Non-linear tests for identifying differentially expressed genes or genetic networks

MOTIVATION: One of the recently developed statistics for identifying differentially expressed genetic networks is Hotelling T2 statistic, which is a quadratic form of difference in linear functions of means of gene expressions between two types of tissue samples, and so their power is limited. RESULTS: To improve the power of test statistics, a general statistical framework for construction of non-linear tests is presented, and two specific non-linear test statistics that use non-linear transformations of means are developed. Asymptotical distributions of the non-linear test statistics under the null and alternative hypothesis are derived. It has been proved that under some conditions the power of the non-linear test statistics is higher than that of the T2 statistic. Besides theory, to evaluate in practice the performance of the non-linear test statistics, they are applied to two real datasets. The preliminary results demonstrate that the P-values of the non-linear statistics for testing differential expressions of the genetic networks are much smaller than those of the T2 statistic. And furthermore simulations show the Type I errors of the non-linear statistics agree with the threshold used and the statistics fit the chi2 distribution. SUPPLEMENTARY INFORMATION: Supplementary data are available on Bioinformatics online.     

A conservative test for multiple comparison based on highly correlated test statistics

In genetics, we often encounter a large number of highly correlated test statistics. The most famous conservative bound for multiple comparison is Bonferroni's bound, which is suitable when the test statistics are independent but not when the test statistics are highly correlated. This article proposes a new conservative bound that is easily calculated without multiple integration and is a good approximation when the test statistics are highly correlated. The performance of the proposed method is evaluated by simulation and real data analysis.     

Weighted Logrank Testing Procedures for the Simple Tree Alternatives

Weighted logrank testing procedures for comparing r treatments with a control when some of the data are randomly censored are discussed. Four kinds of test statistics for the simple tree alternatives are considered. The weighted logrank statistics based on pairwise ranking scheme is proposed and the covariances of the test statistics are explicitly obtained. This class of test statistics can be viewed as the general statistics of constructing the test procedures for various order restricted alternatives by modifying weights. Four kinds of weighted logrank tests are illustrated with an example. Simulation studies are performed to compare the sizes and the powers of the considered tests with the other.     

A Note on Combining Dependent Tests of Significance

In combining several tests of significance the individual test statistics are allowed to be stochastically dependent. By choosing the weighted inverse normal method for the combination, the dependency of the original test statistics is then characterized by a correlation of the transformed statistics. For this correlation a confidence region, an unbiased estimator and an unbiased estimate of its variance are derived. The combined test statistic is extended to include the case of possibly dependent original test statistics. Simulation studies show the performance of the actual significance level.     

Exploiting excess sharing: a more powerful test of linkage for affected sib pairs than the transmission/disequilibrium test

The transmission/disequilibrium test (TDT) is a popular, simple, and powerful test of linkage, which can be used to analyze data consisting of transmissions to the affected members of families with any kind pedigree structure, including affected sib pairs (ASPs). Although it is based on the preferential transmission of a particular marker allele across families, it is not a valid test of association for ASPs. Martin et al. devised a similar statistic for ASPs, Tsp, which is also based on preferential transmission of a marker allele but which is a valid test of both linkage and association for ASPs. It is, however, less powerful than the TDT as a test of linkage for ASPs. What I show is that the differences between the TDT and Tsp are due to the fact that, although both statistics are based on preferential transmission of a marker allele, the TDT also exploits excess sharing in identity-by-descent transmissions to ASPs. Furthermore, I show that both of these statistics are members of a family of "TDT-like" statistics for ASPs. The statistics in this family are based on preferential transmission but also, to varying extents, exploit excess sharing. From this family of statistics, we see that, although the TDT exploits excess sharing to some extent, it is possible to do so to a greater extent-and thus produce a more powerful test of linkage, for ASPs, than is provided by the TDT. Power simulations conducted under a number of disease models are used to verify that the most powerful member of this family of TDT-like statistics is more powerful than the TDT for ASPs.     

Statistical Tests for Signals in Categorial Temporal Data

This paper describes methods for using categorical temporal data to detect differences in behavior between a treated group and a control group. The first-level output from the data is typically a set of many different correlated test statistics comparing the two groups. In previous work, a decision was made by counting the number of significant individual tests and calibrating with bootstrap simulation. This article goes further, suggesting two possible alternative statistics: the sum of the squared individual test statistics and a Wald-like combination of the individual test statistics. All three overall comparison statistics are defined and a method for computing critical values from simulated distributions using a bootstrap method is given. The use of all three methods is then demonstrated on each of three data sets. Finally, a simulated power study reveals that the Wald-like statistic is much better than the other two, leading to the suggestion of its use in place of the other two statistics.     

Nonlinear tests for genomewide association studies

As millions of single-nucleotide polymorphisms (SNPs) have been identified and high-throughput genotyping technologies have been rapidly developed, large-scale genomewide association studies are soon within reach. However, since a genomewide association study involves a large number of SNPs it is therefore nearly impossible to ensure a genomewide significance level of 0.05 using the available statistics, although the multiple-test problems can be alleviated, but not sufficiently, by the use of tagging SNPs. One strategy to circumvent the multiple-test problem associated with genome-wide association tests is to develop novel test statistics with high power. In this report, we introduce several nonlinear tests, which are based on nonlinear transformation of allele or haplotype frequencies. We investigate the power of the nonlinear test statistics and demonstrate that under certain conditions, some nonlinear test statistics have much higher power than the standard chi2-test statistic. Type I error rates of the nonlinear tests are validated using simulation studies. We also show that a class of similarity measure-based test statistics is based on the quadratic function of allele or haplotype frequencies, and thus they belong to nonlinear tests. To evaluate their performance, the nonlinear test statistics are also applied to three real data sets. Our study shows that nonlinear test statistics have great potential in association studies of complex diseases.     

An Alternative Method for Meta-Analysis

In many fields of applications, test statistics are obtained by combining estimates from several experiments, studies or centres of a multi-centre trial. The commonly used test procedure to judge the evidence of a common overall effect can result in a considerable overestimation of the significance level, leading to a high rate of too liberal decisions. Alternative test statistics are presented and better approximating test distributions are derived. Explicitly discussed are the methods in the unbalanced heteroscedastic 1-way random ANOVA model and for the probability difference method, including interaction treatment by centres or studies. Numerical results are presented by simulation studies.     

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.     

A Comparison of Weighted Tests for a Slope Change in Simple Regression

Testing for a change in the slope of the simple linear regression model has many applications in bio‐sciences, quality control and survival analysis. This paper compares Anderson‐Darling and Erdós‐Darling type test statistics which are based on the least squares change point process of Sen (1980) with the corresponding Kolmogorov‐Smirnov and Crámer‐von Mises type test statistics. We estimated the limiting critical values of these test statistics and conducted Monte Carlo simulation studies to compare their powers.     

Tests for Equality of Several Binomial Populations Against an Order Restricted Alternative and Model Selection for One-Dimensional Multinomials

The problem of testing the equality of several binomial population against an order restricted alternative and model selection for one-dimensional multinomials is studied. Test procedures are proposed. The asymptotic distribution of the test statistics are obtained. Comparisons are made with other test statistics including the likelihood ratio test for stochastic ordering. Also alternatives which does not depend on the distribution of test statistic is proposed.     

A comparison of test statistics for assessing the effects of concomitant variables in survival analysis

In data analysis involving the proportional-hazards regression model due to Cox (1972, Journal of the Royal Statistical Society, Series B 34, 187-220), the test criteria commonly used for assessing the partial contribution to survival of subsets of concomitant variables are the classical likelihood ratio (LR) and Wald statistics. This paper presents an investigation of three other test criteria with potentially major computational advantages over the classical tests, especially for stepwise variable selection in moderate to large data sets. The alternative criteria considered are Rao's efficient score statistic and two other score statistics. Under the Cox model, the performance of these tests is examined empirically and compared with the performance of the LR and Wald statistics. Rao's test performs comparably to the LR test in all the cases considered. The performance of the other criteria is competitive in many cases. The use of these statistics is illustrated in a study of coronary artery disease.     

The use of bootstrap resampling to assess the uncertainty of cooper statistics

The predictive abilities of two-group classification models (CMs) are often expressed in terms of their Cooper statistics. These statistics are often reported without any indication of their uncertainty, making it impossible to judge whether the predicted classifications are significantly better than the predictions made by a different CM, or whether the predictive performance of the CM exceeds predefined performance criteria in a statistically significant way. Bootstrap resampling routines are reported that provide a means of expressing the uncertainty associated with Cooper statistics. The usefulness of the bootstrapping routines is illustrated by constructing 95% confidence intervals for the Cooper statistics of four alternative skin-corrosivity tests (the rat skin transcutaneous electrical resistance assay, EPISKIN, Skin(2) and CORROSITEX), and four two-step sequences in which each in vitro test is used in combination with a physicochemical test for skin corrosion based on pH measurements.     

Multipoint linkage detection in the presence of heterogeneity

Linkage heterogeneity is common for complex diseases. It is well known that loss of statistical power for detecting linkage will result if one assumes complete homogeneity in the presence of linkage heterogeneity. To this end, Smith (1963, Annals of Human Genetics 27, 175-182) proposed an admixture model to account for linkage heterogeneity. It is well known that for this model, the conventional chi-squared approximation to the likelihood ratio test for no linkage does not apply even when the sample size is large. By dealing with nuclear families and one marker at a time for genetic diseases with simple modes of inheritance, score-based test statistics (Liang and Rathouz, 1999, Biometrics 55, 65-74) and likelihood-ratio-based test statistics (Lemdani and Pons, 1995, Biometrics 51, 1033-1041) have been proposed which have a simple large-sample distribution under the null hypothesis of linkage. In this paper, we extend their work to more practical situations that include information from multiple markers and multi-generational pedigrees while allowing for a class of general genetic models. Three different approaches are proposed to eliminate the nuisance parameters in these test statistics. We show that all three approaches lead to the same asymptotic distribution under the null hypothesis of no linkage. Simulation results show that the proposed test statistics have adequate power to detect linkage and that the performances of these two classes of test statistics are quite comparable. We have applied the proposed method to a family study of asthma (Barnes et al., 1996), in which the score-based test shows evidence of linkage with p-value <0.0001 in the region of interest on chromosome 12. Additionally, we have implemented this score-based test within the frequently used computer package GENEHUNTER.     

The use of score tests for inference on variance components

Whenever inference for variance components is required, the choice between one-sided and two-sided tests is crucial. This choice is usually driven by whether or not negative variance components are permitted. For two-sided tests, classical inferential procedures can be followed, based on likelihood ratios, score statistics, or Wald statistics. For one-sided tests, however, one-sided test statistics need to be developed, and their null distribution derived. While this has received considerable attention in the context of the likelihood ratio test, there appears to be much confusion about the related problem for the score test. The aim of this paper is to illustrate that classical (two-sided) score test statistics, frequently advocated in practice, cannot be used in this context, but that well-chosen one-sided counterparts could be used instead. The relation with likelihood ratio tests will be established, and all results are illustrated in an analysis of continuous longitudinal data using linear mixed models.     

General Analysis of Covariance Models for the Split Block and Related Block Designs

Exact test statistics and confidence intervals for a general split block ANOCOVA model are derived. With a single covariate, each statistic for testing main effect A, main effect B, and the AxB interaction has one less numerator degree of freedom than its counterpart in the ordinary ANOVA without a covariate. Sufficient conditions on the model parameters which allow these lost numerator degrees of freedom to be regained are given, as are exact statistics and confidence intervals for the corresponding reduced models. A note of caution is offered when constructing test statistics for reduced versions of the general model using the method of generalized least squares. General analysis of covariance models for two other block designs are presented.     

Gene genealogy and properties of test statistics of neutrality under population growth

We consider the Wright-Fisher model with exponential population growth and investigate effects of population growth on the shape of genealogy and the distributions of several test statistics of neutrality. In the limiting case as the population grows rapidly, the rapid-growth-limit genealogy is characterized. We obtained approximate expressions for expectations and variances of test statistics in the rapid-growth-limit genealogy and star genealogy. The distributions in the star genealogy are narrower than those in the cases of the simulated and rapid-growth-limit genealogies. The expectations and variances of the test statistics are monotone decreasing functions of the time length of the expansion, and the higher power of R(2) against population growth is suggested to be due to their smaller variances rather than to change of the expectations. We also investigated by simulation how quickly the distributions of test statistics approach those of the rapid-growth-limit genealogy.     

Testing Equality of Correlated Means in the Presence of Unequal Variances and Missing Values

Several statistics are proposed for testing the hypothesis of equality of the means of bivariate normal distribution with unknown variances and correlation coefficient when observations are missing on both variatea. The null distributions of the statistics are approximated by well-known distributions. The empirical sizes and powers of the statistics are computed and compared with paired t test and some of the known statistics based on available data. The comparisons support the use of two of the statistics proposed in this paper.     

A note on testing separability in spatial-temporal marked point processes

Summary .   In environmental risk analysis, it is common to assume the stochastic independence (or separability) between the marks associated with the random events of a spatial-temporal point process. Schoenberg (2004, Biometrics 60, 471–481) proposed several test statistics for this hypothesis and used simulated data to evaluate their performance. He found that a Cramér-von Mises-type test is powerful to detect gradual departures from separability although it is not uniformly powerful over a large class of alternative models. We present a semiparametric approach to model alternative hypotheses to separability and derive a score test statistic. We show that there is a relationship between this score test and some of the test statistics proposed by Schoenberg. Specifically, all are different versions of weighted Cramér-von Mises-type statistics. This gives some insight into the reasons for the similarities and differences between the test statistics' performance. We also point out some difficulties in controlling the type I error probability in Schoenberg's residual test.     

Tests of proportional hazards and proportional odds models for grouped survival data

In this paper, we derive score test statistics to discriminate between proportional hazards and proportional odds models for grouped survival data. These models are embedded within a power family transformation in order to obtain the score tests. In simple cases, some small-sample results are obtained for the score statistics using Monte Carlo simulations. Score statistics have distributions well approximated by the chi-squared distribution. Real examples illustrate the proposed tests.