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PubMed1986

Sample size for individually matched case-control studies.

The standard formulas used to calculate sample size for an individually matched case-control study assume a constant probability of exposure throughout the pool of possible controls. We propose new formulas that allow for heterogeneity in the probability of exposure among controls in different matched sets. Since matching factors are suspected of being confounders, they are expected to divide the total population into subgroups with different proportions exposed. Thus, the assumption of homogeneity of exposure among controls, made by the currently used formulas, is inconsistent with the assumptions used to design a matched study. The proposed formulas avoid this inconsistency. We present an example to illustrate how heterogeneity can affect the required sample size.

Biometry↗
PubMed1995

Misclassification of a prognostic dichotomous variable: sample size and parameter estimate adjustment.

Under general conditions, Lagakos showed that for an explanatory variable observed with error, the asymptotic relative efficiency (ARE) when using the observed rather than the true values in linear models, logistic models and proportional hazards models for survival is the square of the correlation between the true and observed variables. The result is useful for sample size adjustment when this correlation is estimable. Often, one cannot observe correct values of the explanatory variable under any circumstances. We show, however, that under the models considered by Lagakos for a dichotomous explanatory variable, the ARE equals the kappa statistic in a read-reread protocol. Consequently, one need not know 'truth' in this situation to estimate the ARE and to adjust sample size to maintain desired power; divide the estimated sample size obtained with the assumption of no measurement error by the consistent estimate of the kappa statistic (which is unlikely to be zero or negative). We then develop heuristically an adjusted estimate of the beta parameter in a proportional hazards survival model. The work was motivated by analyses of the Childhood Brain Tumour Consortium database. Examples from this database illustrate the method.

Brain Neoplasms↗
PubMed2005

False discovery rate, sensitivity and sample size for microarray studies.

MOTIVATION: In microarray data studies most researchers are keenly aware of the potentially high rate of false positives and the need to control it. One key statistical shift is the move away from the well-known P-value to false discovery rate (FDR). Less discussion perhaps has been spent on the sensitivity or the associated false negative rate (FNR). The purpose of this paper is to explain in simple ways why the shift from P-value to FDR for statistical assessment of microarray data is necessary, to elucidate the determining factors of FDR and, for a two-sample comparative study, to discuss its control via sample size at the design stage. RESULTS: We use a mixture model, involving differentially expressed (DE) and non-DE genes, that captures the most common problem of finding DE genes. Factors determining FDR are (1) the proportion of truly differentially expressed genes, (2) the distribution of the true differences, (3) measurement variability and (4) sample size. Many current small microarray studies are plagued with large FDR, but controlling FDR alone can lead to unacceptably large FNR. In evaluating a design of a microarray study, sensitivity or FNR curves should be computed routinely together with FDR curves. Under certain assumptions, the FDR and FNR curves coincide, thus simplifying the choice of sample size for controlling the FDR and FNR jointly.

Algorithms↗
PubMed2005

Do polymorphic loci require large sample sizes to estimate genetic distances?

The coefficient of variation of estimates of three genetic distances (standard genetic distance of Nei, chord distance, FST) was examined with computer simulation to determine if large samples (per population) are necessary to precisely estimate genetic distances at loci with high levels of polymorphism. These simulations showed that loci with high mutation rates produce estimates of genetic distance with lower coefficients of variation than loci with lower mutation rates--without requiring larger sample sizes from each population. In addition, the rate at which increasing sample sizes decreases the coefficient of variation of estimates of genetic distances was shown to be approximately determined by the value of FST between the populations being sampled. When FST was greater than 0.05, sampling fewer than 20 individuals (per population) should be sufficient. When FST was less than 0.01, sampling 100 individuals (per population) or more will be useful.

Alleles↗
PubMed1997

Linkage analysis of complex traits using affected sibpairs: effects of single-locus approximations on estimates of the required sample size.

We investigated the power of the affected sibpair method for detecting a disease locus when the disease is inherited through two bi-allelic loci. The power was computed for all possible values of the gene frequencies and penetrances that lead to a given population prevalence and a given sibling relative risk. A method to generate rapidly all possible models that give a specific population prevalence and relative risk is provided. We applied it to the case of a two-locus disease with a prevalence of 10% and a low sibling relative risk of 1.5. For this particular example, regardless of the true underlying model, a sample size (N = 450 for alpha = 0.05, N = 1,500 for alpha = 0.0001) may be determined such that one would expect enough power (0.80) to detect at least one of the two disease genes. In addition to the general case, we examined a special class of models in which the marginal penetrances at each locus are either recessive or dominant. In this instance, the gene frequencies were excellent predictors of the power afforded by a particular sample size. These methods have been implemented in a C program called SIBPOWER which is freely available from the first author. With this program, investigators can perform their own power calculations for any two-locus model of their choice thus avoiding the need to use single-locus approximations that may grossly underestimate the necessary sample size.

Gene Frequency↗
PubMed1995

Detection of genotype-environment interaction in case-control studies of birth defects: how big a sample size?

Detecting interactions between risk factors in case-control studies of birth defects and other conditions usually requires increasing the sample size beyond that needed to detect marginal effects. A special case of such interaction is genotype-environment interaction in which the effects of an exposure on disease risk are modified by genetic susceptibility. When case-control studies are designed to detect marginal effects of an exposure (i.e., in the whole population), under many plausible interaction schemes, no additional case and control subjects are needed to detect genotype-environment interaction. On the contrary, inclusion of genotypic information generally can improve the statistical power of the original study. Using the example of oral clefts, maternal cigarette smoking, and genetic variation at the transforming growth factor alpha gene, we illustrate sample size and power issues in designing case-control studies when prior information is available on both the marginal effects of the exposure and the genetic factor.

Case-Control Studies↗
PubMed2003

Bayesian techniques for sample size determination in clinical trials: a short review.

The aim of this paper is to review some key techniques of Bayesian methods of sample size determination. The approach is to cover a small number of simple problems, such as estimating the mean of a normal distribution. The methods considered are in two groups: inferential and decision theoretic. In the inferential Bayesian methods of sample size determination, we are solely concerned with the inference about the parameter(s) of interest. The fully Bayesian or decision theoretic approach treats the problem as a decision problem and employs a loss or utility function.

Bayes Theorem↗
PubMed1988

Sample size formulae for intervention studies with the cluster as unit of randomization.

This paper presents sample size formulae for both continuous and dichotomous endpoints obtained from intervention studies that use the cluster as the unit of randomization. The formulae provide the required number of clusters or the required number of individuals per cluster when the other number is given. The proposed formulae derive from Student's t-test with use of cluster summary measures and a variance that consists of within and between cluster components. Power contours are provided to help in the design of intervention studies that use cluster randomization. Sample size formulae for designs with and without stratification of clusters appear separately.

Cardiovascular Diseases↗
PubMedDate not supplied

Monitoring the impact of Bt maize on butterflies in the field: estimation of required sample sizes.

The monitoring of genetically modified organisms (GMOs) after deliberate release is important in order to assess and evaluate possible environmental effects. Concerns have been raised that the transgenic crop, Bt maize, may affect butterflies occurring in field margins. Therefore, a monitoring of butterflies was suggested accompanying the commercial cultivation of Bt maize. In this study, baseline data on the butterfly species and their abundance in maize field margins is presented together with implications for butterfly monitoring. The study was conducted in Bavaria, South Germany, between 2000-2002. A total of 33 butterfly species was recorded in field margins. A small number of species dominated the community, and butterflies observed were mostly common species. Observation duration was the most important factor influencing the monitoring results. Field margin size affected the butterfly abundance, and habitat diversity had a tendency to influence species richness. Sample size and statistical power analyses indicated that a sample size in the range of 75 to 150 field margins for treatment (transgenic maize) and control (conventional maize) would detect (power of 80%) effects larger than 15% in species richness and the butterfly abundance pooled across species. However, a much higher number of field margins must be sampled in order to achieve a higher statistical power, to detect smaller effects, and to monitor single butterfly species.

Agriculture↗
PubMed2002

Statistics review 4: sample size calculations.

The present review introduces the notion of statistical power and the hazard of under-powered studies. The problem of how to calculate an ideal sample size is also discussed within the context of factors that affect power, and specific methods for the calculation of sample size are presented for two common scenarios, along with extensions to the simplest case.

Clinical Trials as Topic↗
PubMed2004

[Sample size determination given data of preliminary experiment for student's t-test, ANOVA and Tukey's multiple comparison].

The purpose of this article is to calculate the probability that a null hypothesis is rejected using data of preliminary experiment, and to determine an appropriate sample size based on that probability. The procedure to calculate that probability is as follows: (a)generate parameters from the posterior distribution of parameters given data of preliminary experiment; (b) generate test statistics from the conditional distribution given those parameters; (c) count how many of the generated test statistics exceed a critical value. Once the probability of rejecting null hypothesis is estimated, researchers can decide sample size corresponding to the desirable probability. Examples of the procedure are provided and compared with the traditional methods in the case of Student's t-test.

Analysis of Variance↗
PubMed2000

Penny-wise and pound-foolish: the impact of measurement error on sample size requirements in clinical trials.

BACKGROUND: Clinical research studies must compensate for measurement error by increasing the number of subjects that are studied, thereby increasing the financial costs of research and exposing greater numbers of subjects to study risks. In this article, we model the relationship between reliability and sample-size requirements and consider the potential tangible cost savings resulting from the decreased number of subjects needed when reliability of raters is improved or multiple ratings are used. METHODS: Standard methods are used to model reliability based on the intraclass correlation coefficient (R) and to perform power calculations. The impact of multiple raters on reliability for a given baseline level of reliability is modeled according to the Spearman Brown formula. RESULTS: Our models demonstrate that meaningful reductions in sample size requirements are gained from improvements in reliability. For example, improving reliability from R = .7 to R = .9 will decreases sample size requirements by 22%. Reliability is improved by training and by the use of the mean of multiple ratings. For example, if the reliability of a single rating is 0.7, the reliability of the mean of two ratings will be 0.8. CONCLUSIONS: The costs to improve reliability either through rater training efforts or use of the mean of multiple ratings is cost effective because of the consequent reduction in number of subjects needed. Efforts to improve reliability and thus reduce subject requirements in a study also may lead to fewer patients bearing the burden of research participation and to a shortening of the duration of studies.

Clinical Trials as Topic↗
PubMed2005

Bayesian sample-size determination for inference on two binomial populations with no gold standard classifier.

We consider the impact of test properties on the required sample size for the Bayesian design problem for comparing two proportions with error-prone data. Specifically, we examine four cases: a single diagnostic test and two independent diagnostic tests, both when the test properties are identical across populations and when they differ. Interval-based and moment-based sample-size determination criteria are contrasted using Monte Carlo simulation methods. We consider an application in which Strongyloides infections are compared in two populations.

Animals↗
PubMed1982

Sample size for short-term trials of antihypertensive drugs.

1 Controlled trials of antihypertensive drugs published in the British Journal of Clinical Pharmacology during 1979 and 1980 were examined. Studies comparing two or more active drugs or dosage regimens nearly always failed to separate and treatments significantly. The sample size (mean 19 patients) and power of these studies were too low. 2 When planning such studies the aims should be a power of at least 0.8; significance 0.05 or less; and to detect a difference between treatments of 10/5 mmHg, or 6.7 mmHg mean arterial pressure (MAP). The sample size needed can be derived readily from a nomogram if the standard deviation of differences (SDD) between BP measurements under trial conditions is known. 3 In five studies the SDDs were fairly constant despite different observers, patient groups and measuring devices, at approximately 14 mmHg systolic, 9 mmHg diastolic, and 9 mmHg MAP. Use of three BP measurements at each visit reduced the SDD by about 1 mmHg, and would reduce the sample size required by about 20%. Replicate BP measurements at separate visits would be expected to have a larger effect on the power of the study. 4 Published studies with negative results should give an estimate of the power of the study.

Antihypertensive Agents↗
PubMed1994

Sample size considerations for assessing individual bioequivalence based on the method of tolerance intervals.

This is the consideration of sample sizes for assessing individual bioequivalence based on the use of tolerance intervals. The sample size procedures discussed include a direct distribution-free method, indirect parametric method and a direct parametric method. Design considerations are discussed based on the results of the direct parametric method. Tables are provided for easy access of results.

Biological Availability↗
PubMed1998

Trials which randomize practices II: sample size.

BACKGROUND: When practices are randomized in a trial and observations are made on the patients to assess the relative effectiveness of the different interventions, sample size calculations need to estimate the number of practices required, not just the total number of patients. OBJECTIVE: Our aims were to introduce the methodology for appropriate sample size calculation and discuss the implications for power. METHOD: A worked example from general practice is used. DISCUSSION: Designs which randomize practices are less powerful than designs which randomize patients to intervention groups, particularly where a large number of patients is recruited from each practice. Studies which randomize few practices should be avoided if possible, as the loss of power is considerable and simple randomization may not ensure comparability of intervention groups.

Family Practice↗
PubMed1998

Changes in sample size and length of follow-up to maintain power in the coronary artery bypass graft (CABG) patch trial.

The CABG Patch Trial is testing the hypothesis that prophylactic use of implantable cardiac defibrillators (ICDs) will improve survival in high-risk coronary heart disease patients undergoing CABG surgery. The original design called for 800 patients to be randomized to ICD prophylaxis or to no therapy and followed for 2 to 6.5 years (average, 40 months) to a common termination date. Since the ICD pulse generators used in this trial lasted about 42 months, the original design required ICD replacement in many patients. At its first two meetings in 1993, the Data and Safety Monitoring Board (DSMB) formalized a plan to adjust sample size in October 1994 if the control group mortality rate was lower than expected. In June 1994, an unanticipated and unique event--a subpoena from the Office of the Inspector General (OIG)--made it impossible to replace about half of the ICD generators and threatened to shorten follow-up substantially. If follow-up had been stopped on the date originally planned, but without replacing ICDs, the average follow-up would have fallen from 40 months to about 33 months. Also, in October 1994, the control group mortality rate was found to be somewhat lower than expected. Together, the abbreviated follow-up and lower control group mortality threatened to reduce power substantially. The DSMB reviewed several options for restoring power. Because mortality rates in the first month after CABG surgery were about seven times as high as thereafter and because ICD therapy did not reduce surgical mortality (death during the first 30 days), extending the follow-up benefits power more than does increasing the sample size. However, the limit on extending follow-up was 42 months (the expected battery life of the ICD). Data from the ICD-treated group was not reviewed or considered in making the decision. After reviewing many options for restoring power, the DSMB recommended that the sample size be increased from 800 to 900 patients and that almost all patients be followed for 42 months. This recommendation extended follow-up for 2 years beyond the original termination date planned for the trial and dictated that patients close out after 42 months rather than on a common termination date.

Cause of Death↗
PubMed2004

Sample size for identifying differentially expressed genes in microarray experiments.

Microarray technology allows simultaneous comparison of expression levels of thousands of genes under each condition. This paper concerns sample size calculation in the identification of differentially expressed genes between a control and a treated sample. In a typical experiment, only a fraction of genes (altered genes) is expected to be differentially expressed between two samples. Sample size determination depends on a number of factors including the specified significance level (alpha), the desired statistical power (1-beta), the fraction (eta) of truly altered genes out of the total g genes studied, and the effect sizes (Delta) for the altered genes. This paper proposes a method to calculate the number of arrays required to detect at least 100lambda % (where 0 < lambda < or = 1) of the truly altered genes under the model of an equal effect size for all altered genes. The required numbers of arrays are tabulated for various values of alpha, beta, Delta, eta, and lambda for the one-sample and two-sample t-tests for g = 10,000. Based on the proposed approach, to identify up to 90% of truly altered genes among the unknown number of truly altered genes, the estimated numbers of arrays needed appear to be manageable. For instance, when the standardized effect size is at least 2.0, the number of arrays needed is less than or equal to 14 for the two-sample t-test and is less than or equal to 10 for the one-sample t-test. As the cost per array declines, such array numbers become practical. The proposed method offers a simple, intuitive, and practical way to determine the number of arrays needed in microarray experiments in which the true correlation structure among the genes under investigation cannot be reasonably assumed. An example dataset is used to illustrate the use of the proposed approach to plan microarray experiments.

Animals↗