Sample Size Calculation inClinical TrialsDr. Bhaswat S. Chakraborty
Contexts of Sample Size Calculation One fundamental question facing the organizers of any clinical trialat the planning stage is “How many patients do we need?” Statistical methods can be used to determine the required numberof patients to meet the trial’s principal scientific objectives and with agiven statistical power. However, such an approach can only be used as a guideline sincepractical matters such as the availability of patients and resourcesand the ethical need to prevent any patient receiving an inferiortreatment must be taken into account.
Statisticfal Methods of Sample SizeCalculation Although practical and ethical issues need to beconsidered, one’s initial reasoning when determining trialsize should focus on the scientific requirements. To this end there is one standard statistical approach,often called power calculations, which can be applied toa wide range of clinical trials. I will introduce the line ofreasoning by using one example. This is followed by a more general description of thestatistical formulae.
General Formula• Say σ, μ2, μ1 and f (α,β) are 1.8, 9.5(mg/ml), 9.0 (mg/ml), and 13,respectively,then,• N = [2 x (1.8)2x 13] / (0.5)2= 337 patients in each arm.N (each arm) =2σ2(μ2 - μ1)2X f (α,β)
Understanding Basics• μ0 and μA• Means under Null & Alternate Hypotheses• σ02and σA2• Variances under Null & Alternate Hypotheses (may be the same)• N0 and NA• Sample Sizes in two groups (may be the same)• H0: Null Hypothesis• μ0 – μA = 0• HA: Alternate Hypothesis• μ0 – μA = δ• Type I Error (α): False +ve• Probability of rejecting a true H0• Type II Error (β): False –ve• Probability of rejecting a true HA• Power (1-β): True +ve• Probability of accepting a true HA
From the previous graph, we have0+Z1-α/2σ√(2/N) = δ–Z1-βσ√(2/N)Upon simplification,N =2 σ2[Z1-α/2 + Z1-β/2]2δ 2
Planning Statistical Analysis:Answer those Five Key Questions1. What is the main purpose of the trial?2. What is the principal measure of patient outcome?3. How will the data be analysed to detect a treatmentdifference?4. What type of results does one anticipate with standardtreatment?5. How small a treatment difference is it important todetect and with what degree of certainty?Stuart Pocock in Clinical Trials, Wiley Int.
Some Examples: Sample Sizes for tTest, Survival & Case Control Studies
Sample Size for a t TestInput variables you will needαThe Type I error probability for a two sided test.nFor independent t-tests n is the number of experimental subjects. For pair test n isthe number of pairs.powerFor independent tests power is probability of correctly rejecting the null hypothesis ofequal population meansδA difference in population meansσFor independent tests σ is the within group standard deviation. For paired designs itis the standard deviation of difference in the response of matched pairs.mFor independent tests m is the ratio of control to experimental patients. m is notdefined for paired studies.
Sample Size for a t Test• A study is being planned with a continuous response variablefrom independent control and experimental subjects with 1control(s) per experimental subject.• In a previous study the response within each subject group wasnormally distributed with standard deviation 20.• SAMPLE SIZE: If the true difference in the experimental andcontrol means is 15, we will need to study 38 experimentalsubjects and 38 control subjects to be able to reject the nullhypothesis that the population means of the experimental andcontrol groups are equal with probability (power) 0.9.• The Type I error probability associated with this test of this nullhypothesis is 0.05.
Sample Size for a 2 Survival TimesInput variables you will needαThe Type I error probability for a two sided test.powerThe probability of correctly rejecting the null hypothesis of equal treatment survivaltimesnThe number of patients who receive the experimental treatment.m1The median survival time on control treatment.m2The median survival time on experimental treatment.RHazard ratio (relative risk) of the control treatment relative to the experimentaltreatment. If the hazard is constant in each group then R=m2 / m1.AAccrual time during which patients are recruited.FAdditional follow-up time after end of recruitment.mRatio of control to experimental patients.
Sample Size for a 2 Survival Times• This is a study with 1 control: 1 experimental subject, an accrualinterval of 60 months, and additional follow-up after the accrualinterval of 12 months.• Prior data indicate that the median survival time on the controltreatment is 45 months.• If the true median survival times on the control and experimentaltreatments are 45 and 60 months, respectively.• SAMPLE SIZE: we need to study 462 experimental subjectsand 462 control subjects to be able to reject the null hypothesisthat the experimental and control survival curves are equal withprobability (power) 80%.• The Type I error probability associated with this test of this nullhypothesis is 0.05.
Sample Size for a Case Control StudiesInput variables you will needαThe Type I error probability for a two sided test.nFor case-control studies n is the number of case patients. (For prospective studies n isthe number of patients receiving the experimental treatment.)powerthe probability of correctly rejecting the null hypothesis that the relative risk (odds ratio)equals 1 given n case patients, m control patients per experimental patient, and a Type Ierror probability α.p0For case-control studies, p0 is the probability of exposure in controls. In prospectivestudies, p0 is the probability of the outcome for a control patient.p1For case-control studies, p1 is the probability of exposure in cases. In prospective studies,p1 is the probability of the outcome in an experimental subject.mRatio of control to experimental subjects.RRelative risk of failure for experimental subjects relative to controls.fFor case-control studies f is the correlation coefficient for exposure between matchedcases and controls.
Sample Size for a Case Control Studies We are planning a study of independent cases and controls with 1control(s) per case. Prior data indicate that the probability of exposure among controls is0.5. SAMPLE SIZE: If the true odds ratio for disease in exposed subjectsrelative to unexposed subjects is 0.6, we will need to study 246 casepatients and 246 control patients to be able to reject the nullhypothesis that this odds ratio equals 1 with probability (power) 0.8. The Type I error probability associated with this test of this nullhypothesis is 0.05. We will use an uncorrected chi-squared statistic to evaluate this nullhypothesis.