CALCULATION OF SAMPLE SIZE IN SURVIVAL ANALYSIS WITH GROUP SEQUENTIAL DESIGNS AND
ADAPTIVE DESIGNS
E. Terzi1, M. Yildiz2 and Y. Terzi1
1 Department of Statistics, OndokuzMayis University, 55139, Samsun,Turkey
2 Department of Statistics, AnadoluUniversity, 26000, Eskişehir,Turkey
Email of corresponding author:
ABSTRACT
Calculation of sample size and determination of event numbers are huge problems for survival analysis since it is important for the reliability of studies. The aim of this study is to calculate cumulative sample size and event numbers obtained using the group sequential designs and the adaptive designs. Simulation studies were made and two groups of patient were compared at different stages, event ratios and Pocock’s and O’Brien &Fleming’s boundaries.Keywords: Sample size, Adaptive design, Survival analysis, Group sequential design
INTRODUCTION
Sample size calculation is often an important and usually difficult step in a planning a statistical study.It must be planned carefully to ensure that the research time, patient effort and support cost invested in a clinical trial are not wasted. Ina experiment involving human or animal subjects, sample size a pivotal issue for ethical reasons. An under-sized experiment, an unnecessary number of subjects are exposed to a potentially harmful treatment.
Most clinical trials are longitudinal in nature. In practice, it is almost impossible to enrol and randomize all required subjects at the same time. Clinical data are accumulated sequentially over time.As a result, it is of interest to monitor the information for management of the study.Sequential methods were originally developed in order to obtain economic benefits.
However, sequential methods are not useful for clinical experiments because every couple of data need to be evaluated continuously. Instead of this, group sequential test methods that provide making analysis on time scales analyst specified are used. If two treatments are compared in a clinical study for which treatment gives better result, can be fixed by group sequential test methods without waiting the end of the study.
So, after a particular step, patients can be canalized other treatment group and their survival time may be extensive. Recent times, that is developed another method on the same subject is adaptive designs. An adaptive design allows adaptations to trial or statistical procedures that give many benefits to investigator. Use of adaptive design in clinical trials; costs reduce, time of the study reduce, right drug is given to right patient at right time [1].
Adaptive design method has a flexible form and it allows modification in the characteristics of the study. By adaptive design, sample size can be changed and the study time can be stopped early. Studies of clinical researches and developments, adaptive designs application’s can be made easily and effect of any adaptation may make the trial or statistical procedure important for before study, after study or on going study[2].
Calculation of sample size and determination of event numbers is a big problem for survival analysis, because it is important for reliability of studies. In clinical trials, group sequential designs and adaptive designs are very new and they provide many advantages to researchers. The major advantage of adaptive designs over the usual group sequential methods is the flexibility of extending the sample size, study duration, or information in general, in addition to possible early termination [3].
Terzi et al, (2007) gives the applications of group sequential test methods in survival analysis for real data set [4]. In this study we focus on the comparisons of group sequential designs and adaptive designs.
The aim of this study is to calculate cumulative sample size and event numbers for group sequential designs and adaptive designs. For this aim, simulation studieswere made and two groups of patient were compared at different stages, event ratios andPocock’sand O’Brien & Fleming’sboundaries using Adaptive Designs–Plans and Analyses (ADDPLAN) program [5,6].
MATERIALS AND METHODS
One of the most important issues in the planning of a trial is to calculate the necessary sample size for a desired power [7,8]. Studies about calculating the total number of events and sample sizes help researches in the schema of the trial.
Let denotes the combined probability of an event in the two treatment groups.
(1)
where is the mortality of first group and is the mortality of second group.Sample size for the first treatment group is calculated as in equation (2).
(2)
In addition,the sample size for the second group is calculated as in equation (3).
(3)
If we want to calculate the total number of events for both treatment groups, there are two formulas. One of them is Freedman formula and the other is Schoenfield’s formula. Freedman formula is given as follow in equation (4).
(4)
Wheredis the total number of events for both treatment groups and r is the allocation ratio for sample size, is the expected hazard ratio and theis inverse cumulative standard normal distribution function.
Schoenfield’s formula, which uses the same parameters as in equation (4), is given in equation (5).
(5)
For simulation, we prefer using ADDPLAN program. In ADDPLAN there are to options for formulas. One is Schoenfield’s formulaand the otheris Fredman’s formula. It is noted that formulas give the different results. For the balanced designs, Schoenfield’s formula gives smaller number of events required than Freedman’s formula [7].
The expected event number is calculated by equation (6) for group sequential designs.
Where HR shows the hazard ratio, S1 is the first patient group’s survival ratio and S2 is the second group’s survival ratio. Total sample size for group sequential design is calculated as in equation (7)[9, 10].
(7)
SIMULATION STUDY
In recent years, survival analysis studies are commonly used method in clinical trials.As mentioned before, calculation of the sample size and determination of the event numbers is a big problem for survival analysis since it is important for the reliability of studies.
In this study,we focus on calculating the cumulative sample size and the event numbers for the group sequential designs and the adaptive designs. For this aim, simulation studies were performed and two groups of patient were compared at different stages (K=3,5), event ratios (1, 2) and Pocock’s and O’Brien & Fleming’s boundaries.Table 1-2 show the cumulative sample sizes and the event numbers for adaptive designs at each stage in survival analysis studies.
Table 1.Cumulative sample size and event numbers for K=3, =0.05, 1-=0.80(adaptive designs)
Table 2.Cumulative sample size and event numbers for K=5, =0.05, 1-=0.80
(adaptivedesigns)
Table 3-4denote the cumulative sample sizes and the event numbers for group sequential designs at each stagein survival analysis studies.
Table 3.Cumulative sample size and event numbers for K=3, =0.05, 1-=0.80
(groupsequential designs)
1 / / O'Brien-Fleming's / Pocock'sEvent / Cumulativen / Event / Cumulativen
0.1 / 0.2 / 62 / 413 / 72 / 477
0.1 / 0.4 / 19 / 74 / 21 / 85
0.1 / 0.6 / 13 / 37 / 15 / 42
0.2 / 0.3 / 150 / 600 / 173 / 693
0.2 / 0.5 / 30 / 87 / 35 / 100
0.3 / 0.4 / 252 / 720 / 291 / 832
0.3 / 0.6 / 41 / 92 / 48 / 106
0.4 / 0.5 / 347 / 771 / 401 / 891
0.4 / 0.7 / 49 / 89 / 57 / 103
0.5 / 0.6 / 414 / 752 / 478 / 869
0.5 / 0.8 / 51 / 78 / 59 / 90
Table 4.Cumulative sample size and event numbers for K=5, =0.05, 1-=0.80
(groupsequential designs)
1 / / O'Brien-Fleming's / Pocock'sEvent / Cumulativen / Event / Cumulativen
0.1 / 0.2 / 63 / 418 / 74 / 494
0.1 / 0.4 / 19 / 75 / 22 / 88
0.1 / 0.6 / 13 / 37 / 15 / 44
0.2 / 0.3 / 152 / 607 / 180 / 718
0.2 / 0.5 / 31 / 88 / 36 / 104
0.3 / 0.4 / 255 / 728 / 302 / 862
0.3 / 0.6 / 42 / 93 / 50 / 110
0.4 / 0.5 / 351 / 780 / 415 / 923
0.4 / 0.7 / 50 / 90 / 58 / 106
0.5 / 0.6 / 419 / 761 / 496 / 901
0.5 / 0.8 / 51 / 79 / 60 / 93
CONCLUSION AND DISCUSSION
This study shows that there is no difference for the number of sample size at any stage in both adaptive and group sequential designs for any number of stages. It means that the number of stage is not important in a study for cumulative sample size.Freedman formula requires more cumulative sample size than Schoenfeld formula in adaptive designsat different stages.It is easily said that the sample sizecalculated forO’Brien&Fleming’s critical boundaries isless than the sample size calculated forPocock’s critical boundaries for both designs.
When two groups event ratios close to each other for instance 1=0.1 and 2=0.2, the sample size and the event numbersare considerably high. When the amount of difference of two group’s event ratios is high for instance 1=0.1 and 2=0.4, the sample size and the event numbersare relatively low.When two groups event ratiosare high and close to each other (as 1=0.5 and 2=0.6), required the events and the sample sizeare high for both adaptive and group sequential designs.
When all conditions are taken into account, the most important result of this study is that the adaptive designs require less the cumulative sample size and the event numbers than the group sequential designs.Therefore, researchers expect less sample size and events with adaptive designs. In survival analysis for clinical trial, adaptive designs need less sample size than group sequential designs.
REFERENCES
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