Bayesian sample size determination using commensurate priors to leverage preexperimental data

This paper develops Bayesian sample size formulae for experiments comparing two groups, where relevant preexperimental information from multiple sources can be incorporated in a robust prior to support both the design and analysis. We use commensurate predictive priors for borrowing of information and further place Gamma mixture priors on the precisions to account for preliminary belief about the pairwise (in)commensurability between parameters that underpin the historical and new experiments. Averaged over the probability space of the new experimental data, appropriate sample sizes are found according to criteria that control certain aspects of the posterior distribution, such as the coverage probability or length of a defined density region. Our Bayesian methodology can be applied to circumstances that compare two normal means, proportions, or event times. When nuisance parameters (such as variance) in the new experiment are unknown, a prior distribution can further be specified based on preexperimental data. Exact solutions are available based on most of the criteria considered for Bayesian sample size determination, while a search procedure is described in cases for which there are no closed-form expressions. We illustrate the application of our sample size formulae in the design of clinical trials, where pretrial information is available to be leveraged. Hypothetical data examples, motivated by a rare-disease trial with an elicited expert prior opinion, and a comprehensive performance evaluation of the proposed methodology are presented.     

Sample size calculations for noninferiority trials with Poisson distributed count data

Clinical trials with Poisson distributed count data as the primary outcome are common in various medical areas such as relapse counts in multiple sclerosis trials or the number of attacks in trials for the treatment of migraine. In this article, we present approximate sample size formulae for testing noninferiority using asymptotic tests which are based on restricted or unrestricted maximum likelihood estimators of the Poisson rates. The Poisson outcomes are allowed to be observed for unequal follow‐up schemes, and both the situations that the noninferiority margin is expressed in terms of the difference and the ratio are considered. The exact type I error rates and powers of these tests are evaluated and the accuracy of the approximate sample size formulae is examined. The test statistic using the restricted maximum likelihood estimators (for the difference test problem) and the test statistic that is based on the logarithmic transformation and employs the maximum likelihood estimators (for the ratio test problem) show favorable type I error control and can be recommended for practical application. The approximate sample size formulae show high accuracy even for small sample sizes and provide power values identical or close to the aspired ones. The methods are illustrated by a clinical trial example from anesthesia.     

Sample size for gene expression microarray experiments

MOTIVATION: Microarray experiments often involve hundreds or thousands of genes. In a typical experiment, only a fraction of genes are expected to be differentially expressed; in addition, the measured intensities among different genes may be correlated. Depending on the experimental objectives, sample size calculations can be based on one of the three specified measures: sensitivity, true discovery and accuracy rates. The sample size problem is formulated as: the number of arrays needed in order to achieve the desired fraction of the specified measure at the desired family-wise power at the given type I error and (standardized) effect size. RESULTS: We present a general approach for estimating sample size under independent and equally correlated models using binomial and beta-binomial models, respectively. The sample sizes needed for a two-sample z-test are computed; the computed theoretical numbers agree well with the Monte Carlo simulation results. But, under more general correlation structures, the beta-binomial model can underestimate the needed samples by about 1-5 arrays. CONTACT: jchen@nctr.fda.gov.     

Statistical properties of Teng and Risch's sibship type tests for detecting an association between disease and a candidate allele

Risch and Teng [Genome Res 1998;8:1273-1288] and Teng and Risch [Genome Res 1999;9:234-241] proposed a class of transmission/disequilibrium test-like statistical tests based on the difference between the estimated allele frequencies in the affected and control populations. They evaluated the power of a variety of family-based and nonfamily-based designs for detecting an association between a candidate allele and disease. Because they were concerned with diseases with low penetrances, their power calculations assumed that unaffected individuals can be treated as a random sample from the population. They predicted that this assumption rendered their sample size calculations slightly conservative. We generalize their partial ascertainment conditioning by including the status of the unaffected sibs in the calculations of the distribution and power of the statistic used to compare the allele frequency in affected offspring to the estimated frequency in the parents, based on sibships with genotyped affected and unaffected sibs. Sample size formulas for our full ascertainment methods are presented. The sample sizes for our procedure are compared to those of Teng and Risch. The numerical results and simulations indicate that the simplifying assumption used in Teng and Risch can produce both conservative and anticonservative results. The magnitude of the difference between the sample sizes needed by their partial ascertainment approximation and the full ascertainment is small in the circumstances they focused on but can be appreciable in others, especially when the baseline penetrances are moderate. Two other statistics, using different estimators for the variance of the basic statistic comparing the allele frequencies in the affected and unaffected sibs are introduced. One of them incorporates an estimate of the null variance obtained from an auxiliary sample and appears to noticeably decrease the sample sizes required to achieve a prespecified power.     

On Approximate and Exact Sample Sizes of Equivalence Tests for Binomial Proportions

There exist several formulae for sample sizes for testing the equivalence of binomial proportions which are based on approximations by the normal distribution. Quite often these formulae produce drastically different results. In this paper the validity of the approximate sample sizes is investigated with respect to the exact distributions.     

Confidence intervals for genotype relative risks and allele frequencies from the case parent trio design for candidate-gene studies

OBJECTIVES: Confidence intervals for genotype relative risks, for allele frequencies and for the attributable risk in the case parent trio design for candidate-gene studies are proposed which can be easily calculated from the observed familial genotype frequencies. METHODS: Likelihood theory and the delta method were used to derive point estimates and confidence internals. We used Monte Carlo simulations to show the validity of the formulae for a variety of given modes of inheritance and allele frequencies and illustrated their usefulness by applying them to real data. RESULTS: Generally these formulae were found to be valid for 'sufficiently large' sample sizes. For smaller sample sizes the estimators for genotype relative risks tended to be conservative whereas the estimator for attributable risk was found to be anti-conservative for moderate to high allele frequencies. CONCLUSIONS: Since the proposed formulae provide quantitative information on the individual and epidemiological relevance of a genetic variant they might be a useful addition to the traditional statistical significance level of TDT results.     

An analytical formula to estimate confidence interval of QTL location with a saturated genetic map as a function of experimental design

Analytical formulae are derived for the confidence interval for location of a quantitative trait locus (QTL) using a saturated genetic map, as a function of the experimental design, the QTL allele substitution effect, and the number of individuals genotyped and phenotyped. The formulae are derived assuming evenly spaced recombination events, rather than the actual unevenly spaced distribution. The formulae are useful for determining desired sample size when designing a wide variety of QTL mapping experiments, and for evaluating a priori the potential of a given mapping population for defining the location of a QTL. The formulae do not take into account the finite number of recombination events in a given sample.     

Sample size formulae for two-stage randomized trials with survival outcomes

Two-stage randomized trials are growing in importance in developing adaptive treatment strategies, i.e. treatment policies or dynamic treatment regimes. Usually, the first stage involves randomization to one of the several initial treatments. The second stage of treatment begins when an early nonresponse criterion or response criterion is met. In the second-stage, nonresponding subjects are re-randomized among second-stage treatments. Sample size calculations for planning these two-stage randomized trials with failure time outcomes are challenging because the variances of common test statistics depend in a complex manner on the joint distribution of time to the early nonresponse criterion or response criterion and the primary failure time outcome. We produce simple, albeit conservative, sample size formulae by using upper bounds on the variances. The resulting formulae only require the working assumptions needed to size a standard single-stage randomized trial and, in common settings, are only mildly conservative. These sample size formulae are based on either a weighted Kaplan-Meier estimator of survival probabilities at a fixed time-point or a weighted version of the log-rank test.     

Hypothesis Testing and Power Calculations for Taxonomic-Based Human Microbiome Data

This paper presents new biostatistical methods for the analysis of microbiome data based on a fully parametric approach using all the data. The Dirichlet-multinomial distribution allows the analyst to calculate power and sample sizes for experimental design, perform tests of hypotheses (e.g., compare microbiomes across groups), and to estimate parameters describing microbiome properties. The use of a fully parametric model for these data has the benefit over alternative non-parametric approaches such as bootstrapping and permutation testing, in that this model is able to retain more information contained in the data. This paper details the statistical approaches for several tests of hypothesis and power/sample size calculations, and applies them for illustration to taxonomic abundance distribution and rank abundance distribution data using HMP Jumpstart data on 24 subjects for saliva, subgingival, and supragingival samples. Software for running these analyses is available.     

Sample size determination in clinical proteomic profiling experiments using mass spectrometry for class comparison

Mass spectrometric profiling approaches such as MALDI‐TOF and SELDI‐TOF are increasingly being used in disease marker discovery, particularly in the lower molecular weight proteome. However, little consideration has been given to the issue of sample size in experimental design. The aim of this study was to develop a protocol for the use of sample size calculations in proteomic profiling studies using MS. These sample size calculations can be based on a simple linear mixed model which allows the inclusion of estimates of biological and technical variation inherent in the experiment. The use of a pilot experiment to estimate these components of variance is investigated and is shown to work well when compared with larger studies. Examination of data from a number of studies using different sample types and different chromatographic surfaces shows the need for sample‐ and preparation‐specific sample size calculations.     

Group Sequential Monitoring with Arbitrary Inspection Times

The classical group sequential test procedures that were proposed by Pocock (1977) and O'Brien and Fleming (1979) rest on the assumption of equal sample sizes between the interim analyses. Regarding this it is well known that for most situations there is not a great amount of additional Type I error if monitoring is performed for unequal sample sizes between the stages. In some cases, however, problems can arise resulting in an unacceptable liberal behavior of the test procedure. In this article worst case scenarios in sample size imbalancements between the inspection times are considered. Exact critical values for the Pocock and the O'Brien and Fleming group sequential designs are derived for arbitrary and for varying but bounded sample sizes. The approach represents a reasonable alternative to the flexible method that is based on the Type I error rate spending function. The SAS syntax for performing the calculations is provided. Using these procedures, the inspection times or the sample sizes in the consecutive stages need to be chosen independently of the data observed so far.     

The PowerAtlas: a power and sample size atlas for microarray experimental design and research

Background  

Microarrays permit biologists to simultaneously measure the mRNA abundance of thousands of genes. An important issue facing investigators planning microarray experiments is how to estimate the sample size required for good statistical power. What is the projected sample size or number of replicate chips needed to address the multiple hypotheses with acceptable accuracy? Statistical methods exist for calculating power based upon a single hypothesis, using estimates of the variability in data from pilot studies. There is, however, a need for methods to estimate power and/or required sample sizes in situations where multiple hypotheses are being tested, such as in microarray experiments. In addition, investigators frequently do not have pilot data to estimate the sample sizes required for microarray studies.     

Sample size determination

Scientists who use animals in research must justify the number of animals to be used, and committees that review proposals to use animals in research must review this justification to ensure the appropriateness of the number of animals to be used. This article discusses when the number of animals to be used can best be estimated from previous experience and when a simple power and sample size calculation should be performed. Even complicated experimental designs requiring sophisticated statistical models for analysis can usually be simplified to a single key or critical question so that simple formulae can be used to estimate the required sample size. Approaches to sample size estimation for various types of hypotheses are described, and equations are provided in the Appendix. Several web sites are cited for more information and for performing actual calculations     

Sample size recalculation in internal pilot study designs: a review

The adequacy of sample size is important to clinical trials. In the planning phase of a trial, however, the investigators are often quite uncertain about the sizes of parameters which are needed for sample size calculations. A solution to this problem is mid-course recalculation of the sample size during the ongoing trial. In internal pilot study designs, nuisance parameters are estimated on the basis of interim data and the sample size is adjusted accordingly. This review attempts to give an overview on the available methods. It is written not only for biometricians who are already familar with the the topic and wish to update their knowledge but also for users new to the subject.     

Homogeneity test of rate ratios in stratified matched-pair studies

This paper investigates homogeneity test of rate ratios in stratified matched-pair studies on the basis of asymptotic and bootstrap-resampling methods. Based on the efficient score approach, we develop a simple and computationally tractable score test statistic. Several other homogeneity test statistics are also proposed on the basis of the weighted least-squares estimate and logarithmic transformation. Sample size formulae are derived to guarantee a pre-specified power for the proposed tests at the pre-given significance level. Empirical results confirm that (i) the modified score statistic based on the bootstrap-resampling method performs better in the sense that its empirical type I error rate is much closer to the pre-specified nominal level than those of other tests and its power is greater than those of other tests, and is hence recommended, whilst the statistics based on the weighted least-squares estimate and logarithmic transformation are slightly conservative under some of the considered settings; (ii) the derived sample size formulae are rather accurate in the sense that their empirical powers obtained from the estimated sample sizes are very close to the pre-specified nominal powers. A real example is used to illustrate the proposed methodologies.     

Sample sizes in the multivariate analysis of repeated measurements

Determination of sample sizes for comparing two or more treatments in repeated measurements experiments is considered. Multivariate normality of the individual's vector of repeated measures is assumed. Particular emphasis is placed on applications wherein the error variance-covariance matrix is arbitrary positive-definite. Sample size determination is based on power considerations associated with Hotelling's T2 test and the desire to detect a specified difference between any pair of treatment means. Tabulated sample sizes are given in the case of an equal variance-unequal covariance structure. The utility of these sample sizes is also demonstrated for a more general variance-covariance structure. Applications of these sample sizes are illustrated with two examples.     

Correcting eggshell indices of raptor eggs for hole size and eccentricity

Eggshell thickness is often expressed by means of an index based on the length, breadth and weight of the shell. The effect of the blow-hole and egg eccentricity on Ratcliffe's shell thickness index was investigated in a sample of 585 eggs from six raptor species. Corrections for the size of the hole and egg eccentricity are proposed, as is a combined formula to correct both sources of error at the same time. It is shown that by using these formulae, considerable improvements in estimates of shell thinning are obtained. These may be especially useful when sample sizes are small, which is often the case when working with species that have been reduced in numbers.     

Bayesian sample size calculations for comparing two strategies in SMART studies

In the management of most chronic conditions characterized by the lack of universally effective treatments, adaptive treatment strategies (ATSs) have grown in popularity as they offer a more individualized approach. As a result, sequential multiple assignment randomized trials (SMARTs) have gained attention as the most suitable clinical trial design to formalize the study of these strategies. While the number of SMARTs has increased in recent years, sample size and design considerations have generally been carried out in frequentist settings. However, standard frequentist formulae require assumptions on interim response rates and variance components. Misspecifying these can lead to incorrect sample size calculations and correspondingly inadequate levels of power. The Bayesian framework offers a straightforward path to alleviate some of these concerns. In this paper, we provide calculations in a Bayesian setting to allow more realistic and robust estimates that account for uncertainty in inputs through the ‘two priors’ approach. Additionally, compared to the standard frequentist formulae, this methodology allows us to rely on fewer assumptions, integrate pre-trial knowledge, and switch the focus from the standardized effect size to the MDD. The proposed methodology is evaluated in a thorough simulation study and is implemented to estimate the sample size for a full-scale SMART of an internet-based adaptive stress management intervention on cardiovascular disease patients using data from its pilot study conducted in two Canadian provinces.     

Sample size determination for classifiers based on single-nucleotide polymorphisms

Single-nucleotide polymorphisms (SNPs), believed to determine human differences, are widely used to predict risk of diseases. Typically, clinical samples are limited and/or the sampling cost is high. Thus, it is essential to determine an adequate sample size needed to build a classifier based on SNPs. Such a classifier would facilitate correct classifications, while keeping the sample size to a minimum, thereby making the studies cost-effective. For coded SNP data from 2 classes, an optimal classifier and an approximation to its probability of correct classification (PCC) are derived. A linear classifier is constructed and an approximation to its PCC is also derived. These approximations are validated through a variety of Monte Carlo simulations. A sample size determination algorithm based on the criterion, which ensures that the difference between the 2 approximate PCCs is below a threshold, is given and its effectiveness is illustrated via simulations. For the HapMap data on Chinese and Japanese populations, a linear classifier is built using 51 independent SNPs, and the required total sample sizes are determined using our algorithm, as the threshold varies. For example, when the threshold value is 0.05, our algorithm determines a total sample size of 166 (83 for Chinese and 83 for Japanese) that satisfies the criterion.     

Adjusting batch effects in microarray expression data using empirical Bayes methods

Non-biological experimental variation or "batch effects" are commonly observed across multiple batches of microarray experiments, often rendering the task of combining data from these batches difficult. The ability to combine microarray data sets is advantageous to researchers to increase statistical power to detect biological phenomena from studies where logistical considerations restrict sample size or in studies that require the sequential hybridization of arrays. In general, it is inappropriate to combine data sets without adjusting for batch effects. Methods have been proposed to filter batch effects from data, but these are often complicated and require large batch sizes ( > 25) to implement. Because the majority of microarray studies are conducted using much smaller sample sizes, existing methods are not sufficient. We propose parametric and non-parametric empirical Bayes frameworks for adjusting data for batch effects that is robust to outliers in small sample sizes and performs comparable to existing methods for large samples. We illustrate our methods using two example data sets and show that our methods are justifiable, easy to apply, and useful in practice. Software for our method is freely available at: http://biosun1.harvard.edu/complab/batch/.