Cost-effective designs for linkage disequilibrium mapping of complex traits

The current development of densely spaced collections of single nucleotide polymorphisms (SNPs) will lead to genomewide association studies for a wide range of diseases in many different populations. Determinations of the appropriate number of SNPs to genotype involve a balancing of power and cost. Several variables are important in these determinations. We show that there are different combinations of sample size and marker density that can be expected to achieve the same power. Within certain bounds, investigators can choose between designs with more subjects and fewer markers or those with more markers and fewer subjects. Which designs are more cost-effective depends on the cost of phenotyping versus the cost of genotyping. We show that, under the assumption of a set cost for genotyping, one can calculate a "threshold cost" for phenotyping; when phenotyping costs per subject are less than this threshold, designs with more subjects will be more cost-effective than designs with more markers. This framework for determining a cost-effective study will aid in the planning of studies, especially if there are choices to be made with respect to phenotyping methods or study populations.     

A general model of site-dependent population regulation: population-level regulation without individual-level interactions

In this paper, we use a modeling approach to explore the population regulatory consequences of individual choices for where to breed in heterogeneous environments. In contrast to standard models, we focus on individuals that interact only indirectly through their choices of breeding sites (i.e., individuals preempt the occupation of a breeding site by others when they choose to breed there). We consider the consequences of individuals choosing breeding sites either randomly or sequentially from best to worst. Our analysis shows that average per-capita fecundity of the population is independent of the number of occupied breeding sites if individuals choose sites at random and that variation in average per-capita fecundity increases as population size declines. In contrast, if individuals choose breeding sites sequentially from highest to lowest quality, then as population size increases average per-capita fecundity declines and variation in average per-capita fecundity increases. Consequently, aggregate population-level demographic rates can change in ways that generate population regulation, even when change in population size does not change the demographic performance of any individual on any particular breeding site. However, such regulation occurs only when individuals make adaptive choices of where to breed. Because variation in average per-capita fecundity decreases when population size declines, populations regulated in a site-dependent manner should be much less susceptible to the vicissitudes of small population size than those which choose breeding sites at random.     

R/qtlDesign: inbred line cross experimental design

An investigator planning a QTL (quantitative trait locus) experiment has to choose which strains to cross, the type of cross, genotyping strategies, and the number of progeny to raise and phenotype. To help make such choices, we have developed an interactive program for power and sample size calculations for QTL experiments, R/qtlDesign. Our software includes support for selective genotyping strategies, variable marker spacing, and tools to optimize information content subject to cost constraints for backcross, intercross, and recombinant inbred lines from two parental strains. We review the impact of experimental design choices on the variance attributable to a segregating locus, the residual error variance, and the effective sample size. We give examples of software usage in real-life settings. The software is available at .     

A linkage strategy for detection of human quantitative-trait loci. II. Optimization of study designs based on extreme sib pairs and generalized relative risk ratios.

We are concerned here with practical issues in the application of extreme sib-pair (ESP) methods to quantitative traits. Two important factors-namely, the way extreme trait values are defined and the proportions in which different types of ESPs are pooled, in the analysis-are shown to determine the power and the cost effectiveness of a study design. We found that, in general, combining reasonable numbers of both extremely discordant and extremely concordant sib pairs that were available in the sample is more powerful and more cost effective than pursuing only a single type of ESP. We also found that dividing trait values with a less extreme threshold at one end or at both ends of the trait distribution leads to more cost-effective designs. The notion of generalized relative risk ratios (the lambda methods, as described in the first part of this series of two articles) is used to calculate the power and sample size for various choices of polychotomization of trait values and for the combination of different types of ESPs. A balance then can be struck among these choices, to attain an optimum design.     

Harder than Expected: Increased Conflict in Clearly Disadvantageous Delayed Choices in a Computer Game

When choosing between immediate and temporally delayed goods, people sometimes decide disadvantageously. Here, we aim to provide process-level insight into differences between individually determined advantageous and disadvantageous choices. Participants played a computer game, deciding between two different rewards of varying size and distance by moving an agent towards the chosen reward. We calculated individual models of advantageous choices and characterized the decision process by analyzing mouse movements. The larger amount of participants’ choices was classified as advantageous and the disadvantageous choices were biased towards choosing sooner/smaller rewards. The deflection of mouse movements indicated more conflict in disadvantageous choices compared with advantageous choices when the utilities of the options differed clearly. Further process oriented analysis revealed that disadvantageous choices were biased by a tendency for choice-repetition and an undervaluation of the value information in favour of the delay information, making rather simple choices harder than could be expected from the properties of the decision situation.     

Compliance subsampling designs for comparative research: estimation and optimal planning

For studies with treatment noncompliance, analyses have been developed recently to better estimate treatment efficacy. However, the advantage and cost of measuring compliance data have implications on the study design that have not been as systematically explored. In order to estimate better treatment efficacy with lower cost, we propose a new class of compliance subsampling (CSS) designs where, after subjects are assigned treatment, compliance behavior is measured for only subgroups of subjects. The sizes of the subsamples are allowed to relate to the treatment assignment, the assignment probability, the total sample size, the anticipated distributions of outcome and compliance, and the cost parameters of the study. The CSS design methods relate to prior work (i) on two-phase designs in which a covariate is subsampled and (ii) on causal inference because the subsampled postrandomization compliance behavior is not the true covariate of interest. For each CSS design, we develop efficient estimation of treatment efficacy under binary outcome and all-or-none observed compliance. Then we derive a minimal cost CSS design that achieves a required precision for estimating treatment efficacy. We compare the properties of the CSS design to those of conventional protocols in a study of patient choices for medical care at the end of life.     

Combining counts and incidence data: an efficient approach for estimating the log-normal species abundance distribution and diversity indices

Obtaining accurate estimates of diversity indices is difficult because the number of species encountered in a sample increases with sampling intensity. We introduce a novel method that requires that the presence of species in a sample to be assessed while the counts of the number of individuals per species are only required for just a small part of the sample. To account for species included as incidence data in the species abundance distribution, we modify the likelihood function of the classical Poisson log-normal distribution. Using simulated community assemblages, we contrast diversity estimates based on a community sample, a subsample randomly extracted from the community sample, and a mixture sample where incidence data are added to a subsample. We show that the mixture sampling approach provides more accurate estimates than the subsample and at little extra cost. Diversity indices estimated from a freshwater zooplankton community sampled using the mixture approach show the same pattern of results as the simulation study. Our method efficiently increases the accuracy of diversity estimates and comprehension of the left tail of the species abundance distribution. We show how to choose the scale of sample size needed for a compromise between information gained, accuracy of the estimates and cost expended when assessing biological diversity. The sample size estimates are obtained from key community characteristics, such as the expected number of species in the community, the expected number of individuals in a sample and the evenness of the community.     

A mixed approach for proving non-inferiority in clinical trials with binary endpoints

When a new treatment is compared to an established one in a randomized clinical trial, it is standard practice to statistically test for non-inferiority rather than for superiority. When the endpoint is binary, one usually compares two treatments using either an odds-ratio or a difference of proportions. In this paper, we propose a mixed approach which uses both concepts. One first defines the non-inferiority margin using an odds-ratio and one ultimately proves non-inferiority statistically using a difference of proportions. The mixed approach is shown to be more powerful than the conventional odds-ratio approach when the efficacy of the established treatment is known (with good precision) and high (e.g. with more than 56% of success). The gain of power achieved may lead in turn to a substantial reduction in the sample size needed to prove non-inferiority. The mixed approach can be generalized to ordinal endpoints.     

The relative efficiency of the Hardy-Weinberg equilibrium-likelihood and the conditional on parental genotype-likelihood methods for candidate-gene association studies.

Selecting a control group that is perfectly matched for ethnic ancestry with a group of affected individuals is a major problem in studying the association of a candidate gene with a disease. This problem can be avoided by a design that uses parental data in place of nonrelated controls. Schaid and Sommer presented two new methods for the statistical analysis using this approach: (1) a likelihood method (Hardy-Weinberg equilibrium [HWE] method), which rests on the assumption that HWE holds, and (2) a conditional likelihood method (conditional on parental genotype [CPG] method) appropriate when HWE is absent. Schaid and Sommer claimed that the CPG method can be more efficient than the HWE method, even when equilibrium holds. It can be shown, however that in the equilibrium situation the HWE method is always more efficient than the CPG method. For a dominant disease, the differences are slim. But for a recessive disease, the CPG method requires a much larger sample size to achieve a prescribed power than the HWE method. Additionally, we show how the relative risks for the various candidate-gene genotypes can be estimated without relying on iterative methods. For the CPG method, we represent an asymptotic power approximation that is sufficiently precise for planning the sample size of an association study.     

Modeling unobserved sources of heterogeneity in animal abundance using a Dirichlet process prior

Summary .   In surveys of natural populations of animals, a sampling protocol is often spatially replicated to collect a representative sample of the population. In these surveys, differences in abundance of animals among sample locations may induce spatial heterogeneity in the counts associated with a particular sampling protocol. For some species, the sources of heterogeneity in abundance may be unknown or unmeasurable, leading one to specify the variation in abundance among sample locations stochastically. However, choosing a parametric model for the distribution of unmeasured heterogeneity is potentially subject to error and can have profound effects on predictions of abundance at unsampled locations. In this article, we develop an alternative approach wherein a Dirichlet process prior is assumed for the distribution of latent abundances. This approach allows for uncertainty in model specification and for natural clustering in the distribution of abundances in a data-adaptive way. We apply this approach in an analysis of counts based on removal samples of an endangered fish species, the Okaloosa darter. Results of our data analysis and simulation studies suggest that our implementation of the Dirichlet process prior has several attractive features not shared by conventional, fully parametric alternatives.     

A correlation method for detecting and estimating linkage between a marker locus and a quantitative trait locus using inbred lines

The advent of molecular genetic markers has stimulated interest in detecting linkage between a marker locus and a quantitative trait locus (QTL) because the marker locus, even without direct effect on the quantitative trait, could be useful in increasing the response to selection. A correlation method for detecting and estimating linkage between a marker locus and a QTL is described using selfing and sib-mating populations. Computer simulations were performed to estimate the power of the method, the sample size (N) needed to detect linkage, and the recombination value (r). The power of this method was a function of the expected recombination value E(r), the standardized difference (d) between the QTL genotypic means, and N. The power was highest at complete linkage, decreased with an increase in E(r), and then increased at E(r)=0.5. A larger d and N led to a higher power. The sample size needed to detect linkage was dependent upon E(r) and d. The sample size had a minimum value at E(r)=0, increased with an increase in E(r) and a decrease in d. In general, the r was overestimated. With an increase in d, the r was closer to its expectation. Detection of linkage by the proposed method under incomplete linkage was more efficient than estimation of recombination values. The correlation method and the method of comparison of marker-genotype means have a similar power when there is linkage, but the former has a slightly higher power than the latter when there is no linkage.     

Approaches for Calculating Power for Case–Cohort Studies

Summary .   An approach for determining the power of a case–cohort study for a single binary exposure variable and a low failure rate was recently proposed by Cai and Zeng (2004, Biometrics 60, 1015–1024). In this article, we show that computing power for a case–cohort study using a standard case–control method yields nearly identical levels of power. An advantage of the case–control approach is that existing sample size software can be used for the calculations. We also propose an additional formula for computing the power of a case–cohort study for the situation when the event is not rare.     

Investigating maximum power losses in survival studies with nonstratified randomization

In studies with nonstratified randomization, imbalances may occur which lead to power loss. We investigate how much power may be lost at worst, that is, with small but nonnegligible probability. The situation considered is a two-treatment trial with censored survival data, and three to five strata. We also derive asymptotic results, as the sample size and/or number of strata increase. In addition, we consider how much a projected sample size needs to be increased to be confident (at a certain probability level) that the intended power will be retained, conditionally on the unbalanced configuration.     

Design and Analysis of Simple Choice Surveys for Natural Resource Management

Abstract: We used a simple yet powerful method for judging public support for management actions from randomized surveys. We asked respondents to rank choices (representing management regulations under consideration) according to their preference, and we then used discrete choice models to estimate probability of choosing among options (conditional on the set of options presented to respondents). Because choices may share similar unmodeled characteristics, the multinomial logit model, commonly applied to discrete choice data, may not be appropriate. We introduced the nested logit model, which offers a simple approach for incorporating correlation among choices. This forced choice survey approach provides a useful method of gathering public input; it is relatively easy to apply in practice, and the data are likely to be more informative than asking constituents to rate attractiveness of each option separately.     

Efficiency Considerations for Group Sequential Designs with Adaptive Unblinded Sample Size Re-assessment

Clinical trials with adaptive sample size re-assessment, based on an analysis of the unblinded interim results (ubSSR), have gained in popularity due to uncertainty regarding the value of \(\delta \) at which to power the trial at the start of the study. While the statistical methodology for controlling the type-1 error of such designs is well established, there remain concerns that conventional group sequential designs with no ubSSR can accomplish the same goals with greater efficiency. The precise manner in which this efficiency comparison can be objectified has been difficult to quantify, however. In this paper, we present a methodology for making this comparison in a standard, well-accepted manner by plotting the unconditional power curves of the two approaches while holding constant their expected sample size, at each value of \(\delta \) in the range of interest. It is seen that under reasonable decision rules for increasing sample size (conservative promising zones, and no more than a 50% increase in sample size) there is little or no loss of efficiency for the adaptive designs in terms of unconditional power. The two approaches, however, have very different conditional power profiles. More generally, a methodology has been provided for comparing any design with ubSSR relative to a comparable group sequential design with no ubSSR, so one can determine whether the efficiency loss, if any, of the ubSSR design is offset by the advantages it confers for re-powering the study at the time of the interim analysis.     

Heterogeneous treatment effects in stratified clinical trials with time‐to‐event endpoints

When analyzing clinical trials with a stratified population, homogeneity of treatment effects is a common assumption in survival analysis. However, in the context of recent developments in clinical trial design, which aim to test multiple targeted therapies in corresponding subpopulations simultaneously, the assumption that there is no treatment‐by‐stratum interaction seems inappropriate. It becomes an issue if the expected sample size of the strata makes it unfeasible to analyze the trial arms individually. Alternatively, one might choose as primary aim to prove efficacy of the overall (targeted) treatment strategy. When testing for the overall treatment effect, a violation of the no‐interaction assumption renders it necessary to deviate from standard methods that rely on this assumption. We investigate the performance of different methods for sample size calculation and data analysis under heterogeneous treatment effects. The commonly used sample size formula by Schoenfeld is compared to another formula by Lachin and Foulkes, and to an extension of Schoenfeld's formula allowing for stratification. Beyond the widely used (stratified) Cox model, we explore the lognormal shared frailty model, and a two‐step analysis approach as potential alternatives that attempt to adjust for interstrata heterogeneity. We carry out a simulation study for a trial with three strata and violations of the no‐interaction assumption. The extension of Schoenfeld's formula to heterogeneous strata effects provides the most reliable sample size with respect to desired versus actual power. The two‐step analysis and frailty model prove to be more robust against loss of power caused by heterogeneous treatment effects than the stratified Cox model and should be preferred in such situations.     

Assessing statistical power of SNPs for population structure and conservation studies

Single nucleotide polymorphisms (SNPs) have been proposed by some as the new frontier for population studies, and several papers have presented theoretical and empirical evidence reporting the advantages and limitations of SNPs. As a practical matter, however, it remains unclear how many SNP markers will be required or what the optimal characteristics of those markers should be in order to obtain sufficient statistical power to detect different levels of population differentiation. We use a hypothetical case to illustrate the process of designing a population genetics project, and present results from simulations that address several issues for maximizing statistical power to detect differentiation while minimizing the amount of effort in developing SNPs. Results indicate that (i) while ~30 SNPs should be sufficient to detect moderate (F_ST = 0.01) levels of differentiation, studies aimed at detecting demographic independence (e.g. F_ST < 0.005) may require 80 or more SNPs and large sample sizes; (ii) different SNP allele frequencies have little affect on power, and thus, selection of SNPs can be relatively unbiased; (iii) increasing the sample size has a strong effect on power, so that the number of loci can be minimized when sample number is known, and increasing sample size is almost always beneficial; and (iv) power is increased by including multiple SNPs within loci and inferring haplotypes, rather than trying to use only unlinked SNPs. This also has the practical benefit of reducing the SNP ascertainment effort, and may influence the decision of whether to seek SNPs in coding or noncoding regions.