Model fusion and multiple testing in the likelihood paradigm: shrinkage and evidence supporting a point null hypothesis

ABSTRACT

According to the general law of likelihood, the strength of statistical evidence for a hypothesis as opposed to its alternative is the ratio of their likelihoods, each maximized over the parameter of interest. Consider the problem of assessing the weight of evidence for each of several hypotheses. Under a realistic model with a free parameter for each alternative hypothesis, this leads to weighing evidence without any shrinkage toward a presumption of the truth of each null hypothesis. That lack of shrinkage can lead to many false positives in settings with large numbers of hypotheses. A related problem is that point hypotheses cannot have more support than their alternatives. Both problems may be solved by fusing the realistic model with a model of a more restricted parameter space for use with the general law of likelihood. Applying the proposed framework of model fusion to data sets from genomics and education yields intuitively reasonable weights of evidence.     

Stepwise likelihood ratio statistics in sequential studies

Summary.  It is well known that in a sequential study the probability that the likelihood ratio for a simple alternative hypothesis H ₁ versus a simple null hypothesis H ₀ will ever be greater than a positive constant c will not exceed 1/ c under H ₀. However, for a composite alternative hypothesis, this bound of 1/ c will no longer hold when a generalized likelihood ratio statistic is used. We consider a stepwise likelihood ratio statistic which, for each new observation, is updated by cumulatively multiplying the ratio of the conditional likelihoods for the composite alternative hypothesis evaluated at an estimate of the parameter obtained from the preceding observations versus the simple null hypothesis. We show that, under the null hypothesis, the probability that this stepwise likelihood ratio will ever be greater than c will not exceed 1/ c . In contrast, under the composite alternative hypothesis, this ratio will generally converge in probability to ∞. These results suggest that a stepwise likelihood ratio statistic can be useful in a sequential study for testing a composite alternative versus a simple null hypothesis. For illustration, we conduct two simulation studies, one for a normal response and one for an exponential response, to compare the performance of a sequential test based on a stepwise likelihood ratio statistic with a constant boundary versus some existing approaches.     

Likelihood ratio tests based on subglobal optimization: A power comparison in exponential mixture models

The paper compares several versions of the likelihood ratio test for exponential homogeneity against mixtures of two exponentials. They are based on different implementations of the likelihood maximization algorithm. We show that global maximization of the likelihood is not appropriate to obtain a good power of the LR test. A simple starting strategy for the EM algorithm, which under the null hypothesis often fails to find the global maximum, results in a rather powerful test. On the other hand, a multiple starting strategy that comes close to global maximization under both the null and the alternative hypotheses leads to inferior power.     

Testing homogeneity in a multivariate mixture model

Testing homogeneity is a fundamental problem in finite mixture models. It has been investigated by many researchers and most of the existing works have focused on the univariate case. In this article, the authors extend the use of the EM‐test for testing homogeneity to multivariate mixture models. They show that the EM‐test statistic asymptotically has the same distribution as a certain transformation of a single multivariate normal vector. On the basis of this result, they suggest a resampling procedure to approximate the P‐value of the EM‐test. Simulation studies show that the EM‐test has accurate type I errors and adequate power, and is more powerful and computationally efficient than the bootstrap likelihood ratio test. Two real data sets are analysed to illustrate the application of our theoretical results. The Canadian Journal of Statistics 39: 218–238; 2011 © 2011 Statistical Society of Canada     

Advantages of Variance Stabilization

Abstract. Variance stabilization is a simple device for normalizing a statistic. Even though its large sample properties are similar to those of studentizing, many simulation studies of confidence interval procedures show that variance stabilization works better for small samples. We investigated this question in the context of testing a null hypothesis involving a single parameter. We provide support for a measure of evidence for an alternative hypothesis that is simple to compute, calibrate and interpret. It has applications in most routine problems in statistics, and leads to more accurate confidence intervals, estimated power and hence sample size calculations than standard asymptotic methods. Such evidence is readily combined when obtained from different studies. Connections to other approaches to statistical evidence are described, with a notable link to Kullback–Leibler symmetrized divergence.     

Two‐sample empirical likelihood ratio tests for medians in application to biomarker evaluations

The median is a commonly used parameter to characterize biomarker data. In particular, with two vastly different underlying distributions, comparing medians provides different information than comparing means; however, very few tests for medians are available. We propose a series of two‐sample median‐specific tests using empirical likelihood methodology and investigate their properties. We present the technical details of incorporating the relevant constraints into the empirical likelihood function for in‐depth median testing. An extensive Monte Carlo study shows that the proposed tests have excellent operating characteristics even under unfavourable occasions such as non‐exchangeability under the null hypothesis. We apply the proposed methods to analyze biomarker data from Western blot analysis to compare normal cells with bronchial epithelial cells from a case–control study. The Canadian Journal of Statistics 39: 671–689; 2011. © 2011 Statistical Society of Canada     

Likelihood Ratio Tests for Dependent Data with Applications to Longitudinal and Functional Data Analysis

This paper introduces a general framework for testing hypotheses about the structure of the mean function of complex functional processes. Important particular cases of the proposed framework are as follows: (1) testing the null hypothesis that the mean of a functional process is parametric against a general alternative modelled by penalized splines; and (2) testing the null hypothesis that the means of two possibly correlated functional processes are equal or differ by only a simple parametric function. A global pseudo‐likelihood ratio test is proposed, and its asymptotic distribution is derived. The size and power properties of the test are confirmed in realistic simulation scenarios. Finite‐sample power results indicate that the proposed test is much more powerful than competing alternatives. Methods are applied to testing the equality between the means of normalized δ‐power of sleep electroencephalograms of subjects with sleep‐disordered breathing and matched controls.     

Empirical phi-divergence test statistics for testing simple and composite null hypotheses

The main purpose of this paper is to introduce first a new family of empirical test statistics for testing a simple null hypothesis when the vector of parameters of interest is defined through a specific set of unbiased estimating functions. This family of test statistics is based on a distance between two probability vectors, with the first probability vector obtained by maximizing the empirical likelihood (EL) on the vector of parameters, and the second vector defined from the fixed vector of parameters under the simple null hypothesis. The distance considered for this purpose is the phi-divergence measure. The asymptotic distribution is then derived for this family of test statistics. The proposed methodology is illustrated through the well-known data of Newcomb's measurements on the passage time for light. A simulation study is carried out to compare its performance with that of the EL ratio test when confidence intervals are constructed based on the respective statistics for small sample sizes. The results suggest that the ‘empirical modified likelihood ratio test statistic’ provides a competitive alternative to the EL ratio test statistic, and is also more robust than the EL ratio test statistic in the presence of contamination in the data. Finally, we propose empirical phi-divergence test statistics for testing a composite null hypothesis and present some asymptotic as well as simulation results for evaluating the performance of these test procedures.     

Robust inference from multiple test statistics via permutations: a better alternative to the single test statistic approach for randomized trials

Formal inference in randomized clinical trials is based on controlling the type I error rate associated with a single pre‐specified statistic. The deficiency of using just one method of analysis is that it depends on assumptions that may not be met. For robust inference, we propose pre‐specifying multiple test statistics and relying on the minimum p‐value for testing the null hypothesis of no treatment effect. The null hypothesis associated with the various test statistics is that the treatment groups are indistinguishable. The critical value for hypothesis testing comes from permutation distributions. Rejection of the null hypothesis when the smallest p‐value is less than the critical value controls the type I error rate at its designated value. Even if one of the candidate test statistics has low power, the adverse effect on the power of the minimum p‐value statistic is not much. Its use is illustrated with examples. We conclude that it is better to rely on the minimum p‐value rather than a single statistic particularly when that single statistic is the logrank test, because of the cost and complexity of many survival trials. Copyright © 2013 John Wiley & Sons, Ltd.     

Hypothesis testing via a penalized-likelihood approach

It is illustrated in this paper that hypothesis testing procedures can be derived based on the penalized likelihood approach. Based on this point of view, many traditional hypothesis tests, including the two-sample mean test, score test, and Hotelling’s $T^2$ test are revisited under the penalized likelihood framework. Similar framework is also applicable to the empirical likelihood.     

Multiple testing using the posterior probabilities of directional alternatives,with application to genomic studies

This paper aims to connect Bayesian analysis and frequentist theory in the context of multiple comparisons. The authors show that when testing the equality of two sample means, the posterior probability of the one‐sided alternative hypothesis, defined as a half‐space, shares with the frequentist P‐value the property of uniformity under the null hypothesis. Ultimately, the posterior probability may thus be used in the same spirit as a P‐value in the Benjamini‐Hochberg procedure, or in any of its extensions.     

Estimation of the slope in linear non-normal structural relationships

This paper studies the sensitivity to nonnormality of the normal-theory test for the null hypothesis that the slope is a specific value against a two-sided alternative. Edgeworth expansion and thus the asymptotic variance for the normal-theory maximum likelihood estimator of the slope are derived.     

Weighing the Evidence for Hypotheses with Small Samples of Right-censored Exponential Data

The p-value evidence for an alternative to a null hypothesis regarding the mean lifetime can be unreliable if based on asymptotic approximations when there is only a small sample of right-censored exponential data. However, a guarded weight of evidence for the alternative can always be obtained without approximation, no matter how small the sample, and has some other advantages over p-values. Weights of evidence are defined as estimators of 0 when the null hypothesis is true and 1 when the alternative is true, and they are judged on the basis of the ensuing risks, where risk is mean squared error of estimation. The evidence is guarded in that a preassigned bound is placed on the risk under the hypothesis. Practical suggestions are given for choosing the bound and for interpreting the magnitude of the weight of evidence. Acceptability profiles are obtained by inversion of a family of guarded weights of evidence for two-sided alternatives to point hypotheses, just as confidence intervals are obtained from tests; these profiles are arguably more informative than confidence intervals, and are easily determined for any level and any sample size, however small. They can help understand the effects of different amounts of censoring. They are found for several small size data sets, including a sample of size 12 for post-operative cancer patients. Both singly Type I and Type II censored examples are included. An examination of the risk functions of these guarded weights of evidence suggests that if the censoring time is of the same magnitude as the mean lifetime, or larger, then the risks in using a guarded weight of evidence based on a likelihood ratio are not much larger than they would be if the parameter were known.     

A monte carlo investigation of a statistic for a bivariate missing data problem

Testing the equal means hypothesis of a bivariate normal distribution with homoscedastic varlates when the data are incomplete is considered. If the correlational parameter, ρ, is known, the well-known theory of the general linear model is easily employed to construct the likelihood ratio test for the two sided alternative. A statistic, T, for the case of ρ unknown is proposed by direct analogy to the likelihood ratio statistic when ρ is known. The null and nonnull distribution of T is investigated by Monte Carlo techniques. It is concluded that T may be compared to the conventional t distribution for testing the null hypothesis and that this procedure results in a substantial increase in power-efficiency over the procedure based on the paired t test which ignores the incomplete data. A Monte Carlo comparison to two statistics proposed by Lin and Stivers (1974) suggests that the test based on T is more conservative than either of their statistics.     

Relative risk regression for current status data in case‐cohort studies

We propose using the weighted likelihood method to fit a general relative risk regression model for the current status data with missing data as arise, for example, in case‐cohort studies. The missingness probability is either known or can be reasonably estimated. Asymptotic properties of the weighted likelihood estimators are established. For the case of using estimated weights, we construct a general theorem that guarantees the asymptotic normality of the M‐estimator of a finite dimensional parameter in a class of semiparametric models, where the infinite dimensional parameter is allowed to converge at a slower than parametric rate, and some other parameters in the objective function are estimated a priori. The weighted bootstrap method is employed to estimate the variances. Simulations show that the proposed method works well for finite sample sizes. A motivating example of the case‐cohort study from an HIV vaccine trial is used to demonstrate the proposed method. The Canadian Journal of Statistics 39: 557–577; 2011. © 2011 Statistical Society of Canada     

Empirical likelihood‐based inferences for a low income proportion

Low income proportion is an important index in comparisons of poverty in countries around the world. The stability of a society depends heavily on this index. An accurate and reliable estimation of this index plays an important role for government's economic policies. In this paper, the authors study empirical likelihood‐based inferences for a low income proportion under the simple random sampling and stratified random sampling designs. It is shown that the limiting distributions of the empirical likelihood ratios for the low income proportion are the scaled chi‐square distributions. The authors propose various empirical likelihood‐based confidence intervals for the low income proportion. Extensive simulation studies are conducted to evaluate the relative performance of the normal approximation‐based interval, bootstrap‐based intervals, and the empirical likelihood‐based intervals. The proposed methods are also applied to analyzing a real economic survey income dataset. The Canadian Journal of Statistics 39: 1–16; 2011 ©2011 Statistical Society of Canada     

Three enigmatic examples and inference from likelihood

Statistics has many inference procedures for examining a model with data to obtain information concerning the value of a parameter of interest. If these give different results for the same model and data, one can reasonably want a satisfactory explanation. Over the last eighty years, three very simple examples have appeared intermittently in the literature, often with contradictory or misleading results; these enigmatic examples come from Cox, Behrens, and Box & Cox. The procedures in some generality begin with an observed likelihood function, which is known to provide just first order accuracy unless there is additional information that calibrates the parameter. In particular, default Bayes analysis seeks such calibration in the form of a model‐based prior; such a prior with second order accuracy is examined for the Behrens problem, but none seems available for the Box and Cox problem. Alternatively, higher‐order likelihood theory obtains such information by examining likelihood at and near the data and achieves third order accuracy. We examine both Bayesian and frequentist procedures in the context of the three enigmatic examples; simulations support the indicated accuracies. The Canadian Journal of Statistics © 2009 Statistical Society of Canada     

Reduce computation in profile empirical likelihood method

Since its introduction by Owen (1988, 1990), the empirical likelihood method has been extensively investigated and widely used to construct confidence regions and to test hypotheses in the literature. For a large class of statistics that can be obtained via solving estimating equations, the empirical likelihood function can be formulated from these estimating equations as proposed by Qin and Lawless (1994). If only a small part of parameters is of interest, a profile empirical likelihood method has to be employed to construct confidence regions, which could be computationally costly. In this article the authors propose a jackknife empirical likelihood method to overcome this computational burden. This proposed method is easy to implement and works well in practice. The Canadian Journal of Statistics 39: 370–384; 2011 © 2011 Statistical Society of Canada     

Likelihood ratio test against simple stochastic ordering among several multinomial populations

Stochastic ordering between probability distributions has been widely studied in the past 50 years. Because it is often easy to make valuable judgments when such orderings exist, it is desirable to recognize their existence and to model distributional structures under them. Likelihood ratio test is the most commonly used method to test hypotheses involving stochastic orderings. Among the various formally defined notions of stochastic ordering, the least stringent is simple stochastic ordering. In this paper, we consider testing the hypothesis that all multinomial populations are identically distributed against the alternative that they are in simple stochastic ordering. We construct likelihood ratio test statistic for this hypothesis test problem, provide limit form of the objective function corresponding to the test statistic and show that the test statistic is asymptotically distributed as a mixture of chi-squared distributions, i.e., a chi-bar-squared distribution.     

The score test for the two‐sample occupancy model

The score test statistic from the observed information is easy to compute numerically. Its large sample distribution under the null hypothesis is well known and is equivalent to that of the score test based on the expected information, the likelihood‐ratio test and the Wald test. However, several authors have noted that under the alternative hypothesis this no longer holds and in particular the score statistic from the observed information can take negative values. We extend the anthology on the score test to a problem of interest in ecology when studying species occurrence. This is the comparison of two zero‐inflated binomial random variables from two independent samples under imperfect detection. An analysis of eigenvalues associated with the score test in this setting assists in understanding why using the observed information matrix in the score test can be problematic. We demonstrate through a combination of simulations and theoretical analysis that the power of the score test calculated under the observed information decreases as the populations being compared become more dissimilar. In particular, the score test based on the observed information is inconsistent. Finally, we propose a modified rule that rejects the null hypothesis when the score statistic is computed using the observed information is negative or is larger than the usual chi‐square cut‐off. In simulations in our setting this has power that is comparable to the Wald and likelihood ratio tests and consistency is largely restored. Our new test is easy to use and inference is possible. Supplementary material for this article is available online as per journal instructions.