The normal distribution in some constrained sample spaces

Phenomena with a constrained sample space appear frequently in practice. This is the case e.g. with strictly positive data, or with compositional data, like percentages or proportions. If the natural measure of difference is not the absolute one, simple algebraic properties show that it is more convenient to work with a geometry different from the usual Euclidean geometry in real space, and with a measure different from the usual Lebesgue measure, leading to alternative models which better fit the phenomenon under study. The general approach is presented and illustrated using the normal distribution, both on the positive real line and on the D-part simplex. The original ideas of McAlister in his introduction to the lognormal distribution in 1879, are recovered and updated

Cannabis use and other illicit drug use: Do subjective experiences during first cannabis use increase the probability of using illicit drug use?

Background and aims: Few studies have examined whether subjective experiences during first cannabis use are related to other illicit drug (OID) use. This study investigated this topic. Methods: Baseline data from a representative sample of young Swiss men was obtained from an ongoing Cohort Study on Substance Use Risk Factors (N ¼ 5753). Logistic regressions were performed to examine the relationships between cannabis use and of subjective experiences during first cannabis use with 15 OID. Results: Positive experiences increased the likelihood of using hallucinogens (hallucinogens, salvia divinorum, spice; p50.015), stimulants (speed, ecstasy, cocaine, amphetamines/methamphetamines; p50.006) and also poppers, research chemicals, GHB/GBL, and crystal meth (p50.049). Sniffed drugs (poppers, solvents for sniffing) and ''hard'' drugs (heroin, ketamine, research chemicals, GHB/GBL and crystal meth) were more likely to be used by participants who experienced negative feelings on first use of cannabis (p50.034). Conclusion: Subjective feelings seemed to amplify the association of cannabis with OID. The risk increased for drugs with effects resembling feelings experienced on first cannabis use. Negative experiences should also be a concern, as they were associated with increased risk of using the ''hardest'' illicit drugs.

The normal distribution in some constrained sample spaces

Phenomena with a constrained sample space appear frequently in practice. This is the case e.g. with strictly positive data, or with compositional data, like percentages or proportions. If the natural measure of difference is not the absolute one, simple algebraic properties show that it is more convenient to work with a geometry different from the usual Euclidean geometry in real space, and with a measure different from the usual Lebesgue measure, leading to alternative models which better fit the phenomenon under study. The general approach is presented and illustrated using the normal distribution, both on the positive real line and on the D-part simplex. The original ideas of McAlister in his introduction to the lognormal distribution in 1879, are recovered and updated

Simulation-Based Finite and Large Sample Tests in Multivariate Regressions.

In the context of multivariate linear regression (MLR) models, it is well known that commonly employed asymptotic test criteria are seriously biased towards overrejection. In this paper, we propose a general method for constructing exact tests of possibly nonlinear hypotheses on the coefficients of MLR systems. For the case of uniform linear hypotheses, we present exact distributional invariance results concerning several standard test criteria. These include Wilks' likelihood ratio (LR) criterion as well as trace and maximum root criteria. The normality assumption is not necessary for most of the results to hold. Implications for inference are two-fold. First, invariance to nuisance parameters entails that the technique of Monte Carlo tests can be applied on all these statistics to obtain exact tests of uniform linear hypotheses. Second, the invariance property of the latter statistic is exploited to derive general nuisance-parameter-free bounds on the distribution of the LR statistic for arbitrary hypotheses. Even though it may be difficult to compute these bounds analytically, they can easily be simulated, hence yielding exact bounds Monte Carlo tests. Illustrative simulation experiments show that the bounds are sufficiently tight to provide conclusive results with a high probability. Our findings illustrate the value of the bounds as a tool to be used in conjunction with more traditional simulation-based test methods (e.g., the parametric bootstrap) which may be applied when the bounds are not conclusive.

Exact Nonparametric Two-Sample Homogeneity Tests for Possibly Discrete Distributions.

In this paper, we study several tests for the equality of two unknown distributions. Two are based on empirical distribution functions, three others on nonparametric probability density estimates, and the last ones on differences between sample moments. We suggest controlling the size of such tests (under nonparametric assumptions) by using permutational versions of the tests jointly with the method of Monte Carlo tests properly adjusted to deal with discrete distributions. We also propose a combined test procedure, whose level is again perfectly controlled through the Monte Carlo test technique and has better power properties than the individual tests that are combined. Finally, in a simulation experiment, we show that the technique suggested provides perfect control of test size and that the new tests proposed can yield sizeable power improvements.

A jackknife variance estimator for unequal probability sampling

The jackknife method is often used for variance estimation in sample surveys but has only been developed for a limited class of sampling designs.We propose a jackknife variance estimator which is defined for any without-replacement unequal probability sampling design. We demonstrate design consistency of this estimator for a broad class of point estimators. A Monte Carlo study shows how the proposed estimator may improve on existing estimators.

Adjusted jackknife for imputation under unequal probability sampling without replacement

Imputation is commonly used to compensate for item non-response in sample surveys. If we treat the imputed values as if they are true values, and then compute the variance estimates by using standard methods, such as the jackknife, we can seriously underestimate the true variances. We propose a modified jackknife variance estimator which is defined for any without-replacement unequal probability sampling design in the presence of imputation and non-negligible sampling fraction. Mean, ratio and random-imputation methods will be considered. The practical advantage of the method proposed is its breadth of applicability.

Probability density function estimation using orthogonal forward regression

Using the classical Parzen window estimate as the target function, the kernel density estimation is formulated as a regression problem and the orthogonal forward regression technique is adopted to construct sparse kernel density estimates. The proposed algorithm incrementally minimises a leave-one-out test error score to select a sparse kernel model, and a local regularisation method is incorporated into the density construction process to further enforce sparsity. The kernel weights are finally updated using the multiplicative nonnegative quadratic programming algorithm, which has the ability to reduce the model size further. Except for the kernel width, the proposed algorithm has no other parameters that need tuning, and the user is not required to specify any additional criterion to terminate the density construction procedure. Two examples are used to demonstrate the ability of this regression-based approach to effectively construct a sparse kernel density estimate with comparable accuracy to that of the full-sample optimised Parzen window density estimate.

Estimating reliability and resolution of probability forecasts through decomposition of the empirical score

References (20)Cited By (1)Export CitationAboutAbstract Proper scoring rules provide a useful means to evaluate probabilistic forecasts. Independent from scoring rules, it has been argued that reliability and resolution are desirable forecast attributes. The mathematical expectation value of the score allows for a decomposition into reliability and resolution related terms, demonstrating a relationship between scoring rules and reliability/resolution. A similar decomposition holds for the empirical (i.e. sample average) score over an archive of forecast–observation pairs. This empirical decomposition though provides a too optimistic estimate of the potential score (i.e. the optimum score which could be obtained through recalibration), showing that a forecast assessment based solely on the empirical resolution and reliability terms will be misleading. The differences between the theoretical and empirical decomposition are investigated, and specific recommendations are given how to obtain better estimators of reliability and resolution in the case of the Brier and Ignorance scoring rule.

Forecast encompassing tests and probability forecasts

We consider tests of forecast encompassing for probability forecasts, for both quadratic and logarithmic scoring rules. We propose test statistics for the null of forecast encompassing, present the limiting distributions of the test statistics, and investigate the impact of estimating the forecasting models' parameters on these distributions. The small-sample performance is investigated, in terms of small numbers of forecasts and model estimation sample sizes. We show the usefulness of the tests for the evaluation of recession probability forecasts from logit models with different leading indicators as explanatory variables, and for evaluating survey-based probability forecasts.

Small-Sample Corrections for Score Tests in Birnbaum-Saunders Regressions

In this article, we deal with the issue of performing accurate small-sample inference in the Birnbaum-Saunders regression model, which can be useful for modeling lifetime or reliability data. We derive a Bartlett-type correction for the score test and numerically compare the corrected test with the usual score test and some other competitors.

Use of the probability of stone formation (PSF) score to assess stone forming risk and treatment response in a cohort of Brazilian stone formers

Introduction: The aim was to confirm that PSF (probability of stone formation) changed appropriately following medical therapy on recurrent stone formers.Materials and Methods: Data were collected on 26 Brazilian stone-formers. A baseline 24-hour urine collection was performed prior to treatment. Details of the medical treatment initiated for stone-disease were recorded. A PSF calculation was performed on the 24 hour urine sample using the 7 urinary parameters required: voided volume, oxalate, calcium, urate, pH, citrate and magnesium. A repeat 24-hour urine sample was performed for PSF calculation after treatment. Comparison was made between the PSF scores before and during treatment.Results: At baseline, 20 of the 26 patients (77%) had a high PSF score (> 0.5). Of the 26 patients, 17 (65%) showed an overall reduction in their PSF profiles with a medical treatment regimen. Eleven patients (42%) changed from a high risk (PSF > 0.5) to a low risk (PSF < 0.5) and 6 patients reduced their risk score but did not change risk category. Six (23%) patients remained in a high risk category (> 0.5) during both assessments.Conclusions: The PSF score reduced following medical treatment in the majority of patients in this cohort.

Gender differences in drinking patterns and alcohol-related problems in a community sample in Sao Paulo, Brazil

OBJECTIVE: To investigate drinking patterns and gender differences in alcohol-related problems in a Brazilian population, with an emphasis on the frequency of heavy drinking. METHODS: A cross-sectional study was conducted with a probability adult household sample (n = 1,464) in the city of Sao Paulo, Brazil. Alcohol intake and ICD-10 psychopathology diagnoses were assessed with the Composite International Diagnostic Interview 1.1. The analyses focused on the prevalence and determinants of 12-month non-heavy drinking, heavy episodic drinking (4-5 drinks per occasion), and heavy and frequent drinking (heavy drinking at least 3 times/week), as well as associated alcohol-related problems according to drinking patterns and gender. RESULTS: Nearly 22% (32.4% women, 8.7% men) of the subjects were lifetime abstainers, 60.3% were non-heavy drinkers, and 17.5% reported heavy drinking in a 12-month period (26.3% men, 10.9% women). Subjects with the highest frequency of heavy drinking reported the most problems. Among subjects who did not engage in heavy drinking, men reported more problems than did women. A gender convergence in the amount of problems was observed when considering heavy drinking patterns. Heavy and frequent drinkers were twice as likely as abstainers to present lifetime depressive disorders. Lifetime nicotine dependence was associated with all drinking patterns. Heavy and frequent drinking was not restricted to young ages. CONCLUSIONS: Heavy and frequent episodic drinking was strongly associated with problems in a community sample from the largest city in Latin America. Prevention policies should target this drinking pattern, independent of age or gender. These findings warrant continued research on risky drinking behavior, particularly among persistent heavy drinkers at the non-dependent level.

Parameter recovery for a skew-normal IRT model under a Bayesian approach: hierarchical framework, prior and kernel sensitivity and sample size

Item response theory (IRT) comprises a set of statistical models which are useful in many fields, especially when there is an interest in studying latent variables (or latent traits). Usually such latent traits are assumed to be random variables and a convenient distribution is assigned to them. A very common choice for such a distribution has been the standard normal. Recently, Azevedo et al. [Bayesian inference for a skew-normal IRT model under the centred parameterization, Comput. Stat. Data Anal. 55 (2011), pp. 353-365] proposed a skew-normal distribution under the centred parameterization (SNCP) as had been studied in [R. B. Arellano-Valle and A. Azzalini, The centred parametrization for the multivariate skew-normal distribution, J. Multivariate Anal. 99(7) (2008), pp. 1362-1382], to model the latent trait distribution. This approach allows one to represent any asymmetric behaviour concerning the latent trait distribution. Also, they developed a Metropolis-Hastings within the Gibbs sampling (MHWGS) algorithm based on the density of the SNCP. They showed that the algorithm recovers all parameters properly. Their results indicated that, in the presence of asymmetry, the proposed model and the estimation algorithm perform better than the usual model and estimation methods. Our main goal in this paper is to propose another type of MHWGS algorithm based on a stochastic representation (hierarchical structure) of the SNCP studied in [N. Henze, A probabilistic representation of the skew-normal distribution, Scand. J. Statist. 13 (1986), pp. 271-275]. Our algorithm has only one Metropolis-Hastings step, in opposition to the algorithm developed by Azevedo et al., which has two such steps. This not only makes the implementation easier but also reduces the number of proposal densities to be used, which can be a problem in the implementation of MHWGS algorithms, as can be seen in [R.J. Patz and B.W. Junker, A straightforward approach to Markov Chain Monte Carlo methods for item response models, J. Educ. Behav. Stat. 24(2) (1999), pp. 146-178; R. J. Patz and B. W. Junker, The applications and extensions of MCMC in IRT: Multiple item types, missing data, and rated responses, J. Educ. Behav. Stat. 24(4) (1999), pp. 342-366; A. Gelman, G.O. Roberts, and W.R. Gilks, Efficient Metropolis jumping rules, Bayesian Stat. 5 (1996), pp. 599-607]. Moreover, we consider a modified beta prior (which generalizes the one considered in [3]) and a Jeffreys prior for the asymmetry parameter. Furthermore, we study the sensitivity of such priors as well as the use of different kernel densities for this parameter. Finally, we assess the impact of the number of examinees, number of items and the asymmetry level on the parameter recovery. Results of the simulation study indicated that our approach performed equally as well as that in [3], in terms of parameter recovery, mainly using the Jeffreys prior. Also, they indicated that the asymmetry level has the highest impact on parameter recovery, even though it is relatively small. A real data analysis is considered jointly with the development of model fitting assessment tools. The results are compared with the ones obtained by Azevedo et al. The results indicate that using the hierarchical approach allows us to implement MCMC algorithms more easily, it facilitates diagnosis of the convergence and also it can be very useful to fit more complex skew IRT models.

Gender differences in drinking patterns and alcohol-related problems in a community sample in São Paulo, Brazil

OBJECTIVE: To investigate drinking patterns and gender differences in alcohol-related problems in a Brazilian population, with an emphasis on the frequency of heavy drinking. METHODS: A cross-sectional study was conducted with a probability adult household sample (n = 1,464) in the city of São Paulo, Brazil. Alcohol intake and ICD-10 psychopathology diagnoses were assessed with the Composite International Diagnostic Interview 1.1. The analyses focused on the prevalence and determinants of 12-month nonheavy drinking, heavy episodic drinking (4-5 drinks per occasion), and heavy and frequent drinking (heavy drinking at least 3 times/week), as well as associated alcohol-related problems according to drinking patterns and gender. RESULTS: Nearly 22% (32.4% women, 8.7% men) of the subjects were lifetime abstainers, 60.3% were non-heavy drinkers, and 17.5% reported heavy drinking in a 12-month period (26.3% men, 10.9% women). Subjects with the highest frequency of heavy drinking reported the most problems. Among subjects who did not engage in heavy drinking, men reported more problems than did women. A gender convergence in the amount of problems was observed when considering heavy drinking patterns. Heavy and frequent drinkers were twice as likely as abstainers to present lifetime depressive disorders. Lifetime nicotine dependence was associated with all drinking patterns. Heavy and frequent drinking was not restricted to young ages. CONCLUSIONS: Heavy and frequent episodic drinking was strongly associated with problems in a community sample from the largest city in Latin America. Prevention policies should target this drinking pattern, independent of age or gender. These findings warrant continued research on risky drinking behavior, particularly among persistent heavy drinkers at the non-dependent level.