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Results for “Likelihood Functions”

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PubMed2005

Genetic components of functional connectivity in the brain: the heritability of synchronization likelihood.

Cognitive functions require the integrated activity of multiple specialized, distributed brain areas. Such functional coupling depends on the existence of anatomical connections between the various brain areas as well as physiological processes whereby the activity in one area influences the activity in another area. Recently, the Synchronization Likelihood (SL) method was developed as a general method to study both linear and nonlinear aspects of coupling. In the present study the genetic architecture of the SL in different frequency bands was investigated. Using a large genetically informative sample of 569 subjects from 282 extended twin families we found that the SL is moderately to highly heritable (41-67%) especially in the alpha frequency (8-13 Hz) range. This index of functional connectivity of the brain has been associated with a number of pathological states of the brain. The significant heritability found here suggests that SL can be used to examine the genetic susceptibility to these conditions.

Alpha Rhythm↗

PubMed1997

Estimating the proportion of treatment effect explained by a surrogate marker.

In this paper, we measure the extent to which a biological marker is a surrogate endpoint for a clinical event by the proportional reduction in the regression coefficient for the treatment indicator due to the inclusion of the marker in the Cox regression model. We estimate this proportion by applying the partial likelihood function to two Cox models postulated on the same failure time variable. We show that the resultant estimator is asymptotically normal with a simple variance estimator. One can construct confidence intervals for the proportion by using the direct normal approximation to the point estimator or by using Fieller's theorem. Extensive simulation studies demonstrate that the proposed methods are appropriate for practical use. We provide applications to HIV/AIDS clinical trials.

Acquired Immunodeficiency Syndrome↗

PubMed1998

Evaluating effects of exposures on embryo viability and uterine receptivity in in vitro fertilization.

We consider models for the occurrence of pregnancy following in vitro fertilization. In this clinical protocol, implantation depends on two factors: the receptivity of the uterus and the viability of at least one of the embryos transferred to the uterus. This work is motivated by the need to identify reliable bio-markers for these two factors, in order to enhance the success rate for couples undergoing this procedure. We present a general latent variable structure model for outcomes that take either one of two possible forms: as summed Bernoullis, based on an ultrasound count of gestational sacs, or as aggregated Bernoullis, based only on the outcome of a biochemical pregnancy test. We allow both uterine receptivity and embryo viability to be influenced by covariates. The proposed latent variable structure allows us to utilize the existing statistical packages to maximize an otherwise intractable likelihood function. The method is sufficiently flexible to permit any valid choice of link function. We illustrate by applying the method to a recent study of in vitro fertilization carried out in North Carolina. The number of cells at transfer is evidently a marker for embryo viability.

Adult↗

PubMed1999

A GEE moving average analysis of the relationship between air pollution and mortality for asthma in Barcelona, Spain.

Several studies have assessed the association between air pollution and hospital admissions or emergency room visits for asthma. Because of both the presence of missing data and the small number of observations, the relationship between air pollution and mortality for respiratory causes has been rarely analysed, and when it has, the results are very inconclusive or even inconsistent. The objective of this study is to assess the relation between levels of air pollutants (black smoke, sulphur dioxide, nitrogen dioxide and ozone), meteorological variables (24th average temperature and relative humidity) and daily mortality for asthma (ICD-9 493, 2 to 45 years old) in Barcelona, Spain, during the period 1986-1989. Since the range of daily mortality for asthma (2 to 45 years old) during the period 1986-1989 was 0-1), we have preferred to consider this variable as dichotomous. First, the relationship between air pollutants, meteorological variables and daily mortality (controlled for the occurrence of asthma epidemics) was estimated using logistic regression models. As was expected, the residuals from this regression were autocorrelated, showing a complex moving average (MA) structure. If covariates were not time dependent the so-called generalized linear mixed models, could be applied. In our case the covariates vary. As a consequence the likelihood is numerically intractable because it involves the evaluation of n-fold integral. An alternative method that avoids these numerical problems is the generalized estimating equations method (GEE). It is a multivariate analogue of quasi-likelihood estimation. In the absence of a likelihood function the parameters can be estimated by solving a multivariate analogue of the quasi score function. We have modified the GEE method in this paper, allowing for a different structure in the error covariance matrix (MA). Both air pollutants and meteorological variables are related with the occurrence of a death for asthma. In this sense, nitrogen dioxide, NO(2) (ss=0.037, p<0. 05), ozone, O(3) (ss=0.021, p<0.06) and high temperature (the ss's were in the range (0.098-0.182), p<0.05) increased the probability of dying for asthma in Barcelona during the period 1986-1989.

Adolescent↗

PubMed1982

Analyses of carcinogenesis dose-response relations with dichotomous data: implications for carcinogenic risk assessment.

Dose-response data from experimental and epidemiologic carcinogenicity studies were analyzed in attempts to resolve basic questions in extrapolating from high to low doses and assessing human risk. Four models (Weibull, Mantel-Bryan, Marshall-Groer, and Mancuso-Stewart) were fit to 46 sets of experimental and 4 sets of epidemiologic data by maximizing the likelihood function with a Rosenbrock hill-climbing algorithm. The models were compared as to their adequacy in describing the data and analyzed to determine the effect of carcinogen and breeding category, species, and spontaneous tumor incidence. The shapes of the dose-response curves were analyzed, and errors in risk estimation from linear extrapolation through the origin were calculated. All models were shown to fit the data and to be comparable in accuracy. The dose-response curves were generally "stretched out," particularly for outbred strains, with one or two orders of magnitude of dose increase required to increase the proportion of tumor responders from 10% to 70%. Linear extrapolation through the origin generally underestimated the response at low doses, frequently by several orders of magnitude. The power dependence of tumor incidence on dose was generally found to be of the order of unity, substantially less than assumed in most mathematical models.

Carcinogens↗

PubMed1991

Familial aggregation of lipids and lipoproteins in families ascertained through random and nonrandom probands in the Stanford Lipid Research Clinics Family Study.

We examined the familial aggregation of lipids [total cholesterol (CH) and triglyceride (TG)] and lipoproteins [high-density lipoprotein cholesterol (HDL) and low-density lipoprotein cholesterol (LDL)] in families ascertained through random and nonrandom probands in the Stanford Lipid Research Clinics Family Study. Nonrandom probands were selected because their lipid levels at a prior screening visit exceeded a certain prespecified threshold. The statistical method is based on selection through indirect truncation on a correlated trait (in which the likelihood function is conditioned on the actual event that the proband's value is beyond the threshold). This method allows for estimation of the path model parameters in randomly and nonrandomly ascertained families jointly and separately, thus enabling tests of heterogeneity between the two types of samples. The results suggest that the multifactorial transmission is homogeneous in the random and hyperlipidemic samples for CH. However, the evidence for heterogeneity is moderate for LDL, marked HDL, and mixed for TG. The general pattern of observed results is for somewhat higher genetic heritabilities in the random than nonrandom samples, which is compatible with a higher prevalence in the random sample of certain dyslipoproteinemias associated with nonelevated lipids. Substantial genetic heritability is found for CH, HDL, and LDL, with somewhat lower estimates for TG. Cultural heritability is low but significant for all four traits. Little or no spouse resemblance or nontransmitted shared sibship effects are seen. In contrast to the findings from previous studies, little or no parental cultural transmission is seen.

Adult↗

PubMed2005

Exact tests for one sample correlated binary data.

In this paper we developed exact tests for one sample correlated binary data whose cluster sizes are at most two. Although significant progress has been made in the development and implementation of the exact tests for uncorrelated data, exact tests for correlated data are rare. Lack of a tractable likelihood function has made it difficult to develop exact tests for correlated binary data. However, when cluster sizes of binary data are at most two, only three parameters are needed to characterize the problem. One parameter is fixed under the null hypothesis, while the other two parameters can be removed by both conditional and unconditional approaches, respectively, to construct exact tests. We compared the exact and asymptotic p-values in several cases. The proposed method is applied to real-life data.

Anti-Bacterial Agents↗

PubMed1989

Summary measures for evaluating the evidence for linkage.

Two summary measures for evaluating the evidence for linkage are the maximum lod score and the average of the likelihood function (antilod score). The average antilod score can be used to calculate an exact probability of there being no linkage. Use of subjective prior probabilities in the calculation will only influence the final conclusion in any important way if evidence from the sample is not particularly conclusive. It is shown how results from two-point analyses for linkage of the test locus with each marker locus can be easily combined to give an approximate probability of there being no linkage. Some examples using bipolar affective disorder data and marker loci are presented.

Bipolar Disorder↗

PubMedDate not supplied

A new threshold dose-response model including random effects for data from developmental toxicity studies.

Usually, in teratological dose finding studies, there are not only threshold effects but also extra variations that cannot be accounted for by the beta-binomial model alone. The beta-binomial model assumes correlation between fetuses in the same litter. The general random effect threshold (RE) model allows the additional variability that arises due to correlation and between litter variability to be modeled, in combination with threshold in the model. The goal of this research was to investigate a threshold dose-response model with random effects (RE) to model the variability that exists between litters of animals in studies of toxic agents. Data from a developmental toxicity study of a toxic agent were analysed, using the proposed RE threshold dose-response model, which is an extension of logit in form. Also, an approximate likelihood function was used to derive parameter estimates from this model, and tests were performed to determine the significance of the model parameters, in particular, the RE parameter. A simulation study was conducted to assess the performance of the RE threshold model in estimating the model parameters.

Algorithms↗

PubMed2003

Comparison of case-cohort estimators based on data on premature death of adult adoptees.

A case-cohort sample of adoptees was collected to investigate genetic and environmental influences on premature death, which motivated us to supplement existing simulation results to explore the performance of various estimators proposed for case-cohort samples of survival data. We studied six regression coefficients estimators, which differ with regard to the weighting scheme used in a pseudo-likelihood function, and two different estimators of their variances. Compared to earlier simulation studies, we changed the following conditions: type of explanatory variable, the distribution of lifetimes, and the percentage of deaths in the full cohort. The latter condition affected the performance of the estimated variances of the regression coefficients, where we found a systematic bias of the estimator, proposed by Self and Prentice, dependent on the percentages of deaths. This dependence of percentages of death was different for different sizes of case-cohort studies. A robust variance estimator showed a better overall performance. The estimators of regression coefficients compared did not differ much, the estimators proposed by Kalbfleisch and Lawless and by Prentice performing very well. Results of the case-cohort data of adoptees were not in conflict with earlier findings of a moderate genetic influence on premature death in adulthood.

Adoption↗

PubMed2005

Evaluating technologies for classification and prediction in medicine.

Modern technologies promise to provide new ways of diagnosing disease, detecting subclinical disease, predicting prognosis, selecting patient specific treatment, identifying subjects at risk for disease, and so forth. Advances in genomics, proteomics and imaging modalities in particular hold great potential for assisting with classification/prediction in medicine. Before a classifier can be adopted for routine use in health care, its classification accuracy must be determined. Standards for evaluating new clinical classifiers however, lag far behind the well established standards that exist for evaluating new clinical treatments. In this paper, we discuss a phased approach to developing a new classifier (or biomarker). It mirrors the internationally established phase 1-2-3 paradigm for therapeutic drugs. The defined phases lead to a logical sequence of studies for classifier development. We emphasize that evaluating classification accuracy is fundamentally different from simply establishing association with outcome. Therefore, study objectives and designs differ from the familiar methods of clinical trials. We discuss these briefly for each phase.Finally, we argue that classifier development requires some rethinking of traditional data analysis techniques. As an example we show that maximizing the likelihood function to fit a logistic regression model to multiple predictors, can yield a poor classifier. Instead we demonstrate that an approach that maximizes an alternative objective function characterizing classification accuracy performs better.

Diagnostic Techniques and Procedures↗

PubMed2007

Estimating incidence of HIV infection in childbearing age African women using serial prevalence data from antenatal clinics.

Ades and Medley provided the first flexible method for estimating age- and time-specific HIV incidence using HIV prevalence data collected among pregnant women and adjusting for the effect of differential selection between infected and uninfected women. This paper extends the approach proposed by these authors. We used a parametric model that allows the relative inclusion rate to depend on both age, calendar time, and duration of HIV infection. We developed a two dimensional penalized log-likelihood approach for estimating time- and age-specific incidence using a binomial likelihood function and a quadratic roughness penalty which allows smoothing over both age and time. Identifiability of the model parameters and effect of sample size are studied through simulations. The method is illustrated using prenatal HIV testing data recorded from 1995 to 2002 in Abidjan, Côte d'Ivoire, to estimate the HIV annual incidence rate among women aged 12-40 year old, from the beginning of the epidemic to 2002. We show that estimated incidence rates are highly dependent on hypotheses made to model the relative inclusion rate. Despite this dependency, the application of the method leads to new and accurate findings on HIV incidence qualitative features in Abidjan. We highlight the relevance of such a method in monitoring the dynamics of HIV epidemic in Africa which is essential for planning vaccine trials and future treatment needs, and for assessment of prevention policy.

Adolescent↗

PubMed1992

Using an autoregressive model to detect departures from steady states in unequally spaced tumour biomarker data.

A new method, based on a continuous time autoregressive [CAR(1)] model of time series data, is provided for detecting departures of tumour markers from steady states in breast cancer patients following surgery. A Kalman filter recursive algorithm is used to calculate the likelihood function arising from the CAR(1) model and to calculate recursive residuals, which are monitored by a Shewhart-Cusum scheme. This approach can be used to monitor the serial marker data of large numbers of patients even when the series are short and the data are serially correlated and unequally spaced. Further, the methodology can be used to recommend appropriate testing intervals.

Algorithms↗

PubMed1995

Identification of protein coding regions in genomic DNA.

We have developed a computer program, GeneParser, which identifies and determines the fine structure of protein genes in genomic DNA sequences. The program scores all subintervals in a sequence for content statistics indicative of introns and exons, and for sites that identify their boundaries. This information is weighted by a neural network to approximate the log-likelihood that each subinterval exactly represents an intron or exon (first, internal or last). A dynamic programming algorithm is then applied to this data to find the combination of introns and exons that maximizes the likelihood function. Using this method, we can rapidly generate ranked suboptimal solutions, each of which is the optimum solution containing a given intron-exon junction. We have tested the system on a large collection of human genes. On sequences not used in training, we achieved a correlation coefficient for exon nucleotide prediction of 0.89. For a subset of G + C-rich genes, a correlation coefficient of 0.94 was achieved. We have also quantified the robustness of the method to substitution and frame-shift errors and show how the system can be optimized for performance on sequences with known levels of sequencing errors.

Base Composition↗

PubMed1983

Affinity spectra: a novel way for the evaluation of equilibrium binding experiments.

For equilibrium binding isotherms of radioreceptor assays, the affinity spectrum is defined as a plot of the number of binding sites against their corresponding dissociation constants. A numerical procedure for direct calculation of affinity spectra from untransformed binding data is presented and illustrated with experimental values. The advantage of the new method in comparison to non-linear regression analysis is the fact that no starting values and mathematical models have to be supplied and that statistical assessment of the results is straightforward from a detailed graphical display of a likelihood function. Affinity spectra thus show directly all information formerly obtained by means of both graphical plots and regression analysis.

Animals↗

PubMed1989

On the maximum likelihood method for estimating molecular trees: uniqueness of the likelihood point.

Studies are carried out on the uniqueness of the stationary point on the likelihood function for estimating molecular phylogenetic trees, yielding proof that there exists at most one stationary point, i.e., the maximum point, in the parameter range for the one parameter model of nucleotide substitution. The proof is simple yet applicable to any type of tree topology with an arbitrary number of operational taxonomic units (OTUs). The proof ensures that any valid approximation algorithm be able to reach the unique maximum point under the conditions mentioned above. An algorithm developed incorporating Newton's approximation method is then compared with the conventional one by means of computer simulation. The results show that the newly developed algorithm always requires less CPU time than the conventional one, whereas both algorithms lead to identical molecular phylogenetic trees in accordance with the proof.

Algorithms↗

PubMed2007

Bayesian analysis of paired survival data using a bivariate exponential distribution.

We consider a Bayesian analysis method of paired survival data using a bivariate exponential model proposed by Moran (1967, Biometrika 54:385-394). Important features of Moran's model include that the marginal distributions are exponential and the range of the correlation coefficient is between 0 and 1. These contrast with the popular exponential model with gamma frailty. Despite these nice properties, statistical analysis with Moran's model has been hampered by lack of a closed form likelihood function. In this paper, we introduce a latent variable to circumvent the difficulty in the Bayesian computation. We also consider a model checking procedure using the predictive Bayesian P-value.

Bayes Theorem↗

PubMed2006

Model selection and mixed-effects modeling of HIV infection dynamics.

We present an introduction to a model selection methodology and an application to mathematical models of in vivo HIV infection dynamics. We consider six previously published deterministic models and compare them with respect to their ability to represent HIV-infected patients undergoing reverse transcriptase mono-therapy. In the creation of the statistical model, a hierarchical mixed-effects modeling approach is employed to characterize the inter- and intra-individual variability in the patient population. We estimate the population parameters in a maximum likelihood function formulation, which is then used to calculate information theory based model selection criteria, providing a ranking of the abilities of the various models to represent patient data. The parameter fits generated by these models, furthermore, provide statistical support for the higher viral clearance rate c in Louie et al. [AIDS 17:1151-1156, 2003]. Among the candidate models, our results suggest which mathematical structures, e.g., linear versus nonlinear, best describe the data we are modeling and illustrate a framework for others to consider when modeling infectious diseases.

CD4 Lymphocyte Count↗