Sample size determination in microarray experiments for class comparison and prognostic classification

Determining sample sizes for microarray experiments is important but the complexity of these experiments, and the large amounts of data they produce, can make the sample size issue seem daunting, and tempt researchers to use rules of thumb in place of formal calculations based on the goals of the experiment. Here we present formulae for determining sample sizes to achieve a variety of experimental goals, including class comparison and the development of prognostic markers. Results are derived which describe the impact of pooling, technical replicates and dye-swap arrays on sample size requirements. These results are shown to depend on the relative sizes of different sources of variability. A variety of common types of experimental situations and designs used with single-label and dual-label microarrays are considered. We discuss procedures for controlling the false discovery rate. Our calculations are based on relatively simple yet realistic statistical models for the data, and provide straightforward sample size calculation formulae.     

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.     

The Planning of Sample Sizes for Maximum Likelihood Estimation of the Parameters of the Weibull Distribution

Based on maximum likelihood estimators the problem of setting confidence limits for the scale and shape parameters of the Weibull distribution with two parameters is considered. Approximation formulae for the determination of sample sizes for the parameter estimation are given for complete and type II censored samples.     

Sample size calculations based on ranking and selection in microarray experiments

Summary .   We develop formulae to calculate sample sizes for ranking and selection of differentially expressed genes among different clinical subtypes or prognostic classes of disease in genome-wide screening studies with microarrays. The formulae aim to control the probability that a selected subset of genes with fixed size contains enough truly top-ranking informative genes, which can be assessed on the basis of the distribution of ordered statistics from independent genes. We provide strategies for conservative designs to cope with issues of unknown number of informative genes and unknown correlation structure across genes. Application of the formulae to a clinical study for multiple myeloma is given.     

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.     

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 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.     

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.     

自交作物群体的遗传组成分析

用数学方法推导出了自交作物群体遗传组成的分析公式。根据所推出的公式,可以计算出任何自交世代群体的基因型频率和同一表现型中各基因型的频率。对自交群体遗传组成的分析可用于质量性状的选择,确定选择时需要的有效群体大小和选出有效群体的大小,亦有助于确定选择的最佳世代。     

Design and analysis of experiments on mutagenicity. II. Assays involving microorganisms.

The design and statistical analysis of mutagenicity experiments involving microorganisms and a single dose of mutagen are discussed. Test statistics are derived for use in determining the mutagenicity of a chemical when survival data are available and also when such data are not available. One's likelihood (power) of correctly concluding a chemical is mutagenic is examined, and minimum total sample sizes required for 95% power are presented. It is found that one generally has greater power when survival data are available. Required precision is estimating survival is discussed in reference to type-1 and type-2 errors. The proper use of the formulae and figures presented is illustrated by examples.     

Design and analysis of experiments on mutagenicity: II. Assays involving microorganisms

The design and statistical analysis of mutagenicity experiments involving microorganisms and a single dose of mutagen are discussed. Test statistics are derived for use in determining the mutagenicity of a chemica; when survival data are available and also when such data are not available. One's likelihood (power) of correctly concluding a chemical is mutagenic is examined, and minimum total sample sizes required for 95% power are presented. It is found that one generally has greater power when survival data are available. Required precision is estimating survival is discussed in reference to type-1 and type-2 errors. The proper use of the formulae and figures presented is illustrated by examples.     

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.     

Comparing Accuracy in an Unpaired Post‐market Device Study with Incomplete Disease Assessment

The sensitivity and specificity of a new medical device are often compared relative to that of an existing device by calculating ratios of sensitivities and specificities. Although it would be ideal for all study subjects to receive the gold standard so true disease status was known for all subjects, it is often not feasible or ethical to obtain disease status for everyone. This paper proposes two unpaired designs where each subject is only administered one of the devices and device results dictate which subjects are to receive disease verification. Estimators of the ratio of accuracy and corresponding confidence intervals are proposed for these designs as well as sample size formulae. Simulation studies are performed to investigate the small sample bias of the estimators and the performance of the variance estimators and sample size formulae. The sample size formulae are applied to the design of a cervical cancer study to compare the accuracy of a new device with the conventional Pap smear.     

Further Results in Estimating the Classification Error in Discriminance Analysis

Basing on the approach by McLachlan (1977) a procedure for the conditional and common error estimation of the classification error in discriminance analysis is described for k ≧ 2 classes. As a rapid procedure for large sample sizes and feature numbers, a modification of the resubstitution method is proposed being favourable with respect to computing time. Both methods provide useful estimations for the probability of misclassification. In calculating the weighting function w, deviations from preconditions known from the MANOVA such as the skewness, the truncation or the inequality of the covariance matrices, hardly play any role; it appears that only a variation of the sample sizes of the classes substantially influences the weighting functions. The error rates of the tested error estimation methods likewise in effect depend on the sample sizes of the classes. Violations of the mentioned preconditions in the form described above result in different variations of the error estimates, depending on these sample sizes. A comparison between error estimation and allocation relative to a simulated population demonstrates the goodness of the used error estimation procedures.     

Inferring Coalescence Times from DNA Sequence Data

The paper is concerned with methods for the estimation of the coalescence time (time since the most recent common ancestor) of a sample of intraspecies DNA sequences. The methods take advantage of prior knowledge of population demography, in addition to the molecular data. While some theoretical results are presented, a central focus is on computational methods. These methods are easy to implement, and, since explicit formulae tend to be either unavailable or unilluminating, they are also more useful and more informative in most applications. Extensions are presented that allow for the effects of uncertainty in our knowledge of population size and mutation rates, for variability in population sizes, for regions of different mutation rate, and for inference concerning the coalescence time of the entire population. The methods are illustrated using recent data from the human Y chromosome.     

Dynamics of gene differentiation between incompletely isolated populations of unequal sizes

The progress of gene differentiation between two populations of unequal sizes is studied by taking into account the effects of mutation and migration. Explicit formulae for the eventual rate of approach to equilibrium of the probabilities of identity of genes by descent within and between populations are worked out for the case of small migration. Formulae for the equilibrium values of identity probabilities are also given for some special cases. It is shown that in the case of large migration the Nei—Feldman formulae for the temporal changes and equilibrium values of identity probabilities approximately hold, unless the sizes of the two populations are extremely different. The theories developed are applied to study the evolution of cave fish populations in Astyanax mexicanus. The approximate time after divergence between the cave and river populations of this fish has been estimated to be 525,000–710,000 years, which agrees well with the geological data on cave formation.     

Reduction of Total Sample Size and Cost of Sampling in Tests of Hypothesis by Using Unequal Group Sizes

The standard approach in calculating the appropriate sample size required to detect a specified difference in the means of two populations for given type I and II errors assumes equal sample sizes in each group. It is shown here that equal sample sizes are usually sub-optimal with respect to total sample size and with respect to total cost of the experiment. This is may be important in experiments involving animals when costs are measured not only in monetary terms but also in terms of animal suffering or inconvenience.     

An Approximate Solution to the Behrens-Fisher Problem

An approximate and practical solution is proposed for the Behrens-Fisher problem. This solution is compared to the solutions considered by Mehta and Srinivasan (1970) and Welch's (1937) approximate t-test in terms of the stability of the size and magnitude of the power. It is shown that the stability of the size of the new test is better than that of Welch's t when at least one of the sample sizes is small. When the sample sizes are moderately large or large the sizes and powers of all the recommended tests are almost the same.     

The frequency spectrum of a mutation,and its age,in a general diffusion model

General formulae are derived for the probability density and expected age of a mutation of frequency x in a population, and similarly for a mutation with b copies in a sample of n genes. A general formula is derived for the frequency spectrum of a mutation in a sample. Variable population size models are included. Results are derived in two frameworks: diffusion process models for the frequency of the mutation; and birth and death process models. The coalescent structure within the mutant gene group and the non-mutant group is considered.     

Sample size determination for matched-pair equivalence trials using rate ratio

In this article, we compare Wald-type, logarithmic transformation, and Fieller-type statistics for the classical 2-sided equivalence testing of the rate ratio under matched-pair designs with a binary end point. These statistics can be implemented through sample-based, constrained least squares estimation and constrained maximum likelihood (CML) estimation methods. Sample size formulae based on the CML estimation method are developed. We consider formulae that control a prespecified power or confidence width. Our simulation studies show that statistics based on the CML estimation method generally outperform other statistics and methods with respect to actual type I error rate and average width of confidence intervals. Also, the corresponding sample size formulae are valid asymptotically in the sense that the exact power and actual coverage probability for the estimated sample size are generally close to their prespecified values. The methods are illustrated with a real example from a clinical laboratory study.