Meta-Analysis of Hazard Ratios

Transcription

1 NCSS Statstcal Softare Chapter 458 Meta-Analyss of Hazard Ratos Introducton Ths module performs a meta-analyss on a set of to-group, tme to event (survval), studes n hch some data may be censored. These studes have a treatment group and a control group. Each study s result may be summarzed by the log hazard rato and ts standard error. 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 and radal 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.. 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. (000) and Thompson (998). Refer to those sources for more detals about ho to conduct a meta-analyss. As for the partcular topc of combnng hazard rato studes n a meta-analyss, the boo by Parmar and Machn (995) and the paper by Parmar et al.(998) are essental readng. The paper provdes nstructons on ho to obtan estmates of the hazard rato and ts standard error from trals that do not report these tems explctly (a stuaton that s common) NCSS, LLC. All Rghts Reserved.

2 NCSS Statstcal Softare Meta-Analyss of Hazard Ratos Treatment Effect Hazard Rato The most recommended sngle summary statstc for quantfyng the treatment effect n studes usng survval data s the (log) hazard rate. Ths statstc s chosen because t can be calculated from tme-to-event data th censorng and because t measures the sze of the dfference beteen to Kaplan-Meer curves. The Cox-Mantel estmate of the hazard rato s formed by dvdng the hazard rate under treatment by the hazard rate under control. Thus, t measures the change n rs of treatment versus control over the follo-up perod. Snce the dstrbuton of the log hazard rato s nearly normal, the log transformaton s appled. The formula for the hazard rate s HR CM here O s the observed number of events (deaths) n group, E s the expected number of events (deaths) n group, and H s the overall hazard rate for the th group. The calculaton of the E s explaned n Parmar and Machn (995). H H O O A confdence nterval for HR s found by frst transformng to the log scale hch s better approxmated by the normal dstrbuton, calculatng the lmts, and then transformng bac to the orgnal scale. The calculaton s made usng here ln T C T C / E / E ( HR ) ± z ( SE ) SE CM α / ln HR + CM ET T C ln HR CM An alternatve estmate of HR that s sometmes used s the Mantel-Haenszel estmator hch s calculated usng O HRMH exp here V s the hypergeometrc varance. For further detals, see Parmar and Machn (995). A confdence nterval for HR s found by frst transformng to the log scale hch s better approxmated by the normal dstrbuton, calculatng the lmts, and then transformng bac to the orgnal scale. The calculaton s made usng here ln T E E V C T ( HR ) ± z ( SE ) MH SE α / ln HR MH V ln HR MH If the log hazard rato and ts standard error are not reported n a partcular study t ll have to be estmated from the logran test statstc, p-value, or from the Kaplan-Meer curves. Detals of ho to do ths are presented n Parmar et al. (998). Suppose you have obtaned the results for studes, labeled,,. Each study conssts of a treatment group (T) and a control group (C). The results of each study are summarzed by to statstcs: ln ( HR ) the log hazard rato. SE ln ( HR ) the standard error of the log hazard rato NCSS, LLC. All Rghts Reserved.

3 NCSS Statstcal Softare Meta-Analyss of Hazard Ratos It ll be useful n the sequel to mae the follong defnton of the eghts. v ( ) SE ln HR / v Hypothess Tests In the dscusson belo, e let θ represent ln HR. 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 H : θ θ,, 0 Nondrectonal Test The nondrectonal alternatve hypothess that at least one θ 0 may be tested by comparng the quantty th a χ dstrbuton. X ND θ Drectonal Test A test of the more nterestng drectonal alternatve hypothess that θ θ 0 for all may be tested by comparng the quantty X D θˆ th a χ dstrbuton. Note that ths tests the hypothess that all effects are equal to the same nonzero quantty. 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 NCSS, LLC. All Rghts Reserved.

4 NCSS Statstcal Softare here Meta-Analyss of Hazard Ratos Q ˆ θ ( ˆ θ θˆ ) ˆ θ The test s conducted by comparng Q to a χ dstrbuton. Fxed versus Random Effects Combned Confdence Interval If the effects are 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 (000) page 58 s used to calculate the confdence nterval for θ. The formulas used are ˆ θ ± ˆ z α / V here z α / s the approprate percentage pont from the standardzed normal dstrbuton and ˆ θ ( ˆ) Vˆ θ ˆ θ ( ˆ θ ) Random Effects Model The random effects model assumes that the ndvdual θ come from a random dstrbuton th fxed mean θ and varance σ. Sutton (000) page 74 presents the formulas necessary to conduct a random effects analyss usng the eghted method. The formulas used are ˆ θ ± z ˆ ˆ α / V θ here z α / s the approprate percentage pont from the standardzed normal dstrbuton and NCSS, LLC. All Rghts Reserved.

5 NCSS Statstcal Softare Meta-Analyss of Hazard Ratos ˆ θ Vˆ ˆ θ ˆ θ +τˆ Q + f > ˆ τ U Q 0 otherse Q ( ˆ θ θˆ ) U ( ) s s 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. Data Structure The data are entered nto a dataset usng one ro per study. To varables are requred to hold the log hazard rato and ts standard error. In addton to these, an addtonal varable s usually used to hold a short (3 or 4 character) label. Another varable may be used to hold a groupng varable. As an example, e ll use a dataset gvng the results for survval studes. The results of these studes are recorded n the MetaHR dataset. You should load ths database to see ho the data are arranged NCSS, LLC. All Rghts Reserved.

6 NCSS Statstcal Softare Meta-Analyss of Hazard Ratos 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. Varables Log(Hazard Rato) Varable Specfy the varable contanng the log hazard rato of each study. Each ro of data represents a separate study. Note that the base of the logarthm (e or 0) s arbtrary. Hoever, t must be consstent throughout the dataset. S.E. Log(Hazard Rato) Varable Specfy the varable contanng the standard error of the log hazard rato of each study. Each ro of data represents a separate study. Note that the base of the logarthm (e or 0) s arbtrary. Hoever, t must be consstent throughout the dataset. 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. Combne Studes Method 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 NCSS, LLC. All Rghts Reserved.

7 NCSS Statstcal Softare Meta-Analyss of Hazard Ratos Reports Tab The optons on ths screen control the appearance of the reports. Select Reports 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. Varable Names Ths opton lets you select hether to dsplay only varable names, varable labels, or both. Report Optons Decmal Places Probablty Values Z 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 and the radal plot. Select Plots Forest Plot Radal 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 NCSS, LLC. All Rghts Reserved.

8 NCSS Statstcal Softare Data Storage Optons Meta-Analyss of Hazard Ratos 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. Data Storage Optons Select Items to Store th the Dataset Log(HR) - 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 Hazard Ratos Ths secton presents an example of ho to analyze the data contaned n the MetaHR dataset. Ths dataset contans data for sxteen randomzed clncal trals th survval endponts. 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 Hazard Ratos ndo. Open the MetaHR dataset. From the Fle menu of the NCSS Data ndo, select Open Example Data. Clc on the fle MetaHR.NCSS. Clc Open. Open the Meta-Analyss of Hazard Ratos ndo. On the menus, select Analyss, then Meta-Analyss, then Meta-Analyss of Hazard Ratos. The Meta- Analyss of Hazard Ratos procedure ndo ll be dsplayed. On the menus, select Fle, then Ne Template. Ths ll fll the procedure th the default template NCSS, LLC. All Rghts Reserved.

9 NCSS Statstcal Softare Meta-Analyss of Hazard Ratos 3 Select the varables. Select the Varables tab. Set the Log(Hazard Rato) Varable to LogHR. Set the S.E. Log(Hazard Rato) Varable to SELogHR. Set the Label Varable to Study. 4 Specfy the reports. Select the Reports tab. 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. 5 Run the procedure. From the Run menu, select Run Procedure. Alternatvely, just clc the green Run button. Run Summary Secton Parameter Value Parameter Value Log HR Varable LogHR SE(Log HR) Varable SELogHR Group Varable None Number Groups Ro Label Varable Study Ros Processed 6 Ths report records the varables that ere used and the number of ros that ere processed. Numerc Summary Secton Study Log HR SE(Log HR) S S S S S S S S S S S S S S S S [Combned] Average Ths report shos the nput data. You should scan t for any mstaes. Note that the Average lne provdes the estmated group average NCSS, LLC. All Rghts Reserved.

10 NCSS Statstcal Softare Meta-Analyss of Hazard Ratos Nondrectonal Zero-Effect Test Outcome Prob Ros Measure Ch-Square DF Level Combned Log(Hazard 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 H : θ 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 Ros Measure Ch-Square DF Level Combned Log(Hazard 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 H : θ θ,, 0 The alternatve hypothess s that θ 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. 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 Treatment Measure Q DF Level Combned Log(Hazard Rato) NCSS, LLC. All Rghts Reserved.

11 NCSS Statstcal Softare Meta-Analyss of Hazard Ratos 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. Log(Hazard Rato) Detal Secton 95.0% 95.0% Percent Loer Upper Random Log Hazard Standard Confdence Confdence Effects Study Rato Error Lmt Lmt Weght S S S S S S S S S S S S S S S S [Combned] Average Ths report dsplays results for the log hazard rato. 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.

12 NCSS Statstcal Softare Forest Plot Meta-Analyss of Hazard Ratos 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 the mean dfference. The lnes represent the confdence ntervals about the log hazard 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 0.0) NCSS, LLC. All Rghts Reserved.

13 NCSS Statstcal Softare Radal Plot Meta-Analyss of Hazard Ratos 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.