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PubMed2002

Joint modeling of event time and nonignorable missing longitudinal data.

Survival studies usually collect on each participant, both duration until some terminal event and repeated measures of a time-dependent covariate. Such a covariate is referred to as an internal time-dependent covariate. Usually, some subjects drop out of the study before occurrence of the terminal event of interest. One may then wish to evaluate the relationship between time to dropout and the internal covariate. The Cox model is a standard framework for that purpose. Here, we address this problem in situations where the value of the covariate at dropout is unobserved. We suggest a joint model which combines a first-order Markov model for the longitudinally measured covariate with a time-dependent Cox model for the dropout process. We consider maximum likelihood estimation in this model and show how estimation can be carried out via the EM-algorithm. We state that the suggested joint model may have applications in the context of longitudinal data with nonignorable dropout. Indeed, it can be viewed as generalizing Diggle and Kenward's model (1994) to situations where dropout may occur at any point in time and may be censored. Hence we apply both models and compare their results on a data set concerning longitudinal measurements among patients in a cancer clinical trial.

Algorithms↗

PubMed1999

A pharmacokinetic model for tenidap in normal volunteers and rheumatoid arthritis patients.

PURPOSE: To develop a pharmacokinetic model for tenidap and to identify important relationships between the pharmacokinetic parameters and available covariates. METHODS: Plasma concentration data from several phase I and phase II studies were used to develop a pharmacokinetic model for tenidap, a novel anti-rheumatic drug. An appropriate pharmacokinetic model was selected on the basis of individual nonlinear regression analyses and an EM algorithm was used to perform a nonlinear mixed-effects analysis. Scatter plots of posterior individual pharmacokinetic parameters were used to identify possible covariate effects. RESULTS: Predicted responses were in good agreement with the observed data. A bi-exponential model with zero order absorption was subsequently used to develop the mixed-effects model. Covariate relationships selected on the basis of differences in the objective function, although statistically significant, were not particularly strong. CONCLUSIONS: The pharmacokinetics of tenidap can be described by a bi-exponential model with zero order absorption. Based on differences in the log-likelihood, significant covariate-parameter relationships were identified between smoking and CL, and between gender and Vss and CLd. Simulated sparse data analyses indicated that the model would be robust for the analysis of sparse data generated in observational studies.

Adult↗

PubMed2002

An additive genetic gamma frailty model for linkage analysis of diseases with variable age of onset using nuclear families.

Many late-onset complex diseases exhibit variable age of onset. Efficiently incorporating age of onset information into linkage analysis can potentially increase the power of dissecting complex diseases. In this paper, we treat age of onset as a genetic trait with censored observations. We use multiple markers to infer the inheritance vector at the disease susceptibility (DS) locus in order to extract information about the inheritance pattern of the disease allele in a pedigree. Given the inheritance distribution at the DS locus, we define the genetic frailty for each individual within a nuclear family as the sum of frailties due to a putative major disease gene and a polygenic effect due to any remaining DS loci. Conditioning on these frailties we use the proportional hazards model for the risk of developing disease. We show that a test of linkage can be formulated as a test of zero variance due to a specific locus of the additive gamma frailties. Maximum likelihood estimation, using the EM algorithm, and likelihood ratio tests are employed for parameter estimation and tests of linkage. A simulation study presented indicates that the proposed method is well behaved and can be more powerful than the currently available allele-sharing based linkage methods. A breast cancer data example is used for illustration.

Adult↗

PubMed2003

Maximum penalized likelihood estimation in a gamma-frailty model.

The shared frailty models allow for unobserved heterogeneity or for statistical dependence between observed survival data. The most commonly used estimation procedure in frailty models is the EM algorithm, but this approach yields a discrete estimator of the distribution and consequently does not allow direct estimation of the hazard function. We show how maximum penalized likelihood estimation can be applied to nonparametric estimation of a continuous hazard function in a shared gamma-frailty model withright-censored and left-truncated data. We examine the problem of obtaining variance estimators for regression coefficients, the frailty parameter and baseline hazard functions. Some simulations for the proposed estimation procedure are presented. A prospective cohort (Paquid) with grouped survival data serves to illustrate the method which was used to analyze the relationship between environmental factors and the risk of dementia.

Algorithms↗

PubMed2003

Correcting the bias in estimation of genetic variances contributed by individual QTL.

In addition to locating chromosomal positions of quantitative trait loci (QTL), estimating the sizes of identified QTL is also an important component in QTL mapping. The size of a QTL is usually measured by the proportion of the phenotypic variance contributed by the QTL. However, the genetic variance may be overestimated in a small line crossing experiment. In this study, we investigate this bias and develop a simple method to correct the bias. The bias correction, however, requires the error of the estimated genetic effect, which is not trivial if the genetic effect is estimated using the Expectation and Maximization (EM) algorithm. Therefore, we also develop a simple method to estimate the standard error of the estimated genetic effect, which is subsequently used to correct the bias in the variance estimate.

Algorithms↗

PubMed2002

Two exonic single nucleotide polymorphisms in the microsomal epoxide hydrolase gene are jointly associated with preeclampsia.

This study determined whether genetic variability in exons 3 and 4 of the microsomal epoxide hydrolase gene jointly modifies individual preeclampsia risk. The study also determined whether genetic variability in the gene encoding for microsomal epoxide hydrolase (EPHX) contributes to individual differences in susceptibility to the development of preeclampsia. The study involved 133 preeclamptic and 115 healthy control pregnant women who were genotyped for two single nucleotide polymorphisms (SNPs), T-->C (Tyr113His) in exon 3 and A-->G (His139Arg) in exon 4, in the EPHX gene. Chi-square analysis was used to assess genotype and allele frequency differences between the preeclamptic and control groups. In addition, single-point analysis was expanded to pair of loci haplotype analysis to examine the estimated haplotype frequencies of the two SNPs, of unknown phase, among the preeclamptic and control groups. Estimated haplotype frequencies were assessed using the maximum-likelihood method, employing an expectation-maximization (EM) algorithm. Single-point allele and genotype distributions in exons 3 and 4 of the EPHX gene were not statistically different between the groups. However, according to the haplotype estimation analysis, we observed a significantly elevated frequency of haplotype T-A (Tyr113-His139) among the preeclampsia group vs the control group (P=0.01). The odds ratio for preeclampsia associated with the high-activity haplotype T-A (Tyr113-His139) was 1.61 (95% CI: 1.12-2.32). The use of two intragenic SNPs jointly in haplotype analysis of association demonstrated that the genetically determined high-activity haplotype T-A (Tyr113-His139) was significantly associated with preeclampsia.

Adult↗

PubMed2003

Genetic variability of the marine mussel Mytilus galloprovincialis assessed using two-dimensional electrophoresis.

Two-dimensional electrophoresis (2-DE) has been used to measure the degree of genetic variability of the marine mussel Mytilus galloprovincialis. Genetic polymorphisms were detected in 33 of a total of 86 polypeptides scored among the most abundant proteins from foot samples in 38 individuals. Estimates of average heterozygosity were 0.101+/-0.018 and 0.114+/-0.021 in a natural and a cultured population, respectively, from the NW of the Iberian Peninsula. These are the highest estimates of average heterozygosity reported by 2-DE in an animal species to date. We consider that these data throw open the question of the level of genetic variability detectable by two-dimensional electrophoresis. Multilocus genotype data were used to infer haplotypic frequencies by means of the EM algorithm in order to detect linkage disequilibrium between loci coding abundant proteins. Significant associations were found in 22.7% of the 406 two-locus pairs analysed. Also, clusters of loci in which all pairwise combinations exhibit statistically significant associations were detected and physical linkage between some of these loci is postulated from the linkage disequilibrium data.

Animals↗

PubMed1999

The additive genetic gamma frailty model for linkage analysis of age-of-onset variation.

Age of onset is a key factor in the linkage analysis of many complex diseases. Current methods in nonparametric linkage analysis are mainly concentrated on the affected relative pairs or affected family members with age of onset information either ignored or taken into account by specifying age-dependent penetrances for liability classes. On the other hand, gamma frailty models were developed in the biostatistics literature to model familial aggregation of age of onset. However, these frailty models cannot be used directly for linkage analysis. This paper extends the gamma frailty model by incorporating inheritance vector information and provides a semiparametric approach for linkage testing. For a given inheritance vector at the putative disease locus, we construct an additive genetic gamma frailty for each individual within a nuclear family and use the Cox proportional hazard model to model age of onset. We derive the conditional hazard ratio parameter for sib pairs and define a likelihood ratio based LOD score statistic under our model. The EM algorithm is used for estimating the parameters and the maximum likelihood functions. Simulated data sets are used to illustrate these new statistical methods.

Age Factors↗

PubMed2003

Accelerated gene counting for haplotype frequency estimation.

Current implementations of the EM algorithm for estimating haplotype frequencies from genotypes on proximal loci require computational resources that grow as nh2k, where n is the number of individuals genotyped and h is the number of haplotypes possible on k loci. For diallelic loci hk=2k. We present an approach whose computational requirement grows as n2t where t is the largest number of loci at which an individual in the sample is heterozygous. The method is illustrated by haplotype frequency estimation from a sample of 45 individuals genotyped at 26 single nucleotide polymorphisms in the PIK3R1 gene.

1-Phosphatidylinositol 4-Kinase↗

PubMed2003

Association of single nucleotide polymorphisms of the bile salt export pump gene with intrahepatic cholestasis of pregnancy.

BACKGROUND: We determined whether genetic variability in the gene encoding the bile salt export pump (BSEP) contributes to individual differences in susceptibility to the development of intrahepatic cholestasis of pregnancy (ICP). METHODS: The study involved 57 affected and 115 healthy control pregnant women who were genotyped for two single nucleotide polymorphisms (SNPs) in the BSEP gene. Chi-square analysis was used to assess genotype and allele frequency differences between the cholestatic and control groups. In addition, single locus analysis was expanded to pair of loci haplotype analysis to examine the estimated haplotype frequencies of the two SNPs, of unknown phase, among the cholestatic and control groups. Estimated haplotype frequencies were assessed using the maximum-likelihood method, employing an expectation-maximization (EM) algorithm. RESULTS: The genotype and allele frequency distribution of the two intragenic SNPs in the ICP and control groups revealed significant evidence of association with the exon 28 SNP (P=0.04 and P=0.02, respectively). In addition, a borderline allele association was noted with the intron 19 SNP (P=0.08). Although the overall distribution of estimated haplotypes of intron 19 and exon 28 SNPs did not differ between the ICP and control groups, the most common haplotype, A-G, was significantly overrepresented in the ICP group (P=0.02), at an odds ratio of 1.73 (95% CI: 1.08-2.74). CONCLUSIONS: The use of two intragenic SNPs in both single locus and haplotype analyses of association suggests that the BSEP gene is a susceptibility gene in intrahepatic cholestasis of pregnancy.

ATP Binding Cassette Transporter, Subfamily B, Mem↗

PubMed1995

Estimation of the shelf-life of drugs with mixed effects models.

This paper proposes a normal mixed effects model for stability analysis. An EM algorithm is developed to compute the maximum likelihood estimates of regression coefficients of the fixed effects and random effects, and variance components. The likelihood ratio test is used for the preliminary testing of batch-to-batch variation. An example from a marketing stability study is given to illustrate the proposed procedure.

Algorithms↗

PubMed2003

A likelihood-based, counterfactual approach to accounting for treatment failures in clinical trials.

Consider a two-armed, placebo-controlled trial in which subjects may experience treatment failure. For ethical reasons, it is necessary to administer emergency or rescue medications for such subjects. However, the emergency medications may bias the set of response measurements. When analyzing the data from clinical trials, the standard approach is to perform an intent-to-treat (ITT) analysis, wherein the data are analyzed according to treatment assignment. Secondary statistical analyses that supplement the ITT analysis can be performed to account for the impact of treatment failures and emergency medications. A likelihood-based, counterfactual approach to supplemental analyses uses the expectation maximization (EM) algorithm for parameter estimation and a likelihood ratio test to test the equality of the placebo and experimental treatment means for subjects who would not fail under either treatment assignment. A simulation study is performed to assess the operating characteristics of the likelihood ratio test. An example from the Asthma Clinical Research Network (ACRN) is used to draw comparisons between the standard ITT procedure and the developed supplemental analysis.

Clinical Trials as Topic↗

PubMed2000

Segregation analysis in Shwachman-Diamond syndrome: evidence for recessive inheritance.

Shwachman-Diamond syndrome is a rare disorder of unknown cause. Reports have indicated the occurrence of affected siblings, but formal segregation analysis has not been performed. In families collected for genetic studies, the mean paternal age and mean difference in parental ages were found to be consistent with the general population. We determined estimates of segregation proportion in a cohort of 84 patients with complete sibship data under the assumption of complete ascertainment, using the Li and Mantel estimator, and of single ascertainment with the Davie modification. A third estimate was also computed with the expectation-maximization (EM) algorithm. All three estimates supported an autosomal recessive mode of inheritance, but complete ascertainment was found to be unlikely. Although there are no overt signs of disease in adult carriers (parents), the use of serum trypsinogen levels to indicate exocrine pancreatic dysfunction was evaluated as a potential measure for heterozygote expression. No consistent differences were found in levels between parents and a normal control population. Although genetic heterogeneity cannot be excluded, our results indicate that simulation and genetic analyses of Shwachman-Diamond syndrome should consider a recessive model of inheritance.

Abnormalities, Multiple↗

PubMed1991

Parallel simulated annealing for emission tomography.

A method for implementing simulated annealing in parallel to speed up the execution of emission tomography (ET) image reconstruction is presented. A high degree of parallelism can be attained by using a parallel-acceptance partitioning strategy, in which perturbations to subsets of the estimate are evaluated in parallel. However because the point spread function in ET imaging systems is globally dependent, processors cannot update the current estimate independently. Consequently, processors must be synchronized each time a perturbation is accepted to avoid introducing error. This can produce excessive communications overhead, especially when the acceptance rate is high. In this paper an energy function is constructed to reduce the synchronization requirements by using a reformulation of the log-likelihood function from the expectation maximization (EM) algorithm. The approach is to change the global dependence in the energy function from the current estimate to the estimate generated during the last iteration. The synchronization requirements for guaranteed convergence are then significantly reduced from once per acceptance to once per iteration. This parallel implementation on 54 Inmos T800 transputers connected in a ring topology resulted in execution times that were almost 50 times faster than on a VAX 8600.

Algorithms↗

PubMed1992

Truncation artifact suppression in cone-beam radionuclide transmission CT using maximum likelihood techniques: evaluation with human subjects.

Transverse image truncation can be a serious problem for human imaging using cone-beam transmission CT (CB-CT) implemented on a conventional rotating gamma camera. If this problem can be solved, CB-CT will be useful for attenuation compensation of SPECT images. This paper presents a reconstruction method to reduce or eliminate the artifacts resulting from the truncation. The method uses a previously published transmission maximum likelihood EM algorithm, adapted to the cone-beam geometry. The reconstruction method is evaluated qualitatively using three human subjects of various dimensions and various degrees of truncation. For the two smaller subjects, with moderate truncation, the maximum likelihood method is very successful, nearly eliminating the artifacts seen with conventional filtered backprojection of truncated geometries. The use of an expanded reconstructed space, which contains the entire transverse slice of the subject, is necessary for optimal truncation removal. For the largest subject investigated, the truncation was substantial, and the artifacts were only partially removed by the maximum likelihood reconstruction. Nonetheless, the images were qualitatively superior to those obtained with filtered backprojection. An added elliptical support prior moderately increased the rate of convergence, and helped to force a reasonable body contour.

Adult↗

PubMed1993

The convergence of object dependent resolution in maximum likelihood based tomographic image reconstruction.

Study of the maximum likelihood by EM algorithm (ML) with a reconstruction kernel equal to the intrinsic detector resolution and sieve regularization has demonstrated that any image improvements over filtered backprojection (FBP) are a function of image resolution. Comparing different reconstruction algorithms potentially requires measuring and matching the image resolution. Since there are no standard methods for describing the resolution of images from a nonlinear algorithm such as ML, we have defined measures of effective local Gaussian resolution (ELGR) and effective global Gaussian resolution (EGGR) and examined their behaviour in FBP images and in ML images using two different measurement techniques. For FBP these two resolution measures are equal and exhibit the standard convolution behaviour of linear systems. For ML, the FWHM of the ELGR monotonically increased with decreasing Gaussian object size due to slower convergence rates for smaller objects. For the simple simulated phantom used, this resolution dependence is independent of object position. With increasing object size, number of iterations and sieve size the object size dependence of the ELGR decreased. The FWHM of the EGGR converged after approximately 200 iterations, masking the fact that the ELGR for small objects was far from convergence. When FBP is compared to a nonlinear algorithm such as ML, it is recommended that at least the EGGR be matched; for ML this requires more than the number of iterations (e.g., < 100) that are typically run to minimize the mean square error or to satisfy a feasibility or similar stopping criterion. For many tasks, matching the EGGR of ML to FBP images may be insufficient and >> 200 iterations may be needed, particularly for small objects in the ML image because their ELGR has not yet converged.

Algorithms↗

PubMed1995

A Monte Carlo investigation of dual-energy-window scatter correction for volume-of-interest quantification in 99Tcm SPECT.

Using Monte Carlo simulation of 99Tcm single-photon-emission computed tomography (SPECT), we investigate the effects of tissue-background activity, tumour location, patient size, uncertainty of energy windows, and definition of tumour region on the accuracy of quantification. The dual-energy-window method of correction for Compton scattering is employed and the multiplier which yields correct activity for the VI as a whole calculated. The model is usually a sphere containing radioactive water located within a cylinder filled with a more dilute solution of radioactivity. Two simulation codes are employed. Reconstruction is by ML-EM algorithm with attenuation compensation. The scatter multiplier depends only slightly on the sphere location or the cylinder diameter. It also depends little on whether correction is before or after reconstruction. At low background level, it changes with VOI size, but not at higher background. For a geometrical VOI, it is 1.25 at zero background, decreases sharply to 0.56 for equal concentrations, and is 0.44 when the background concentration is very large. Quantification is accurate (less than 9% error) if the test background is reasonably close to that used in setting the universal scatter-multiplier value, or if the test backgrounds are always large and so is the universal-value background, but not if the test backgrounds cover a large range of values including zero. Results largely agree with those from experiment after the experimental data with background is re-evaluated with prejudice.

Humans↗

PubMed1997

Validation of the central-ray approximation for attenuated depth-dependent convolution in quantitative SPECT reconstruction.

In order to model photon attenuation and detector resolution variation as a depth-dependent convolution for efficient reconstruction of quantitative SPECT, a central-ray approximation is necessary. This work investigates the impact of the approximation upon reconstruction accuracy and computational efficiency. A patient chest CT image was acquired and converted into an object-specific attenuation map. From a segmentation of the map, an emission thorax phantom was constructed with a cardiac insert. To generate a system-specific resolution-variant kernal, a point source was measured at several depths from the surface of a low-energy, high-resolution, parallel-hole collimator of a SPECT system. Projections of parallel-beam geometry were simulated from the phantom, the map, and the kernel on an elliptical orbit. Reconstruction was performed by the ML-EM algorithm with and without the central-ray approximation. The approximation cuts down dramatically (more than 100 fold) the computing time with a negligible loss (less than 1%) of reconstruction accuracy.

Algorithms↗