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At least 19 recordsLinked to original sources
PubMed2005

Comparison of maximum statistics for hypothesis testing when a nuisance parameter is present only under the alternative.

In many practical problems, a hypothesis testing involves a nuisance parameter which appears only under the alternative hypothesis. Davies (1977, Biometrika 64, 247-254) proposed the maximum of the score statistics over the whole range of the nuisance parameter as a test statistic for this type of hypothesis testing. Freidlin, Podgor, and Gastwirth (1999, Biometrics 55, 883-886) studied two other simpler maximum test statistics, the maximum of the score statistics at two extreme points of the nuisance parameter, and the maximum of the score statistics at three points of the nuisance parameter including the two extreme points. In this article, we compare the powers of these three maximum-type statistics in the context of three genetic problems.

Biometry↗
PubMed2003

Fisher Lecture: the 2002 R. A. Fisher lecture: dedicated to the memory of Shanti S. Gupta. Variances are not always nuisance parameters.

In classical problems, e.g., comparing two populations, fitting a regression surface, etc., variability is a nuisance parameter. The term "nuisance parameter" is meant here in both the technical and the practical sense. However, there are many instances where understanding the structure of variability is just as central as understanding the mean structure. The purpose of this article is to review a few of these problems. I focus in particular on two issues: (a) the determination of the validity of an assay; and (b) the issue of the power for detecting health effects from nutrient intakes when the latter are measured by food frequency questionnaires. I will also briefly mention the problems of variance structure in generalized linear mixed models, robust parameter design in quality technology, and the signal in microarrays. In these and other problems, treating variance structure as a nuisance instead of a central part of the modeling effort not only leads to inefficient estimation of means, but also to misleading conclusions.

Analysis of Variance↗
PubMed2005

Two-stage sample size re-estimation based on a nuisance parameter: a review.

Sample size calculations are important and difficult in clinical trails because they depend on the nuisance parameter and treatment effect. Recently, much attention has been focused on two-stage methods whereby the first stage constitutes an internal pilot study used to estimate parameters and revise the final sample size. This paper reviews two-stage methods based on estimation of nuisance parameters in either a continuous or dichotomous outcome setting.

Algorithms↗
PubMed2000

Linkage analysis in the presence of errors III: marker loci and their map as nuisance parameters.

In linkage and linkage disequilibrium (LD) analysis of complex multifactorial phenotypes, various types of errors can greatly reduce the chance of successful gene localization. The power of such studies-even in the absence of errors-is quite low, and, accordingly, their robustness to errors can be poor, especially in multipoint analysis. For this reason, it is important to deal with the ramifications of errors up front, as part of the analytical strategy. In this study, errors in the characterization of marker-locus parameters-including allele frequencies, haplotype frequencies (i.e., LD between marker loci), recombination fractions, and locus order-are dealt with through the use of profile likelihoods maximized over such nuisance parameters. It is shown that the common practice of assuming fixed, erroneous values for such parameters can reduce the power and/or increase the probability of obtaining false positive results in a study. The effects of errors in assumed parameter values are generally more severe when a larger number of less informative marker loci, like the highly-touted single nucleotide polymorphisms (SNPs), are analyzed jointly than when fewer but more informative marker loci, such as microsatellites, are used. Rather than fixing inaccurate values for these parameters a priori, we propose to treat them as nuisance parameters through the use of profile likelihoods. It is demonstrated that the power of linkage and/or LD analysis can be increased through application of this technique in situations where parameter values cannot be specified with a high degree of certainty.

Alleles↗
PubMed2001

Efficiency gain from auxiliary data requiring additional nuisance parameters.

In a mark-recapture study of an animal population, live-recapture information may be supplemented by resightings from marked animals obtained throughout the period of the study and the two types of data analyzed simultaneously. The resighting data can only contribute to estimates of survival probability if they are jointly modeled with the live-recapture data and require the inclusion of additional nuisance parameters. We show that, under quite general conditions, estimates of the original parameters are estimated with the same, or improved, precision despite the inclusion of the nuisance parameters. Banding data from female goldeneye ducks (Bucephala clangula) are used as an illustration.

Animals↗
PubMed1991

Nuisance parameters in the assessment of noncognitive manifestations of dementia.

Measures of noncognitive disturbances in dementia typically rely on ratings by informants. In this study, method variance in measures of affect, behavior, and functional competence was evaluated by comparing ratings from 2 types of informants (spouse vs. child) in patients with probable Alzheimer's disease. No differences were observed on 5 variables; for 2 measures of behavior disturbance and 1 measure of functional disability, ratings obtained from child informants were significantly higher than spouse ratings for patients at comparable levels of dementia severity. Although the effect of the rater dimension was relatively small and selective, the findings suggest the need for further research on potential sources of bias in the ratings of noncognitive features of the dementia syndrome.

Activities of Daily Living↗
PubMed2004

The size of the chi-square test for the Hardy-Weinberg law.

Many scientific problems can be formulated in terms of a statistical model indexed by parameters, only some of which are of scientific interest and the other parameters, called nuisance parameters, are not of interest in themselves. For testing the Hardy-Weinberg law, a relation among genotype and allele probabilities is of interest and allele probabilities are of no interest and now nuisance parameters. In this paper we investigate how the size (the maximum of the type I error rate over the nuisance parameter space) of the chi-square test for the Hardy-Weinberg law is affected by the nuisance parameters. Whether the size is well controlled or not under the nominal level has been frequently investigated as basic components of statistical tests. The size represents the type I error rate at the worst case. We prove that the size is always greater than the nominal level as the sample size increases. Extensive computations show that the size of the chi-squared test (worst type I error rate over the nuisance parameter space) deviates more upwardly from the nominal level as the sample size gets larger. The value at which the maximum of the type I error rate was found moves closer to the edges of the the nuisance parameter space with increasing sample size. An exact test is recommended as an alternative when the type I error is inflated.

Alleles↗
PubMed2003

Monte carlo conditional inference for log-linear and logistic models: a survey of current methodology.

Nuisance parameters are parameters that are not of immediate interest to the experimenter. For log-linear and logistic models the null distribution of (most) statistics of interest depends on such parameters. Traditionally, nuisance parameters were eliminated by performing inference with respect to the chi-squared limiting distribution of common test statistics. An alternative solution is to eliminate the nuisance parameters by conditioning on their minimal sufficient statistics. The support of the resulting conditional distribution is often intractable, making null probability calculations challenging. An often feasible way to avoid complete enumeration of this support is to approximate conditional probabilities using Monte Carlo methods. In this article we survey recent developments in Monte Carlo conditional analysis for log-linear and logistic models focusing on the algorithms proposed by Booth and Butler, Diaconis and Sturmfels, Smith et al., and Mehta et al. We illustrate these algorithms with simple motivating examples.

Algorithms↗
PubMed2005

Genome scans with gene-covariate interaction.

Genetic models for gene-covariate interactions are described. Methods of linkage analysis that utilize special features of these models and the corresponding score statistics are derived. Their power is compared with that of simple genome scans that ignore these special features, and substantial gains in power are observed when the gene-covariate interaction is strong. Quantitative trait mapping in randomly ascertained sibships and affected sibpair mapping are discussed. For the latter case, a simpler statistic is proposed that has similar performance to the score statistic, but does not require the estimation of nuisance parameters. Since the nuisance parameters are not estimable solely from affected sib-pair data, this statistic would be much easier to apply in practice. Similarities with linkage analysis of models for longitudinal data and multivariate phenotypes are also briefly discussed. Approximations for the P-value and power are derived under the framework of local alternatives.

Chromosome Mapping↗
PubMed2007

Maximum-likelihood methods in wavefront sensing: stochastic models and likelihood functions.

Maximum-likelihood (ML) estimation in wavefront sensing requires careful attention to all noise sources and all factors that influence the sensor data. We present detailed probability density functions for the output of the image detector in a wavefront sensor, conditional not only on wavefront parameters but also on various nuisance parameters. Practical ways of dealing with nuisance parameters are described, and final expressions for likelihoods and Fisher information matrices are derived. The theory is illustrated by discussing Shack-Hartmann sensors, and computational requirements are discussed. Simulation results show that ML estimation can significantly increase the dynamic range of a Shack-Hartmann sensor with four detectors and that it can reduce the residual wavefront error when compared with traditional methods.

Electrons↗
PubMed2001

Effects of implicit parameters in segregation analysis.

In human genetic analysis, data are collected through the so-called 'ascertainment procedure'. Statistically this sampling scheme can be thought of as a multistage sampling method. At the first stage, one or several probands are ascertained. At the subsequent stages, a sequential sampling scheme is applied. Sampling in such a way is virtually a nonrandom procedure, which, in most cases, causes biased estimation which may be intractable. This paper focuses on the underlying causes of the intractability problem of ascertained genetic data. Three types of parameters, i.e. target, design and nuisance parameters, are defined as the essences to formulate the true likelihood of a set of data. These parameters are also classified into explicit or implicit parameters depending on whether they can be expressed explicity in the likelihood function. For ascertained genetic data, a sequential scheme is regarded as an implicit design parameter, and a true pedigree structure as an implicit nuisance parameter. The intractability problem is attributed to loss of information of any implicit parameter in likelihood formulation. Several approaches to build a likelihood for estimation of the segregation ratio when only an observed pedigree structure is available are proposed.

Female↗
PubMed1992

A Gibbs sampling approach to linkage analysis.

We present a Monte Carlo approach to estimation of the recombination fraction theta and the profile likelihood for a dichotomous trait and a single marker gene with 2 alleles. The method is an application of a technique known as 'Gibbs sampling', in which random samples of each of the unknowns (here genotypes, theta and nuisance parameters, including the allele frequencies and the penetrances) are drawn from their posterior distributions, given the data and the current values of all the other unknowns. Upon convergence, the resulting samples derive from the marginal distribution of all the unknowns, given only the data, so that the uncertainty in the specification of the nuisance parameters is reflected in the variance of the posterior distribution of theta. Prior knowledge about the distribution of theta and the nuisance parameters can be incorporated using a Bayesian approach, but adoption of a flat prior for theta and point priors for the nuisance parameters would correspond to the standard likelihood approach. The method is easy to program, runs quickly on a microcomputer, and could be generalized to multiple alleles, multipoint linkage, continuous phenotypes and more complex models of disease etiology. The basic approach is illustrated by application to data on cholesterol levels and an a low-density lipoprotein receptor gene in a single large pedigree.

Animals↗
PubMed1998

Inference using conditional logistic regression with missing covariates.

When there are many nuisance parameters in a logistic regression model, a popular method for eliminating these nuisance parameters is conditional logistic regression. Unfortunately, another common problem in a logistic regression analysis is missing covariate data. With many nuisance parameters to eliminate and missing covariates, many investigators exclude any subject with missing covariates and then use conditional logistic regression, often called a complete-case analysis. In this article, we derive a modified conditional logistic regression that is appropriate with covariates that are missing at random. Performing a conditional logistic regression with only the complete cases is convenient with existing statistical packages, but it may give bias if missingness is not completely at random.

Bias↗
PubMed1993

Two-stage tests for studying monotherapy and combination therapy in two-by-two factorial trials.

Two-stage testing involves a preliminary test of a nuisance parameter prior to testing a main hypothesis. In a two-by-two factorial trial, the treatment interaction is the nuisance to the inference about the efficacy of one of the treatments given alone. In comparing a combination therapy to both of its component therapies, the nuisance parameter is the difference in the component effects. When the preliminary test is an integral part of inference about the main parameter, the actual level of significance for the two-stage test procedure can be much higher than the desired nominal level. If one places no restriction on the value of the nuisance parameter, then any two-stage test with its significance level properly controlled has undesirable properties. This applies to comparative studies of combination agents relative to the component agents. When the interaction with an ineffective treatment is null, two-stage testing may have some power advantage for assessing monotherapy efficacy.

Clinical Trials as Topic↗
PubMed1999

Likelihood methods for incomplete longitudinal binary responses with incomplete categorical covariates.

We consider longitudinal studies in which the outcome observed over time is binary and the covariates of interest are categorical. With no missing responses or covariates, one specifies a multinomial model for the responses given the covariates and uses maximum likelihood to estimate the parameters. Unfortunately, incomplete data in the responses and covariates are a common occurrence in longitudinal studies. Here we assume the missing data are missing at random (Rubin, 1976, Biometrika 63, 581-592). Since all of the missing data (responses and covariates) are categorical, a useful technique for obtaining maximum likelihood parameter estimates is the EM algorithm by the method of weights proposed in Ibrahim (1990, Journal of the American Statistical Association 85, 765-769). In using the EM algorithm with missing responses and covariates, one specifies the joint distribution of the responses and covariates. Here we consider the parameters of the covariate distribution as a nuisance. In data sets where the percentage of missing data is high, the estimates of the nuisance parameters can lead to highly unstable estimates of the parameters of interest. We propose a conditional model for the covariate distribution that has several modeling advantages for the EM algorithm and provides a reduction in the number of nuisance parameters, thus providing more stable estimates in finite samples.

Affect↗
PubMed2001

Estimation for paired binomial data with application to radiation therapy.

We compare and contrast several different methods for estimating the effect of treatment when responses are paired binomial observations. The ratio of binomial probabilities is the parameter of interest, while the binomial probabilities are nuisance parameters which may vary between pairs. The application is a meta-analysis of the treatment of rectal cancer, with observations in each study indicating the number of recurrences of the cancer in each of two groups, one with radiation therapy and one without. The ratio of the probabilities of recurrence in the radiation to non-radiation groups is of substantive interest, and is modelled as a logistic or complementary log-log function of an unknown linear combination of the covariates. The three methods we consider are maximum likelihood, a Bayesian approach and an approach based on estimating equations. For the MLE and Bayesian approach the potentially large number of nuisance parameters are estimated together with the parameters of interest, whereas for the estimating equation approach only the parameters of interest are estimated. A simulation study is performed to compare the methods and evaluate the impact of overdispersion.

Bayes Theorem↗
PubMed2005

Maximum information designs.

BACKGROUND: Expressions to determine the sample size N needed to provide power 1-beta to detect a difference between groups, say delta, involve other nuisance parameters, such as the variance of the observations for a test or means, or the control group probability for a test of proportions or the control group hazard rate for a logrank test of event-times. Designs where N is fixed are called maximum N or duration designs because the sample size and required duration of the study can be fixed, or estimated. However, such designs are expected, but not guaranteed, to provide the desired level of power to detect the specified difference delta at the study end because the true or estimated values of the nuisance parameters are unknown. Thus, the actual information to be accrued and the associated level of power are random variables with sample variation. METHODS: Expressions are developed to determine the amount of information needed to provide the desired level of power, regardless of the values of the nuisance parameters. RESULTS: The amount of information (I) in the observed data is readily quantified and can often be expressed as the inverse of the variance of the test statistic. Also, the total amount of information required to provide the desired level of power 1-beta to detect a difference delta with a test at level alpha is readily determined, designated as Ialpha,beta,delta. Under a maximum information design, the study is continued until the required total amount of information is accrued, or I = Ialpha,beta,delta. In this case, the sample size or duration are random variables, but each can be estimated under various projections about the underlying parameters (variance, probability, hazard) in advance. The implementation of a maximum information design for two and multiple group trials is described for a test of means, proportions and event-times using the logrank test. Application to other methods of analysis is described. CONCLUSIONS: A maximum information design provides greater assurance that the desired level of power will be attained. However, the exact study duration is unknown. Issues related to the implementation of such a design are discussed.

Clinical Trials as Topic↗
PubMed2001

Mid-trial design reviews for sequential clinical trials.

When sequential clinical trials are conducted by plotting a statistic measuring treatment difference against another measuring information, power is guaranteed regardless of nuisance parameters. However, values need to be assigned to nuisance parameters in order to gain an impression of the sample size distribution. Each interim analysis provides an opportunity to re-evaluate the relationship between sample size and information. In this paper we discuss such mid-trial design reviews. In the special cases of trials with a relatively short recruitment phase followed by a longer period of follow-up, and of normally distributed responses, mid-trial design reviews are particularly important. Examples are given of the various situations considered, and extensive simulations are reported demonstrating the validity of the review procedure in the case of normally distributed responses.

Cardiovascular Diseases↗