An efficient sample size adaptation strategy with adjustment of randomization ratio

In clinical trials, sample size reestimation is a useful strategy for mitigating the risk of uncertainty in design assumptions and ensuring sufficient power for the final analysis. In particular, sample size reestimation based on unblinded interim effect size can often lead to sample size increase, and statistical adjustment is usually needed for the final analysis to ensure that type I error rate is appropriately controlled. In current literature, sample size reestimation and corresponding type I error control are discussed in the context of maintaining the original randomization ratio across treatment groups, which we refer to as “proportional increase.” In practice, not all studies are designed based on an optimal randomization ratio due to practical reasons. In such cases, when sample size is to be increased, it is more efficient to allocate the additional subjects such that the randomization ratio is brought closer to an optimal ratio. In this research, we propose an adaptive randomization ratio change when sample size increase is warranted. We refer to this strategy as “nonproportional increase,” as the number of subjects increased in each treatment group is no longer proportional to the original randomization ratio. The proposed method boosts power not only through the increase of the sample size, but also via efficient allocation of the additional subjects. The control of type I error rate is shown analytically. Simulations are performed to illustrate the theoretical results.     

Stratified Fisher's exact test and its sample size calculation

Chi‐squared test has been a popular approach to the analysis of a 2 × 2 table when the sample sizes for the four cells are large. When the large sample assumption does not hold, however, we need an exact testing method such as Fisher's test. When the study population is heterogeneous, we often partition the subjects into multiple strata, so that each stratum consists of homogeneous subjects and hence the stratified analysis has an improved testing power. While Mantel–Haenszel test has been widely used as an extension of the chi‐squared test to test on stratified 2 × 2 tables with a large‐sample approximation, we have been lacking an extension of Fisher's test for stratified exact testing. In this paper, we discuss an exact testing method for stratified 2 × 2 tables that is simplified to the standard Fisher's test in single 2 × 2 table cases, and propose its sample size calculation method that can be useful for designing a study with rare cell frequencies.     

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 and sample size calculation of comparative diagnostic accuracy studies with multiple correlated test results

We consider the power and sample size calculation of diagnostic studies with normally distributed multiple correlated test results. We derive test statistics and obtain power and sample size formulas. The methods are illustrated using an example of comparison of CT and PET scanner for detecting extra-hepatic disease for colorectal cancer.     

A sample size adjustment procedure for clinical trials based on conditional power

When designing clinical trials, researchers often encounter the uncertainty in the treatment effect or variability assumptions. Hence the sample size calculation at the planning stage of a clinical trial may also be questionable. Adjustment of the sample size during the mid-course of a clinical trial has become a popular strategy lately. In this paper we propose a procedure for calculating additional sample size needed based on conditional power, and adjusting the final-stage critical value to protect the overall type-I error rate. Compared to other previous procedures, the proposed procedure uses the definition of the conditional type-I error directly without appealing to an extra special function for it. It has better flexibility in setting up interim decision rules and the final-stage test is a likelihood ratio test.     

Ratios as a size adjustment in morphometrics

Simple ratios in which a measurement variable is divided by a size variable are commonly used but known to be inadequate for eliminating size correlations from morphometric data. Deficiencies in the simple ratio can be alleviated by incorporating regression coefficients describing the bivariate relationship between the measurement and size variables. Recommendations have included: 1) subtracting the regression intercept to force the bivariate relationship through the origin (intercept-adjusted ratios); 2) exponentiating either the measurement or the size variable using an allometry coefficient to achieve linearity (allometrically adjusted ratios); or 3) both subtracting the intercept and exponentiating (fully adjusted ratios). These three strategies for deriving size-adjusted ratios imply different data models for describing the bivariate relationship between the measurement and size variables (i.e., the linear, simple allometric, and full allometric models, respectively). Algebraic rearrangement of the equation associated with each data model leads to a correctly formulated adjusted ratio whose expected value is constant (i.e., size correlation is eliminated). Alternatively, simple algebra can be used to derive an expected value function for assessing whether any proposed ratio formula is effective in eliminating size correlations. Some published ratio adjustments were incorrectly formulated as indicated by expected values that remain a function of size after ratio transformation. Regression coefficients incorporated into adjusted ratios must be estimated using least-squares regression of the measurement variable on the size variable. Use of parameters estimated by any other regression technique (e.g., major axis or reduced major axis) results in residual correlations between size and the adjusted measurement variable. Correctly formulated adjusted ratios, whose parameters are estimated by least-squares methods, do control for size correlations. The size-adjusted results are similar to those based on analysis of least-squares residuals from the regression of the measurement on the size variable. However, adjusted ratios introduce size-related changes in distributional characteristics (variances) that differentially alter relationships among animals in different size classes. © 1993 Wiley-Liss, Inc.     

Quick calculation for sample size while controlling false discovery rate with application to microarray analysis

MOTIVATION: Sample size calculation is important in experimental design and is even more so in microarray or proteomic experiments since only a few repetitions can be afforded. In the multiple testing problems involving these experiments, it is more powerful and more reasonable to control false discovery rate (FDR) or positive FDR (pFDR) instead of type I error, e.g. family-wise error rate (FWER). When controlling FDR, the traditional approach of estimating sample size by controlling type I error is no longer applicable. RESULTS: Our proposed method applies to controlling FDR. The sample size calculation is straightforward and requires minimal computation, as illustrated with two sample t-tests and F-tests. Based on simulation with the resultant sample size, the power is shown to be achievable by the q-value procedure. AVAILABILITY: A Matlab code implementing the described methods is available upon request.     

Gene expression * Quick calculation for sample size while controlling false discovery rate with application to microarray analysis

Vol. 23, No. 6, 2007, pp. 739–746 doi:10.1093/bioinformatics/btl664 The calculation of the sample sizes using the method of Poundsand Cheng     

Time to the MRCA of a sample in a Wright-Fisher model with variable population size

Determining the expected distribution of the time to the most recent common ancestor of a sample of individuals may deliver important information about the genetic markers and evolution of the population. In this paper, we introduce a new recursive algorithm to calculate the distribution of the time to the most recent common ancestor of the sample from a population evolved by any conditional multinomial sampling model. The most important advantage of our method is that it can be applied to a sample of any size drawn from a population regardless of its size growth pattern. We also present a very efficient method to implement and store the genealogy tree of the population evolved by the Galton–Watson process. In the final section we present results applied to a simulated population with a single bottleneck event and to real populations of known size histories.     

Interim analysis and sample size reassessment

This article deals with sample size reassessment for adaptive two-stage designs based on conditional power arguments utilizing the variability observed at the first stage. Fisher's product test for the p-values from the disjoint samples at the two stages is considered in detail for the comparison of the means of two normal populations. We show that stopping rules allowing for the early acceptance of the null hypothesis that are optimal with respect to the average sample size may lead to a severe decrease of the overall power if the sample size is a priori underestimated. This problem can be overcome by choosing designs with low probabilities of early acceptance or by midtrial adaptations of the early acceptance boundary using the variability observed in the first stage. This modified procedure is negligibly anticonservative and preserves the power.     

Test statistic and sample size for a two-sample McNemar test

McNemar's (1947, Psychometrika 12, 153-157) test of marginal homogeneity is generalized to a two-sample situation where the hypothesis of interest is that the marginal changes in each of two independently sampled tables are equal. This situation is especially applicable to two cohorts (a control and an intervention cohort), each measured at baseline and after the intervention on a binary outcome variable. Some assumptions often realistic in this situation simplify the calculation of sample size. The calculation of sample size in a study designed to increase utilization of breast cancer screening is demonstrated.     

Variance estimation in clinical studies with interim sample size re-estimation

We consider clinical studies with a sample size re-estimation based on the unblinded variance estimation at some interim point of the study. Because the sample size is determined in such a flexible way, the usual variance estimator at the end of the trial is biased. We derive sharp bounds for this bias. These bounds have a quite simple form and can help for the decision if this bias is negligible for the actual study or if a correction should be done. An exact formula for the bias is also provided. We discuss possibilities to get rid of this bias or at least to reduce the bias substantially. For this purpose, we propose a certain additive correction of the bias. We see in an example that the significance level of the test can be controlled when this additive correction is used.     

Blinded sample size reestimation in non-inferiority trials with binary endpoints

Sample size calculations in the planning of clinical trials depend on good estimates of the model parameters involved. When the estimates of these parameters have a high degree of uncertainty attached to them, it is advantageous to reestimate the sample size after an internal pilot study. For non-inferiority trials with binary outcome we compare the performance of Type I error rate and power between fixed-size designs and designs with sample size reestimation. The latter design shows itself to be effective in correcting sample size and power of the tests when misspecification of nuisance parameters occurs with the former design.     

Power and sample size estimation in microarray studies

Background  

Before conducting a microarray experiment, one important issue that needs to be determined is the number of arrays required in order to have adequate power to identify differentially expressed genes. This paper discusses some crucial issues in the problem formulation, parameter specifications, and approaches that are commonly proposed for sample size estimation in microarray experiments. Common methods for sample size estimation are formulated as the minimum sample size necessary to achieve a specified sensitivity (proportion of detected truly differentially expressed genes) on average at a specified false discovery rate (FDR) level and specified expected proportion (π ₁) of the true differentially expression genes in the array. Unfortunately, the probability of detecting the specified sensitivity in such a formulation can be low. We formulate the sample size problem as the number of arrays needed to achieve a specified sensitivity with 95% probability at the specified significance level. A permutation method using a small pilot dataset to estimate sample size is proposed. This method accounts for correlation and effect size heterogeneity among genes.     

Analytic power and sample size calculation for the genotypic transmission/disequilibrium test in case‐parent trio studies

Case‐parent trio studies considering genotype data from children affected by a disease and their parents are frequently used to detect single nucleotide polymorphisms (SNPs) associated with disease. The most popular statistical tests for this study design are transmission/disequilibrium tests (TDTs). Several types of these tests have been developed, for example, procedures based on alleles or genotypes. Therefore, it is of great interest to examine which of these tests have the highest statistical power to detect SNPs associated with disease. Comparisons of the allelic and the genotypic TDT for individual SNPs have so far been conducted based on simulation studies, since the test statistic of the genotypic TDT was determined numerically. Recently, however, it has been shown that this test statistic can be presented in closed form. In this article, we employ this analytic solution to derive equations for calculating the statistical power and the required sample size for different types of the genotypic TDT. The power of this test is then compared with the one of the corresponding score test assuming the same mode of inheritance as well as the allelic TDT based on a multiplicative mode of inheritance, which is equivalent to the score test assuming an additive mode of inheritance. This is, thus, the first time the power of these tests are compared based on equations, yielding instant results and omitting the need for time‐consuming simulation studies. This comparison reveals that these tests have almost the same power, with the score test being slightly more powerful.     

A genome-wide study of preferential amplification/hybridization in microarray-based pooled DNA experiments

Microarray-based pooled DNA methods overcome the cost bottleneck of simultaneously genotyping more than 100000 markers for numerous study individuals. The success of such methods relies on the proper adjustment of preferential amplification/hybridization to ensure accurate and reliable allele frequency estimation. We performed a hybridization-based genome-wide single nucleotide polymorphisms (SNPs) genotyping analysis to dissect preferential amplification/hybridization. The majority of SNPs had less than 2-fold signal amplification or suppression, and the lognormal distributions adequately modeled preferential amplification/hybridization across the human genome. Comparative analyses suggested that the distributions of preferential amplification/hybridization differed among genotypes and the GC content. Patterns among different ethnic populations were similar; nevertheless, there were striking differences for a small proportion of SNPs, and a slight ethnic heterogeneity was observed. To fulfill appropriate and gratuitous adjustments, databases of preferential amplification/hybridization for African Americans, Caucasians and Asians were constructed based on the Affymetrix GeneChip Human Mapping 100 K Set. The robustness of allele frequency estimation using this database was validated by a pooled DNA experiment. This study provides a genome-wide investigation of preferential amplification/hybridization and suggests guidance for the reliable use of the database. Our results constitute an objective foundation for theoretical development of preferential amplification/hybridization and provide important information for future pooled DNA analyses.