Complementary Log–Log Regression for the Estimation of Covariate‐Adjusted Prevalence Ratios in the Analysis of Data from Cross‐Sectional Studies

We assessed complementary log–log (CLL) regression as an alternative statistical model for estimating multivariable‐adjusted prevalence ratios (PR) and their confidence intervals. Using the delta method, we derived an expression for approximating the variance of the PR estimated using CLL regression. Then, using simulated data, we examined the performance of CLL regression in terms of the accuracy of the PR estimates, the width of the confidence intervals, and the empirical coverage probability, and compared it with results obtained from log–binomial regression and stratified Mantel–Haenszel analysis. Within the range of values of our simulated data, CLL regression performed well, with only slight bias of point estimates of the PR and good confidence interval coverage. In addition, and importantly, the computational algorithm did not have the convergence problems occasionally exhibited by log–binomial regression. The technique is easy to implement in SAS (SAS Institute, Cary, NC), and it does not have the theoretical and practical issues associated with competing approaches. CLL regression is an alternative method of binomial regression that warrants further assessment.     

Interval mapping of quantitative trait loci in autotetraploid species.

This article presents a method for QTL interval mapping in autotetraploid species for a full-sib family derived by crossing two parents. For each offspring, the marker information on each chromosome is used to identify possible configurations of chromosomes inherited from the two parents and the locations of crossovers on these chromosomes. A branch and bound algorithm is used to identify configurations with the minimum number of crossovers. From these configurations, the conditional probability of each possible QTL genotype for a series of positions along the chromosome can be estimated. An iterative weighted regression is then used to relate the trait values to the QTL genotype probabilities. A simulation study is performed to assess this approach and to investigate the effects of the proportion of codominant to dominant markers, the heritability, and the population size. We conclude that the method successfully locates QTL and estimates their parameters accurately, and we discuss different modes of action of the QTL that may be modeled.     

An Empirical Method for Establishing Positional Confidence Intervals Tailored for Composite Interval Mapping of QTL

Background

Improved genetic resolution and availability of sequenced genomes have made positional cloning of moderate-effect QTL realistic in several systems, emphasizing the need for precise and accurate derivation of positional confidence intervals (CIs) for QTL. Support interval (SI) methods based on the shape of the QTL likelihood curve have proven adequate for standard interval mapping, but have not been shown to be appropriate for use with composite interval mapping (CIM), which is one of the most commonly used QTL mapping methods.

Results

Based on a non-parametric confidence interval (NPCI) method designed for use with the Haley-Knott regression method for mapping QTL, a CIM-specific method (CIM-NPCI) was developed to appropriately account for the selection of background markers during analysis of bootstrap-resampled data sets. Coverage probabilities and interval widths resulting from use of the NPCI, SI, and CIM-NPCI methods were compared in a series of simulations analyzed via CIM, wherein four genetic effects were simulated in chromosomal regions with distinct marker densities while heritability was fixed at 0.6 for a population of 200 isolines. CIM-NPCIs consistently capture the simulated QTL across these conditions while slightly narrower SIs and NPCIs fail at unacceptably high rates, especially in genomic regions where marker density is high, which is increasingly common for real studies. The effects of a known CIM bias toward locating QTL peaks at markers were also investigated for each marker density case. Evaluation of sub-simulations that varied according to the positions of simulated effects relative to the nearest markers showed that the CIM-NPCI method overcomes this bias, offering an explanation for the improved coverage probabilities when marker densities are high.

Conclusions

Extensive simulation studies herein demonstrate that the QTL confidence interval methods typically used to positionally evaluate CIM results can be dramatically improved by accounting for the procedural complexity of CIM via an empirical approach, CIM-NPCI. Confidence intervals are a critical measure of QTL utility, but have received inadequate treatment due to a perception that QTL mapping is not sufficiently precise for procedural improvements to matter. Technological advances will continue to challenge this assumption, creating even more need for the current improvement to be refined.     

Constructing Confidence Intervals for Qtl Location

We describe a method for constructing the confidence interval of the QTL location parameter. This method is developed in the local asymptotic framework, leading to a linear model at each position of the putative QTL. The idea is to construct a likelihood ratio test, using statistics whose asymptotic distribution does not depend on the nuisance parameters and in particular on the effect of the QTL. We show theoretical properties of the confidence interval built with this test, and compare it with the classical confidence interval using simulations. We show in particular, that our confidence interval has the correct probability of containing the true map location of the QTL, for almost all QTLs, whereas the classical confidence interval can be very biased for QTLs having small effect.     

Confidence Intervals in Qtl Mapping by Bootstrapping

The determination of empirical confidence intervals for the location of quantitative trait loci (QTLs) was investigated using simulation. Empirical confidence intervals were calculated using a bootstrap resampling method for a backcross population derived from inbred lines. Sample sizes were either 200 or 500 individuals, and the QTL explained 1, 5, or 10% of the phenotypic variance. The method worked well in that the proportion of empirical confidence intervals that contained the simulated QTL was close to expectation. In general, the confidence intervals were slightly conservatively biased. Correlations between the test statistic and the width of the confidence interval were strongly negative, so that the stronger the evidence for a QTL segregating, the smaller the empirical confidence interval for its location. The size of the average confidence interval depended heavily on the population size and the effect of the QTL. Marker spacing had only a small effect on the average empirical confidence interval. The LOD drop-off method to calculate empirical support intervals gave confidence intervals that generally were too small, in particular if confidence intervals were calculated only for samples above a certain significance threshold. The bootstrap method is easy to implement and is useful in the analysis of experimental data.     

Use of regression analysis in the quantitative evaluation of the activity of biological preparations]

Use of regression analysis in the assessment of the activity of biological preparations under experimental conditions permitted not only to assess the quantitative effect (ED50) more strictly, but also to find other parameters of importance for the results of comparison, for example with the standard, i.e. in standardization. To these belong regression coefficient, parallelism of regressions, and the relative potency. By the presence of a parallelism one can judge the similarity between the activity mechanism of the active principle of the preparations being compared. Relative potency characterizes the activity of the preparation in the relative values in comparison with the standard, with a statistical evaluation of this value with the aid of the confidence interval. The authors suggest a program for Mir-2 computer facilitating the calculations in using the analystical method which is more objective than the graphic method of assessment of the linear dosage-response curve.     

A Monte Carlo maximum likelihood method for estimating uncertainty arising from shared errors in exposures in epidemiological studies of nuclear workers

Errors in the estimation of exposures or doses are a major source of uncertainty in epidemiological studies of cancer among nuclear workers. This paper presents a Monte Carlo maximum likelihood method that can be used for estimating a confidence interval that reflects both statistical sampling error and uncertainty in the measurement of exposures. The method is illustrated by application to an analysis of all cancer (excluding leukemia) mortality in a study of nuclear workers at the Oak Ridge National Laboratory (ORNL). Monte Carlo methods were used to generate 10,000 data sets with a simulated corrected dose estimate for each member of the cohort based on the estimated distribution of errors in doses. A Cox proportional hazards model was applied to each of these simulated data sets. A partial likelihood, averaged over all of the simulations, was generated; the central risk estimate and confidence interval were estimated from this partial likelihood. The conventional unsimulated analysis of the ORNL study yielded an excess relative risk (ERR) of 5.38 per Sv (90% confidence interval 0.54-12.58). The Monte Carlo maximum likelihood method yielded a slightly lower ERR (4.82 per Sv) and wider confidence interval (0.41-13.31).     

Simple method for calculating confidence limits around inbreeding rate and effective population size estimates from pedigrees

An important concept in population genetics is effective population size (N_e), which describes the expected rate of loss of genetic variability from a population. One way to estimate N_e is using a pedigree. However, there are no methods for comparing the N_e estimated from a pedigree with that expected from life-history models. In the paper we show how N_e can be estimated from the change in inbreeding rate (f) estimated from a pedigree. The mean individual inbreeding rate in a population at a given time must be calculated from averaged values for males and females, where each age class is weighted by its reproductive value. We show an exact method for placing confidence intervals around f and N_e using a binomial distribution, and present a method for approximating this interval for large N_es using a Poisson distribution. These confidence intervals can be used to compare f and N_e from a pedigree to expected values from demographic models, and to compare N_es of two populations.     

Confidence intervals for directly standardized rates using mid‐p gamma intervals

Directly standardized rates continue to be an integral tool for presenting rates for diseases that are highly dependent on age, such as cancer. Statistically, these rates are modeled as a weighted sum of Poisson random variables. This is a difficult statistical problem, because there are k observed Poisson variables and k unknown means. The gamma confidence interval has been shown through simulations to have at least nominal coverage in all simulated scenarios, but it can be overly conservative. Previous modifications to that method have closer to nominal coverage on average, but they do not achieve the nominal coverage bound in all situations. Further, those modifications are not central intervals, and the upper coverage error rate can be substantially more than half the nominal error. Here we apply a mid‐p modification to the gamma confidence interval. Typical mid‐p methods forsake guaranteed coverage to get coverage that is sometimes higher and sometimes lower than the nominal coverage rate, depending on the values of the parameters. The mid‐p gamma interval does not have guaranteed coverage in all situations; however, in the (not rare) situations where the gamma method is overly conservative, the mid‐p gamma interval often has at least nominal coverage. The mid‐p gamma interval is especially appropriate when one wants a central interval, since simulations show that in many situations both the upper and lower coverage error rates are on average less than or equal to half the nominal error rate.     

ESTIMATED POPULATION SIZE OF THE CALIFORNIA GRAY WHALE

Abstract: The 1987-1988 counts of gray whales passing Monterey are reanalyzed to provide a revised population size estimate. The double count data are modeled using iterative logistic regression to allow for the effects of various covariates on probability of detection, and a correction factor is introduced for night rate of travel. The revised absolute population size estimate is 20,869 animals, with CV = 4.37% and 95% confidence interval (19,200, 22,700). In addition the series of relative population size estimates from 1967-1968 to 1987-1988 is scaled to pass through this estimate and modeled to provide variance estimates from interannual variation in population size estimates. This method yields an alternative population size estimate for 1987-1988 of 21,296 animals, with CV = 6.05% and 95% confidence interval (18,900, 24,000). The average annual rate of increase between 1967-1968 and 1987-1988 was estimated to be 3.29% with standard error 0.44%.     

Improved confidence intervals in quantitative trait loci mapping by permutation bootstrapping

The nonparametric bootstrap approach is known to be suitable for calculating central confidence intervals for the locations of quantitative trait loci (QTL). However, the distribution of the bootstrap QTL position estimates along the chromosome is peaked at the positions of the markers and is not tailed equally. This results in conservativeness and large width of the confidence intervals. In this study three modified methods are proposed to calculate nonparametric bootstrap confidence intervals for QTL locations, which compute noncentral confidence intervals (uncorrected method I), correct for the impact of the markers (weighted method I), or both (weighted method II). Noncentral confidence intervals were computed with an analog of the highest posterior density method. The correction for the markers is based on the distribution of QTL estimates along the chromosome when the QTL is not linked with any marker, and it can be obtained with a permutation approach. In a simulation study the three methods were compared with the original bootstrap method. The results showed that it is useful, first, to compute noncentral confidence intervals and, second, to correct the bootstrap distribution of the QTL estimates for the impact of the markers. The weighted method II, combining these two properties, produced the shortest and less biased confidence intervals in a large number of simulated configurations.     

Image based weighted center of proximity versus directly measured knee contact location during simulated gait

To understand the mechanical consequences of knee injury requires a detailed analysis of the effect of that injury on joint contact mechanics during activities of daily living. Three-dimensional (3D) knee joint geometric models have been combined with knee joint kinematics to dynamically estimate the location of joint contact during physiological activities—using a weighted center of proximity (WCoP) method. However, the relationship between the estimated WCoP and the actual location of contact has not been defined. The objective of this study was to assess the relationship between knee joint contact location as estimated using the image-based WCoP method, and a directly measured weighted center of contact (WCoC) method during simulated walking. To achieve this goal, we created knee specific models of six human cadaveric knees from magnetic resonance imaging. All knees were then subjected to physiological loads on a knee simulator intended to mimic gait. Knee joint motion was captured using a motion capture system. Knee joint contact stresses were synchronously recorded using a thin electronic sensor throughout gait, and used to compute WCoC for the medial and lateral plateaus of each knee. WCoP was calculated by combining knee kinematics with the MRI-based knee specific model. Both metrics were compared throughout gait using linear regression. The anteroposterior (AP) location of WCoP was significantly correlated with that of WCoC on both tibial plateaus in all specimens (p<0.01, 95% confidence interval of Pearson?s coefficient r>0), but the correlation was not significant in the mediolateral (ML) direction for 4/6 knees (p>0.05). Our study demonstrates that while the location of joint contact obtained from 3D knee joint contact model, using the WCoP method, is significantly correlated with the location of actual contact stresses in the AP direction, that relationship is less certain in the ML direction.     

Power analysis of statistical methods for comparing treatment differences from limiting dilution assays

Summary Six different statistical methods for comparing limiting dilution assays were evaluated, using both real data and a power analysis of simulated data. Simulated data consisted of a series of 12 dilutions for two treatment groups with 24 cultures per dilution and 1,000 independent replications of each experiment. Data within each replication were generated by Monte Carlo simulation, based on a probability model of the experiment. Analyses of the simulated data revealed that the type I error rates for the six methods differed substantially, with only likelihood ratio and Taswell's weighted mean methods approximating the nominal 5% significance level. Of the six methods, likelihood ratio and Taswell's minimum Chi-square exhibited the best power (least probability of type II errors). Taswell's weighted mean test yielded acceptable type I and type II error rates, whereas the regression method was judged unacceptable for scientific work.     

Effects of genotyping errors,missing values and segregation distortion in molecular marker data on the construction of linkage maps

A simulation study was performed to investigate the effects of missing values, typing errors and distorted segregation ratios in molecular marker data on the construction of genetic linkage maps, and to compare the performance of three locus-ordering criteria (weighted least squares, maximum likelihood and minimum sum of adjacent recombination fractions criteria) in the presence of such effects. The study was based upon three linkage groups of 10 loci at 2, 6, and 10 cM spacings simulated from a doubled-haploid population of size 150. Criteria performance were assessed using the number of replicates with correctly estimated orders, the mean rank correlation between the estimated and the true order and the mean total map length. Bootstrap samples from replicates in the maximum likelihood analysis produced a measure of confidence in the estimated locus order. The effects of missing values and/or typing errors in the data are to reduce the proportion of correctly ordered maps, and this problem worsens as the distances between loci decreases. The maximum likelihood criterion is most successful at ordering loci correctly, but gives estimated map lengths, which are substantially inflated when typing errors are present. The presence of missing values in the data produces shorter map lengths for more widely spaced markers, especially under the weighted least-squares criterion. Overall, the presence of segregation distortion has little effect on this population.     

Confidence Intervals for Population Allele Frequencies: The General Case of Sampling from a Finite Diploid Population of Any Size

The estimation of population allele frequencies using sample data forms a central component of studies in population genetics. These estimates can be used to test hypotheses on the evolutionary processes governing changes in genetic variation among populations. However, existing studies frequently do not account for sampling uncertainty in these estimates, thus compromising their utility. Incorporation of this uncertainty has been hindered by the lack of a method for constructing confidence intervals containing the population allele frequencies, for the general case of sampling from a finite diploid population of any size. In this study, we address this important knowledge gap by presenting a rigorous mathematical method to construct such confidence intervals. For a range of scenarios, the method is used to demonstrate that for a particular allele, in order to obtain accurate estimates within 0.05 of the population allele frequency with high probability (%), a sample size of is often required. This analysis is augmented by an application of the method to empirical sample allele frequency data for two populations of the checkerspot butterfly (Melitaea cinxia L.), occupying meadows in Finland. For each population, the method is used to derive % confidence intervals for the population frequencies of three alleles. These intervals are then used to construct two joint % confidence regions, one for the set of three frequencies for each population. These regions are then used to derive a % confidence interval for Jost''s D, a measure of genetic differentiation between the two populations. Overall, the results demonstrate the practical utility of the method with respect to informing sampling design and accounting for sampling uncertainty in studies of population genetics, important for scientific hypothesis-testing and also for risk-based natural resource management.     

Estimating genetic diversity across the neutral genome with the use of dense marker maps

Background

With the advent of high throughput DNA typing, dense marker maps have become available to investigate genetic diversity on specific regions of the genome. The aim of this paper was to compare two marker based estimates of the genetic diversity in specific genomic regions lying in between markers: IBD-based genetic diversity and heterozygosity.

Methods

A computer simulated population was set up with individuals containing a single 1-Morgan chromosome and 1665 SNP markers and from this one, an additional population was produced with a lower marker density i.e. 166 SNP markers. For each marker interval based on adjacent markers, the genetic diversity was estimated either by IBD probabilities or heterozygosity. Estimates were compared to each other and to the true genetic diversity. The latter was calculated for a marker in the middle of each marker interval that was not used to estimate genetic diversity.

Results

The simulated population had an average minor allele frequency of 0.28 and an LD (r²) of 0.26, comparable to those of real livestock populations. Genetic diversities estimated by IBD probabilities and by heterozygosity were positively correlated, and correlations with the true genetic diversity were quite similar for the simulated population with a high marker density, both for specific regions (r = 0.19-0.20) and large regions (r = 0.61-0.64) over the genome. For the population with a lower marker density, the correlation with the true genetic diversity turned out to be higher for the IBD-based genetic diversity.

Conclusions

Genetic diversities of ungenotyped regions of the genome (i.e. between markers) estimated by IBD-based methods and heterozygosity give similar results for the simulated population with a high marker density. However, for a population with a lower marker density, the IBD-based method gives a better prediction, since variation and recombination between markers are missed with heterozygosity.     

Adaptive design: estimation and inference with censored data in a semiparametric model

In this article, we provide a method of estimation for the treatment effect in the adaptive design for censored survival data with or without adjusting for risk factors other than the treatment indicator. Within the semiparametric Cox proportional hazards model, we propose a bias-adjusted parameter estimator for the treatment coefficient and its asymptotic confidence interval at the end of the trial. The method for obtaining an asymptotic confidence interval and point estimator is based on a general distribution property of the final test statistic from the weighted linear rank statistics at the interims with or without considering the nuisance covariates. The computation of the estimates is straightforward. Extensive simulation studies show that the asymptotic confidence intervals have reasonable nominal probability of coverage, and the proposed point estimators are nearly unbiased with practical sample sizes.     

Interval estimation of genetic susceptibility for retrospective case-control studies

Background

This article describes classical and Bayesian interval estimation of genetic susceptibility based on random samples with pre-specified numbers of unrelated cases and controls.

Results

Frequencies of genotypes in cases and controls can be estimated directly from retrospective case-control data. On the other hand, genetic susceptibility defined as the expected proportion of cases among individuals with a particular genotype depends on the population proportion of cases (prevalence). Given this design, prevalence is an external parameter and hence the susceptibility cannot be estimated based on only the observed data. Interval estimation of susceptibility that can incorporate uncertainty in prevalence values is explored from both classical and Bayesian perspective. Similarity between classical and Bayesian interval estimates in terms of frequentist coverage probabilities for this problem allows an appealing interpretation of classical intervals as bounds for genetic susceptibility. In addition, it is observed that both the asymptotic classical and Bayesian interval estimates have comparable average length. These interval estimates serve as a very good approximation to the "exact" (finite sample) Bayesian interval estimates. Extension from genotypic to allelic susceptibility intervals shows dependency on phenotype-induced deviations from Hardy-Weinberg equilibrium.

Conclusions

The suggested classical and Bayesian interval estimates appear to perform reasonably well. Generally, the use of exact Bayesian interval estimation method is recommended for genetic susceptibility, however the asymptotic classical and approximate Bayesian methods are adequate for sample sizes of at least 50 cases and controls.     

Spreadsheet Method for Evaluation of Biochemical Reaction Rate Coefficients and Their Uncertainties by Weighted Nonlinear Least-Squares Analysis of the Integrated Monod Equation

A convenient method for evaluation of biochemical reaction rate coefficients and their uncertainties is described. The motivation for developing this method was the complexity of existing statistical methods for analysis of biochemical rate equations, as well as the shortcomings of linear approaches, such as Lineweaver-Burk plots. The nonlinear least-squares method provides accurate estimates of the rate coefficients and their uncertainties from experimental data. Linearized methods that involve inversion of data are unreliable since several important assumptions of linear regression are violated. Furthermore, when linearized methods are used, there is no basis for calculation of the uncertainties in the rate coefficients. Uncertainty estimates are crucial to studies involving comparisons of rates for different organisms or environmental conditions. The spreadsheet method uses weighted least-squares analysis to determine the best-fit values of the rate coefficients for the integrated Monod equation. Although the integrated Monod equation is an implicit expression of substrate concentration, weighted least-squares analysis can be employed to calculate approximate differences in substrate concentration between model predictions and data. An iterative search routine in a spreadsheet program is utilized to search for the best-fit values of the coefficients by minimizing the sum of squared weighted errors. The uncertainties in the best-fit values of the rate coefficients are calculated by an approximate method that can also be implemented in a spreadsheet. The uncertainty method can be used to calculate single-parameter (coefficient) confidence intervals, degrees of correlation between parameters, and joint confidence regions for two or more parameters. Example sets of calculations are presented for acetate utilization by a methanogenic mixed culture and trichloroethylene cometabolism by a methane-oxidizing mixed culture. An additional advantage of application of this method to the integrated Monod equation compared with application of linearized methods is the economy of obtaining rate coefficients from a single batch experiment or a few batch experiments rather than having to obtain large numbers of initial rate measurements. However, when initial rate measurements are used, this method can still be used with greater reliability than linearized approaches.     

Shared uncertainty in measurement error problems, with application to Nevada Test Site fallout data

In radiation epidemiology, it is often necessary to use mathematical models in the absence of direct measurements of individual doses. When complex models are used as surrogates for direct measurements to estimate individual doses that occurred almost 50 years ago, dose estimates will be associated with considerable error, this error being a mixture of (a) classical measurement error due to individual data such as diet histories and (b) Berkson measurement error associated with various aspects of the dosimetry system. In the Nevada Test Site(NTS) Thyroid Disease Study, the Berkson measurement errors are correlated within strata. This article concerns the development of statistical methods for inference about risk of radiation dose on thyroid disease, methods that account for the complex error structure inherence in the problem. Bayesian methods using Markov chain Monte Carlo and Monte-Carlo expectation-maximization methods are described, with both sharing a key Metropolis-Hastings step. Regression calibration is also considered, but we show that regression calibration does not use the correlation structure of the Berkson errors. Our methods are applied to the NTS Study, where we find a strong dose-response relationship between dose and thyroiditis. We conclude that full consideration of mixtures of Berkson and classical uncertainties in reconstructed individual doses are important for quantifying the dose response and its credibility/confidence interval. Using regression calibration and expectation values for individual doses can lead to a substantial underestimation of the excess relative risk per gray and its 95% confidence intervals.