Power Computations Research Articles

Power of genome‐wide association studies in the presence of interacting loci

Though multiple interacting loci are likely involved in the etiology of complex diseases, early genome-wide association studies (GWAS) have depended on the detection of the marginal effects of each locus. Here, we evaluate the power of GWAS in the presence of two linked and potentially associated causal loci for several models of interaction between them and find that interacting loci may give rise to marginal relative risks that are not generally considered in a one-locus model. To derive power under realistic situations, we use empirical data generated by the HapMap ENCODE project for both allele frequencies and linkage disequilibrium (LD) structure. The power is also evaluated in situations where the causal single nucleotide polymorphisms (SNPs) may not be genotyped, but rather detected by proxy using a SNP in LD. A common simplification for such power computations assumes that the sample size necessary to detect the effect at the tSNP is the sample size necessary to detect the causal locus directly divided by the LD measure r(2) between the two. This assumption, which we call the "proportionality assumption", is a simplification of the many factors that contribute to the strength of association at a marker, and has recently been criticized as unreasonable (Terwilliger and Hiekkalinna [2006] Eur J Hum Genet 14(4):426-437), in particular in the presence of interacting and associated loci. We find that this assumption does not introduce much error in single locus models of disease, but may do so in so in certain two-locus models.

The Coupling of Regression Model and Differential Equation Model in GM (1, 1) Modeling and Extended GM (1, 1) Models

The GM (1,1) model is a small-sample based coupled data-assimilation approach with the advantages of highly predictive power and easy computations. However, in the standard GM (1,1) model building exercises we often face a statistical-grey inconsistency problem. Therefore, in this paper we examine the GM(1,1) model from its component-level and try to reveal an interactive coupling nature of differential equation model and corresponding regression model constituting of a GM(1,1) model. Based our analysis, we state a coupling principle for establishing an extended GM(1,1) model and further explore certain families of extended GM(1,1) models with statistical-grey consistency.

The Power of the Test in Three-Level Designs

Field experiments that involve nested structures may assign treatment conditions either to entire groups (such as classrooms or schools), or individuals within groups (such as students). Since field experiments involve clustering, key aspects of their design include knowledge of the intraclass correlation structure and the sample sizes necessary to achieve adequate power to detect the treatment effect. This study provides methods for computing power in three-level designs, where for example, students are nested within classrooms and classrooms are nested within schools. The power computations take into account clustering effects at the classroom and at the school level, sample size effects (e.g., number of students, classrooms, and schools), and covariate effects (e.g., pre-treatment measures). The methods are generalizable to quasi-experimental studies that examine group differences in an outcome, or associations between predictors and outcomes.

Power and sample size computations in simultaneous tests for non-inferiority based on relative margins

In this paper, we address the problem of calculating power and sample sizes associated with simultaneous tests for non-inferiority. We consider the case of comparing several experimental treatments with an active control. The approach is based on the ratio view, where the common non-inferiority margin is chosen to be some percentage of the mean of the control treatment. Two power definitions in multiple hypothesis testing, namely, complete power and minimal power, are used in the computations. The sample sizes associated with the ratio-based inference are also compared with that of a comparable inference based on the difference of means for various scenarios. It is found that the sample size required for ratio-based inferences is smaller than that of difference-based inferences when the relative non-inferiority margin is less than one and when large response values indicate better treatment effects. The results are illustrated with examples.

USING NORMAL APPROXIMATION ON TESTING AND DETERMINING SAMPLE SIZE FOR C pk

ABSTRACT Process capability indices are widely used to measure the ability of the process to manufacture products meeting established specification limits. Simply reporting and then making decision from the calculated estimate of the capability index along is not reliable since sampling errors are ignored. However, a more appropriate estimate would be provided by a hypothesis testing. Using a Bayesian-like estimator of C pk index, this paper presents a generalized approach for testing C pk index and power computations determining whether a process meets the capability requirement. Applying Hamaker's approximation, the testing procedure can be adequately simplified as a normal approximation and that the more complicated use of the non-central t distribution can be avoided. It allows for testing at any α-level and any sample size. Additionally, an analysis is provided to offer the practitioner the sampling requirements to obtain a fixed power.

Using Confidence Intervals to Compare Process Capability Indices

Process capability indices are widely used to measure process performance. In situations such as selecting a supplier and assessing process improvement, it is of interest to compare capability indices for two different processes or the same process before and after an adjustment. In this paper, we consider several methods for performing this comparison on the indices Cpk and Cpm . The methods are compared using a computer simulation. Recommendations are provided for selecting an appropriate method based on power and test size computations.

Power estimation for the Dorfman-Berbaum-Metz method 1

The authors describe how to perform power and sample size computations for the Dorfman-Berbaum-Metz method for analyzing multireader receiver operating characteristic (ROC) studies when data from a pilot study or from a previous study similar to the planned study are available. Power and sample size computations are described in a step-by-step procedure. To illustrate the power computation method we treat 2 studies that did not show a significant difference between modalities as pilot studies for planning future studies. For the future studies we plan to compare the modalities with respect to the mean of the treatment-reader area under the curve (AUC) estimates. We show how to estimate the reader and case sample sizes for each future study to have 80% power to detect a specified difference in modality AUCs. Application of the power and sample size computation procedure for the Dorfman-Berbaum-Metz method is straightforward. For a given effect size there will be several different reader and case sample size combinations that yield the desired power. It is important that the pilot study or previous study used for the computations be comparable to the planned study with respect to the modalities, reader expertise, case selection, and ratio of normal to abnormal cases.

Association Studies in Candidate Genes: Strategies to Select SNPs to Be Tested

Objective: When numerous single nucleotide polymorphisms (SNPs) have been identified in a candidate gene, a relevant and still unanswered question is to determine how many and which of these SNPs should be optimally tested to detect an association with the disease. Testing them all is expensive and often unnecessary. Alleles at different SNPs may be associated in the population because of the existence of linkage disequilibrium, so that knowing the alleles carried at one SNP could provide exact or partial knowledge of alleles carried at a second SNP. We present here a method to select the most appropriate subset of SNPs in a candidate gene based on the pairwise linkage disequilibrium between the different SNPs. Method: The best subset is identified through power computations performed under different genetic models, assuming that one of the SNPs identified is the disease susceptibility variant. Results: We applied the method on two data sets, an empirical study of the APOE gene region and a simulated study concerning one of the major genes (MG1) from the Genetic Analysis Workshop 12. For these two genes, the sets of SNPs selected were compared to the ones obtained using two other methods that need the reconstruction of multilocus haplotypes in order to identify haplotype-tag SNPs (htSNPs). We showed that with both data sets, our method performed better than the other selection methods.

ERODIBILITY OF URBAN BEDROCK AND ALLUVIAL CHANNELS, NORTH TEXAS1

ABSTRACT: Major erosion of urban stream channels is found in smaller basins in the North Texas study area with contributing drainage areas of less than ten square miles. Within these basins, four basic channel types are identified based on bed and bank lithologies: alluvial banks and bottoms, alluvial banks and gravel bottoms, alluvial banks with rock bottoms, and rock banks with rock bottoms. Most channels (75 percent) have alluvial banks with gravel or rock bottoms. Channel slopes are steep (.38 to.76 percent). Rock consists predominantly of shale and limestone. Channel cross sections are divided into the following four zones based on weathering, scour and entrainment mechanisms: soil zone, slake zone, rock zone and bed material zone. Erodibility of the channels is determined using multiple techniques including reach hydraulics and stream power computations, submerged jet testing, slab entrainment thresholds, and slake durability rates. Procedures are based on both empirical and modeled time series estimates of channel erosion. Field and modeled results support rates of erosion of up to four inches per year. Rates are tied to flow regime, climate, and type of channel bed and banks.

Optimally tuned vibration absorbers to control sound transmission

A design optimization method is proposed for controlling broadband vibration of a structure and it concomitant acoustic radiation using multiple-tuned absorbers. A computationally efficient model of a structure is developed and coupled with a nonlinear optimization search algorithm. The eigenvectors of the original structure are used as repeated basis functions in the analysis of the structural dynamic re-analysis problem. The re-analysis time for acoustic power computations is reduced by calculating and storing modal radiation resistance matrices at discrete frequencies. The matrices are then interpolated within the optimization loop for eigenvalues that fall between stored frequencies. The method is demonstrated by applying multiple-tuned vibration absorbers to an acoustically-excited composite panel. The absorber parameters are optimized with an objective of maximizing the panel’s sound power transmission loss. It is shown that in some cases the optimal solution includes vibration absorbers that are tuned very closely in frequency, thus acting effectively as a broadband vibration absorber (BBVA). The numerical model and design optimization method are validated experimentally, and the BBVA is found to be an effective noise abatement tool.

Testing homogeneity of control and treatment populations — local optimality and related issues

Durairajan and Raman (1996 a, b) studied the robustness of Locally most powerful invariant (LMPI) tests for compound normal model in control and treatment populations. In the present paper, the Locally most powerful (LMP) tests are constructed for no contamination in normal mixture model through testing the parameter of mixture of distributions and the mixing proportion. The expected performance of LMP tests are compared using Efron’s Statistical Curvature on the lines of Sen Gupta and Pal (1991). The Locally most powerful similar (LMPS) tests for the equality of control and treatment populations in the presence of nuisance parameters are also constructed. Further, the null and non-null distributions of the test statistics are derived and some power computations are made.

Fast acoustic power computations for structural optimization problems

Boundary element analyses in acoustics are often computationally inefficient because a large complex-valued matrix has to be solved at every frequency for which a solution is desired. This problem can be alleviated somewhat by using appropriate mesh densities for both the structural acoustic analyses, as discussed previously by the author. To achieve further enhancements, a computer program has been written to interpolate the acoustic resistance matrix over a frequency band from values calculated at sparse intervals. A preliminary analysis is performed to map out the resistance matrix as a function of frequency, requiring specification of only the frequency band of interest and the desired error tolerance. Automatic refinement is performed in regions where the resistance varies rapidly, as it does near acoustic resonances. A simple interpolation scheme is then used to compute the resistance matrix at any arbitrary frequency knowing the precomputed matrices, resulting in an order of magnitude reduction in the computation times when compared to a direct calculation. Because the acoustic analysis is independent of structural parameters, this formulation is ideal for problems where structural parameters are to be optimized without changing the radiating surface’s shape. Examples are given to demonstrate the formulation’s accuracy and computational efficiency.

Computations for Group Sequential Boundaries Using the Lan-DeMets Spending Function Method

We describe an interactive Fortran program which performs computations related to the design and analysis of group sequential clinical trials using Lan-DeMets spending functions. Many clinical trials include interim analyses of accumulating data and rely on group sequential methods to avoid consequent inflation of the type I error rate. The computations are appropriate for interim test statistics whose distribution or limiting distribution is multivariate normal with independent increments. Recent theoretical results indicate that virtually any design likely to be used in a clinical trial will fall into this category. Interim analyses need not be equally spaced, and their number need not be specified in advance. In addition to determining sequential boundaries using an alpha spending function, the program can perform power computations, compute probabilities associated with a given set of boundaries, and generate confidence intervals. Control Clin Trials 2000;21:190–207

Design and statistical analysis for the Pathways study

We report the design, rationale, and statistical procedures used in Pathways, a randomized, school-based intervention for the primary prevention of obesity in American Indian children. The intervention, which is now being implemented in 7 American Indian communities around the country, includes a health-promotion curriculum, a physical education program, a school meal program, and a family involvement component. Forty-one schools serving American Indian children were randomly assigned to be either intervention or control groups. The intervention will begin in the third grade and continue through the end of the fifth grade. Efficacy of intervention will be assessed by differences in mean percentage body fat, calculated by a prediction equation, between intervention and control schools at the end of the fifth grade. Power computations indicate that the study has power to detect a mean difference of 2.8% in body fat. Data analysis will use intention-to-treat concepts and the mixed linear model. The study will be completed in 2000.

Exact Power and Sample-Size Computations for the Cochran-Armitage Trend Test

The Cochran-Armitage test of trend is commonly used to determine if a dose-response relationship exists in a wide variety of biomedical settings including clinical trials, carcinogenicity studies, and toxicological risk assessment. For small, sparse, or unbalanced data sets, one generally adopts the exact version of the Cochran-Armitage test of trend for which numerical algorithms and software are readily available. No corresponding algorithms or software exist for the exact power and samplesize computations that are needed at design time, prior to gathering the data. This paper develops a network algorithm for computing the exact power of the Cochran-Armitage test of trend and applies it to several examples, thereby demonstrating that the corresponding asymptotic power computations can be rather misleading.

Statistical Power Computations for Detecting Dichotomous Moderator Variables with Moderated Multiple Regression

A revised and improved version of Aguinis, Pierce, and Stone-Romero's (1994) program for estimating the statistical power of moderated multiple regression to detect dichotomous moderator variables is described. The Quick BASIC program runs on IBM and IBM-compatible personal computers and estimates power based on user-provided values for (a) total sample size, (b) sample sizes across the two moderator-based subgroups, (c) correlation coefficients between the predictor and criterion for each of the two moderator-based subgroups, (d) correlation coefficient between the predictor and hypothesized moderator, and (e) sample and population standard deviations for the predictor. Program-generated power estimates for typical research situations in education, psychology, and management indicate that hypothesis tests of moderating effects are typically conducted at insufficient levels of statistical power.

Problem-based learning: measurable outcomes.

Problem-based learning (PBL) could potentially contribute to four key objectives in the education of doctors. (1) Motivating learning. Three studies show that students studying PBL problems choose fewer topics to study than those identified by the faculty, but one study of a critical care rotation showed that students were motivated to learn over a wider range of basic science topics than had been included in the basic science curriculum. (2) Developing clinical reasoning. One study compared methods of solving problems in PBL and conventional track curricula and suggested that PBL students work backwards from clinical information to theory, while conventional curriculum students tend to reason forward from theory. One study showed that computer searches provide knowledge for helping solve some PBL problems, and another study showed that specific knowledge in emergency medicine correlated with test scores. (3) Structuring knowledge in clinical contexts. A few studies show that PBL students perform less well on basic science examinations but better on clinical examinations. Educational outcomes, however, have been assessed quantitatively mainly by the U.S. National Boards of Medical Examiners Examinations or by clinical examinations with small samples. Only one study includes a power computation to assess type II error. There are no studies that examine how much variance occurs between PBL programmes in their curricular methods and outcomes. (4) Developing self learning skills. PBL students use a much wider range and number of resources than conventional track students. There is only one study comparing the knowledge of doctors trained by PBL and conventional curricula after the doctors have been in practice for a substantial number of years, and no studies of patient outcomes. Patient outcomes need to be assessed with randomized controlled trials, and sample sizes should be determined by power computations to avoid Type II error. Four possible methods of improving PBL would be to derive national and internationally accepted PBL curricula; to organize internationally accepted and psychometrically validated methods of evaluation; to develop attitudes among students and tutors to facilitate co-operative PBL teamwork; and to teach group process diagnostic skills.

A computer program for sample size and power calculations in the design of multi-arm and factorial clinical trials with survival time endpoints

This paper presents a computer program for use in the design of long-term clinical trials with multiple treatment arms in which the primary outcome variables are censored survival times. The treatment arms may be structured as a one-way or multi-way factorial design. It is assumed that patients are entered and randomized to a treatment arm during an accrual period. The patients are then followed for a fixed period during which there may be dropouts. Various distributional assumptions can be used to model the survival times. These include an option in which there is an effect of treatment duly after a lag or delay time. The program then computes the power of various statistical tests of hypotheses concerning treatment differences, interactions and trends. The power computations are “exact” in that they use the Monte Carlo method to obtain Type I and II error probabilities. However the program also outputs the normal approximations for comparison, although they are typically not accurate in these situations. Fisher's LSD method is used to adjust for the multiple comparisons. By comparing the power for various sets of design parameters, such as sample size, numbers of factor levels, patient accrual rate, and length of follow-up, an appropriate design can be constructed. Two examples are provided. The first is a simple one-way layout with multiple treatment arms; the second a two-way factorial design for a proposed large scale cancer chemoprevention trial.

Interreplicate Variance and Statistical Power of Electrofishing Data from Low-Gradient Streams in the Southeastern United States

I assessed the relationship between interreplicate variance and the mean for electrofishing catch per unit effort (defined as number of fish per 100 m2 of stream surface area) from 56 sites in 22 first- through fourth-order streams on the coastal plain of South Carolina. Variance and mean were strongly and positively related. This relationship was unaffected by type and quantity of instream structure and was similar among schooling and nonschooling species. Interreplicate variance was unaffected by replicate reach lengths ranging from 60 to 130 m. Replicate lengths near the lower end of this range required less total effort (number of replicates times replicate length) than longer replicates for comparable precision. Statistical power computations indicated that large numbers of samples (>25) were required to detect small (<20%) differences among means, and that the required number of samples was greater when catch per unit effort was low, indicating that power can be increased if multiple passes are used to increase the catch. However, it required less total time to use a relatively large number of single-pass replicates than a smaller number of multiple-pass replicates to achieve a given level of precision. Investigation of several types of data transformations indicated that the log transformation was most efficient at reducing the correlation between the variance and the mean, as required for parametric statistical analysis.

Statistical tests for random effects in staggered nested designs

In this paper, we present certain statistical tests under a staggered nested design set-up, for the hypotheses that certain variance components are zero. To do so, the particular variance-covariance, structure induced by the staggering is exploited and certain results of multivariate analysis are used. In most problems, the test statistics can be easily computed. An example is provided for illustration and some power computations for comparison of test statistics are shown.