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.     

Effects of net, pump and bottle sampling on the abundance estimates of planktonic Rotifera

Eighty-seven freshwater plankton samples collected by meansof two different nets, a submersible centrifugal pump and a5.81 water bottle were compared with respect to their abilityto catch the rotifer Keratella cochlearis. Sample size, illumination(day/night), the presence of bridles ahead of the net mouths(versus unbridled nets), and different mouth diameters (0.2and 0.5 m) did not affect abundance estimates. Slight differenceswere found between the yields of pushed nets versus towed nets;these are probably due to uneven distribution of the animalsin the 0–3 m layer. Both pump and bottle volumes stronglyaffected abundance estimates (K. cochlearisl^–1 in pumpsamples=164.84 volume of water filtered^{–0 817}; K. cochlearisl^–1in bottle samples=84.74+2336.6 volume of water filtered^–1)Net sample results were always considerably higher (4.6–12.3times) than pump and bottle estimates; these differences aremost probably due to evading reactions of the rotifer.     

A Fast Estimate for the Population Recombination Rate Based on Regression

Recombination is a fundamental evolutionary force. Therefore the population recombination rate ρ plays an important role in the analysis of population genetic data; however, it is notoriously difficult to estimate. This difficulty applies both to the accuracy of commonly used estimates and to the computational efforts required to obtain them. Some particularly popular methods are based on approximations to the likelihood. They require considerably less computational efforts than the full-likelihood method with not much less accuracy. Nevertheless, the computation of these approximate estimates can still be very time consuming, in particular when the sample size is large. Although auxiliary quantities for composite likelihood estimates can be computed in advance and stored in tables, these tables need to be recomputed if either the sample size or the mutation rate θ changes. Here we introduce a new method based on regression combined with boosting as a model selection technique. For large samples, it requires much less computational effort than other approximate methods, while providing similar levels of accuracy. Notably, for a sample of hundreds or thousands of individuals, the estimate of ρ using regression can be obtained on a single personal computer within a couple of minutes while other methods may need a couple of days or months (or even years). When the sample size is smaller (n ≤ 50), our new method remains computational efficient but produces biased estimates. We expect the new estimates to be helpful when analyzing large samples and/or many loci with possibly different mutation rates.     

Satellite Telemetry in Avian Research and Management: Sample Size Considerations

Abstract: Satellite tracking is currently used to make inferences to avian populations. Cost of transmitters and logistical challenges of working with some species can limit sample size and strength of inferences. Therefore, careful study design including consideration of sample size is important. We used simulations to examine how sample size, population size, and population variance affected probability of making reliable inferences from a sample and the precision of estimates of population parameters. For populations of >100 individuals, a sample >20 birds was needed to make reliable inferences about questions with simple outcomes (i.e., 2 possible outcomes). Sample size demands increased rapidly for more complex problems. For example, in a problem with 3 outcomes, a sample of >75 individuals will be needed for proper inference to the population. Combining data from satellite telemetry studies with data from surveys or other types of sampling may improve inference strength.     

Population size estimation with interval censored counts and external information: Prevalence of multiple sclerosis in Rome

We discuss Bayesian log-linear models for incomplete contingency tables with both missing and interval censored cells, with the aim of obtaining reliable population size estimates. We also discuss use of external information on the censoring probability, which may substantially reduce uncertainty. We show in simulation that information on lower bounds and external information can each improve the mean squared error of population size estimates, even when the external information is not completely accurate. We conclude with an original example on estimation of prevalence of multiple sclerosis in the metropolitan area of Rome, where five out of six lists have interval censored counts. External information comes from mortality rates of multiple sclerosis patients.     

Calculating sample size for studies with expected all-or-none nonadherence and selection bias

Summary .  We develop sample size formulas for studies aiming to test mean differences between a treatment and control group when all-or-none nonadherence (noncompliance) and selection bias are expected. Recent work by Fay, Halloran, and Follmann (2007, Biometrics 63, 465–474) addressed the increased variances within groups defined by treatment assignment when nonadherence occurs, compared to the scenario of full adherence, under the assumption of no selection bias. In this article, we extend the authors' approach to allow selection bias in the form of systematic differences in means and variances among latent adherence subgroups. We illustrate the approach by performing sample size calculations to plan clinical trials with and without pilot adherence data. Sample size formulas and tests for normally distributed outcomes are also developed in a Web Appendix that account for uncertainty of estimates from external or internal pilot data.     

Sample size determination for testing whether an identified treatment is best

Laska and Meisner (1989, Biometrics 45, 1139-1151) dealt with the problem of testing whether an identified treatment belonging to a set of k + 1 treatments is better than each of the other k treatments. They calculated sample size tables for k = 2 when using multiple t-tests or Wilcoxon-Mann-Whitney tests, both under normality assumptions. In this paper, we provide sample size formulas as well as tables for sample size determination for k > or = 2 when t-tests under normality or Wilcoxon-Mann-Whitney tests under general distribution assumptions are used.     

Sample Size Requirements of a Mixture Analysis Method with Applications in Systematic Biology

The available information on sample size requirements of mixture analysis methods is insufficient to permit a precise evaluation of the potential problems facing practical applications of mixture analysis. We use results from Monte Carlo simulation to assess the sample size requirements of a simple mixture analysis method under conditions relevant to biological applications of mixture analysis. The mixture model used includes two univariate normal components with equal variances but assumes that the researcher is ignorant as to the equality of the variances. The method used relies on the EM algorithm to compute the maximum likelihood estimates of the mixture parameters, and the likelihood ratio test to assess the number of components in the mixtures. Our results suggest that sample sizes close to 500 or 1000 data may be required to adequately solve mixtures commonly found in biology. Sample sizes of 500 or 1000 are difficult to achieve. However, use of this MA method may be a reasonable option when the researcher deals with problems which are intractable by other means. Copyright 1999 Academic Press.     

Sample Size Requirements in Studies of the Etiologic Fraction

This paper outlines methods of determining sample size for epidemiologic research in studies of the etiologic fraction. The basic model with a dichotomous disease and a single dichotomous exposure factor is considered. To determine sample size, the researcher must specify: the magnitude of the etiologic fraction ε to be detected as statistically significant, the level of significance α, the power 1 - β of the test, p the proportion of the population exposed to the risk factor and R the proportion of the population with the disease. Sample size formulas and tables are presented for the case-control, cohort and cross-sectional designs. Optimal allocation considerations are examined to minimize cost for a specified power. Extensive use is made of Walter's results concerning the asymptotic variance of the maximum likelihood estimator of the etiologic fraction for the three epidemiologic study designs.     

Mendelian randomization: can genetic epidemiology help redress the failures of observational epidemiology?

Establishing causal relationships between environmental exposures and common diseases is beset with problems of unresolved confounding, reverse causation and selection bias that may result in spurious inferences. Mendelian randomization, in which a functional genetic variant acts as a proxy for an environmental exposure, provides a means of overcoming these problems as the inheritance of genetic variants is independent of—that is randomized with respect to—the inheritance of other traits, according to Mendel’s law of independent assortment. Examples drawn from exposures and outcomes as diverse as milk and osteoporosis, alcohol and coronary heart disease, sheep dip and farm workers’ compensation neurosis, folate and neural tube defects are used to illustrate the applications of Mendelian randomization approaches in assessing potential environmental causes of disease. As with all genetic epidemiology studies there are problems associated with the need for large sample sizes, the non-replication of findings, and the lack of relevant functional genetic variants. In addition to these problems, Mendelian randomization findings may be confounded by other genetic variants in linkage disequilibrium with the variant under study, or by population stratification. Furthermore, pleiotropy of effect of a genetic variant may result in null associations, as may canalisation of genetic effects. If correctly conducted and carefully interpreted, Mendelian randomization studies can provide useful evidence to support or reject causal hypotheses linking environmental exposures to common diseases.     

Improved Estimation of Human Lipoprotein Kinetics with Mixed Effects Models

Context

Mathematical models may help the analysis of biological systems by providing estimates of otherwise un-measurable quantities such as concentrations and fluxes. The variability in such systems makes it difficult to translate individual characteristics to group behavior. Mixed effects models offer a tool to simultaneously assess individual and population behavior from experimental data. Lipoproteins and plasma lipids are key mediators for cardiovascular disease in metabolic disorders such as diabetes mellitus type 2. By the use of mathematical models and tracer experiments fluxes and production rates of lipoproteins may be estimated.

Results

We developed a mixed effects model to study lipoprotein kinetics in a data set of 15 healthy individuals and 15 patients with type 2 diabetes. We compare the traditional and the mixed effects approach in terms of group estimates at various sample and data set sizes.

Conclusion

We conclude that the mixed effects approach provided better estimates using the full data set as well as with both sparse and truncated data sets. Sample size estimates showed that to compare lipoprotein secretion the mixed effects approach needed almost half the sample size as the traditional method.     

Sample size determination for confidence intervals on the population mean and on the difference between two population means

Sample size determination is usually based on the premise that a hypothesis test is to be used. A confidence interval can sometimes serve better than a hypothesis test. In this paper a method is presented for sample size determination based on the premise that a confidence interval for a simple mean, or for the difference between two means, with normally distributed data is to be used. For this purpose, a concept of power relevant to confidence intervals is given. Some useful tables giving required sample size using this method are also presented.     

Sample Size Determination in Comparing Two Population Variances with Paired‐data: Application to Bilirubin Tests

Sample size needed by using paired‐data to compare two population variances is much smaller than that of using usual independent samples. In this paper, we proposed three parametric methods to compare two population variances with paired‐data. After comparing the powers of all three methods, we furnish the tables to indicate the least sample sizes required in various parameter values for the method with the largest power. The bilirubin example is used to illustrate the usefulness of the tables.     

A simple method for the analysis of clustered binary data.

A simple method for comparing independent groups of clustered binary data with group-specific covariates is proposed. It is based on the concepts of design effect and effective sample size widely used in sample surveys, and assumes no specific models for the intracluster correlations. It can be implemented using any standard computer program for the analysis of independent binary data after a small amount of preprocessing. The method is applied to a variety of problems involving clustered binary data: testing homogeneity of proportions, estimating dose-response models and testing for trend in proportions, and performing the Mantel-Haenszel chi-squared test for independence in a series of 2 x 2 tables and estimating the common odds ratio and its variance. Illustrative applications of the method are also presented.     

An application of multivariate ratio methods for the analysis of a longitudinal clinical trial with missing data.

This paper presents an analysis of a longitudinal multi-center clinical trial with missing data. It illustrates the application, the appropriateness, and the limitations of a straightforward ratio estimation procedure for dealing with multivariate situations in which missing data occur at random and with small probability. The parameter estimates are computed via matrix operators such as those used for the generalized least squares analysis of catetorical data. Thus, the estimates may be conveniently analyzed by asymptotic regression methods within the same computer program which computes the estimates, provided that the sample size is sufficiently computer program which computes the estimates, provided that the sample size is sufficiently large.     

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.     

Software for the development and evaluation of probabilistic identification matrices

A number of programs are described for the development and evaluationof probabilistic identification matrices for use with computer-assistedidentification. The program BEST reads an initial matrix ofper cent probabilities for all the binary characters examinedduring a cluster analysis and determines the most useful setof tests for distinguishing taxa in the matrix. Program RESORTcreates from the initial matrix an identification matrix inwhich the order of the tests may be different or the numberof tests reduced. A printed version of a matrix can be producedby MATPRJNT, which creates tables giving the per cent probabilityof a positive test result and the test results presented as‘–’, ‘v’ and ‘+’ dependingon a user-specified threshold. Program IDSC evaluates an identificationmatrix by calculating the best identification score for eachtaxon in the matrix using the expected result for each test.The programs were written in FORTRAN 77 and can be run on anymicrocomputer using the PC/MS-DOS operating system. Received on April 27, 1990; accepted on November 5, 1990     

THE POWER OF SENSORY DISCRIMINATION METHODS

Difference testing methods are extensively used in a variety of applications from small sensory evaluation tests to large scale consumer tests. A central issue in the use of these tests is their statistical power, or the probability that if a specified difference exists it will be demonstrated as a significant difference in a difference test. A general equation for the power of any discrimination method is given. A general equation for the sample size required to meet Type I and Type II error specifications is also given. Sample size tables for the 2-alternative forced choice (2-AFC), 3-AFC, the duo-trio and the triangular methods are given. Tables of the psychometric functions for the 2-AFC, 3-AFC, triangular and duo-trio methods are also given.     

Exploiting collider bias to apply two-sample summary data Mendelian randomization methods to one-sample individual level data

Over the last decade the availability of SNP-trait associations from genome-wide association studies has led to an array of methods for performing Mendelian randomization studies using only summary statistics. A common feature of these methods, besides their intuitive simplicity, is the ability to combine data from several sources, incorporate multiple variants and account for biases due to weak instruments and pleiotropy. With the advent of large and accessible fully-genotyped cohorts such as UK Biobank, there is now increasing interest in understanding how best to apply these well developed summary data methods to individual level data, and to explore the use of more sophisticated causal methods allowing for non-linearity and effect modification.In this paper we describe a general procedure for optimally applying any two sample summary data method using one sample data. Our procedure first performs a meta-analysis of summary data estimates that are intentionally contaminated by collider bias between the genetic instruments and unmeasured confounders, due to conditioning on the observed exposure. These estimates are then used to correct the standard observational association between an exposure and outcome. Simulations are conducted to demonstrate the method’s performance against naive applications of two sample summary data MR. We apply the approach to the UK Biobank cohort to investigate the causal role of sleep disturbance on HbA1c levels, an important determinant of diabetes.Our approach can be viewed as a generalization of Dudbridge et al. (Nat. Comm. 10: 1561), who developed a technique to adjust for index event bias when uncovering genetic predictors of disease progression based on case-only data. Our work serves to clarify that in any one sample MR analysis, it can be advantageous to estimate causal relationships by artificially inducing and then correcting for collider bias.     

A comparison of estimates of productivity and consumption by zooplankton for planktonic ciliates in Lake Ontario

A multiple regression equation predicting growth rate for ciliatesfrom cell size and temperature was combined with measurementsof biomass to estimate the productivity of ciliates in the epilimnionof Lake Ontario. This method predicts daily production to biomassvalues for ciliates of up to 5 day^–1 and leads to theconclusion that ciliate production could equal half of the carbonfixation by phototrophs. Consumption of ciliates by metazoanzooplankton was estimated by incubating samples passed through44 µm screens, and determining the increase in abundanceof ciliates over 24 h. These rates are much lower, >1 day^–1and often near zero. Production estimates based on these latterrates would be 3–4% of primary production Possible explanationsfor this discrepancy include both predation within the microzooplanktoncommunity and food limitation, as well as bottle effects However,the lower production estimates are still compatible with ciliatesplaying a major role as grazers in this ecosystem