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PubMed1997

Maximum likelihood estimation of aggregated Markov processes.

We present a maximum likelihood method for the modelling of aggregated Markov processes. The method utilizes the joint probability density of the observed dwell time sequence as likelihood. A forward-backward recursive procedure is developed for efficient computation of the likelihood function and its derivatives with respect to the model parameters. Based on the calculated forward and backward vectors, analytical formulae for the derivatives of the likelihood function are derived. The method exploits the variable metric optimizer for search of the likelihood space. It converges rapidly and is numerically stable. Numerical examples are given to show the effectiveness of the method.

Markov Chains↗
PubMed2001

Phase specific analysis of herpes zoster associated pain data: a new statistical approach.

Herpes zoster or shingles is a frequent occurrence in both elderly individuals and immunocompromised hosts. The pain associated with herpes zoster is the most debilitating complication of the disease. It can be described as acute pain and post-herpetic neuralgia or zoster associated pain (ZAP). The latter definition encompasses pain from the onset of disease through its resolution and provides a convenient analytic tool for evaluation of antiviral therapy. A heuristic examination of ZAP historical data suggests the existence of three phases of pain resolution: the acute, subacute and chronic phases. The subacute and chronic phases comprise the post-herpetic neuralgia (PHN) stage. Common analytic methods, such as a Kaplan-Meier survival function or a Cox's model, have been used to assess the pain. However, such approaches do not adequately allow for phase comparison. Notably, in the clinical trial setting the comparison of specific treatment effects on the latter stages of pain are of the greatest medical relevance since this is the most debilitating phase of the illness. In order to incorporate the phase-specific information in the modelling of time to cessation of ZAP, we assumed the hazard function was a stepwise constant. Utilizing the full likelihood function, we obtained the maximum likelihood estimate for the transition times (that is, change-points), and other parameters of medical importance. The standard error of the change-point estimates were obtained through a bootstrapping method. The asymptotic properties of the parameter estimates are also discussed. Hence, the rates of pain resolution across all phases can be examined in order to precisely define the existence of multiple phases. In addition, the covariates effect can be examined across phases and populations, thereby allowing us to translate potential efficacy of a standard therapy to different populations. These results can be utilized in the design of clinical trials or in targeting the outcome for a specific phase while controlling for the effect of other variables.

Acute Disease↗
PubMed2006

A Bayesian model of the disambiguation of gravitoinertial force by visual cues.

The otoliths are stimulated in the same fashion by gravitational and inertial forces, so otolith signals are ambiguous indicators of self-orientation. The ambiguity can be resolved with added visual information indicating orientation and acceleration with respect to the earth. Here we present a Bayesian model of the statistically optimal combination of noisy vestibular and visual signals. Likelihoods associated with sensory measurements are represented in an orientation/acceleration space. The likelihood function associated with the otolith signal illustrates the ambiguity; there is no unique solution for self-orientation or acceleration. Likelihood functions associated with other sensory signals can resolve this ambiguity. In addition, we propose two priors, each acting on a dimension in the orientation/acceleration space: the idiotropic prior and the no-acceleration prior. We conducted experiments using a motion platform and attached visual display to examine the influence of visual signals on the interpretation of the otolith signal. Subjects made pitch and acceleration judgments as the vestibular and visual signals were manipulated independently. Predictions of the model were confirmed: (1) visual signals affected the interpretation of the otolith signal, (2) less variable signals had more influence on perceived orientation and acceleration than more variable ones, and (3) combined estimates were more precise than single-cue estimates. We also show that the model can explain some well-known phenomena including the perception of upright in zero gravity, the Aubert effect, and the somatogravic illusion.

Acceleration↗
PubMed2004

Degeneracy, frequency response and filtering in IMRT optimization.

This paper attempts to provide an answer to some questions that remain either poorly understood, or not well documented in the literature, on basic issues related to intensity modulated radiation therapy (IMRT). The questions examined are: the relationship between degeneracy and frequency response of optimizations, effects of initial beamlet fluence assignment and stopping point, what does filtering of an optimized beamlet map actually do and how could image analysis help to obtain better optimizations? Two target functions are studied, a quadratic cost function and the log likelihood function of the dynamically penalized likelihood (DPL) algorithm. The algorithms used are the conjugate gradient, the stochastic adaptive simulated annealing and the DPL. One simple phantom is used to show the development of the analysis tools used and two clinical cases of medium and large dose matrix size (a meningioma and a prostate) are studied in detail. The conclusions reached are that the high number of iterations that is needed to avoid degeneracy is not warranted in clinical practice, as the quality of the optimizations, as judged by the DVHs and dose distributions obtained, does not improve significantly after a certain point. It is also shown that the optimum initial beamlet fluence assignment for analytical iterative algorithms is a uniform distribution, but such an assignment does not help a stochastic method of optimization. Stopping points for the studied algorithms are discussed and the deterioration of DVH characteristics with filtering is shown to be partially recoverable by the use of space-variant filtering techniques.

Algorithms↗
PubMed2003

A latent variable model approach for assembling and scoring screening tests for dementia.

In dementia studies, the diagnosis of dementia often relies on results of screening tests aimed at measuring various dimensions of cognitive functions. The current practice of scoring a screening test involves simply summing the correct responses from each item. However, this method may be imprecise and inefficient in the predictive power of the score for dementia. We propose a latent variable model approach for the scoring and item selection of such tests. We model the item responses to be random variables based on latent variables. We also model the disease outcomes to be a function of the latent variables. Maximum likelihood estimates are obtained by maximizing the joint likelihood functions of disease and the item responses over a specified distribution function for the latent variables. Variances of model parameters are estimated using a nonparametric bootstrap method. We illustrate the approach using a screening test for dementia from a community-based study.

Aged↗
PubMed2025

Accounting for contact tracing in epidemiological birth-death models.

Phylodynamics bridges the gap between classical epidemiology and pathogen genome sequence data by estimating epidemiological parameters from time-scaled pathogen phylogenetic trees. The models used in phylodynamics typically assume that the sampling procedure is independent between infected individuals. However, this assumption does not hold for many epidemics, in particular for such sexually transmitted infections as HIV-1, for which contact tracing schemes are included in health policies of many countries. We extended phylodynamic multi-type birth-death (MTBD) models with contact tracing (CT), and developed a simulator to generate trees under MTBD and MTBD-CT models. We proposed a non-parametric test for detecting contact tracing in pathogen phylogenetic trees. Its application to simulated data showed that it is both highly specific and sensitive. For the simplest representative of the MTBD-CT family, the BD-CT(1) model, where only the last contact can be notified, we solved the differential equations and proposed a closed form solution for the likelihood function. We implemented a maximum-likelihood program, which estimates the BD-CT(1) model parameters and their confidence intervals from phylogenetic trees. It performed accurate parameter inference on BD and BD-CT(1) simulated data, and detected contact tracing in HIV-1 B epidemics in Zurich and the UK. Importantly, we showed that not accounting for contact tracing when it is present, leads to bias in parameter estimation with the BD model (overestimation of the becoming-non-infectious rate). This bias is also present, but greatly reduced, when the BD-CT(1) model is used on data where multiple contacts can be notified. Our CT test, MTBD-CT tree simulator and BD-CT(1) parameter estimator are freely available at GitHub (evolbioinfo/treesimulator and evolbioinfo/bdct).

Contact Tracing↗
PubMed2004

A shared random effect parameter approach for longitudinal dementia data with non-ignorable missing data.

A significant source of missing data in longitudinal epidemiologic studies on elderly individuals is death. It is generally believed that these missing data by death are non-ignorable to likelihood based inference. Inference based on data only from surviving participants in the study may lead to biased results. In this paper we model both the probability of disease and the probability of death using shared random effect parameters. We also propose to use the Laplace approximation for obtaining an approximate likelihood function so that high dimensional integration over the distributions of the random effect parameters is not necessary. Parameter estimates can be obtained by maximizing the approximate log-likelihood function. Data from a longitudinal dementia study will be used to illustrate the approach. A small simulation is conducted to compare parameter estimates from the proposed method to the 'naive' method where missing data is considered at random.

Aged↗
PubMed2004

Contour definition and tracking in cardiac imaging through the integration of knowledge and image evidence.

The main contribution of this paper is the use of simple processing techniques, incorporated in a new multistage approach, to automatically delineate left ventricle contours. Another contribution is the proposal of the centerline distances for contour comparison, which promises a more accurate measurement than the common method, based on the distance to the closest point. Edges are detected by Gaussian filtering at coarse and fine scale. The region of interest is defined as a binary map where coarse edges are extracted throughout image sequence. A contour template is matched against the gradient of the first image. Candidate boundary points are instantiated by scanning the coarse edge map perpendicularly to the matched template. A candidate contour is estimated from these points by maximizing an edge likelihood function. A region growing algorithm gives another candidate contour. Both edge and region candidate contours are then integrated with the edge map computed at fine scale by maximizing another likelihood function. Evaluation was carried out on 12 echocardiographic and 4 angiocardiographic sequences (for a total of 289 frames). Distances between computer-generated contours and the contours traced by three experts were within interobserver variability, unlike the results obtained by Acoustic Quantification and by a general-purpose deformable model.

Artificial Intelligence↗
PubMed2003

Modeling gene expression measurement error: a quasi-likelihood approach.

BACKGROUND: Using suitable error models for gene expression measurements is essential in the statistical analysis of microarray data. However, the true probabilistic model underlying gene expression intensity readings is generally not known. Instead, in currently used approaches some simple parametric model is assumed (usually a transformed normal distribution) or the empirical distribution is estimated. However, both these strategies may not be optimal for gene expression data, as the non-parametric approach ignores known structural information whereas the fully parametric models run the risk of misspecification. A further related problem is the choice of a suitable scale for the model (e.g. observed vs. log-scale). RESULTS: Here a simple semi-parametric model for gene expression measurement error is presented. In this approach inference is based an approximate likelihood function (the extended quasi-likelihood). Only partial knowledge about the unknown true distribution is required to construct this function. In case of gene expression this information is available in the form of the postulated (e.g. quadratic) variance structure of the data. As the quasi-likelihood behaves (almost) like a proper likelihood, it allows for the estimation of calibration and variance parameters, and it is also straightforward to obtain corresponding approximate confidence intervals. Unlike most other frameworks, it also allows analysis on any preferred scale, i.e. both on the original linear scale as well as on a transformed scale. It can also be employed in regression approaches to model systematic (e.g. array or dye) effects. CONCLUSIONS: The quasi-likelihood framework provides a simple and versatile approach to analyze gene expression data that does not make any strong distributional assumptions about the underlying error model. For several simulated as well as real data sets it provides a better fit to the data than competing models. In an example it also improved the power of tests to identify differential expression.

Calibration↗
PubMed2005

Simulation Procedure in Periodic Cancer Screening Trials.

A general simulation procedure is described to validate model fitting algorithms for complex likelihood functions that are utilized in periodic cancer screening trials. Although screening programs have existed for a few decades, there are still many unsolved problems, such as how age or hormone affects the screening sensitivity, the sojourn time in the preclinical state, and the transition probability from disease-free state to the preclinical state. Simulations are needed to check reliability or validity of the likelihood function combined with the associated effect functions. One bottleneck in the simulation procedure is the very time consuming calculations of the maximum likelihood estimates (MLE) from generated data. A practical procedure is presented, along with results for when both sensitivity and transition probability into the preclinical state are age-dependent. The procedure is also suitable for other applications.

Journal Article↗
PubMed1996

Implementing the Bayesian paradigm: reporting research results over the World-Wide Web.

For decades, statisticians, philosophers, medical investigators and others interested in data analysis have argued that the Bayesian paradigm is the proper approach for reporting the results of scientific analyses for use by clients and readers. To date, the methods have been too complicated for non-statisticians to use. In this paper we argue that the World-Wide Web provides the perfect environment to put the Bayesian paradigm into practice: the likelihood function of the data is parsimoniously represented on the server side, the reader uses the client to represent her prior belief, and a downloaded program (a Java applet) performs the combination. In our approach, a different applet can be used for each likelihood function, prior belief can be assessed graphically, and calculation results can be reported in a variety of ways. We present a prototype implementation, BayesApplet, for two-arm clinical trials with normally-distributed outcomes, a prominent model for clinical trials. The primary implication of this work is that publishing medical research results on the Web can take a form beyond or different from that currently used on paper, and can have a profound impact on the publication and use of research results.

Bayes Theorem↗
PubMed2004

Sibship reconstruction from genetic data with typing errors.

Likelihood methods have been developed to partition individuals in a sample into full-sib and half-sib families using genetic marker data without parental information. They invariably make the critical assumption that marker data are free of genotyping errors and mutations and are thus completely reliable in inferring sibships. Unfortunately, however, this assumption is rarely tenable for virtually all kinds of genetic markers in practical use and, if violated, can severely bias sibship estimates as shown by simulations in this article. I propose a new likelihood method with simple and robust models of typing error incorporated into it. Simulations show that the new method can be used to infer full- and half-sibships accurately from marker data with a high error rate and to identify typing errors at each locus in each reconstructed sib family. The new method also improves previous ones by adopting a fresh iterative procedure for updating allele frequencies with reconstructed sibships taken into account, by allowing for the use of parental information, and by using efficient algorithms for calculating the likelihood function and searching for the maximum-likelihood configuration. It is tested extensively on simulated data with a varying number of marker loci, different rates of typing errors, and various sample sizes and family structures and applied to two empirical data sets to demonstrate its usefulness.

Algorithms↗
PubMed2000

Empirical Bayes procedure for estimating genetic distance between populations and effective population size.

We developed an empirical Bayes procedure to estimate genetic distances between populations using allele frequencies. This procedure makes it possible to describe the skewness of the genetic distance while taking full account of the uncertainty of the sample allele frequencies. Dirichlet priors of the allele frequencies are specified, and the posterior distributions of the various composite parameters are obtained by Monte Carlo simulation. To avoid overdependence on subjective priors, we adopt a hierarchical model and estimate hyperparameters by maximizing the joint marginal-likelihood function. Taking advantage of the empirical Bayesian procedure, we extend the method to estimate the effective population size using temporal changes in allele frequencies. The method is applied to data sets on red sea bream, herring, northern pike, and ayu broodstock. It is shown that overdispersion overestimates the genetic distance and underestimates the effective population size, if it is not taken into account during the analysis. The joint marginal-likelihood function also estimates the rate of gene flow into island populations.

Algorithms↗
PubMed2004

Analysis of the relationship between type traits and functional survival in Canadian Holsteins using a Weibull proportional hazards model.

The aim of this study was to explore the impact of type traits on the functional survival of Canadian Holstein cows using a Weibull proportional hazards model. The data set consisted of 1,130,616 registered cows from 13,606 herds calving from 1985 to 2003. Functional survival was defined as the number of days from first calving to culling, death, or censoring. Type information consisted of phenotypic type scores for 8 composite traits (with 18 classes of each) and 23 linear descriptive traits (with 9 classes of each). The statistical model included the effects of stage of lactation, season of production, the annual change in herd size, type of milk recording supervision, age at first calving, effects of milk, fat and protein yields calculated within herd-year-parity deviations, herd-year-season of calving, each type trait, and the sire. Analysis was done one at a time for each of 31 type traits. The relative culling risk was calculated for animals in each class after accounting for the previously mentioned effects. Among the composite type traits with the greatest contribution to the likelihood function were final score, mammary system, and feet and legs, all having a strong relationship with functional survival. Cows with low scores for these traits had higher risk of culling compared with higher scores. For instance, cows classified as poor plus 1 vs. excellent plus 1 have a relative risk of culling 3.66 and 0.28, respectively. The corresponding figures for mammary system are 4.19 and 0.46 and for feet and legs are 2.34 and 0.50. Linear type traits with the greatest contribution to the likelihood function were fore udder attachment, udder texture, udder depth, rear udder attachment height, and rear udder attachment width. Stature and size had no strong relationship with functional survival.

Animals↗
PubMed1989

An interpretation of cortical maps in echolocating bats.

Target parameters such as reflectivity, range, velocity, and angular position are represented by ordered maps of tuned cortical neurons in insectivorous bats. It is suggested that the response of each neuron in such a map is determined by a hypothesis test conditioned on a particular value of the mapped parameter. The excitation of each neuron is then interpreted as a sample value of a conditional log-likelihood ratio or a log-likelihood function. Interpolation between the samples, which is needed to find the parameter that maximizes the mapped function (e.g., the maximum likelihood parameter estimate), can be accomplished with overlapped tuning curves. An attempt to portray a sharp peak by a weighted sum of relatively broad neuronal tuning curves or interpolation functions results in excitatory center/inhibitory surround behavior. Facilitation or antifacilitation of neurons that are likely to be excited by succeeding observations can be used for sequential detection and tracking. Interpolation and pulse-to-pulse data storage capability are required to explain range jitter sensitivity and to allow for moving target indication in bat sonar. If a cortical map represents an ordered array of hypothesis tests, then many such tests are implemented in parallel when target parameters are unknown. Detection performance is then degraded relative to the idealized situation in which all parameters are specified. Performance in noise may thus appear to be much worse than that of an ideal detector, even if each hypothesis test is optimally implemented.

Animals↗
PubMed2001

A piecewise Markov process for analysing survival from breast cancer in different risk groups.

A study of the relapse and survival times for 300 breast cancer patients submitted to post-surgical treatments is presented. After surgery, these patients were given three treatments: chemotherapy; radiotherapy; hormonal therapy and a combination of them. From the data set, a non-homogeneous Markov model is selected as suitable for the evolution of the disease. The model is applied considering two time periods during the observation of the cohort where the disease is well differentiated with respect to death and relapse. The effect of the treatments on the patients is introduced into the model via the transition intensity functions. A piecewise Markov process is applied, the likelihood function is built and the parameters are estimated, following a parametric methodological procedure. As a consequence, a survival table for different treatments is given, and survival functions for different treatments are plotted and compared with the corresponding empirical survival function. The fit of the different curves is good, and predictions can be made on the survival probabilities to post-surgical treatments for different risk groups.

Breast Neoplasms↗
PubMed1984

Evaluation of path analysis through computer simulation: effect of incorrectly assuming independent distribution of familial correlations.

Path analysis of family data has been widely applied to resolve genetic and environmental patterns of familial resemblance. A prevalent statistical approach in path analysis has been, first, to estimate the familial correlations and, second, by assuming these estimates to be independently distributed, define a likelihood function from which maximum likelihood estimates of model parameters can be obtained and likelihood ratio tests of hypotheses performed. Although it is generally known that the independence assumption does not hold when multiple familial correlations are estimated from the same family data, this statistical method has still been used in these situations owing, in part, to the lack of any viable alternatives and, in part, to the lack of any knowledge about the specific quantitative effects of not meeting the assumption of independence. Here, using computer-simulation methods, we evaluate the robustness of this statistical method to deviations from the assumption of independence. In general, we found that the failure to meet the assumption of independence leads to a conservative test of the goodness-of-fit of the path model, although likelihood ratio tests of specific null hypotheses were at times liberal, at times conservative, and at times nearly exact. Although the test statistics were found to be distorted, the parameter estimates using this method were nearly unbiased.

Biometry↗
PubMed2003

Markov chain Monte Carlo without likelihoods.

Many stochastic simulation approaches for generating observations from a posterior distribution depend on knowing a likelihood function. However, for many complex probability models, such likelihoods are either impossible or computationally prohibitive to obtain. Here we present a Markov chain Monte Carlo method for generating observations from a posterior distribution without the use of likelihoods. It can also be used in frequentist applications, in particular for maximum-likelihood estimation. The approach is illustrated by an example of ancestral inference in population genetics. A number of open problems are highlighted in the discussion.

Algorithms↗