Edgar Brunner

The Nonparametric Behrens‐Fisher Problem: Asymptotic Theory and a Small‐Sample Approximation

A generalization of the Behrens-Fisher problem for two samples is examined in a nonparametric model. It is not assumed that the underlying distribution functions are continuous so that data with arbitrary ties can be handled. A rank test is considered where the asymptotic variance is estimated consistently by using the ranks over all observations as well as the ranks within each sample. The consistency of the estimator is derived in the appendix. For small samples (n 1 , n 2 > 10), a simple approximation by a central t-distribution is suggested where the degrees of freedom are taken from the Satterthwaite-Smith-Welch approximation in the parametric Behrens-Fisher problem. It is demonstrated by means of a simulation study that the Wilcoxon-Mann-Whitney-test may be conservative or liberal depending on the ratio of the sample sizes and the variances of the underlying distribution functions. For the suggested approximation, however, it turns out that the nominal level is maintained rather accurately. The suggested nonparametric procedure is applied to a data set from a clinical trial. Moreover, a confidence interval for the nonparametric treatment effect is given.

Association of ADHD and conduct disorder – brain electrical evidence for the existence of a distinct subtype

BACKGROUND To evaluate the impact of psychopathological comorbidity with oppositional defiant/conduct disorder (ODD/CD) on brain electrical correlates in children with attention deficit hyperactivity disorder (ADHD) and to study the pathophysiological background of comorbidity of ADHD+ODD/CD. METHOD Event-related potentials (ERPs) were recorded during a cued continuous performance test (CPT-A-X) in children (aged 8 to 14 years) with ICD-10 diagnoses of either hyperkinetic disorder (HD; n = 15), hyperkinetic conduct disorder (HCD; n = 16), or ODD/CD (n = 15) and normal children (n = 18). HD/HCD diagnoses in all children were fully concordant with the DSM-IV diagnosis of ADHD-combined type. ERP-microstates, i.e., time segments with stable brain electrical map topography were identified by adaptive segmentation. Their characteristic parameters and behavioral measures were further analyzed. RESULTS Children with HD but not comorbid children showed slower and more variable reaction times compared to control children. Children with HD and ODD/CD-only but not comorbid children displayed reduced P3a amplitudes to cues and certain distractors (distractor-X) linked to attentional orienting. Correspondingly, global field power of the cue-CNV microstate related to anticipation and preparation was reduced in HD but not in HCD. Topographical alterations of the HD occurred already in the cue-P2/N2 microstate. In sum, the comorbid group was less deviant than both the HD-group and the ODD/CD-group. CONCLUSIONS The findings suggest that HD children (ADHD-combined type without ODD/CD) suffer from a more general deficit (e.g., suboptimal energetical state regulation) including deficits of attentional orienting and response preparation than just a responseinhibitory deficit, backing the hypothesis of an involvement of a dysregulation of the central noradrenergic networks. The results contradict the hypothesis that ADHD+ODD/CD represents an additive co-occurrence of ADHD and ODD/CD and strongly suggest that it represents a separate pathological entity as considered in the ICD-10 classification system, which differs from both HD and ODD/CD-only.

Nonparametric Hypotheses and Rank Statistics for Unbalanced Factorial Designs

Abstract Factorial designs are studied with independent observations, fixed number of levels, and possibly unequal number of observations per factor level combination. In this context, the nonparametric null hypotheses introduced by Akritas and Arnold are considered. New rank statistics are derived for testing the nonparametric hypotheses of no main effects, no interaction, and no factor effects in unbalanced crossed classifications. The formulation of all results includes tied observations. Extensions of these procedures to higher-way layouts are given, and the efficacies of the test statistics against nonparametric alternatives are derived. A modification of the test statistics and approximations to their finite-sample distributions are also given. The small-sample performance of the procedures for two factors is examined in a simulation study. As an illustration, a real dataset with ordinal data is analyzed.

Box-Type Approximations in Nonparametric Factorial Designs

Abstract Linear rank statistics in nonparametric factorial designs are asymptotically normal and, in general, heteroscedastic. In a comprehensive simulation study, the asymptotic chi-squared law of the corresponding quadratic forms is shown to be a rather poor approximation of the finite-sample distribution. Motivated by this problem, we propose simple finite-sample size approximations for the distribution of quadratic forms in factorial designs under a normal heteroscedastic error structure. These approximations are based on an F distribution with estimated degrees of freedom that generalizes ideas of Patnaik and Box. Simulation studies show that the nominal level is maintained with high accuracy and in most cases the power is comparable to the asymptotic maximin Wald test. Data-driven guidelines are given to select the most appropriate test procedure. These ideas are finally transferred to nonparametric factorial designs where the same quadratic forms as in the parametric case are applied to the vector ...

Questioning inhibitory control as the specific deficit of ADHD – evidence from brain electrical activity

Summary.Objective: To investigate motor response control during a cued continuous performance test (CPT-A-X) by performance and ERP parameters in children with hyperkinetic disorder (HD), hyperkinetic conduct disorder (HCD) or oppositional deviant/conduct disorder (ODD/CD), to examine the evidence for an inhibition-specific deficit as indicated by these parameters, and to analyze whether possible deviations are specific for HD/HCD. Method: Behavioral parameters and event-related potentials (ERPs) were recorded during a CPT-A-X-task in children (aged 8 to 14 years) with either HD (n=15), HCD (n=16), or ODD/CD (n=15) and normal children (n=18) and analysed. ICD-10 diagnoses of HD/HCD diagnoses in all children were fully concordant with the DSM-IV diagnosis of ADHD-combined type. Results: Children with HCD committed more dyscontrol errors and differed most from normal children on ERP measures of motor response control, while children with HD-only were more impaired during processing of the warning stimuli for motor preparation. ERP measures specific for response inhibition were not different between the groups. Conclusions: The results show that ADHD cannot be fully explained by an inhibition-specific deficit and implicate impaired response execution processes as well. This indicates that comorbid children suffer from a reduced ability to control their prepared motor responses. Further, they seem to have difficulties in timely switching attention from monitoring the sensory input stream to the monitoring of own responses and actions.

Nonparametric analysis of ordered categorical data in designs with longitudinal observations and small sample sizes

For designs with longitudinal observations of ordered categorical data, a nonparametric model is considered where treatment effects and interactions are defined by means of the marginal distributions. These treatment effects are estimated consistently by ranking methods. The hypotheses in this nonparametric setup are formulated by means of the distribution functions. The asymptotic distribution of the estimators for the nonparametric effects are given under the hypotheses. For small samples, a rather accurate approximation is suggested. A clinical trial with ordered categorical data is used to motivate the ideas and to explain the procedures which are extensions of the Wilcoxon-Mann-Whitney test to factorial designs with longitudinal observations. The application of the procedures requires only some trivial regularity assumptions.

The Heat Shock Protein HSP70 Promotes Mouse NK Cell Activity against Tumors That Express Inducible NKG2D Ligands

The stress-inducible heat shock protein (HSP) 70 is known to function as an endogenous danger signal that can increase the immunogenicity of tumors and induce CTL responses. We show in this study that HSP70 also activates mouse NK cells that recognize stress-inducible NKG2D ligands on tumor cells. Tumor size and the rate of metastases derived from HSP70-overexpressing human melanoma cells were found to be reduced in T and B cell-deficient SCID mice, but not in SCID/beige mice that lack additionally functional NK cells. In the SCID mice with HSP70-overexpressing tumors, NK cells were activated so that they killed ex vivo tumor cells that expressed NKG2D ligands. In the tumors, the MHC class I chain-related (MIC) A and B molecules were found to be expressed. Interestingly, a counter selection was observed against the expression of MICA/B in HSP70-overexpressing tumors compared with control tumors in SCID, but not in SCID/beige mice, suggesting a functional relevance of MICA/B expression. The melanoma cells were found to release exosomes. HSP70-positive exosomes from the HSP70-overexpressing cells, in contrast to HSP70-negative exosomes from the control cells, were able to activate mouse NK cells in vitro to kill YAC-1 cells, which express NKG2D ligands constitutively, or the human melanoma cells, in which MICA/B expression was induced. Thus, HSP70 and inducible NKG2D ligands synergistically promote the activation of mouse NK cells resulting in a reduced tumor growth and suppression of metastatic disease.

A unified approach to rank tests for mixed models

Abstract The nonparametric version of the classical mixed model is considered and the common hypotheses of (parametric) main effects and interactions are reformulated in a nonparametric setup. To test these nonparametric hypotheses, the asymptotic distributions of quadratic forms of rank statistics are derived in a general framework which enables the derivation of the statistics for the nonparametric hypotheses of the fixed treatment effects and interactions in an arbitrary mixed model. The procedures given here are not restricted to semiparametric models or models with additive effects. Moreover, they are robust to outliers since only the ranks of the observations are needed. They are also applicable to pure ordinal data and since no continuity of the distribution functions is assumed, they can also be applied to data with ties. Some approximations for small sample sizes are suggested and analyzed in a simulation study. The application of the statistics and the interpretation of the results is demonstrated in several worked-out examples where some data sets given in the literature are re-analyzed.

Rank-based multiple test procedures and simultaneous confidence intervals

We study simultaneous rank procedures for unbalanced designs with independent observations. The hypotheses are formulated in terms of purely nonparametric treatment effects. In this context, we derive rankbased multiple contrast test procedures and simultaneous confidence intervals which take the correlation between the test statistics into account. Hereby, the individual test decisions and the simultaneous confidence intervals are compatible. This means, whenever an individual hypothesis has been rejected by the multiple contrast test, the corresponding simultaneous confidence interval does not include the null, i.e. the hypothetical value of no treatment effect. The procedures allow for testing arbitrary purely nonparametric multiple linear hypotheses (e.g. many-to-one, all-pairs, changepoint, or even average comparisons). We do not assume homogeneous variances of the data; in particular, the distributions can have different shapes even under the null hypothesis. Thus, a solution to the multiple nonparametric Behrens-Fisher problem is presented in this unified framework.

A studentized permutation test for the non-parametric Behrens-Fisher problem

For the non-parametric Behrens-Fisher problem a permutation test based on the studentized rank statistic of Brunner and Munzel is proposed. This procedure is applicable to count or ordered categorical data. By applying the central limit theorem of Janssen, it is shown that the asymptotic permutational distribution of this test statistic is a standard normal distribution. For very small and very different sample sizes, frequently occurring in medical and biological applications, an extensive simulation study suggests that this permutation test works well for data from several underlying distributions. The proposed test is applied to data from a clinical trial.