may be tested by comparng the quantty th a χ 2 dstrbuton. X ND = θ 2 = Drectonal Test A test of the more nterestng drectonal alternatve hypothess that θ comparng the quantty = θ 0 for all may be tested by X D = = θ = 2 th a χ 2 dstrbuton. Note that ths tests the hypothess that all effects are equal to the same nonzero quantty NCSS, LLC. All Rghts Reserved.

4 NCSS Statstcal Softare Meta-Analyss of Correlated Proportons Effect-Equalty (Heterogenety) Test When the overall null hypothess s rejected, the next step s to test hether all effects are equal, that s, hether the effects are homogeneous. Specfcally, the hypothess s H : θ = θ =,, 0 versus the alternatve that at least one effect s dfferent, that s, that the effects are heterogeneous. Ths may also be nterpreted as a test of the study-by-treatment nteracton. Ths hypothess s tested usng Cochran s Q test hch s gven by here ( ) Q = θ θ 2 θ = = = = θ 2 The test s conducted by comparng Q to a χ dstrbuton. Fxed, Versus Random, Effects Combned Confdence Interval If the effects are be assumed to be equal (homogeneous), ether through testng or from other consderatons, a fxed effects model may be used to construct a combned confdence nterval. Hoever, f the effects are heterogeneous, a random effects model should be used to construct the combned confdence nterval. Fxed Effects Model The fxed effects model assumes homogenety of study results. That s, t assumes that θ = θ for all. Ths assumpton may not be realstc hen combnng studes th dfferent patent pools, protocols, follo-up strateges, doses, duratons, etc. If the fxed effects model s adopted, the nverse varance-eghted method as descrbed by Sutton (2000) page 58 s used to calculate the confdence nterval for θ. The formulas used are θ ± z α / 2 ( θ ) V here z α / 2 s the approprate percentage pont from the standardzed normal dstrbuton and θ = = ( ) V θ = = θ = NCSS, LLC. All Rghts Reserved.

5 NCSS Statstcal Softare Meta-Analyss of Correlated Proportons Random Effects Model The random effects model assumes that the ndvdual θ come from a random dstrbuton th fxed mean θ and varance σ 2. Sutton (2000) page 74 presents the formulas necessary to conduct a random effects analyss usng the eghted method. The formulas used are θ ± z α / 2 ( θ ) V here z α / 2 s the approprate percentage pont from the standardzed normal dstrbuton and θ = = ( ) V θ = = = θ = 2 + τ Q + τ 2 f Q > = U 0 otherse s ( ) Q = θ θ 2 = U ( ) s 2 = = = = = Graphcal Dsplays A number of plots have been devsed to dsplay the nformaton n a meta-analyss. These nclude the forest plot, the radal plot, and the L Abbe plot. More ll be sad about each of these plots n the Output secton NCSS, LLC. All Rghts Reserved.

6 NCSS Statstcal Softare Meta-Analyss of Correlated Proportons Data Structure The data are entered nto a dataset usng one ro per study. The four counts of the study s 2-by-2 table are entered nto four columns. In addton to these, an addtonal varable s usually used to hold a short (3 or 4 character) label. Another varable may be needed to hold a groupng varable. As an example, e ll use the METACPROP dataset hch presents the results of 24 matched case-control studes that ere conducted to study the effectveness of a certan treatment. The goal of each study as to compare the proporton of cases that respondng th a Yes to the correspondng proporton of control responses th a Yes. The studes ere grouped nto to dets, but these ere not ther man focus. These data are contaned n the METACPROP database. You should load ths database to see ho the data are arranged. Procedure Optons Ths secton descrbes the optons avalable n ths procedure. Varables Tab The optons on ths screen control the varables that are used n the analyss. Data and Varables N Count (A) Varable Specfy the varable contanng the count of the number of subjects that responded th a (Yes) to both varables. In a matched case control study, ths varable contans the number of case-control pars that both shoed the event of nterest. N0 Count (B) Varable Specfy the varable contanng the count of the number of subjects that responded th a (Yes) to frst varable and a 0 (No) to the second. In a matched case control study, ths varable contans the number of casecontrol pars that had a postve case and a negatve control. N0 Count (C) Varable Specfy the varable contanng the count of the number of subjects that responded th a 0 (No) to frst varable and a (Yes) to the second. In a matched case control study, ths varable contans the number of case-control pars that had a negatve case and a postve control. N00 Count (D) Varable Specfy the varable contanng the count of the number of subjects that responded th a 0 (No) to both varables. In a matched case control study, ths varable contans the number of case-control pars that ere negatve for both the case and the control. Data and Varables Optonal Varables Label Varable Specfy an optonal varable contanng a label for each study (ro) n the database. Ths label should be short (< 8 letters) so that t can ft on the plots. Group Varable Specfy an optonal varable contanng a group dentfcaton value. Each unque value of ths varable ll receve ts on plottng symbol on the forest plots. Some reports are sorted by these group values NCSS, LLC. All Rghts Reserved.

7 NCSS Statstcal Softare Combne Studes Method Meta-Analyss of Correlated Proportons Combne Studes Usng Specfy the method used to combne treatment effects. Use the Fxed Effects method hen you do not ant to account for the varaton beteen studes. Use the Random Effects method hen you ant to account for the varaton beteen studes as ell as the varaton thn the studes. Zero Counts Change Zero Counts To (Delta) Ths s the value added to each cell to avod havng zero cell counts. Outcome measures le the odds rato and rs rato are not defned hen certan counts are zero. By addng a small amount to each cell count, ths opton lets you analyze data th zero counts. You mght consder runnng your analyss a couple of tmes th to or three dfference delta values to determne f the delta value s mang a bg dfference n the outcome (t should not). If all cells n all ros are non-zero, enter 0. Otherse, use 0.5 or (Recent smulaton studes have shon that 0.25 produces better results n some stuatons than the more tradtonal 0.5.) Reports Tab The optons on ths screen control the appearance of the reports. Select Reports Odds Rato Reports/Plots - Rs Dfference Reports/Plots Indcate hether to dsplay reports and plots about ths outcome measure. You must chec at least one of the three outcome measures. Summary Report - Outcome Detal Reports Indcate hether to dsplay the correspondng report. Alpha Level Ths settng controls the confdence coeffcent used n the confdence lmts. Note that 00 x ( - alpha)% confdence lmts ll be calculated. Ths must be a value beteen 0 and 0.5. The most common choce s 0.05, hch results n 95% confdence ntervals. Report Optons Sho Notes Indcate hether to sho the notes at the end of reports. Although these notes are helpful at frst, they may tend to clutter the output. Ths opton lets you omt them. Precson Specfy the precson of numbers n the report. A sngle-precson number ll sho seven-place accuracy, hle a double-precson number ll sho thrteen-place accuracy. Note that the reports are formatted for sngle precson. If you select double precson, some numbers may run nto others. Also note that all calculatons are performed n double precson regardless of hch opton you select here. Sngle precson s for reportng purposes only NCSS, LLC. All Rghts Reserved.

8 NCSS Statstcal Softare Meta-Analyss of Correlated Proportons Varable Names Ths opton lets you select hether to dsplay only varable names, varable labels, or both. Report Optons Decmal Places Probablty Values Rato Values Ths settng controls the number of dgts to the rght of the decmal place that are dsplayed hen shong ths tem. Plots Tab The optons on ths panel control the ncluson and the appearance of the forest plot, radal plot and L Abbe plot. Select Plots Forest Plot L Abbe Plot Indcate hether to dsplay the correspondng plot. Clc the plot format button to change the plot settngs. Storage Tab These optons let you specfy f, and here on the dataset, varous statstcs are stored. Warnng: Any data already n these columns are replaced by the ne data. Be careful not to specfy columns that contan mportant data. Data Storage Optons Storage Opton Ths opton controls hether the values ndcated belo are stored on the dataset hen the procedure s run. Do not store data No data are stored even f they are checed. Store n empty columns only The values are stored n empty columns only. Columns contanng data are not used for data storage, so no data can be lost. Store n desgnated columns Begnnng at the Frst Storage Varable, the values are stored n ths column and those to the rght. If a column contans data, the data are replaced by the storage values. Care must be used th ths opton because t cannot be undone. Store Frst Item In The frst tem s stored n ths column. Each addtonal tem that s checed s stored n the columns mmedately to the rght of ths varable. Leave ths value blan f you ant the data storage to begn n the frst blan column on the rght-hand sde of the data. Warnng: any exstng data n these columns s automatcally replaced, so be careful NCSS, LLC. All Rghts Reserved.

9 NCSS Statstcal Softare Meta-Analyss of Correlated Proportons Data Storage Optons Select Items to Store th the Dataset P Rs Dff. Weghts Indcate hether to store these ro-by-ro values, begnnng at the column ndcated by the Store Frst Item In opton. Example Meta-Analyss of Correlated Proportons Ths secton presents an example of ho to analyze the data contaned n the MetaCProp dataset. Ths dataset contans data for 24 matched case-control studes. The response of each case subject as compared to the response of a matched control subject. You may follo along here by mang the approprate entres or load the completed template Example by clcng on Open Example Template from the Fle menu of the Meta-Analyss of Correlated Proportons ndo. Open the MetaCProp dataset. From the Fle menu of the NCSS Data ndo, select Open Example Data. Clc on the fle MetaCProp.NCSS. Clc Open. 2 Open the Meta-Analyss of Correlated Proportons ndo. Usng the Analyss menu or the Procedure Navgator, fnd and select the Meta-Analyss of Correlated Proportons procedure. On the menus, select Fle, then Ne Template. Ths ll fll the procedure th the default template. 3 Select the varables. Select the Varables tab. Set the N Count (A) Varable to CaseYes. Set the N0 Count (B) Varable to CaseNo. Set the N0 Count (C) Varable to ControlYes. Set the N00 Count (D) Varable to ControlNo. Set the Label Varable to Study. Set the Group Varable to Det. Set Combne Studes Usng to Random Effects Method. Set the Change Zero Counts To (Delta) to Specfy the reports. Select the Reports tab. Chec the Odds Rato Reports/Plots opton box. Chec the Summary Report opton box. Chec the Heterogenety Tests opton box. Chec the Outcome Detal Reports opton box. On the Plots tab, chec the Forest Plot opton box. Chec Radal Plot opton box. Chec the L Abbe Plot opton box. 5 Run the procedure. From the Run menu, select Run Procedure. Alternatvely, just clc the green Run button NCSS, LLC. All Rghts Reserved.

10 NCSS Statstcal Softare Run Summary Secton Meta-Analyss of Correlated Proportons Parameter Value Parameter Value N Count (A) Varable CaseYes Ros Processed 24 N0 Count (B) Varable CaseNo Number Groups 2 N0 Count (C) Varable ControlYes Delta Value 0 N00 Count (D) Varable ControlNo Ro Label Varable Study Group Varable Det Ths report records the varables that ere used and the number of ros that ere processed. Numerc Summary Secton [Treatment] Odds Rs Rs StudyId Data P P2 Rato Rato Dfference [A] S 25/43 6/ S2 44/79 5/ S4 26/5 0/ S7 26/73 0/ S0 23/48 8/ S3 28/66 6/ S6 25/42 0/ S9 29/46 0/ S20 44/76 8/ S22 25/43 8/ S24 75/23 5/ Average [B] S3 53/72 2/ S5 73/08 49/ S6 58/97 37/ S8 42/74 8/ S9 56/98 4/ S 7/2 2/ S2 60/08 28/ S4 46/8 5/ S5 58/77 2/ S7 74/26 3/ S8 62/0 3/ S2 58/77 4/ S23 7/58 / Average [Combned] Average Note: Ths report shos the nput data and the three outcomes for each study n the analyss. The 'Average' values are actually eghted averages th eghts based on the effects model that as selected. Ths report summarzes the nput data. You should scan t for any mstaes. Note that the Average lnes provde the estmated group averages. The values depend on your selecton of hether the Random Effects model or Fxed Effects model as used. The Combned lne provdes the combned results of all studes. Data These are the count values that ere read from the database. P Ths s the estmated event proporton for varable (the cases). P2 Ths s the estmated event proporton for varable 2 (the controls) NCSS, LLC. All Rghts Reserved.

11 NCSS Statstcal Softare Meta-Analyss of Correlated Proportons Odds Rato Ths s the estmated value of the odds rato. Note that t depends not only on the data, but also on the delta value used. Rs Rato Ths s the estmated value of the rs rato. Note that t depends not only on the data, but also on the delta value used. Rs Dfference Ths s the estmated value of the rs dfference. Note that t depends not only on the data, but also on the delta value used. Nondrectonal Zero-Effect Test Outcome Prob Det Measure Ch-Square DF Level A Odds Rato B Odds Rato Combned Odds Rato Ths reports the results of the nondrectonal zero-effect ch-square test desgned to test the null hypothess that all treatment effects are zero. The null hypothess s rtten H0: θ = 0 =,, The alternatve hypothess s that at least one θ 0, that s, at least one study had a statstcally sgnfcant result. Ch-Square Ths s the computed ch-square value for ths test. The formula as presented earler. DF Ths s the degrees of freedom. For ths test, the degrees of freedom s equal to the number of studes. Prob Level Ths s the sgnfcance level of the test. If ths value s less than the nomnal value of alpha (usually 0.05), the test s statstcally sgnfcant and the alternatve s concluded. If the value s larger than the specfed value of alpha, no concluson can be dran other than that you do not have enough evdence to reject the null hypothess. Drectonal Zero-Effect Test Outcome Prob Det Measure Ch-Square DF Level A Odds Rato B Odds Rato Combned Odds Rato Ths reports the results of the drectonal zero-effect ch-square test desgned to test the overall null hypothess that all treatment effects are zero. The null hypothess s rtten The alternatve hypothess s that θ = H0: θ = 0 =,, θ 0 for all, that s, that all effects are equal to the same, non-zero value. Ch-Square Ths s the computed ch-square value for ths test. The formula as presented earler NCSS, LLC. All Rghts Reserved.

12 NCSS Statstcal Softare Meta-Analyss of Correlated Proportons DF Ths s the degrees of freedom. For ths test, the degrees of freedom s equal one. Prob Level Ths s the sgnfcance level of the test. If ths value s less than the specfed value of alpha (usually 0.05), the test s statstcally sgnfcant and the alternatve s concluded. If the value s larger than the specfed value of alpha, no concluson can be dran other than that you do not have enough evdence to reject the null hypothess. Effect-Equalty (Heterogenety) Test Outcome Cochran s Prob Det Measure Q DF Level A Odds Rato B Odds Rato Combned Odds Rato Ths reports the results of the effect-equalty (homogenety) test. Ths ch-square test as desgned to test the null hypothess that all treatment effects are equal. The null hypothess s rtten H : θ = θ =,, 0 The alternatve s that at least one effect s dfferent, that s, that the effects are heterogeneous. Ths may also be nterpreted as a test of the study-by-treatment nteracton. Ths test may help you determne hether to use a Fxed Effects model (used for homogeneous effects) or a Random Effects model (heterogeneous effects). Cochran s Q Ths s the computed ch-square value for Cochran s Q statstc. The formula as presented earler. DF Ths s the degrees of freedom. For ths test, the degrees of freedom s equal to the number of studes mnus one.. Prob Level Ths s the sgnfcance level of the test. If ths value s less than the specfed value of alpha (usually 0.05), the test s statstcally sgnfcant and the alternatve s concluded. If the value s larger than the specfed value of alpha, no concluson can be dran other than that you do not have enough evdence to reject the null hypothess NCSS, LLC. All Rghts Reserved.

13 NCSS Statstcal Softare Meta-Analyss of Correlated Proportons Odds Rato Detal Secton 95.0% 95.0% Percent Loer Upper Random [Det] Odds Confdence Confdence Effects Study P P2 Rato Lmt Lmt Weght [A] S S S S S S S S S S S Average [B] S S S S S S S S S S S S S Average [Combned] Average Ths report dsplays results for the odds rato outcome measure. You can obtan a smlar report for the rs rato and the rs dfference. The report gves you the Confdence Lmts These are the loer and upper confdence lmts (the formulas ere gven earler n ths chapter). Weghts The last column gves the relatve (percent) eght used n creatng the eghted average. Usng these values, you can decde ho much nfluence each study has on the eghted average NCSS, LLC. All Rghts Reserved.

14 NCSS Statstcal Softare Forest Plot Meta-Analyss of Correlated Proportons Ths plot presents the results for each study on one plot. The sze of the plot symbol s proportonal to the sample sze of the study. The ponts on the plot are sorted by group and by the odds rato. The lnes represent the confdence ntervals about the odds ratos. Note that the narroer the confdence lmts, the better. By studyng ths plot, you can determne the man conclusons that can be dran from the set of studes. For example, you can determne ho many studes ere sgnfcant (the confdence lmts do not ntersect the vertcal lne at.0). You can see f there ere dfferent conclusons for the dfferent groups. The results of the combnng the studes are dsplayed at the end of each group NCSS, LLC. All Rghts Reserved.

15 NCSS Statstcal Softare Radal Plot Meta-Analyss of Correlated Proportons The radal (or Galbrath) plot shos the z-statstc (outcome dvded by standard error) on the vertcal axs and a measure of eght on the horzontal axs. Studes that have the largest eght are closest to the Y axs. Studes thn the lmts are nterpreted as homogeneous. Studes outsde the lmts may be outlers NCSS, LLC. All Rghts Reserved.

16 NCSS Statstcal Softare L Abbe Plot Meta-Analyss of Correlated Proportons The L Abbe plot dsplays the varable (case) proporton on vertcal axs versus the varable 2 (control) proporton on the horzontal axs. Homogenous studes ll be arranged along the dagonal lne. Ths plot s especally useful n determnng f the relatonshp beteen the to varables s the same for all values of varable NCSS, LLC. All Rghts Reserved.