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PubMed2001

Power comparisons between the TDT and two likelihood-based methods.

We compare the statistical power of the transmission disequilibrium test (TDT) with that of two likelihood-based linkage tests, the classical LOD score and a modified LOD score in which a linkage disequilibrium (LD) parameter is incorporated into the likelihood (LD-LOD). We hypothesize that, when LD is present, the LD-LOD will have the greatest power of the three tests because the TDT breaks a multiplex pedigree into triads, and the LOD score has previously been shown to have lower power when LD is present but not accounted for. We test this hypothesis using a simulation study in which we generate affected sib-pair (ASP) pedigrees under a range of genetic models, varying the genotypic relative risk (GRR) from 6 to 16. Because the likelihood-based tests require that a genetic model be specified, we compare the tests under two scenarios. First, we assume the true genetic model in the analysis, and second, we compare the tests when the LD-LOD (LOD) is maximized over two wrong genetic models. For the generating models we considered, we find that the LD-LOD has greater power than the TDT even when the genetic models is mis-specified and the results corrected for multiple tests. Extreme differences occur under the multiplicative and dominant models, for which the difference in power is as high as 40% at complete LD. The LOD score provides the lowest power in the presence of LD for the range of GRR considered here.

Adult↗

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↗

PubMed2006

The admixture maximum likelihood test: a novel experiment-wise test of association between disease and multiple SNPs.

Disease association studies often test large numbers of markers, and various methods have been proposed to correct for multiple testing. In this paper, we propose an admixture maximum likelihood approach that estimates both the proportion of associated single nucleotide polymorphisms (SNPs) and their typical effect size. We assessed this method and compared it with several previously proposed approaches by simulation. The maximum likelihood approach performed similarly to or better than all other tests across a wide range of alternative hypotheses. The rank truncated product method also had good power, though somewhat inferior to the maximum likelihood approach in most cases. A simple Bonferroni correction performed best only when the number of associated SNPs was small.

Genetic Predisposition to Disease↗

PubMed2002

Likelihood methods for measuring statistical evidence.

Focused on interpreting data as statistical evidence, the evidential paradigm uses likelihood ratios to measure the strength of statistical evidence. Under this paradigm, re-examination of accumulating evidence is encouraged because (i) the likelihood ratio, unlike a p-value, is unaffected by the number of examinations and (ii) the probability of observing strong misleading evidence is naturally low, even for study designs that re-examine the data with each new observation. Further, the controllable probabilities of observing misleading and weak evidence provide assurance that the study design is reliable without affecting the strength of statistical evidence in the data. This paper illustrates the ideas and methods associated with using likelihood ratios to measure statistical evidence. It contains a comprehensive introduction to the evidential paradigm, including an overview of how to quantify the probability of observing misleading evidence for various study designs. The University Group Diabetes Program (UGDP), a classic and still controversial multi-centred clinical trial, is used as an illustrative example. Some of the original UGDP results, and subsequent re-analyses, are presented for comparison purposes.

Biometry↗

PubMed2003

Missing data in the 2 x 2 table: patterns and likelihood-based analysis for cross-sectional studies with supplemental sampling.

Standard measures of crude association in the context of a cross-sectional study are the risk difference, relative risk and odds ratio as derived from a 2x 2 table. Most such studies are subject to missing data on disease, exposure, or both, introducing bias into the usual complete-case analysis. We describe several scenarios distinguished by the manner in which missing data arise, and for each we adjust the natural multinomial likelihood to properly account for missing data. The situations presented allow for increasing levels of generality with regard to the missing data mechanism. The final case, quite conceivable in epidemiologic studies, assumes that the probability of missing exposure depends on true exposure and disease status, as well as upon whether disease status is missing (and conversely for the probability of missing disease information). When parameters relating to the missing data process are inestimable without strong assumptions, we propose maximum likelihood analysis subsequent to collecting supplemental data in the spirit of a validation study. Analytical results give insight into the bias inherent in complete-case analysis for each scenario, and numerical results illustrate the performance of likelihood-based point and interval estimates in the most general case. Adjustment for potential confounders via stratified analysis is also discussed.

Bias↗

PubMed2006

Likelihood based tests for spatial randomness.

Many different methods have been proposed to test the spatial randomness of a point pattern adjusting for an inhomogeneous background population. These tests can be classified into cluster detection tests, concerned with the detection and inference of local clusters, and global clustering tests, which collect evidence for clustering throughout the study region. This paper is mainly concerned about global clustering tests. Some tests for spatial randomness are based on likelihoods, which include the spatial and space-time scan statistics with variable window size and Gangnon and Clayton's weighted average likelihood ratio tests. Both of these tests perform well compared to other tests for cluster detection and global clustering, respectively. In this study, we develop other likelihood based tests for global clustering and we explore the use of different weight functions with these tests. The power of these tests is evaluated using simulated data set and compared with existing methods.

Cluster Analysis↗

PubMed1993

Relative risk, risk difference and rate difference models for sparse stratified data: a pseudo likelihood approach.

We consider a relative risk and a risk difference model for binomial data, and a rate difference model for Poisson (person year) data. It is assumed that the data are stratified in a large number of small strata. If each stratum has its own parameter in the model, then, due to the large number of parameters, straightforward maximum likelihood leads to inconsistent estimates of the relevant parameters. By contrast to the logistic model, conditioning on the number of events per stratum does not help in eliminating the stratum nuisance parameters. We propose a pseudo likelihood method to overcome these consistency problems. The resulting pseudo maximum likelihood estimates can easily be computed with standard statistical software. Our approach gives a more general framework for the Mantel-Haenszel type estimators proposed in the literature. In the special case of a series of 2 x 2 tables, for the risk and rate difference models, our approach yields exactly these ad hoc Mantel-Haenszel estimators, while for the relative risk model it gives a close approximation of the Mantel-Haenszel relative risk estimator. For the regression models corresponding to the association measures relative risk, risk difference and rate difference, our method provides analogues of conditional logistic regression, which were not previously available.

Analysis of Variance↗

PubMedDate not supplied

Penalized likelihood in Cox regression.

In a Cox regression model, instability of the estimated regression coefficients can be reduced by maximizing a penalized partial log-likelihood, where a penalty function of the regression coefficients is substracted from the partial log-likelihood. In this paper, we choose the optimal weight of the penalty function by maximizing the predictive value of the model, as measured by the crossvalidated partial log-likelihood. Our methods are illustrated by a study of ovarian cancer survival and by a study of centre-effects in kidney graft survival.

Clinical Trials as Topic↗

PubMed1996

Instability of quartet analyses of molecular sequence data by the maximum likelihood method: the Cetacea/Artiodactyla relationships.

By using the maximum likelihood method, we study the reliability of phylogeny based on only four taxa, which have been studied by Philippe and Douzery with the maximum parsimony and neighbor-joining methods. Amino acid sequences of cytochrome b were provided for the analysis to examine the relationships among Ruminantia, Suiformes, and Cetacea with out-group mammals by using one representative species from each group. In accord with Philippe and Douzery's analysis, it was shown that we can find a quartet of species which provides a high bootstrap proportion for each of the three possible trees and that the quartet analyses with only four species can be misleading even with the maximum likelihood. However, if the confidence level of the inferred tree is estimated adequately by the maximum likelihood method and we take a conservative attitude, we can avoid the danger of concluding an erroneous tree by using quartet analysis, although the problem may remain unresolved in many cases. In order to obtain a reliable tree, it is important to carry out analyses based on as many different genes and as many relevant species as possible and to synthesize the results.

Animals↗

PubMed1995

Approximate solutions for the maximum-likelihood estimates in models of univariate human twin data.

We present numerical results concerning the accuracy of approximate maximum-likelihood estimators of variance components for several models of univariate human twin data. The approximations are obtained via a spectral decomposition of the twin model covariance matrix. The results apply to likelihood functions for univariate twin data based on either the Wishart distribution or the bivariate normal distribution. For sample sizes of 100 twin pairs for each zygosity group, if the difference of the traces of the sample covariance matrices is 10% or less of the sum of the traces, the approximate solutions can be used as the maximum-likelihood estimators for some models.

Analysis of Variance↗

PubMed2004

Joining genetic linkage maps using a joint likelihood function.

We present an efficient method to join genetic maps derived from different crosses, which is especially appropriate for dominant markers. In contrast to the "JoinMap" algorithm, which estimates information about recombination in a given cross from the LOD values and then combines estimates among crosses assuming a binomial sampling distribution, we construct a joint likelihood function that combines information across all crosses, to obtain a joint estimate of recombination. Simulations indicated that the difference between these two approaches is small when codominant markers are used, but that the joint likelihood approach shows substantially improved estimates when dominant or a mixture of dominant and codominant markers are used. This is because the joint likelihood implicitly finds the optimal weights among different classes of data, while the former method does not accurately predict the information from crosses involving dominant markers. Application of our method is illustrated by the construction of a linkage map for Linanthus using both backcrosses and the F2 of a cross between Linanthus jepsonii and L. bicolor, assayed with amplified fragment length polymorphisms (AFLP).

Chromosome Mapping↗

PubMedDate not supplied

Genetic algorithm-based maximum-likelihood analysis for molecular phylogeny.

A heuristic approach to search for the maximum-likelihood (ML) phylogenetic tree based on a genetic algorithm (GA) has been developed. It outputs the best tree as well as multiple alternative trees that are not significantly worse than the best one on the basis of the likelihood criterion. These near-optimum trees are subjected to further statistical tests. This approach enables ones to infer phylogenetic trees of over 20 taxa taking account of the rate heterogeneity among sites on practical time scales on a PC cluster. Computer simulations were conducted to compare the efficiency of the present approach with that of several likelihood-based methods and distance-based methods, using amino acid sequence data of relatively large (5-24) taxa. The superiority of the ML method over distance-based methods increases as the condition of simulations becomes more realistic (an incorrect model is assumed or many taxa are involved). This approach was applied to the inference of the universal tree based on the concatenated amino acid sequences of vertically descendent genes that are shared among all genomes whose complete sequences have been reported. The inferred tree strongly supports that Archaea is paraphyletic and Eukarya is specifically related to Crenarchaeota. Apart from the paraphyly of Archaea and some minor disagreements, the universal tree based on these genes is largely consistent with the universal tree based on SSU rRNA.

Algorithms↗

PubMed1993

A Monte Carlo investigation of the robustness of the Wald, likelihood Ratio, and Median tests with specified symmetric and asymmetric marginal distributions.

The effect of nonnormality on the Type I (tau) error when comparing two independent binomial proportions (P) or the nonparametric alternatives, the Median (Me), Wald (W), and Likelihood Ratio (LR), has not been investigated. If these selected tests are overly conservative the implied loss of power would moderate their practical use. The purpose of the present study was to investigate the impact of nonnormality on small to moderate sample sizes on the estimated tau for alpha = 0.10, 0.05, and 0.01 for the P, Me, W, and LR tests. Samples were generated from nine long-tailed symmetric and asymmetric distributions using a multiplicative congruential generator. For each marginal distribution and for a variety of sample sizes, the proportion of samples for which the test statistic exceeded the 10, 5, and 1 percentage points was tabulated. For data that mimic a symmetric distribution, the median test uniformly yields an empirical alpha considerably less than tau, while the likelihood ratio test consistently overestimates tau for small samples (n < or = 15) over all symmetric distributions and empirical alpha levels. For asymmetric distributions, the median test again yields an empirical alpha significantly less than tau. Similar underestimates of tau were found for the chi-square (2 df), chi-square (4 df) log normal, and gamma (2, 1) distributions. The likelihood ratio test consistently overestimates tau for small samples (n < or = 15) over all asymmetric distributions and empirical alpha levels. The independent proportions test produces an empirical alpha closest to tau for n = 10 for all asymmetric distributions.(ABSTRACT TRUNCATED AT 250 WORDS)

Data Interpretation, Statistical↗

PubMed2006

Classifying patients suspected of appendicitis with regard to likelihood.

BACKGROUND: We sought to develop a clinical predictive model for acute appendicitis and contrast it with current clinical practice. METHODS: A prospective observational study of patients presenting with signs or symptoms consistent with acute appendicitis. Random-partition modeling was used to develop an appendicitis likelihood model (ALM). RESULTS: Four hundred thirty-nine patients were enrolled, 101 with appendicitis, and 338 with other diagnoses. The ALM classified patients as "low likelihood" if they had a white blood cell count <9,500 and either no right lower-quadrant tenderness or a neutrophil count <54%. Patients were classified as "high likelihood" if they had a white blood cell count >13,000 with rebound tenderness or both voluntary guarding and neutrophil count >82%. The ALM outperformed actual clinical practice with regard to "missed" appendicitis, negative laparotomies, and total number of imaging studies. CONCLUSION: The ALM may permit more judicious use of advanced radiographic imaging with lower nontherapeutic laparotomy rates.

Adolescent↗

PubMed2006

Pediatric adenoidectomy: what is the effect of obstructive symptoms on the likelihood of future surgery?

OBJECTIVE: Determine if pediatric patients undergoing adenoidectomy for obstruction have an increased likelihood of undergoing eventual tonsillectomy and/or second adenoidectomy over extended follow-up. METHODS: Retrospective cohort study within the ambulatory surgery component of a tertiary children's hospital. Exclusion criteria consisted of the presence of significant co-morbid conditions (obesity, craniofacial syndrome, cerebral palsy, etc.) and less than 1 year of post-adenoidectomy follow-up. The study population included 100 children (mean age=3.9 years, range 0.9-15 years) randomly selected from a financial billing database undergoing adenoidectomy either with pre-operative obstructive symptoms (OB n=52) or without pre-operative obstructive symptoms (NOB n=48) with a minimum of 1 year of retrospective follow-up. Electronic records were searched for subsequent surgery (tonsillectomy and/or revision adenoidectomy). RESULTS: Overall, 29 of 100 children (29%) underwent subsequent surgery (tonsillectomy and/or revision adenoidectomy) over an average of 3.46 years (range 1.0-6.59 years) retrospective follow-up. Children in the OB group were three times more likely (age, sex adjusted odds ratio=3.03, 95% confidence interval =1.18-7.78 p=0.021) than children in the NOB group to require tonsillectomy or second adenoidectomy. Age less than 2 years at time of initial adenoidectomy also suggested an increased likelihood of subsequent surgery. CONCLUSION: Children undergoing adenoidectomy with obstructive symptoms are more likely to require eventual tonsillectomy and/or second adenoidectomy than those undergoing adenoidectomy without obstruction. However, the low absolute likelihood of future surgery precludes a generalized recommendation for tonsillectomy in these children.

Adenoidectomy↗

PubMed2004

Likelihood ratio and a Bayesian approach were superior to standard noninferiority analysis when the noninferiority margin varied with the control event rate.

OBJECTIVE: To present and compare three statistical approaches for analyzing a noninferiority trial when the noninferiority margin depends on the control event rate. STUDY DESIGN AND SETTING: In noninferiority trials with a binary outcome, the noninferiority margin is often defined as a fixed delta, the largest clinically acceptable difference in event rates between treatment groups. An alternative and more flexible approach is to allow delta to vary according to the true event rate in the control group. The appropriate statistical method for evaluating noninferiority with a variable noninferiority margin is not apparent. Three statistical approaches are proposed and compared: an observed event rate (OER) approach based on equating the true control rate to the observed rate, a Bayesian approach, and a likelihood ratio test. RESULTS AND CONCLUSIONS: Simulations studies indicate that the proportion of trials in which noninferiority was erroneously demonstrated was higher for the OER approach than with the Bayesian and likelihood ratio approaches. In some cases, the Type I error rate exceeded 10% for the OER approach. The OER approach is not recommended for the analysis of noninferiority trials with a variable margin of equivalence. The Bayesian and likelihood ratio methods yielded better operating characteristics.

Bayes Theorem↗

PubMed2005

New patient-oriented diagnostic test characteristics analogous to the likelihood ratios conveyed information on trustworthiness.

OBJECTIVE: Suggesting patient-oriented ratio measures of diagnostic tests analogous to the likelihood ratios. STUDY DESIGN AND SETTING: Algebraic equation based on Bayes' theorem. RESULTS: Two new measures are suggested: the positive predictive ratio (PPR), relating the positive predictive value (PPV) to the false negative rate in the patient population, and the negative predictive ratio (NPR), relating the negative predictive value (NPV) to the false positive rate in the patient population. The new measures are analogous to likelihood ratios. These measures are calculable from the familiar 2x2 table in a selected study population or in a target (general) population. Examples are given, using published data. CONCLUSION: PPR adds to the information given by the PPV, and NPR adds to the information given by the NPV. PPR and NPR may be more appropriate in the clinical setting than likelihood ratios.

Bayes Theorem↗

PubMed2005

Maximum-likelihood multi-reference refinement for electron microscopy images.

A maximum-likelihood approach to multi-reference image refinement is presented. In contrast to conventional cross-correlation refinement, the new approach includes a formal description of the noise, implying that it is especially suited to cases with low signal-to-noise ratios. Application of this approach to a cryo-electron microscopy dataset revealed two major classes for projections of simian virus 40 large T-antigen in complex with an asymmetric DNA-probe, containing the origin of simian virus 40 replication. Strongly bent projections of dodecamers showed density that may be attributed to the complexed double-stranded DNA, while almost straight projections revealed a twist in the relative orientation of the hexameric subunits. This new level of detail for large T-antigen projections was not detected using conventional techniques. For a negative stain dataset, maximum-likelihood refinement yielded results that were practically identical to those obtained using conventional multi-reference refinement. Results obtained using simulated data suggest that the efficiency of the maximum-likelihood approach may be further enhanced by explicitly incorporating the microscope contrast transfer function in the image formation model.

Algorithms↗