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Results for “Likelihood Functions”

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PubMed1994

Leukemia in patients with breast cancer following adjuvant chemotherapy and/or postoperative radiation therapy.

We investigated the possible etiological role of adjuvant chemotherapy and postoperative radiation therapy in the development of leukemia. A case-control design with four controls matched to each case of leukemia from a cohort of women who had been treated for breast cancer during the years from 1970 to 1985 was used. Thirteen (0.23%) of the women in this cohort developed leukemia over varying lengths of follow-up time, ranging from 3 to 14 years. A higher percentage of the leukemia cases previously had adjuvant chemotherapy compared to their matched controls (54% versus 13%). The relative odds estimate of developing leukemia after chemotherapy compared to no chemotherapy was 14.8 (95% C.I. (1.8; 125.3) p < 0.01). This estimate and the test of statistical significance was based on the likelihood function for matched sets with one case and more than one control. Approximately the same percentage of leukemia cases as their controls had received postoperative regional radiation therapy (28% versus 23%). No significant association was found between postoperative radiation therapy and development of leukemia. A combination of adjuvant chemotherapy and postoperative radiation therapy was found more frequently in the leukemia cases than in their matched controls (33% versus 9%). The leukemia developing in patients having received adjuvant chemotherapy was frequently therapy resistant, resulting in a short survival.

Aged↗

PubMed2005

A tree-based model for homogeneous groupings of multinomials.

The motivation of this paper is to provide a tree-based method for grouping multinomial data according to their classification probability vectors. We produce an initial tree by binary recursive partitioning whereby multinomials are successively split into two subsets and the splits are determined by maximizing the likelihood function. If the number of multinomials k is too large, we propose to order the multinomials, and then build the initial tree based on a dramatically smaller number k-1 of possible splits. The tree is then pruned from the bottom up. The pruning process involves a sequence of hypothesis tests of a single homogeneous group against the alternative that there are two distinct, internally homogeneous groups. As pruning criteria, the Bayesian information criterion and the Wilcoxon rank-sum test are proposed. The tree-based model is illustrated on genetic sequence data. Homogeneous groupings of genetic sequences present new opportunities to understand and align these sequences.

Animals↗

PubMed1999

Bootstrap approach for constructing confidence intervals for population pharmacokinetic parameters. I: A use of bootstrap standard error.

In population pharmacokinetic studies, one of the main objectives is to estimate population pharmacokinetic parameters specifying the population distributions of pharmacokinetic parameters. Confidence intervals for population pharmacokinetic parameters are generally estimated by assuming the asymptotic normality, which is a large-sample property, that is, a property which holds for the cases where sample sizes are large enough. In actual clinical trials, however, sample sizes are limited and not so large in general. Likelihood functions in population pharmacokinetic modelling include a multiple integral and are quite complicated. We hence suspect that the sample sizes of actual trials are often not large enough for assuming the asymptotic normality and that the asymptotic confidence intervals underestimate the uncertainties of the estimates of population pharmacokinetic parameters. As an alternative to the asymptotic normality approach, we can employ a bootstrap approach. This paper proposes a bootstrap standard error approach for constructing confidence intervals for population pharmacokinetic parameters. Comparisons between the asymptotic and bootstrap confidence intervals are made through applications to a simulated data set and an actual phase I trial.

Adult↗

PubMed2003

Fertility assessment through heterospermic insemination of flow-sorted sperm in cattle.

The ability to assess fertility of bovine sperm accurately and rapidly would be very useful for research and applications to the cattle industry. Sperm motility and other in vitro tests of sperm normality are only partially correlated with fertility, and lengthy breeding trials are expensive and time consuming. Heterospermic insemination by mixing sperm from more than one male provides an in vivo method to assess relative fertility among bulls that can be economical and rapid. Sperm that had been flow-sorted and cryopreserved from four groups of four bulls were inseminated in all combinations of three bulls within groups into nonsuperovulated heifers or superovulated heifers. Embryos were collected nonsurgically between d 13.5 and 20 following estrus and evaluated for paternity by genotyping. Following determination of paternity, a heterospermic index was created for each bull using a maximum likelihood function. These indices ranged from 0.22 +/- 0.15 to 2.43 +/- 0.43 (mean = 1.00, with a higher value indicative of greater fertility). In all four groups, either the high- or low-fertility bull was identified (P < 0.05) using a total of 25 to 36 genotypable embryos from nonsuperovulated heifers. The heterospermic rankings of bulls were similar for single and superovulated heifers for one group of bulls, but dissimilar for a second group. Heterospermic insemination followed by genotyping of embryos proved to be efficacious for rapidly ranking fertility of flow-sorted sperm from bulls when females were not superovulated, but results were less clear when females were superovulated.

Animals↗

PubMed1993

Mathematical models of marrow cell kinetics: differential effects of protracted irradiations on stromal and stem cells in mice.

UNLABELLED: It is known that hematopoiesis is supported by bone-marrow stem cells, but those cells must seed and grow on a stromal microenvironment. Typically, studies have shown that a surviving fraction of about 30 hematopoietic stem cells (HSCs) (i.e., about 0.04%) correspond to the LD50, although other studies have shown that marrow can repopulate from a single viable cell under strong regiments of antibiotics and infusions of irradiated blood elements. PURPOSE: This paper describes comparisons between our results (from maximum-likelihood estimation techniques for cellular damage, repair, and compensatory repopulation) and published experimental data on marrow stromal cells. METHODS AND MATERIALS: After biophysical consideration of the rate constants that were derived by maximizing the likelihood function (a consideration necessary to extend the model to cell populations not indicated by the model as "critical" for recovery), the rate constants for cellular damage to stem cells are fitted to experimental data. Rate constants for repair and proliferation of stem cells are assigned based on published data on repair/proliferation half-times, and these assignments affect the evaluation of the rate constants for cellular damage. From the two models, that is one for "critical" cells (having radiosensitive and repopulation characteristics similar to stromal cells) and another for stem cells, effects on two cell populations of different radiosensitivities and repopulation rates can be demonstrated for complex schedules of protracted irradiations which could reduce either cell population below a critical need for marrow repopulation. RESULTS: Our analysis of animal mortality data has indicated that recovery of an animal from potentially lethal irradiation is dominantly regulated by cells with survival and repopulation characteristics similar to those of stroma cells. CONCLUSION: In contrast to the surviving fraction of hematopoietic stem cells, it appears that the probability of an animal's recovery is high if the "critical" population of cells is above 1% (our "best" maximum likelihood estimate, from mouse data, with the corresponding lower confidence bound at about 0.2%). Of course, a few stem cells--perhaps only one--must maintain a potential for repopulation of blood and marrow.

Animals↗

PubMed2004

Bayesian estimation of stimulus responses in Poisson spike trains.

A Bayesian method is developed for estimating neural responses to stimuli, using likelihood functions incorporating the assumption that spike trains follow either pure Poisson statistics or Poisson statistics with a refractory period. The Bayesian and standard estimates of the mean and variance of responses are similar and asymptotically converge as the size of the data sample increases. However, the Bayesian estimate of the variance of the variance is much lower. This allows the Bayesian method to provide more precise interval estimates of responses. Sensitivity of the Bayesian method to the Poisson assumption was tested by conducting simulations perturbing the Poisson spike trains with noise. This did not affect Bayesian estimates of mean and variance to a significant degree, indicating that the Bayesian method is robust. The Bayesian estimates were less affected by the presence of noise than estimates provided by the standard method.

Action Potentials↗

PubMed2003

Examining outlying subjects and outlying records in bioequivalence trials.

The problem of detecting outliers in bioequivalence trials is considered. We formulate the problem as a hypothesis-testing problem under a mean-shift model and propose a test procedure based on the likelihood function. The test statistic has two components: one is to detect whether a specific pharmacokinetic measurement of a subject for certain formulation/drug product is an outlying value; the other is to test whether a subject as a whole is an outlying subject (with unusual high or low bioavailability for all formulations/drug products). Under normality assumption, the proposed procedure is most powerful. The small sample distribution of the proposed test statistic is derived. A numerical example illustrates the use of the procedure. The proposed test is then compared in a simulation study against the Hotelling T2 test, recommended by Liu and Weng (1991) for the use of outlier detection in bioequivalence studies. The results from the simulation study show that the proposed test is more powerful than the Hotelling T2 test.

Biological Availability↗

PubMed1996

Use of the Weibull hazard model to estimate age-specific probability of permanent tooth loss.

In analysis of the probability of tooth loss with age, the exact time of tooth loss is often unknown, although it is clear whether a tooth remains or has been lost. That is, left censoring is inevitable during data sampling. This may provide a biased estimate if such data are dealt with by the product-limit method, which is a common method of survival analysis. To reduce such a bias in estimating the age-specific probability of tooth loss, we developed a survival analysis method taking left censoring into consideration. Four hundred and forty-six panoramic radiographs obtained in a daily clinical practice were used. The frequency of tooth loss with age was assumed to follow the Weibull hazard model, and a likelihood function taking left censoring into consideration was defined to estimate the probability of tooth loss. The estimate obtained from our method was compared with that from the product-limit method to examine whether the effect of left censoring was reduced. We found that the probability of tooth loss estimated by the product-limit method was biased by left-censored data, and that the bias was reduced when our method was used. A Monte Carlo simulation study, in which the true tooth loss time was given, also showed that our method provided an estimate closer to the true value. Our method is considered to be more accurate in estimating the probability of tooth loss, since it reduces the bias caused by left-censored data.

Adolescent↗

PubMed1994

Multiway sequential hypothesis testing for tachyarrhythmia discrimination.

A multiway sequential hypothesis testing (M-SHT) algorithm is proposed for simultaneous discrimination of cardiac tachyarrhythmias--supraventricular tachycardia (SVT) and ventricular tachycardia (VT)--from normal sinus rhythm (NSR). The M-SHT algorithm calculates a likelihood function from atrio-ventricular delay measurements, and compares this function with thresholds derived from specified error probabilities for the arrhythmias to be discriminated. Performance of this algorithm was evaluated on dual channel endocardial electrograms recorded in the cardiac electrophysiology laboratory. Two databases were developed, one for development of the algorithm and another for evaluation. The M-SHT algorithm accurately classified 26 out of 28 NSR (2 misclassified as SVT), 31 out of 31 cases of SVT, and 41 out of 43 VT (2 misclassified as NSR). The average length of time taken for classification of the three rhythms was: 3.6 s for NSR, 5.0 s for SVT, and 1.6 s for VT. Unique features of this algorithm are that acceptable error rates for each arrhythmia are independently specified and accuracy can be traded off for a faster detection time, and vice versa.

Adult↗

PubMed2003

The EIM algorithm in the joint segregation analysis of quantitative traits.

In this article, a new algorithm for obtaining the maximum likelihood estimators (MLEs) of parameters in the joint segregation analysis (JSA) of multiple generations of P1, F1, P2, F2 and F2:3 (MG5) for quantitative traits was set up. Firstly, owing to the fact that the component variance of the heterogeneous genotype in F2:3 included both the first-order genetic parameters (denoted by the means of distributions) and the second-order parameters, a simple closed form for the MLEs of the means of component distributions did not exist while the expectation and maximization (EM) algorithm was used. To simplify the estimation of parameters, the first partial derivative of the above variance on the mean in the sample log-likelihood function was omitted. However, this would be remedied by the iterated method. Then, variances of component distributions for segregating populations were partitioned into major-gene, polygenic and environmental variances so that the generally iterated formulae for estimating the means as well as polygenic and environmental variances of component distributions in the maximization step (M-step) of the EM algorithm were obtained. Therefore, the EM algorithm for estimating parameters in the JSA model for the MG5 was simplified. This is called the expectation and iterated maximization (EIM) algorithm. Finally, an example of the inheritance of the resistance of soybean to beanfly showed that the results of mixed inheritance analysis in this paper coincided with those in both Wang & Gai (2001) and Wei et al. (1989), so the EIM algorithm was appropriate.

Algorithms↗

PubMed2003

Fisher's information in maximum-likelihood macromolecular crystallographic refinement.

Fisher's information is a statistical quantity related to maximum-likelihood theory. It is a matrix defined as the expected value of the squared gradient of minus the log-likelihood function. This matrix is positive semidefinite for any parameter value. Fisher's information is used in the quasi-Newton scoring method of minimization to calculate the shift vectors of model parameters. If the matrix is non-singular, the scoring-minimization step is always downhill. In this article, it is shown how the scoring method can be applied to macromolecular crystallographic refinement. It is also shown how the computational costs involved in calculation of the Fisher's matrix can be efficiently reduced. Speed is achieved by assuming a continuous distribution of reciprocal-lattice points. Matrix elements calculated with this method agree very well with those calculated analytically. The scoring algorithm has been implemented in the program REFMAC5 of the CCP4 suite. The Fisher's matrix is used in its sparse approximation. Tests indicate that the algorithm performs satisfactorily.

Algorithms↗

PubMed2006

A classical likelihood based approach for admixture mapping using EM algorithm.

Several disease-mapping methods have been proposed recently, which use the information generated by recent admixture of populations from historically distinct geographic origins. These methods include both classic likelihood and Bayesian approaches. In this study we directly maximize the likelihood function from the hidden Markov Model for admixture mapping using the EM algorithm, allowing for uncertainty in model parameters, such as the allele frequencies in the parental populations. We determined the robustness of the proposed method by examining the ancestral allele frequency estimate and individual marker-location specific ancestry when the data were generated by different population admixture models and no learning sample was used. The proposed method outperforms a widely used Bayesian MCMC strategy for data generated from various population admixture models. The multipoint information content for ancestry was derived based on the map provided by Smith et al. (2004) and the associated statistical power was calculated. We examined the distribution of admixture LD across the genome for both real and simulated data and established a threshold for genome wide significance applicable to admixture mapping studies. The software ADMIXPROGRAM for performing admixture mapping is available from authors.

Algorithms↗

PubMed1998

High-resolution 3D Bayesian image reconstruction using the microPET small-animal scanner.

A Bayesian method is described for reconstruction of high-resolution 3D images from the microPET small-animal scanner. Resolution recovery is achieved by explicitly modelling the depth dependent geometric sensitivity for each voxel in combination with an accurate detector response model that includes factors due to photon pair non-collinearity and inter-crystal scatter and penetration. To reduce storage and computational costs we use a factored matrix in which the detector response is modelled using a sinogram blurring kernel. Maximum a posteriori (MAP) images are reconstructed using this model in combination with a Poisson likelihood function and a Gibbs prior on the image. Reconstructions obtained from point source data using the accurate system model demonstrate a potential for near-isotropic FWHM resolution of approximately 1.2 mm at the center of the field of view compared with approximately 2 mm when using an analytic 3D reprojection (3DRP) method with a ramp filter. These results also show the ability of the accurate system model to compensate for resolution loss due to crystal penetration producing nearly constant radial FWHM resolution of 1 mm out to a 4 mm radius. Studies with a point source in a uniform cylinder indicate that as the resolution of the image is reduced to control noise propagation the resolution obtained using the accurate system model is superior to that obtained using 3DRP at matched background noise levels. Additional studies using pie phantoms with hot and cold cylinders of diameter 1-2.5 mm and 18FDG animal studies appear to confirm this observation.

Animals↗

PubMed2002

Efficiency of haplotype frequency estimation when nuclear family information is included.

In genetic studies the haplotype structure of the regarded population is expected to carry important information. Experimental methods to derive haplotypes, however, are expensive and none of them has yet become standard methodology. On the other hand, maximum likelihood haplotype estimation from unphased individual genotypes may incur inaccuracies. We therefore investigated the relative efficiency of haplotype frequency estimation when nuclear family information is included compared to estimation from experimentally derived haplotypes. Efficiency was measured in terms of variance ratios of the estimates. The variances were derived from the binomial distribution for experimentally derived haplotypes, and from the Fisher information matrix corresponding to the general likelihood function of the haplotype frequency parameters, including family information. We subsequently compared these variance ratios to the variance ratios for the case of estimation from individual genotypes. We found that the information gained from a single child compensates missing phase information to a high degree, resulting in estimates almost as reliable as those derived from observed haplotypes. Thus, if children have already been genotyped for other reasons, it is highly recommendable to include them into the estimation. If child information is not already present, it depends on the number of loci and the haplotype diversity if it is useful to genotype a single child just to reduce phase ambiguity. In general, if the number of loci is less than or equal to three or if the number of haplotypes with a frequency >5% is less than or equal to four, haplotype estimation from individuals is quite good already and the improvement gained from a single child can not compensate the genotyping effort for it. On the other hand, under scenarios with many loci and high haplotype diversity, haplotype frequency estimation from trios can be more efficient than haplotype frequency estimation from individuals also on a per genotype base.

Data Interpretation, Statistical↗

PubMed2006

Analysis of longitudinal multinomial outcome data.

Analysis of categorical outcomes in a longitudinal study has been an important statistical issue. Continuous outcome in a similar study design is commonly handled by the mixed effects model. The longitudinal binary or Poisson-like outcome analysis is often handled by the generalized estimation equation (GEE) method. Neither method is appropriate for analyzing a multinomial outcome in a longitudinal study, although the cross-sectional multinomial outcome is often analyzed by generalized linear models. One reason that these methods are not used is that the correlation structure of two multinomial variables can not be easily specified. In addition, methods that rely upon GEE or mixed effects models are unsuitable in instances when the focus of a longitudinal study is on the rate of moving from one category to another. In this research, a longitudinal model that has three categories in the outcome variable will be examined. A continuous-time Markov chain model will be used to examine the transition from one category to another. This model permits an unbalanced number of measurements collected on individuals and an uneven duration between pairs of consecutive measurements. In this study, the explicit expression for the transition probability is derived that provides an algebraic form of the likelihood function and hence allows the implementation of the maximum likelihood method. Using this approach, the instantaneous transition rate that is assumed to be a function of the linear combination of independent variables can be estimated. For a comparison between two groups, the odds ratios of occurrence at a particular category and their confidence intervals can be calculated. Empirical studies will be performed to compare the goodness of fit of the proposed method with other available methods. An example will also be used to demonstrate the application of this method.

Biometry↗

PubMed2006

Prior specification in Bayesian statistics: three cautionary tales.

One of the most important differences between Bayesian and traditional techniques is that the former combines information available beforehand-captured in the prior distribution and reflecting the subjective state of belief before an experiment is carried out-and what the data teach us, as expressed in the likelihood function. Bayesian inference is based on the combination of prior and current information which is reflected in the posterior distribution. The fast growing implementation of Bayesian analysis techniques can be attributed to the development of fast computers and the availability of easy to use software. It has long been established that the specification of prior distributions should receive a lot of attention. Unfortunately, flat distributions are often (inappropriately) used in an automatic fashion in a wide range of types of models. We reiterate that the specification of the prior distribution should be done with great care and support this through three examples. Even in the absence of strong prior information, prior specification should be done at the appropriate scale of biological interest. This often requires incorporation of (weak) prior information based on common biological sense. Very weak and uninformative priors at one scale of the model may result in relatively strong priors at other levels affecting the posterior distribution. We present three different examples intuïvely illustrating this phenomenon indicating that this bias can be substantial (especially in small samples) and is widely present. We argue that complete ignorance or absence of prior information may not exist. Because the central theme of the Bayesian paradigm is to combine prior information with current data, authors should be encouraged to publish their raw data such that every scientist is able to perform an analysis incorporating his/her own (subjective) prior distributions.

Animals↗

PubMed2006

Combining predictors for classification using the area under the receiver operating characteristic curve.

No single biomarker for cancer is considered adequately sensitive and specific for cancer screening. It is expected that the results of multiple markers will need to be combined in order to yield adequately accurate classification. Typically, the objective function that is optimized for combining markers is the likelihood function. In this article, we consider an alternative objective function-the area under the empirical receiver operating characteristic curve (AUC). We note that it yields consistent estimates of parameters in a generalized linear model for the risk score but does not require specifying the link function. Like logistic regression, it yields consistent estimation with case-control or cohort data. Simulation studies suggest that AUC-based classification scores have performance comparable with logistic likelihood-based scores when the logistic regression model holds. Analysis of data from a proteomics biomarker study shows that performance can be far superior to logistic regression derived scores when the logistic regression model does not hold. Model fitting by maximizing the AUC rather than the likelihood should be considered when the goal is to derive a marker combination score for classification or prediction.

Biomarkers, Tumor↗

PubMedDate not supplied

Assessment of stratum-covariate interactions in Cox's proportional hazards regression model.

We assess stratum (e.g. treatment) interactions with covariates and with the baseline hazard function in the proportional hazards (PH) regression model for lifetime data. We consider models incorporating stratum interactions both with and without stratification of the risk sets in the likelihood function, and describe likelihood ratio statistics for tests of the presence of these interactions. We also present step-down methods for building reduced models which include stratum-specific parameters corresponding to covariates which interact with treatment. We apply PH models with such interactions to a clinical trial of DES in the treatment of prostate cancer to determine optimal treatment conditional on each patient's covariates.

Analysis of Variance↗