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In this two-part series, the main designs in clinical research are considered and, using examples, sample size calculation is explained. The information required in order to calculate the sample size for clinical research is highlighted to enable researchers to spend their time most efficiently with statisticians.
The necessary number of subjects to be included in a clinical trial depends on different factors. Exemplified by clinical trials on gastroduodenal tolerability of NSAID, we show how the sample size increases when the significance level decreases, the detection level increases, or the clinically relevant difference decreases. The sample size also increases when the trial design is changed from a cross-over to a parallel-group design. It is concluded from an ethical and statistical point of view that the sample size must be calculated when planning a study.
A new sample size formula is derived herein for matched-pair case-control studies. This result is based on an unconditional approach to the use of McNemar's statistic. We contrast this method for determining sample size and power with Schlesselman's conditional approach and compare the results numerically in a Monte Carlo study.
Sample size formulas are developed to estimate the repeatability and reproducibility standard deviations (Sr and S(R)) such that the actual error in (Sr and S(R)) relative to their respective true values, sigmar and sigmaR, are at predefined levels. The statistical consequences associated with AOAC INTERNATIONAL required sample size to validate an analytical method are discussed. In addition, formulas to estimate the uncertainties of (Sr and S(R)) were derived and are provided as supporting documentation. Formula for the Number of Replicates Required for a Specified Margin of Relative Error in the Estimate of the Repeatability Standard Deviation.
The number of abnormal glomeruli present in a renal biopsy can be viewed as a binomial distribution. Recognition of this fact permits a quantitative assessment of the effect of biopsy sample size in renal biopsy interpretation. If the percent of glomerular involvement in a biopsy is used to determine the severity of a focal glomerular lesion, a small biopsy sample size will lead to considerable misclassification of disease severity. In addition, a small biopsy sample size will make the exclusion of focal disease difficult.
Category-based induction involves making decisions about some member(s) of a category based on information concerning other category members. Recent studies indicate that although adults make use of information concerning sample size (larger samples are a stronger basis of inference than smaller samples) and sample diversity (more diverse samples are better than more homogeneous samples) when making category-based inductive judgments, children do not do so until age 8 or 9 and even then to only a limited degree. This research however, was conducted at the superordinate level of categorization, and it is unclear if general difficulty with this category level may have masked children's ability to use size and diversity, or if these results represent a more entrenched conceptual difficulty in using this information. We therefore conducted three studies that investigate both 8- and 9-year-olds' and adults' ability to use sample size and diversity within basic level categories. Our results indicate that children's difficulty with this information is independent of category level, and may be based on preferences for other strategies concerning category membership and perceptual similarity.
Dry air exercise challenges are frequently used to screen medications that have potential utility in the management of exercise-induced bronchoconstriction (EIB). The purpose of this study was to determine the reproducibility of three outcome measurements made using such challenges, and sample size requirements for drug evaluation studies based on these outcomes. Forty adult subjects with asthma, who tested positively on a screening exercise challenge, were subjected to two further identical challenges, separated by 1 to >35 days. Outcome measurements included the maximum per cent fall in forced expiratory volume in one second (FEV1), after exercise (% fallmax), and the area under the per cent fall in FEV1/time curve for 30 min (AUC30) and 60 min (AUC60) after exercise. The reproducibility of these outcomes, as assessed by intraclass correlation coefficients was 0.72, 0.53 and 0.35 for % fallmax, AUC30 and AUC60 measurements, respectively. The sample size requirements to demonstrate an attenuation of EIB equivalent to a 50% reduction in % fallmax was 9, 14 and 19 subjects for the % fallmax, AUC30 and AUC60 responses, respectively (90% power). It is concluded that the maximum percentage fall in forced expiratory volume in one second has greater reproducibility and results in greater power in clinical trials than area under the curve measurements. Sample size calculation curves are provided which may be used in study design and interpretation of published studies.
BACKGROUND: When the efficacy of a treatment in a randomized controlled trial is required for multiple primary endpoints, trial design and analysis differ from trial requiring efficacy in only one of the multiple endpoints. METHODS: We consider a two-arm clinical trial requiring efficacy analysis for multiple primary endpoints, formulating the appropriate null and alternative hypotheses for the test of treatment efficacy. We study the significance level/statistical power of an intersection-union test (IUT) in this situation. We compare IUT with the intuitive approach (selecting the maximum sample size over those obtained from testing individual primary endpoints one by one) for determination of sample size. RESULTS: The proposed IUT reserves the same Type I error rate as shared by all endpoint-specific tests. The statistical power of the proposed IUT is no more than the minimum from the individual tests. The maximum sample size from multiple endpoint-specific tests is often inadequate for the test of treatment efficacy, especially when the standardized effect sizes are similar. Finally, the IUT can be applied to Alzheimer's disease treatment trials in which two primary endpoints are typically used. CONCLUSIONS: The IUT is a valid method for use in the design and analysis of clinical trials requiring efficacy at multiple primary endpoints.
This note explores the effect of within-strain sample sizes on the correlations between a phenotype and a molecular-genetic marker in a battery of inbred strains. It is shown that the maximum correlation possible between a molecular marker and a behavioral or neuronal phenotype equals the additive-genetic correlation. How close the strain correlation will approach the additive-genetic correlation depends only on heritability and within-strain sample sizes. The equations derived can be used to optimize designs of studies attempting to localize Quantitative Trait Loci utilizing Recombinant Inbred Strains, provided information about the heritability of the character under study is available.
In nonrandomized trials, patients are not randomly assigned to treatment groups with equal probability. Instead, the probability of assignment varies from patient to patient depending on patients baseline covariates. This often results in a non-comparable treatment groups due to treatment imbalance. As a result, the United States Food and Drug Administration (FDA) recommended that the method of propensity score analysis be employed to overcome this problem. In this note, a formula for sample size calculation is developed based on a proposed weighted Mantel-Haenszel test on the strata defined by the propensity score analysis. It was shown that the sample size formula derived by Nam (1998) based on the test statistic proposed by Gart (1985) is a special case of the sample size formula derived in this note.
We examine the asymptotic properties of the Tarone and Ware and Harrington and Fleming classes of test statistics under alternative hypotheses when there are comparisons between more than two survival distributions in the presence of arbitrary right censoring. When we assume equal censoring distributions across treatment groups and proportional hazards, we derive the sample size formula for testing the equality of k > or = 2 survival distributions using the logrank test. This work extends Schoenfeld's derivation for comparing two survival distributions and also generalizes the results of Makuch and Simon. We also derive the sample size formula for testing monotone dose-response using Tarone's trend test. We then investigate the practicality of the formula in various situations by presenting empirical power with use of Monte Carlo simulations. In addition, with stratification present, we derive the sample size formula for the stratified logrank test, which is an extension of Palta and Amini.
The purpose of this review is to describe the statistical methods available to determine sample size and power analysis in clinical trials. The information was obtained from standard textbooks and personal experience. Equations are provided for the calculations and suggestions are made for the use of power tables. It is concluded that sample size calculations and power analysis can be performed with the information provided and that the validity of clinical investigation would be improved by greater use of such analyses.
A cancer prevention trial may seek to test the effectiveness of an intervention in disease-free individuals, but the study population may include subjects with preexisting (but undiagnosed) disease. If sample size calculations assume all events are incident cases, the study will have less power than anticipated because preexisting cases cannot be expected to benefit from the intervention. Sample size can be increased appropriately by using revised event rates that include both preexisting and incident cases. These rates incorporate screening parameters and are applicable to the situation where subjects are screened before randomization. A simple cost model is given that permits examination of the tradeoffs involved in prescreening subjects versus increasing total sample size.
In clinical trials, one of the main questions that is being asked is how many additional observations, if any, are needed beyond those originally planned. In a two-treatment double-blind clinical experiment, one is interested in testing the null hypothesis of equality of the means against one-sided alternative when the common variance sigma2 is unknown. We wish to determine the required total sample size when the error probabilities alpha and beta are specified at a predetermined alternative. Shih provided a two-stage procedure which is an extension of Stein's one-sample procedure, assuming normal response. He estimates sigma2 by the method of maximum likelihood via the EM algorithm and carries out a simulation study in order to evaluate the effective level of significance and the power. The author proposed a closed-form estimator for sigma2 and showed analytically that the difference between the effective and nominal levels of significance is negligible and that the power exceeds 1-beta when the initial sample size is large. Here we consider responses from arbitrary distributions in which the mean and the variance are not functionally related and show that when the initial sample size is large, the conclusions drawn previously by the author still hold. The effective coverage probability of a fixed-width interval is also evaluated. Proofs of certain assertions are deferred to the Appendix.
A general model is formulated that allows for time-dependent dropout and event rates in the determination of sample sizes for long-term medical trials when a therapy group and a control group are to be compared. The need for time-dependent event (dropout) rate is illustrated by using the Framingham Heart Study Mortality data to estimate sample size for the NHLBI (National Heart, Lung, and Blood Institute) Multiple Risk Factor Intervention Trial (MRFIT). A further generalization of the model allows for participants who drop out of the control group (e.g., because of therapeutic measures prescribed by their own physicians) to return (or not return) to the control group at a later date. The question of unequal sample sizes for the therapy and the control groups is also discussed.
This paper discusses a simple method to compute sample sizes for testing interactions in analysis of variance. It uses orthogonal contrasts to determine the power. For the 2 X 2 table this gives a single degree of freedom contrast for consideration. For larger designs, one must consider a set of contrasts. The use of Bonferroni or Scheffé critical values is discussed. This method allows the user to compute easily a sample size without recourse to the non-central F distribution.
INTRODUCTION: Detection of change during bone mineral density (BMD) monitoring is affected by test precision. The International Society of Clinical Densitometry (ISCD) recommends that each center determine precision error using repeat measurements in 30 subjects (or an equivalent method providing 30 degrees of freedom). METHODS: We hypothesized that this sample size may be too small for a robust precision estimate, which could affect the performance of BMD monitoring in clinical practice. Replicate measurements of the spine and total hip (198 spine and 193 hip scan pairs) were obtained (interval 6+/-5 days). The sample was randomly divided into six groups of 30 patients each. Root mean square standard deviation (RMS-SD in g/cm(2)) and coefficient of variation (RMS-CV in %) precision errors and corresponding 95% least significant change (LSC) were calculated for each group and the pooled sample. LSC cutoffs were applied to 1,420 individuals from the Manitoba Bone Density Program who had follow-up measurements on the same instrument (interval 21+/-9 months). While the pooled spine RMS-SD was 0.017 and pooled hip RMS-SD was 0.009 g/cm(2), sample sizes of 30 gave a range of RMS-SD point estimates from 0.012 to 0.021 for the spine and from 0.008 to 0.012 for the hip. RESULTS: When the respective LSC cutoffs were applied to the 1,420 follow-up scan pairs, the fraction of patients categorized with significant change in the spine varied from 20.7% to 46.0%; four of the six LSCs based upon 30 subjects gave fractions significantly different from the pooled LSC of 30.7%. Significant change fractions for the hip varied from 31.1% to 51.1%; two of the six LSCs based upon 30 subjects gave fractions significantly different from the pooled LSC of 40.1%. Similar results were obtained using relative precision errors. CONCLUSION: BMD precision studies using a sample size of 30 are insufficient to reliably characterize precision error or change during clinical monitoring.
Recent development of multi-unit recording techniques such as optical recording and multi-electrode arrays makes it possible to record neuronal activities from tens or hundreds of neurons simultaneously. To analyze functional connections between these neurons, cross-correlation analysis has been most commonly applied to the hundreds to thousands of pairs of these neurons. However, conventional cross-correlation data needs statistical tests for significance especially when the sample size of recorded spike trains is small. Here, a multiple hypergeometric model based on a transformation of the cross-correlogram data to a 2 x J table has been suggested. The exact p value for significance can be obtained by the generalized Fisher's method with small sample size and a cross-correlation coefficient for the strength of cross-correlation can be obtained based on the R-square analogue for nominal data. For large sample size, chi 2 test can be applied based on the same transformation. Examples of real spike train data set and simulation show that the methods are applicable to the data of multi-unit activity with only tens of spikes. These methods are especially useful when thousands of cross-correlograms need to be screened quickly and automatically.