Sample size calculation for microarray experiments with blocked one-way design

Background  

One of the main objectives of microarray analysis is to identify differentially expressed genes for different types of cells or treatments. Many statistical methods have been proposed to assess the treatment effects in microarray experiments.     

Sample size for FDR-control in microarray data analysis

We consider identifying differentially expressing genes between two patient groups using microarray experiment. We propose a sample size calculation method for a specified number of true rejections while controlling the false discovery rate at a desired level. Input parameters for the sample size calculation include the allocation proportion in each group, the number of genes in each array, the number of differentially expressing genes and the effect sizes among the differentially expressing genes. We have a closed-form sample size formula if the projected effect sizes are equal among differentially expressing genes. Otherwise, our method requires a numerical method to solve an equation. Simulation studies are conducted to show that the calculated sample sizes are accurate in practical settings. The proposed method is demonstrated with a real study.     

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 requirements for addressing the population genetic issues of forensic use of DNA typing.

DNA typing offers a unique opportunity to identify individuals for medical and forensic purposes. Probabilistic inference regarding the chance occurrence of a match between the DNA type of an evidentiary sample and that of an accused suspect, however, requires reliable estimation of genotype and allele frequencies in the population. Although population-based data on DNA typing at several hypervariable loci are being accumulated at various laboratories, a rigorous treatment of the sample size needed for such purposes has not been made from population genetic considerations. It is shown here that the loci that are potentially most useful for forensic identification of individuals have the intrinsic property that they involve a large number of segregating alleles, and a great majority of these alleles are rare. As a consequence, because of the large number of possible genotypes at the hypervariable loci that offer the maximum potential for individualization, the sample size needed to observe all possible genotypes in a sample is large. In fact, the size is so large that even if such a huge number of individuals could be sampled, it could not be guaranteed that such a sample was drawn from a single homogeneous population. Therefore adequate estimation of genotypic probabilities must be based on allele frequencies, and the sample size needed to represent all possible alleles is far more reasonable. Further economization of sample size is possible if one wants to have representation of only the frequent alleles in the sample, so that the rare allele frequencies can be approximated by an upper bound for forensic applications.     

Sample size calculation for multiple testing in microarray data analysis

Microarray technology is rapidly emerging for genome-wide screening of differentially expressed genes between clinical subtypes or different conditions of human diseases. Traditional statistical testing approaches, such as the two-sample t-test or Wilcoxon test, are frequently used for evaluating statistical significance of informative expressions but require adjustment for large-scale multiplicity. Due to its simplicity, Bonferroni adjustment has been widely used to circumvent this problem. It is well known, however, that the standard Bonferroni test is often very conservative. In the present paper, we compare three multiple testing procedures in the microarray context: the original Bonferroni method, a Bonferroni-type improved single-step method and a step-down method. The latter two methods are based on nonparametric resampling, by which the null distribution can be derived with the dependency structure among gene expressions preserved and the family-wise error rate accurately controlled at the desired level. We also present a sample size calculation method for designing microarray studies. Through simulations and data analyses, we find that the proposed methods for testing and sample size calculation are computationally fast and control error and power precisely.     

Sample size requirements for studies of treatment effects on beta-cell function in newly diagnosed type 1 diabetes

Preservation of β-cell function as measured by stimulated C-peptide has recently been accepted as a therapeutic target for subjects with newly diagnosed type 1 diabetes. In recently completed studies conducted by the Type 1 Diabetes Trial Network (TrialNet), repeated 2-hour Mixed Meal Tolerance Tests (MMTT) were obtained for up to 24 months from 156 subjects with up to 3 months duration of type 1 diabetes at the time of study enrollment. These data provide the information needed to more accurately determine the sample size needed for future studies of the effects of new agents on the 2-hour area under the curve (AUC) of the C-peptide values. The natural log(x), log(x+1) and square-root (√x) transformations of the AUC were assessed. In general, a transformation of the data is needed to better satisfy the normality assumptions for commonly used statistical tests. Statistical analysis of the raw and transformed data are provided to estimate the mean levels over time and the residual variation in untreated subjects that allow sample size calculations for future studies at either 12 or 24 months of follow-up and among children 8-12 years of age, adolescents (13-17 years) and adults (18+ years). The sample size needed to detect a given relative (percentage) difference with treatment versus control is greater at 24 months than at 12 months of follow-up, and differs among age categories. Owing to greater residual variation among those 13-17 years of age, a larger sample size is required for this age group. Methods are also described for assessment of sample size for mixtures of subjects among the age categories. Statistical expressions are presented for the presentation of analyses of log(x+1) and √x transformed values in terms of the original units of measurement (pmol/ml). Analyses using different transformations are described for the TrialNet study of masked anti-CD20 (rituximab) versus masked placebo. These results provide the information needed to accurately evaluate the sample size for studies of new agents to preserve C-peptide levels in newly diagnosed type 1 diabetes.     

Sample size calculation, power analysis and randomization: research project design in Windows

Single estimates of sample size for a study may be easily obtainedby use of a hand calculator or from published tables. In contrast,performing multiple calculations is a tedious and time-consumingtask, which is greatly simplified by a computer program. Thecomputer program presented here assists the investigator incalculating sample size estimates, determining statistical powerand creating randomization tables for a study. The program isdesigned primarily for clinical trials and thus includes somefeatures not found in other software packages performing similartasks. Sample size calculation and power analysis are performedfor dichotomous, continuous Coammetric and non–parametrictests) and time–to–failure (exponential distributionand log-rank test) response variables, and for correlation coefficients.Sample size estimates and significance levels may be adjustedfor multiple participating centers, non compliance, interimanalyses and multiple testing. The randomization subroutinegenerates tables for studies with up to nine treatment armsand with any valid block size. As a Windows application, theprogram runs in a multitasking environment, allowing switchingbetween programs and easy pasting of results into word–processingdocuments and other applications. It is very simple to use,with a completely menu–driven interface and sufficientbuilt–in help to obviate the use of a manual.     

Sample size for beginners.

The common failure to include an estimation of sample size in grant proposals imposes a major handicap on applicants, particularly for those proposing work in any aspect of research in the health services. Members of research committees need evidence that a study is of adequate size for there to be a reasonable chance of a clear answer at the end. A simple illustrated explanation of the concepts in determining sample size should encourage the faint hearted to pay more attention to this increasingly important aspect of grantsmanship.     

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.     

Sexual selection: harem size and the variance in male reproductive success

Sexual selection is potentially stronger than natural selection when the variance in male reproductive fitness exceeds all other components of fitness variance combined. However, measuring the variance in male reproductive fitness is difficult when nonmating males are absent, inconspicuous, or otherwise difficult to find. Omitting the nonmating males inflates estimates of average male reproductive success and diminishes the variance, leading to underestimates of the potential strength of sexual selection. We show that, in theory, the proportion of the total variance in male fitness owing to sexual selection is approximately equal to H, the mean harem size, as long as H is large and females are randomly distributed across mating males (i.e., Vharem=H). In this case, mean harem size not only provides an easy way to estimate the potential strength of sexual selection but also equals the opportunity for sexual selection, I(mates). In nature, however, females may be overdispersed with VharemH. We show that H+(k-1) is a good measure of the opportunity for sexual selection, where k is the ratio Vharem/H. A review of mating system data reveals that in nature the median ratio for Vharem/H is 1.04, but as H increases, females tend to become more aggregated across mating males with V(harem) two to three times larger than H.