Sample size for individually matched case-control studies

The standard formulas used to calculate sample size for an individually matched case-control study assume a constant probability of exposure throughout the pool of possible controls. We propose new formulas that allow for heterogeneity in the probability of exposure among controls in different matched sets. Since matching factors are suspected of being confounders, they are expected to divide the total population into subgroups with different proportions exposed. Thus, the assumption of homogeneity of exposure among controls, made by the currently used formulas, is inconsistent with the assumptions used to design a matched study. The proposed formulas avoid this inconsistency. We present an example to illustrate how heterogeneity can affect the required sample size.     

Sample size for case-control studies using Cochran's statistic

Sample size determination for case-control studies of chronic disease are often based on the simple 2 X 2 tabular cross-classification of exposure and disease, thereby ignoring stratification which may be considered in the analysis. One consequence of this approach is that the sample size may be inadequate to attain a specified power and size when performing a statistical analysis on J 2 X 2 tables using Cochran's (1954, Biometrics 10, 417-451) statistic or the Mantel-Haenszel (1959, Journal of the National Cancer Institute 22, 719-748) statistic. A sample size formula is derived from Cochran's statistic and it is compared with the corresponding one derived when the data are treated as unstratified, and also with two other formulas proposed for stratified data analysis. The formula developed yields values slightly higher than one recently proposed by Mu?oz and Rosner (1984, Biometrics 40, 995-1004), which assumes that both margins of each 2 X 2 table are fixed, while the present study considers only the case-control margin to be fixed.     

Sample size and sampling error in geometric morphometric studies of size and shape

Geometric morphometric studies are increasingly becoming common in systematics and palaeontology. The samples in such studies are often small, due to the paucity of material available for analysis. However, very few studies have tried to assess the impact of sampling error on analytical results. Here, this issue is addressed empirically using repeated randomized selection experiments to build progressively smaller samples from an original dataset of ∼400 vervet monkey (Cercopithecus aethiops) skulls. Size and shape parameters (including mean size and shape, size and shape variances, angles of allometric trajectories) that are commonly used in geometric morphometric studies, are estimated first in the original sample and then in the random subsamples. Estimates are then compared to give an indication of what is the minimum desirable sample size for each parameter. Mean size, standard deviation of size and variance of shape are found to be fairly accurate even in relatively small samples. In contrast, mean shapes and angles between static allometric trajectories are strongly affected by sampling error. If confirmed in other groups, our findings may have substantial implications for studies of morphological variation in present and fossil species. By performing rarefaction analyses like those presented in our study, morphometricians can be easily provided with important clues on how a simple but crucial factor like sample size can alter results of their studies.     

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 and replication in 2D gel electrophoresis studies

Two-dimensional gel electrophoresis can be an expensive technology, and many studies are based on a modest number of replicates. It is important that the statistical power is sufficient to detect protein expression differences of interest. This paper reviews the application of power calculations and considers how other issues affect the choice of sample size. The important distinction between biological and technical replication is made, and the superiority of the former is emphasized.     

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 for a phylogenetic inference.

The objective of this work is to describe sample-size calculations for the inference of a nonzero central branch length in an unrooted four-species phylogeny. Attention is restricted to independent binary characters, such as might be obtained from an alignment of the purine-pyrimidine sequences of a nucleic acid molecule. A statistical test based on a multinomial model for character-state configurations is described. The importance of including invariable sites in models for sequence change is demonstrated, and their effect on sample size is quantified. The methods are applied to a four-species alignment of small-subunit rRNA sequences derived from two archaebacteria, a eubacteria and a eukaryote. We conclude that the information in these sequences is not sufficient to resolve the branching order of this tree. Estimates of the number of aligned nucleotide positions required to provide a reasonably powerful test are given.     

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.     

Balancing population size and genetic information in parentage analysis studies

In parentage analysis studies, the parameters of interest typically are not the parent assignments themselves, but population parameters such as variance in fertility, self-pollination rate, or average dispersal distances. The precision of parameter estimates is affected by two factors: the number of offspring under consideration, and the precision with which the offspring can be assigned to parents. When assignment of parents is based on genetic information, the confidence in assignments is affected by the number and polymorphism of the loci considered, and by the number of potential parents in the population. Studying larger populations may yield higher numbers of offspring, but since larger populations contain more potential parents, more (or more highly polymorphic) loci are necessary to attain a given level of confidence in the parent assignments. This article addresses how to relate the size of the population and the number of loci when designing a study. It is shown that the number of loci needed to assign all offspring unambiguously is proportional to the logarithm of the population size. In some cases, the constant of proportionality can be determined, eliminating the need for simulation-based projections. Population-wide measures of uncertainty in parent assignments are also introduced, and it is shown that holding uncertainty "steady" as the population size increases also requires increasing the number of loci proportional to the logarithm of the population size. Data from a study of self-pollination are used to illustrate the techniques suggested.     

Sample size determination for case-control studies and the comparison of stratified and unstratified analyses.

Woolson, Bean, and Rojas (1986, Biometrics 42, 927-932) present a simple approximation of sample size for Cochran's (1954, Biometrics 10, 417-451) test for detecting association between exposure and disease. It is useful in the design of case-control studies. We derive a sample size formula for Cochran's statistic with continuity correction which guarantees that the actual Type I error rate of the test does not exceed the nominal level. The corrected sample size is necessarily larger than the uncorrected one given by Woolson et al. and the relative difference between the two sample sizes is considerable. Allocation of equal number of cases and controls within each stratum is asymptotically optimal when the costs per case and control are the same. When any effect of stratification is absent, Cochran's stratified test, although valid, is less efficient than the unstratified one except for the important case of a balanced design.     

Actuarial considerations on genetic testing.

In the UK the majority of life insurers employ relatively liberal underwriting standards so that people can easily gain access to life assurance cover. Up to 95% of applicants are accepted at standard terms. If genetic testing becomes widespread then the buying habits of the public may change. Proportionately more people with a predisposition to major types of disease may take life assurance cover while people with no predisposition may take proportionately less. A model is used to show the possible effect. However, the time-scales are long and the mortality of assured people is steadily improving. The change in buying habits may result in the rate of improvement slowing down. In the whole population, the improvement in mortality is likely to continue and could improve faster if widespread genetic testing results in earlier diagnosis and treatment. Life insurers would not call for genetic tests and need not see the results of previous tests except for very large sums assured. In the UK, life insurers are unlikely to change their underwriting standards, and are extremely unlikely to bring in basic premium rating systems that give discounts on the premium or penalty points according to peoples genetic profile. The implications of widespread genetic testing on medical insurance and some health insurance covers may be more extreme.     

Sample size and statistical power considerations in high-dimensionality data settings: a comparative study of classification algorithms

Background  

Data generated using 'omics' technologies are characterized by high dimensionality, where the number of features measured per subject vastly exceeds the number of subjects in the study. In this paper, we consider issues relevant in the design of biomedical studies in which the goal is the discovery of a subset of features and an associated algorithm that can predict a binary outcome, such as disease status. We compare the performance of four commonly used classifiers (K-Nearest Neighbors, Prediction Analysis for Microarrays, Random Forests and Support Vector Machines) in high-dimensionality data settings. We evaluate the effects of varying levels of signal-to-noise ratio in the dataset, imbalance in class distribution and choice of metric for quantifying performance of the classifier. To guide study design, we present a summary of the key characteristics of 'omics' data profiled in several human or animal model experiments utilizing high-content mass spectrometry and multiplexed immunoassay based techniques.     

Sample size for identifying differentially expressed genes in microarray experiments.

Microarray technology allows simultaneous comparison of expression levels of thousands of genes under each condition. This paper concerns sample size calculation in the identification of differentially expressed genes between a control and a treated sample. In a typical experiment, only a fraction of genes (altered genes) is expected to be differentially expressed between two samples. Sample size determination depends on a number of factors including the specified significance level (alpha), the desired statistical power (1-beta), the fraction (eta) of truly altered genes out of the total g genes studied, and the effect sizes (Delta) for the altered genes. This paper proposes a method to calculate the number of arrays required to detect at least 100lambda % (where 0 < lambda < or = 1) of the truly altered genes under the model of an equal effect size for all altered genes. The required numbers of arrays are tabulated for various values of alpha, beta, Delta, eta, and lambda for the one-sample and two-sample t-tests for g = 10,000. Based on the proposed approach, to identify up to 90% of truly altered genes among the unknown number of truly altered genes, the estimated numbers of arrays needed appear to be manageable. For instance, when the standardized effect size is at least 2.0, the number of arrays needed is less than or equal to 14 for the two-sample t-test and is less than or equal to 10 for the one-sample t-test. As the cost per array declines, such array numbers become practical. The proposed method offers a simple, intuitive, and practical way to determine the number of arrays needed in microarray experiments in which the true correlation structure among the genes under investigation cannot be reasonably assumed. An example dataset is used to illustrate the use of the proposed approach to plan microarray experiments.     

Phosphoglucomutase genetic polymorphism and human fertility.

We studied the phosphoglucomutase phenotype in relation to fertility parameters in a consecutive series of 204 women who had delivered a normal live-born child in Rome. A highly significant association was found between age of the women and phosphoglucomutase phenotype, suggesting a reduced rate of reproduction among women of phosphoglucomutase Type 1. Previous spontaneous abortion appears related to both age and phosphoglucomutase enzymatic type. An increased incidence of abortion in women of older ages was observed only in phosphoglucomutase Type 1. Gestational duration and fetal intrauterine growth rate are also significantly associated with maternal phosphoglucomutase phenotype. The pattern is complex, but also in this instance the influence of maternal age was evident. Considered altogether, the data suggest that phosphoglucomutase may have an important role in zygote development and survival through the whole span of intrauterine life.