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PubMed2005

Comparing the likelihood ratios of two binary diagnostic tests in the presence of partial verification.

The comparison of the efficiency of two binary diagnostic tests requires one to know the disease status for all patients in the sample, by applying a gold standard. In two-phase studies the gold standard is not applied to all patients in a sample, and the problem of partial verification of the disease arises. At present, one of the approaches most used for comparing two binary diagnostic tests are the likelihood ratios. In this study, the maximum likelihood estimators of likelihood ratios are obtained. The tests of hypothesis to compare the likelihood ratios of two binary diagnostic tests when both are applied to the same random sample in the presence of verification bias are deduced, and simulation experiments are performed in order to investigate the asymptotic behaviour of the tests of hypothesis. The results obtained have been applied to the study of Alzheimer's disease.

Algorithms↗

PubMed2004

Estimating penetrance from family data using a retrospective likelihood when ascertainment depends on genotype and age of onset.

In diseases caused by deleterious gene mutations, knowledge of age-specific cumulative risks is necessary for medical management of mutation carriers. When pedigrees are ascertained through several affected persons, ascertainment bias can be corrected by using a retrospective likelihood. This likelihood is a function of the genotypes of pedigree members given their phenotypes and provides unbiased estimates of penetrance without modeling the selection process, provided that selection is independent of genotypes. However, since mutation testing is offered only to relatives of mutation carriers, the genotypes of family members are available only in mutated families and selection does depend on genotype. In the present study, we quantified the bias due to selection on genotype using simulations. We found that this bias depended on the true penetrance value: the lower the penetrance, the higher the bias (risk by age 80 estimated to be 46% for a true penetrance value of 20%). When age of onset is added to the selection criteria, as usually done, we showed that the bias was even higher. We modified the conditioning in the retrospective likelihood, what we call "genotype restricted likelihood" (GRL). Using simulations, we show that this method provided unbiased parameter estimates under all the selection designs considered.

Age of Onset↗

PubMed2005

Maximum likelihood estimation of haplotype effects and haplotype-environment interactions in association studies.

The associations between haplotypes and disease phenotypes offer valuable clues about the genetic determinants of complex diseases. It is highly challenging to make statistical inferences about these associations because of the unknown gametic phase in genotype data. We describe a general likelihood-based approach to inferring haplotype-disease associations in studies of unrelated individuals. We consider all possible phenotypes (including disease indicator, quantitative trait, and potentially censored age at onset of disease) and all commonly used study designs (including cross-sectional, case-control, cohort, nested case-control, and case-cohort). The effects of haplotypes on phenotype are characterized by appropriate regression models, which allow various genetic mechanisms and gene-environment interactions. We present the likelihood functions for all study designs and disease phenotypes under Hardy-Weinberg disequilibrium. The corresponding maximum likelihood estimators are approximately unbiased, normally distributed, and statistically efficient. We provide simple and efficient numerical algorithms to calculate the maximum likelihood estimators and their variances, and implement these algorithms in a freely available computer program. Extensive simulation studies demonstrate that the proposed methods perform well in realistic situations. An application to the Carolina Breast Cancer Study reveals significant haplotype effects and haplotype-smoking interactions in the development of breast cancer.

Algorithms↗

PubMed2004

Advantages to transforming the receiver operating characteristic (ROC) curve into likelihood ratio co-ordinates.

Traditionally, the receiver operating characteristic (ROC) curve for a diagnostic test plots true positives (sensitivity) against false positives (one minus specificity). However, this representation brings with it several drawbacks. A transformation to positive and negative likelihood ratio co-ordinates, scaled by base-ten logarithms, offers several advantages. First we motivate the use of positive and negative likelihood ratios, emphasizing their relationship to modification of the odds ratio. Then we highlight properties of likelihood ratios using the traditional ROC axes. Finally, we demonstrate ROC curves and their properties after conversion to likelihood ratio co-ordinates. These graphs do not waste space for tests lacking diagnostic power, and offer a simple visual assessment of a test's impact on the odds ratio.

Diagnostic Tests, Routine↗

PubMed2005

Bias-corrected maximum likelihood estimator of the intraclass correlation parameter for binary data.

A popular model to analyse over/under-dispersed proportions is to assume the extended beta-binomial model with dispersion (intraclass correlation) parameter phi and then to estimate this parameter by maximum likelihood. However, it is well known that maximum likelihood estimate (MLE) may be biased when the sample size n or the total Fisher information is small. In this paper we obtain a bias-corrected maximum likelihood (BCML) estimator of the intraclass correlation parameter and compare it, by simulation, in terms of bias and efficiency, with the MLE, an estimator Q(2) based on optimal quadratic estimating equations of Crowder and recommended by Paul et al. and a double extended quasi-likelihood (DEQL) estimator proposed by Lee. The BCML estimator has superior bias and efficiency properties in most instances. Analyses of a set of toxicological data from Paul and a set of medical data pertaining to chromosomal abnormalities among survivors of the atomic bomb in Hiroshima from Otake and Prentice show, in general, much improvement in standard errors of the BCML estimates over the other three estimates.

Animals↗

PubMed2006

Likelihood approaches to the non-parametric two-sample problem for right-censored data.

The classical two-sample problem with random right-censoring is considered. We show that non- parametric likelihood techniques can be used to obtain tests for either the identity hypothesis or the non-parametric Behrens-Fisher hypothesis (NBFH). In the case of the identity hypothesis, a special imputed permutation distribution is used to estimate the distribution under the null hypothesis. In the case of the NBFH, simulation from the constrained non-parametric maximum likelihood estimate is used. Simulation shows that the tests using either approximation have excellent control of the type I error rate, even with quite small sample sizes. Further, for Lehmann-type alternatives the likelihood-based methods have similar power to the logrank test, while for the non-Lehmann-type alternatives tried here the likelihood-based methods have superior power.

Computer Simulation↗

PubMed2006

A maximum likelihood method for studying gene-environment interactions under conditional independence of genotype and exposure.

Given the biomedical interest in gene-environment interactions along with the difficulties inherent in gathering genetic data from controls, epidemiologists need methodologies that can increase precision of estimating interactions while minimizing the genotyping of controls. To achieve this purpose, many epidemiologists suggested that one can use case-only design. In this paper, we present a maximum likelihood method for making inference about gene-environment interactions using case-only data. The probability of disease development is described by a logistic risk model. Thus the interactions are model parameters measuring the departure of joint effects of exposure and genotype from multiplicative odds ratios. We extend the typical inference method derived under the assumption of independence between genotype and exposure to that under a more general assumption of conditional independence. Our maximum likelihood method can be applied to analyse both categorical and continuous environmental factors, and generalized to make inference about gene-gene-environment interactions. Moreover, the application of this method can be reduced to simply fitting a multinomial logistic model when we have case-only data. As a consequence, the maximum likelihood estimates of interactions and likelihood ratio tests for hypotheses concerning interactions can be easily computed. The methodology is illustrated through an example based on a study about the joint effects of XRCC1 polymorphisms and smoking on bladder cancer. We also give two simulation studies to show that the proposed method is reliable in finite sample situation.

DNA-Binding Proteins↗

PubMed2007

Statistical evidence for GLM regression parameters: a robust likelihood approach.

When a likelihood ratio is used to measure the strength of evidence for one hypothesis over another, its reliability (i.e. how often it produces misleading evidence) depends on the specification of the working model. When the working model happens to be the 'true' or 'correct' model, the probability of observing strong misleading evidence is low and controllable. But this is not necessarily the case when the working model is misspecified. Royall and Tsou (J. R. Stat. Soc., Ser. B 2003; 65:391-404) show how to adjust working models to make them robust to misspecification. Likelihood ratios derived from their 'robust adjusted likelihood' are just as reliable (asymptotically) as if the working model were correctly specified in the first place. In this paper, we apply and extend these ideas to the generalized linear model (GLM) regression setting. We provide several illustrations (both from simulated data and real data concerning rates of parasitic infection in Philippine adolescents), show how the required adjustment factor can be obtained from standard statistical software, and draw some connections between this approach and the 'sandwich estimator' for robust standard errors of regression parameters. This substantially broadens the availability and the viability of likelihood methods for measuring statistical evidence in regression settings.

Adolescent↗

PubMed1993

A comparison of the generalized estimating equation approach with the maximum likelihood approach for repeated measurements.

Liang and Zeger proposed an extension of generalized linear models to the analysis of longitudinal data. Their approach is closely related to quasi-likelihood methods and can handle both normal and non-normal outcome variables such as Poisson or binary outcomes. Their approach, however, has been applied mainly to non-normal outcome variables. This is probably due to the fact that there is a large class of multivariate linear models available for normal outcomes such as growth models and random-effects models. Furthermore, there are many iterative algorithms that yield maximum likelihood estimators (MLEs) of the model parameters. The multivariate linear model approach, based on maximum likelihood (ML) estimation, specifies the joint multivariate normal distribution of outcome variables while the approach of Liang and Zeger, based on the quasi-likelihood, specifies only the marginal distributions. In this paper, I compare the approach of Liang and Zeger and the ML approach for the multivariate normal outcomes. I show that the generalized estimating equation (GEE) reduces to the score equation only when the data do not have missing observations and the correlation is unstructured. In more general cases, however, the GEE estimation yields consistent estimators that may differ from the MLEs. That is, the GEE does not always reduce to the score equation even when the outcome variables are multivariate normal. I compare the small sample properties of the GEE estimators and the MLEs by means of a Monte Carlo simulation study.

Clinical Trials as Topic↗

PubMed2005

Development of an integrated genetic map of a sugarcane (Saccharum spp.) commercial cross, based on a maximum-likelihood approach for estimation of linkage and linkage phases.

Sugarcane (Saccharum spp.) is a clonally propagated outcrossing polyploid crop of great importance in tropical agriculture. Up to now, all sugarcane genetic maps had been developed using either full-sib progenies derived from interspecific crosses or from selfing, both approaches not directly adopted in conventional breeding. We have developed a single integrated genetic map using a population derived from a cross between two pre-commercial cultivars ('SP80-180' x 'SP80-4966') using a novel approach based on the simultaneous maximum-likelihood estimation of linkage and linkage phases method specially designed for outcrossing species. From a total of 1,118 single-dose markers (RFLP, SSR and AFLP) identified, 39% derived from a testcross configuration between the parents segregating in a 1:1 fashion, while 61% segregated 3:1, representing heterozygous markers in both parents with the same genotypes. The markers segregating 3:1 were used to establish linkage between the testcross markers. The final map comprised of 357 linked markers, including 57 RFLPs, 64 SSRs and 236 AFLPs that were assigned to 131 co-segregation groups, considering a LOD score of 5, and a recombination fraction of 37.5 cM with map distances estimated by Kosambi function. The co-segregation groups represented a total map length of 2,602.4 cM, with a marker density of 7.3 cM. When the same data were analyzed using JoinMap software, only 217 linked markers were assigned to 98 co-segregation groups, spanning 1,340 cM, with a marker density of 6.2 cM. The maximum-likelihood approach reduced the number of unlinked markers to 761 (68.0%), compared to 901 (80.5%) using JoinMap. All the co-segregation groups obtained using JoinMap were present in the map constructed based on the maximum-likelihood method. Differences on the marker order within the co-segregation groups were observed between the two maps. Based on RFLP and SSR markers, 42 of the 131 co-segregation groups were assembled into 12 putative homology groups. Overall, the simultaneous maximum-likelihood estimation of linkage and linkage phases was more efficient than the method used by JoinMap to generate an integrated genetic map of sugarcane.

Chromosome Mapping↗

PubMed2004

Estimating the expectation of the log-likelihood with censored data for estimator selection.

A criterion for choosing an estimator in a family of semi-parametric estimators from incomplete data is proposed. This criterion is the expected observed log-likelihood (ELL). Adapted versions of this criterion in case of censored data and in presence of explanatory variables are exhibited. We show that likelihood cross-validation (LCV) is an estimator of ELL and we exhibit three bootstrap estimators. A simulation study considering both families of kernel and penalized likelihood estimators of the hazard function (indexed on a smoothing parameter) demonstrates good results of LCV and a bootstrap estimator called ELL(bboot). We apply the ELL(bboot) criterion to compare the kernel and penalized likelihood estimators to estimate the risk of developing dementia for women using data from a large cohort study.

Bias↗

PubMed1995

The likelihood of recurrence in bipolar affective disorder: the importance of episode recency.

These analyses used a high-intensity follow-up of of patients with bipolar affective disorder to describe the immediate and long-term risks for recurrence and the importance of sustained recovery to those risks. At the baseline evaluation, all patients were in episodes of Research Diagnostic Criteria major depressive disorder, mania or schizoaffective disorder (excluding the mainly schizophrenic subtype); those who were depressed at intake had a history of mania or schizoaffective mania. Raters re-evaluated these patients at 6-month intervals for 5 years and annually for the remainder of a 10-year follow-up. The following report describes relapse risks for the 186 patients observed to recover from their index episodes. Survival analyses quantified the likelihood of relapse over time, beginning after symptom-free periods of 4 months and 1, 2 and 3 years. Further survival analyses used treatment status as a censoring variable to estimate the eventual likelihood of recurrence among those who reported sustained compliance with lithium prophylaxis; the prophylaxis group remained under observation until they relapsed, were lost to follow-up or ceased taking lithium. Progressively longer symptom-free periods were clearly associated with lower relapse risks over the subsequent 4 years. Thereafter, however, this effect dissipitated. 7 years after recovery, the cumulative likelihood of recurrence was four in five for all bipolar patients and two in three for those whose index episode had been followed by at least 3 years without symptoms. Even with sustained lithium prophylaxis, the likelihood of at least one recurrence exceeded 70% within 5 years of recovery.(ABSTRACT TRUNCATED AT 250 WORDS)

Adult↗

PubMed2005

Impact of C-reactive protein on the likelihood of peripheral arterial disease in United States adults with the metabolic syndrome, diabetes mellitus, and preexisting cardiovascular disease.

We sought to determine, in United States (US) patients with the metabolic syndrome (MS), diabetes mellitus (DM), or preexisting cardiovascular disease, whether higher levels of C-reactive protein (CRP) would identify those with an increased likelihood of peripheral arterial disease (PAD). In a cross-sectional evaluation of the National Health and Nutrition Examination Survey (NHANES), 1999 to 2000, of 1,600 adults (representing a US population of 62.9 million) aged > or =40 years who had valid ankle-brachial index measurements available, subjects were categorized as having MS (without DM), DM, preexisting cardiovascular disease, or none of these conditions. The presence of PAD was defined as an ankle-brachial index <0.9. Subjects were also divided into groups according to CRP levels that were low (<1 mg/L), intermediate (1 to 3 mg/L), and elevated (>3.0 mg/L). Weighted multiple logistic regression analysis examined the odds of PAD by CRP group and disease category compared with the reference group of subjects who did not have MS, DM, or cardiovascular disease and had a CRP level of <1 mg/L. Those with MS (including DM) had an increased likelihood of PAD (odds ratio 4.8, 95% confidence interval 1.4 to 16.1, p = 0.01) as did those with MS without diabetes and an elevated CRP level (odds ratio 3.9, 95% confidence interval 1.1 to 14.6, p = 0.04); those with DM and an elevated CRP had the highest likelihood of PAD (odds ratio 8.6, 95% confidence interval 2.2 to 34.0, p = 0.001). In conclusion, the likelihood of PAD in US adults with MS and DM is enhanced by elevated CRP levels.

Adult↗

PubMed2005

Adaptive penalty likelihood for reconstruction of multidimensional confocal microscopy images.

In this paper we devise a penalty likelihood with noise constraints method to restore 2D and 3D confocal microscope images. Regularization is a commonly used technique in image restoration to balance restored image quality and noise suppression, but despite this noise is usually amplified. Taking into account common confocal imaging system degradation, we develop an algorithm by using a gradient descent method (PLGDA) to approach the minimum solution of the penalty likelihood equation. A Lagrange parameter controls the balance between the penalty and likelihood terms and is estimated using an adaptive method. We show that the a priori information is key to the regularization and Lagrange parameter estimation. The convergence characteristics are analysed and discussed. PLGDA and a traditional maximum likelihood expectation maximization are used to restore 2D and 3D confocal images. The point spread function (PSF), used to restore the data is collected from an experiment and modelled by bi-cubic splines to give an accurate noise free representation. Our experimental results show that the restored images are significantly improved by PLGDA.

Algorithms↗

PubMed2004

The likelihood of adverse outcomes in triplet pregnancies estimated by pregravid maternal characteristics.

OBJECTIVE: To estimate the likelihood of adverse outcomes in triplet pregnancies by a score comprising pregravid maternal characteristics. DESIGN: A cross-sectional study. SETTING: Triplets database collected by Matria Healthcare, Inc. PATIENT(S): A scoring system was constructed, assigning 1 point for the presence of a risk factor (nulliparity, stature <165 cm, and age <35 years) and 0 for the absence of a risk factor. Data related to 2,887 triplet sets were analyzed. INTERVENTION(S): None. MAIN OUTCOME MEASURE(S): Total triplet birth weight <4,500 and delivery at 27-32 weeks. RESULT(S): We identified 18% of triplets' mothers (score 3) in whom the likelihood for adverse results is 50%-90% higher and the likelihood for optimal results is 40% to 70% lower than background rates. CONCLUSION(S): A pregravid maternal profile could estimate the likelihood of adverse outcomes and be used for consulting patients at risk of having or carrying a triplet pregnancy.

Birth Weight↗

PubMed2006

A note on the accuracy of PAC-likelihood inference with microsatellite data.

Stephens and Donnelly have introduced a simple yet powerful importance sampling scheme for computing the likelihood in population genetic models. Fundamental to the method is an approximation to the conditional probability of the allelic type of an additional gene, given those currently in the sample. As noted by Li and Stephens, the product of these conditional probabilities for a sequence of draws that gives the frequency of allelic types in a sample is an approximation to the likelihood, and can be used directly in inference. The aim of this note is to demonstrate the high level of accuracy of "product of approximate conditionals" (PAC) likelihood when used with microsatellite data. Results obtained on simulated microsatellite data show that this strategy leads to a negligible bias over a wide range of the scaled mutation parameter theta. Furthermore, the sampling variance of likelihood estimates as well as the computation time are lower than that obtained with importance sampling on the whole range of theta. It follows that this approach represents an efficient substitute to IS algorithms in computer intensive (e.g. MCMC) inference methods in population genetics.

Gene Frequency↗

PubMed1998

On the use of linear regression and maximum likelihood for QTL mapping in half-sib designs.

Methods of identification of quantitative trait loci (QTL) using a half-sib design are generally based on least-squares or maximum likelihood approaches. These methods differ in the genetical model considered and in the information used. Despite these differences, the power of the two methods in a daughter design in very similar. Using an analogy with a one-way analysis of variance, we propose an equation connecting the two test-statistics (F ratio for regression and likelihood ratio test in the case of the maximum likelihood). The robustness of this relationship is tested by simulation for different single QTL models. In general, the correspondence between the two statistics is good under both the null hypothesis and the alternative hypothesis of a single QTL segregating. Practical implications are discussed with particular emphasis on the theoretical distribution of the likelihood ratio test.

Chromosome Mapping↗

PubMed2001

A profile conditional likelihood approach for the semiparametric transformation regression model with missing covariates.

We propose a profile conditional likelihood approach to handle missing covariates in the general semiparametric transformation regression model. The method estimates the marginal survival function by the Kaplan-Meier estimator, and then estimates the parameters of the survival model and the covariate distribution from a conditional likelihood, substituting the Kaplan-Meier estimator for the marginal survival function in the conditional likelihood. This method is simpler than full maximum likelihood approaches, and yields consistent and asymptotically normally distributed estimator of the regression parameter when censoring is independent of the covariates. The estimator demonstrates very high relative efficiency in simulations. When compared with complete-case analysis, the proposed estimator can be more efficient when the missing data are missing completely at random and can correct bias when the missing data are missing at random. The potential application of the proposed method to the generalized probit model with missing continuous covariates is also outlined.

Humans↗