Single-step Procedures Research Articles

Three p‐value consistent procedures for multiple comparisons with a control in direction‐mixed families

In clinical studies, it is common to compare several treatments with a control. In such cases, the most popular statistical technique is the Dunnett procedure. However, the Dunnett procedure is designed to deal with particular families of inferences in which all hypotheses are either one sided or two sided. Recently, based on the minimization of average simultaneous confidence interval width, a single-step procedure was derived to handle more general inferential families that contained a mixture of one- and two-sided inferences. But that single-step procedure is unable to guarantee the condition of p-value consistency which means that when a hypothesis with a certain p-value is rejected, all other hypotheses with smaller p-values are also rejected. In this paper, we present a single-step procedure and two stepwise procedures which are p-value consistent. The two proposed stepwise procedures provide more powerful testing methods when compared with single-step procedures. The extent of their superiority is demonstrated with a simulation study of average power. Selected critical values are tabulated for the implementation of the three proposed procedures. Additional simulation studies provide evidence that the new stepwise procedures are robust to moderate changes in the underlying probability distributions, and the proposed step-up procedure is uniformly more powerful than the resampling-based Hochberg step-up approach in all considered distribution models. Finally, we provide a practical example with sample data extracted from a medical experiment.

A Generalized Sidak-Holm Procedure and Control of Generalized Error Rates under Independence

For testing multiple null hypotheses, the classical approach to dealing with the multiplicity problem is to restrict attention to procedures that control the familywise error rate (FWER), the probability of even one false rejection. In many applications, one might be willing to tolerate more than one false rejection provided the number of such cases is controlled, thereby increasing the ability of the procedure to detect false null hypotheses. This suggests replacing control of the FWER by controlling the probability of k or more false rejections, which is called the k-FWER. In Hommel and Hoffmann (1987) and Lehmann and Romano (2005a), single step and stepdown procedures are derived that control the k-FWER, without making any assumptions concerning the dependence structure of the p-values of the individual tests. However, if the p-values are mutually independent, one can improve the procedures. In fact, Sarkar (2005) provided such an improvement. However, we show other improvements are possible which appear to be generally much better, and are sometimes unimprovable. When k=1, the procedure reduces to the classical method of Sidak, and the stepdown procedure is unimprovable and strictly dominates that of Sarkar. Under a monotonicity condition, an unimprovable procedure is obtained. In the case k=2, the monotonicity condition is satisfied, and the condition can be checked numerically in general. We then develop a stepdown method that controls the false discovery proportion. Except for the case of k-FWER control with k=1, the gains are surprisingly dramatic, and theoretical and numerical evidence is given.

Simple CVD Routes Towards Infiltration of Mesoporous TiO2.

AbstractFor Abstract see ChemInform Abstract in Full Text.

Generalizations of the familywise error rate

Consider the problem of simultaneously testing null hypotheses H_1,...,H_s. The usual approach to dealing with the multiplicity problem is to restrict attention to procedures that control the familywise error rate (FWER), the probability of even one false rejection. In many applications, particularly if s is large, one might be willing to tolerate more than one false rejection provided the number of such cases is controlled, thereby increasing the ability of the procedure to detect false null hypotheses. This suggests replacing control of the FWER by controlling the probability of k or more false rejections, which we call the k-FWER. We derive both single-step and stepdown procedures that control the k-FWER, without making any assumptions concerning the dependence structure of the p-values of the individual tests. In particular, we derive a stepdown procedure that is quite simple to apply, and prove that it cannot be improved without violation of control of the k-FWER. We also consider the false discovery proportion (FDP) defined by the number of false rejections divided by the total number of rejections (defined to be 0 if there are no rejections). The false discovery rate proposed by Benjamini and Hochberg [J. Roy. Statist. Soc. Ser. B 57 (1995) 289-300] controls E(FDP). Here, we construct methods such that, for any \gamma and \alpha, P{FDP>\gamma}\le\alpha. Two stepdown methods are proposed. The first holds under mild conditions on the dependence structure of p-values, while the second is more conservative but holds without any dependence assumptions.

Simple CVD Routes Towards Infiltration of Mesoporous TiO2

There is considerable interest in developing simple, single-step chemical procedures for controlled infiltration of mesoporous materials in order to create novel structures with practical applications. In this paper, we demonstrate the use of low-pressure metal-organic (MO)CVD and aerosol assisted (AA)CVD methods for infiltration and growth of compound semiconductors (CdS and CuInS 2 ) within mesoporous TiO 2 thin films using single-source dithiocarbamate precursors. The relative rates of surface reaction and diffusion of precursors can be controlled through simple modification of our existing single-step CVD processes. Films have been characterized by X-ray diffraction (XRD), scanning electron microscopy (SEM). energy dispersive X-ray (EDX), and electronic spectroscopy. Promising transient absorption data have been obtained for mesostructured CuInS 2 / TIO 2 films, which suggest the AACVD-grown ternary has desirable properties for employment as a solar sensitizer in solid-state inorganic photovoltaic cells. The simplicity of the approach may also find practical applications in the production of inverse opals for photonic crystals and thin films.

Multiple Comparisons with the Best Control in a One-Way Layout

In many settings, one wishes to show that one or more experimental treatments are better than the best of more than one control or standard treatments in a one-way layout. One-sided, normal theory and distribution-free, single-step and step-down simultaneous testing procedures are developed in this article. These tests are intersection-union tests and hence use existing critical points. The single-step tests are inverted to form simultaneous lower confidence bounds on the amount of improvement that each experimental treatment offers relative to the best control or standard treatment. The single-step, normal theory procedures can be improved if one assumes that the control expectations do not differ by more than a predetermined number of standard deviations.

9,9-Dimethyl-8,10-dioxapentacyclo[5.3.0.0 2,5.0 3,5.0 3,6]decane and naphthotetracyclo[5.1.0.0 1,6.0 2,7]oct-3-ene: new substituted [1.1.1]propellanes as precursors to 1,2,3,4-tetrafunctionalized bicyclo[1.1.1]pentanes

Two new substituted [1.1.1]propellanes have been generated from the corresponding bicyclo[1.1.0]butanes in either single-step ( 1a ) or four-step procedures ( 1b ). The observed degree of double lithiation of the bicyclo[1.1.0]butanes is discussed in the context of DFT computational results. Addition reactions across the central C(1)–C(3) bonds of the propellanes were studied. Only the propellane 1b gave the biacetyl addition product.

FUZZY C-MEANS CLUSTERING MODELS FOR MULTIVARIATE TIME-VARYING DATA: DIFFERENT APPROACHES

The classification of multivariate time-varying data finds application in several fields, such as economics, finance, marketing research, psychometrics, bioinformatics, medicine, signal processing, pattern recognition, etc. In this paper, by considering an exploratory formalization, we propose different unsupervised clustering models for multivariate data time arrays (objects×quantitative variables×times). These models can be classified in two different approaches: the cross sectional and the longitudinal approach. In the first case, after the objects, observed at each time, have been classified, comparison among the classifications made in different time instants will be done. In the second approach, we cluster the time trajectories of the objects; then, we obtain only one classification by comparing the instantaneous and evolutive features of the trajectories of the objects. In particular, in this work, the second approach is analyzed in detail, with reference to the so-called single and double step procedures. Geometric, correlative, instantaneous, evolutive and trend characteristics of the multivariate time arrays are taken into account in the different proposed clustering models. Furthermore, the fuzzy approach, that is particularly suitable in the dynamic classification problem, has been considered. Extensions of a cluster-validity criterion for the proposed fuzzy dynamic clustering models are also suggested. A socio-economic example concludes the paper.

Multiple Testing. Part I. Single-Step Procedures for Control of General Type I Error Rates

The present article proposes general single-step multiple testing procedures for controlling Type I error rates defined as arbitrary parameters of the distribution of the number of Type I errors, such as the generalized family-wise error rate. A key feature of our approach is the test statistics null distribution (rather than data generating null distribution) used to derive cut-offs (i.e., rejection regions) for these test statistics and the resulting adjusted p-values. For general null hypotheses, corresponding to submodels for the data generating distribution, we identify an asymptotic domination condition for a null distribution under which single-step common-quantile and common-cut-off procedures asymptotically control the Type I error rate, for arbitrary data generating distributions, without the need for conditions such as subset pivotality. Inspired by this general characterization of a null distribution, we then propose as an explicit null distribution the asymptotic distribution of the vector of null value shifted and scaled test statistics. In the special case of family-wise error rate (FWER) control, our method yields the single-step minP and maxT procedures, based on minima of unadjusted p-values and maxima of test statistics, respectively, with the important distinction in the choice of null distribution. Single-step procedures based on consistent estimators of the null distribution are shown to also provide asymptotic control of the Type I error rate. A general bootstrap algorithm is supplied to conveniently obtain consistent estimators of the null distribution. The special cases of t- and F-statistics are discussed in detail. The companion articles focus on step-down multiple testing procedures for control of the FWER (van der Laan et al., 2004b) and on augmentations of FWER-controlling methods to control error rates such as tail probabilities for the number of false positives and for the proportion of false positives among the rejected hypotheses (van der Laan et al., 2004a). The proposed bootstrap multiple testing procedures are evaluated by a simulation study and applied to genomic data in the fourth article of the series (Pollard et al., 2004).

Testing for Treatment Effects on Subsets of EndpointsThe view presented in this paper are those of authors and not necessary representing those of the U.S. Food and Drug Administration

Multiple endpoints are tested to assess an overall treatment effect and also to identify which endpoints or subsets of endpoints contributed to treatment differences. The conventional p-value adjustment methods, such as single-step, step-up, or step-down procedures, sequentially identify each significant individual endpoint. Closed test procedures can also detect individual endpoints that have effects via a step-by-step closed strategy. This paper proposes a global-based statistic for testing an a priori number, say, r of the k endpoints, as opposed to the conventional approach of testing one (r = 1) endpoint. The proposed test statistic is an extension of the single-step p-value-based statistic based on the distribution of the smallest p-value. The test maintains strong control of the FamilyWise Error (FWE) rate under the null hypothesis of no difference in any (sub)set of r endpoints among all possible combinations of the k endpoints. After rejecting the null hypothesis, the individual endpoints in the sets that are rejected can be tested further, using a univariate test statistic in a second step, if desired. However, the second step test only weakly controls the FWE. The proposed method is illustrated by application to a psychosis data set.

Multiple hypotheses testing and expected number of type I. errors

The performance of multiple test procedures with respect to error control is an old issue. Assuming that all hypotheses are true we investigate the behavior of the expected number of type I errors (ENE) as a characteristic of certain multiple tests controlling the familywise error rate (FWER) or the false discovery rate (FDR) at a prespecified level. We derive explicit formulas for the distribution of the number of false rejections as well as for the ENE for single-step, step-down and step-up procedures based on independent $p$-values. Moreover, we determine the corresponding asymptotic distributions of the number of false rejections as well as explicit formulae for the ENE if the number of hypotheses tends to infinity. In case of FWER-control we mostly obtain Poisson distributions and in one case a geometric distribution as limiting distributions; in case of FDR control we obtain limiting distributions which are apparently not named in the literature. Surprisingly, the ENE is bounded by a small number regardless of the number of hypotheses under consideration. Finally, it turns out that in case of dependent test statistics the ENE behaves completely differently compared to the case of independent test statistics.

Multiplicity, directional (Type III) errors, and the Null Hypothesis.

L. V. Jones and J. W. Tukey (2000) pointed out that the usual 2-sided, equal-tails null hypothesis test at level alpha can be reinterpreted as simultaneous tests of 2 directional inequality hypotheses, each at level alpha/2, and that the maximum probability of a Type I error is alpha/2 if the truth of the null hypothesis is considered impossible. This article points out that in multiple testing with familywise error rate controlled at alpha, the directional error rate (assuming all null hypotheses are false) is greater than alpha/2 and can be arbitrarily close to alpha. Single-step, step-down, and step-up procedures are analyzed, and other error rates, including the false discovery rate, are discussed. Implications for confidence interval estimation and hypothesis testing practices are considered.

Producing "self-plasticizing" ion-selective membranes.

Polymer membranes have been explored for the analysis of ions that do not require plasticizers and with photocurable properties. This work was focused on investigating the viability of the methacrylic-acrylic copolymers as new self-plasticizing membrane matrixes for ion-selective electrodes or other ion-sensor applications. Copolymers with glass transition temperatures ranging from -20 to -44 degrees C could be prepared without added plasticizer and were found to be functional as ion-selective membranes. Both free-radical solution polymerization and photopolymerization could be used, and "self-plasticizing" behavior of copolymers was observed with a high alkylacrylate (R = C4) content. This was found to be compatible with most commercially available ionophores, and sensors for potassium, sodium, calcium and pH were fabricated entirely by photocure procedures; single-step procedures for the immobilization of benzo-15-crown-5 ionophore on these self-plasticizing copolymer matrixes were also developed. Even though the ionophore was immobilized, potentiometric studies revealed that the ionophore remained functional, and thus, these copolymers have the advantage of suffering neither leaching of ionophore nor plasticizers. All these sensors exhibited a Nernstian or near Nernstian response with selectivity comparable to plasticized PVC membranes or other plasticized and photocurable polymer membranes. The long-term response of the potassium sensor with immobilized ionophore and the sodium sensor showed little deterioration for as long as one month and three months, respectively, under continuous use.

Conformational and Packing Modeling of Optically Active Polyesters. 1. Planar Zigzag Structure of Isotactic Polylactone

The optically active and crystalline poly(α-methyl-α-ethyl-β-propiolactone) (PMEPL) shows conformational changes and polymorphic behavior depending on the preparation method of the sample. The melt-crystallized isotactic PMEPL exhibits a monoclinic lattice and an extended planar zigzag conformation. Using computational modeling, we have solved and refined its crystal structure. Both the two-step and single-step procedures were used to evaluate packing energies. The results indicate that two chains with the same chirality and opposite directions pass through a monoclinic unit cell having lattice dimensions a = 9.10, b = 7.44, c(fiber period) = 4.84 A and β = 83.1°. The space group is P21 with unique axis b. The final crystal structure was confirmed by comparison with electron diffraction data and X-ray powder diffraction spectra. Both procedures result in crystal structures in good agreement with the experiments. The final discrepancy R factors are 0.16 for the flexible model and 0.20 for the rigid model w...

A new approach to variable-topology shape design using a constraint on perimeter

This paper introduces a method for variable-topology shape optimization of elastic structures called theperimeter method. An upper-bound constraint on the perimeter of the solid part of the structure ensures a well-posed design problem. The perimeter constraint allows the designer to control the number of holes in the optimal design and to establish their characteristic length scale. Finite element implementations generate practical designs that are convergent with respect to grid refinement. Thus, an arbitrary level of geometric resolution can be achieved, so single-step procedures for topology design and detailed shape design are possible. The perimeter method eliminates the need for relaxation, thereby circumventing many of the complexities and restrictions of other approaches to topology design.

The Effects of Purification of Matching Criterion on the Identification of DIF Using the Mantel-Haenszel Procedure

To distinguish between differential item functioning. (DIF) and item impact, the Mantel-Haenszel procedure compares the odds for success between groups after conditioning on ability. This conditioning requires a valid criterion, Because an appropriate external criterion is not routinely available, practitioners have typically used the total test score as the matching criterion. Holland and Thayer recommended a potential means of improving that criterion by using a two-step form of the Mantel-Haenszel procedure in which items identified as displaying DIF on an initial implementation are removed from the matching criterion for a subsequent analysis. This study examines the utility of their recommendation by comparing the results of the single-step and two-step procedures in the context of a simulated data set. The results show the two-step procedure to be equal or superior to the single-step procedure in identifying simulated DIF items across all conditions examined. This advantage is achieved without apparent inflation in the Type I error rate. A comparison of the single-step and two-step forms of the procedure is also carried out using an actual data set. The results show changes in identification rate between the two steps of the order of magnitude of those observed with simulated data.

Aromatic Isothiazolopyridines: New Direct Synthetic Approaches

Abstract Alternative synthetic routes to the title ring systems were examined: the isothiazolopyridines 5a,b and 10 were obtained by single step procedures from pyridine derivatives.

Processable conductive polypyrrole/poly(alkyl methacrylate) composites prepared by an emulsion pathway

Electrically conductive polymer composites of polypyrrole and poly(alkyl methacrylate) have been prepared using a two-step procedure. First, an emulsion is generated by dispersing a chloroform solution of poly(alkyl methacrylate) and pyrrole in a small amount of an aqueous surfactant solution. The surfactant is adsorbed upon the interface between the two phases and ensures, via double-layer repulsion, the stability of the emulsion. Second, the pyrrole present in the emulsion is polymerized and doped by introducing with stirring an aqueous solution of an oxidant in the emulsion. The polypyrrole deposits on the host polymer, and the composite formed is precipitated using a suitable non-solvent. Smooth, lustrous films or other shaped objects with good mechanical properties have been prepared by hot-pressing the obtained materials. The relation between conductivity and the polypyrrole content of the composite exhibits a percolation behaviour. The electrical conductivity can reach values as high as 6–7 S cm −1. The mechanical properties of the material depend on the nature of the host polymer employed and on the content of polypyrrole in the composite. The effect of the length of the alkyl chain on the mechanical properties is investigated. The polypyrrole-poly(ethyl methacrylate) composites appear to be the most suitable because they have enough flexibility and also sufficient strength. Compared to cold-pressing, hot-pressing improves the mechanical characteristics, but decreases somewhat the conductivity. Two single-step procedures have also been employed, but they are less efficient than the two-step procedure and lead to very low conductivities.

Von Willebrand disease family studies: comparison of three methods of analysis of the von Willebrand factor gene polymorphism related to a variable number tandem repeat sequence in intron 40

A region with a variable number of tandem ATCT repeats (VNTR) has previously been localized within intron 40 of the von Willebrand factor (vWF) gene. In the present report we describe the use of this polymorphism as a genetic marker to study the inheritance pattern in five families affected with various types of von Willebrand disease (vWD): types I, IIA, IIB, IIC and the newly characterized variant with totally defective FVIII binding. Three means of investigation previously reported, all using polymerase chain reaction (PCR) amplification of this vWF gene region, were compared in terms of informativeness. The two direct single-step procedures analysing only partial sequences of the VNTR region turned out to be less informative (three studies informative out of five) than the third method characterizing the variability of the whole VNTR sequence. This latter approach, based on the analysis of the Alu I restriction pattern of the VNTR region, was informative in all the families investigated, therefore avoiding the need to combine it with other genetic marker studies for efficient gene tracking. In conclusion, this two-step (PCR and digestion) method is the most informative for the characterization of the inheritance of the different subtypes of vWD and for the prenatal diagnosis of its severe forms.

A step-down heteroscedastic multiple comparison procedure

All-pairs power in a one-way ANOVA is the probability of detecting all true differences between pairs of means. Ramsey (1978) found that for normal distributions having equal variances, step-down multiple comparison procedures can have substantially more all-pairs power than single-step procedures, such as Tukey’s HSD, when equal sample sizes are randomly sampled from each group. This paper suggests a step-down procedure for the case of unequal variances and compares it to Dunnett's T3 technique. The new procedure is similar in spirit to one of the heteroscedastic procedures described by Hochberg and Tamhane (1987), but it has certain advantages that are discussed in the paper. Included are results on unequal sample sizes.