SEARCH · Search PubMed

Results for “Nuisance parameters”

Search indexed PubMed citations on genomics, clinical trials, systematic reviews and public health. Explore titles, authors and supplied subject terms, then open the PubMed record.

SourceResearch connectionFrom yearThrough year

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 73 records · Page 4Linked to original sources

PubMed1986

The statistics of quantifiable homeostasis. I. Simple linear homeostasis.

The statistical properties are explored of the least-squares estimators of the parameters of a deterministic linear model for homeostasis with random, normally distributed technical errors. The parameters are: the amplitude of perturbation (A), the homing value (H), the restoration constant (B), and the lag time (L). The former two, although important in their own right, tell us nothing about the processes of biologic restitution and are thus considered "nuisance" parameters; the latter two are the "business" parameters, which tell us about the processes of biologic adjustment. The properties studied include bias and precision and their relationships to the number of data points and to the size of the technical error. Efforts have been concentrated on data points equally spaced over six lag times (which corresponds roughly to the period of observation used in loading studies, such as glucose tolerance tests). The distributions of these four estimators are studied in sets of 100 Monte Carlo simulations, each with 15, 30, and 100 data points, respectively. The estimators are virtually unbiased and all satisfactorily close to Gaussian. The residual mean-square errors (the divisor being four less than the number of data points, since four parameters are estimated) seem to be unbiased estimators of the variances of the errors. Moreover, when suitably scaled, the residual sums of squares have distributions close to the chi-square distribution for the appropriate degrees of freedom and with the corresponding means and variances. The correlations among the estimators are modest and, except where A is involved, small.

Homeostasis↗

PubMed1996

Inferring patterns of migration from gene frequencies under equilibrium conditions.

A new maximum likelihood method to simultaneously estimate the parameters of any migration pattern from gene frequencies in stochastic equilibrium is developed, based on a model of multivariate genetic drift in a subdivided population. Motivated by simulations of this process in the simplified case of two subpopulations, problems related to the nuisance parameter q, the equilibrium gene frequency, are eliminated by conditioning on the observed mean gene frequency. The covariance matrix of this conditional distribution is calculated by constructing an abstract process that mimics the behavior of the original process in the subspace of interest. The approximation holds as long as there is limited differentiation between subpopulations. The bias and variance of estimates of long-range and short-range migration in a finite stepping stone model are evaluated by fitting the model to simulated data with known values of the parameters. Possible ecological extensions of the model are discussed.

Animals↗

PubMed1989

Modelling paired survival data with covariates.

The objective of this paper is to consider the parametric analysis of paired censored survival data when additional covariate information is available, as in the Diabetic Retinopathy Study, which assessed the effectiveness of laser photocoagulation in delaying loss of visual acuity. Our first approach is to extend the fully parametric model of Clayton (1978, Biometrika 65, 141-151) to incorporate covariate information. Our second approach is to obtain parameter estimates from an independence working model together with robust variance estimates. The approaches are compared in terms of efficiency and computational considerations. A fundamental consideration in choosing a strategy for the analysis of paired survival data is whether the correlation within a pair is a nuisance parameter or a parameter of intrinsic scientific interest. The approaches are illustrated with the Diabetic Retinopathy Study.

Actuarial Analysis↗

PubMed2005

A permutation test for inference in logistic regression with small- and moderate-sized data sets.

Inference based on large sample results can be highly inaccurate if applied to logistic regression with small data sets. Furthermore, maximum likelihood estimates for the regression parameters will on occasion not exist, and large sample results will be invalid. Exact conditional logistic regression is an alternative that can be used whether or not maximum likelihood estimates exist, but can be overly conservative. This approach also requires grouping the values of continuous variables corresponding to nuisance parameters, and inference can depend on how this is done. A simple permutation test of the hypothesis that a regression parameter is zero can overcome these limitations. The variable of interest is replaced by the residuals from a linear regression of it on all other independent variables. Logistic regressions are then done for permutations of these residuals, and a p-value is computed by comparing the resulting likelihood ratio statistics to the original observed value. Simulations of binary outcome data with two independent variables that have binary or lognormal distributions yield the following results: (a) in small data sets consisting of 20 observations, type I error is well-controlled by the permutation test, but poorly controlled by the asymptotic likelihood ratio test; (b) in large data sets consisting of 1000 observations, performance of the permutation test appears equivalent to that of the asymptotic test; and (c) in small data sets, the p-value for the permutation test is usually similar to the mid-p-value for exact conditional logistic regression.

Clinical Trials as Topic↗

PubMed2001

Interpretation of variation across marker loci as evidence of selection.

Population structure and history have similar effects on the genetic diversity at all neutral loci. However, some marker loci may also have been strongly influenced by natural selection. Selection shapes genetic diversity in a locus-specific manner. If we could identify those loci that have responded to selection during the divergence of populations, then we may obtain better estimates of the parameters of population history by excluding these loci. Previous attempts were made to identify outlier loci from the distribution of sample statistics under neutral models of population structure and history. Unfortunately these methods depend on assumptions about population structure and history that usually cannot be verified. In this article, we define new population-specific parameters of population divergence and construct sample statistics that are estimators of these parameters. We then use the joint distribution of these estimators to identify outlier loci that may be subject to selection. We found that outlier loci are easier to recognize when this joint distribution is conditioned on the total number of allelic states represented in the pooled sample at each locus. This is so because the conditional distribution is less sensitive to the values of nuisance parameters.

Animals↗

PubMed1979

Analysis of life tables with grouping and withdrawals.

A number of individuals is observed at the beginning of a period. At the end of the period the number is surviving, the number who have died and the number who have withdrawn are noted. From these three numbers it is required to estimate the death rate for the period. All relevant quantities are supposed independent and identically distributed for the individuals. The likelihood is calculated and found to depend on two parameters, other than the death rate, and to be unidenttifiable so that no consistent estimators exist. For large numbers, the posterior distribution of the death rate is approximated by a normal distribution whose mean is the root of a quadratic equation and whose variance is the sum of two terms; the first is proportional to the reciprocal of the number of individuals, as usually happens with a consistent estimator; the second does not tend to zero and depends on initial opinions about one of the nuisance parameters. The paper is a simple exercise in the routine use of coherent, Bayesian methodology. Numerical calucations illustrate the results.

Actuarial Analysis↗

PubMed2005

A case-cohort design for assessing covariate effects in longitudinal studies.

The case-cohort design for longitudinal data consists of a subcohort sampled at the beginning of the study that is followed repeatedly over time, and a case sample that is ascertained through the course of the study. Although some members in the subcohort may experience events over the study period, we refer to it as the "control-cohort." The case sample is a random sample of subjects not in the control-cohort, who have experienced at least one event during the study period. Different correlations among repeated observations on the same individual are accommodated by a two-level random-effects model. This design allows consistent estimation of all parameters estimable in a cohort design and is a cost-effective way to study the effects of covariates on repeated observations of relatively rare binary outcomes when exposure assessment is expensive. It is an extension of the case-cohort design (Prentice, 1986, Biometrika73, 1-11) and the bidirectional case-crossover design (Navidi, 1998, Biometrics54, 596-605). A simulation study compares the efficiency of the longitudinal case-cohort design to a full cohort analysis, and we find that in certain situations up to 90% efficiency can be obtained with half the sample size required for a full cohort analysis. A bootstrap method is presented that permits testing for intra-subject homogeneity in the presence of unidentifiable nuisance parameters in the two-level random-effects model. As an illustration we apply the design to data from an ongoing study of childhood asthma.

Animals↗

PubMed1999

Monte Carlo EM for missing covariates in parametric regression models.

We propose a method for estimating parameters for general parametric regression models with an arbitrary number of missing covariates. We allow any pattern of missing data and assume that the missing data mechanism is ignorable throughout. When the missing covariates are categorical, a useful technique for obtaining parameter estimates is the EM algorithm by the method of weights proposed in Ibrahim (1990, Journal of the American Statistical Association 85, 765-769). We extend this method to continuous or mixed categorical and continuous covariates, and for arbitrary parametric regression models, by adapting a Monte Carlo version of the EM algorithm as discussed by Wei and Tanner (1990, Journal of the American Statistical Association 85, 699-704). In addition, we discuss the Gibbs sampler for sampling from the conditional distribution of the missing covariates given the observed data and show that the appropriate complete conditionals are log-concave. The log-concavity property of the conditional distributions will facilitate a straightforward implementation of the Gibbs sampler via the adaptive rejection algorithm of Gilks and Wild (1992, Applied Statistics 41, 337-348). We assume the model for the response given the covariates is an arbitrary parametric regression model, such as a generalized linear model, a parametric survival model, or a nonlinear model. We model the marginal distribution of the covariates as a product of one-dimensional conditional distributions. This allows us a great deal of flexibility in modeling the distribution of the covariates and reduces the number of nuisance parameters that are introduced in the E-step. We present examples involving both simulated and real data.

Algorithms↗

PubMed2000

Bootstrap confidence intervals for relative risk parameters in affected-sib-pair data.

In affected-sib-pair (ASP) studies, parameters such as the locus-specific sibling relative risk, lambda(s), may be estimated and used to decide whether or not to continue the search for susceptibility genes. Typically, a maximum likelihood point estimate of lambda(s) is given, but since this estimate may have substantial variability, it is of interest to obtain confidence limits for the true value of lambda(s). While a variety of methods for doing this exist, there is considerable uncertainty over their reliability. This is because the discrete nature of ASP data and the imposition of genetic "possible triangle" constraints during the likelihood maximization mean that asymptotic results may not apply. In this paper, we use simulation to evaluate the reliability of various asymptotic and simulation-based confidence intervals, the latter being based on a resampling, or bootstrap approach. We seek to identify, from the large pool of methods available, those methods that yield short intervals with accurate coverage probabilities for ASP data. Our results show that many of the most popular bootstrap confidence interval methods perform poorly for ASP data, giving coverage probabilities much lower than claimed. The test-inversion, profile-likelihood, and asymptotic methods, however, perform well, although some care is needed in choice of nuisance parameter. Overall, in simulations under a variety of different genetic hypotheses, we find that the asymptotic methods of confidence interval evaluation are the most reliable, even in small samples. We illustrate our results with a practical application to a real data set, obtaining confidence intervals for the sibling relative risks associated with several loci involved in type 1 diabetes.

Confidence Intervals↗

PubMed1996

Association within twin pairs for a dichotomous trait.

We propose an odds-ratio measure of twin association for a dichotomous trait. This odds ratio can be estimated without arbitrarily specifying an index twin and without estimating additional nuisance parameters. Tests of association and confidence intervals may be computed easily, in contrast to those proposed previously [Donner et al. (1995) Genet Epidemiol 12:267-277] for a correlation measure of association. For testing homogeneity of association in two samples of twins, monozygotic and dizygotic, we propose a large-sample test and an exact test, both based on the odds-ratio parameterization. Large-sample tests of homogeneity are slightly anticonservative when some cell counts are small, and the exact test may be preferred in these situations. We also propose a long-linear parameterization useful for modeling more complex data sets.

Attention↗

PubMed1997

Alternative test for linkage between two loci.

When using the maximum likelihood method to test for linkage it is traditionally assumed that the female (theta f) and male (theta m) recombination fractions are equal. However, this assumption is not always realistic. In this paper we present a testing for linkage that does not require this assumption. Specifically, we propose testing for linkage by testing the null hypothesis H0: theta f + theta m = 1 vs. the alternative HA: theta f + theta m < 1, treating theta f - theta m as a nuisance parameter. This leads to a likelihood ratio test statistic that is asymptotically distributed as a chi-square with one degree of freedom. By examining the expected values of the maximum lod scores, we show that for data from phase-known meioses this test can provide a more powerful test for linkage-especially so when theta f not equal to theta m-unless the recombination fraction is zero. For data from phase-unknown meioses, the proposed test is a more powerful test for linkage only for large values of theta f - theta m; otherwise, the traditional test has higher power. Thus, the proposed test can lead to gain in power for detecting linkage when either phase is known for most meioses or when there is a large absolute difference between the male and female recombination fractions.

Chromosome Mapping↗

PubMed1997

Genotype relative-risks and association tests for nuclear families with missing parental data.

The development of a new method for testing the association of genetic markers with disease is presented. This approach is applicable when sampling nuclear families with one or more affected siblings and when neither, one, or both parents are missing marker genotype data. All siblings, affected and not affected, are used to probabilistically infer the missing parental marker data. A likelihood ratio statistic, which treats marker allele frequencies as nuisance parameters, is presented to test whether all marker relative risks are equal to one (i.e., no marker association). This approach offers a solution to test for marker associations when parents are difficult to obtain.

Algorithms↗

PubMed2000

Analysis of a cluster randomized trial with binary outcome data using a multi-level model.

The use of multi-level logistic regression models was explored for the analysis of data from a cluster randomized trial investigating whether a training programme for general practitioners' reception staff could improve women's attendance at breast screening. Twenty-six general practices were randomized with women nested within them, requiring a two-level model which allowed for between-practice variability. Comparisons were made with fixed effect (FE) and random effects (RE) cluster summary statistic methods, ordinary logistic regression and a marginal model based on generalized estimating equations with robust variance estimates. An FE summary statistic method and ordinary logistic regression considerably understated the variance of the intervention effect, thus overstating its statistical significance. The marginal model produced a higher statistical significance for the intervention effect compared to that obtained from the RE summary statistic method and the multi-level model. Because there was only a moderate number of practices and these had unbalanced cluster sizes, reliable asymptotic properties for the robust standard errors used in the marginal model may not have been achieved. While the RE summary statistic method cannot handle multiple covariates easily, marginal and multi-level models can do so. In contrast to multi-level models however, marginal models do not provide direct estimates of variance components, but treat these as nuisance parameters. Estimates of the variance components were of particular interest in this example. Additionally, parametric bootstrap methods within the multi-level model framework provide confidence intervals for these variance components, as well as a confidence interval for the effect of intervention which allows for the imprecision in the estimated variance components. The assumption of normality of the random effects can be checked, and the models extended to investigate multiple sources of variability.

Breast Neoplasms↗

PubMed2003

Bayesian analysis of a previously published genome screen for panic disorder reveals new and compelling evidence for linkage to chromosome 7.

This article presents a Bayesian re-analysis of a linkage study of panic disorder Crowe et al. [2001: Am J Med Genet (Neuropsychiatr Genet) 105:105-109]. In the initial analysis Crowe et al. failed to find compelling evidence for linkage based on either LOD scores or NPL scores anywhere in the genome. The maximum LOD score was 2.23 on chromosome 7 at marker D7S2846 (57.79 cM according to Marshfield). Over the past several years we have been developing a Bayesian alternative approach to linkage analysis, based on direct measurement of the posterior probability of linkage (PPL), and have shown elsewhere that this approach has several advantages over the available alternatives for mapping complex-disease genes Vieland [1998: Am J Med Genet 63:947-954]; Wang et al. [1999: Genet Epidemiol 17(Suppl 1):S749-S754]; Wang et al. [2000: Ann Hum Genet 64:533-553]; and Vieland et al. [2001: Hum Hered 51:199-208]. One limitation of this approach in previous applications has been that it required the investigator to specify a fixed genetic model for the trait. We employ a new implementation of the PPL that treats the unknown trait model as a vector of nuisance parameters, which is integrated out of the PPL equation. When we apply this new model-integrated version of the PPL to the data of Crowe et al. [2001: Am J Med Genet (Neuropsychiatr Genet) 105:105-109] we obtain much clearer evidence than previously reported for a locus on chromosome 7, with an 80% probability of linkage to marker D7S521. A second location is also identified on chromosome 16 near marker D16S749 (PPL = 24%). The results for the remainder of the genome are consistently low. The two loci identified here are also supported by independent evidence from other studies.

Bayes Theorem↗

PubMed2005

Some remarks about the analysis of active control studies.

In an active-controlled trial, the experimental treatment can be declared to be non-inferior to the control if the confidence interval for the difference excludes a fixed pre-specified margin. Recently, some articles have discussed an alternative method where the data from the current study and placebo-controlled studies for the active control are combined together into a single test statistic to test whether a fixed fraction of the effect of the active control is preserved. It has been shown that, conditional on nuisance parameters from the active-controlled study, a fixed margin can be defined that will be operationally equivalent to this latter method. In this article, we will discuss statistical properties associated with these approaches. Specifically, the interim monitoring boundaries and level of evidence will be considered.

Controlled Clinical Trials as Topic↗

PubMed2006

Practical use of modified maximum likelihoods for stratified data.

Stratified data arise in several settings, such as longitudinal studies or multicenter clinical trials. Between-strata heterogeneity is usually addressed by random effects models, but an alternative approach is given by fixed effects models, which treat the incidental nuisance parameters as fixed unknown quantities. This approach presents several advantages, like computational simplicity and robustness to confounding by strata. However, maximum likelihood estimates of the parameter of interest are typically affected by incidental parameter bias. A remedy to this is given by the elimination of stratum-specific parameters by exact or approximate conditioning. The latter solution is afforded by the modified profile likelihood, which is the method applied in this paper. The aim is to demonstrate how the theory of modified profile likelihoods provides convenient solutions to various inferential problems in this setting. Specific procedures are available for different kinds of response variables, and they are useful both for inferential purposes and as a diagnostic method for validating random effects models. Some examples with real data illustrate these points.

Animals↗

PubMed2004

Case/pseudocontrol analysis in genetic association studies: A unified framework for detection of genotype and haplotype associations, gene-gene and gene-environment interactions, and parent-of-origin effects.

Estimation and testing of genetic effects (genotype relative risks) are often performed conditionally on parental genotypes, using data from case-parent trios. This strategy avoids having to estimate nuisance parameters such as parental mating type frequencies, and also avoids generating spurious results due to confounding causes of association such as population stratification. For effects at a single locus, the resulting analysis is equivalent to matched case/control analysis via conditional logistic regression, using the case and three "pseudocontrols" derived from the untransmitted parental alleles. We previously showed that a similar approach can be used for analyzing genotype and haplotype effects at a set of closely linked loci, but with a required adjustment to the conditioning argument that results in varying numbers of pseudocontrols, depending on the disease model that is to be fitted. Here we extend this method to include the analysis of epistatic effects (gene-gene interactions) at unlinked loci, to include parent-of-origin effects at one or more loci, and to allow additional incorporation of gene-environment interactions. The conditional logistic approach provides a natural and flexible framework for incorporating these additional effects. By relaxing the conditioning on parental genotypes to allow exchangeability of parental genotypes, we show how the power of this approach can be increased when studying parent-of-origin effects. Simulations suggest that there is limited power to distinguish between parent-of-origin effects and effects due to interaction between genotypes of mother and child.

Algorithms↗

PubMed2002

Analysis of the ratio of marginal probabilities in a matched-pair setting.

Statistical methods for testing and interval estimation of the ratio of marginal probabilities in the matched-pair setting are considered in this paper. We are especially interested in the situation where the null value is not one, as in one-sided equivalence trials. We propose a Fieller-type statistic based on constrained maximum likelihood (CML) estimation of nuisance parameters. For a series of examples, the significance level of the CML test is satisfactorily close to the nominal level, while a Wald-type test is anticonservative for reasonable sample sizes. We present formulae for approximate power and sample size for the CML and Wald tests. The matched design is seen to have a clear advantage over the unmatched design in terms of asymptotic efficiency when the two responses of the pair are highly positively correlated. We recommend the CML method over the Wald method, especially for small or moderate sample sizes.

Animals↗