Likelihood Ratio Test For Independence Research Articles

Testing independence between discrete random variables

In this paper we develop a likelihood ratio test for independence between multivariate discrete random vectors whose components may be dependent or may have different marginal distributions. A special case that tests independence between two discrete random variables is also discussed. The model under the null hypothesis is constructed based on the marginal distributions of the random variables, while the model under the alternative is built by adding a common random effect which is not restricted to follow the same distribution as those of the other variables. The test statistic is shown to admit an asymptotic distribution of a weighted sum of chi-square random variables, each with one degree of freedom. In order to avoid the high computational cost of finding the parameters of the asymptotic distribution, a permutation-based procedure is introduced. It is shown in simulations that the permutation-based realization yields empirical level close to the nominal level, while achieving power comparable to or higher than that of existing competitors in the literature. The application of the test is illustrated with a real data set.

On the distribution of the likelihood ratio test of independence for random sample size — a computational approach

The test of independence of two groups of variables is addressed in the case where the sample size N is considered randomly distributed. This assumption may lead to a more realist testing procedure since in many situations the sample size is not known in advance. Three sample schemes are considered where N may have a Poisson, Binomial or Hypergeometric distribution. For the case of two groups with p1 and p2 variables, it is shown that when either p1 or p2 (or both) are even the exact distribution corresponds to a finite or an infinite mixture of Exponentiated Generalized Integer Gamma distributions. In these cases a computational module is made available for the cumulative distribution function of the test statistic. When both p1 and p2 are odd, the exact distribution of the test statistic may be represented as a finite or an infinite mixture of products of independent Beta random variables whose density and cumulative distribution functions do not have a manageable closed form. Therefore, a computational approach for the evaluation of the cumulative distribution function is provided based on a numerical inversion formula originally developed for Laplace transforms. When the exact distribution is represented through infinite mixtures, an upper bound for the error of truncation of the cumulative distribution function is provided. Numerical studies are developed in order to analyze the precision of the results and the accuracy of the upper bounds proposed. A simulation study is provided in order to assess the power of the test when the sample size N is considered randomly distributed. The results are compared with the ones obtained for the fixed sample size case.

Limiting distributions of likelihood ratio test for independence of components for high-dimensional normal vectors

Consider a p-variate normal random vector. We are interested in the limiting distributions of likelihood ratio test (LRT) statistics for testing the independence of its grouped components based on a random sample of size n. In classical multivariate analysis, the dimension p is fixed or relatively small, and the limiting distribution of the LRT is a chi-square distribution. When p goes to infinity, the chi-square approximation to the classical LRT statistic may be invalid. In this paper, we prove that the LRT statistic converges to a normal distribution under quite general conditions when p goes to infinity. We propose an adjusted test statistic which has a chi-square limit in general. Our comparison study indicates that the adjusted test statistic outperforms among the three approximations in terms of sizes. We also report some numerical results to compare the performance of our approaches and other methods in the literature.

Use of Direct Oral Anticoagulants in Patients with Sickle Cell Disease and Venous Thromboembolism Is Associated with a Significant Decrease in Incidence of Bleeding Compared to Vitamin K Antagonists and Low-Molecular-Weight Heparins

Background: Venous thromboembolism (VTE) is a recognized complication of sickle cell disease (SCD). Several studies confirm that SCD itself is an independent risk factor for development of VTE. However, the optimal pharmacologic anticoagulant remains unknown.Methods: This retrospective single-institution cohort study was exempt by the Institutional Review Board. Data were collected via review of electronic medical records including ambulatory, emergency department, general floor, and intensive care unit encounters. Patients with SCD were identified spanning 1/2009-7/2017 using ICD 9/10 codes. Inclusion criteria were age ≥18 years at time of VTE diagnosis, imaging confirming VTE, and documented compliance based on INR values and/or provider/pharmacy documentation. VTE diagnosis included deep vein thrombosis (DVT) at any location and pulmonary embolism based on documented imaging and ICD 9/10 codes. Anticoagulants included direct oral anticoagulants (DOACs), vitamin K antagonists (VKA), and low-molecular-weight heparin (LMWH). The DOACs used in this study were rivaroxaban, apixaban and dabigatran. Exclusion criteria were known active malignancy, confirmed hypercoagulable risk factors beyond SCD, atrial fibrillation and/or history of major bleeding prior to anticoagulation. Due to low event rates, a log likelihood ratio test of independence was calculated for associations between drug type and two endpoints: bleeding rate and rate of VTE recurrence. Rate of VTE recurrence was defined as a newly diagnosed VTE within 6 months of initiation of anticoagulation. Bleeding rate was defined using International Society on Thrombosis and Hemostasis criteria: bleeding event into a critical site and/or a ≥2 point decrease in baseline hemoglobin.Results: A total of 109 patients with SCD met inclusion criteria. 66 patients (60%) were female. SCD genotypes represented included HbSS in 91 patients (83%), HbSC in 12 (11%) and HbS β+ thalassemia in 4 (4%). There were no patients with HbS-β0 thalassemia. VTEs consisted of 69 DVTs and 43 pulmonary emboli. 31 out of 109 VTEs were provoked, including 30 catheter-related incidents. After initial VTE event, 32 patients received a VKA, 34 received LMWH, and 43 received a DOAC. Within the class of DOACs, 31 patients received rivaroxaban, 5 received apixaban, and 7 received dabigatran.Sixteen of 109 patients (15%) experienced a clinically significant bleeding event, including 8 on VKA, 6 on LMWH, and 2 on a DOAC. Bleeding incidence was least with the DOAC class [0.22 CI (0.04-0.84) p < 0.05], greatest with warfarin [1.55 CI (0.57-4.33) p < 0.05] and slightly less with LMWH [0.64 CI (0.23-1.73) p < 0.05]. There was a significant decrease in incidence of bleeding events in patients receiving a DOAC for anticoagulation, compared to a VKA or LMWH (p = 0.033).At a median follow-up of 11.8 months (range, 3.4 - 60 months), 33 patients had a recurrent VTE, including 10 on VKA, 10 on LMWH and 13 taking a DOAC (p = 0.833). An association between VTE and SCD genotype could not be identified due to small numbers of patients with non-HbSS genotypes.Conclusion: In patients with SCD and VTE, there was a significant decrease in incidence of bleeding events in patients receiving a DOAC for anticoagulation, compared to a VKA or LMWH (p = 0.033). There was no difference between VTE recurrence rate and choice of initial anticoagulation. [Display omitted] DisclosuresNo relevant conflicts of interest to declare.

A Simple Density-Based Empirical Likelihood Ratio Test for Independence

We develop a novel nonparametric likelihood ratio test for independence between two random variables using a technique that is free of the common constraints of defining a given set of specific dependence structures. Our methodology revolves around an exact density-based empirical likelihood ratio test statistic that approximates in a distribution-free fashion the corresponding most powerful parametric likelihood ratio test. We demonstrate that the proposed test is very powerful in detecting general structures of dependence between two random variables, including nonlinear and/or random-effect dependence structures. An extensive Monte Carlo study confirms that the proposed test is superior to the classical nonparametric procedures across a variety of settings. The real-world applicability of the proposed test is illustrated using data from a study of biomarkers associated with myocardial infarction. Supplementary materials for this article are available online.

A likelihood ratio test of quasi-independence for sparse two-way contingency tables

We consider a likelihood ratio test of independence for large two-way contingency tables having both structural (non-random) and sampling (random) zeros in many cells. The solution of this problem is not available using standard likelihood ratio tests. One way to bypass this problem is to remove the structural zeroes from the table and implement a test on the remaining cells which incorporate the randomness in the sampling zeros; the resulting test is a test of quasi-independence of the two categorical variables. This test is based only on the positive counts in the contingency table and is valid when there is at least one sampling (random) zero. The proposed (likelihood ratio) test is an alternative to the commonly used ad hoc procedures of converting the zero cells to positive ones by adding a small constant. One practical advantage of our procedure is that there is no need to know if a zero cell is structural zero or a sampling zero. We model the positive counts using a truncated multinomial distribution. In fact, we have two truncated multinomial distributions; one for the null hypothesis of independence and the other for the unrestricted parameter space. We use Monte Carlo methods to obtain the maximum likelihood estimators of the parameters and also the p-value of our proposed test. To obtain the sampling distribution of the likelihood ratio test statistic, we use bootstrap methods. We discuss many examples, and also empirically compare the power function of the likelihood ratio test relative to those of some well-known test statistics.

The sizes of the three popular asymptotic tests for testing homogeneity of two binomial proportions

In statistical hypothesis testing it is important to ensure that the type I error rate is preserved under the nominal level. This paper addresses the sizes and the type I errors rates of the three popular asymptotic tests for testing homogeneity of two binomial proportions: the chi-square test with and without continuity correction, the likelihood ratio test. Although it has been recognized that, based on limited simulation studies, the sizes of the tests are inflated in small samples, it has been thought that the sizes are well preserved under the nominal level when the sample size is sufficiently large. But, Loh [1989. Bounds on the size of the χ 2 test of independence in a contingency table. Ann. Statist. 17, 1709–1722], and Loh and Yu [1993. Bounds on the size of the likelihood ratio test of independence in a contingency table. J. Multivariate Anal. 45, 291–304] showed theoretically that the sizes are always greater than or equal to the nominal level when the sample size is infinite. In this paper, we confirm their results by computing the large-sample lower bounds of the sizes numerically. Applying complete enumeration which does not have any error, we confirm again the results by computing the sizes precisely on computer in moderate sample sizes. When the sample sizes are unbalanced, the peaks of the type I error rates occur at the extremes of the nuisance parameter. But, the type I error rates of the three tests are close to the nominal level in most values of the nuisance parameter except the extremes. We also find that, when the sample sizes are severely unbalanced and the value of the nuisance parameter is very small, the size of the chi-square test with continuity correction can exceed the nominal level excessively (for instance, the size could be at least 0.877 at 5% nominal level in some cases).

Inferences based on a bivariate distribution with von Mises marginals

There is very little literature concerning modeling the correlation between paired angular observations. We propose a bivariate model with von Mises marginal distributions. An algorithm for generating bivariate angles from this von Mises distribution is given. Maximum likelihood estimation is then addressed. We also develop a likelihood ratio test for independence in paired circular data. Application of the procedures to paired wind directions is illustrated. Employing simulation, using the proposed model, we compare the power of the likelihood ratio test with six existing tests of independence.

Bootstrap tests for independence in two-way ordinal contingency tables

For the analysis of an r× c contingency table having ordered row categories and ordered column categories, a bootstrap method is applied for the model-based likelihood ratio test for independence. A model-based likelihood ratio chi-square statistic and the statistic of the maximum eigenvalue of a Wishart matrix are also discussed. A simulation study is performed to compare the proposed method with existing ones. A real data example is included.

Exact likelihood ratio test of independence of binary responses within clusters

We consider the exact likelihood ratio test of independence of binary responses within clusters developed by George and Kodell (J. Amer. Statist. Assoc., 1996, 91, 1602–1611) conditional on the sufficient statistics for the nuisance parameter under the null hypothesis. Only an exchangeability of responses within clusters is assumed for the distribution of the observed vector. Two formulas for the exact P-values are obtained. An algorithm is proposed to avoid unnecessary enumeration when the cluster size is two. We compare the exact and asymptotic P-values in several cases.

Updating exact P- values for the conditional likelihood ratio test of independence

We consider the exact likelihood ratio test of independence conditioned on row and column margins in an r × c contingency table with multinomial sampling. We develop an update algorithm to compute the exact Pvalue of the test and show it is better than the network algorithm in terms of computing speed. In the algorithm the P-value is reduced to a sum of probabilities for 2 × 2 contingency tables, which we compute using the hyper-geometric distribution. The same algorithm can also be used for testing homogeneity of independent multinomial populations.

Bounds on the Size of the Likelihood Ratio Test of Independence in a Contingency Table

Bounds are obtained on the limiting size of the nominal level-α likelihood ratio test of independence in a r × c contingency table. The situations considered include sampling with both marginal totals random and with one margin fixed. Upper and lower bounds are obtained. The limiting size is greater than α when some marginal probabilities are small. As the degrees of freedom increase, the limiting size tends to 1 for all α-values.

On a likelihood ratio test for independence

The likelihood-ratio (LR) statistic is derived for testing the hypothesis of independence between k subvectors of a normally distributed random vector X against the alternative that exactly s, 2 ⩽ s ⩽ k, subvectors are dependent. A null asymptotic distribution for the LR statistic is also presented.

Likelihood ratio test for independence with partial multivariate normal data

This article derives the likelihood ratio statistic to test the independence between (X 1,…,X r ) and (X r+1,…,X k ) under the assumption that (X 1,…,X k ) has a multivariate normal distribution and that a sample of size n is available, where for N observation vectors all components are available, while for M = (n + N) observation vectors, the data on the last q components, (Xk-q+1,…,X k ) are missing (k+q≥r).

Statistical inference for censored bivariate normal distributions based on induced order statistics

SUMMARY For a censored sample from a bivariate normal distribution consisting of the first k of n order statistics of one variable and the induced order statistics of the other variable, the maximum likelihood estimators of the parameters and the associated large sample covariance matrix are derived. The likelihood ratio test for independence is also given and its power properties studied. These methods are useful in selection problems or in life testing situations in which concomitant variates are observable only for the uncensored primary variates.

A Model for Cylindrical Variables with Applications

Summary A suitable model when two random variables take values on a cylinder is proposed and its main properties are discussed. The maximum likelihood estimators for the parameters of the model are obtained and a likelihood ratio test of independence is derived. A numerical example is given.