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A method is developed to calculate the approximate number of subjects required to obtain an exact confidence interval of desired width for certain types of intraclass correlations in one-way and two-way ANOVA models. The sample size approximation is shown to be very accurate.
We derive the asymptotic power function of the score test for detecting a common relative risk greater than unity from multiple 2 x 2 tables and formulate methods of sample size determination for use when designing stratified prospective studies. The stratified score test is more efficient than the unstratified test when the latter is unbiased.
Kang et al. [Genet Epidemiol 2004;26:132-141] addressed the question of which genotype misclassification errors are most costly, in terms of minimum percentage increase in sample size necessary (%MSSN) to maintain constant asymptotic power and significance level, when performing case/control studies of genetic association in a genetic model-free setting. They answered the question for single nucleotide polymorphisms (SNPs) using the 2 x 3 chi2 test of independence. We address the same question here for a genetic model-based framework. The genetic model parameters considered are: disease model (dominant, recessive), genotypic relative risk, SNP (marker) and disease allele frequency, and linkage disequilibrium. %MSSN coefficients of each of the six possible error rates are determined by expanding the non-centrality parameter of the asymptotic distribution of the 2 x 3 chi2 test under a specified alternative hypothesis to approximate %MSSN using a linear Taylor series in the error rates. In this work we assume errors misclassifying one homozygote as another homozygote are 0, since these errors are thought to rarely occur in practice. Our findings are that there are settings of the genetic model parameters that lead to large total %MSSN for both dominant and recessive models. As SNP minor allele approaches 0, total %MSSN increases without bound, independent of other genetic model parameters. In general, %MSSN is a complex function of the genetic model parameters. Use of SNPs with small minor allele frequency requires careful attention to frequency of genotyping errors to insure that power specifications are met. Software to perform these calculations for study design is available, and an example of its use to study a disease is given.
Three methods for incorporating cost considerations into sample size determination for cohort and case-control studies are considered. It is found that when the costs of obtaining exposed and nonexposed subjects or of obtaining case and control subjects differ, an improvement in design can be achieved.
As genetic markers become more available, case-control studies will be increasingly important in defining the role of genetic factors in disease causality. The authors estimate the minimum sample size needed to assure adequate statistical power to detect gene-environment interaction. One assumption is made: the prevalence of exposure is independent of marker genotypes among controls. Given the assumption, six parameters (three odds ratios, the prevalence of exposure, the proportion of those with the susceptible genotype, and the ratio of controls to cases) dictate the expected cell sizes in a 2 x 2 x 2 table contrasting genetic susceptibility, exposure, and disease. The three odds ratios reflect the association between disease and 1) exposure among non-susceptibles; 2) susceptible genotypes among nonexposed individuals; and 3) the gene-environment interaction itself, respectively. Given these parameters, the number of cases and controls needed to assure any particular Type I and Type II error rates can be estimated. Results presented here demonstrate that case-control designs can be used to detect gene-environment interaction when there is both a common exposure and a highly polymorphic marker of susceptibility.
We derive formulae for estimating sample size and power for detecting the effect of an unrestricted covariate on survival time. These are useful in designing survival studies with different patterns of recruitment and follow-up when survival time is exponentially distributed. We use the asymptotic covariance matrix, conditional expectation and Taylor's expansion techniques to develop these formulae. Computer simulations indicate that the asymptotic approximations used in developing the formulae are good over a range of parameter values and different patterns of recruitment and follow-up that are relevant to survival studies.
Gene expression analysis using high-throughput microarray technology has become a powerful approach to study systems biology. The exponential growth in microarray experiments has spawned a number of investigations into the reliability and reproducibility of this type of data. However, the sample size requirements necessary to obtain statistically significant results has not had as much attention. We report here statistical methods for the determination of the sufficient number of subjects necessary to minimize the false discovery rate while maintaining high power to detect differentially expressed genes. Two experimental designs were considered: 1) a comparison between two groups at a single time point, and 2) a comparison of two experimental groups with sequential time points. Computer programs are available for the methods discussed in this paper and are adaptable to more complicated situations.
This paper discusses some general methods for determining approximate power, sample size, and smallest detectable effect for studies of multiple risk factors. These methods are based on standard large-sample formulae for determining the power of chi-square tests, and emphasis is given to determinations for Pearson chi 2 tests in multiway contingency tables. The methods are illustrated in application to the design of a clinical trial of the preventive effect of alpha-tocopherol, ascorbic acid and beta-carotene on colon polyp recurrence, and a case-control study of the joint effect of smoking and asbestos exposure on lung cancer incidence.
The presence of linkage disequilibrium between closely linked loci can aid in the fine mapping of disease loci. We investigate the power of several designs for sampling individuals with different disease genotypes. As expected, haplotype data provide the greatest power for detecting disequilibrium, but, in the absence of parental information to resolve the phase of double heterozygotes, the most powerful design samples only individuals homozygous at the trait locus. For rare diseases, such a scheme is generally not feasible, and we also provide power and sample-size calculations for designs that sample heterozygotes. The results provide information useful in planning disequilibrium studies.
Large sample sizes of uranyl ions are eluted on a strenedivinylbenzene copolymer phase and an octadecyl phase column, respectively, using alpha-hydroxyisobutyric acid (alpha-HiBA) as an eluent. Chromatograms are obtained from variations of the uranyl sample amounts, eluent concentrations, concentrations of the sample matrix, and the pH of the sample solution for both columns, respectively. Column capacities are estimated from the loading factors measured from the retention times of the peaks. Bandwidths of the peaks and apparent column efficiencies are measured as a function of the loading factor and calculated using the equations derived from the assumptions of a Langmuir isotherm for a single solute. Comparison between the experiment and the calculation reveals that the former showed a broader bandwidth and worse column efficiency than the latter for both columns. The two columns are compared with regards to the retention time, peak shape, column capacity, column efficiency, etc. The PRP-1 column shows a rectangular-, triangle-type peak shape, longer retention time, lower column capacity, and better column efficiency, and the LC-18 column shows a distorted Gaussian curve, shorter retention time, higher column capacity, and worse column efficiency. Column capacity, peak shape, and retention time are dependent on the eluent concentration rather than the alpha-HiBA concentration in the sample solutions.
I provide sample size formulae and tables for the design of studies that compare two or more coefficients of inter-observer agreement or concordance. Such studies may arise, for example, when interest centres on assessment of how measures of inter-observer agreement vary across different patient subgroups or treatment centres. I consider cases of both a continuous and a dichotomous outcome measure. Three examples illustrate the results.
In early phase clinical trials of a new medical treatment, patients are treated to decide whether there is sufficient promise to justify additional studies. A decision theoretic approach is proposed to help determine the number of patients that should be treated. The optimal sample size is obtained by maximizing a utility function which incorporates both the number of 'gained successes' and the costs of treatment. The method extends work of Sylvester and Staquet, and adopts a Bayesian formulation. Numbers of patients in later studies and in eventual routine use of the treatment are taken into account. We allow for the possibility that a later study might lead to an erroneous conclusion. The effects of these various influences on the recommended sampling plan for the early phase clinical trial are explored.
Clinical trials often determine the sample size based on the use of statistical methods such as analysis of variance, t tests, and rank sum tests, which compare mean or median values. The resulting studies rarely are big enough to show that the method is based on mistaken assumptions. Data from a recent clinical trial of nephrotoxicity associated with the use of contrast agents during angiography found a significant difference on the order of the difference that previous studies had intended to detect. It also showed that a central assumption does not apply to changes in serum creatinine. As a result, the previous studies had considerably lower power than believed. Their lack of significance reflected only the mistaken assumption. Analysis of variance, t tests, and rank sum tests may be just as invalid for other clinical parameters. Lack of significance cannot be automatically taken to imply a small treatment effect.
To increase change between groups, randomised clinical trials (RCT) often include patients with high risk for a particular outcome, by inclusion criteria that select predictors for that outcome. This increases the statistical power, and fewer patients are required for that RCT. The way in which patient selection influences the power, and thus sample size required, depends on how an intervention reduces the individual risk: by an absolute or relative risk reduction model.
Four different nomograms were devised to obtain a necessary, minimum sample size at a 95% confidence rate when data were distributed normally. They corresponded to four different cases, the population of which was either infinite or finite and the permitted error of which was either mu--m or mu--m/s, where mu was population mean, m sample mean and s sample standard deviation. They could also be used for obtaining the confidence limits of the population mean from the data after having carried out a work.
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We have undertaken a series of experiments to examine several issues that directly affect design of gene expression studies using Affymetrix GeneChip arrays: probe-level analysis, need for technical replication, relative contribution of various sources of variability, and utility of pooling RNA from different samples. Probe-level data were analyzed by Affymetrix MAS 5.0, and three model-based methods, PM-MM and PM-only models by dChip, and the RMA model by Bioconductor, with the latter two providing the best performance. We found that replicate chips of the same RNA have limited value in reducing total variability, and for relatively highly expressed genes in this biologically homogeneous animal model of aging, about 11% of total variation is due to day effects and the remainder is approximately equally split between sample and residual sources. We also found that pooling samples is neither advantageous nor detrimental. Finally we suggest a strategy for sample size calculations using formulas appropriate when coefficients of variation are known, target effects are expressed as fold changes, and data can be assumed to be approximately lognormally distributed.
To test the effect of a therapeutic or prognostic factor on the occurrence of a particular cause of failure in the presence of other causes, the interest has shifted in some studies from the modelling of the cause-specific hazard to that of the subdistribution hazard. We present approximate sample size formulas for the proportional hazards modelling of competing risk subdistribution, considering either independent or correlated covariates. The validity of these approximate formulas is investigated through numerical simulations. Two illustrations are provided, a randomized clinical trial, and a prospective prognostic study.