Tests for Homogeneity of Variances Using Robust Weighted Likelihood Estimates

The classical normal-theory tests for testing the null hypothesis of common variance and the classical estimates of scale have long been known to be quite nonrobust to even mild deviations from normality assumptions for moderate sample sizes. Levene (1960) suggested a one-way ANOVA type statistic as a robust test. Brown and Forsythe (1974) considered a modified version of Levene's test by replacing the sample means with sample medians as estimates of population locations, and their test is computationally the simplest among the three tests recommended by Conover , Johnson , and Johnson (1981) in terms of robustness and power. In this paper a new robust and powerful test for homogeneity of variances is proposed based on a modification of Levene's test using the weighted likelihood estimates (Markatou , Basu , and Lindsay , 1996) of the population means. For two and three populations the proposed test using the Hellinger distance based weighted likelihood estimates is observed to achieve better empirical level and power than Brown-Forsythe's test in symmetric distributions having a thicker tail than the normal, and higher empirical power in skew distributions under the use of F distribution critical values.     

Bootstrap method of interior-branch test for phylogenetic trees

Statistical properties of the bootstrap test of interior branch lengths of phylogenetic trees have been studied and compared with those of the standard interior-branch test in computer simulations. Examination of the properties of the tests under the null hypothesis showed that both tests for an interior branch of a predetermined topology are quite reliable when the distribution of the branch length estimate approaches a normal distribution. Unlike the standard interior-branch test, the bootstrap test appears to retain this property even when the substitution rate varies among sites. In this case, the distribution of the branch length estimate deviates from a normal distribution, and the standard interior-branch test gives conservative confidence probability values. A simple correction method was developed for both interior- branch tests to be applied for testing the reliability of tree topologies estimated from sequence data. This correction for the standard interior-branch test appears to be as effective as that obtained in our previous study, though it is much simpler. The bootstrap and standard interior-branch tests for estimated topologies become conservative as the number of sequence groups in a star-like tree increases.      

Locating regions of differential variability in DNA and protein sequences.

In the comparison of DNA and protein sequences between species or between paralogues or among individuals within a species or population, there is often some indication that different regions of the sequence are divergent or polymorphic to different degrees, indicating differential constraint or diversifying selection operating in different regions of the sequence. The problem is to test statistically whether the observed regional differences in the density of variant sites represent real differences and then to estimate as accurately as possible the location of the differential regions. A method is given for testing and locating regions of differential variation. The method consists of calculating G(x(k)) = k/n - x(k)/N, where x(k) is the position of the kth variant site along the sequence, n is the total number of variant sites, and N is the total sequence length. The estimated region is the longest stretch of adjacent sequence for which G(x(k)) is monotonically increasing (a hot spot) or decreasing (a cold spot). Critical values of this length for tests of significance are given, a sequential method is developed for locating multiple differential regions, and the power of the method against various alternatives is explored. The method locates the endpoints of hot spots and cold spots of variation with high accuracy.     

Properties of Statistical Tests of Neutrality for DNA Polymorphism Data

A class of statistical tests based on molecular polymorphism data is studied to determine size and power properties. The class includes TAJIMA''s D statistic as well as the D* and F* tests proposed by FU and LI. A new method of constructing critical values for these tests is described. Simulations indicate that TAJIMA''s test is generally most powerful against the alternative hypotheses of selective sweep, population bottleneck, and population subdivision, among tests within this class. However, even TAJIMA''s test can detect a selective sweep or bottleneck only if it has occurred within a specific interval of time in the recent past or population subdivision only when it has persisted for a very long time. For greatest power against the particular alternatives studied here, it is better to sequence more alleles than more sites.     

Usefulness of molecular markers for detecting population bottlenecks via monitoring genetic change

It is important to detect population bottlenecks in threatened and managed species because bottlenecks can increase the risk of population extinction. Early detection is critical and can be facilitated by statistically powerful monitoring programs for detecting bottleneck-induced genetic change. We used Monte Carlo computer simulations to evaluate the power of the following tests for detecting genetic changes caused by a severe reduction in a population's effective size ( N _e): a test for loss of heterozygosity, two tests for loss of alleles, two tests for change in the distribution of allele frequencies, and a test for small N _e based on variance in allele frequencies (the 'variance test'). The variance test was most powerful; it provided an 85% probability of detecting a bottleneck of size N _e = 10 when monitoring five microsatellite loci and sampling 30 individuals both before and one generation after the bottleneck. The variance test was almost 10-times more powerful than a commonly used test for loss of heterozygosity, and it allowed for detection of bottlenecks before 5% of a population's heterozygosity had been lost. The second most powerful tests were generally the tests for loss of alleles. However, these tests had reduced power for detecting genetic bottlenecks caused by skewed sex ratios. We provide guidelines for the number of loci and individuals needed to achieve high-power tests when monitoring via the variance test. We also illustrate how the variance test performs when monitoring loci that have widely different allele frequency distributions as observed in five wild populations of mountain sheep ( Ovis canadensis ).     

A new statistical test for linkage heterogeneity.

A new, statistical test for linkage heterogeneity is described. It is a likelihood-ratio test based on a beta distribution for the prior distribution of the recombination fraction among families (or individuals). The null distribution for this statistic (called the B-test) is derived under a broad range of circumstances. Two other heterogeneity test statistics--the admixture test or A-test first described by Smith and Morton's test (here referred to as the K-test)--are also examined. The probability distribution for the K-test statistic is very sensitive to family size, whereas the other two statistics are not. All three statistics are somewhat sensitive to the magnitude of the recombination fraction theta. Critical values for each of the test statistics are given. A conservative approximation for both the A-test and B-test is given by a chi 2 distribution when P/2 instead of P is used for the observed significance level. In terms of power, the B-test performs best among the three tests over a broad range of alternate heterogeneity hypotheses--except for the specific case of admixture with loose linkage, in which the A-test performs best. Overall, the difference in power among the three tests is not large. An application to some recently published data on the fragile-X syndrome and X-chromosome markers is given.     

Interior-branch and bootstrap tests of phylogenetic trees

We have compared statistical properties of the interior-branch and bootstrap tests of phylogenetic trees when the neighbor-joining tree- building method is used. For each interior branch of a predetermined topology, the interior-branch and bootstrap tests provide the confidence values, PC and PB, respectively, that indicate the extent of statistical support of the sequence cluster generated by the branch. In phylogenetic analysis these two values are often interpreted in the same way, and if PC and PB are high (say, > or = 0.95), the sequence cluster is regarded as reliable. We have shown that PC is in fact the complement of the P-value used in the standard statistical test, but PB is not. Actually, the bootstrap test usually underestimates the extent of statistical support of species clusters. The relationship between the confidence values obtained by the two tests varies with both the topology and expected branch lengths of the true (model) tree. The most conspicuous difference between PC and PB is observed when the true tree is starlike, and there is a tendency for the difference to increase as the number of sequences in the tree increases. The reason for this is that the bootstrap test tends to become progressively more conservative as the number of sequences in the tree increases. Unlike the bootstrap, the interior-branch test has the same statistical properties irrespective of the number of sequences used when a predetermined tree is considered. Therefore, the interior-branch test appears to be preferable to the bootstrap test as long as unbiased estimators of evolutionary distances are used. However, when the interior-branch is applied to a tree estimated from a given data set, PC may give an overestimate of statistical confidence. For this case, we developed a method for computing a modified version (P'C) of the PC value and showed that this P'C tends to give a conservative estimate of statistical confidence, though it is not as conservative as PB. In this paper we have introduced a model in which evolutionary distances between sequences follow a multivariate normal distribution. This model allowed us to study the relationships between the two tests analytically.      

An Exact Unconditional test for the Hardy-Weinberg Equilibrium

An exact test based on the unconditional distribution of a test statistic for the Hardy-Weinberg equilibrium is introduced. This test is usually more powerful and requires less extensive tabulations compared with the (ordinary) exact conditional test. Tables of critical values are given for N = 5(1)100 and α =·10, ·05, ·01. Sample sizes required to attain a given power with the unconditional and conditional exact tests are provided.     

A Comparison of Methods to Deteet Adulteration in Natural Products Under a Restricted Alternative

Multivariate statistical tests used to assess purity of natural food products, such as fruit juices, do not assume prior knowledge about the most likely causes of adulteration. We describe a likelihood ratio test that has a directional alternate hypothesis so that prior knowlege about the most likely method of adulteration can be utilized. Critical regions for this statistic are based on a nonparametric tolerance interval, which makes it insensitive to an assumption of multivariate normality. This new test is compared to two tests currently used for identifying adulterated samples. It is shown that the new test will have increased power relative to the other tests in the direction of the predicted adulteration.     

Variability of Nuclear SSU-rDNA Group Introns Within Septoria Species: Incongruence with Host Sequence Phylogenies

We report structural features and distribution patterns of 26 different group I introns located at three distinct nucleotide positions in nuclear small subunit ribosomal DNA (SSU-rDNA) of 10 Septoria and 4 other anamorphic species related to the teleomorphic genus Mycosphaerella. Secondary structure and sequence characteristics assigned the introns to the common IC1 and IE groups. Intron distribution patterns and phylogenetic relationships strongly suggested that some horizontal transfer events have occurred among the closely related fungal species sampled. To test this hypothesis, we used a comparative approach of intron- and rDNA-based phylogenies through MP- and ML-based topology tests. Our results showed two statistically well-supported major incongruences between the intron and the equivalent internal transcribed spacer (ITS) tree comparisons made. Such absence of a co-evolutive history between group I introns and host sequences is discussed relatively to the intron structures, the mechanisms of intron movement, and the biology of the Mycosphaerella pathogenic fungi. Electronic Supplementary Material The online version of this article (doi:) contains supplementary material, which is available to authorized users. Reviewing Editor: Debashish Bhattacharya     

A Fourier characteristic of coding sequences: origins and a non-Fourier approximation.

The 3-base periodicity, identified as a pronounced peak at the frequency N/3 (N is the length of the DNA sequence) of the Fourier power spectrum of protein coding regions, is used as a marker in gene-finding algorithms to distinguish protein coding regions (exons) and noncoding regions (introns) of genomes. In this paper, we reveal the explanation of this phenomenon which results from a nonuniform distribution of nucleotides in the three coding positions. There is a linear correlation between the nucleotide distributions in the three codon positions and the power spectrum at the frequency N/3. Furthermore, this study indicates the relationship between the length of a DNA sequence and the variance of nucleotide distributions and the average Fourier power spectrum, which is the noise signal in gene-finding methods. The results presented in this paper provide an efficient way to compute the Fourier power spectrum at N/3 and the noise signal in gene-finding methods by calculating the nucleotide distributions in the three codon positions.     

Beyond Bonferroni: less conservative analyses for conservation genetics

Studies in conservation genetics often attempt to determine genetic differentiation between two or more temporally or geographically distinct sample collections. Pairwise p-values from Fisher’s exact tests or contingency Chi-square tests are commonly reported with a Bonferroni correction for multiple tests. While the Bonferroni correction controls the experiment-wise α, this correction is very conservative and results in greatly diminished power to detect differentiation among pairs of sample collections. An alternative is to control the false discovery rate (FDR) that provides increased power, but this method only maintains experiment-wise α when none of the pairwise comparisons are significant. Recent modifications to the FDR method provide a moderate approach to determining significance level. Simulations reveal that critical values of multiple comparison tests with both the Bonferroni method and a modified FDR method approach a minimum asymptote very near zero as the number of tests gets large, but the Bonferroni method approaches zero much more rapidly than the modified FDR method. I compared pairwise significance from three published studies using three critical values corresponding to Bonferroni, FDR, and modified FDR methods. Results suggest that the modified FDR method may provide the most biologically important critical value for evaluating significance of population differentiation in conservation genetics.␣Ultimately, more thorough reporting of statistical significance is needed to allow interpretation of biological significance of genetic differentiation among populations.An erratum to this article can be found at     

Length variation and secondary structure of introns in the Mlc1 gene in six species of Drosophila

A nearly universal feature of intron sequences is that even closely related species exhibit a large number of insertion/deletion differences. The goal of the analysis described here is to test whether the observed pattern of insertion/deletion events in the genealogy of the myosin alkali light chain (Mlc1) gene is consistent with neutrality, and if not, to determine the underlying forces of evolutionary change. Mlc1 pre-mRNA is alternatively spliced, and one constraint is that signals necessary for tissue-specificity of directed splicing must be conserved. If the total length of an intron is functionally constrained, then the distribution of indels on branches of the gene genealogy should reflect a departure from randomness. Here we perform a phylogenetic analysis, inferring ancestral states wherever possible on a phylogeny of 29 alleles of Mlc1 from six species of Drosophila. Observed patterns of indels on the genealogy were compared to those from simulated data, with the result that we cannot reject the null hypothesis of neutrality. A clear departure from a neutral prediction was seen in the excess folding free energy predicted for the introns flanking the alternatively spliced exon. Relative rate tests also suggest a retardation in the rate of Mlc1 sequence evolution in the simulans clade.      

A theoretical treatment of interval mapping of a disease gene using transmission disequilibrium tests

The genetic basis of the transmission disequilibrium test (TDT) for two-marker loci is explored from first principles. In this case, parents doubly heterozygous for a given haplotype at the pair of marker loci that are each in linkage disequilibrium with the disease gene with the further possibility of a second-order linkage disequilibrium are considered. The number of times such parents transmit the given haplotype to their affected offspring is counted and compared with the frequencies of haplotypes that are not transmitted. This is done separately for the coupling and repulsion phases of doubly heterozygous genotypes. Expectations of the counts for each of the sixteen cells possible with four-marker gametic types (transmitted vs not transmitted) are derived. Based on a test of symmetry in a square 4 × 4 contingency table, chi-square tests are proposed for the null hypothesis of no linkage between the markers and the disease gene. The power of the tests is discussed in terms of the corresponding non-centrality parameters for the alternative hypothesis that both the markers are linked with the disease locus. The results indicate that the power increases with the decrease in recombination probability and that it is higher for a lower frequency of the disease gene. Taking a pair of markers in an interval for exploring the linkage with the disease gene seems to be more informative than the single-marker case since the values of the non-centrality parameters tend to be consistently higher than their counterparts in the single-marker case. Limitations of the proposed test are also discussed.     

A mixture-model approach for parallel testing for unequal variances

Testing for unequal variances is usually performed in order to check the validity of the assumptions that underlie standard tests for differences between means (the t-test and anova). However, existing methods for testing for unequal variances (Levene's test and Bartlett's test) are notoriously non-robust to normality assumptions, especially for small sample sizes. Moreover, although these methods were designed to deal with one hypothesis at a time, modern applications (such as to microarrays and fMRI experiments) often involve parallel testing over a large number of levels (genes or voxels). Moreover, in these settings a shift in variance may be biologically relevant, perhaps even more so than a change in the mean. This paper proposes a parsimonious model for parallel testing of the equal variance hypothesis. It is designed to work well when the number of tests is large; typically much larger than the sample sizes. The tests are implemented using an empirical Bayes estimation procedure which `borrows information' across levels. The method is shown to be quite robust to deviations from normality, and to substantially increase the power to detect differences in variance over the more traditional approaches even when the normality assumption is valid.     

Phylogenetic test of the molecular clock and linearized trees

To estimate approximate divergence times of species or species groups with molecular data, we have developed a method of constructing a linearized tree under the assumption of a molecular clock. We present two tests of the molecular clock for a given topology: two-cluster test and branch-length test. The two-cluster test examines the hypothesis of the molecular clock for the two lineages created by an interior node of the tree, whereas the branch-length test examines the deviation of the branch length between the tree root and a tip from the average length. Sequences evolving excessively fast or slow at a high significance level may be eliminated. A linearized tree will then be constructed for a given topology for the remaining sequences under the assumption of rate constancy. We have used these methods to analyze hominoid mitochondrial DNA and drosophilid Adh gene sequences.      

SYMMETRY VERSUS ASYMMETRY IN SEXUAL ISOLATION EXPERIMENTS

Two hypotheses predicting the ancestral or derived status of populations and based on asymmetrical mate discrimination (Kaneshiro, 1976; Watanabe and Kawanishi, 1979) were tested using nine laboratory populations of D. simulans, a highly outcrossed ancestral population, and eight populations derived from it via founder-flush-crash cycles. The data from individual mating tests using pairwise combinations of these populations fit the Kaneshiro hypothesis reasonably well, rejecting the Watanabe-Kawanishi hypothesis. However, more powerful tests rejected the Kaneshiro hypothesis for the data we analyzed. The values for derived females predicted by the Kaneshiro hypothesis were biased: they were consistently high for derived males and consistently low for ancestral males. We propose a hypothesis, based on variation in mating propensities and symmetrical mate discrimination. We assessed the power of Kaneshiro's and our hypotheses to predict the number of matings between derived females and derived males by plotting predicted vs. observed values and fitting these points to the expected line of unit slope passing through the origin. Predictions of our hypothesis explained more of the variance (r² = 0.87) than predictions of the Kaneshiro model (r² = 0.63). While asymmetrical sexual isolation undoubtedly occurs between some species, its existence cannot be determined simply by measuring mating frequencies in a single experiment.     

Tests Against Qualitative Interaction: Exact Critical Values and Robust Tests

Consider a study to evaluate treatment A with a placebo in two or more groups of patients. If treatment A is beneficial to one group of patients and harmful to another, then we say that there is qualitative interaction or crossover interaction between patient groups and the treatments. Gail and Simon (1985, Biometrics 41, 361-372) developed a large-sample procedure for this testing problem. Their test has received favorable coverage in the literature. In this article, we obtain corresponding exact finite sample results for normal error distribution and provide a table of critical values. The test statistic is similar to the familiar F-ratio, and its p-value is equal to a weighted sum of tail areas of F-distributions. The computations to implement this are simple. A simulation study shows that the exact critical values provided here for normal error distribution are preferable to the asymptotic critical values for a wide range of error distributions. We also develop tests that are power robust against long-tailed error distributions. Our robust test uses M-estimators instead of the least squares estimators. We show that the efficiency robustness of the M-estimator translates to power robustness of the corresponding test. Therefore, our robust tests are better if outliers are expected. A simulation study illustrates the substantial power advantages of our robust tests.