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PubMed1987

Sample size for testing differences in proportions for the paired-sample design.

Miettinen (1968, Biometrics 24, 339-352) presented an approximation for power and sample size for testing the differences between proportions in the matched-pair case. Duffy (1984, Biometrics 40, 1005-1015) gave the exact power for this case and showed that Miettinen's approximation tends to slightly overestimate the power or underestimate the sample size necessary for the design power. A simple alternative approximation that is more conservative is presented here. In many cases, the sample size for the independent-sample case provides a conservative approximation for the matched-pair design.

Research Design↗
PubMed1994

Estimating sample sizes for repeated measurement designs.

Formulas for estimating sample sizes that are required to provide specified power for analysis of variance (ANOVA) tests of significance in a two-group repeated measurements design are presented and evaluated. Power and sample size requirements depend on the pattern of treatment effects and the pattern of correlations among the repeated measurements, as well as on parameters common to sample size estimation for cross-sectional comparisons of treatment effects in simple randomized designs. Simplifying assumptions permit generation of these numerous parameter estimates from predictions of the magnitude of the standardized "effect size" at end of trial and the single correlation between the baseline and endpoint measurements. Monte Carlo methods are used to verify the actual power of different tests of significance for treatment effects in repeated measurement designs using sample sizes estimated by the formulas. The sample size implications of different patterns of treatment effects, levels of correlation, and numbers of repeated measurements are evaluated.

Analysis of Variance↗
PubMedDate not supplied

Sample size recalculation using conditional power.

The sample size required to achieve a given power at a prespecified absolute difference in mean response may depend on one or more nuisance parameters, which are usually unknown. Proposed methods for using an internal pilot to recalculate the sample size using estimates of these parameters have been well studied. Most of these methods ignore the fact that data on the parameter of interest from within this internal pilot will contribute towards the value of the final test statistic. We propose a method which involves recalculating the target sample size by computing the number of further observations required to maintain the probability of rejecting the null hypothesis at the end of the study under the prespecified absolute difference in mean response conditional on the data observed so far. We do this within the framework of a two-group error-spending sequential test, modified so as to prevent inflation of the type I error rate.

Breast Neoplasms↗
PubMed2004

Increasing the sample size when the unblinded interim result is promising.

Increasing the sample size based on unblinded interim result may inflate the type I error rate and appropriate statistical adjustments may be needed to control the type I error rate at the nominal level. We briefly review the existing approaches which allow early stopping due to futility, or change the test statistic by using different weights, or adjust the critical value for final test, or enforce rules for sample size recalculation. The implication of early stopping due to futility and a simple modification to the weighted Z-statistic approach are discussed. In this paper, we show that increasing the sample size when the unblinded interim result is promising will not inflate the type I error rate and therefore no statistical adjustment is necessary. The unblinded interim result is considered promising if the conditional power is greater than 50 per cent or equivalently, the sample size increment needed to achieve a desired power does not exceed an upper bound. The actual sample size increment may be determined by important factors such as budget, size of the eligible patient population and competition in the market. The 50 per cent-conditional-power approach is extended to a group sequential trial with one interim analysis where a decision may be made at the interim analysis to stop the trial early due to a convincing treatment benefit, or to increase the sample size if the interim result is not as good as expected. The type I error rate will not be inflated if the sample size may be increased only when the conditional power is greater than 50 per cent. If there are two or more interim analyses in a group sequential trial, our simulation study shows that the type I error rate is also well controlled.

Anti-HIV Agents↗
PubMed1997

Using aspects of study design in sample size estimation.

The basis of sample size calculations is usually needed in protocols for clinical trials and when publishing results in respected journals. Although a large amount of research has been undertaken on sample size estimation for different trial designs, in practice the methods are rarely used. This paper describes some useful theory that has practical relevance.

Clinical Trials as Topic↗
PubMed2005

Sample size computations for PK/PD population models.

We describe an accurate, yet simple and fast sample size computation method for hypothesis testing in population PK/PD studies. We use a first order approximation to the nonlinear mixed effects model and chi-square distributed Wald statistic to compute the minimum sample size to achieve given degree of power in rejecting a null hypothesis in population PK/PD studies. The method is an extension of Rochon's sample size computation method for repeated measurement experiments. We compute sample sizes for PK and PK/PD models with different conditions, and use Monte Carlo simulation to show that the computed sample size retrieves the required power. We also show the effect of different sampling strategies, such as minimal, i.e., as many observations per individual as parameters in the model, and intensive on sample size. The proposed sample size computation method can produce estimates of minimum sample size to achieve the desired power in hypothesis testing in a greatly reduced time than currently available simulation-based methods. The method is rapid and efficient for sample size computation in population PK/PD study using nonlinear mixed effect models. The method is general and can accommodate any type of hierarchical models. Simulation results suggest that intensive sampling allows the reduction of the number of patients enrolled in a clinical study.

Algorithms↗
PubMed2002

Estimating allelic richness: effects of sample size and bottlenecks.

Although differences in sampling intensity can bias comparisons of allelic richness (A) among populations, investigators often fail to correct estimates of A for differences in sample size. Methods that standardize A on the basis of the size of the smallest number of samples in a comparison are preferable to other approaches. Rarefaction and repeated random subsampling provide unbiased estimates of A with the greatest precision and thus provide greatest statistical power to detect differences in variation. Less promising approaches, in terms of bias or precision, include single random subsampling, eliminating very small samples, using sample size as a covariate or extrapolating estimates obtained from small samples to a larger number of individuals.

Alleles↗
PubMed2004

How many do I need? Basic principles of sample size estimation.

BACKGROUND: In conducting randomized trials, formal estimations of sample size are required to ensure that the probability of missing an important difference is small, to reduce unnecessary cost and to reduce wastage. Nevertheless, this aspect of research design often causes confusion for the novice researcher. AIM: This paper attempts to demystify the process of sample size estimation by explaining some of the basic concepts and issues to consider in determining appropriate sample sizes. METHOD: Using a hypothetical two group, randomized trial as an example, we examine each of the basic issues that require consideration in estimating appropriate sample sizes. Issues discussed include: the ethics of randomized trials, the randomized trial, the null hypothesis, effect size, probability, significance level and type I error, and power and type II error. The paper concludes with examples of sample size estimations with varying effect size, power and alpha levels. CONCLUSION: Health care researchers should carefully consider each of the aspects inherent in sample size estimations. Such consideration is essential if care is to be based on sound evidence, which has been collected with due consideration of resource use, clinically important differences and the need to avoid, as far as possible, types I and II errors. If the techniques they employ are not appropriate, researchers run the risk of misinterpreting findings due to inappropriate, unrepresentative and biased samples.

Data Interpretation, Statistical↗
PubMedDate not supplied

Planning and revising the sample size for a trial.

The sample size for a trial depends on the type I and type II error rates and on the minimum relevant clinical difference, all of which are known, and on the anticipated, but unknown, value of a measure of variation for the key response. This measure is the overall response rate when the key response is binomially distributed, or the residual variance in each treatment group when the key response is continuous and normally distributed. Since the true value of the measure is unknown, it must be guessed or estimated from previous trials. We describe approaches to determine an appropriate value for it, both before the trial begins and after it has begun, for use in calculating the final sample size. These approaches differ from previously described 'internal pilot' methods in not requiring unblinding of the treatment assignments in the trial. They preserve the power and do not affect the type I error rate materially. The approaches can be applied to longitudinal studies where the rate of change over time is the response of interest, and to group sequential trials.

Algorithms↗
PubMed1978

A graphical aid for determining sample size when comparing two independent proportions.

Standard sample size calculations for n, the number of observations per group when comparing two independent proportions, P1 and P2, require the specification of four quantities: P1, one of the two proportions of interest; delta = P2 - P1, the smallest difference which it is important to detect; alpha, the significance level; and beta, the chance of failing to detect a difference as large as delta. In terms of these four quantities, the graphical aid is a series of charts showing isographs of sample size for selected values of n ranging from 35 to 500. The isographs, i.e. curves connecting points of equal sample size, are based on the asymptotic arc sine approximation and are plotted on the grid formed by P1 on the abscissa and delta on the ordinate. Eight separate charts are available for different choices of alpha and beta. These charts are especially useful in situations where the feasible sample size is roughly known, in which case the detectable difference, delta, can be read directly from the graph.

Mathematics↗
PubMed2001

Uniform power method for sample size calculation in historical control studies with binary response.

Makuch and Simon gave a sample size calculation formula for historical control (HC) studies that assumed that the observed response rate in the control group is the true response rate. We dropped this assumption and computed the expected power and expected sample size to evaluate the performance of the procedure under the omniscient model. When there is uncertainty in the HC response rate but this uncertainty is not considered, Makuch and Simon's method produces a sample size that gives a considerably lower power than that specified. Even the larger sample size obtained from the randomized design formula and applied to the HC setting does not guarantee the advertised power in the HC setting. We developed a new uniform power method to search for the sample size required for the experimental group to yield an exact power without relying on the estimated HC response rate being perfectly correct. The new method produces the correct uniform predictive power for all permissible response rates. The resulting sample size is closer to the sample size needed for the randomized design than Makuch and Simon's method, especially when there is a small difference in response rates or a limited sample size in the HC group. HC design may be a viable option in clinical trials when the patient selection bias and the outcome evaluation bias can be minimized. However, the common perception of the extra sample size savings is largely unjustified without the strong assumption that the observed HC response rate is equal to the true control response rate. Generally speaking, results from HC studies need to be confirmed by studies with concurrent controls and cannot be used for making definitive decisions.

Bias↗
PubMed2001

Sample size determination for equivalence test using rate ratio of sensitivity and specificity in paired sample data.

Before implementing a new diagnostic test, we may wish to study whether this test is noninferior to a reference test with respect to the sensitivity and/or the specificity. This paper discusses sample size determination for one-sided equivalence (or noninferiority) testing of the rate ratio using paired-sample data. Using large sample theory, this paper derives asymptotic sample size formulae for the required number of subjects giving a desired power 100(1 - beta)% at a specified alpha-level. To evaluate the accuracy of these formulae, this paper considers several test statistics and uses Monte Carlo simulation to estimate the corresponding type I error and power with the given resulting sample sizes in a variety of situations. Finally, this paper notes those situations for which the asymptotic sample size formulae developed here are of limited use and suggests a simple empirical adjustment to alleviate this limitation.

Controlled Clinical Trials as Topic↗
PubMed1998

Estimating sample size for epidemiologic studies: the impact of ignoring exposure measurement uncertainty.

Sample size requirements for epidemiologic studies are usually determined on the basis of the desired level of statistical power. Suppose, however, that one is planning a study in which the participants' true exposure levels are unobservable. Instead, the analysis will be based on an imprecise surrogate measure that differs from true exposure by some non-negligible amount of measurement error. Sample size estimates for tests of association between the surrogate exposure measure and the outcome of interest may be misleading if they are based solely on the anticipated characteristics of the distribution of surrogate measures in the study population. We examine the accuracy of sample size estimates for cohort studies in which a continuous surrogate exposure measure is subject to either classical or Berkson measurement error. In particular, we evaluate the consequences of not adjusting the sample size estimation procedure for tests based on imprecise exposure measurements to account for anticipated differences between the distributions of the true exposure and the surrogate measure in the study population. As expected, failure to adjust for classical measurement error can lead to underestimation of the required sample size. Disregard of Berkson measurement error, however, can result in sample size estimates that exceed the actual number of participants required for tests of association between the outcome and the surrogate exposure measure. We illustrate this Berkson error effect by estimating sample size for a hypothetical cohort study that examines an association between childhood exposure to radioiodine and the development of thyroid neoplasms.

Adolescent↗
PubMed2001

Use of sample size for estimating efficacy of a vaccine against an infectious disease.

OBJECTIVE: To determine the sample size necessary to evaluate the efficacy of a vaccine in a population. PROCEDURE: An equation was coded into a computer spreadsheet to compare the traditional sample size calculation with that needed when evaluating the efficacy of a vaccine applied in a population. RESULTS: The traditional approach used to conservatively estimate sample size necessary to detect a given difference in group proportions potentially greatly underestimates the number of animals needed for vaccine efficacy (VE) trials. In VE trials, it is necessary to estimate the effect of population-level vaccination prior to estimating sample size. In VE trials, as incidence proportion in the population or herd decreases or VE decreases, necessary sample size increases. CONCLUSIONS AND CLINICAL RELEVANCE: In designing a clinical or field trial, such as one to evaluate the efficacy of a vaccine against an infectious disease in a population, one needs to approach sample size calculations in a nontraditional manner. The proportion of the population vaccinated, disease transmission dynamics, and VE will affect the incidence in the nonvaccinated and vaccinated groups and, hence, sample size. Thus, estimation of the effect of the vaccination on the population must be made prior to calculating sample size. Otherwise, sample size and the power to identify VE will be insufficient.

Animal Diseases↗
PubMed1992

Comparing sample size formulae for trials with unbalanced allocation using the logrank test.

This paper compares the sample size formulae given by Schoenfeld, Freedman, Hsieh and Shuster for unbalanced designs. Freedman's formula predicts the highest power for the logrank test when the sample size ratio of the two groups equals the reciprocal of the hazard ratio. The other three formulae predict highest powers when sample sizes in the two groups are equal. Results of Monte Carlo simulations performed for the power of the logrank test with various sample size ratios show that the power curve of the logrank test is almost flat between a sample size ratio of one and a sample size ratio close to the reciprocal of the hazard ratio. An equal sample-size allocation may not maximize the power of the logrank test. Monte Carlo simulations also show that, under an exponential model, when the sample size ratio is toward the reciprocal of the hazard ratio, Freedman's formula predicts more accurate powers. Schoenfeld's formula, however, seems best for predicting powers with equal sample size.

Clinical Trials as Topic↗
PubMed2002

Sample size calculation for a proof of concept study.

Sample size calculation is vital for a confirmatory clinical trial since the regulatory agencies require the probability of making Type I error to be significantly small, usually less than 0.05 or 0.025. However, the importance of the sample size calculation for studies conducted by a pharmaceutical company for internal decision making, e.g., a proof of concept (PoC) study, has not received enough attention. This article introduces a Bayesian method that identifies the information required for planning a PoC and the process of sample size calculation. The results will be presented in terms of the relationships between the regulatory requirements, the probability of reaching the regulatory requirements, the goalpost for PoC, and the sample size used for PoC.

Amyotrophic Lateral Sclerosis↗
PubMed1995

Estimating sample sizes for a two-stage sampling survey of seroprevalence of pseudorabies virus (PRV)-infected swine at a regional level in The Netherlands.

In the European Union, vaccination campaigns against Pseudorabies virus (PRV) in swine have been started to eradicate PRV. Specific sampling designs are needed to monitor PRV seroprevalence at a regional level. This paper demonstrates how sampling theory can be applied to design a disease seroprevalence survey, using PRV as an example. In the spring of 1994, the four regions in the Netherlands covered by the regional Animal Health Services were monitored with respect to PRV seroprevalence. Per region, blood samples from approximately 1400 herds, with two animals per herd, were collected. The sampling design accounted for stratification by fattening pig and sow population within each region. The regional PRV seroprevalence of swine in the Southern region was the highest (24.9%), closely followed by the PRV seroprevalence of swine in the Eastern region (20.5%). These regions have the highest density of swine in the Netherlands. The PRV seroprevalence in the Western and Central region (11.7%) was about half of the seroprevalence in the Southern and Eastern regions; the lowest regional PRV seroprevalence was observed in the Northern region (3.5%). The Northern part also has the lowest pig density. The PRV seroprevalence was approximately two times higher in sows than in fattening pigs.

Animals↗
PubMed2005

A Bayesian approach on sample size calculation for comparing means.

In clinical research, parameters required for sample size calculation are usually unknown. A typical approach is to use estimates from some pilot studies as the true parameters in the calculation. This approach, however, does not take into consideration sampling error. Thus, the resulting sample size could be misleading if the sampling error is substantial. As an alternative, we suggest a Bayesian approach with noninformative prior to reflect the uncertainty of the parameters induced by the sampling error. Based on the informative prior and data from pilot samples, the Bayesian estimators based on appropriate loss functions can be obtained. Then, the traditional sample size calculation procedure can be carried out using the Bayesian estimates instead of the frequentist estimates. The results indicate that the sample size obtained using the Bayesian approach differs from the traditional sample size obtained by a constant inflation factor, which is purely determined by the size of the pilot study. An example is given for illustration purposes.

Aged↗