Vine Regression with Bayes Nets: A Critical Comparison with Traditional Approaches Based on a Case Study on the Effects of Breastfeeding on IQ.

Regular vines (R-vines) copulas build high dimensional joint densities from arbitrary one-dimensional margins and (conditional) bivariate copula densities. Vine densities enable the computation of all conditional distributions, though the calculations can be numerically intensive. Saturated continuous nonparametric Bayes nets (CNPBN) are regular vines. Computing regression functions from the vine copula density is termed vine regression. The epicycles of regression-including/excluding covariates, interactions, higher order terms, multicollinearity, model fit, transformations, heteroscedasticity, bias-are dispelled. One simply computes the regressions from the vine copula density. Only the question of finding an adequate vine copula remains. Vine regression is applied to a data set from the National Longitudinal Study of Youth relating breastfeeding to IQ. The expected effects of breastfeeding on IQ depend on IQ, on the baseline level of breastfeeding, on the duration of additional breastfeeding and on the values of other covariates. A child given two weeks breastfeeding can expect to increase his/her IQ by 1.5-2 IQ points by adding 10 weeks of breastfeeding, depending on values of other covariates. A child given two years breastfeeding can expect to gain from 0.48-0.65 IQ points from 10 additional weeks. Adding 10 weeks breastfeeding to each of the 3,179 children in this data set has a net present value $50,700,000 according to the Bayes net, compared to $29,000,000 according to the linear regression.

Copula diagnostics for asymmetries and conditional dependence.

Vine copulas are constructed from a sequence of trees to represent dependence and conditional dependence, and a set of bivariate copulas that are applied to univariate distributions in tree 1 and to conditional univariate distributions in subsequent trees. Diagnostic methods based on measures of dependence and tail asymmetry are proposed to guide the choice of parametric bivariate copula families assigned to the edges of the trees in the vine and to assess whether a copula is constant over the conditioning value(s) for trees 2 and higher. The measures are conditional measures applied to bivariate conditional distributions in trees 2 and higher. If the diagnostic methods suggest the existence of reflection asymmetry, permutation asymmetry and possible asymmetric tail dependence, then three- or four-parameter bivariate copula families might be needed. Moreover, if the conditional dependence measures or asymmetry measures in trees 2 and up are not constant over the conditioning value(s), then non-constant copulas should be considered. We illustrate the use of the diagnostic methods for a gamma factor model and two real datasets. The examples show that better models are attained by using asymmetric and non-constant copulas under the guidance of the diagnostic tools.

Untangling serially dependent underreported count data for gender-based violence.

Underreporting in gender-based violence data is a worldwide problem leading to the underestimation of the magnitude of this social and public health concern. This problem deteriorates the data quality, providing poor and biased results that lead society to misunderstand the actual scope of this domestic violence issue. The present work proposes time series models for underreported counts based on a latent integer autoregressive of order 1 time series with Poisson distributed innovations and a latent underreporting binary state, that is, a first-order Markov chain. Relevant theoretical properties of the models are derived, and the moment-based and maximum-based methods are presented for parameter estimation. The new time series models are applied to the quarterly complaints of domestic violence against women recorded in some judicial districts of Galicia (Spain) between 2007 and 2017. The models allow quantifying the degree of underreporting. A comprehensive discussion is presented, studying how the frequency and intensity of underreporting in this public health concern are related to some interesting socioeconomic and health indicators of the provinces of Galicia (Spain).

Clinical and molecular predictors of mortality in neurofibromatosis 2: a UK national analysis of 1192 patients.

BACKGROUND: Neurofibromatosis 2 (NF2) is an autosomal-dominant tumour predisposition syndrome characterised by bilateral vestibular schwannomas, considerable morbidity and reduced life expectancy. Although genotype-phenotype correlations are well established in NF2, little is known about effects of mutation type or location within the gene on mortality. Improvements in NF2 diagnosis and management have occurred, but their effect on patient survival is unknown. METHODS: We evaluated clinical and molecular predictors of mortality in 1192 patients (771 with known causal mutations) identified through the UK National NF2 Registry. Kaplan-Meier survival and Cox regression analyses were used to evaluate predictors of mortality, with jackknife adjustment of parameter SEs to account for the strong intrafamilial phenotypic correlations that occur in NF2. RESULTS: The study included 241 deaths during 10 995 patient-years of follow-up since diagnosis. Early age at diagnosis and the presence of intracranial meningiomas were associated with increased mortality, and having a mosaic, rather than non-mosaic, NF2 mutation was associated with reduced mortality. Patients with splice-site or missense mutations had lower mortality than patients with truncating mutations (OR 0.459, 95% CI 0.213 to 0.990, and OR 0.196, 95% CI 0.213 to 0.990, respectively). Patients with splice-site mutations in exons 6-15 had lower mortality than patients with splice-site mutations in exons 1-5 (OR 0.333, 95% CI 0.129 to 0.858). The mortality of patients with NF2 diagnosed in more recent decades was lower than that of patients diagnosed earlier. CONCLUSIONS: Continuing advances in molecular diagnosis, imaging and treatment of NF2-associated tumours offer hope for even better survival in the future.

Factor copula models for item response data.

Factor or conditional independence models based on copulas are proposed for multivariate discrete data such as item responses. The factor copula models have interpretations of latent maxima/minima (in comparison with latent means) and can lead to more probability in the joint upper or lower tail compared with factor models based on the discretized multivariate normal distribution (or multidimensional normal ogive model). Details on maximum likelihood estimation of parameters for the factor copula model are given, as well as analysis of the behavior of the log-likelihood. Our general methodology is illustrated with several item response data sets, and it is shown that there is a substantial improvement on existing models both conceptually and in fit to data.

Assessing Approximate Fit in Categorical Data Analysis.

A family of Root Mean Square Error of Approximation (RMSEA) statistics is proposed for assessing the goodness of approximation in discrete multivariate analysis with applications to item response theory (IRT) models. The family includes RMSEAs to assess the approximation up to any level of association of the discrete variables. Two members of this family are RMSEA2, which uses up to bivariate moments, and the full information RMSEAn. The RMSEA2 is estimated using the M2 statistic of Maydeu-Olivares and Joe (2005, 2006), whereas for maximum likelihood estimation, RMSEAn is estimated using Pearson's X(2) statistic. Using IRT models, we provide cutoff criteria of adequate, good, and excellent fit using the RMSEA2. When the data are ordinal, we find a strong linear relationship between the RMSEA2 and the Standardized Root Mean Squared Residual goodness-of-fit index. We are unable to offer cutoff criteria for the RMSEAn as its population values decrease as the number of variables and categories increase.

Empirical development of improved diagnostic criteria for neurofibromatosis 2.

PURPOSE: Four sets of clinical diagnostic criteria have been proposed for neurofibromatosis 2, but all have low sensitivity at the time of initial clinical assessment for the disease among patients with a negative family history who do not present with bilateral vestibular schwannomas. We have empirically developed and tested an improved set of diagnostic criteria that uses current understanding of the natural history and genetic characteristics of neurofibromatosis 2 to increase sensitivity while maintaining very high specificity. METHODS: We used data from the UK Neurofibromatosis 2 Registry and Kaplan-Meier curves to estimate frequencies of clinical features at various ages among patients with or without unequivocal neurofibromatosis 2. On the basis of this analysis, we developed the Baser criteria, a new diagnostic system that incorporates genetic testing and gives more weight to the most characteristic features and to those that occur before 30 years of age. RESULTS: In an independent validation subset of patients with unequivocal neurofibromatosis 2, the Baser criteria increased diagnostic sensitivity to 79% (9-15% greater than previous sets of criteria) while maintaining 100% specificity at the age at onset of the first characteristic sign of neurofibromatosis 2. CONCLUSION: The Baser criteria permit early diagnosis in a greater proportion of patients with neurofibromatosis 2 than previous sets of diagnostic criteria.

Weighted scores method for regression models with dependent data.

There are copula-based statistical models in the literature for regression with dependent data such as clustered and longitudinal overdispersed counts, for which parameter estimation and inference are straightforward. For situations where the main interest is in the regression and other univariate parameters and not the dependence, we propose a "weighted scores method", which is based on weighting score functions of the univariate margins. The weight matrices are obtained initially fitting a discretized multivariate normal distribution, which admits a wide range of dependence. The general methodology is applied to negative binomial regression models. Asymptotic and small-sample efficiency calculations show that our method is robust and nearly as efficient as maximum likelihood for fully specified copula models. An illustrative example is given to show the use of our weighted scores method to analyze utilization of health care based on family characteristics.

Generalized Poisson distribution: the property of mixture of Poisson and comparison with negative binomial distribution.

We prove that the generalized Poisson distribution GP(theta, eta) (eta > or = 0) is a mixture of Poisson distributions; this is a new property for a distribution which is the topic of the book by Consul (1989). Because we find that the fits to count data of the generalized Poisson and negative binomial distributions are often similar, to understand their differences, we compare the probability mass functions and skewnesses of the generalized Poisson and negative binomial distributions with the first two moments fixed. They have slight differences in many situations, but their zero-inflated distributions, with masses at zero, means and variances fixed, can differ more. These probabilistic comparisons are helpful in selecting a better fitting distribution for modelling count data with long right tails. Through a real example of count data with large zero fraction, we illustrate how the generalized Poisson and negative binomial distributions as well as their zero-inflated distributions can be discriminated.

Genotype-phenotype correlations for nervous system tumors in neurofibromatosis 2: a population-based study.

Neurofibromatosis 2 (NF2) is an autosomal dominant disease that is characterized by tumors on the vestibular branch of the VIII cranial nerve, but other types of nervous system tumors usually occur as well. Genotype-phenotype correlations are well documented for overall NF2 disease severity but have not been definitively evaluated for specific types of non-VIII nerve tumors. We evaluated genotype-phenotype correlations for various types of non-VIII nerve tumors in 406 patients from the population-based United Kingdom NF2 registry, using regression models with the additional covariates of current age and type of treatment center (specialty or nonspecialty). The models also permitted consideration of intrafamilial correlation. We found statistically significant genotype-phenotype correlations for intracranial meningiomas, spinal tumors, and peripheral nerve tumors. People with constitutional NF2 missense mutations, splice-site mutations, large deletions, or somatic mosaicism had significantly fewer tumors than did people with constitutional nonsense or frameshift NF2 mutations. In addition, there were significant intrafamilial correlations for intracranial meningiomas and spinal tumors, after adjustment for the type of constitutional NF2 mutation. The type of constitutional NF2 mutation is an important determinant of the number of NF2-associated intracranial meningiomas, spinal tumors, and peripheral nerve tumors.

Analysis of neurofibromatosis 1 (NF1) lesions by body segment.

Café-au-lait spots and neurofibromas are defining features of neurofibromatosis 1 (NF1), but they vary greatly in number, size, and clinical importance from patient to patient. The cause of this variability is unknown. We tested the hypotheses that development of these lesions is influenced by local or familial factors. The presence or absence of café-au-lait spots, cutaneous neurofibromas, and diffuse plexiform neurofibromas was recorded for each of 10 divisions of the body surface in 547 NF1 patients, including 117 affected individuals in 52 families. We used stratified Mantel-Haenszel tests to look for local associations between the presence of diffuse plexiform neurofibromas, cutaneous neurofibromas, and café-au-lait spots in individual body segments of NF1 patients. We used a random effects model to obtain intrafamilial correlation coefficients for the age-adjusted number of body divisions affected with each of the three lesions. No significant association was observed between the occurrence of cutaneous and diffuse plexiform neurofibromas, between café-au-lait spots and cutaneous neurofibromas, or between café-au-lait spots and plexiform neurofibromas in the same body segment. The correlation among relatives in the number of body segments affected with café-au-lait spots was 0.45 (95% confidence interval [CI] = 0.18-0.71), with cutaneous neurofibromas, 0.37 (95% CI = 0.15-0.55), and with plexiform neurofibromas, 0.35 (95% CI = 0.15-0.57). We conclude that the development of café-au-lait spots, cutaneous neurofibromas, and plexiform neurofibromas are spatially independent in NF1 patients but that the development of all three lesions is influenced by familial factors.

Exploring the "two-hit hypothesis" in NF2: tests of two-hit and three-hit models of vestibular schwannoma development.

Neurofibromatosis 2 (NF2) is a genetic disease that occurs in approximately 1 in 40,000 live births. Almost all affected individuals develop bilateral tumors of Schwann cells that surround the vestibular nerves; these tumors are known as vestibular schwannomas (VS). Evidence from molecular genetic studies suggests that at least two mutations are involved in formation of VS in patients with NF2. Several authors proposed probabilistic models for this process in other tumors, and showed that such models are consistent with incidence data. We evaluated two different probabilistic models for a "2-hit" hypothesis for VS development in NF2 patients, and we present results from fitting these models to incidence data. Molecular evidence does not exclude the possibility that additional hits are necessary for the development of VS, and we also assessed a "3-hit" model for tumor formation. The "3-hit" model fits the data marginally better than one of the "2-hit" models and much better than the other "2-hit" model. Our findings suggest that more than two mutations may be necessary for VS development in NF2 patients.

Predictors of the risk of mortality in neurofibromatosis 2.

To evaluate clinical and molecular predictors of the risk of mortality in people with neurofibromatosis 2 (NF2), we analyzed the mortality experience of 368 patients from 261 families in the United Kingdom NF2 registry, using the Cox proportional-hazards model and the jackknife method. Age at diagnosis, intracranial meningiomas, and type of treatment center were informative predictors of the risk of mortality. In Cox models, the relative risk of mortality increased 1.13-fold per year decrease in age at diagnosis (95% confidence interval [CI] 1.08-1.18) and was 2.51-fold greater in people with meningiomas compared with those without meningiomas (95% CI 1.38-4.57). The relative risk of mortality in patients treated at specialty centers was 0.34 compared with those treated at nonspecialty centers (95% CI 0.12-0.98). In a separate model, the relative risk of mortality in people with constitutional NF2 missense mutations was very low compared with those with other types of mutations (nonsense or frameshift mutations, splice-site mutations, and large deletions), but the CI could not be well quantified because there was only one death among people with missense mutations. We conclude that age at diagnosis, the strongest single predictor of the risk of mortality, is a useful index for patient counseling and clinical management (as are intracranial meningiomas). To ensure optimal care, we recommend that people with NF2 be referred to specialty treatment centers.

Multivariate Extreme Value Distributions and Coverage of Ranking Probabilities.

The main result in this paper is that the class of multivariate extreme value distributions, when used as random utility models, can approximate all ranking probability distributions. This extends Theorem 1 of Dagsvik (1995). Copyright 2001 Academic Press.