Multiple Comparisons of Polynomial Distributions

Asymptotically correct 90 and 95 percentage points are given for multiple comparisons with control and for all pair comparisons of several independent samples of equal size from polynomial distributions. Test statistics are the maxima of the X²-statistics for single comparisons. For only two categories the asymptotic distributions of these test statistics result from DUNNETT'S many-one tests and TUKEY'S range test (cf. MILLER, 1981). The percentage points for comparisons with control are computed from the limit distribution of the test statistic under the overall hypothesis H₀. To some extent the applicability of these bounds is investigated by simulation. The bounds can also be used to improve Holm's sequentially rejective Bonferroni test procedure (cf. HOLM, 1979). The percentage points for all pair comparisons are obtained by large simulations. Especially for 3×3-tables the limit distribution of the test statistic under H₀ is derived also for samples of unequal size. Also these bounds can improve the corresponding Bonferroni-Holm procedure. Finally from SKIDÁK's probability inequality for normal random vectors (cf. SKIDÁK, 1967) a similar inequality is derived for dependent X²-variables applicable to simultaneous X²-tests.     

Monte Carlo Comparison of Multiple Comparison Procedures

Seven procedures of multiple comparisons: Tukey, Scheffé, Bonferroni, Studentized Maximum Modulus, Duncan, Newman-Keuls and F are compared with respect to the probability of the correct decision. Monte Carlo simulation shows that there is no the best procedure. AMS 1985 Subject Classification: 62 J 15.     

Pairwise Comparisons Using Trimmed Means or M-Estimators when Working with Dependent Groups

For J dependent groups, let θ_j, j = 1, …, J, be some measure of location associated with the jth group. A common goal is computing confidence intervals for the pairwise differences, θ_j — θ_k, j < k, such that the simultaneous probability coverage is 1 — α. If means are used, it is well known that slight departures from normality (as measured by the Kolmogorov distance) toward a heavy-tailed distribution can substantially inflate the standard error of the sample mean, which in turn can result in relatively low power. Also, when distributions differ in shape, or when sampling from skewed distributions with relatively light tails, practical problems arise when the goal is to obtain confidence intervals with simultaneous probability coverage reasonably close to the nominal level. Extant theoretical and simulation results suggest replacing means with trimmed means. The Tukey-McLaughlin method is easily adapted to the problem at hand via the Bonferroni inequality, but this paper illustrates that practical concerns remain. Here, the main result is that the percentile t bootstrap method, used in conjunction with trimmed means, gives improved probability coverage and substantially better power. A method based on a one-step M-estimator is also considered but found to be less satisfactory.     

Multiple comparisons distortions of parameter estimates

In experiments involving many variables, investigators typically use multiple comparisons procedures to determine differences that are unlikely to be the result of chance. However, investigators rarely consider how the magnitude of the greatest observed effect sizes may have been subject to bias resulting from multiple testing. These questions of bias become important to the extent investigators focus on the magnitude of the observed effects. As an example, such bias can lead to problems in attempting to validate results, if a biased effect size is used to power a follow-up study. An associated important consequence is that confidence intervals constructed using standard distributions may be badly biased. A bootstrap approach is used to estimate and adjust for the bias in the effect sizes of those variables showing strongest differences. This bias is not always present; some principles showing what factors may lead to greater bias are given and a proof of the convergence of the bootstrap distribution is provided.     

Multiple comparison of entropies with application to dinosaur biodiversity

Summary. Did the biodiversity of dinosaurs decline, or did it remain more or less constant before their mass extinction 65 million years ago? Sheehan et al. (1991, Science, 835–839) reported that the biodiversity of families of dinosaur species remained more or less constant preceding their extinction, suggesting extinction due to a cataclysmic event such as an asteroid strike. But that claim was based on the incorrect interpretation that a large p value associated with a test of null hypothesis of equality supports that null hypothesis. To assess whether there is a basis for such a claim, we formulate the problem as one of practical equivalence, in analogy to bioequivalence. We then develop reliable practical equivalence confidence intervals for differences of entropies by applying the bootstrap-t technique to a nearly pivotal quantity. Confidence intervals for changes in the biodiversity of dinosaurs are then computed, allowing the reader to assess whether there is evidence of near constancy of dinosaur biodiversity before extinction.     

Estimation and Multiple Comparisons When There Are Missing Values with an Application in Immunology

A method is suggested for handling multiple comparisons in repeated measurement situations with completely random missing values. Exact results are obtained for the situation with normally distributed observations in the case of compound symmetry. The method uses grouping with respect to the positions of the missing values. It is most efficient and best suited when there are not too many measurement occasions in the longitudinal investigation.     

Obtaining Critical Values for Simultaneous Confidence Intervals and Multiple Testing

There are many situations where it is desired to make simultaneous tests or give simultaneous confidence intervals for linear combinations (contrasts) of population or treatment means. Somerville (1997, 1999) developed algorithms for calculating the critical values for a large class of simultaneous tests and simultaneous confidence intervals. Fortran 90 and SAS‐IML batch programs and interactive programs were developed. These programs calculate the critical values for 15 different simultaneous confidence interval procedures (and the corresponding simultaneous tests) and for arbitrary procedures where the user specifies a combination of one and two sided contrasts. The programs can also be used to obtain the constants for “step‐down” testing of multiple hypotheses. This paper gives examples of the use of the algorithms and programs and illustrates their versatility and generality. The designs need not be balanced, multiple covariates may be present and there may be many missing values. The use of multiple regression and dummy variables to obtain the required variance covariance matrix is illustrated. Under weak normality assumptions the methods are “exact” and make the use of approximate methods or “simulation” unnecessary.     

Simultaneous Comparisons of Multiple Treatments to Two (or More) Control

DUNNETT (1955) developed a procedure simultaneously comparing k treatments to one control with an exact overall type I error of α when all sampling distributions are normal. Sometimes it is desirable to compare k treatments to m≧2 controls, in particular to two controls. For instance, several new therapies (e.g., pain relievers) could be compared to two standard therapies (e.g., Aspirin and Tylenol). Alternatively, a standard therapy could be very expensive, difficult to apply and/or have bad side effects, making it useful to compare each new therapy to both standard therapy and no therapy (Placebo). Dunnett's method is expanded here to give comparisons of mean values for k treatments to mean values for m≧2 controls at an exact overall type I error of α when all sampling distributions are normal. Tabled values needed to make exact simultaneous comparisons at α = .05 are given for m = 2. An application is made to an example from the literature.     

Comparisons of creatine kinase primary structures

Comparisons of nine creatine kinase sequences show that 67% of the protein sequence is identical among rabbit, rat, mouse, and chicken muscle, rabbit, rat, and chicken brain, and electric organ sequences from two species of electric ray(Torpedo). The extensive homology precludes a facile prediction of active-site residues based on sequence conservation. The sequences are more similar within isozyme types than are the different isozymes from any one species. There are 35 positions in the muscle and brain sequence pairs for three species which differentiate the two forms. TheTorpedo sequences do not fall completely into either of these patterns. Except for homology with partial sequences of other ATP-guanidino phosphotransferases, no significant homology with other protein or nucleic acid sequences in available databases was found. Preliminary secondary structural predictions suggest that the C-terminal half of the protein is likely an /-type protein. Placement in the sequence of two peptides found in previous cross-linking studies reveals two stretches of primary structure that are presumably close in space to the reactive Cys-283 and hence close to the active site.     

Generalized Maximum Range Tests for Pairwise Comparisons of Several Populations

A multiple comparison procedure (MCP) is proposed for the comparison of all pairs of several independent samples. This MCP is essentially the closed procedure with union-intersection tests based on given single tests Q_ij for the minimal hypotheses H_ij. In such cases where the α-levels of the nominal tests associated with the MCP can be exhausted, this MCP has a uniformly higher all pair power than any refined Bonferroni test using the same Q_ij. Two different general algorithms are described in section 3. A probability inequality for ranges of i.i.d. random variables which is useful for some algorithms is proved in section 4. Section 5 contains the application to independent normally distributed estimates and section 6 the comparisons of polynomial distributions by multivariate ranges. Further applications are possible. Tables of the 0.05-bounds for the tests of section 5 and 6 are enclosed.     

Multiple Comparisons of Treatments with Stable Multivariate Tests in a Two‐Stage Adaptive Design,Including a Test for Non‐Inferiority

The application of stabilized multivariate tests is demonstrated in the analysis of a two‐stage adaptive clinical trial with three treatment arms. Due to the clinical problem, the multiple comparisons include tests of superiority as well as a test for non‐inferiority, where non‐inferiority is (because of missing absolute tolerance limits) expressed as linear contrast of the three treatments. Special emphasis is paid to the combination of the three sources of multiplicity – multiple endpoints, multiple treatments, and two stages of the adaptive design. Particularly, the adaptation after the first stage comprises a change of the a‐priori order of hypotheses.     

Fractionation in Triad Comparisons

A statistical procedure for comparing the performance of a new product with the existing products on the basis of sensory characters has been developed. A test statistic was evolved for testing the null hypothesis of equality of treatment effects in the case of fractional triad comparisons. The null distribution of the test statistic has been obtained and it is found that it has a χ²-distribution for large number of observations. The procedure is quite simple and is based on a distribution-free test requiring only ordinal scale measurement. The method of analysis has been explained by a numerical example.     

Multiple McNemar Tests

Summary Methods for performing multiple tests of paired proportions are described. A broadly applicable method using McNemar's exact test and the exact distributions of all test statistics is developed; the method controls the familywise error rate in the strong sense under minimal assumptions. A closed form (not simulation‐based) algorithm for carrying out the method is provided. A bootstrap alternative is developed to account for correlation structures. Operating characteristics of these and other methods are evaluated via a simulation study. Applications to multiple comparisons of predictive models for disease classification and to postmarket surveillance of adverse events are given.     

Three Multiple Comparison Procedures for Trimmed Means

Two common goals when choosing a method for performing all pairwise comparisons of J independent groups are controlling experiment wise Type I error and maximizing power. Typically groups are compared in terms of their means, but it has been known for over 30 years that the power of these methods becomes highly unsatisfactory under slight departures from normality toward heavy-tailed distributions. An approach to this problem, well-known in the statistical literature, is to replace the sample mean with a measure of location having a standard error that is relatively unaffected by heavy tails and outliers. One possibility is to use the trimmed mean. This paper describes three such multiple comparison procedures and compares them to two methods for comparing means.     

Sample Sizes for Comparisons of k Treatments with a Control Based on Different Definitions of the Power

The determination of sample sizes for the comparison of k treatments against a control by means of the test of Dunnett (1955, 1964) as well as by means of the multiple t-test will be considered. The power in multiple comparisons can be defined in different ways, see Hochberg and Tamhane (1987). We will derive formulas for the per-pair power, the any-pair power and the all-pairs power for both one- and two-sided comparisons. Tables will be provided that allow sample sizes to be determined for preassigned values of the power.     

An undersirable property of Hill's diversity indexN 2

Summary A desirable property of a diversity index is that it be a convex function of the proportions of different species that constitute a community.N ₂, a diversity index belonging to the Hill's series of diversity indices and which is equivalent to the reciprocal of Simpson's diversity index violates this requirement when the proportion of one of the species is close to 1. Recommendations ofN ₂ as the best possible index of diversity therefore need to be examined judiciously.     

Biodiversity of Living,Non-marine,Thrombolites of Lake Clifton,Western Australia

Lake Clifton in Western Australia is recognized as a critically endangered ecosystem and the only thrombolite reef in the southern hemisphere. There have been concerns that increases in salinity and nutrient run-off have significantly impacted upon the thrombolite microbial community. Here we used cultivation-independent molecular approaches to characterize the microbial diversity of the thrombolites at Lake Clifton. The most dominant phyla currently represented are the Proteobacteria with significant populations of Bacteroidetes and Firmicutes. Cyanobacteria, previously invoked as the main drivers of thrombolite growth, represent only a small fraction (～1–3% relative abundance) of the microbial community. We report an increase in salinity and nitrogen levels at Lake Clifton that may be contributing to a change in dominant microbial populations. This heightens concerns about the long-term health of the Lake Clifton thrombolites; future work is needed to determine if phyla now dominating this system are capable of the required mineral precipitation for continued thrombolite growth.     

Biodiversity of constructed wetlands for wastewater treatment

Constructed wetlands are often built for wastewater treatment to mitigate the adverse effects of organic pollution in streams and rivers caused by inputs of municipal wastewater. However, there has been little analysis of biodiversity and related factors influencing the ecosystem functioning of constructed wetlands. The purpose of this study was to evaluate the biodiversity of two free-water-surface integrated constructed wetlands in subtropical Taiwan by analyzing the water quality, habitat characteristics, and biotic communities of algae, macrophytes, birds, fish, and aquatic macroinvertebrates in the treatment cells. Our results indicated that the two integrated constructed wetlands (Hsin-Hai II and Daniaopi Constructed Wetlands) achieved good performance in reducing the concentrations of total nitrogen (TN) and total phosphorus (TP), and loadings of biochemical oxygen demand (BOD) and chemical oxygen demand (COD) from municipal sewage. In total, 58 bird species, 7 fish species, and 34 aquatic macroinvertebrate taxa were recorded in the two wetlands. The results of stepwise multiple regressions showed that the richness, abundance, and diversity of birds increased with wetland area. Fish richness and abundance respectively increased with wetland area and dissolved oxygen, while the diversity decreased with increases in TP concentrations. The richness and density of aquatic macroinvertebrates increased with the cover of aquatic macrophytes, while the diversity increased with wetland area. Ordination analyses indicated that variations in the community structures of birds, fishes, and aquatic macroinvertebrates were respectively best explained by water temperature, wetland area, and species richness of fish. Our results suggest that wetland area, cover of aquatic macrophytes, and water quality were the most important factors governing the diversity in the constructed wetlands, and that the factors influencing community structures varied among different taxonomic groups. In addition to improving water quality, this study implied that the biodiversity of constructed wetlands for wastewater treatment can be enhanced through proper design and management.     

A comparison of species diversity estimators

Although having been much criticized, diversity indices are still widely used in animal and plant ecology to evaluate, survey, and conserve ecosystems. It is possible to quantify biodiversity by using estimators for which statistical characteristics and performance are, as yet, poorly defined. In the present study, four of the most frequently used diversity indices were compared: the Shannon index, the Simpson index, the Camargo eveness index, and the Pielou regularity index. Comparisons were performed by simulating the Zipf–Mandelbrot parametric model and estimating three statistics of these indices, i.e., the relative bias, the coefficient of variation, and the relative root-mean-squared error. Analysis of variance was used to determine which of the factors contributed most to the observed variation in the four diversity estimators: abundance distribution model or sample size. The results have revealed that the Camargo eveness index tends to demonstrate a high bias and a large relative root-mean-squared error whereas the Simpson index is least biased and the Shannon index shows a smaller relative root-mean-squared error, regardless of the abundance distribution model used and even when sample size is small. Shannon and Pielou estimators are sensitive to changes in species abundance pattern and present a nonnegligible bias for small sample sizes (<1000 individuals). Received: May 8, 1998 / Accepted: May 6, 1999