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Simple sample size calculation for cluster-randomized trials.

BACKGROUND: Cluster-randomized trials, in which health interventions are allocated randomly to intact clusters or communities rather than to individual subjects, are increasingly being used to evaluate disease control strategies both in industrialized and in developing countries. Sample size computations for such trials need to take into account between-cluster variation, but field epidemiologists find it difficult to obtain simple guidance on such procedures. METHODS: In this paper, we provide simple formulae for sample size determination for both unmatched and pair-matched trials. Outcomes considered include rates per person-year, proportions and means. For simplicity, formulae are expressed in terms of the coefficient of variation (SD/mean) of cluster rates, proportions or means. Guidance is also given on the estimation of this value, with or without the use of prior data on between-cluster variation. CASE STUDIES: The methods are illustrated using two case studies: an unmatched trial of the impact of impregnated bednets on child mortality in Kenya, and a pair-matched trial of improved sexually-transmitted disease (STD) treatment services for HIV prevention in Tanzania.

Child, Preschool↗

Sample size, power calculations, and their implications for the cost of thorough studies of drug induced QT interval prolongation.

Regulatory authorities require new drugs to be investigated using a so-called "thorough QT/QTc study" to identify compounds with a potential of influencing cardiac repolarization in man. Presently drafted regulatory consensus requires these studies to be powered for the statistical detection of QTc interval changes as small as 5 ms. Since this translates into a noticeable drug development burden, strategies need to be identified allowing the size and thus the cost of thorough QT/QTc studies to be minimized. This study investigated the influence of QT and RR interval data quality and the precision of heart rate correction on the sample sizes of thorough QT/QTc studies. In 57 healthy subjects (26 women, age range 19-42 years), a total of 4,195 drug-free digital electrocardiograms (ECG) were obtained (65-84 ECGs per subject). All ECG parameters were measured manually using the most accurate approach with reconciliation of measurement differences between different cardiologists and aligning the measurements of corresponding ECG patterns. From the data derived in this measurement process, seven different levels of QT/RR data quality were obtained, ranging from the simplest approach of measuring 3 beats in one ECG lead to the most exact approach. Each of these QT/RR data-sets was processed with eight different heart rate corrections ranging from Bazett and Fridericia corrections to the individual QT/RR regression modelling with optimization of QT/RR curvature. For each combination of data quality and heart rate correction, standard deviation of individual mean QTc values and mean of individual standard deviations of QTc values were calculated and used to derive the size of thorough QT/QTc studies with an 80% power to detect 5 ms QTc changes at the significance level of 0.05. Irrespective of data quality and heart rate corrections, the necessary sample sizes of studies based on between-subject comparisons (e.g., parallel studies) are very substantial requiring >140 subjects per group. However, the required study size may be substantially reduced in investigations based on within-subject comparisons (e.g., crossover studies or studies of several parallel groups each crossing over an active treatment with placebo). While simple measurement approaches with ad-hoc heart rate correction still lead to requirements of >150 subjects, the combination of best data quality with most accurate individualized heart rate correction decreases the variability of QTc measurements in each individual very substantially. In the data of this study, the average of standard deviations of QTc values calculated separately in each individual was only 5.2 ms. Such a variability in QTc data translates to only 18 subjects per study group (e.g., the size of a complete one-group crossover study) to detect 5 ms QTc change with an 80% power. Cost calculations show that by involving the most stringent ECG handling and measurement, the cost of a thorough QT/QTc study may be reduced to approximately 25%-30% of the cost imposed by the simple ECG reading (e.g., three complexes in one lead only).

Adult↗

Independence estimating equations for controlled clinical trials with small sample sizes--interval estimation.

OBJECTIVES: The application of independence estimating equations (IEE) for controlled clinical trials (CCTs) has recently been discussed, and recommendations for its use have been derived for testing hypotheses. The robust estimator of variance has been shown to be liberal for small sample sizes. Therefore a series of modifications has been proposed. In this paper we systematically compare confidence intervals (CIs) proposed in the literature for situations that are common in CCTs. METHODS: Using Monte-Carlo simulation studies, we compared the coverage probabilities of CIs and non-convergence probabilities for the parameters of the mean structure for small samples using modifications of the variance estimator proposed by Mancl and de Rouen [7], Morel et al. [8] and Pan [3]. RESULTS: None of the proposed modifications behave well in each investigated situation. For parallel group designs with repeated measurements and binary response the method proposed by Pan maintains the nominal level. We observed non-convergence of the IEE algorithm in up to 10% of the replicates depending on response probabilities in the treatment groups. For comparing slopes with continuous responses, the approach of Morel et al. can be recommended. CONCLUSIONS: Results of non-convergence probabilities show that IEE should not be used in parallel group designs with binary endpoints and response probabilities close to 0 or 1. Modifications of the robust variance estimator should be used for sample sizes up to 100 clusters for CI estimation.

Algorithms↗

Establishing equivalence of two treatments and sample size requirements in matched-pairs design.

Statistical methods for testing the null hypothesis of a nonzero difference between two treatments and the sample size determination for matched-pairs studies are investigated. A Wald-type test proposed by Lu and Bean (1995, Statistics in Medicine 14, 1831-1839) is anticonservative, i.e., its false positive error rate is greater than specified. Score method and normal deviate based on a restricted maximum likelihood estimation are presented. These two test statistics are shown to be algebraically equal. Their actual type I error probabilities are satisfactorily close to a nominal level. Numerical examinations demonstrate that the sample size formulas using these alternative methods are reasonable while that by Lu and Bean is not. The efficiency of matching in equivalence studies is positively related to intercorrelation or the kappa coefficient of agreements. We recommend the score or ML methods to establish equivalence of two treatments for individually matched samples.

Autopsy↗

Sample size determination based on Fisher's Exact Test for use in 2 x 2 comparative trials with low event rates.

A collection of sample size tables are presented for designing comparative trials when the event rates p1 and p2 are low. The tables are based on exact power calculations for Fisher's Exact Test. Both one-sided and two-sided alternative hypotheses are considered. A comparison is made between these sample sizes and those obtained by using popular asymptotic approximations.

Clinical Trials as Topic↗

Comparison of several regression procedures for method comparison studies and determination of sample sizes. Application of linear regression procedures for method comparison studies in Clinical Chemistry, Part II.

In part I of this series (H. Passing & W. Bablok (1983), J. Clin. Chem. Clin. Biochem. 21, 709-720) we described a new biometrical procedure for the evaluation of method comparison studies. In part II we now discuss its properties and compare them with those of other established procedures by means of a simulation study. We demonstrate that the reliability of the results not only depends on the sample size but also on the sampling distribution, the precision of the methods, and the concentration range covered by the samples. Linear regression and principal component procedures are either inadequate or not as reliable as our new procedure. The appropriate sample size is discussed and recommendations are given.

Chemistry, Clinical↗

Sample size and power for McNemar's test with clustered data.

McNemar's test is used to compare the distribution of two paired binary random variables. When the data are clustered adjustment is needed to ensure that it is still a valid test. This article presents two approximations for calculating the power and sample size for the adjusted McNemar's test for clustered data, working with a particular adjustment. A simulation study is conducted to demonstrate the accuracy of these approximations. The method is also applied to the design of a study involving positron emission tomography in detecting metastatic colorectal cancer and sensitivity of sample size computations to the design parameters are explored in this context.

Cluster Analysis↗

The effect of sample size and MLP architecture on Bayesian learning for cancer prognosis--a case study.

In this paper we investigate the independent effects of training sample size and multilayer perceptron (MLP) architecture on Bayesian learning to build prognostic models for metastatic breast cancer. We trained two types of Bayesian neural networks on a data set of 1477 metastatic breast cancer patients followed at the Institut Curie using disjoint training sets of sizes k = 50, 100, 200, 300, and 450. The learning performance as measured by an expected loss appeared independent of the two architectures modelling the log hazard function under either proportional or non proportional hazard assumptions, thus indicating that no other sources of nonlinearity besides interactions are present. We found a performance breakdown at k = 50, and no sample size effect for k > or = 100.

Bayes Theorem↗

The quest for "power": contradictory hypotheses and inflated sample sizes.

To have the "power" of avoiding undersized clinical trials, the customary statistical strategy used in the past few decades is aimed at rejecting both a null stochastic hypothesis and a contradictory alternative hypothesis. This approach gives a trial the "power" to confirm the "insignificance" of differences much smaller than the large value of delta desired in trials done to show efficacy. In many instances, however, a prime problem is that the current "double-significance" approach produces sample sizes 2-3 times larger than needed for stochastic confirmation of large differences (> or =delta). The inflated sample sizes and consequent problems can be avoided if a realistic value for delta is chosen and maintained thereafter, and if an adequate "capacity" is calculated for "single significance."

Bias↗

Sample size calculation for multicenter randomized trial: taking the center effect into account.

In multicenter trials, data from the same center are more similar than those from different centers. These similarities induce a correlation between data, known as the center effect, which is assessed by the intraclass correlation coefficient (ICC). Here, we derive a sample size formula for continuous data that takes into account this center effect. Our analytical developments lead to an elementary formula different from the classical one by a (1-rho) factor, where rho is the ICC. This work allows for adjusting and reducing the sample size according to the magnitude of the center effect and leads to a better consistency in the conduct of multicenter randomized trials.

Humans↗

Sample size requirements for stratified prospective studies with null hypothesis of non-unity relative risk using the score test.

The standard test of the null hypothesis of unity of the relative risk seeks to determine if two treatments differ. It does not apply when the requirement is either to establish the equivalence of two treatments or to determine whether the relative risk is less than a specified value other than one. This paper presents the asymptotic power function of the score test for the null hypothesis of a specified value of a common relative risk for stratified prospective studies and proposes an approximate formula for the sample size required for a specific power of the test. One can obtain a sample size formula for stratified studies with the standard null hypothesis of unity relative risk as a special case of this formula.

Child, Preschool↗

Estimating sample size for continuous outcomes, comparing more than two parallel groups with unequal sizes.

This paper contains a short generalization of a known method for sample size determination in the case of more than two parallel groups. The term 'set of allocation ratios' corresponding to the allocation ratio from the two-group design is defined. A formula using these ratios to determine the non-centrality parameter of the F distribution is deduced. It is shown that in case of more than two groups, equal group numbers does not constitute an optimal design. Two worked examples are presented.

Analysis of Variance↗

A two-stage sample size recalculation procedure for placebo- and active-controlled non-inferiority trials.

Many non-inferiority trials of a test treatment versus an active control may also, if ethical, incorporate a placebo arm. Inclusion of a placebo arm enables a direct assessment of assay sensitivity. It also allows construction of a non-inferiority test that avoids the problematic specification of an absolute non-inferiority margin, and instead evaluates whether the test treatment preserves a pre-specified proportion of the effect of the active control over placebo. We describe a two-stage procedure for sample size recalculation in such a setting that maintains the desired power more closely than a fixed sample approach when the magnitude of the effect of the active control differs from that anticipated. We derive an allocation rule for randomization under which the procedure preserves the type I error rate, and show that this coincides with that previously presented for optimal allocation of the sample size among the three treatment arms.

Computer Simulation↗

Signal optimisation in cw-laser crossed-beam photothermal spectrometry: influence of the chopping frequency, sample size and flow rate.

Optimisation of the optical design for cw-laser crossed-beam thermal lens spectrometry in infinite and finite samples has been investigated using different excitation beam waists and various lens combinations. The characteristics of the photothermal signal depending on the position of the sample with respect to the probe beam waist, the chopping frequency, the sample size and the flow rate have been considered. Depending on the irradiation duration, the size of the thermal element at the measurement time can be much greater than the waist of the excitation beam. As a result, the optimum sample position is closely related to the probe beam to thermal element size ratio and therefore depends on the chopping frequency and of the sample size. At low frequencies, the size of the thermal element is almost independent of the degree of focusing of the excitation beam because a smaller beam waist induces a faster thermal expansion. As a result, the amplitude of the optimum signal does not depend on the waist of the excitation beam. In contrast, at high frequency, the size of the thermal element remains closer to the size of the excitation beam and the signal is inversely proportional to the waist of the excitation beam as previously demonstrated under pulsed-laser excitation. Moreover, at moderate flow velocities, the signal is significantly enhanced because the negative effect produced by the displacement of the thermal element across the probe beam axis is more than compensated by a decrease of the effective thermal time constant due to radial mixing.

Lasers↗

Sample size calculation should be performed for design accuracy in diagnostic test studies.

BACKGROUND AND OBJECTIVES: Guidelines for conducting studies and reading medical literature on diagnostic tests have been published: Requirements for the selection of cases and controls, and for ensuring a correct reference standard are now clarified. Our objective was to provide tables for sample size determination in this context. STUDY DESIGN AND SETTING: In the usual situation, where the prevalence Prev of the disease of interest is <0.50, one first determines the minimal number Ncases of cases required to ensure a given precision of the sensitivity estimate. Computations are based on the binomial distribution, for user-specified type I and type II error levels. The minimal number N(controls) of controls is then derived so as to allow for representativeness of the study population, according to Ncontrols=Ncases [(1-Prev)/Prev]. RESULTS: Tables give the values of Ncases corresponding to expected sensitivities from 0.60 to 0.99, acceptable lower 95% confidence limits from 0.50 to 0.98, and 5% probability of the estimated lower confidence limit being lower than the acceptable level. CONCLUSION: When designing diagnostic test studies, sample size calculations should be performed in order to guarantee the design accuracy.

Biomedical Research↗

Sample size estimation in occupational mortality studies with use of confidence interval theory.

In occupational epidemiology, the standardized mortality ratio is firmly entrenched as an analytic tool. Sample size determinations have usually been made in a hypothesis testing framework. However, since interval estimation is the usual goal of analysis, it may be more appropriate to use confidence interval theory to estimate the required sample size. This paper gives the method and some results for determining the required "expected number" of deaths, based on specifications concerning the confidence interval for the true standardized mortality ratio.

Epidemiologic Methods↗

Estimation of a parameter and its exact confidence interval following sequential sample size reestimation trials.

For confirmatory trials of regulatory decision making, it is important that adaptive designs under consideration provide inference with the correct nominal level, as well as unbiased estimates, and confidence intervals for the treatment comparisons in the actual trials. However, naive point estimate and its confidence interval are often biased in adaptive sequential designs. We develop a new procedure for estimation following a test from a sample size reestimation design. The method for obtaining an exact confidence interval and point estimate is based on a general distribution property of a pivot function of the Self-designing group sequential clinical trial by Shen and Fisher (1999, Biometrics55, 190-197). A modified estimate is proposed to explicitly account for futility stopping boundary with reduced bias when block sizes are small. The proposed estimates are shown to be consistent. The computation of the estimates is straightforward. We also provide a modified weight function to improve the power of the test. Extensive simulation studies show that the exact confidence intervals have accurate nominal probability of coverage, and the proposed point estimates are nearly unbiased with practical sample sizes.

Biometry↗

Relationship between sample size and the definition of equivalence in non-inferiority drug studies.

Statistical testing of clinical trial data leads to acceptance of a hypothesis if a test of the opposite (null) hypothesis (H0) fails to reach a critical probability value. The usual aim is to demonstrate that a new treatment is superior to a comparator, whence H0 is that the two treatments are the same. By contrast, in studies designed to show that a new treatment is equivalent to an existing therapy, the same principle is satisfied by an amended null hypothesis, that the treatments differ by more than a defined amount. This reversal entails subtle but important logical and practical problems which affect particularly the calculation of sample size. The choice of the limits used to define equivalence is critical to the calculation of sample size in a manner not previously discussed, and in the interpretation of data in relation to the probability of Type I and Type II errors. Investigators, regulatory bodies and institutional ethics committees must ensure that the range of values chosen to indicate equivalence is clinically appropriate and be aware of the effect of this decision on possible errors in accepting or rejecting H0.

Clinical Trials as Topic↗