Small-sample inference for the comparison of means of log-normal distributions

We propose a likelihood-based test for comparing the means of two or more log-normal distributions, with possibly unequal variances. A modification to the likelihood ratio test is needed when sample sizes are small. The performance of the proposed procedures is compared with the F-ratio test using Monte Carlo simulations.     

On the finite sample distribution of the likelihood ratio statistic for testing heterogeneity in meta-analysis

In meta-analysis, hypothesis testing is one of the commonly used approaches for assessing whether heterogeneity exists in effects between studies. The literature concluded that the Q-statistic is clearly the best choice and criticized the performance of the likelihood ratio test in terms of the type I error control and power. However, all the criticism for the likelihood ratio test is based on the use of a mixture of two chi-square distributions with 0 and 1 degrees of freedom, which is justified only asymptotically. In this study, we develop a novel method to derive the finite sample distribution of the likelihood ratio test and restricted likelihood ratio test statistics for testing the zero variance component in the random effects model for meta-analysis. We also extend this result to the heterogeneity test when metaregression is applied. A numerical study shows that the proposed statistics have superior performance to the Q-statistic, especially when the number of studies collected for meta-analysis is small to moderate.     

Testing for Poisson Zero Inflation in Disease Mapping

When analyzing mortality data due to rare diseases in small areas, it is common to find several health zones with no mortality cases. In these circumstances, the classical homogeneous model based on the Poisson distribution used to estimate the relative risks within each area may encounter lack of fit due to a disproportionately large frequency of zeros. To cope with these zeros, the zero inflated Poisson model can be used. In this paper, we propose a test for detecting zero inflation in the context of disease mapping which is based on bootstrap techniques. The test is illustrated using male mortality data due to brain cancer in Navarra, Spain. In addition, comparisons with other tests for Poisson zero inflation such as the score test and the likelihood ratio test are carried out in terms of empirical power and size using the brain cancer scenario. The proposed bootstrap test has good power and size and works well when detecting the excess of zeros in small area data sets. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Testing the non-unity of rate ratio under inverse sampling

Inverse sampling is considered to be a more appropriate sampling scheme than the usual binomial sampling scheme when subjects arrive sequentially, when the underlying response of interest is acute, and when maximum likelihood estimators of some epidemiologic indices are undefined. In this article, we study various statistics for testing non-unity rate ratios in case-control studies under inverse sampling. These include the Wald, unconditional score, likelihood ratio and conditional score statistics. Three methods (the asymptotic, conditional exact, and Mid-P methods) are adopted for P-value calculation. We evaluate the performance of different combinations of test statistics and P-value calculation methods in terms of their empirical sizes and powers via Monte Carlo simulation. In general, asymptotic score and conditional score tests are preferable for their actual type I error rates are well controlled around the pre-chosen nominal level, and their powers are comparatively the largest. The exact version of Wald test is recommended if one wants to control the actual type I error rate at or below the pre-chosen nominal level. If larger power is expected and fluctuation of sizes around the pre-chosen nominal level are allowed, then the Mid-P version of Wald test is a desirable alternative. We illustrate the methodologies with a real example from a heart disease study.     

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     

Is the Exact Test Better than the Asymptotic Test for Testing Marginal Homogeneity in 2 × 2 Tables?

McNemar test is commonly used to test for the marginal homogeneity in 2 × 2 contingency tables. McNemar test is an asymptotic test based either on standard normal distribution or on the chi‐square distribution. When the total sample size is small, an exact version of McNemar test is available based on the binomial probabilities. The example in the paper came from a clinical study to investigate the effect of epidermal growth factor for children who had microvillus inclusion diseases. There were only six observations available. The test results differ between the exact test and the asymptotic test. It is a common belief that with this small sample size the exact test be used. However, we claim that McNemar test performs better than the exact test even when the sample size is small. In order to investigate the performances of McNemar test and the exact test, we identify the parameters that affect the test results and then perform sensitivity analysis. In addition, through Monte Carlo simulation studies we compare the empirical sizes and powers of these tests as well as other asymptotic tests such as Wald test and the likelihood ratio test.     

Comparison of properties of tests for assessing tumor clonality

SUMMARY: In a recent article Begg et al. (2007, Biometrics 63, 522-530) proposed a statistical test to determine whether or not a diagnosed second primary tumor is biologically independent of the original primary tumor, by comparing patterns of allelic losses at candidate genetic loci. The proposed concordant mutations test is a conditional test, an adaptation of Fisher's exact test, that requires no knowledge of the marginal mutation probabilities. The test was shown to have generally good properties, but is susceptible to anticonservative bias if there is wide variation in mutation probabilities between loci, or if the individual mutation probabilities of the parental alleles for individual patients differ substantially from each other. In this article, a likelihood ratio test is derived in an effort to address these validity issues. This test requires prespecification of the marginal mutation probabilities at each locus, parameters for which some information will typically be available in the literature. In simulations this test is shown to be valid, but to be considerably less efficient than the concordant mutations test for sample sizes (numbers of informative loci) typical of this problem. Much of the efficiency deficit can be recovered, however, by restricting the allelic imbalance parameter estimate to a prespecified range, assuming that this parameter is in the prespecified range.     

The Wald Statistic in Proportional Hazards Hypothesis Testing

In survivorship modelling using the proportional hazards model of Cox (1972, Journal of the Royal Statistical Society, Series B, 34, 187–220), it is often desired to test a subset of the vector of unknown regression parameters β in the expression for the hazard rate at time t. The likelihood ratio test statistic is well behaved in most situations but may be expensive to calculate. The Wald (1943, Transactions of the American Mathematical Society 54, 426–482) test statistic is easier to calculate, but has some drawbacks. In testing a single parameter in a binomial logit model, Hauck and Donner (1977, Journal of the American Statistical Association 72, 851–853) show that the Wald statistic decreases to zero the further the parameter estimate is from the null and that the asymptotic power of the test decreases to the significance level. The Wald statistic is extensively used in statistical software packages for survivorship modelling and it is therefore important to understand its behavior. The present work examines empirically the behavior of the Wald statistic under various departures from the null hypothesis and under the presence of Type I censoring and covariates in the model. It is shown via examples that the Wald statistic's behavior is not as aberrant as found for the logistic model. For the single parameter case, the asymptotic non-null distribution of the Wald statistic is examined.     

Quantifying the change of melanoma incidence by Breslow thickness

Melanoma incidence has increased throughout the world over the past 25 years. A surrogate for the severity of melanoma is the Breslow thickness of the lesions. Data on melanoma, including Breslow thickness, were collected in 1978-1980 and 1988-1990 from the Tasmania Tumor Registry. We use a density ratio model to quantify the change of melanoma by Breslow thickness. In this model, the ratio of two densities is assumed to have a known form up to a parameter, but the underlying densities are not modeled. This model includes the length bias sampling model as a special case. The Kolmogorov-Smirnov test statistic is used to test the correctness of the density ratio model. Model-based cumulative distribution estimation is studied. Methodology developed in this article is applied to the Tasmania Tumor Registry data.     

Statistical Inference For Risk Difference in an Incomplete Correlated 2 × 2 Table

In some infectious disease studies and 2‐step treatment studies, 2 × 2 table with structural zero could arise in situations where it is theoretically impossible for a particular cell to contain observations or structural void is introduced by design. In this article, we propose a score test of hypotheses pertaining to the marginal and conditional probabilities in a 2 × 2 table with structural zero via the risk/rate difference measure. Score test‐based confidence interval will also be outlined. We evaluate the performance of the score test and the existing likelihood ratio test. Our empirical results evince the similar and satisfactory performance of the two tests (with appropriate adjustments) in terms of coverage probability and expected interval width. Both tests consistently perform well from small‐ to moderate‐sample designs. The score test however has the advantage that it is only undefined in one scenario while the likelihood ratio test can be undefined in many scenarios. We illustrate our method by a real example from a two‐step tuberculosis skin test study.     

A note on permutation tests for variance components in multilevel generalized linear mixed models

In many applications of generalized linear mixed models to multilevel data, it is of interest to test whether a random effects variance component is zero. It is well known that the usual asymptotic chi-square distribution of the likelihood ratio and score statistics under the null does not necessarily hold. In this note we propose a permutation test, based on randomly permuting the indices associated with a given level of the model, that has the correct Type I error rate under the null. Results from a simulation study suggest that it is more powerful than tests based on mixtures of chi-square distributions. The proposed test is illustrated using data on the familial aggregation of sleep disturbance.     

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.