Sample size reestimation by Bayesian prediction

We review a Bayesian predictive approach for interim data monitoring and propose its application to interim sample size reestimation for clinical trials. Based on interim data, this approach predicts how the sample size of a clinical trial needs to be adjusted so as to claim a success at the conclusion of the trial with an expected probability. The method is compared with predictive power and conditional power approaches using clinical trial data. Advantages of this approach over the others are discussed.     

Sample Size Calculations for the Mean in a Two Component Nonstandard Mixture Distribution

We are concerned with calculating the sample size required for estimating the mean of the continuous distribution in the context of a two component nonstandard mixture distribution (i.e., a mixture of an identifiable point degenerate function F at a constant with probability P and a continuous distribution G with probability 1 – P). A common ad hoc procedure of escalating the naïve sample size n (calculated under the assumption of no point degenerate function F) by a factor of 1/(1 – P), has about 0.5 probability of achieving the pre‐specified statistical power. Such an ad hoc approach may seriously underestimate the necessary sample size and jeopardize inferences in scientific investigations. We argue that sample size calculations in this context should have a pre‐specified probability of power ≥1 – β set by the researcher at a level greater than 0.5. To that end, we propose an exact method and an approximate method to calculate sample size in this context so that the pre‐specified probability of achieving a desired statistical power is determined by the researcher. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Modification of sample size in group sequential clinical trials

In group sequential clinical trials, sample size reestimation can be a complicated issue when it allows for change of sample size to be influenced by an observed sample path. Our simulation studies show that increasing sample size based on an interim estimate of the treatment difference can substantially inflate the probability of type I error in most practical situations. A new group sequential test procedure is developed by modifying the weights used in the traditional repeated significance two-sample mean test. The new test has the type I error probability preserved at the target level and can provide a substantial gain in power with the increase of sample size. Generalization of the new procedure is discussed.     

Heterogeneous treatment effects in stratified clinical trials with time‐to‐event endpoints

When analyzing clinical trials with a stratified population, homogeneity of treatment effects is a common assumption in survival analysis. However, in the context of recent developments in clinical trial design, which aim to test multiple targeted therapies in corresponding subpopulations simultaneously, the assumption that there is no treatment‐by‐stratum interaction seems inappropriate. It becomes an issue if the expected sample size of the strata makes it unfeasible to analyze the trial arms individually. Alternatively, one might choose as primary aim to prove efficacy of the overall (targeted) treatment strategy. When testing for the overall treatment effect, a violation of the no‐interaction assumption renders it necessary to deviate from standard methods that rely on this assumption. We investigate the performance of different methods for sample size calculation and data analysis under heterogeneous treatment effects. The commonly used sample size formula by Schoenfeld is compared to another formula by Lachin and Foulkes, and to an extension of Schoenfeld's formula allowing for stratification. Beyond the widely used (stratified) Cox model, we explore the lognormal shared frailty model, and a two‐step analysis approach as potential alternatives that attempt to adjust for interstrata heterogeneity. We carry out a simulation study for a trial with three strata and violations of the no‐interaction assumption. The extension of Schoenfeld's formula to heterogeneous strata effects provides the most reliable sample size with respect to desired versus actual power. The two‐step analysis and frailty model prove to be more robust against loss of power caused by heterogeneous treatment effects than the stratified Cox model and should be preferred in such situations.     

Comparison of Two New Robust Parameter Estimation Methods for the Power Function Distribution

Estimation of any probability distribution parameters is vital because imprecise and biased estimates can be misleading. In this study, we investigate a flexible power function distribution and introduced new two methods such as, probability weighted moments, and generalized probability weighted methods for its parameters. We compare their results with L-moments, trimmed L-moments by a simulation study and a real data example based on performance measures such as, mean square error and total deviation. We concluded that all the methods perform well in the case of large sample size (n>30), however, the generalized probability weighted moment method performs better for small sample size.     

A Bayesian basket trial design that borrows information across strata based on the similarity between the posterior distributions of the response probability

Basket trials simultaneously evaluate the effect of one or more drugs on a defined biomarker, genetic alteration, or molecular target in a variety of disease subtypes, often called strata. A conventional approach for analyzing such trials is an independent analysis of each of the strata. This analysis is inefficient as it lacks the power to detect the effect of drugs in each stratum. To address these issues, various designs for basket trials have been proposed, centering on designs using Bayesian hierarchical models. In this article, we propose a novel Bayesian basket trial design that incorporates predictive sample size determination, early termination for inefficacy and efficacy, and the borrowing of information across strata. The borrowing of information is based on the similarity between the posterior distributions of the response probability. In general, Bayesian hierarchical models have many distributional assumptions along with multiple parameters. By contrast, our method has prior distributions for response probability and two parameters for similarity of distributions. The proposed design is easier to implement and less computationally demanding than other Bayesian basket designs. Through a simulation with various scenarios, our proposed design is compared with other designs including one that does not borrow information and one that uses a Bayesian hierarchical model.     

Power calculations for the transmission/disequilibrium and affected sib pair tests using elementary probability methods

The transmission/disequilibrium test (TDT) and the affected sib pair test (ASP) both test for the association of a marker allele with some conditions. Here, we present methods for calculating the probability of detecting the association (power) for a study examining a fixed number of families for suitability for the study and for calculating the number of such families to be examined. Both calculations use a genetic model for the association. The model considered posits a bi-allelic marker locus that is linked to a bi-allelic disease locus with a possibly nonzero recombination fraction between the loci. The penetrance of the disease is an increasing function of the number of disease alleles. The TDT tests whether the transmission by a heterozygous parent of a particular allele at a marker locus to an affected offspring occurs with probability greater than 0.5. The ASP tests whether transmission of the same allele to two affected sibs occurs with probability greater than 0.5. In either case, evidence that the probability is greater than 0.5 is evidence for association between the marker and the disease. Study inclusion criteria (IC) can greatly affect the necessary sample size of a TDT or ASP study. IC considered by us include a randomly selected parent at least one parent or both parents required to be heterozygous. It also allows a specified minimum number of affected offspring to be required (TDT only). We use elementary probability calculations rather than complex mathematical manipulations or asymptotic methods (large sample size approximations) to compute power and requisite sample size for a proposed study. The advantages of these methods are simplicity and generality.     

Estimating species' absence, colonization and local extinction in patchy landscapes: an application of occupancy models with rodents

Making an inference on the absence of a species in a site is often problematic, due to detection probability being, in most cases, <1. Inference is more complicated if detection probability, together with distribution patterns, vary during the year, since the possibility of inferring a species absence, at reasonable costs, may be possible only in certain periods. Our aim here is to show how such challenging situations can be by tackled by applying some recently developed occupancy models combined with sample size (number of repeated surveys) estimation. We thus analysed the distribution of two rodents Myodes glareolus and Mus musculus domesticus in a fragmented landscape in central Italy pointing out how it is possible to identify true absences, non-detections, extinctions/colonizations and determine seasonal values of detection probability.     

Test equality and sample size calculation based on risk difference in a randomized clinical trial with noncompliance and missing outcomes

In a randomized clinical trial (RCT), noncompliance with an assigned treatment can occur due to serious side effects, while missing outcomes on patients may happen due to patients' withdrawal or loss to follow up. To avoid the possible loss of power to detect a given risk difference (RD) of interest between two treatments, it is essentially important to incorporate the information on noncompliance and missing outcomes into sample size calculation. Under the compound exclusion restriction model proposed elsewhere, we first derive the maximum likelihood estimator (MLE) of the RD among compliers between two treatments for a RCT with noncompliance and missing outcomes and its asymptotic variance in closed form. Based on the MLE with tanh(-1)(x) transformation, we develop an asymptotic test procedure for testing equality of two treatment effects among compliers. We further derive a sample size calculation formula accounting for both noncompliance and missing outcomes for a desired power 1 - beta at a nominal alpha-level. To evaluate the performance of the test procedure and the accuracy of the sample size calculation formula, we employ Monte Carlo simulation to calculate the estimated Type I error and power of the proposed test procedure corresponding to the resulting sample size in a variety of situations. We find that both the test procedure and the sample size formula developed here can perform well. Finally, we include a discussion on the effects of various parameters, including the proportion of compliers, the probability of non-missing outcomes, and the ratio of sample size allocation, on the minimum required sample size.     

Power analysis to determine sample size for monitoring vegetation change in salt marsh habitats

Numerous initiatives are underway throughout New England and elsewhere to quantify salt marsh vegetation change, mostly in response to habitat restoration, sea level rise, and nutrient enrichment. To detect temporal changes in vegetation at a marsh or to compare vegetation among different marshes with a degree of statistical certainty an adequate sample size is required. Based on sampling 1 m² vegetation plots from 11 New England salt marsh data sets, we conducted a power analysis to determine the minimum number of samples that were necessary to detect change between vegetation communities. Statistical power was determined for sample sizes of 5, 10, 15, and 20 vegetation plots at an alpha level of 0.05. Detection of subtle differences between vegetation data sets (e.g., comparing vegetation in the same marsh over two consecutive years) can be accomplished using a sample size of 20 plots with a reasonable probability of detecting a difference when one truly exists. With a lower sample size, and thus lower power, there is an increased probability of not detecting a difference when one exists (e.g., Type II error). However, if investigators expect to detect major changes in vegetation (e.g., such as those between an un-impacted and a highly impacted marsh) then a sample size of 5, 10, or 15 plots may be appropriate while still maintaining adequate power. Due to the relative ease of collecting vegetation data, we suggest a minimum sample size of 20 randomly located 1 m² plots when developing monitoring designs to detect vegetation community change of salt marshes. The sample size of 20 plots per New England salt marsh is appropriate regardless of marsh size or permanency (permanent or non-permanent) of the plots.     

Two-stage case–control designs for rare genetic variants

The search for the association of rare genetic variants with common diseases is of high interest, yet challenging because of cost considerations. We present an efficient two-stage design that uses diseased cases to first screen for rare variants at stage-1. If too few cases are found to carry any variants, the study stops. Otherwise, the selected variants are screened at stage-2 in a larger set of cases and controls, and the frequency of variants is compared between cases and controls by an exact test that corrects for the stage-1 ascertainment. Simulations show that our new method provides conservative Type-I error rates, similar to the conservative aspect of Fisher’s exact test. We show that the probability of stopping at stage-1 increases with a smaller number of cases screened at stage-1, a larger stage-1 continuation threshold, or a smaller carrier probability. Our simulations also show how these factors impact the power at stage-2. To balance stopping early when there are few variant carriers versus continuation to stage-2 when the variants have a reasonable effect size on the phenotype, we provide guidance on designing an optimal study that minimizes the expected sample size when the null hypothesis is true, yet achieves the desired power.     

Designing Genome-Wide Association Studies: Sample Size, Power, Imputation, and the Choice of Genotyping Chip

Genome-wide association studies are revolutionizing the search for the genes underlying human complex diseases. The main decisions to be made at the design stage of these studies are the choice of the commercial genotyping chip to be used and the numbers of case and control samples to be genotyped. The most common method of comparing different chips is using a measure of coverage, but this fails to properly account for the effects of sample size, the genetic model of the disease, and linkage disequilibrium between SNPs. In this paper, we argue that the statistical power to detect a causative variant should be the major criterion in study design. Because of the complicated pattern of linkage disequilibrium (LD) in the human genome, power cannot be calculated analytically and must instead be assessed by simulation. We describe in detail a method of simulating case-control samples at a set of linked SNPs that replicates the patterns of LD in human populations, and we used it to assess power for a comprehensive set of available genotyping chips. Our results allow us to compare the performance of the chips to detect variants with different effect sizes and allele frequencies, look at how power changes with sample size in different populations or when using multi-marker tags and genotype imputation approaches, and how performance compares to a hypothetical chip that contains every SNP in HapMap. A main conclusion of this study is that marked differences in genome coverage may not translate into appreciable differences in power and that, when taking budgetary considerations into account, the most powerful design may not always correspond to the chip with the highest coverage. We also show that genotype imputation can be used to boost the power of many chips up to the level obtained from a hypothetical “complete” chip containing all the SNPs in HapMap. Our results have been encapsulated into an R software package that allows users to design future association studies and our methods provide a framework with which new chip sets can be evaluated.     

The Ranking Probability Approach and Its Usage in Design and Analysis of Large-Scale Studies

In experiments with many statistical tests there is need to balance type I and type II error rates while taking multiplicity into account. In the traditional approach, the nominal -level such as 0.05 is adjusted by the number of tests, , i.e., as 0.05/. Assuming that some proportion of tests represent “true signals”, that is, originate from a scenario where the null hypothesis is false, power depends on the number of true signals and the respective distribution of effect sizes. One way to define power is for it to be the probability of making at least one correct rejection at the assumed -level. We advocate an alternative way of establishing how “well-powered” a study is. In our approach, useful for studies with multiple tests, the ranking probability is controlled, defined as the probability of making at least correct rejections while rejecting hypotheses with smallest P-values. The two approaches are statistically related. Probability that the smallest P-value is a true signal (i.e., ) is equal to the power at the level , to an excellent approximation. Ranking probabilities are also related to the false discovery rate and to the Bayesian posterior probability of the null hypothesis. We study properties of our approach when the effect size distribution is replaced for convenience by a single “typical” value taken to be the mean of the underlying distribution. We conclude that its performance is often satisfactory under this simplification; however, substantial imprecision is to be expected when is very large and is small. Precision is largely restored when three values with the respective abundances are used instead of a single typical effect size value.     

Is There a Size Limit for Cosmopolitan Distribution in Free-Living Microorganisms? A Biogeographical Analysis of Testate Amoebae from Polar Areas

A long-standing debate in microbial ecology is the extent to which free-living microorganisms exhibit cosmopolitan distributions. We use a comparison of testate amoebae communities in cold “polar” locations (Arctic, Antarctic, and Tibet) to investigate how a microorganism’s size affects its probability of having a cosmopolitan distribution. We show that the probability a given taxa being reported in all three locations increases as testate size decreases. Likewise, excluding those testates found only in Tibet, very small testates (<20 μm) are more likely to occur in both the Arctic and Antarctic than in either of these poles alone. Attempting to correct for phylogeny reduces the number of statistically significant relationships—both because of decreased sample size and potentially real phylogenetic patterns, although some size-dependent effects were still apparent. In particular, taxa found in both the Arctic and Antarctic poles were significantly smaller than congeneric taxa found only in Tibet. This pattern may in part be due to habitat effects, with the Tibetan samples being more likely to have come from aquatic sites which may be more suitable for larger taxa. Overall, our analysis suggests that, at least within testate amoebae, a cosmopolitan distribution becomes increasingly common as median taxon size decreases.     

Authenticity of ancient-DNA results: a statistical approach

Although there have been several papers recommending appropriate experimental designs for ancient-DNA studies, there have been few attempts at statistical analysis. We assume that we cannot decide whether a result is authentic simply by examining the sequence (e.g., when working with humans and domestic animals). We use a maximum-likelihood approach to estimate the probability that a positive result from a sample is (either partly or entirely) an amplification of DNA that was present in the sample before the experiment began. Our method is useful in two situations. First, we can decide in advance how many samples will be needed to achieve a given level of confidence. For example, to be almost certain (95% confidence interval 0.96-1.00, maximum-likelihood estimate 1.00) that a positive result comes, at least in part, from DNA present before the experiment began, we need to analyze at least five samples and controls, even if all samples and no negative controls yield positive results. Second, we can decide how much confidence to place in results that have been obtained already, whether or not there are positive results from some controls. For example, the risk that at least one negative control yields a positive result increases with the size of the experiment, but the effects of occasional contamination are less severe in large experiments.     

On the Use of the Upper Confidence Limit for the Variance from a Pilot Sample for Sample Size Determination

In a recent paper, Browne (1995) investigated the use of a pilot sample for sample size calculation. Monte Carlo simulations indicated that using a 100 · (1 — γ) per cent upper one-sided confidence limit on the population varíance σ² leads to a sample size that guarantees the planned power with a probability of at least 1 ? γ. The purpose of this paper is to get further insight into the results of Browne by analytical considerations. Furthermore, the expected power is investigated when applying the strategy and recommendations for the choice of the pilot sample size are given.     

Bayesian Adaptive Randomization and Trial Monitoring with Predictive Probability for Time-to-Event Endpoint

There has been much development in Bayesian adaptive designs in clinical trials. In the Bayesian paradigm, the posterior predictive distribution characterizes the future possible outcomes given the currently observed data. Based on the interim time-to-event data, we develop a new phase II trial design by combining the strength of both Bayesian adaptive randomization and the predictive probability. By comparing the mean survival times between patients assigned to two treatment arms, more patients are assigned to the better treatment on the basis of adaptive randomization. We continuously monitor the trial using the predictive probability for early termination in the case of superiority or futility. We conduct extensive simulation studies to examine the operating characteristics of four designs: the proposed predictive probability adaptive randomization design, the predictive probability equal randomization design, the posterior probability adaptive randomization design, and the group sequential design. Adaptive randomization designs using predictive probability and posterior probability yield a longer overall median survival time than the group sequential design, but at the cost of a slightly larger sample size. The average sample size using the predictive probability method is generally smaller than that of the posterior probability design.     

A note on comparing exposure data to a regulatory limit in the presence of unexposed and a limit of detection

In some occupational health studies, observations occur in both exposed and unexposed individuals. If the levels of all exposed individuals have been detected, a two-part zero-inflated log-normal model is usually recommended, which assumes that the data has a probability mass at zero for unexposed individuals and a continuous response for values greater than zero for exposed individuals. However, many quantitative exposure measurements are subject to left censoring due to values falling below assay detection limits. A zero-inflated log-normal mixture model is suggested in this situation since unexposed zeros are not distinguishable from those exposed with values below detection limits. In the context of this mixture distribution, the information contributed by values falling below a fixed detection limit is used only to estimate the probability of unexposed. We consider sample size and statistical power calculation when comparing the median of exposed measurements to a regulatory limit. We calculate the required sample size for the data presented in a recent paper comparing the benzene TWA exposure data to a regulatory occupational exposure limit. A simulation study is conducted to investigate the performance of the proposed sample size calculation methods.     

Sample Size Determination for Designing a Strata-Matched Case-Control Study to Detect Multiple Risk Factors

Investigations of sample size for planning case-control studies have usually been limited to detecting a single factor. In this paper, we investigate sample size for multiple risk factors in strata-matched case-control studies. We construct an omnibus statistic for testing M different risk factors based on the jointly sufficient statistics of parameters associated with the risk factors. The statistic is non-iterative, and it reduces to the Cochran statistic when M = 1. The asymptotic power function of the test is a non-central chi-square with M degrees of freedom and the sample size required for a specific power can be obtained by the inverse relationship. We find that the equal sample allocation is optimum. A Monte Carlo experiment demonstrates that an approximate formula for calculating sample size is satisfactory in typical epidemiologic studies. An approximate sample size obtained using Bonferroni's method for multiple comparisons is much larger than that obtained using the omnibus test. Approximate sample size formulas investigated in this paper using the omnibus test, as well as the individual tests, can be useful in designing case-control studies for detecting multiple risk factors.     

Direct inference of SNP heterozygosity rates and resolution of LOH detection

Single nucleotide polymorphisms (SNPs) have been increasingly utilized to investigate somatic genetic abnormalities in premalignancy and cancer. LOH is a common alteration observed during cancer development, and SNP assays have been used to identify LOH at specific chromosomal regions. The design of such studies requires consideration of the resolution for detecting LOH throughout the genome and identification of the number and location of SNPs required to detect genetic alterations in specific genomic regions. Our study evaluated SNP distribution patterns and used probability models, Monte Carlo simulation, and real human subject genotype data to investigate the relationships between the number of SNPs, SNP HET rates, and the sensitivity (resolution) for detecting LOH. We report that variances of SNP heterozygosity rate in dbSNP are high for a large proportion of SNPs. Two statistical methods proposed for directly inferring SNP heterozygosity rates require much smaller sample sizes (intermediate sizes) and are feasible for practical use in SNP selection or verification. Using HapMap data, we showed that a region of LOH greater than 200 kb can be reliably detected, with losses smaller than 50 kb having a substantially lower detection probability when using all SNPs currently in the HapMap database. Higher densities of SNPs may exist in certain local chromosomal regions that provide some opportunities for reliably detecting LOH of segment sizes smaller than 50 kb. These results suggest that the interpretation of the results from genome-wide scans for LOH using commercial arrays need to consider the relationships among inter-SNP distance, detection probability, and sample size for a specific study. New experimental designs for LOH studies would also benefit from considering the power of detection and sample sizes required to accomplish the proposed aims.