confidence intervals for an effect size measure based on the mann whitney statistic part 1 general issues and tail area based methods for two random variables x and y pr y x pr y x is advocated as a general measure of effect size to characterize the degree of separation of their distributions it is estimated by u mn a generalization of the mann whitney u statistic derived by dividing u by the product of the two sample sizes it is equivalent to the area under the receiver operating characteristic curve it is readily visualized in terms of two gaussian distributions with appropriately separated peaks the effect of discretization of a continuous variable is explored tail area based confidence interval methods are developed which can be applied to very small samples or extreme outcomes copyright 2005 john wiley sons ltd

exact nonparametric inference when doses are measured with random errors studies that estimate the effects of exposure to a possibly harmful agent often compare exposed subjects who received varied doses with matched controls who received zero dose if the doses are measured with error then one may wish to use the fallible doses to estimate a linear relationship between the unobserved true dose and the observed response if one is willing to assume that the dose errors for exposed subjects are symmetrically distributed about 0 that is the dose errors are pure errors and not say systematic underreporting of exposure then the presence of zero dose controls is all that is needed to obtain exact distribution free confidence intervals and tests and consistent point estimates the method is simpler for matched pairs than for matched sets with two or more matched subjects and it is illustrated using two studies one of each kind with matched pairs as in this first example the method uses wilcoxon s signed rank test as the basis for inference when there are several zero dos

error probability analysis of bit interleaved coded modulation this correspondence presents a simple method to accurately compute the error probability of bit interleaved coded modulation bicm thanks to the binary input output symmetric bios nature of the channel the pairwise error probability pep is equal to the tail probability of a sum of random variables with a particular distribution this probability is in turn computed with a saddlepoint approximation its precision is numerically validated for coded transmission over standard gaussian noise and fully interleaved fading channels for both convolutional and turbo like codes

comparing the areas under two or more correlated receiver operating characteristic curves a nonparametric approach methods of evaluating and comparing the performance of diagnostic tests are of increasing importance as new tests are developed and marketed when a test is based on an observed variable that lies on a continuous or graded scale an assessment of the overall value of the test can be made through the use of a receiver operating characteristic roc curve the curve is constructed by varying the cutpoint used to determine which values of the observed variable will be considered abnormal and then plotting the resulting sensitivities against the corresponding false positive rates when two or more empirical curves are constructed based on tests performed on the same individuals statistical analysis on differences between curves must take into account the correlated nature of the data this paper presents a nonparametric approach to the analysis of areas under correlated roc curves by using the theory on generalized u statistics to generate an estimated covariance matrix

exemplary data set sample size calculation for wilcoxon mann whitney tests zhao rahardja and qu consider sample size calculation for wilcoxon mann whitney wmw tests for data with ties and present a straightforward formula we observe that the exemplary data set approach usually applied in more complex situations has a close relationship to the zhao rahardja qu method for wmw sample size estimation and they are asymptotically equivalent therefore the exemplary data set approach can be used to easily obtain estimates similar to those that the closed formula gives we illustrate application of both methods for a wmw sample size estimation example and also extend the simulation study presented by zhao et al we find that the zhao rahardja qu formula and by extension the exemplary data set method can give estimates just as accurate as those obtained using either the kolassa approach via nquery advisor or the o brien castelloe approach via sas 9 2 proc power for 1 1 and 1 2 allocation ratios however the latter two methods can be more accurate for a ratio of 1 4 or 1 19 finally we note the general utility of the exemplary data set approach for sample size estimation even in other situations where closed form sample size formulae exist

asymptotic bep and sep of quadratic diversity combining receivers in correlated ricean fading non gaussian noise and interference in this paper we study the asymptotic behavior of the bit error probability bep and the symbol error probability sep of quadratic diversity combining schemes such as coherent maximum ratio combining mrc differential equal gain combining egc and noncoherent combining nc in correlated ricean fading and non gaussian noise which in our definition also includes interference we provide simple and easy to evaluate asymptotic bep and sep expressions which show that at high signal to noise ratios snrs the performance of the considered combining schemes depends on certain moments of the noise and interference impairing the transmission we derive general rules for calculation of these moments and we provide closed form expressions for the moments of several practically important types of noise such as spatially dependent and spatially independent gaussian mixture noise correlated synchronous and asynchronous co channel interference and correlated gaussian interference from our asymptotic results we conclude that a the asymptotic performance loss of binary frequency shift keying bfsk with nc compared to binary phase shift keying bpsk with mrc is always 6 db independent of the type of noise and the number of diversity branches b the asymptotic performance loss of differential egc compared to mrc is always 3 db for additive white gaussian noise but depends on the number of diversity branches and may be larger or smaller than 3 db for other types of noise and c not only fading correlation but also noise correlation negatively affects the performance of quadratic diversity combiners

sample size estimation for the van elteren test a stratified wilcoxon mann whitney test the van elteren test is a type of stratified wilcoxon mann whitney test for comparing two treatments accounting for strata in this paper we study sample size estimation methods for the asymptotic version of the van elteren test assuming that the stratum fractions ratios of each stratum size to the total sample size and the treatment fractions ratios of each treatment size to the stratum size are known in the study design in particular we develop three large sample sample size estimation methods and present a real data example to illustrate the necessary information in the study design phase in order to apply the methods simulation studies are conducted to compare the performance of the methods and recommendations are made for method choice finally sample size estimation for the van elteren test when the stratum fractions are unknown is also discussed

robust l_p norm decoding for bicm based secondary user systems the frequency bands used by secondary user systems such as cognitive radio and ultra wideband uwb systems are subject to various forms of non gaussian noise and interference including co channel and narrowband interference caused by the primary user and other secondary users as well as man made impulsive noise for secondary user systems employing popular bit interleaved coded modulation bicm we propose a robust lp norm metric for viterbi decoding to overcome the harmful effects of non gaussian noise and interference for the resulting system we derive an approximate upper bound on the bit error rate ber and a general analytical expression for the asymptotic ber valid for all practically relevant types of fading including rayleigh ricean nakagami m nakagami q and weibull fading and all types of noise and interference with finite moments simulation and analytical results show that in non gaussian noise and interference the proposed lp norm metric yields large performance gains over the conventional l2 norm metric and outperforms erasure decoding

tight error bounds for space time orthogonal block codes under slow rayleigh flat fading the performance of space time orthogonal block stob codes over slow rayleigh fading channels and maximum likelihood ml decoding is investigated two bonferroni type bounds one upper bound and one lower bound for the symbol error rate ser and bit error rate ber of the system are obtained the bounds are expressed in terms of the pairwise error probabilities peps and the two dimensional pairwise error probabilities 2 d peps of the transmitted symbols furthermore the bounds can be efficiently evaluated and they hold for arbitrary nonstandard signaling schemes and mappings numerical results demonstrate that the bounds are very accurate in estimating the performance of stob codes in particular the upper and lower bounds often coincide even at low channel signal to noise ratios large constellation sizes and large diversity orders

determining the appropriate sample size for nonparametric tests for location shift for the case in which two independent samples arc to be compared using a nonparametric test for location shift we propose a bootstrap technique for estimating the sample sizes required to achieve a specified power the estimator called boot uses information from a small pilot experiment for the special case of the wilcoxon test a simulation study is conducted to compare boot to two other sample size estimators one method called anpv is based on the assumption that the underlying distribution is normal with a variance estimated from the pilot data the other method called noether adapts the sample size formula of noether for use with a location shift alternative the boot and noether sample size estimators are particularly appropriate for this nonparametric setting because they do not require assumptions about the shape of the underlying continuous probability distribution the simulation study shows that a sample size estimates can have large uncertainty b boot is at least as accurate as an

asymptotical performance of m ary and binary signals over multipath multichannel rayleigh and rician fading this paper presents a general framework for computing the asymptotic error probability i e at high average snrs of m ary and binary signaling schemes over rician and rayleigh fading diversity channels a general theorem theorem 1 relates the asymptotic error rate of multipath and multichannel receivers over awgn isi free channels to the multidimensional integral of the conditional error probability two other theorems are presented for the particular cases where the conditional error probability is a function of the sum of received snrs theorem 2 or received amplitudes theorem 3 theorems 2 and 3 are related for linear coherent systems and closed form expressions are obtained for equal gain combining systems detection structures for typical diversity schemes coherent noncoherent maximal ratio and equal gain combining and quadratic noncoherent combining are considered we analyze the asymptotic error rates of some m ary signaling schemes mpsk mpam with kth order diversity and orthogonal signals with k 1 and with coherent and noncoherent detection binary signaling is also considered in our study

performance analysis for bicm transmission over gaussian mixture noise fading channels bit interleaved coded modulation bicm has been adopted in many systems and standards for spectrally efficient coded transmission the analytical evaluation of bicm performance parameters in particular bit error rate ber has received considerable attention in the recent past in this paper we derive ber approximations for bicm transmission over general fading channels impaired by gaussian mixture noise gmn to this end we build upon the saddlepoint approximation of the pairwise error probability pep and a recently established approximation for the probability density function pdf of bit wise reliability metrics for nonfading additive white gaussian noise awgn channels we extend this pdf approximation to the case of gmn and obtain closed form expressions for its laplace transform for fading gmn channels the latter allows us to express the pep and thus ber via the saddlepoint approximation for the special case of fading awgn channels the presented approximations are closed form since the saddlepoint is well approximated by 1 2 for bicm decoding furthermore we derive closed form pep expressions also for gmn channels in the high signal to noise ratio regime and establish the diversity and coding gain for bicm transmission over fading gmn channels selected numerical results for the ber of convolutional coded bicm highlight the usefulness of the proposed approximations and the differences between awgn and gmn channels

approximation of achievable rates in additive gaussian mixture noise channels in this paper we detail effective methods to approximate the achievable rates of channels with additive gaussian mixture gm noise for both real and complex channels to achieve any desired level of accuracy attention is paid to a gaussian input a discrete real input and a complex input with discrete amplitude and independent uniform phase such discrete inputs represent a wide range of input distributions and they include the capacity achieving inputs as special cases at first we propose a simple technique to accurately calculate the noise entropy specifically when the noise level is high a lower bound on the integrand of the entropy is established and the noise entropy can be estimated using a closed form solution in the low noise region the piecewise linear curve fitting pwlcf method is applied we then extend this result to calculate the achievable rate when the input is gaussian distributed which is shown to be asymptotically optimal next we propose a simple pwlcf based method to approximate the output entropy for a real gm channel when the input is discrete and for a complex gm channel when the input is discrete in amplitude with independent uniform phase in particular for the real channel the output entropy is evaluated by examining the output in high and low regions of amplitude using a lower bound on the integrand of the output entropy and pwlcf respectively for the complex channel the output entropy is approximated a similar manner but using polar coordinates and the kernel function it is demonstrated that the output entropy and consequently the achievable rates can be computed to achieve any given accuracy level

spaces of continuous maps on a class of noncompact metric spaces for a metric space x let c x be the family of hypographs of all con tinuous maps from x to i 0 1 endowed with the topology induced by the hausdorff metric of the metric space x i in the present paper the following result is proved if x i s a noncompact locally compact totally bounded separable metric space then c x i s home omorphic to c0 if and only if the set of all isolated points of x is not dense in x w here c0 xn n n 1 1 s up xn 1 and limn xn 0 specially for the open interval 0 1 with the usual metric c 0 1 is homeomorphic to c0

probabilistic index an intuitive non parametric approach to measuring the size of treatment effects effect sizes es tell the magnitude of the difference between treatments and ideally should tell clinicians how likely their patients will benefit from the treatment currently used es are expressed in statistical rather than in clinically useful terms and may not give clinicians the appropriate information we restrict our discussion to studies with two groups one with n patients receiving a new treatment and the other with m patients receiving the usual or no treatment the standardized mean difference e g cohen s d is a well known index for continuous outcomes there is some intuitive value to d but measuring improvement in standard deviations sd is a statistical concept that may not help a clinician how much improvement is a half sd a more intuitive and simple to calculate es is the probability that the response of a patient given the new treatment x is better than the one for a randomly chosen patient given the old or no treatment y i e p x y larger values meaning better outcomes this probability has an immediate identity with the area under the curve auc measure in procedures for receiver operator characteristic roc curve comparing responses to two treatments it also can be easily calculated from the mann whitney u wilcoxon or kendall statistics we describe the characteristics of an ideal es we propose p x y as an alternative index summarize its correspondence with well known non parametric statistics compare it to the standardized mean difference index and illustrate with clinical data copyright 2005 john wiley sons ltd

sets of filter convergence of sequences of continuous functions abstract we consider sets of filter convergence and divergence to infinity of sequences of continuous functions for borel filters we characterize the sets of filter convergence of sequences of continuous functions on the real line for borel filters we also give sufficient and necessary conditions for results involving sets of filter divergence to infinity in particular we give the full description of such sets for the statistical convergence for metric spaces

power and sample size calculations for discrete bounded outcome scores we consider power and sample size calculations for randomized trials with a bounded outcome score bos as primary response adjusted for a priori chosen covariates we define bos to be a random variable restricted to a finite interval typically a bos has a j or u shaped distribution hindering traditional parametric methods of analysis when no adjustment for covariates is needed a non parametric test could be chosen however there is still a problem with calculating the power since the common location shift alternative does not hold in general for a bos in this paper we consider a parametric approach and assume that the observed bos is a coarsened version of a true bos which has a logit normal distribution in each treatment group allowing correction for covariates a two step procedure is used to calculate the power firstly the power function is defined conditionally on the covariate values secondly the marginal power is obtained by averaging the conditional power with respect to an assumed distribution for the covariates using monte carlo integration a simulation study evaluates the performance of our method which is also applied to the ecass 1 stroke study

log concavity property of the error probability with application to local bounds for wireless communications a clear understanding of the behavior of error probability ep as a function of signal to noise ratio snr and other system parameters is fundamental for assessing the design of digital wireless communication systems we propose an analytical framework based on the log concavity property of the ep which we prove for a wide family of multidimensional modulation formats in the presence of gaussian disturbances and fading based on this property we construct a class of local bounds for the ep that improve known generic bounds in a given region of the snr and are invertible as well as easily tractable for further analysis this concept is motivated by the fact that communication systems often operate with performance in a certain region of interest roi and thus it may be advantageous to have tighter bounds within this region instead of generic bounds valid for all snrs we present a possible application of these local bounds but their relevance is beyond the example made in this paper

power and sample size estimation for the clustered wilcoxon test the wilcoxon rank sum test is widely used for two group comparisons of nonnormal data an assumption of this test is independence of sampling units both within and between groups which will be violated in the clustered data setting such as in ophthalmological clinical trials where the unit of randomization is the subject but the unit of analysis is the individual eye for this purpose we have proposed the clustered wilcoxon test to account for clustering among multiple subunits within the same cluster rosner glynn and lee 2003 biometrics 59 1089 1098 2006 biometrics 62 1251 1259 however power estimation is needed to plan studies that use this analytic approach we have recently published methods for estimating power and sample size for the ordinary wilcoxon rank sum test rosner and glynn 2009 biometrics 65 188 197 in this article we present extensions of this approach to estimate power for the clustered wilcoxon test simulation studies show a good agreement between estimated and empirical power these methods are illustrated with examples from randomized trials in ophthalmology enhanced power is achieved with use of the subunit as the unit of analysis instead of the cluster using the ordinary wilcoxon rank sum test

estimating the power of the two sample wilcoxon test for location shift summary traditional methods for calculating the power of a statistical test for location shift require knowledge of the shape of the underlying probability distribution the distribution shape however may be unknown this paper describes a bootstrap method for using observed data or pilot data to approximate the power no assumptions need be made about the shape of the underlying continuous probability distribution simulation evidence shows that when applied to the wilcoxon two sample test for location shift the suggested method is reliable the evidence also shows that it is more accurate than a benchmark traditional approach the bootstrap method is applied to a real data example the analysis demonstrates how the method can be used to determine sample sizes and how to choose the more powerful of two alternative tests for location shift

power and sample size estimation for the wilcoxon rank sum test with application to comparisons of c statistics from alternative prediction models the wilcoxon mann whitney wmw u test is commonly used in nonparametric two group comparisons when the normality of the underlying distribution is questionable there has been some previous work on estimating power based on this procedure lehmann 1998 nonparametrics in this article we present an approach for estimating type ii error which is applicable to any continuous distribution and also extend the approach to handle grouped continuous data allowing for ties we apply these results to obtaining standard errors of the area under the receiver operating characteristic curve auroc for risk prediction rules under h 1 and for comparing auroc between competing risk prediction rules applied to the same data set these results are based on sas callable functions to evaluate the bivariate normal integral and are thus easily implemented with standard software

calculation of power and sample size with bounded outcome scores the two sample wilcoxon rank sum test is the most popular non parametric test for the comparison of two samples when the underlying distributions are not normal although the underlying distributions need not be known in detail to calculate the null distribution of the test statistic parametric assumptions are often made to determine the power of the test or the sample size we encountered difficulties with this approach in the planning of a recent clinical trial in stroke patients it is shown that for power and sample size estimation it can be dangerous to apply the classical formulae routinely especially with outcome scores having a u shaped or a j shaped distribution as an example we have taken the barthel index a quality of life outcome measure in stroke patients further we have investigated alternative methods by means of monte carlo simulation the distributional characteristics of the estimated powers were compared our findings suggest more appropriate computer software is necessary for the calculation of power and sample size when efficacy is measured by a non parametric method

new analytical probability of error expressions for classes of orthogonal signals in rayleigh fading new exact expressions for the symbol error rate and bit error rate of coherent 3 ary and 4 ary orthogonal and transorthogonal signaling in slowly fading rayleigh channels are derived new exact error probability expressions for coherent 6 ary and 8 ary biorthogonal signaling in slow rayleigh fading are also presented the use of these exact expressions as accurate approximations for the error rates of m ary orthogonal biorthogonal and transorthogonal signaling with arbitrary m is illustrated

rank sum tests for clustered data the wilcoxon rank sum test is widely used to test the equality of two populations because it makes fewer distributional assumptions than parametric procedures such as the t test however the wilcoxon rank sum test can be used only if data are independent when data are clustered tests based on generalized estimating equations gees that generalize the t test have been proposed here we develop a rank sum test that can be used when data are clustered as an application we use our rank sum test to develop a nonparametric test of association between a genetic marker and a quantitative trait locus we also give a rank sum test for equivalence of three or more populations that generalizes the kruskal wallis test to situations with clustered data unlike previous rank tests for clustered data our proposal is valid when members of the same cluster belong to different groups or when the correlation between cluster members differs across groups

combining several ordinal measures in clinical studies in medical research it is rare that a single variable is sufficient to represent all relevant aspects of epidemiological risk genomic activity adverse events or clinical response since biological systems tend to be neither linear nor hierarchical in nature the assumptions of traditional multivariate statistical methods based on the linear model can often not be justified on theoretical grounds establishing concept validity through empirical validation is not only problematic but also time consuming this paper proposes the use of u statistics for scoring multivariate ordinal data and a family of simple non parametric tests for analysis the scoring method is demonstrated to be applicable to scoring clinical response profiles in the treatment of psoriasis and then to identifying genomic pathways that best correlate with these profiles

a u statistics based approach to sample size planning of two arm trials with discrete outcome criterion aiming to establish either superiority or noninferiority in current practice the most frequently applied approach to the handling of ties in the mann whitney wilcoxon mww test is based on the conditional distribution of the sum of mid ranks given the observed pattern of ties starting from this conditional version of the testing procedure a sample size formula was derived and investigated by zhao et al stat med 2008 in contrast the approach we pursue here is a nonconditional one exploiting explicit representations for the variances of and the covariance between the two u statistics estimators involved in the mann whitney form of the test statistic the accuracy of both ways of approximating the sample sizes required for attaining a prespecified level of power in the mww test for superiority with arbitrarily tied data is comparatively evaluated by means of simulation the key qualitative conclusions to be drawn from these numerical comparisons are as follows with the sample sizes calculated by means of the respective formula both versions of the test maintain the level and the prespecified power with about the same degree of accuracy despite the equivalence in terms of accuracy the sample size estimates obtained by means of the new formula are in many cases markedly lower than that calculated for the conditional test perhaps a still more important advantage of the nonconditional approach based on u statistics is that it can be also adopted for noninferiority trials copyright 2016 john wiley sons ltd

confidence intervals for an effect size measure based on the mann whitney statistic part 2 asymptotic methods and evaluation several asymptotic confidence interval methods for u mn the mann whitney u statistic divided by the product of the two sample sizes are developed and evaluated alongside published methods novel methods derived from gaussian and beta models perform well though overall a modification of the hanley mcneil approach method 5 in this evaluation performed better and is recommended for calculating confidence intervals for u mn for continuously distributed data

generalized pairwise comparisons of prioritized outcomes in the two sample problem this paper extends the idea behind the u statistic of the wilcoxon mann whitney test to perform generalized pairwise comparisons between two groups of observations the observations are outcomes captured by a single variable possibly repeatedly measured or by several variables of any type e g discrete continuous time to event when several outcomes are considered they must be prioritized we show that generalized pairwise comparisons extend well known non parametric tests and illustrate their interest using data from two randomized clinical trials we also show that they lead to a general measure of the difference between the groups called the proportion in favor of treatment denoted which is related to traditional measures of treatment effect for a single variable

recent advances of stretched gaussian distribution underlying hausdorff fractal distance and its applications in fitting stretched gaussian noise abstract this paper summarizes the latest advances of the third author s research group on stretched gaussian distribution underlying the hausdorff fractal theory and its applications in fitting stretched gaussian noise firstly the hausdorff fractal metrics are introduced as an extension of non euclidean distance based on the fractal scaling the hausdorff derivative is derived which can describe the problems of interest to construct non integer differential equations secondly we introduce the hausdorff derivatives in tackling partial differential equations and the fundamental solution of hausdorff derivative diffusion equation is stretched gaussian distribution thirdly we analyze the feasibility of the least square method for stretched gaussian noise which obeys stretched gaussian distribution the least square method is inapplicable when the noise level is larger than 5 finally by using the hausdorff fractal distance we introduce the stretched least square method which improves the traditional least square method to fit stretched gaussian noise

power and sample size calculations for the wilcoxon mann whitney test in the presence of death censored observations we consider a clinical trial of a potentially lethal disease in which patients are randomly assigned to two treatment groups and are followed for a fixed period of time a continuous endpoint is measured at the end of follow up for some patients however death or severe disease progression may preclude measurement of the endpoint a statistical analysis that includes only patients with endpoint measurements may be biased an alternative analysis includes all randomized patients with rank scores assigned to the patients who are available for the endpoint measurement on the basis of the magnitude of their responses and with worst rank scores assigned to those patients whose death precluded the measurement of the continuous endpoint the worst rank scores are worse than all observed rank scores the treatment effect is then evaluated using the wilcoxon mann whitney test in this paper we derive closed form formulae for the power and sample size of the wilcoxon mann whitney test when missing measurements of the continuous endpoints because of death are replaced by worst rank scores we distinguish two approaches for assigning the worst rank scores in the tied worst rank approach all deaths are weighted equally and the worst rank scores are set to a single value that is worse than all measured responses in the untied worst rank approach the worst rank scores further rank patients according to their time of death so that an earlier death is considered worse than a later death which in turn is worse than all measured responses in addition we propose four methods for the implementation of the sample size formulae for a trial with expected early death we conduct monte carlo simulation studies to evaluate the accuracy of our power and sample size formulae and to compare the four sample size estimation methods

performance of bicm sc and bicm ofdm systems with diversity reception in non gaussian noise and interference in this paper we present a general mathematical framework for performance analysis of single carrier sc and orthogonal frequency division multiplexing ofdm systems employing popular bit interleaved coded modulation bicm and multiple receive antennas the proposed analysis is applicable to bicm systems impaired by general types of fading including rayleigh ricean nakagami m nakagami q and weibull fading and general types of noise and interference with finite moments such as additive white gaussian noise awgn additive correlated gaussian noise gaussian mixture noise co channel interference narrowband interference and ultra wideband interference we present an approximate upper bound for the bit error rate ber and an accurate closed form approximation for the asymptotic ber at high signal to noise ratios for viterbi decoding with the standard euclidean distance branch metric for the standard rate 1 2 convolutional code the proposed approximate upper bound and the asymptotic approximation become tight at bers of 10 6 and 10 12 respectively however if the code is punctured to higher rates e g 2 3 or 3 4 the asymptotic approximation also becomes tight at a ber of 10 6 exploiting the asymptotic ber approximation we show that the diversity gain of bicm systems only depends on the free distance of the code the type of fading and the number of receive antennas but not on the type of noise in contrast their coding gain strongly depends on the noise moments our asymptotic analysis shows that as long as the standard euclidean distance branch metric is used for viterbi decoding bicm systems optimized for awgn are also optimum for any other type of noise and interference with finite moments

performance and optimization of network coded cooperative diversity systems in this paper we study network coded cooperative diversity nccd systems comprising multiple sources one relay and one destination where the relay detects the packets received from all sources and performs galois field gf network coding over rm gf 2 m before forwarding a single packet to the destination assuming independent rayleigh fading for all links of the network we derive simple and accurate closed form approximations for the asymptotic symbol and bit error rates of nccd systems the derived error rate expressions are valid for arbitrary numbers of sources arbitrary modulation schemes and arbitrary constellation mappings and provide significant insight into the impact of various system and channel parameters on performance moreover these expressions can be exploited for optimization of the constellation mapping as well as for formulation of various nccd system optimization problems including optimal power allocation relay selection and relay placement

infinite series representations of the bivariate rayleigh and nakagami m distributions despite the usefulness of the bivariate correlated nakagami m cumulative distribution function cdf in communications systems analysis an infinite series for the computation of this function is not known in this letter an infinite series representation of the bivariate cdf is derived bounds on the error resulting from truncation of the infinite series are also derived

the use of bootstrap methods for estimating sample size and analysing health related quality of life outcomes health related quality of life hrqol measures are increasingly used in trials as primary outcome measures investigators are now asking statisticians for advice on how to plan and analyse studies using such outcomes hrqol outcomes like the sf 36 are usual measured on an ordinal scale although most investigators assume that there exists an underlying continuous latent variable and that the actual measured outcomes the ordered categories reflect contiguous intervals along this continuum the ordinal scaling of hrqol measures means they tend to generate data that have discrete bounded and skewed distributions thus standard methods of analysis that assume normality and constant variance may not be appropriate for this reason conventional statistical advice would suggest non parametric methods be used to analyse hrqol data the bootstrap is one such computer intensive non parametric method for estimating sample sizes and analysing data we describe three methods of estimating sample sizes for two group cross sectional comparisons of hrqol outcomes we then compared the power of the three methods for a two group cross sectional study design using bootstrap simulation the results showed that under the location shift alternative hypothesis conventional methods of sample size estimation performed well particularly whitehead s method whitehead s method is recommended if the hrqol outcome has a limited number of discrete values 7 and or the expected proportion of cases at either of the bounds is high if a pilot data set is readily available then bootstrap simulation will provide a more accurate and reliable estimate than conventional methods finally we used the bootstrap for hypothesis testing and the estimation of standard errors and confidence intervals for parameters in an example data set we then compared and contrasted the bootstrap with standard methods of analysing hrqol outcomes in the data set studied with the sf 36 outcome the use of the bootstrap for estimating sample sizes and analysing hrqol data produces results similar to conventional statistical methods these results suggest that bootstrap methods are not more appropriate for analysing hrqol outcome data than standard methods

sample size determination for diagnostic accuracy studies involving binormal roc curve indices receiver operating characteristic roc curves and their associated indices are valuable tools for the assessment of the accuracy of diagnostic tests the area under the roc curve is a popular summary measure of the accuracy of a test the full area under the roc curve however has been criticized because it gives equal weight to all false positive error rates alternative indices include the area under the roc curve in a particular range of false positive rates partial area and the sensitivity of the test for a single fixed false positive rate fpr we present a unified approach for computing sample size for binormal roc curves and their indices our method uses taylor series expansions to derive approximate large sample estimates of the variance and covariance of binormal roc curve parameters several examples from diagnostic radiology illustrate the proposed method

design sensitivity in observational studies summary outside the field of statistics the literature on observational studies offers advice about research designs or strategies for judging whether or not an association is causal such as multiple operationalism or a dose response relationship these useful suggestions are typically informal and qualitative a quantitative measure design sensitivity is proposed for measuring the contribution such strategies make in distinguishing causal effects from hidden biases several common strategies are then evaluated in terms of their contribution to design sensitivity a related method for computing the power of a sensitivity analysis is also developed

sample size determination for testing whether an identified treatment is best laska and meisner 1989 biometrics 45 1139 1151 dealt with the problem of testing whether an identified treatment belonging to a set of k 1 treatments is better than each of the other k treatments they calculated sample size tables for k 2 when using multiple t tests or wilcoxon mann whitney tests both under normality assumptions in this paper we provide sample size formulas as well as tables for sample size determination for k or 2 when t tests under normality or wilcoxon mann whitney tests under general distribution assumptions are used

on the inverse symbol error probability for diversity reception this paper addresses the problem of finding the inverse symbol error probability sep expression for coherent detection of m ary phase shift keying with multichannel reception and maximal ratio combining in rayleigh fading to this aim we derive upper and lower bounds on sep that are simply invertible and uniformly tight for all values of signal to noise ratio this enables us to obtain tight bounds on the inverse sep and on the symbol error outage seo i e sep based outage probability as an example of application to digital mobile radio the seo in a log normal shadowing environment is analyzed

performance of optimum and suboptimum receivers in the presence of impulsive noise modeled as an alpha stable process impulsive noise bursts in communication systems are traditionally handled by incorporating in the receiver a limiter which clips the received signal before integration an empirical justification for this procedure is that it generally causes the signal to noise ratio to increase recently very accurate models of impulsive noise were presented based on the theory of symmetric spl alpha stable probability density functions we examine the performance of optimum receivers designed to detect signals embedded in impulsive noise which is modeled as an infinite variance symmetric spl alpha stable process and compare it against the performance of several suboptimum receivers as a measure of receiver performance we compute an asymptotic expression for the probability of error for each receiver and compare it to the probability of error calculated by extensive monte carlo simulation

extension of the rank sum test for clustered data two group comparisons with group membership defined at the subunit level the wilcoxon rank sum test is widely used for two group comparisons for nonnormal data an assumption of this test is independence of sampling units both between and within groups in ophthalmology data are often collected on two eyes of an individual which are highly correlated in ophthalmological clinical trials randomization is usually performed at the subject level but the unit of analysis is the eye if the eye is used as the unit of analysis then a modification to the usual wilcoxon rank sum variance formula must be made to account for the within cluster dependence for some clustered data designs where the unit of analysis is the subunit group membership may be defined at the subunit level for example in some randomized ophthalmologic clinical trials different treatments may be applied to fellow eyes of some patients while the same treatment may be applied to fellow eyes of other patients in general binary eye specific covariates may be present scored as exposed or unexposed and one wishes to compare nonnormally distributed outcomes between exposed and unexposed eyes using the wilcoxon rank sum test while accounting for the clustering in this article we present a corrected variance formula for the wilcoxon rank sum statistic in the setting of eye subunit specific covariates we apply it to compare ocular itching scores in ocular allergy patients between eyes treated with active versus placebo eye drops where some patients receive the same eye drop in both eyes while other patients receive different eye drops in fellow eyes we also present comparisons between the clustered wilcoxon test and each of the signed rank tests and mixed model approaches and show dramatic differences in power in favor of the clustered wilcoxon test for some designs

comments on bit interleaved coded modulation caire taricco and biglieri presented a detailed analysis of bit interleaved coded modulation bicm a simple and popular technique used to improve system performance especially in the context of fading channels they derived an upper bound to the probability of error called the expurgated bound in this correspondence the proof of the expurgated bound is shown to be flawed a new upper bound is also derived it is not known whether the original expurgated bound is valid for the important special case of square qam with gray labeling but the new bound is very close to and slightly tighter than the original bound for a numerical example

power and sample size evaluation for the cochran mantel haenszel mean score wilcoxon rank sum test and the cochran armitage test for trend the power of a chi square test and thus the required sample size are a function of the noncentrality parameter that can be obtained as the limiting expectation of the test statistic under an alternative hypothesis specification herein we apply this principle to derive simple expressions for two tests that are commonly applied to discrete ordinal data the wilcoxon rank sum test for the equality of distributions in two groups is algebraically equivalent to the mann whitney test the kruskal wallis test applies to multiple groups these tests are equivalent to a cochran mantel haenszel mean score test using rank scores for a set of c discrete categories although various authors have assessed the power function of the wilcoxon and mann whitney tests herein it is shown that the power of these tests with discrete observations that is with tied ranks is readily provided by the power function of the corresponding cochran mantel haenszel mean scores test for two and r 2 groups these expressions yield results virtually identical to those derived previously for rank scores and also apply to other score functions the cochran armitage test for trend assesses whether there is an monotonically increasing or decreasing trend in the proportions with a positive outcome or response over the c ordered categories of an ordinal independent variable for example dose herein it is shown that the power of the test is a function of the slope of the response probabilities over the ordinal scores assigned to the groups that yields simple expressions for the power of the test copyright 2011 john wiley sons ltd

achieving full frequency and space diversity in wireless systems via bicm ofdm stbc and viterbi decoding orthogonal frequency division multiplexing ofdm is known as an efficient technique to combat frequency selective channels in this paper we show that the combination of bit interleaved coded modulation bicm and ofdm achieves the full frequency diversity offered by a frequency selective channel with any kind of power delay profile pdp conditioned on the minimum hamming distance dfree of the convolutional code this system has a simple viterbi decoder with a modified metric we then show that by combining such a system with space time block coding stbc one can achieve the full space and frequency diversity of a frequency selective channel with n transmit and m receive antennas bicm stbc ofdm achieves the maximum diversity order of nml over l tap frequency selective channels regardless of the pdp of the channel this latter system also has a simple viterbi decoder with a properly modified metric we verify our analytical results via simulations including channels employed in the ieee 802 11 standards

sample size determination for some common nonparametric tests abstract the article discusses the problem of determining the number of observations required by some common nonparametric tests so that the tests have power at least 1 against alternatives that differ sufficiently from the hypothesis being tested it is shown that the number of observations depends on certain simple probabilities a method is suggested for fixing the value of the appropriate probability when determining sample size

sample size calculation for the wilcoxon mann whitney test adjusting for ties in this paper we study sample size calculation methods for the asymptotic wilcoxon mann whitney test for data with or without ties the existing methods are applicable either to data with ties or to data without ties but not to both cases while the existing methods developed for data without ties perform well the methods developed for data with ties have limitations in that they are either applicable to proportional odds alternatives or have computational difficulties we propose a new method which has a closed form formula and therefore is very easy to calculate in addition the new method can be applied to both data with or without ties simulations have demonstrated that the new sample size formula performs very well as the corresponding actual powers are close to the nominal powers

asymptotic error analysis of diversity schemes on arbitrarily correlated rayleigh channels asymptotic error rate expressions are derived for multi branch equal gain combining and selection combining operating on arbitrarily correlated rayleigh fading channels these closed form solutions are used to provide rapid and accurate estimation of the error rates in large signal to noise ratio regions more importantly they reveal additional insights into the transmission characteristics of linear diversity combining schemes operating on correlated rayleigh fading channels it is shown that the asymptotic error rates over correlated branches can be obtained by scaling the asymptotic error rates over independent branches with a factor det m where det m is the determinant of the normalized channel correlation matrix this relationship is valid for both coherent and noncoherent signalings a similar relationship is also established for the outage probabilities of fading wireless systems employing multi branch diversity reception

unified asymptotic analysis of linearly modulated signals in fading non gaussian noise and interference in this paper we present a unified asymptotic symbol error rate ser analysis of linearly modulated signals impaired by fading and possibly non gaussian noise which in our definition also includes interference the derived asymptotic closed form results are valid for a large class of fading and noise processes our analysis also encompasses diversity reception with equal gain and selection combining and is extended to binary orthogonal modulation we show that for high signal to noise ratios snrs the ser of linearly modulated signals depends on the mellin transform of the probability density function pdf of the noise since the mellin transform can be readily obtained for all commonly encountered noise pdfs the provided ser expressions are easy and fast to evaluate furthermore we show that the diversity gain only depends on the fading statistic and the number of diversity branches whereas the combining gain depends on the modulation format the type of fading the number of diversity branches and the type of noise an exception are systems with a diversity gain of one since their combining gain and asymptotic ser are independent of the type of noise however in general in a log log scale for high snr the ser curves for different types of noise are parallel but not identical and their relative shift depends on the mellin transforms of the noise pdfs

on properties of the support of capacity achieving distributions for additive noise channel models with input cost constraints we study the classical problem of characterizing the channel capacity and its achieving distribution in a generic fashion we derive a simple relation between three parameters the input output function the input cost function and the noise probability density function one which dictates the type of the optimal input in layman terms we prove that the support of the optimal input is bounded whenever the cost grows faster than a cutoff growth rate equal to the logarithm of the inverse of the noise probability density function evaluated at the input output function furthermore we prove a converse statement that says whenever the cost grows slower than the cutoff rate the optimal input has necessarily an unbounded support in addition we show how the discreteness of the optimal input is guaranteed whenever the triplet satisfy some analyticity properties we argue that a suitable cost function to be imposed on the channel input is one that grows similarly to the cutoff rate our results are valid for any cost function that is super logarithmic they summarize a large number of previous channel capacity results and give new ones for a wide range of communication channel models such as gaussian mixtures generalized gaussians and heavy tailed noise models that we state along with numerical computations

analysis of a clinical trial involving a combined mortality and adherence dependent interval censored endpoint clinical trials often involve a variety of clinical and laboratory measures that are used as endpoints and sometimes two of these measures are combined in one endpoint when the individual components of such a combined endpoint are time to event measurements the analysis is straightforward if each of the components is measured frequently and regularly over time however the analysis of the combined endpoint is more difficult when one component of the endpoint is right censored and the other is interval censored this paper describes a statistic based on a rank ordering of events for such a combined measure the power of the test statistic is explored

global rank tests for multiple possibly censored outcomes clinical trials often collect multiple outcomes on each patient as the treatment may be expected to affect the patient on many dimensions for example a treatment for a neurological disease such as als is intended to impact several dimensions of neurological function as well as survival the assessment of treatment on the basis of multiple outcomes is challenging both in terms of selecting a test and interpreting the results several global tests have been proposed and we provide a general approach to selecting and executing a global test the tests require minimal parametric assumptions are flexible about weighting of the various outcomes and are appropriate even when some or all of the outcomes are censored the test we propose is based on a simple scoring mechanism applied to each pair of subjects for each endpoint the pairwise scores are then reduced to a summary score and a rank sum test is applied to the summary scores this can be seen as a generalization of previously proposed nonparametric global tests e g o brien 1984 we discuss the choice of optimal weighting schemes based on power and relative importance of the outcomes as the optimal weights are generally unknown in practice we also propose an adaptive weighting scheme and evaluate its performance in simulations we apply the methods to analyze the impact of a treatment on neurological function and death in an als trial

error probability bounds for bit interleaved space time trellis coding over block fading channels in this correspondence we investigate the error probability bounds of a bit interleaved space time trellis coding bisttc scheme which concatenates bit interleaved coded modulation bicm with a space time trellis code sttc we focus on general block fading channels wherein each data packet or frame spans a number of independent fading blocks bicm applied to such channel environments can effectively exploit both time and frequency selectivity while a sttc maximizes the spacial diversity order the exact pairwise error probability pep and weight enumeration function wef of bisttc are used to evaluate the error bound due to the concatenation of an outer error correction code ecc and a sttc the overall wef of bisttc is obtained by combining the wef of the outer code with that of the sttc through an uniform interleaver the main challenge here is to compute the wef of a sttc for block fading channels with reasonable complexity we rely on constructing a composite state transition matrix based on a number of single step virtual trellises each corresponding to an independent fading block within a frame we discuss how this approach reduces storage and computational requirements in the bound analysis compared to the existing method of obtaining the state transition matrix through accumulation of single step transition matrices the derived bound is applicable to both spatially uncorrelated and correlated channels as well as to both flat and frequency selective block fading channels the bound is shown to provide a reasonably close estimate of the simulated performance based on the turbo equalizer like iterative processing of soft information between the sttc decoder and the ecc decoder

applying coordinate products to the topological identification of normed spaces using the l 2 products we find pre hilbert spaces that are absorbing sets for all borelian classes of order 1 we also show that the following spaces are homeomorphic to the countable product of the space x n r x n is bounded 1 every coordinate product c h n of normed spaces h n in the sense of a banach space c where each h n is an absolute f set and infinitely many of the h n s are z spaces 2 every function space l p p p l p with the l q topology 0 q p 3 every sequence space l p p p l p with the l q topology 0 p q we also note that each additive and multiplicative borelian class of order 2 each projective class and the class of nonprojective spaces contain uncountably many topologically different pre hilbert spaces which are z spaces

a note on sample size determination for a nonparametric test of location sample size determination methods are considered for hypothesis testing about location parameters of two continuous distributions using independent random samples and the mann whitney wilcoxon mww test for a specified shift size and power these methods are based on the so called noether s formula derived from a normal approximation to the power of the mww test and a pilot sample from each of the distributions compared with alternate methods including the one using bootstrap similar to that of hamilton and collings 1991 the new methods are shown to be at least as good in terms of accuracy and substantially more efficient in terms of speed and variability the simpler method using linearly smoothed empirical cdf s of the pilot samples and requiring no bootstrapping is recommended for practical use extensions and adaptations for obtaining an upper confidence bound on the sample size and to general linear rank tests are indicated

on crb for parameter estimation in two component gaussian mixtures and the impact of misspecification a closed form expression for the cramer rao lower bound crb for parameter estimation in the presence of a two component gaussian mixture noise model is derived it is further shown that this closed form expression can be lower bounded by a simple two term expression closed form expressions are also derived for the variance of the maximum likelihood estimator mle when the parameters of the gaussian mixture model are misspecified it is then shown that the mle can handle a significant amount of misspecification of the parameters of the gaussian mixture model and yet maintain a variance close to the crb

estimating shannon and constrained capacities of bernoulli gaussian impulsive noise channels in rayleigh fading this paper presents a novel approach to tightly estimate the ergodic shannon and constrained capacities of an additive bernoulli gaussian bg impulsive noise channel in rayleigh fading environments where channel gains are known at the receiver but not at the transmitter we first show that the differential entropy of the bg impulsive noise can be established in closed form using gaussian hypergeometric function 2 f 1 1 1 cdot cdot the shannon capacity is then calculated via upper and lower bounds specifically we derive in closed form two upper bounds on the shannon capacity using the assumption of a gaussian output and using full knowledge of noise state respectively under the assumption of a gaussian input we propose a novel approach to calculate a lower bound by examining the instantaneous output entropy in two regions of channel gains in the high gain region the lower bound is evaluated via the upper bound obtained under the gaussian output assumption in the other region we apply the piecewise linear curve fitting pwlcf method to estimate the lower bound it is then demonstrated that the lower bound can be calculated with a predetermined accuracy by establishing the difference between the lower bound and the two upper bounds we show that the lower bound can be used to effectively estimate the shannon capacity finally we detail a pwlcf based method to estimate the constrained capacity for a finite alphabet constellation to this end we first propose a numerical technique to calculate the instantaneous entropy of the output using 2 dimensional 2 d gauss hermite quadrature formulas the average output entropy is then obtained using the pwlcf method combined with the closed form expression of the entropy of the bg impulsive noise the constrained capacity can be effectively estimated

on sample size of the kruskal wallis test with application to a mouse peritoneal cavity study summary as the nonparametric generalization of the one way analysis of variance model the kruskal wallis test applies when the goal is to test the difference between multiple samples and the underlying population distributions are nonnormal or unknown although the kruskal wallis test has been widely used for data analysis power and sample size methods for this test have been investigated to a much lesser extent this article proposes new power and sample size calculation methods for the kruskal wallis test based on the pilot study in either a completely nonparametric model or a semiparametric location model no assumption is made on the shape of the underlying population distributions simulation results show that in terms of sample size calculation for the kruskal wallis test the proposed methods are more reliable and preferable to some more traditional methods a mouse peritoneal cavity study is used to demonstrate the application of the methods

optimal sample size planning for the wilcoxon mann whitney test there are many different proposed procedures for sample size planning for the wilcoxon mann whitney test at given type i and type ii error rates and respectively most methods assume very specific models or types of data to simplify calculations eg ordered categorical or metric data location shift alternatives etc we present a unified approach that covers metric data with and without ties count data ordered categorical data and even dichotomous data for that we calculate the unknown theoretical quantities such as the variances under the null and relevant alternative hypothesis by considering the following synthetic data approach we evaluate data whose empirical distribution functions match the theoretical distribution functions involved in the computations of the unknown theoretical quantities then well known relations for the ranks of the data are used for the calculations in addition to computing the necessary sample size n for a fixed allocation proportion t n1 n where n1 is the sample size in the first group and n n1 n2 is the total sample size we provide an interval for the optimal allocation rate t which minimizes the total sample size n it turns out that for certain distributions a balanced design is optimal we give a characterization of such distributions furthermore we show that the optimal choice of t depends on the ratio of the two variances which determine the variance of the wilcoxon mann whitney statistic under the alternative this is different from an optimal sample size allocation in case of the normal distribution model

incorporation of clustering effects for the wilcoxon rank sum test a large sample approach the wilcoxon rank sum test is frequently used in statistical practice for the comparison of measures of location when the underlying distributions are far from normal or not known in advance an assumption of the ordinary rank sum test is that individual sampling units are independent in many ophthalmologic clinical trials the early treatment for diabetic retinopathy scale etdrs is a principal endpoint used for measuring the level of diabetic retinopathy this is an ordinal scale and it is natural to consider the wilcoxon rank sum test for the comparison of the level of diabetic retinopathy between treatment groups however under this design unlike the usual wilcoxon rank sum test the subject is the unit of randomization but the eye is the unit of analysis furthermore a person will tend to have different but correlated etdrs scores for fellow eyes thus we propose a correction to the variance of the wilcoxon rank sum statistic that accounts for clustering effects and that can be used for both balanced same number of subunits per cluster or unbalanced different number of subunits per cluster data both in the presence or absence of ties with p value adjusted accordingly in this article we present large sample theory and simulation results for this test procedure and apply it to diabetic retinopathy data from type i diabetics in the sorbinil retinopathy trial

on topological classification of function spaces _ of low borel complexity we prove that if x is a countable nondiscrete completely regular space such that the function space cp x is an absolute fa set then cp x is homeomorphic to v where a xi e r xi o for all but finitely many i as an application we answer in the negative some problems of a v arhangel skil by giving examples of countable completely regular spaces x and y such that x fails to be a br space and a k space and hence x is not a kc space and not a sequential space and y fails to be an 80 space while the function spaces cp x and cp y are homeomorphic to cp x for the compact metric space 3s 0 u n1 n 1 2

feasibility study on the least square method for fitting non gaussian noise data abstract this study is to investigate the feasibility of least square method in fitting non gaussian noise data we add different levels of the two typical non gaussian noises levy and stretched gaussian noises to exact value of the selected functions including linear equations polynomial and exponential equations and the maximum absolute and the mean square errors are calculated for the different cases levy and stretched gaussian distributions have many applications in fractional and fractal calculus it is observed that the non gaussian noises are less accurately fitted than the gaussian noise but the stretched gaussian cases appear to perform better than the levy noise cases it is stressed that the least squares method is inapplicable to the non gaussian noise cases when the noise level is larger than 5