1,721,114 research outputs found
Optimum allocation of treatments for Welch's test in equivalence assessment
No full textAs an extension of Welch's test to equivalence trials in a matched pair design, we determine the sample sizes n(1) and n(2) that maximize the power at given alternatives for a given total sample size n = n(1) + n(2). Although the optimal allocation is obtained asymptotically when the ratio of the standard deviations in both treatment groups equals the ratio of the sample sizes, numerical investigations show that this result does not hold for sample sizes where n less than or equal to 25. For convenience, we provide tables containing the optimal combinations for the bioequivalence problem as well as for the problem of testing a clinically relevant difference. Results indicate that for small sample sizes, the optimum allocations of the treatments differ significantly in both testing problems, although they are asymptotically identical. In addition, we provide simple approximate equations that can be used for the determination of the required sample sizes to control a preassigned probability of type II error
Some methodological aspects of validation of models in nonparametric regression
No full textIn this paper we describe some general methods for constructing goodness of fit tests in nonparametric regression models. Our main concern is the development of statisticial methodology for the assessment (validation) of specific parametric models M as they arise in various fields of applications. The fundamental idea which underlies all these methods is the investigation of certain goodness of fit statistics (which may depend on the particular problem and may be driven by different criteria) under the assumption that a specified model (which has to be validated) holds true as well as under a broad range of scenaria, where this assumption is violated. This is motivated by the fact that outcomes of tests for the classical hypothesis: "The model M holds true" (and their associated p values) bear various methodological flaws. Hence, our suggestion is always to accompany such a test by an analysis of the type II error, which is in goodness of fit problems often the more serious one. We give a careful description of the methodological aspects, the required asymptotic theory, and illustrate the main principles in the problem of testing model assumptions such as a specific parametric form or homoscedasticity in nonparametric regression models
Testing heteroscedasticity in nonparametric regression
No full textThe importance of being able to detect heteroscedasticity in regression is widely recognized because efficient inference for the regression function requires that heteroscedasticity is taken into account. In this paper a simple consistent test for heteroscedasticity is proposed in a nonparametric regression set-up. The test is based on an estimator for the best L-2-approximation of the variance function by a constant. Under mild assumptions asymptotic normality of the corresponding test statistic is established even under arbitrary fixed alternatives. Confidence intervals are obtained for a corresponding measure of heteroscedasticity. The finite sample performance and robustness of these procedures are investigated in a simulation study and Box-type corrections are suggested for small sample sizes
A simple goodness-of-fit test for linear models under a random design assumption
No full textLet (X,Y) denote a random vector with decomposition Y = f(X)+ epsilon where f(x) = E[Y X = x] is the regression of Y on X. In this paper we propose a test for the hypothesis that f is a linear combination of given linearly independent regression functions gl:..., gd. The test is based on an estimator of the minimal L-2-distance between f and the subspace spanned by the regression functions. More precisely, the method is based on the estimation of certain integrals of the regression function and therefore does not require an explicit estimation of the regression. For this reason the test proposed in this paper does not depend on the subjective choice of a smoothing parameter. Differences between the problem of regression diagnostics in the nonrandom and random design case are also discussed
Sign regularity of a generalized Cauchy kernel with applications
No full textThis note provides a complete determination of the sign regularity properties of the 'Cauchy kernel' C-n(x,omega) = (ax+b omega+c)(n), which unifies and generalizes numerous results of sign regular-properties of various distribution families in the literature. As special case the total positivity of Pearson-type distributions including the family of Beta- and Pareto-densities is established. A further application is the construction of invariant UMP tests for interval hypotheses for the ratio of the variances of two normal distributions and the scaling parameters of two Gamma-or exponential-distributions. Finally, some more applications are given in the theory of optimal experimental designs
An extension of Welch's approximate t-solution to comparative bioequivalence trials
No full textWe consider the problem of proving statistically the equivalence of two treatments with normally distributed observations. The size and the power of the commonly-used test procedures which assume equality of the variances are investigated when this assumption is violated. On the one hand the preassigned level is considerably exceeded, and on the other hand the power if no treatment difference exists drops heavily for differing sample sizes. We propose an extension of Welch's approximate t-solution for the Behrens-Fisher problem in bioequivalence assessment, which is asymptotically optimal in a certain subclass of tests. The behaviour of the proposed test is investigated in various finite sample situations. The results show that the extension of Welch's approximate t-solution should be preferred for testing bioequivalence of two treatments whenever the population variances cannot be assumed equal and the sample sizes are different. Sometimes the experimenter knows which sample has the larger variance. In this case, the larger sample size should always be assigned to that group with the larger variance in order to optimize the actual level and power of the test
An extension of Welch’s approximative t-solution to comparative bioequivalence trials
No full textWe consider the problem of proving statistically the equivalence of two treatments with normally distributed observations. The size and the power of the commonly-used test procedures which assume equality of the variances are investigated when this assumption is violated. On the one hand the preassigned level is considerably exceeded, and on the other hand the power if no treatment difference exists drops heavily for differing sample sizes. We propose an extension of Welch's approximate t-solution for the Behrens-Fisher problem in bioequivalence assessment, which is asymptotically optimal in a certain subclass of tests. The behaviour of the proposed test is investigated in various finite sample situations. The results show that the extension of Welch's approximate t-solution should be preferred for testing bioequivalence of two treatments whenever the population variances cannot be assumed equal and the sample sizes are different. Sometimes the experimenter knows which sample has the larger variance. In this case, the larger sample size should always be assigned to that group with the larger variance in order to optimize the actual level and power of the test
Box-type approximations in nonparametric factorial designs
No full textLinear 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 of averaged ranks. A simulation study shows that the corresponding nonparametric "F-test" keeps its level with high accuracy and has power comparable to that of the rank version of the likelihood statistic
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