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PubMed2001

The Cox proportional hazards model with a partly linear relative risk function.

The conventional Cox proportional hazards regression model contains a loglinear relative risk function, linking the covariate information to the hazard ratio with a finite number of parameters. A generalization, termed the partly linear Cox model, allows for both finite dimensional parameters and an infinite dimensional parameter in the relative risk function, providing a more robust specification of the relative risk function. In this work, a likelihood based inference procedure is developed for the finite dimensional parameters of the partly linear Cox model. To alleviate the problems associated with a likelihood approach in the presence of an infinite dimensional parameter, the relative risk is reparameterized such that the finite dimensional parameters of interest are orthogonal to the infinite dimensional parameter. Inference on the finite dimensional parameters is accomplished through maximization of the profile partial likelihood, profiling out the infinite dimensional nuisance parameter using a kernel function. The asymptotic distribution theory for the maximum profile partial likelihood estimate is established. It is determined that this estimate is asymptotically efficient; the orthogonal reparameterization enables employment of profile likelihood inference procedures without adjustment for estimation of the nuisance parameter. An example from a retrospective analysis in cancer demonstrates the methodology.

Humans↗
PubMed2003

Exact unconditional tests for testing non-inferiority in matched-pairs design.

The problem of testing non-inferiority in a 2 x 2 matched-pairs sample is considered. Two exact unconditional tests based on the standard and the confidence interval p-values are proposed. Although tests of non-inferiority have two nuisance parameters under the null hypothesis, the exact tests are defined by reducing the dimension of nuisance parameter space from two to one using the monotonicity of the distribution. The exact sizes and powers of these tests and the existing asymptotic test are considered. The exact tests are found to be accurate in view of their size property. In addition, the exact test based on the confidence interval p-value is more powerful than the other exact test. It is shown that the asymptotic test is inaccurate, that is, its size exceeds the claimed nominal level alpha. Therefore, it recommends a cautious approach in use of the asymptotic test for the problem of testing non-inferiority, particularly when sample sizes are small or moderately large.

Algorithms↗
PubMed2006

Information-based monitoring of clinical trials.

When designing a clinical trial to compare the effect of different treatments on response, a key issue facing the statistician is to determine how large a study is necessary to detect a clinically important difference with sufficient power. This is the case whether the study will be analysed only once (single-analysis) or whether it will be monitored periodically with the possibility of early stopping (group-sequential). Standard sample size calculations are based on both the magnitude of difference that is considered clinically important as well as values for the nuisance parameters in the statistical model. For planning purposes, best guesses are made for the value of the nuisance parameters and these are used to determine the sample size. However, if these guesses are incorrect this will affect the subsequent power to detect the clinically important difference. It is argued in this paper that statistical precision is directly related to Statistical Information and that the study should continue until the requisite statistical information is obtained. This is referred to as information-based design and analysis of clinical trials. We also argue that this type of methodology is best suited with group-sequential trials which monitor the data periodically and allow for estimation of the statistical information as the study progresses.

Clinical Trials Data Monitoring Committees↗
PubMed2002

Sample size re-estimation in cluster randomization trials.

Cluster randomization trials in which families are the unit of allocation are commonly adopted for the evaluation of disease prevention interventions. Sample size estimation for cluster randomization trials depends on parameters that quantify the variability within and between clusters and the variability in cluster size. Accurate advance estimates of these nuisance parameters may be difficult to obtain and misspecification may lead to an underpowered study. Since families are typically recruited over time, we propose using a portion of the data to estimate the nuisance parameters and to re-estimate sample size based on the estimates. This extends the standard internal pilot study methods to the setting of cluster randomization trials. The effect of this design on the power, significance level and sample size is analysed via simulation and is shown to provide a flexible and practical approach to cluster randomization trials.

Cluster Analysis↗
PubMed2000

Family-based tests of association and linkage that use unaffected sibs, covariates, and interactions.

We extend the methodology for family-based tests of association and linkage to allow for both variation in the phenotypes of subjects and incorporation of covariates into general-score tests of association. We use standard association models for a phenotype and any number of predictors. We then construct a score statistic, using likelihoods for the distribution of phenotype, given genotype. The distribution of the score is computed as a function of offspring genotypes, conditional on parental genotypes and trait values for offspring and parents. This approach provides a natural extension of the transmission/disequilibrium test to any phenotype and to multiple genes or environmental factors and allows the study of gene-gene and gene-environment interaction. When the trait varies among subjects or when covariates are included in the association model, the score statistic depends on one or more nuisance parameters. We suggest two approaches for obtaining parameter estimates: (1) choosing the estimate that minimizes the variance of the test statistic and (2) maximizing the statistic over a nuisance parameter and using a corrected P value. We apply our methods to a sample of families with attention-deficit/hyperactivity disorder and provide examples of how covariates and gene-environment and gene-gene interactions can be incorporated.

Alleles↗
PubMed2003

Exact unconditional tests for a 2 x 2 matched-pairs design.

The problem of comparing two proportions in a 2 x 2 matched-pairs design with binary responses is considered. We consider one-sided null and alternative hypotheses. The problem has two nuisance parameters. Using the monotonicity of the multinomial distribution, four exact unconditional tests based on p-values are proposed by reducing the dimension of the nuisance parameter space from two to one in computation. The size and power of the four exact tests and two other tests, the exact conditional binomial test and the asymptotic McNemar's test, are considered. It is shown that the tests based on the confidence interval p-value are more powerful than the tests based on the standard p-value. In addition, it is found that the exact conditional binomial test is conservative and not powerful for testing the hypothesis. Moreover, the asymptotic McNemar's test is shown to have incorrect size; that is, its size is larger than the nominal level of the test. Overall, the test based on McNemar's statistic and the confidence interval p-value is found to be the most powerful test with the correct size among the tests in this comparison.

Binomial Distribution↗
PubMed1993

Testing for the existence of a desirable dose combination.

We consider the problem of studying several dose combinations of two drugs for a therapeutic endpoint in a multilevel factorial clinical trial. Two test statistics are constructed to test whether there exists at least one dose combination that is more effective than its component doses. Their distributions involve nuisance parameters quantifying the mean differences among the doses of the two component drugs. It is shown that their power functions achieve maxima as all the nuisance parameters approach infinity in absolute value. The significance levels of the two tests are derived and two alpha-level tests are proposed. Tables are given to provide the alpha-level critical values for these tests and to gain insights into their power performances.

Antihypertensive Agents↗
PubMed1998

Measuring departures from Hardy-Weinberg: a Markov chain Monte Carlo method for estimating the inbreeding coefficient.

Many well-established statistical methods in genetics were developed in a climate of severe constraints on computational power. Recent advances in simulation methodology now bring modern, flexible statistical methods within the reach of scientists having access to a desktop workstation. We illustrate the potential advantages now available by considering the problem of assessing departures from Hardy-Weinberg (HW) equilibrium. Several hypothesis tests of HW have been established, as well as a variety of point estimation methods for the parameter which measures departures from HW under the inbreeding model. We propose a computational, Bayesian method for assessing departures from HW, which has a number of important advantages over existing approaches. The method incorporates the effects-of uncertainty about the nuisance parameters--the allele frequencies--as well as the boundary constraints on f (which are functions of the nuisance parameters). Results are naturally presented visually, exploiting the graphics capabilities of modern computer environments to allow straightforward interpretation. Perhaps most importantly, the method is founded on a flexible, likelihood-based modelling framework, which can incorporate the inbreeding model if appropriate, but also allows the assumptions of the model to he investigated and, if necessary, relaxed. Under appropriate conditions, information can be shared across loci and, possibly, across populations, leading to more precise estimation. The advantages of the method are illustrated by application both to simulated data and to data analysed by alternative methods in the recent literature.

Algorithms↗
PubMed2002

Approximate Bayesian computation in population genetics.

We propose a new method for approximate Bayesian statistical inference on the basis of summary statistics. The method is suited to complex problems that arise in population genetics, extending ideas developed in this setting by earlier authors. Properties of the posterior distribution of a parameter, such as its mean or density curve, are approximated without explicit likelihood calculations. This is achieved by fitting a local-linear regression of simulated parameter values on simulated summary statistics, and then substituting the observed summary statistics into the regression equation. The method combines many of the advantages of Bayesian statistical inference with the computational efficiency of methods based on summary statistics. A key advantage of the method is that the nuisance parameters are automatically integrated out in the simulation step, so that the large numbers of nuisance parameters that arise in population genetics problems can be handled without difficulty. Simulation results indicate computational and statistical efficiency that compares favorably with those of alternative methods previously proposed in the literature. We also compare the relative efficiency of inferences obtained using methods based on summary statistics with those obtained directly from the data using MCMC.

Bayes Theorem↗
PubMed2002

A semiparametric empirical likelihood method for data from an outcome-dependent sampling scheme with a continuous outcome.

Outcome-dependent sampling (ODS) schemes can be a cost effective way to enhance study efficiency. The case-control design has been widely used in epidemiologic studies. However, when the outcome is measured on a continuous scale, dichotomizing the outcome could lead to a loss of efficiency. Recent epidemiologic studies have used ODS sampling schemes where, in addition to an overall random sample, there are also a number of supplemental samples that are collected based on a continuous outcome variable. We consider a semiparametric empirical likelihood inference procedure in which the underlying distribution of covariates is treated as a nuisance parameter and is left unspecified. The proposed estimator has asymptotic normality properties. The likelihood ratio statistic using the semiparametric empirical likelihood function has Wilks-type properties in that, under the null, it follows a chi-square distribution asymptotically and is independent of the nuisance parameters. Our simulation results indicate that, for data obtained using an ODS design, the semiparametric empirical likelihood estimator is more efficient than conditional likelihood and probability weighted pseudolikelihood estimators and that ODS designs (along with the proposed estimator) can produce more efficient estimates than simple random sample designs of the same size. We apply the proposed method to analyze a data set from the Collaborative Perinatal Project (CPP), an ongoing environmental epidemiologic study, to assess the relationship between maternal polychlorinated biphenyl (PCB) level and children's IQ test performance.

Biometry↗
PubMed2003

Some fundamental issues with non-inferiority testing in active controlled trials.

In an active controlled non-inferiority trial without a placebo arm, it is often not entirely clear what the primary objective is. In many cases the considered goal is to demonstrate that the experimental treatment preserves at least some fraction of the effect of the active control. The active control effect is a parameter, the value of which is unknown. To test the hypothesis of effect preservation, the classical confidence interval approach requires specification of a non-inferiority margin which is a function of the unknown active control effect. When the margin is estimated, it is also not clear what is the relevant type I error of making a false assertion about preservation of the active control effect. The statistical uncertainty of the estimated margin arguably needs to be incorporated in evaluation of the type I error. In this paper we discuss these fundamental issues. We show that the classical confidence interval approach cannot attain the target type I error exactly since this error varies as the sample size or as the values of the nuisance parameters in the active controlled trial change. In contrast, the preservation tests, as proposed in literature, can attain the target type I error rate exactly, regardless of the sample size and the values of the nuisance parameters, but can do so only at the price of several strong assumptions holding that may not be directly verifiable. One assumption is the constancy condition holding whereby the effect of the active control in the historical trial populations is assumed to carry to the population of the active control trial. When this condition is violated, both the confidence interval approach and the preservation test method may be problematic.

Confidence Intervals↗
PubMed2004

A novel means of using gene clusters in a two-step empirical Bayes method for predicting classes of samples.

MOTIVATION: The classification of samples using gene expression profiles is an important application in areas such as cancer research and environmental health studies. However, the classification is usually based on a small number of samples, and each sample is a long vector of thousands of gene expression levels. An important issue in parametric modeling for so many gene expression levels is the control of the number of nuisance parameters in the model. Large models often lead to intensive or even intractable computation, while small models may be inadequate for complex data. METHODOLOGY: We propose a two-step empirical Bayes classification method as a solution to this issue. At the first step, we use the model-based cluster algorithm with a non-traditional purpose of assigning gene expression levels to form abundance groups. At the second step, by assuming the same variance for all the genes in the same group, we substantially reduce the number of nuisance parameters in our statistical model. RESULTS: The proposed model is more parsimonious, which leads to efficient computation under an empirical Bayes estimation procedure. We consider two real examples and simulate data using our method. Desired low classification error rates are obtained even when a large number of genes are pre-selected for class prediction.

Algorithms↗
PubMed2004

Testing for association with a case-parents design in the presence of genotyping errors.

Genotyping errors can create a problem for the analysis of case-parents data because some families will exhibit genotypes that are inconsistent with Mendelian inheritance. The problem with correcting Mendelian inconsistent genotype errors by regenotyping or removing families in which they occur is that the remaining unidentified genotype errors can produce excess type I (false positive) error for some family-based tests for association. We address this problem by developing a likelihood ratio test (LRT) for association in a case-parents design that incorporates nuisance parameters for a general genotype error model. We extend the likelihood approach for a single SNP to include short haplotypes consisting of 2 or 3 SNPs. The extension to haplotypes is based on assumptions of random mating, multiplicative penetrances, and at most a single genotype error per family. For a single SNP, we found, using Monte Carlo simulation, that type I error rate can be controlled for a number of genotype error models at different error rates. Simulation results suggest the same is true for 2 and 3 SNPs. In all cases, power declined with increasing genotyping error rates. In the absence of genotyping errors, power was similar whether nuisance parameters for genotype error were included in the LRT or not. The LRT developed here does not require prior specification of a particular model for genotype errors and it can be readily computed using the EM algorithm. Consequently, this test may be generally useful as a test of association with case-parents data in which Mendelian inconsistent families are observed.

Adult↗
PubMed2000

An approximate unconditional test of non-inferiority between two proportions.

This paper investigates an approximate unconditional test for non-inferiority between two independent binomial proportions. The P-value of the approximate unconditional test is evaluated using the maximum likelihood estimate of the nuisance parameter. In this paper, we clarify some differences in defining the rejection regions between the approximate unconditional and conventional conditional or unconditional exact test. We compare the approximate unconditional test with the asymptotic test and unconditional exact test by Chan (Statistics in Medicine, 17, 1403-1413, 1998) with respect to the type I error and power. In general, the type I errors and powers are in the decreasing order of the asymptotic, approximate unconditional and unconditional exact tests. In many cases, the type I errors are above the nominal level from the asymptotic test, and are below the nominal level from the unconditional exact test. In summary, when the non-inferiority test is formulated in terms of the difference between two proportions, the approximate unconditional test is the most desirable, because it is easier to implement and generally more powerful than the unconditional exact test and its size rarely exceeds the nominal size. However, when a test between two proportions is formulated in terms of the ratio of two proportions, such as a test of efficacy, more caution should be made in selecting a test procedure. The performance of the tests depends on the sample size and the range of plausible values of the nuisance parameter. Published in 2000 by John Wiley & Sons, Ltd.

Child↗
PubMed1977

Sequential allocation in clinical trials comparing two exponential survival curves.

The problem of comparing two medical treatments with respect to survival is considered. Treatment outcome is assumed to follow an exponential distribution. The ratio of expected survivals associated with the two treatments is the clinical parameter of interest. A nuisance parameter is present, but it is removed by an invariance reduction and a sequential probability ratio test is applied to the invariant likelihood ratio. A class of data-dependent treatment assignment rules is identified over which the probability of correct treatment selection at the termination of the trial is approximately constant. A cost function, the weighted sum of total patients in the trial and the number assigned to the inferior treatment is introduced, and a treatment allocation rule conjectured to minimize the expected cost is constructed. Both analytic and simulation results show that it is an improvement over rules previously proposed. The methodology contained herein can be used to construct near-optimal rules in other testing contexts.

Biometry↗
PubMed1991

Modelling digit preference in fecundability studies.

Retrospective studies of fecundability, in which women are asked how many cycles they required to become pregnant, are often affected by problems of digit preference. A probability model for such digit preference is proposed in which misreporting favours 6 or 12 (and possibly also 3) cycles. It is assumed that in the absence of misreporting the number of cycles follows a beta-geometric distribution. The model is applied to two data sets, with clear-cut results: The inclusion of additional parameters to model the misreporting can lead to substantial improvements in fit, but causes little change to the estimated parameters of the underlying beta-geometric distribution. In some cases misreporting parameters may be regarded as nuisance parameters, while in others they may be of interest. We have found estimates of these parameters to vary between different categories of women in an interpretable manner. The models may also be used to estimate the percentage of couples in any study that misreport their conception waiting time.

Biometry↗
PubMed2006

Detecting pulsatile hormone secretions using nonlinear mixed effects partial spline models.

Neuroendocrine ensembles communicate with their remote and proximal target cells via an intermittent pattern of chemical signaling. The identification of episodic releases of hormonal pulse signals constitutes a major emphasis of endocrine investigation. Estimating the number, temporal locations, secretion rate, and elimination rate from hormone concentration measurements is of critical importance in endocrinology. In this article, we propose a new flexible statistical method for pulse detection based on nonlinear mixed effects partial spline models. We model pulsatile secretions using biophysical models and investigate biological variation between pulses using random effects. Pooling information from different pulses provides more efficient and stable estimation for parameters of interest. We combine all nuisance parameters including a nonconstant basal secretion rate and biological variations into a baseline function that is modeled nonparametrically using smoothing splines. We develop model selection and parameter estimation methods for the general nonlinear mixed effects partial spline models and an R package for pulse detection and estimation. We evaluate performance and the benefit of shrinkage by simulations and apply our methods to data from a medical experiment.

Animals↗
PubMed2000

Directly modelling matched case-control data.

Matching in case-control studies is a situation in which one wishes to make inferences about a parameter of interest in the presence of nuisance parameters. The usual approach is to apply a conditional likelihood. A bivariate latent class log-linear model for binomial responses is shown to yield a standard likelihood identical to the usual conditional one. This extension of the Rasch model for binary responses gives consistent estimates and a suitable likelihood function for cases matched with any fixed number of controls.

Age Factors↗