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PubMed1986

Sire evaluation for multiple binary responses when information is missing on some traits.

A procedure of sire evaluation for multiple binary responses when information is missing for some traits is described. The genetic model assumes a conceptual underlying multivariate normal distribution rendered discrete by abrupt thresholds. Statistical inferences are made from a posterior distribution consisting of several conditionally independent likelihood functions and a multivariate normal prior distribution. Each likelihood function corresponds to a particular class of information available. Point estimators and predictors of fixed and random effects are the values that maximize the posterior distribution conditionally on heritabilities and genetic correlations. The procedure involves nonlinear maximization. An example involving joint selection for calving ease and skeletal development illustrates the principles. Application of the methodology as a potential means of removing bias due to selection for categorical traits is discussed.

Animals↗
PubMed1996

A note on the combination of estimates of a recombination fraction.

A number of ways of combining two or more independent estimates of the same recombination fraction can be found in the literature. We revisit this topic in the context of human gene mapping, and explore the value of transforming the recombination fraction to a new parameter whose log-likelihood function is more nearly quadratic. It is shown that the arcsine of the cube-root is one such function. These observations lead naturally to a way of summarizing and combining the summarized set of log-likelihood functions of a common recombination fraction. This idea is illustrated using pedigree data concerning six loci on chromosome 10 from the CEPH consortium. A comparison is also made with the method of summarizing and combining using 'equivalent numbers' of recombinants and informative meioses.

Chromosome Mapping↗
PubMed2004

Direct incorporation of experimental phase information in model refinement.

The incorporation of prior phase information into a maximum-likelihood formalism has been shown to strengthen model refinement. However, the currently available likelihood refinement target using prior phase information has shortcomings; the 'phased' refinement target considers experimental phase information indirectly and statically in the form of Hendrickson-Lattman coefficients. Furthermore, the current refinement target implicitly assumes that the prior phase information is independent of the calculated model structure factor. This paper describes the derivation of a multivariate likelihood function that overcomes these shortcomings and directly incorporates experimental phase information from a single-wavelength anomalous diffraction (SAD) experiment. This function, which simultaneously refines heavy-atom and model parameters, has been implemented in the refinement program REFMAC5. The SAD function used in conjunction with the automated model-building procedures of ARP/wARP leads to a successful solution when current likelihood functions fail in a test case shown.

Algorithms↗
PubMed2005

Maximum-likelihood techniques for joint segmentation-classification of multispectral chromosome images.

Traditional chromosome imaging has been limited to grayscale images, but recently a 5-fluorophore combinatorial labeling technique (M-FISH) was developed wherein each class of chromosomes binds with a different combination of fluorophores. This results in a multispectral image, where each class of chromosomes has distinct spectral components. In this paper, we develop new methods for automatic chromosome identification by exploiting the multispectral information in M-FISH chromosome images and by jointly performing chromosome segmentation and classification. We (1) develop a maximum-likelihood hypothesis test that uses multispectral information, together with conventional criteria, to select the best segmentation possibility; (2) use this likelihood function to combine chromosome segmentation and classification into a robust chromosome identification system; and (3) show that the proposed likelihood function can also be used as a reliable indicator of errors in segmentation, errors in classification, and chromosome anomalies, which can be indicators of radiation damage, cancer, and a wide variety of inherited diseases. We show that the proposed multispectral joint segmentation-classification method outperforms past grayscale segmentation methods when decomposing touching chromosomes. We also show that it outperforms past M-FISH classification techniques that do not use segmentation information.

Algorithms↗
PubMed1994

Measuring the safety effect of road measures at junctions.

The effects of road measures implemented at main road junctions in 1984-1986 were studied on the basis of police-reported accidents that had occurred at the junctions in 1983-1987. The data were obtained from an extensive junction inventory performed by the District Offices of the Finnish National Road Administration in 1988. A total of 325 three-arm and 298 four-arm junctions were included in the study. The number and type of accidents were studied at each junction both before and after the implementation of the measure. The effects of the measure were determined by comparing the observed number of accidents after the measure with the number that would have been expected to occur at the junction if the measure had not been implemented. The expected number of accidents was calculated on the basis of the observed number of accidents in the before period and the expected number of accidents as predicted by an accident model, utilizing at the same time the information on the variation of the number of accidents at similar junctions revealed by the model. Likelihood functions determined for the effects of the measures were used for studying the accuracy of the estimates of the effects. The likelihood functions can also be utilized later when new before-and-after studies are performed. Road lighting, stop signs, signal control, and lowering of the speed limit value were found to decrease the number of accidents. Through-flow junction widenings, additional lanes for turning vehicles, and road widenings, however, did not seem to affect the safety at junctions to any marked extent. If we had not accounted for the regression-to-the-mean effect, these measures would have seemed to have a positive effect on safety. The study method also enabled us to quantify the regression-to-the-mean effect. The magnitude of the regression effect was on average 20%, i.e. the number of accidents would have decreased by 20% at the junctions studied even if the measures had not been implemented. The magnitude of the regression-to-the-mean varied greatly between the different measures.

Accidents, Traffic↗
PubMed1999

A random walk model of oligodendrocyte generation in vitro and associated estimation problems.

A branching stochastic process proposed earlier to model oligodendrocyte generation by O-2A progenitor cells under in vitro conditions does not allow invoking the maximum likelihood techniques for estimation purposes. To overcome this difficulty, we propose a partial likelihood function based on an embedded random walk model of clonal growth and differentiation of O-2A progenitor cells. Under certain conditions, the partial likelihood function yields consistent estimates of model parameters. The usefulness of this approach is illustrated with computer simulations and data analyses.

Algorithms↗
PubMed2005

Maximum likelihood Jukes-Cantor triplets: analytic solutions.

Maximum likelihood (ML) is a popular method for inferring a phylogenetic tree of the evolutionary relationship of a set of taxa, from observed homologous aligned genetic sequences of the taxa. Generally, the computation of the ML tree is based on numerical methods, which in a few cases, are known to converge to a local maximum on a tree, which is suboptimal. The extent of this problem is unknown, one approach is to attempt to derive algebraic equations for the likelihood equation and find the maximum points analytically. This approach has so far only been successful in the very simplest cases, of three or four taxa under the Neyman model of evolution of two-state characters. In this paper we extend this approach, for the first time, to four-state characters, the Jukes-Cantor model under a molecular clock, on a tree T on three taxa, a rooted triple. We employ spectral methods (Hadamard conjugation) to express the likelihood function parameterized by the path-length spectrum. Taking partial derivatives, we derive a set of polynomial equations whose simultaneous solution contains all critical points of the likelihood function. Using tools of algebraic geometry (the resultant of two polynomials) in the computer algebra packages (Maple), we are able to find all turning points analytically. We then employ this method on real sequence data and obtain realistic results on the primate-rodents divergence time.

Biological Evolution↗
PubMed2001

Using full probability models to compute probabilities of actual interest to decision makers.

The objective of this paper is to illustrate the advantages of the Bayesian approach in quantifying, presenting, and reporting scientific evidence and in assisting decision making. Three basic components in the Bayesian framework are the prior distribution, likelihood function, and posterior distribution. The prior distribution describes analysts' belief a priori, the likelihood function captures how data modify the prior knowledge; and the posterior distribution synthesizes both prior and likelihood information. The Bayesian approach treats the parameters of interest as random variables, uses the entire posterior distribution to quantify the evidence, and reports evidence in a "probabilistic" manner. Two clinical examples are used to demonstrate the value of the Bayesian approach to decision makers. Using either an uninformative or a skeptical prior distribution, these examples show that the Bayesian methods allow calculations of probabilities that are usually of more interest to decision makers, e.g., the probability that treatment A is similar to treatment B, the probability that treatment A is at least 5% better than treatment B, and the probability that treatment A is not within the "similarity region" of treatment B, etc. In addition, the Bayesian approach can deal with multiple endpoints more easily than the classic approach. For example, if decision makers wish to examine mortality and cost jointly, the Bayesian method can report the probability that a treatment achieves at least 2% mortality reduction and less than $20,000 increase in costs. In conclusion, probabilities computed from the Bayesian approach provide more relevant information to decision makers and are easier to interpret.

Arterial Occlusive Diseases↗
PubMed2005

Conditional likelihood score functions for mixed models in linkage analysis.

In this paper, we develop a general strategy for linkage analysis, applicable for arbitrary pedigree structures and genetic models with one major gene, polygenes and shared environmental effects. Extending work of Whittemore (1996), McPeek (1999) and Hossjer (2003d), the efficient score statistic is computed from a conditional likelihood of marker data given phenotypes. The resulting semiparametric linkage analysis is very similar to nonparametric linkage based on affected individuals. The efficient score S depends not only on identical-by-descent sharing and phenotypes, but also on a few parameters chosen by the user. We focus on (1) weak penetrance models, where the major gene has a small effect and (2) rare disease models, where the major gene has a possibly strong effect but the disease causing allele is rare. We illustrate our results for a large class of genetic models with a multivariate Gaussian liability. This class incorporates one major gene, polygenes and shared environmental effects in the liability, and allows e.g. binary, Gaussian, Poisson distributed and life-length phenotypes. A detailed simulation study is conducted for Gaussian phenotypes. The performance of the two optimal score functions S(wpairs) and S(normdom) are investigated. The conclusion is that (i) inclusion of polygenic effects into the score function increases overall performance for a wide range of genetic models and (ii) score functions based on the rare disease assumption are slightly more powerful.

Alleles↗
PubMed1979

Distribution-free regression analysis of grouped survival data.

Methods based on regression models for logarithmic hazard functions, Cox models, are given for analysis of grouped and censored survival data. By making an approximation it is possible to obtain explicitly a maximum likelihood function involving only the regression parameters. This likelihood function is a convenient analog to Cox's partial likelihood for ungrouped data. The method is applied to data from a toxicological experiment.

Animals↗
PubMed1988

Multifactorial analysis of family data ascertained through truncation: a comparative evaluation of two methods of statistical inference.

When family data are ascertained through single selection based on truncation, a prevailing method of analysis is to condition the likelihood function on the proband's actual phenotypic value. An alternative method conditions the likelihood function on the event that the proband's measurement lies in the truncation region. Both methods are contrasted here by using Monte Carlo simulations; identical sets of data were analyzed using both methods. The results suggest that, under either method, (1) parameter estimates are nearly unbiased and (2) likelihood-ratio tests of null hypotheses are approximately distributed as chi 2. However, conditioning on the proband's actual phenotypic value yields considerably less efficient estimates and reduced power for hypothesis tests. A corresponding result also holds under complete ascertainment. It is argued, therefore, that whenever sufficient information is available on the nature of truncation, the alternative approach should be used.

Computer Simulation↗
PubMed1998

Efficient, robust, and unified method for mapping complex traits (I): two-point linkage analysis.

The completion of a preliminary human genome map and development of molecular methods have enabled researchers to assay a large number of polymorphic markers that are evenly spaced along the entire human genome. Among many applications, marker data are valuable for mapping complex traits through linkage or linkage-disequilibrium analysis, the former of which is the focus of this paper, the first in a series on this subject. Formalizing the concept and computation for linkage analysis, Elston and Stewart [1971; Human Heredity 21:523-542] introduced a likelihood function to capture relevant genetic information and a recursive algorithm for computing the likelihood function. However, the computing burden is prohibitive in processing complex pedigrees. Since that fundamental development, improving the computational algorithm and extending the method has been a dynamic area of research. The primary objective of this communication is to introduce a semiparametric method for linkage analysis. It is a particularly suitable approach with desirable properties for mapping complex traits that may be binary, continuous, and partially observed (i.e., censored). It incorporates candidate genes, environmental factors, and their interactions with the putative gene and is expected to be robust and efficient in comparison with likelihood-based methods. The properties of the estimates have been studied in finite samples with a limited simulation study. This method is illustrated with an application to family data contributed to the Breast Cancer Consortium.

Algorithms↗
PubMed1996

Genome scanning for linkage: an overview.

Several different methods for linkage analysis are shown to arise from a single likelihood function L for the observed allele-sharing data at multiple markers in a chromosomal region. These include classical parametric lod score methods, nonparametric or "model-free" affected pedigree-member (APM) methods, and the Gaussian process method. Setting the methods in the context of the likelihood function L clarifies their underlying assumptions. A test statistic derived from L, the efficient score statistic, is introduced. It is asymptotically equivalent to the lod score, but it can be easier to compute when the penetrances and frequencies of alleles of the trait gene are not known. APM test statistics and the Gaussian lod score are shown to be special cases of efficient score statistics. This unified framework facilitates exploration of a range of models for the effects of a putative trait-predisposing gene, and it facilitates sensitivity analyses to examine the consequences of model misspecification.

Female↗
PubMed1997

Bayesian registration of models using finite element eigenmodes.

This paper is concerned with registering three-dimensional wire-frame organ models. This involves finding correspondences between points on the models of two different examples of the same organ. Such registration is widely used in the processing of medical data; for example in segmentation, or to superimpose functional information on a more detailed structural map. The algorithm described in this paper is based on matching the modes of deformation of organ shapes. Modes with lower spatial frequency characterise large scale organ features whereas small scale variations determine the high frequency modes. First, the organ sizes are normalised using a generalised version of the centroid size metric. The axes of the fundamental frequency modes are then aligned to provide initial rigid-body registration. The registration is refined by matching increasingly high frequency modes using the 'Highest confidence first' algorithm. The matches are evaluated using a Bayesian combination of local prior and likelihood functions. The prior is derived from the Gompertz metric of biological growth and ensures that physically impossible matches are not accepted. The likelihood function is a measure of the similarity between local modal deformation components. The registration algorithm has been applied by the authors in the analysis of three dimensional ultrasound data. Results are presented showing the registration of two liver models derived from 3D ultrasound.

Algorithms↗
PubMed1997

Fully parametric and semi-parametric regression models for common events with covariate measurement error in main study/validation study designs.

The derivation of the likelihood function for binary data from two types of main study/validation study designs where model covariates are measured with error is elaborated. Rather than limiting consideration to a restricted family of models with convenient mathematical properties, we suggest that empirical considerations, customized to the data at hand, should drive model choices. The joint likelihood function for the main study, in which the covariates are measured with error, and the validation study, in which they are not, is maximized, and estimation and inference proceeds using standard theory. Although the choice of the measurement error model is driven by empirical considerations, the relatively small validation study sizes typically seen may lead to misspecification, resulting in bias in estimation and inference about exposure-disease relationships. By using a nonparametric form for the measurement error model, the resulting semi-parametric methods suggested by Robins, Rotnitzky, and Zhao (1994, Journal of the American Statistical Association 89, 864-866) and Robins, Hsieh, and Newey (1995, Journal of the Royal Statistical Society, Series B 57, 409-424) are free from bias due to misspecification of the measurement error model, trading efficiency for robustness as usual. These fully and semi-parametric methods are illustrated with a detailed example from a main study/validation study of the health effects of occupational exposure to chemotherapeutics among pharmacists (Valanis et al., 1993, American Journal of Hospital Pharmacy 50, 455-462). A constant, prevalence ratio model for common binary events, with gamma covariate measurement error, is derived and empirically verified by the available data. A careful reanalysis of the data, taking measurement error fully into account, leads to a threefold increase in the log relative risk and no loss of statistical power. The semi-parametric estimates are consistent with the parametric results, providing reassurance that important bias due to misspecification of the measurement error model is unlikely.

Analysis of Variance↗
PubMedDate not supplied

A review of some extensions to generalized linear models.

Although generalized linear models are reasonably well known, they are not as widely used in medical statistics as might be appropriate, with the exception of logistic, log-linear, and some survival models. At the same time, the generalized linear modelling methodology is decidedly outdated in that more powerful methods, involving wider classes of distributions, non-linear regression, censoring and dependence among responses, are required. Limitations of the generalized linear modelling approach include the need for the iterated weighted least squares (IWLS) procedure for estimation and deviances for inferences; these restrict the class of models that can be used and do not allow direct comparisons among models from different distributions. Powerful non-linear optimization routines are now available and comparisons can more fruitfully be made using the complete likelihood function. The link function is an artefact, necessary for IWLS to function with linear models, but that disappears once the class is extended to truly non-linear models. Restricting comparisons of responses under different treatments to differences in means can be extremely misleading if the shape of the distribution is changing. This may involve changes in dispersion, or of other shape-related parameters such as the skewness in a stable distribution, with the treatments or covariates. Any exact likelihood function, defined as the probability of the observed data, takes into account the fact that all observable data are interval censored, thus directly encompassing the various types of censoring possible with duration-type data. In most situations this can now be as easily used as the traditional approximate likelihood based on densities. Finally, methods are required for incorporating dependencies among responses in models including conditioning on previous history and on random effects. One important procedure for constructing such likelihoods is based on Kalman filtering.

Animals↗
PubMed2004

SAS macro program for non-homogeneous Markov process in modeling multi-state disease progression.

Writing a computer program for modeling multi-state disease process for cancer or chronic disease is often an arduous and time-consuming task. We have developed a SAS macro program for estimating the transition parameters in such models using SAS IML. The program is very flexible and enables the user to specify homogeneous and non-homogeneous (i.e. Weibull distribution, log-logistic, etc.) Markov models, incorporate covariates using the proportional hazards form, derive transition probabilities, formulate the likelihood function, and calculate the maximum likelihood estimate (MLE) and 95% confidence interval within a SAS subroutine. The program was successfully applied to an example of a three-state disease model for the progression of colorectal cancer from normal (disease free), to adenoma (pre-invasive disease), and finally to invasive carcinoma, with or without adjusting for covariates. This macro program can be generalized to other k-state models with s covariates.

Chronic Disease↗
PubMed2005

A comparison of different bivariate correlated frailty models and estimation strategies.

Frailty models are becoming increasingly popular in multivariate survival analysis. Shared frailty models in particular are often used despite their limitations. To overcome their disadvantages numerous correlated frailty models were established during the last decade. In the present study, we examine bivariate correlated frailty models, and especially the behavior of the parameter estimates when using different estimation strategies. We consider three different bivariate frailty models: the gamma model and two versions of the log-normal model. The traditional maximum likelihood procedure of parameter estimation in the gamma case with an explicit available likelihood function is compared with maximum likelihood methods based on numerical integration and a Bayesian approach using MCMC methods with the help of a comprehensive simulation study. We detected a strong dependence between the two parameter estimates (variance and correlation of frailties) in the bivariate correlated frailty model and analyzed this dependence in detail.

Bayes Theorem↗