Comparing diagnostic tests: test of hypothesis for likelihood ratios

Likelihood ratios (LRs) are used to characterize the efficiency of diagnostic tests. In this paper, we use the classical weighted least squares (CWLS) test procedure, which was originally used for testing the homogeneity of relative risks, for comparing the LRs of two or more binary diagnostic tests. We compare the performance of this method with the relative diagnostic likelihood ratio (rDLR) method and the diagnostic likelihood ratio regression (DLRReg) approach in terms of size and power, and we observe that the performances of CWLS and rDLR are the same when used to compare two diagnostic tests, while DLRReg method has higher type I error rates and powers. We also examine the performances of the CWLS and DLRReg methods for comparing three diagnostic tests in various sample size and prevalence combinations. On the basis of Monte Carlo simulations, we conclude that all of the tests are generally conservative and have low power, especially in settings of small sample size and low prevalence.     

Asymptotic hypothesis test to simultaneously compare the weighted kappa coefficients of multiple binary diagnostic tests in the presence of ignorable missing data

The weighted kappa coefficient of a binary diagnostic test is a measure of the beyond-chance agreement between the diagnostic test and the gold standard, and is a measure that allows us to assess and compare the performance of binary diagnostic tests. In the presence of partial disease verification, the comparison of the weighted kappa coefficients of two or more binary diagnostic tests cannot be carried out ignoring the individuals with an unknown disease status, since the estimators obtained would be affected by verification bias. In this article, we propose a global hypothesis test based on the chi-square distribution to simultaneously compare the weighted kappa coefficients when in the presence of partial disease verification the missing data mechanism is ignorable. Simulation experiments have been carried out to study the type I error and the power of the global hypothesis test. The results have been applied to the diagnosis of coronary disease.     

Comparison of the accuracy of multiple binary tests in the presence of partial disease verification

In the presence of partial disease verification, the comparison of the accuracy of binary diagnostic tests cannot be carried out through the paired comparison of the diagnostic tests applying McNemar's test, since for a subsample of patients the disease status is unknown. In this study, we have deduced the maximum likelihood estimators for the sensitivities and specificities of multiple binary diagnostic tests and we have studied various joint hypothesis tests based on the chi-square distribution to compare simultaneously the accuracy of these binary diagnostic tests when for some patients in the sample the disease status is unknown. Simulation experiments were carried out to study the type I error and the power of each hypothesis test deduced. The results obtained were applied to the diagnosis of coronary stenosis.     

Adaptive tests of regression functions via multiscale generalized likelihood ratios

Many applications of nonparametric tests based on curve estimation involve selecting a smoothing parameter. The author proposes an adaptive test that combines several generalized likelihood ratio tests in order to get power performance nearly equal to whichever of the component tests is best. She derives the asymptotic joint distribution of the component tests and that of the proposed test under the null hypothesis. She also develops a simple method of selecting the smoothing parameters for the proposed test and presents two approximate methods for obtaining its P‐value. Finally, she evaluates the proposed test through simulations and illustrates its application to a set of real data.     

A likelihood ratio test of a homoscedastic normal mixture against a heteroscedastic normal mixture

It is generally assumed that the likelihood ratio statistic for testing the null hypothesis that data arise from a homoscedastic normal mixture distribution versus the alternative hypothesis that data arise from a heteroscedastic normal mixture distribution has an asymptotic χ ² reference distribution with degrees of freedom equal to the difference in the number of parameters being estimated under the alternative and null models under some regularity conditions. Simulations show that the χ ² reference distribution will give a reasonable approximation for the likelihood ratio test only when the sample size is 2000 or more and the mixture components are well separated when the restrictions suggested by Hathaway (Ann. Stat. 13:795–800, 1985) are imposed on the component variances to ensure that the likelihood is bounded under the alternative distribution. For small and medium sample sizes, parametric bootstrap tests appear to work well for determining whether data arise from a normal mixture with equal variances or a normal mixture with unequal variances.     

Asymptotic test of mixture model and its applications to QTL interval mapping

Quantitative trait loci (QTL) mapping has been a standard means in identifying genetic regions harboring potential genes underlying complex traits. Likelihood ratio test (LRT) has been commonly applied to assess the significance of a genetic locus in a mixture model content. Given the time constraint in commonly used permutation tests to assess the significance of LRT in QTL mapping, we study the behavior of the LRT statistic in mixture model when the proportions of the distributions are unknown. We found that the asymptotic null distribution is stationary Gaussian process after suitable transformation. The result can be applied to one-parameter exponential family mixture model. Under certain condition, such as in a backcross mapping model, the tail probability of the supremum of the process is calculated and the threshold values can be determined by solving the distribution function. Simulation studies were performed to evaluate the asymptotic results.     

Optimal designs to improve the power of multiresponse hypothesis tests

In this paper we develop a design criterion to improve the power of the multiresponse hypothesis tests, and give a sequential procedure for generating designs which satisfy the criterion.     

A likelihood ratio test for bimodality in two-component mixtures with application to regional income distribution in the EU

We propose a parametric test for bimodality based on the likelihood principle by using two-component mixtures. The test uses explicit characterizations of the modal structure of such mixtures in terms of their parameters. Examples include the univariate and multivariate normal distributions and the von Mises distribution. We present the asymptotic distribution of the proposed test and analyze its finite sample performance in a simulation study. To illustrate our method, we use mixtures to investigate the modal structure of the cross-sectional distribution of per capita log GDP across EU regions from 1977 to 1993. Although these mixtures clearly have two components over the whole time period, the resulting distributions evolve from bimodality toward unimodality at the end of the 1970s.     

Small sample distribution of the likelihood ratio test in the random effects model

In this paper we examine the small sample distribution of the likelihood ratio test in the random effects model which is often recommended for meta-analyses. We find that this distribution depends strongly on the true value of the heterogeneity parameter (between-study variance) of the model, and that the correct p-value may be quite different from its large sample approximation. We recommend that the dependence of the heterogeneity parameter be examined for the data at hand and suggest a (simulation) method for this. Our setup allows for explanatory variables on the study level (meta-regression) and we discuss other possible applications, too. Two data sets are analyzed and two simulation studies are performed for illustration.     

The effect of verification bias on the comparison of predictive values of two binary diagnostic tests

The comparison of the accuracy of two binary diagnostic tests has traditionally required knowledge of the disease status in all of the patients in the sample via the application of a gold standard. In practice, the gold standard is not always applied to all patients in a sample, and the problem of partial verification of the disease arises. The accuracy of a binary diagnostic test can be measured in terms of positive and negative predictive values, which represent the accuracy of a diagnostic test when it is applied to a cohort of patients. In this paper, we deduce the maximum likelihood estimators of predictive values (PVs) of two binary diagnostic tests, and the hypothesis tests to compare these measures when, in the presence of partial disease verification, the verification process only depends on the results of the two diagnostic tests. The effect of verification bias on the naïve estimators of PVs of two diagnostic tests is studied, and simulation experiments are performed in order to investigate the small sample behaviour of hypothesis tests. The hypothesis tests which we have deduced can be applied when all of the patients are verified with the gold standard. The results obtained have been applied to the diagnosis of coronary stenosis.     

The likelihood approach for the comparison of medical diagnostic system with multiple binary tests

Detection (diagnosis) techniques play an important role in clinical medicine. Early detection of diseases could be life-saving, and the consequences of false-positives and false-negatives could be costly. Using multiple measurements strategy is a popular tool to increase diagnostic accuracy. In addition to the new diagnostic technology, recent advances in genomics, proteomics, and other areas have allowed some of these newly developed individual biomarkers measured by non-invasive and inexpensive procedures (e.g. samples from serum, urine or stool) to progress from basic discovery research to assay development. As more tests become commercially available, there is an increasing interest for clinicians to request combinations of various non-invasive and inexpensive tests to increase diagnostic accuracy. Using information regarding individual test sensitivities and specificities, we proposed a likelihood approach to combine individual test results and to approximate or estimate the combined sensitivities and specificities of various tests taking into account the conditional correlations to quantify system performance. To illustrate this approach, we considered an example using various combinations of diagnostic tests to detect bladder cancer.     

A simple alternative to the likelihood ratio test in a logistic discrimination problem

We consider the problem of hypothesis-testing under a logistic model with two dichotomous independent variables. In particular, we consider the case in which the coefficients β₁, and β₂ of these variables are known on an a priori basis to not be of opposite sign. For this situation we show that there exists a simple nonparametric altenative to the likelihood ratio test for testing H₀: β₁ = β₂ = 0 VS.H₁ at least one β₁ = 0. We find the asympotic relative efficiency of this test and show that it exceeds 0.90 under a wide range of conditions. We also given an example.     

Asymptotic properties of likelihood ratio test statistics in affected‐sib‐pair analysis

In an affected‐sib‐pair genetic linkage analysis, identical by descent data for affected sib pairs are routinely collected at a large number of markers along chromosomes. Under very general genetic assumptions, the IBD distribution at each marker satisfies the possible triangle constraint. Statistical analysis of IBD data should thus utilize this information to improve efficiency. At the same time, this constraint renders the usual regularity conditions for likelihood‐based statistical methods unsatisfied. In this paper, the authors study the asymptotic properties of the likelihood ratio test (LRT) under the possible triangle constraint. They derive the limiting distribution of the LRT statistic based on data from a single locus. They investigate the precision of the asymptotic distribution and the power of the test by simulation. They also study the test based on the supremum of the LRT statistics over the markers distributed throughout a chromosome. Instead of deriving a limiting distribution for this test, they use a mixture of chi‐squared distributions to approximate its true distribution. Their simulation results show that this approach has desirable simplicity and satisfactory precision.     

Likelihood procedure for testing changes in skew normal model with applications to stock returns

The skew normal distribution family is an attractive distribution family due to its mathematical tractability and inclusion of the normal distribution as the special case. It has wide applications in many applied fields such as finance, economics, and medical research. Such a distribution family has been studied extensively since it was introduced by Azzalini in 1985 Azzalini, A. (1985). A class of distributions which includes the normal ones. Scandinavian Journal of Statistics 12:171–178. [Google Scholar] for the first time. Yet, few work has been done on the study of change point problem related to this distribution family. In this article, we propose the likelihood ratio test (LRT) to detect changes in the parameters of the skew normal distribution associated with some asymptotic results of the test statistic. Simulations have been conducted under different scenarios to investigate the performance of the proposed method. Comparisons to some other existing method indicate the comparable power of the method in detecting changes in parameters of the skew normal distribution model. Applications on two real data: Brazilian and Tanzanian stock returns illustrate the detection procedure.     

Performance of the shrinkage preliminary test ridge regression estimators based on the conflicting of W,LR and LM tests

The shrinkage preliminary test ridge regression estimators (SPTRRE) based on the Wald (W), the likelihood ratio (LR) and the Lagrangian multiplier (LM) tests are considered in this paper. The bias and the risk functions of the proposed estimators are derived. The regions of optimality of the estimators are determined under the quadratic risk function. Under the null hypothesis, the SPTRRE based on LM test has the smallest risk, followed by the estimators based on LR and W tests. However, the SPTRRE based on W test performs the best followed by the LR and LM based estimators when the parameter moves away from the subspace of the restrictions. The conditions of superiority of the proposed estimator for both ridge and departure parameters are discussed. The optimum choice of the level of significance becomes the traditional choice by using the W test for all non-negative ridge parameters.     

Local power properties of some asymptotic tests in symmetric linear regression models

In this paper we obtain asymptotic expansions, up to order n^−1/2 and under a sequence of Pitman alternatives, for the nonnull distribution functions of the likelihood ratio, Wald, score and gradient test statistics in the class of symmetric linear regression models. This is a wide class of models which encompasses the t model and several other symmetric distributions with longer-than normal tails. The asymptotic distributions of all four statistics are obtained for testing a subset of regression parameters. Furthermore, in order to compare the finite-sample performance of these tests in this class of models, Monte Carlo simulations are presented. An empirical application to a real data set is considered for illustrative purposes.     

Non Null Asymptotic Distributions of Some Criteria in Censored Exponential Regression

In this article, we consider the class of censored exponential regression models which is very useful for modeling lifetime data. Under a sequence of Pitman alternatives, the asymptotic expansions up to order n^{? 1/2} of the non null distribution functions of the likelihood ratio, Wald, Rao score, and gradient statistics are derive in this class of models. The non null asymptotic distribution functions of these statistics are obtained for testing a composite null hypothesis in the presence of nuisance parameters. The power of all four tests, which are equivalent to first order, are compared based on these non null asymptotic expansions. Furthermore, in order to compare the finite-sample performance of these tests in this class of models, we consider Monte Carlo simulations. We also present an empirical application for illustrative purposes.     

A test of the missing data mechanism for repeated measures data

The occurrence of missing data is an often unavoidable consequence of repeated measures studies. Fortunately, multivariate general linear models such as growth curve models and linear mixed models with random effects have been well developed to analyze incomplete normally-distributed repeated measures data. Most statistical methods have assumed that the missing data occur at random. This assumption may include two types of missing data mechanism: missing completely at random (MCAR) and missing at random (MAR) in the sense of Rubin (1976). In this paper, we develop a test procedure for distinguishing these two types of missing data mechanism for incomplete normally-distributed repeated measures data. The proposed test is similar in spiril to the test of Park and Davis (1992). We derive the test for incomplete normally-distribrlted repeated measures data using linear mixed models. while Park and Davis (1992) cleirved thr test for incomplete repeatctl categorical data in the framework of Grizzle Starmer. and Koch (1969). Thr proposed procedure can be applied easily to any other multivariate general linear model which allow for missing data. The test is illustrated using the hip-replacernent patient.data from Crowder and Hand (1990).     

Evaluation of incomplete multiple diagnostic tests,with an application in the colon cancer family registry study

Accurate diagnosis of a molecularly defined subtype of cancer is often an important step toward its effective control and treatment. For the diagnosis of some subtypes of a cancer, a gold standard with perfect sensitivity and specificity may be unavailable. In those scenarios, tumor subtype status is commonly measured by multiple imperfect diagnostic markers. Additionally, in many such studies, some subjects are only measured by a subset of diagnostic tests and the missing probabilities may depend on the unknown disease status. In this paper, we present statistical methods based on the EM algorithm to evaluate incomplete multiple imperfect diagnostic tests under a missing at random assumption and one missing not at random scenario. We apply the proposed methods to a real data set from the National Cancer Institute (NCI) colon cancer family registry on diagnosing microsatellite instability for hereditary non-polyposis colorectal cancer to estimate diagnostic accuracy parameters (i.e. sensitivities and specificities), prevalence, and potential differential missing probabilities for 11 biomarker tests. Simulations are also conducted to evaluate the small-sample performance of our methods.