Use of the Likelihood Ratio Test on the Uniform Distribution

In this article we give a simple procedure to determine the exact distribution of the likelihood ratio test of a statistical hypothesis regarding the parameter of the uniform distribution. The resulting distribution will be shown to serve as an approximation to the distribution of the likelihood ratio statistic for testing the equality of scale parameters of k independent Exponential populations.     

Bootstrap Restricted Likelihood Ratio Test for the Detection of Rare Variants

In this paper the detection of rare variants association with continuous phenotypes of interest is investigated via the likelihood-ratio based variance component test under the framework of linear mixed models. The hypothesis testing is challenging and nonstandard, since under the null the variance component is located on the boundary of its parameter space. In this situation the usual asymptotic chisquare distribution of the likelihood ratio statistic does not necessarily hold. To circumvent the derivation of the null distribution we resort to the bootstrap method due to its generic applicability and being easy to implement. Both parametric and nonparametric bootstrap likelihood ratio tests are studied. Numerical studies are implemented to evaluate the performance of the proposed bootstrap likelihood ratio test and compare to some existing methods for the identification of rare variants. To reduce the computational time of the bootstrap likelihood ratio test we propose an effective approximation mixture for the bootstrap null distribution. The GAW17 data is used to illustrate the proposed test.     

A Likelihood Ratio Test of the Equality of Paired Survival Data with Censoring

A test is developed to determine whether the mean survival times are equal when dealing with paired survival data. We assume the data follow a bivariate exponential distribution for which the variables are conditionally independent. The unconditional distribution is derived in which the distribution of the nuissance variable is general. A method based on the likelihood ratio is derived to obtain the test. The data are allowed to have both left and right censoring.     

The Residual Likelihood Ratio Test for the Variance Ratio in a Linear Model with Two Variance Components

Residual maximum likelihood has proved to be a successful approach to the estimation of variance components. In this paper, its counterpart in testing, the residual likelihood ratio test, is applied to testing the ratio of two variance components. The test is compared with the Wald test and the locally most powerful invariant test.     

Accuracy and Power of the Likelihood Ratio Test for Comparing Evolutionary Rates Among Genes

Sequences for multiple protein-coding genes are now commonly available from several, often closely related species. These data sets offer intriguing opportunities to test hypotheses regarding whether different types of genes evolve under different selective pressures. Although maximum likelihood (ML) models of codon substitution that are suitable for such analyses have been developed, little is known about the statistical properties of these tests. We use a previously developed fixed-sites model and computer simulations to examine the accuracy and power of the likelihood ratio test (LRT) in comparing the nonsynonymous-to-synonymous substitution rate ratio (=dN/dS) between two genes. Our results show that the LRT applied to fixed-sites models may be inaccurate in some cases when setting significance thresholds using a ² approximation. Instead, we use a parametric bootstrap to describe the distribution of the LRT statistic for fixed-sites models and examine the power of the test as a function of sampling variables and properties of the genes under study. We find that the power of the test is high (>80%) even when sampling few taxa (e.g., six species) if sequences are sufficiently diverged and the test is largely unaffected by the tree topology used to simulate data. Our simulations show fixed-sites models are suitable for comparing substitution parameters among genes evolving under even strong evolutionary constraint ( 0.05), although relative rate differences of 25% or less may be difficult to detect.Reviewing Editor: Dr. Rosmus Nielsen     

Real Time QRS Detection Based on M-ary Likelihood Ratio Test on the DFT Coefficients

This paper shows an adaptive statistical test for QRS detection of electrocardiography (ECG) signals. The method is based on a M-ary generalized likelihood ratio test (LRT) defined over a multiple observation window in the Fourier domain. The motivations for proposing another detection algorithm based on maximum a posteriori (MAP) estimation are found in the high complexity of the signal model proposed in previous approaches which i) makes them computationally unfeasible or not intended for real time applications such as intensive care monitoring and (ii) in which the parameter selection conditions the overall performance. In this sense, we propose an alternative model based on the independent Gaussian properties of the Discrete Fourier Transform (DFT) coefficients, which allows to define a simplified MAP probability function. In addition, the proposed approach defines an adaptive MAP statistical test in which a global hypothesis is defined on particular hypotheses of the multiple observation window. In this sense, the observation interval is modeled as a discontinuous transmission discrete-time stochastic process avoiding the inclusion of parameters that constraint the morphology of the QRS complexes.     

A Note on Maximum Likelihood Ratio Test of No Outliers in Regression Models

A derivation of the maximum likelihood ratio test for testing no outliers in regression models is given using the method of WETHERILL (1981, pp. 106–107) for estimating the regression parameters. This method is essentially similar to the one outlined in BARNETT and LEWIS (1978, p. 263), although by our detailed derivation it is easier to see that the maximum likelihood estimate of θ of model (3) under the hypothesis that the i^th observation in an outlier is the same as that obtained from model (1) when the i^th observation is removed.     

Small Sample Distribution and Power of the Binomial Index of Dispersion and Log Likelihood Ratio Tests

This paper considers the exact distribution of the X² index of dispersion and -2 log (likelihood ratio) tests for the hypothesis of homogeneity of c independent samples from a common binomial population. The exact significance levels and power of these tests under ‘logit’ alternatives are compared numerically for the cases: c = 3, 4, 5 and various sample sizes. n₁ = 5,10 for i = 1,…, c.     

An Adjusted Likelihood Ratio Approach Analysing Distribution of Food Products to Assist the Investigation of Foodborne Outbreaks

In order to facilitate foodborne outbreak investigations there is a need to improve the methods for identifying the food products that should be sampled for laboratory analysis. The aim of this study was to examine the applicability of a likelihood ratio approach previously developed on simulated data, to real outbreak data. We used human case and food product distribution data from the Norwegian enterohaemorrhagic Escherichia coli outbreak in 2006. The approach was adjusted to include time, space smoothing and to handle missing or misclassified information. The performance of the adjusted likelihood ratio approach on the data originating from the HUS outbreak and control data indicates that the adjusted approach is promising and indicates that the adjusted approach could be a useful tool to assist and facilitate the investigation of food borne outbreaks in the future if good traceability are available and implemented in the distribution chain. However, the approach needs to be further validated on other outbreak data and also including other food products than meat products in order to make a more general conclusion of the applicability of the developed approach.     

Maximum Likelihood Methods for Fitting the Burr Type XII Distribution to Life Test Data

This paper describes mathematical and computational methodology for estimating the parameters of the Burr Type XII distribution by the method of maximum likelihood. Expressions for the asymptotic variances and covariances of the parameter estimates are given, and the modality of the log-likelihood and conditional log-likelihood functions is analyzed. As a result of this analysis for various a priori known and unknown parameter combinations, conditions are given which guarantee that the parameter estimates obtained will, indeed, be maximum likelihood estimates. An efficient numerical method for maximizing the conditional log-likelihood function is described, and mathematical expressions are given for the various numerical approximations needed to evaluate the expressions given for the asymptotic variances and covariances of the parameter estimates. The methodology discussed is applied in a numerical example to life test data arising in a clinical setting.     

The Exact Distribution of the Anderson-Kannemann Statistic

The Anderson-Kannemann test is a rank test for treatment effects in a randomized block design with K treatments and N blocks. In this paper, an algorithm for computing the exact distribution of the Anderson-Kannemann test statistic under the null hypothesis is deduced. Then, the exact distribution is compared with the asymptotic χ²-distribution, and it is shown that the exact distribution is approximated fairly well by the asymptotic distribution. Tables of the exact distribution are given for K = 3, N = 3(1)15; K = 4, N = 3(1)11; K = 5, N = 3(1)7; and K = 6, N = 3(1)5.     

Comparisons of Some Chi-squared Tests for the Test of Independence in Sparse Two-Way Contingency Tables

We consider in this paper, the behaviour of a class of the CRESSIE READ (1984) power divergence test statistics indexed by parameter λ - I (λ), with the modified X₂ test statistics (LU) proposed by LAWAL and UPTON (1984), for sparse contingency tables ranging from the 3×3 to the 10×10. We present a sample of our results here. The results indicate that the LU test out-performs either the Cressie-Read suggested test I(2/3) or the Pearson's test - I(1). Our results further show that the modification to the likelihood ratio test [Y₂ = I'(0)] proposed by WILLIAMS (1976) performs like the parent Y₂ test, very poorly compared with either the I(2/3), X₂ or the LU test statistics. Power results also indicate that the powers of the LU test are in all cases considered in this study slightly higher than those of X₂ and I(2/3) tests. The LU test is therefore strongly recommended for use with sparse two-way contingency tables because in all of the cases considered, none of the other test statistics consistently out-performs the LU test with respect to attained α level or power.     

Jeffreys' Likelihood Ratio for the 2×2-Table — A Monte Carlo Study —

Jeffreys' approach for analyzing a 2×2-table is discussed via a Monte Carlo study. The main findings are reported.     

The Evaluation and Interpretation of Cervical Cytology: Application of the Likelihood Ratio Concept

The Papanicolaou smear (Pap test), used for the detection and prevention of neoplastic lesions of the cervix, is known to have both false negative and false positive results. Proper handling of the diagnostic uncertainty resulting from these errors demands quantification of flaws. Traditionally, sensitivity, specificity and predictive values are used for that aim. In this study another approach is advocated, namely the use of the likelihood ratio. For cervical cytology this ratio is the quotient of the probability of a Pap class within the diseased population to the probability of that same Pap class within the non-diseased group. This approach enables the characterization of each Pap class separately, and is therefore much better for clinical interpretation of the result. It is also a superior approach for quality assessment.     

Maximum Likelihood Estimation of Exponential Polynomial Rate for Poisson Data

The heterogeneous Poisson process with discretized exponential quadratic rate function is considered. Maximum likelihood estimates of the parameters of the rate function are derived for the case when the data consists of numbers of occurrences in consecutive equal time periods. A likelihood ratio test of the null hypothesis of exponential quadratic rate is presented. Its power against exponential linear rate functions is estimated using Monte Carlo simulation. The maximum likelihood method is compared with a log-linear least squares techniques. An application of the technique to the analysis of mortality rates due to congenital malformations is presented.     

Effect of Intracluster Correlation on the R-Square Statistic

R²-statistic is a popular and very widely used statistic in regression analysis to estimate the square multiple correlation (SMC), ρ², between a response variable Y and p predictor variables, _X1, …, _Xp. Numerous articles are available in the statistical literature on the properties of R² as an estimator of ρ² when the observations are uncorrelated. However, relatively little is known about the behavior of R² when the available observations are correlated such as the data that result from complex sampling schemes. In this paper, we study the behavior R² in the presence of two-stage sampling data. An approximate expressions for the variance and the bias of R² in the presence of two-stage cluster sampling data with positive intracluster correlation (ρ*) are obtained. It is evident from these formulas and from a simulation study that R² is a poor estimator of ρ² except when ρ* is small. As such, we consider several alternative estimators of ρ² and evaluate their theoretical properties and finite sample performance using a simulation study.     

Small Sample Properties of the Mantel-Haenszel Test Statistic for Density Follow-Up Studies

For the analysis of combinations of 2×2 non-contingency tables as obtained from density follow-up studies (relating a number of events to a number of person-years of follow-up) an analogue of the Mantel-Haenszel test for 2×2 contingency tables is widely used. In this paper the small sample properties of this test, both with and without continuity correction, are evaluated. Also the improvement of the test-statistic by using the first four cumulants via the Edgeworth expansion was studied. Results on continuity correction agree with similar studies on the Mantel-Haenszel statistic for 2×2 contingency tables: Continuity correction gives a p-value which approximates the exact p-value better than the p-value obtained without this correction; both the exact test and its approximations show considerable conservatism in small samples; the uncorrected Mantel-Haenszel test statistic gives a p-value that agrees more with the nominal significance level, but can be anti-conservative. The p-value based on the first four cumulants gives a better approximation of the exact p-value than the continuity corrected test, especially when the distribution has marked skewness.     

Confidence Limits for the Ratio of Two Binomial Proportions Based on Likelihood Scores: Non-Iterative Method

The interval estimation of the ratio of two binomial proportions based on the score statistic is superior over other methods. Iterative algorithms for calculating the approximate confidence interval have been provided by, e.g., KOOPMAN (1984, Biometrics 40:513–517) and GART and NAM (1988a, Biometrics 44:323–338). This note presents the analytical solutions for upper and lower confidence limits in a closed form and gives examples for numerical illustration. The non-iterative method is generally more desirable than the iterative method.     

Crockett's Test of Fit for the Bivariate Poisson

An improved version of the test of CROCKETT (1979) for testing for the bivariate Poisson distribution is given and evaluated.