Trendtests with adaptive scoring

The concept of adaptive two‐stage designs is applied to the problem of testing the equality of several normal means against an ordered (monotone) alternative. The likelihood‐ratio‐test proposed by Bartholomew is known to have favorable power properties when testing against a monotonic trend. Tests based on contrasts provide a flexible way to incorporate available information regarding the pattern of the unknown true means through appropriate specification of the scores. The basic idea of the presented concept is the combination of Bartholomew 's test (first stage) with an “adaptive score test” (second stage) which utilizes the information resulting from isotonic regression estimation at the first stage. In a Monte Carlo simulation study the adaptive scoring procedure is compared to the non‐adaptive two‐stage procedure using the Bartholomew test at both stages. We found that adaptive scoring may improve the power of the two stage design, in particular if the sample size at the first stage is considerably larger than at the second stage.     

Common pitfalls when testing additivity of treatment mixtures with chi‐square analyses

Studying interactions of multiple pesticides applied simultaneously in a mixture is a common task in phytopathology. Statistical methods are employed to test whether the treatment components influence each other's efficacy in a promotive or inhibitory way (synergistic or antagonistic interaction) or rather act independent of one another (additivity). The trouble is that widely used procedures based on chi‐square tests are often seriously flawed, either because people apply them in a preposterous way or because the method simply does not fit the problem at hand. Browsing recent volumes of entomological journals, we found that numerous researchers have (in all likelihood unwittingly) analysed their data as if they had had a sample size of 100 or, equally bad, a sample size of one! We show how to avoid such poor practices and further argue that chi‐square testing is, even if applied correctly (meaning that no technical errors are made), a limited purpose tool for assessing treatment interactions.     

Comparing Power Behaviours for Some Goodness-of-Fit Test Statistics in Sparse Multinomials

This paper is concerned with the power behaviour of four goodness-of-fit test statistics in sparse multinomials with k cells. Most previous work has been concerned only with both Pearson's X² and the likelihood ratio test statistics. We consider in this study, two additional test statistics, namely, the Cressie-Read test statistic – I(2/3) and the modified Freeman-Tukey test (FT) statistic. Because k ≥ 10 in this study, a Monte Carlo procedure based on 1000 simulated samples is used to estimate the powers for the four test statistics. Alternatives on various line segments are employed. Results suggest that none of the test statistics completely dominate the other and that the choice of which test to use depends on the nature of the alternative hypothesis. These results are consistent with those obtained by West and Kempthorne (1972), although, the Pearson's χ² test statistic may be preferred because of its closer approximation to the χ² distribution in terms of the attained α levels.     

Notes on Testing Equality in Dichotomous Data with Matched Pairs

In attempting to improve the efficiency of McNemar's test statistic, we develop two test procedures that account for the information on both the discordant and concordant pairs for testing equality between two comparison groups in dichotomous data with matched pairs. Furthermore, we derive a test procedure derived from one of the most commonly‐used interval estimators for odds ratio. We compare these procedures with those using McNemar's test, McNemar's test with the continuity correction, and the exact test with respect to type I error and power in a variety of situations. We note that the test procedures using McNemar's test with the continuity correction and the exact test can be quite conservative and hence lose much efficiency, while the test procedure using McNemar's test can actually perform well even when the expected number of discordant pairs is small. We also find that the two test procedures, which incorporate the information on all matched pairs into hypothesis testing, may slightly improve the power of using McNemar's test without essentially losing the precision of type I error. On the other hand, the test procedure derived from an interval estimator of adds ratio with use of the logarithmic transformation may have type I error much larger than the nominal α‐level when the expected number of discordant pairs is not large and therefore, is not recommended for general use.     

Confidence Intervals of the Simple Difference between the Proportions of a Primary Infection and a Secondary Infection,Given the Primary Infection

This paper discusses interval estimation of the simple difference (SD) between the proportions of the primary infection and the secondary infection, given the primary infection, by developing three asymptotic interval estimators using Wald's test statistic, the likelihood‐ratio test, and the basic principle of Fieller's theorem. This paper further evaluates and compares the performance of these interval estimators with respect to the coverage probability and the expected length of the resulting confidence intervals. This paper finds that the asymptotic confidence interval using the likelihood ratio test consistently performs well in all situations considered here. When the underlying SD is within 0.10 and the total number of subjects is not large (say, 50), this paper further finds that the interval estimators using Fieller's theorem would be preferable to the estimator using the Wald's test statistic if the primary infection probability were moderate (say, 0.30), but the latter is preferable to the former if this probability were large (say, 0.80). When the total number of subjects is large (say, ≥200), all the three interval estimators perform well in almost all situations considered in this paper. In these cases, for simplicity, we may apply either of the two interval estimators using Wald's test statistic or Fieller's theorem without losing much accuracy and efficiency as compared with the interval estimator using the asymptotic likelihood ratio test.     

Use of the Maximum Likelihood Method in the Analysis of Phenotypic Stability

The stability variance is an important estimator of phenotypic stability of genotypes. It may be estimated by method of moments and by maximum likelihood. We demonstrate by Monte Carlo simulation that, given a sufficient number of environments, maximum likelihood estimates (MLE's) are slightly better if ranking of genotypes is the experimenter's major aim. A likelihood ratio test is available for different hypotheses.     

A Test of Equality of Survival Distributions for Two Sample Censored Grouped Survival Data

A common testing problem for a life table or survival data is to test the equality of two survival distributions when the data is both grouped and censored. Several tests have been proposed in the literature which require various assumptions about the censoring distributions. It is shown that if these conditions are relaxed then the tests may no longer have the stated properties. The maximum likelihood test of equality when no assumptions are made about the censoring marginal distributions is derived. The properties of the test are found and it is compared to the existing tests. The fact that no assumptions are required about the censoring distributions make the test a useful initial testing procedure.     

Stratified or Multicentre Analysis with Extended Mantel–Haenszel Statistics

Since its introduction in 1959 the ability of the classical Mantel-Haenszel (M–H) procedure for combining the odds ratios of a set of I 2 × 2 tables has led to its use also in stratified or multicentre type clinical trials. A familiar application is the M–H logrank test in survival analysis. An extension of the M–H procedure covering the case of 2 × K contingency tables (MANTEL , 1963) with ordered levels retains the essential property of pooling the results of I homogeneous tables (i.e. in absence of qualitative interactions). The assignment of some score for the K columns of a table is essential for the use of the method (in comparing 2 treatments). Some possibilities of score assignment are discussed: for clinical outcome variables such as the degree of severity of a disease, pain and so on, the score is at hand in a natural way. A less well-known type of scoring consists in ranking the observations of a continuous variable, leading to cell sizes of 1 or 0. In this case, however, if equidistant ranking was used, the E–M–H procedure appears as an extension of Wilcoxon's rank sum test and represents a powerful non-parametric approach in stratified or multicentre type designs with non normally distributed outcome variables. The results of some Monte-Carlo simulations for 2 possible equidistant ranking procedures are presented, which indicate only a moderate gain in power as compared to Wilcoxon's rank sum test under the common situation of centre effects not exceeding treatment effects. Use of the E–M–H pro?edure is also recommended as a simple method to overcome the potential bias due to unequally distributed prognostic factors among treatment groups.     

Testing for qualitative interactions between treatment effects and patient subsets

Evaluation of evidence that treatment efficacy varies substantially among different subsets of patients is an important feature of the analysis of large clinical trials. Qualitative or crossover interactions are said to occur when one treatment is superior for some subsets of patients and the alternative treatment is superior for other subsets. A non-crossover interaction arises when there is variation in the magnitude, but not in the direction, of treatment effects among subsets. Some authors use the term quantitative interaction to mean non-crossover interaction. Non-crossover interactions are usually of less clinical importance than qualitative interactions, which often have major therapeutic significance. A likelihood ratio test is developed to test for qualitative interactions. Exact critical values are determined and tabulated.     

Powerful Haplotype-Based Hardy-Weinberg Equilibrium Tests for Tightly Linked Loci

Recently, there have been many case-control studies proposed to test for association between haplotypes and disease, which require the Hardy-Weinberg equilibrium (HWE) assumption of haplotype frequencies. As such, haplotype inference of unphased genotypes and development of haplotype-based HWE tests are crucial prior to fine mapping. The goodness-of-fit test is a frequently-used method to test for HWE for multiple tightly-linked loci. However, its degrees of freedom dramatically increase with the increase of the number of loci, which may lack the test power. Therefore, in this paper, to improve the test power for haplotype-based HWE, we first write out two likelihood functions of the observed data based on the Niu''s model (NM) and inbreeding model (IM), respectively, which can cause the departure from HWE. Then, we use two expectation-maximization algorithms and one expectation-conditional-maximization algorithm to estimate the model parameters under the HWE, IM and NM models, respectively. Finally, we propose the likelihood ratio tests LRT and LRT for haplotype-based HWE under the NM and IM models, respectively. We simulate the HWE, Niu''s, inbreeding and population stratification models to assess the validity and compare the performance of these two LRT tests. The simulation results show that both of the tests control the type I error rates well in testing for haplotype-based HWE. If the NM model is true, then LRT is more powerful. While, if the true model is the IM model, then LRT has better performance in power. Under the population stratification model, LRT is still more powerful. To this end, LRT is generally recommended. Application of the proposed methods to a rheumatoid arthritis data set further illustrates their utility for real data analysis.     

Testing the effect of sex differences on sib-sib correlations

Procedures for testing the effect of sex differences on sib-sib correlations are studied. It is demonstrated that a simple procedure for testing the equality of a pair of brother-brother and sister-sister correlations (the modified Z test) is comparable in power to the likelihood ratio test in moderately large samples drawn from sibship-size distributions that are likely to occur in practice. In small samples, the modified Z test is preferable to the likelihood ratio test, since the latter tends to be anti-conservative.     

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.     

Testing whether an identified treatment is best

We consider the problem of testing whether an identified treatment is better than each of K treatments. Suppose there are univariate test statistics Si that contrast the identified treatment with treatment i for i = 1, 2,...., K. The min test is defined to be the alpha-level procedure that rejects the null hypothesis that the identified treatment is not best when, for all i, Si rejects the one-sided hypothesis, at the alpha-level, that the identified treatment is not better than the ith treatment. In the normal case where Si are t statistics the min test is the likelihood ratio test. For distributions satisfying mild regularity conditions, if attention is restricted to test statistics that are monotone nondecreasing functions of Si, then regardless of their covariance structure the min test is an optimal alpha-level test. Tables of the sample size needed to achieve power .5, .8, .90, and .95 are given for the min test when the Si are Student's t and Wilcoxon.     

Detecting Specific Populations in Mixtures

Mixed stock analysis (MSA) estimates the relative contributions of distinct populations in a mixture of organisms. Increasingly, MSA is used to judge the presence or absence of specific populations in specific mixture samples. This is commonly done by inspecting the bootstrap confidence interval of the contribution of interest. This method has a number of statistical deficiencies, including almost zero power to detect small contributions even if the population has perfect identifiability. We introduce a more powerful method based on the likelihood ratio test and compare both methods in a simulation demonstration using a 17 population baseline of sockeye salmon, Oncorhynchus nerka, from the Kenai River, Alaska, watershed. Power to detect a nonzero contribution will vary with the population(s) identifiability relative to the rest of the baseline, the contribution size, mixture sample size, and analysis method. The demonstration shows that the likelihood ratio method is always more powerful than the bootstrap method, the two methods only being equal when both display 100% power. Power declines for both methods as contribution declines, but it declines faster and goes to zero for the bootstrap method. Power declines quickly for both methods as population identifiability declines, though the likelihood ratio test is able to capitalize on the presence of 'perfect identification' characteristics, such as private alleles. Given the baseline-specific nature of detection power, researchers are encouraged to conduct a priori power analyses similar to the current demonstration when planning their applications.     

A Note on the Bias of O'Brien's OLS Test

O'Brien's Ordinary Least Squares (OLS) test is a well known procedure for testing the multivariate one-sided hypothesis when the covariance matrix is unknown. Simulation results have been reported where the actual test level exceeds the nominal one. Here it is shown analytically that the actual level is always larger than the nominal one if the covariance matrix is nonnegative.     

Discussion on “Adaptive enrichment designs with a continuous biomarker” by Nigel Stallard

We congratulate Dr. Nigel Stallard on his stimulating paper on adaptive enrichment designs with a continuous biomarker. Dr. Stallard details a framework for a large and interesting class of enrichment procedures. His work has motivated us to offer some thoughts in response. Dr. Stallard's strategy is to use the maximum of a test statistic over a set of possible threshold values to define the enriched population to be sampled in a second stage. This reminds us of procedures for identifying a change point, a biomarker value beyond which the effect of treatment is increased. For simplicity we focus our comments on Dr. Stallard's Rule 1 for selecting the second-stage sampling threshold. Using this rule, we present the likelihood ratio approach for adaptive testing and compare it to Dr. Stallard's approach for a few scenarios.     

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 random effects in linear mixed‐effects models with serially correlated errors

In linear mixed‐effects models, random effects are used to capture the heterogeneity and variability between individuals due to unmeasured covariates or unknown biological differences. Testing for the need of random effects is a nonstandard problem because it requires testing on the boundary of parameter space where the asymptotic chi‐squared distribution of the classical tests such as likelihood ratio and score tests is incorrect. In the literature several tests have been proposed to overcome this difficulty, however all of these tests rely on the restrictive assumption of i.i.d. measurement errors. The presence of correlated errors, which often happens in practice, makes testing random effects much more difficult. In this paper, we propose a permutation test for random effects in the presence of serially correlated errors. The proposed test not only avoids issues with the boundary of parameter space, but also can be used for testing multiple random effects and any subset of them. Our permutation procedure includes the permutation procedure in Drikvandi, Verbeke, Khodadadi, and Partovi Nia (2013) as a special case when errors are i.i.d., though the test statistics are different. We use simulations and a real data analysis to evaluate the performance of the proposed permutation test. We have found that random slopes for linear and quadratic time effects may not be significant when measurement errors are serially correlated.     

Pervasive adaptive evolution in mammalian fertilization proteins

Mammalian fertilization exhibits species specificity, and the proteins mediating sperm-egg interactions evolve rapidly between species. In this study, we demonstrate that the evolution of seven genes involved in mammalian fertilization is promoted by positive Darwinian selection by using likelihood ratio tests (LRTs). Several of these proteins are sperm proteins that have been implicated in binding the mammalian egg coat zona pellucida glycoproteins, which were shown previously to be subjected to positive selection. Taken together, these represent the major candidates involved in mammalian fertilization, indicating positive selection is pervasive amongst mammalian reproductive proteins. A new LRT is implemented to determine if the d(N)/d(S) ratio is significantly greater than one. This is a more refined test of positive selection than the previous LRTs which only identified if there was a class of sites with a d(N)/d(S) ratio >1 but did not test if that ratio was significantly greater than one.