Chapter 706 No-Iferiority Lograk Tests Itroductio This module computes the sample size ad power for o-iferiority tests uder the assumptio of proportioal hazards. Accrual time ad follow-up time are icluded amog the parameters to be set. The o-iferiority lograk test is used for data aalysis. Sometimes, the objective of a study is to show that a experimetal therapy is ot iferior to (o worse tha) the stadard therapy. The experimetal therapy may be cheaper, less toxic, or have fewer side effects. Such studies are ofte called o-iferiority trials ad have a oe-sided hypothesis. Power ad sample size calculatios for the o-iferiority lograk test have bee developed by Jug et al. (005), ad we use their results. These calculatios assume a uderlyig expoetial survival distributio with a uiform patiet accrual patter durig the accrual period. Techical Details Test Statistic Suppose a cliical trial cosists of two idepedet groups. Desigate group oe as the stadard group with hazard rate h ad sample size. Desigate group two as the experimetal group with hazard rate h ad sample size. The total sample size is N = +. Usually, you would pla to have =. Defie the proportio of the total sample i each group as i Q i = N, i =, Idividuals are recruited durig a accrual period of R years (or moths or days). They are followed for a additioal period of time util a total of T years is reached. Hece, the follow-up period is T-R years. At the ed of the study, the o-iferiority lograk test is coducted at sigificace level α with power β. Uder the proportio hazards assumptio, the hazard ratio HR = h / h is costat across time. For a give o-iferiority margi HR 0 (>) (the maximum ratio of cliical isigificace), the statistical hypotheses tested are H : HR HR vs. H HR < HR 0 0 : 0 706-
Defie the partial score fuctio as W ( HR) = HR i= ad the iformatio fuctio as ( X X ) i j i j= ( X j X i ) + HR I ( X j X i ) I δ j= j= σ N ( HR) I = HR k k = i= δ ki j= i= i j= ( X X ) j ( X j X i ) + HR I ( X j X i ) I δ j= j= ( X j X ki ) I ( X j X ki ) j= ( X j X ki ) + HR I ( X j X ki ) I I j= j= where X ki is the miimum of the survival time, ad the cesorig time, δ ki,is a evet idicator takig if there was a evet or 0 otherwise, ad I(.) is a idicator fuctio. Note that W() is the stadard lograk test statistic. Uder W HR / HR 0 0 σ N 0 is asymptotically ormal with mea 0 ad variace. Reject H 0 i favor of H if W ( HR ) / 0 σ ( HR N 0 ) > z α with oe-sided type I error probability α. H, ( ) ( ) The partial MLE, show that R Ĥ, is obtaied by solvig ( HR) = 0 Ĥ R is asymptotically ormal with mea * A asymptotic 00( α )% W. Let HR * deote the true value of HR. It ca be HR ad variace ( ) I σ N HR *. cofidece iterval for HR is Hˆ R ± z σ ( HR ˆ ) α / N. i Power Calculatios Jug (005) shows that the power of the o-iferiority lograk test ca be expressed as β = Φ ( HR ) 0 DQ Q z Q + Q HR where D is the observed umber of deaths (evets). The total sample size N is obtaied by iflatig D accordig to the relatioship E(d)N = D, where E(d) is the expected death rate for the trial. Followig the proposal of Yatema ad Skee (99) ad the results of Lakatos (988), we compute E(d) usig the Markov Model give i chapter 75 as ( d ) = QS, QS, E + where S, ad S, are the occupacy probabilities for the evet state for the stadard ad experimetal groups, respectively. This formulatio allows the iclusio of loss to follow-up, ocompliace, ad drop-i alog with various accrual patters. α 0 HR 0 706-
Procedure Optios This sectio describes the optios that are specific to this procedure. These are located o the Desig tab. For more iformatio about the optios of other tabs, go to the Procedure Widow chapter. Desig Tab The Desig tab cotais most of the parameters ad optios that you will be cocered with. Solve For Solve For This optio specifies the parameter to be solved for from the other parameters. The parameters that may be selected are Power or Sample Size. Select Sample Size whe you wat to calculate the sample size eeded to achieve a give power ad alpha level. Select Power whe you wat to calculate the power. Power ad Alpha Power This optio specifies oe or more values for power. Power is the probability of rejectig a false ull hypothesis, ad is equal to oe mius Beta. Beta is the probability of a type-ii error, which occurs whe a false ull hypothesis is ot rejected. I this procedure, a type-ii error occurs whe you do ot reject the ull hypothesis that the hazard ratio is greater tha HR0 whe i fact it is. Values must be betwee zero ad oe. Historically, the value of 0.80 (Beta = 0.0) was used for power. Now, 0.90 (Beta = 0.0) is also commoly used. A sigle value may be etered here or a rage of values such as 0.8 to 0.95 by 0.05 may be etered. Alpha This optio specifies oe or more values for the probability of a type-i error. A type-i error occurs whe you reject the ull hypothesis that the hazard ratio is greater tha HR0 whe i fact it is ot. Values of alpha must be betwee zero ad oe. Historically, the value of 0.05 has bee used for alpha. This meas that about oe test i twety will falsely reject the ull hypothesis. You should pick a value for alpha that represets the risk of a type-i error you are willig to take i your experimetal situatio. You may eter a rage of values such as 0.0 0.05 0.0 or 0.0 to 0.0 by 0.0. Sample Size N (Total Sample Size) This is the combied sample size of both groups. This amout is divided betwee the two groups usig the value of the Proportio i Referece Group. Proportio i Referece Group This is the proportio of N i the referece (cotrol) group. If this value is labeled Q, the sample size of group oe is NQ ad the sample size of group two is N NQ. Note that the value of NQ is rouded to the earest iteger. 706-3
Effect Size HR0 (Equivalece) This is the maximum value of the hazard ratio that still results i the coclusio of equivalece. It is referred to as HR0 i this documetatio. Assumig that evets are bad (such as death), the this umber should be > oe. For example, if you eter.0 here, you are sayig that hazard ratios <.0 will result i the coclusio of oiferiority whe H0 is rejected. I other words, hazard ratios up to.0 idicate that the treatmet group is o worse tha the referece group. Estimates of the hazard ratio may be obtaied from media survival times, hazard rates, or from the proportio survivig past a certai time poit. Pressig the Parameter Coversio butto will load a tool for doig this. h (Hazard Rate of Referece Group) Specify oe or more hazard rates (istataeous failure rate) for the referece group. For a expoetial distributio, the hazard rate is the iverse of the mea survival time. A estimate of the hazard rate may be obtaied from the media survival time or from the proportio survivig to a certai time poit. This calculatio is automated by pressig the Parameter Coversio butto. Hazard rates must be greater tha zero. Costat hazard rates are specified by eterig them directly. Variable hazard rates are specified as colums of the spreadsheet. Whe you wat to specify differet hazard rates for differet time periods, you would eter those rates ito a colum of the spreadsheet, oe row per time period. You specify the colum (or colums) by begiig the etry with a equals sig. For example, if you have etered the hazard rates i colum, you would eter = here. Duratio Accrual Time (Itegers Oly) Eter oe or more values for the umber of time periods (moths, years, etc.) durig which subjects are etered ito the study. The total duratio of the study is equal to the Accrual Time plus the Follow-Up Time. These values must be itegers. Accrual times ca rage from 0 to the Total Time. That is, the accrual time must be less tha or equal to the Total Time. Otherwise, the sceario is skipped. Eter 0 whe all subjects begi the study together. Accrual Patter Specify the type of accrual (patiet etry) patter. Two types of etries are possible: Uiform or Equal If you wat to specify a uiform accrual rate for all time periods, eter Equal here. No-Uiform (Spreadsheet Etry) Use this optio whe you wat to specify oe or more accrual patters with differet accrual rates per time period. You will specify the differet accrual rates for each time period i the spreadsheet. 706-4
Accrual Values i Colums Specify the colums of the spreadsheet cotaiig the differet accrual (patiet etry) rates. Oe value per row is etered i spreadsheet cells for each time period. Each value is the proportio of the total umber of subjects that eroll durig the correspodig time period. Sytax Eter a equals sig followed by a list of colums cotaiig the accrual patters. For example, if you have etered two sets of accrual patters i colums ad, you would eter =C C. Stadardized Note that cell values i a colum are stadardized so they sum to oe. Thus, the accrual patters ad 50 5 5 both result i the same accrual patter as 0.50 0.5 0.5. Number of Rows ad Colums The umber of rows i each colum should equal the Accrual Time. The umber of colums is up to you. A separate aalysis is coducted for each colum. Spreadsheet Cells I a specified colum, the proportio of all subjects that are expected to eroll durig the first time period is specified i row oe. The proportio of all subjects that are expected durig the secod time period is specified i row two. Ad so o. For example, if you had specified three accrual-time periods ad you wated to specify double the accrual rate i the first period tha i the other two, the spreadsheet would appear as C Total Time (Itegers Oly) Eter oe or more values for the umber of time periods (moths, years, etc.) i the study. The follow-up time is equal to the Total Time mius the Accrual Time. These values must be itegers. Proportio Lost or Switchig Groups Referece (or Treatmet) Lost This is the proportio of subjects i the referece (treatmet) group that disappear from the study durig a sigle time period (moth, year, etc.). Multiple etries, such as 0.0 0.03 0.05, are allowed. Whe you wat to specify differet proportios for differet time periods, you would eter those rates ito a colum of the spreadsheet, oe row per time period. You specify the colum of the spreadsheet by begiig your etry with a equals sig. For example, if you have etered the proportios i colum 5, you would eter =C5 here. Referece Switchig to Treatmet This is the proportio of subjects i the referece group that chage to a treatmet regime similar i efficacy to the treatmet group durig a sigle time period (moth, year, etc.). This is sometimes referred to as drop i. Multiple etries, such as 0.0 0.03 0.05, are allowed. Whe you wat to specify differet proportios for differet time periods, you would eter those values ito a colum of the spreadsheet, oe row per time period. You specify the colum of the spreadsheet by begiig your etry with a equals sig. For example, if you have etered the proportios i colum, you would eter =C here. 706-5
Treatmet Switchig to Referece This is the proportio of subjects i the treatmet group that chage to a treatmet regime similar i efficacy to the referece group durig a sigle time period (moth, year, etc.). This is sometimes referred to as ocompliace. Multiple etries, such as 0.0 0.03 0.05, are allowed. Whe you wat to specify differet proportios for differet time periods, you would eter those values ito a colum of the spreadsheet, oe row per time period. You specify the colum of the spreadsheet by begiig your etry with a equals sig. For example, if you have etered the proportios i colum, you would eter =C here. Reports Tab The Reports tab cotais additioal settigs for this procedure. Report Colum Width Report Colum Width This optio sets the width of the each colum of the umeric report. The umeric report for this optio ecessarily cotais may colums, so the maximum umber of decimal places that ca be displayed is four. If you try to icrease that umber, the umbers may ru together. You ca icrease the width of each colum usig this optio. The recommeded report colum width for scearios without large umbers of decimal places or extremely large sample sizes is 0.49. Optios Tab The Optios tab cotais additioal settigs for this procedure. Optios Number of Itervals withi a Time Period The algorithm requires that each time period be partitioed ito a umber of equal-width itervals. Each of these subitervals is assumed to follow a expoetial distributio. This optio cotrols the umber of subitervals. All parameters such as hazard rates, loss to follow-up rates, ad ocompliace rates are assumed to be costat withi a subiterval. Lakatos (988) gives little iput as to how the umber of subitervals should be chose. I a private commuicatio, he idicated that 00 ought to be adequate. This seems to work whe the hazard rate is less tha.0. As the hazard rate icreases above.0, this umber must icrease. A value of 000 should be sufficiet as log as the hazard rates (h ad h) are less tha 0. Whe the hazard rates are greater tha 0, you may wat to icrease this value to 5000 or eve 0000. 706-6
Example Fidig the Power A o-iferiority trial is plaed i which the primary aalysis will use the o-iferiority lograk test. After extesive discussio, the researchers have decided that the upper boud o o-iferiority is.3. The trial will iclude a recruitmet period of two-years after which participats will be followed for three more years. It is assumed that patiets will eter the study uiformly over the accrual period. The researcher estimates a loss-to-follow rate of 5% per year i both the referece ad experimetal groups. Past experiece leads to a base lie hazard rate of 0.04. A equal sample allocatio desig will be used with a target power of 0.90 ad sigificace level of 0.05. Setup This sectio presets the values of each of the parameters eeded to ru this example. First, from the PASS Home widow, load the procedure widow by expadig Survival, the Two Survival Curves, the clickig o No-Iferiority, ad the clickig o. You may the make the appropriate etries as listed below, or ope Example by goig to the File meu ad choosig Ope Example Template. Optio Value Desig Tab Solve For... Power Alpha... 0.05 Group Allocatio... Eter total sample size ad percetage i Group Total Sample Size (N)... 000 to 5000 by 000 Percet i Group... 50 HR0....3 h... 0.04 Accrual Time... Accrual Patter... Uiform or Equal Total Time... 5 Refereces Lost... 0.05 Refereces Switch to Treatmet... 0.0 Treatmets Lost... 0.0 Treatmets Switch to Referece... 0.0 Reports Tab Show Detail Numeric Reports... Checked 706-7
Aotated Output Click the Calculate butto to perform the calculatios ad geerate the followig output. Numeric Results Numeric Results i Terms of Sample Size Acc- Equiv Actual Ref rual Haz Haz Haz Acc- Time/ Ref Trt Ratio Ratio Rate rual Total Ref Trt to to Power N N N (HR0) (HR) (h) Pat' Time Loss Loss Trt Ref Alpha Beta 0.4665 500 500 000.30.00 0.0400 Equal / 5 0.0500 0.0000 0.0000 0.0000 0.0500 0.5335 0.7 000 000 000.30.00 0.0400 Equal / 5 0.0500 0.0000 0.0000 0.0000 0.0500 0.889 0.858 500 500 3000.30.00 0.0400 Equal / 5 0.0500 0.0000 0.0000 0.0000 0.0500 0.47 0.98 000 000 4000.30.00 0.0400 Equal / 5 0.0500 0.0000 0.0000 0.0000 0.0500 0.078 0.966 500 500 5000.30.00 0.0400 Equal / 5 0.0500 0.0000 0.0000 0.0000 0.0500 0.0338 Refereces Jug,Si-Ho; Kag, Su J.; McCall, Lida M.; Blumestei, Bret. 005. 'Sample Sizes Computatio for Two-Sample Noiferiority Log-Rak Test', J. of Biopharmaceutical Statistics, Volume 5, pages 969-979. Lakatos, Edward. 988. 'Sample Sizes Based o the Log-Rak Statistic i Complex Cliical Trials', Biometrics, Volume 44, March, pages 9-4. Report Defiitios Power is the probability of rejectig a false ull hypothesis. Power should be close to oe. N N N are the sample sizes of the referece group, treatmet group, ad both groups, respectively. E E E are the umber of evets i the referece group, the treatmet group, ad both groups, respectively. Equivalece Haz Ratio (HR0) is the upper boud for the hazard ratio that still leads to the coclusio of o-iferiority. Actual Haz Ratio (HR) is assumed to be the actual value of the hazard ratio. Ref Haz Rate is the hazard (istataeous failure) rate of the referece group. Its scale is evets per time period. Accrual Time is the umber of time periods (years or moths) durig which accrual takes place. Total Time is the total umber of time periods i the study. Follow-up time = (Total Time) - (Accrual Time). Ref Loss is the proportio of the referece group that is lost (drop out) durig a sigle time period (year or moth). Trt Loss is the proportio of the treatmet group that is lost (drop out) durig a sigle time period (year or moth). Ref to Trt (drop i) is the proportio of the referece group that switch to a group with a hazard rate equal to the treatmet group. Trt to Ref (ocompliace) is the proportio of the treatmet group that switch to a group with a hazard rate equal to the referece group. Alpha is the probability of rejectig a true ull hypothesis. It should be small. Beta is the probability of acceptig a false ull hypothesis. It should be small. This report shows the values of each of the parameters, oe sceario per row. We see that almost 4000 subjects will be required for this study. Next, a report displayig the umber of required evets rather tha the sample size is displayed. Numeric Results i Terms of Evets Acc- Equiv Actual Ref rual Ref Trt Total Haz Haz Haz Acc- Time/ Ref Trt Evts Evts Evts Ratio Ratio Rate rual Total Ref Trt to to Power E E E (HR0) (HR) (h) Pat' Time Loss Loss Trt Ref Alpha Beta 0.4665 66.8 73.8 40.6.30.00 0.0400 Equal / 5 0.0500 0.0000 0.0000 0.0000 0.0500 0.5335 0.7 33.6 47.6 8.3.30.00 0.0400 Equal / 5 0.0500 0.0000 0.0000 0.0000 0.0500 0.889 0.858 00.4.5 4.9.30.00 0.0400 Equal / 5 0.0500 0.0000 0.0000 0.0000 0.0500 0.47 0.98 67.3 95.3 56.5.30.00 0.0400 Equal / 5 0.0500 0.0000 0.0000 0.0000 0.0500 0.078 0.966 334. 369. 703..30.00 0.0400 Equal / 5 0.0500 0.0000 0.0000 0.0000 0.0500 0.0338 Most of this report is idetical to the last report, except that the sample sizes are replaced by the umber of required evets. 706-8
Next, reports displayig the idividual settigs year-by-year for each sceario are displayed. Detailed Iput whe Power=0.450 HR0=.30 HR=.00 N=000 Alpha=0.0500 Accrual/Total Time= / 5 Referece Switch Switch Hazard Percet Referece Treatmet Rate Percet Admi.Referece Treatmet to to Time (H) Accrual Cesored Loss Loss Treatmet Referece Period (0.0400) (Equal) (Calc.) (0.0500) (0.0000) (0.0000) (0.0000) 0.0400 50.00 0.00 0.0500 0.0000 0.0000 0.0000 0.0400 50.00 0.00 0.0500 0.0000 0.0000 0.0000 3 0.0400 0.00 0.00 0.0500 0.0000 0.0000 0.0000 4 0.0400 0.00 50.00 0.0500 0.0000 0.0000 0.0000 5 0.0400 0.00 00.00 0.0500 0.0000 0.0000 0.0000 This report shows the idividual settigs for each time period (year). It becomes very useful whe you wat to documet a study i which these parameters vary from year to year. Next, summary statemets are displayed. Summary Statemets A o-iferiority lograk test with a overall sample size of 000 subjects (500 i the referece group ad 500 i the treatmet group) achieves 45.% power at a 0.050 sigificace level to detect a equivalece hazard ratio of.30 whe the actual hazard ratio is a equivalece hazard ratio of.00 ad the referece group hazard rate is 0.0400. The study lasts for 5 time periods of which subject accrual (etry) occurs i the first time periods. The accrual patter across time periods is uiform (all periods equal). The proportio droppig out of the referece group is 0.0500. The proportio droppig out of the treatmet group is 0.0500. The proportio switchig from the referece group to aother group with a hazard rate equal to the treatmet group is 0.0000. The proportio switchig from the treatmet group to aother group with a hazard rate equal to the referece group is 0.0000. Fially, a scatter plot of the results is displayed. Plots Sectio This plot shows the relatioship betwee sample size ad. Note that for 90% power, a total sample size of about 4000 is required. The exact umber will be foud i Example. 706-9
Example Fidig the Sample Size Cotiuig with the previous example, the researcher wats to ivestigate the sample sizes ecessary to achieve 80% ad 90% power. All other parameters will remai the same as i Example. Setup This sectio presets the values of each of the parameters eeded to ru this example. First, from the PASS Home widow, load the procedure widow by expadig Survival, the Two Survival Curves, the clickig o No-Iferiority, ad the clickig o. You may the make the appropriate etries as listed below, or ope Example by goig to the File meu ad choosig Ope Example Template. Optio Value Desig Tab Solve For... Sample Size Power... 0.80 0.90 Alpha... 0.05 Group Allocatio... Equal (N = N) HR0....3 h... 0.04 Accrual Time... Accrual Patter... Uiform or Equal Total Time... 5 Refereces Lost... 0.05 Refereces Switch to Treatmet... 0.0 Treatmets Lost... 0.05 Treatmets Switch to Referece... 0.0 Output Click the Ru butto to perform the calculatios ad geerate the followig output. Numeric Results Numeric Results i Terms of Sample Size Acc- Equiv Actual Ref rual Haz Haz Haz Acc- Time/ Ref Trt Ratio Ratio Rate rual Total Ref Trt to to Power N N N (HR0) (HR) (h) Pat' Time Loss Loss Trt Ref Alpha Beta 0.8000 344 345 689.30.00 0.0400 Equal / 5 0.0500 0.0500 0.0000 0.0000 0.0500 0.000 0.9000 865 866 373.30.00 0.0400 Equal / 5 0.0500 0.0500 0.0000 0.0000 0.0500 0.000 706-0
Example 3 Validatio usig Jug Jug et al. (005) pages 974-975 preset a example that will be used to validate this procedure. I this article, a 8.8-year trial is preseted i which patiet accrual occurs the first 3.8 years. The baselie hazard rate is 0.0446. The value of HR0 is.3 ad the value of HR is.0. Equal allocatio betwee groups is used ad uiform accrual is assumed. The sigificace level is 0.05 ad the desired power is 0.90. Give these values, the umber of evets is foud to be 499 ad the sample size is 89. Sice this procedure usig iteger values for the accrual ad trial time, the accrual time ad total time will be set to 4 ad 9 years, respectively. Setup This sectio presets the values of each of the parameters eeded to ru this example. First, from the PASS Home widow, load the procedure widow by expadig Survival, the Two Survival Curves, the clickig o No-Iferiority, ad the clickig o. You may the make the appropriate etries as listed below, or ope Example 3 by goig to the File meu ad choosig Ope Example Template. Optio Value Desig Tab Fid (Solve For)... Sample Size Power... 0.90 Alpha... 0.05 Group Allocatio... Equal (N = N) HR0....3 h... 0.0446 Accrual Time... 4 Accrual Patter... Uiform or Equal Total Time... 9 Refereces Lost... 0.0 Refereces Switch to Treatmet... 0.0 Treatmets Lost... 0.0 Treatmets Switch to Referece... 0.0 Output Click the Calculate butto to perform the calculatios ad geerate the followig output. Numeric Results Numeric Results i Terms of Sample Size Acc- Equiv Actual Ref rual Haz Haz Haz Acc- Time/ Ref Trt Ratio Ratio Rate rual Total Ref Trt to to Power N N N (HR0) (HR) (h) Pat' Time Loss Loss Trt Ref Alpha Beta 0.9000 933 933 866.30.00 0.0446 Equal 4 / 9 0.0000 0.0000 0.0000 0.0000 0.0500 0.000 706-
Numeric Results i Terms of Evets Acc- Equiv Actual Ref rual Ref Trt Total Haz Haz Haz Acc- Time/ Ref Trt Evts Evts Evts Ratio Ratio Rate rual Total Ref Trt to to Power E E E (HR0) (HR) (h) Pat' Time Loss Loss Trt Ref Alpha Beta 0.9000 49.3 49.3 498.6.30.00 0.0446 Equal 4 / 9 0.0000 0.0000 0.0000 0.0000 0.0500 0.000 Note that the umber of evets (499) matches Jug s results exactly. The sample size of 866 is slightly less tha Jug s 89. This differece occurs because these results were obtaied for 4 years of accrual, ot 3.8, ad because we used Lakatos method for trasformig the umber of evets ito the sample size. Example 4 Iputtig Time-Depedet Hazard Rates from a Spreadsheet Time-depedet parameters (hazard rates, losses to follow-up, etc) may be etered. See Example 4 of Chapter 75 (Lograk Tests) for a extesive example of how this is doe for the lograk test. 706-