Approximate Likelihood Ratio Test Research Articles

A statistical approach to urban runoff pollution modelling

In evaluating the present or future state of integrated urban water systems, sewer drainage models are often used in calculating the expected return periods of given detrimental acute pollution events and the uncertainty thereof. In the present study an urban runoff pollution model is developed for a Dutch combined sewer catchment. The water quantity model consists, in its final shape, of a combination of linear reservoirs in series and some physical characteristics of the sewer system. Attached to the water quantity model, a model was developed to simulate the suspended chemical oxygen demand (COD) concentration at the point of combined sewer overflow (CSO). The model structure has been developed on the basis of a rainfall input series and a water level and a suspended COD concentration output series. Physically interpretable parameters have been included where this could be statistically justified by the available observation. Model development has been based on an approximate likelihood ratio test, tests for parameter significance, parameter correlation and information criteria.It has been possible to formulate a model to describe the rainfall-runoff process at the Loenen catchment with a coefficient of determination (R2) of 0.83 for the water level at the overflow weir. The best model contained eight estimated parameters of which six were directly interpretable in traditional urban drainage terms. A three parameter suspended COD concentration model was developed with an R2 of 0.44. Though considerably less successful than the water quantity modelling, the results are interesting when viewed in relation to the small number of model parameters. Furthermore the model structure can be linked to existing resuspension theory involving a critical shear stress.

A model for seasonal variation of rainfall at Adelaide and Turen

A rigorous statistical methodology is given to estimate the coefficients of annually periodic functions representing the parameters of a continuous time rainfall model. The methodology is applied to the Rectangular Pulses Poisson model (RPPM) which simulates rainfall occurrences and rainfall amounts in continuous time. Two types of periodic functions, Fourier series and periodic quadratic polynomial splines, are used to represent the seasonal variation of the rainfall model parameters at two locations, Adelaide (Australia) and Turen (Venezuela). The coefficients of the periodic functions representing each model parameter are estimated by minimising a weighted residual sum of squares between observed and theoretical statistics of daily data. The numbers of coefficients of the periodic functions are selected by applying successive approximate likelihood ratio tests, by which the number of required coefficients is increased until no further improvement is gained with respect to a previous fit. Comparisons between the goodness of fit and numbers of selected coefficients show marginally superior performance of the periodic quadratic polynomial splines in comparison with Fourier Series for this particular rainfall model. The methodology is intended to provide an efficient procedure to parameterize the seasonal variability of rainfall data with the smallest number of coefficients, by reducing the original number of degrees of freedom used in the estimation procedure. The coefficients of the periodic functions are amenable to spatial interpolation and interpolated values can be used to simulate rainfall at any point of a particular region, for more detailed climatic impact assessment analyses.

Multivariate tests for multiple endpoints in clinical trials

Clinical trials frequently lack a single definitive endpoint that completely describes treatment efficacy. When a treatment affects a disease in a multitude of ways, several endpoints are necessary to describe efficacy. There is a variety of statistical procedures to provide a single p-value when a treatment affects several endpoints. This paper reviews several procedures including Hotelling's T2 test, an approximate likelihood ratio test (Tang et al.), the weighted version of O'Brien's test, tests involving the maximum of several test statistics, and a test based on the average of the maximum of several endpoints (Wittes). I propose a risk score test whose rejection boundary corresponds to a contour of constant risk. Calculations and simulation studies help to compare the different tests with an emphasis on the effect of non-standard alternatives, and on identifying settings where some tests may lack clinical relevance.

Robust approximate likelihood ratio tests for nonlinear dynamic systems

The paper addresses the problem of determining which one of a finite number of nonlinear dynamic systems generated a given noisy measured signal. An approximate likelihood ratio test (LRT) is proposed that consists of bank of an extended Kalman filters (EKFs) each tuned to one of the candidate signal models. The prediction error sequences of each EKF are used to form the LRT since each is nominally approximately zero mean Gaussian with known covariance if it matches the measured signal. The Gaussian approximation is good at high signal-to-noise ratios (SNRs) and when the signal models are close to linear, but can degrade rapidly as the SNR decreases, or when the system becomes increasingly nonlinear. The paper proposes a robustification of the test that is based on a generalization of Huber's (1965) robust LRTs. Simulations are used to examine the performance of the proposed tests. >

On The Design and Analysis of Randomized Clinical Trials with Multiple Endpoints

This paper considers some methods for reducing the number of significance tests undertaken when analyzing and reporting results of clinical trials. Emphasis is placed on designing and analyzing clinical trials to examine a composite hypothesis concerning multiple endpoints and combining this multiple endpoint methodology with group sequential methodology. Four methods for composite hypotheses are considered: an ordinary least squares and a generalized least squares approach both due to O'Brien (1984, Biometrics 40, 1079-1087), a new modification of these, and an approximate likelihood ratio test, due to Tang, Gnecco, and Geller (1989, Biometrika 76, 577-583). These are extended for group sequential use. In particular, simulation is used to generate critical values and sequences of nominal significance levels for the approximate likelihood ratio test, which is not normally distributed. An example is given and the relative merits of the suggested approaches are discussed.

An Approximate Likelihood Ratio Test for a Normal Mean Vector with Nonnegative Components with Application to Clinical Trials

An Approximate Likelihood Ratio Test for a Normal Mean Vector with Nonnegative Components with Application to Clinical Trials

An approximate likelihood ratio test for a normal mean vector with nonnegative components with application to clinical trials

A new multivariate statistic, designed to be sensitive to the alternative hypothesis that a mean vector lies in the positive orthant, rather than to all alternatives or to alternatives on a line (O'Brien, 1984), is proposed. The statistic is shown to approximate the likelihood ratio test statistic for such alternatives (Kudô, 1963; Perlman, 1969). Its null distribution is a special case of the chi-bar-squared distribution, which allows critical values to be obtained. Such a statistic may be used in two-armed clinical trials where we are interested in showing that one treatment is better than another with respect to several responses. As an example, a clinical trial is reanalyzed, with stronger conclusions than had been drawn originally.

Testing the equality of several intraclass correlation coefficients

Abstract An approximate likelihood ratio test statistic is proposed for testing the equality of several intraclass correlation coefficients when independent samples are drawn from multivariate normal populations. The asymptotic null and nonnull distributions of the proposed statistic are derived for the nonnormal populations. The corresponding results for sampling from multivariate normal and elliptic populations are given. Test procedures based on Fisher's z-transformation are also considered. Monte Carlo experiments are performed to examine the proposed tests.

Linear models with an inverse gaussian poisson error distribution

The inverse Gaussian-Poisson (two-parameter Sichel) distribution is useful in fitting overdispersed count data. We consider linear models on the mean of a response variable, where the response is in the form of counts exhibiting extra-Poisson variation, and assume an IGP error distribution. We show how maximum likelihood estimation may be carried out using iterative Newton-Raphson IRLS fitting, where GLIM is used for the IRLS part of the maximization. Approximate likelihood ratio tests are given.

Testing hypotheses about the common mean of normal distributions

An overview of hypothesis testing for the common mean of independent normal distributions is given. The case of two populations is studied in detail. A number of different types of tests are studied. Among them are a test based on the maximum of the two available t -tests, Fisher's combined test, a test based on Graybill–Deal's estimator, an approximation to the likelihood ratio test, and some tests derived using some Bayesian considerations for improper priors along with intuitive considerations. Based on some theoretical findings and mostly based on a Monte Carlo study the conclusions are that for the most part the Bayes-intuitive type tests are superior and can be recommended. When the variances of the populations are close the approximate likelihood ratio test does best.

Approximate Likelihood Ratio Tests in the Analysis of Beta-Binomial Data

SUMMARY Likelihood ratio tests for comparing logistic-linear models in the analysis of beta-binomially distributed observations are generally tedious to compute. A method of obtaining approximate likelihood ratio tests that uses the GLIM system is proposed.

Parametric tests for agreement amongst groups of judges

SUMMARY Series approximations to the logarithm of the probability of a permutation of the natural numbers, 1, ..., K, using an extension of the Thurstone model, are investigated. The asymptotic bias of the quadratic and cubic approximations using the normal density is shown to be small. Approximate likelihood ratio tests are considered for testing for agreement between and within groups of judges and X2 approximations seem adequate for significance levels.

Tests for correlation matrices

SUMMARY Exact and approximate likelihood ratio tests are derived for certain structures in multivariate normal correlation matrices. Reasonable asymptotic distributions for the approximate tests are proposed and examined by empirical sampling. The powers of these tests are compared with those of previously proposed tests. Tests for certain structures in correlation matrices have been proposed by several authors. Hotelling (1940) proposed a conditional t test for the equality of two correlations in a trivariate normal distribution; Bartlett (1950, 1951), Anderson (1963) and Lawley (1963) considered tests for equality of all correlations in a multivariate normal distribution; and Bartlett & Rajalakshman (1953) and Kullback (1959) proposed a test for a completely specified correlation matrix. The object of the present investigation is to examine likelihood ratio tests, or simple approximations to them, for the above hypotheses. Asymptotic distributions are not available for the approximate tests, but reasonable approximations are proposed and examined by empirical sampling. The powers of these tests are compared with those of previously proposed tests.