inferences on multivariate measures of interclass and intraclass correlations in familial data summary multivariate measures of interclass and intraclass correlations are introduced to assess the degree of resemblance between family members with respect to more than one characteristic unified estimators for the multivariate measures are proposed as the eigenvalues of certain random matrices constructed by the matrices of the weighted sums of squares and products of observations asymptotic distributions of the proposed estimators are derived under the assumption of multivariate normality the asymptotic results produce approximate confidence intervals and test procedures in a simple way

limiting the effects of single member families in the estimation of the intraclass correlation summary because the maximum likelihood estimator for the intraclass correlation p does not exist in closed form for the unbalanced one way random effects model several noniterative estimators have been proposed for p in normal populations estimators that exclude all families of only one member have been recommended in the literature of familial correlations however these estimators lose efficiency appreciably for certain ranges of values of p indeed no single estimator can have the highest efficiency along the full range of the intraclass correlation coefficient three proposed estimators each formed by combining a pair of noniterative estimators of which one constituent estimator excludes families having only one member are shown by simulation to be nearly fully efficient for the full range of values of p a mathematical expression is given that can be used to calculate the large sample standard errors of all known noniterative estimators of the intraclass correlation

constant depth reducibility the purpose of this paper is to study reducibilities that can be computed by combinational logic networks of polynomial size and constant depth containing and s or s and not s with no bound placed on the fan in of and gates and or gates two such reducibilities are defined and reductions and equivalences among several common problems such as parity sorting integer multiplication graph connectivity bipartite matching and network flow are given certain problems are shown to be complete with respect to these reducibilities in the complexity classes deterministic logarithmic space nondeterministic logarithmic space and deterministic polynomial time new upper bounds on the size depth unbounded fan in circuit complexity of symmetric boolean functions are established

parent offspring and sibling correlation estimation based on minque theory summary we derive easily computable expressions for minque estimators of covariance parameters in an unbalanced family data structure used to study traits these estimators are strongly consistent and asymptotically normal simple expressions for limiting sample variances and covariances of minque estimators are provided

each n by n matrix with n 1 is a sum of 5 coninvolutory matrices abstract an n n complex matrix a is called coninvolutory if a a i n and skew coninvolutory if a a i n which implies that n is even we prove that each matrix of size n n with n 1 is a sum of 5 coninvolutory matrices and each matrix of size 2 m 2 m is a sum of 5 skew coninvolutory matrices we also prove that each square complex matrix is a sum of a coninvolutory matrix and a condiagonalizable matrix a matrix m is called condiagonalizable if m s 1 d s in which s is nonsingular and d is diagonal

test for equality of intraclass correlation coefficients under unequal family sizes the likelihood ratio test the large sample z test and the large sample z test have been proposed to test for the equality of two intraclass correlation coefficients under unequal family sizes based on two independent multinomial samples it has been found on the basis of simulation studies that the likelihood ratio test consistently and reliably produces results superior to those of the large sample z test and large sample z test in terms of power for various combinations of intraclass correlation coefficient values

a modified large sample approach to approximate interval estimation for a particular intraclass correlation coefficient we consider the problem of constructing a confidence interval for the intraclass correlation coefficient in an interrater reliability study when both raters and subjects are random effects in a two way random effects model a simulation study is conducted to investigate and compare the interval coverage per cents of three methods of approximation i satterthwaite s two moments the standard approach ii modified three moments a newer approach and iii modified large sample the newest approach for this particular problem one sided lower and two sided bounds are examined for the two moment approach coverage of confidence intervals can be understated for not only one sided bounds but also two sided bounds coverage of confidence intervals for the modified large sample approach is either correct or conservative and provides narrower two sided widths than the modified three moment approach which can understate coverage on one sided and two sided bounds for two or three raters the competing methods are illustrated with data from a reliability study of four raters evaluating the number of decayed missing and filled surfaces in the permanent teeth of ten subjects overall the modified large sample approach provides the most satisfactory performance of the three methods

confidence interval construction for parent offspring correlations summary a method is proposed for constructing a confidence interval about an interclass correlation between a parent score and one or more offspring scores in the analysis of family data the method is based on a modification of fisher s z transformation for the case of dependent observations a monte carlo study is performed comparing the effectiveness of the method to several other procedures that could be adopted

association arrays in assessing forms of dependencies between bivariate random variables abstract the bivariate distribution of pairs of random variables x y is said to be associated with respect to the classes of functions unk and unk if the product moment correlation r x y 0 for all unk and unk in the case in which both unk unk unk consist of all increasing functions then the bivariate distribution of x y is said to be positive quadrant dependent to apply the concept to data i examine the correlations for classes of extremal functions that span by positive combinations the totality of functions unk and unk to investigate whether the pair of random variables x y are associated with respect to unk and unk and to assess the relative degree or strength of association when comparing two sets of random variables x y and z w

confidence interval construction for a difference between two dependent intraclass correlation coefficients inferences for the difference between two dependent intraclass correlation coefficients iccs may arise in studies in which a sample of subjects are each assessed several times with a new device and a standard the icc estimates for the two devices may then be compared using a test of significance however a confidence interval for a difference between two iccs is more informative since it combines point estimation and hypothesis testing into a single inference statement we propose a procedure that uses confidence limits for a single icc to recover variance estimates needed to set confidence limits for the difference an advantage of this approach is that it provides a confidence interval that reflects the underlying sampling distribution simulation results show that this method performs very well in terms of overall coverage percentage and tail errors two data sets are used to illustrate this procedure

signed digit addition and related operations with threshold logic assuming signed digit number representations we investigate the implementation of some addition related operations assuming linear threshold networks we measure the depth and size of the networks in terms of linear threshold gates we show first that a depth 2 network with o n size weight and fan in complexities can perform signed digit symmetric functions consequently assuming radix 2 signed digit representation we show that the two operand addition can be performed by a threshold network of depth 2 having o n size complexity and o 1 weight and fan in complexities furthermore we show that assuming radix 2n 1 signed digit representations the multioperand addition can be computed by a depth 2 network with o n sup 3 size with the weight and fan in complexities being polynomially bounded finally we show that multiplication can be performed by a linear threshold network of depth 3 with the size of o n sup 3 requiring o n sup 3 weights and o n sup 2 log n fan in

estimation of interclass correlations in familial data summary the maximum likelihood estimates of the parent child interclass correlation and other parameters from familial data when the families have unequal numbers of offspring is given also given are two alternative sets of estimates which are easier to compute than the maximum likelihood estimates using these estimates a test for the interclass correlation to be zero requires only the ordinary student s t distribution similarly other tests use only known distributions in the analysis of familial data the degree of mother sib resemblance is measured by the interclass correlation the correlation between two different types of family member several estimates of this correlation have been proposed in the literature see rosner donner hennekens 1977 for details and their monte carlo comparisons of mean squared errors in particular they gave maximum likelihood estimates when the sib sizes are equal however when the sib sizes are not equal rosner 1979 proposed an algorithm for finding the maximum likelihood estimates which involves iterative maximization of an implicit function of two parameters this algorithm is difficult to implement and may not even converge for some sets of data recently mak ng 1981 used the linear model approach of kempthorne tandan 1953 to obtain the maximum likelihood estimates this reduces the problem to iterative maximization of a function of one parameter however nothing is known about the convergence of the procedure another iterative method of finding the maximum likelihood estimates has been given by smith 1980 in this paper an alternative approach is given which requires solving only one equation also two alternative sets of estimates are given these estimates are easier to compute than the maximum likelihood estimates using these estimates a test for the interclass correlation to be zero is given it requires only the ordinary student s t distribution similarly other tests use only known results

a comparison of confidence interval methods for the intraclass correlation coefficient in cluster randomized trials a correction has been published for this article in statistics in medicine 23 18 2004 2935 this study compared different methods for assigning confidence intervals to the analysis of variance estimator of the intraclass correlation coefficient the context of the comparison was the use of to estimate the variance inflation factor when planning cluster randomized trials the methods were compared using monte carlo simulations of unbalanced clustered data and data from a cluster randomized trial of an intervention to improve the management of asthma in a general practice setting the coverage and precision of the intervals were compared for data with different numbers of clusters mean numbers of subjects per cluster and underlying values of the performance of the methods was also compared for data with normal and non normally distributed cluster specific effects results of the simulations showed that methods based upon the variance ratio statistic provided greater coverage levels than those based upon large sample approximations to the standard error of searle s method provided close to nominal coverage for data with normally distributed random effects adjusted versions of searle s method to allow for lack of balance in the data generally did not improve upon it either in terms of coverage or precision analyses of the trial data however showed that limits provided by thomas and hultquist s method may differ from those of the other variance ratio statistic methods when the arithmetic mean differs markedly from the harmonic mean cluster size the simulation results demonstrated that marked non normality in the cluster level random effects compromised the performance of all methods confidence intervals for the methods were generally wide relative to the underlying size of suggesting that there may be great uncertainty associated with sample size calculations for cluster trials where large clusters are randomized data from cluster based studies with sample sizes much larger than those typical of cluster randomized trials are required to estimate with a reasonable degree of precision copyright 2002 john wiley sons ltd

the sum of orthogonal matrices abstract let f r c h let u n f be the set of unitary matrices in m n f and let o n f be the set of orthogonal matrices in m n f suppose n 2 we show that every a m n f can be written as a sum of matrices in u n f and of matrices in o n f let a m n f be given and let k 2 be the least integer that is a least upper bound of the singular values of a when f c we show that a can be written as a sum of k matrices from u n f when f r we show that if k 3 then a can be written as a sum of 6 orthogonal matrices if k 4 we show that a can be written as a sum of k 2 orthogonal matrices

sample size and power calculations for correlations between bivariate longitudinal data the analysis of a baseline predictor with a longitudinally measured outcome is well established and sample size calculations are reasonably well understood analysis of bivariate longitudinally measured outcomes is gaining in popularity and methods to address design issues are required the focus in a random effects model for bivariate longitudinal outcomes is on the correlations that arise between the random effects and between the bivariate residuals in the bivariate random effects model we estimate the asymptotic variances of the correlations and we propose power calculations for testing and estimating the correlations we compare asymptotic variance estimates to variance estimates obtained from simulation studies and compare our proposed power calculations for correlations on bivariate longitudinal data to power calculations for correlations on cross sectional data

comparison of recent estimators of interclass correlation from familial data the asymptotic and finite sample properties of several recent estimators of interclass correlation are compared to more traditional estimators in the case of a variable number of siblings per family it is shown that karlin s family weighted pairwise estimator karlin cameron and williams 1981 proceedings of the national academy of science 78 2664 2668 is virtually equivalent to the ensemble estimator rosner donner and hennekens 1977 applied statistics 26 179 187 thus suggesting an estimator of the former s asymptotic variance further an estimator proposed by srivastava 1984 biometrika 71 177 185 is shown to be identical to the modified sib mean estimator konishi 1982 annals of the institute of statistical mathematics 34 505 515 when the sib sib correlation is estimated by the method of unweighted group means although the estimator due to srivastava has smaller asymptotic variance than the other two the gain in efficiency is slight for familial data both asymptotically and in finite samples

estimation of parent offspring correlation summary when families have different numbers of offspring srivastava 1984 provided an easily computable estimator as an alternative to the maximum likelihood estimator of parentoffspring correlation some properties of this estimator are discussed vis a vis with those of ensemble and pairwise estimators further we derive the small sample distribution of srivastava s estimator under the assumption of null parent offspring correlation in the process we demonstrate that a certain function of the classical sib mean estimator has student s t distribution despite differing sib ship sizes

quantile estimation from repeated measurements abstract quantile estimators for a nonparametric components of variance situation are proposed consistency and asymptotic normality are proved situations with different numbers of measurements for different subjects are considered measurements on separate subjects are assumed to be independent whereas measurements on the same subject have a fixed dependence the estimators are obtained by inverting weighted empirical distribution functions an optimal estimator and a simple estimator based on within subject averages are studied small sample properties are studied by simulation as an illustration the estimators are applied to give normal limits for differential light sensitivity of the human eye

general purpose computation with neural networks a survey of complexity theoretic results we survey and summarize the literature on the computational aspects of neural network models by presenting a detailed taxonomy of the various models according to their complexity theoretic characteristics the criteria of classification include the architecture of the network feedforward versus recurrent time model discrete versus continuous state type binary versus analog weight constraints symmetric versus asymmetric network size finite nets versus infinite families and computation type deterministic versus probabilistic among others the underlying results concerning the computational power and complexity issues of perceptron radial basis function winner take all and spiking neural networks are briefly surveyed with pointers to the relevant literature in our survey we focus mainly on the digital computation whose inputs and outputs are binary in nature although their values are quite often encoded as analog neuron states we omit the important learning issues

cluster robust bootstrap inference in quantile regression models abstractin this article i develop a wild bootstrap procedure for cluster robust inference in linear quantile regression models i show that the bootstrap leads to asymptotically valid inference on the entire quantile regression process in a setting with a large number of small heterogeneous clusters and provides consistent estimates of the asymptotic covariance function of that process the proposed bootstrap procedure is easy to implement and performs well even when the number of clusters is much smaller than the sample size an application to project star data is provided supplementary materials for this article are available online

estimation of interclass correlation under circular covariance summary the maximum likelihood estimates of the parent sib interclass correlation and other parameters are obtained when the between sibs correlation matrix has a circular structure the problem of testing hypotheses about interclass correlation is also considered

confidence intervals for a variance ratio or for heritability in an unbalanced mixed linear model a procedure is presented for constructing an exact confidence interval for the ratio of the two variance components in a possibly unbalanced mixed linear model that contains a single set of m random effects this procedure can be used in animal and plant breeding problems to obtain an exact confidence interval for a heritability the confidence interval can be defined in terms of the output of a least squares analysis it can be computed by a graphical or iterative technique requiring the diagonalization of an m x m matrix or alternatively the inversion of a number of m x m matrices confidence intervals that are approximate can be obtained with much less computational burden using either of two approaches the various confidence interval procedures can be extended to some problems in which the mixed linear model contains more than one set of random effects corresponding to each interval procedure is a significance test and one or more estimators

kleiner s theorem for unitary representations of posets abstract a subspace representation of a poset s s 1 s t is given by a system v v 1 v t consisting of a vector space v and its subspaces v i such that v i v j if s i s j for each real valued vector 1 t with positive components we define a unitary representation of s as a system u u 1 u t that consists of a unitary space u and its subspaces u i such that u i u j if s i s j and satisfies 1 p 1 t p t 1 in which p i is the orthogonal projection onto u i we prove that s has a finite number of unitarily nonequivalent indecomposable representations for each weight if and only if s has a finite number of nonequivalent indecomposable subspace representations that is if and only if s contains any of kleiner s critical posets

simulating threshold circuits by majority circuits we prove that a single threshold gate with arbitrary weights can be simulated by an explicit polynomial size depth 2 majority circuit in general we show that a polynomial size depth d threshold circuit can be simulated uniformly by a polynomial size majority circuit of depth d 1 goldmann hastad and razborov showed in comput complexity 2 1992 pp 277 300 that a nonuniform simulation exists our construction answers two open questions posed by them we give an explicit construction whereas they use a randomized existence argument and we show that such a simulation is possible even if the depth d grows with the number of variables n their simulation gives polynomial size circuits only when d is constant

a comparison of confidence interval methods for the intraclass correlation coefficient summary different methods of obtaining confidence intervals for the intraclass correlation coefficient p in the unbalanced one way random effects model are investigated focusing on applications to family studies methods based on simple modifications of formulas for the case of equal group sizes are found to provide adequate coverage at small to moderate values of p a method based on the large sample standard error of the sample intraclass correlation as derived by smith 1956 annals of human genetics 21 363 373 is shown to provide consistently good coverage at all values of p a method proposed by thomas and hultquist 1978 annals of statistics 6 582 587 also provides consistently good coverage but generates mean interval widths substantially greater than those generated by smith s method at values of p likely to arise in practice

reliability of graph analysis of resting state fmri using test retest dataset from the human connectome project the exploration of brain networks with resting state fmri rs fmri combined with graph theoretical approaches has become popular with the perspective of finding network graph metrics as biomarkers in the context of clinical studies a preliminary requirement for such findings is to assess the reliability of the graph based connectivity metrics in previous test retest trt studies this reliability has been explored using intraclass correlation coefficient icc with heterogeneous results but the issue of sample size has not been addressed using the large trt rs fmri dataset from the human connectome project hcp we computed iccs and their corresponding p values applying permutation and bootstrap techniques and varied the number of subjects from 20 to 100 the scan duration from 400 to 1200 time points the cost and the graph metrics using the anatomic automatic labelling aal parcellation scheme we quantified the reliability of the graph metrics computed both at global and regional level depending at optimal cost on two key parameters the sample size and the number of time points or scan duration in the cost range between 20 to 35 most of the global graph metrics are reliable with 40 subjects or more with long scan duration 14 min 24 s in large samples for instance 100 subjects most global and regional graph metrics are reliable for a minimum scan duration of 7 min 14s finally for 40 subjects and long scan duration 14 min 24 s the reliable regions are located in the main areas of the default mode network dmn the motor and the visual networks

semiparametric bayesian modeling of random genetic effects in family based association studies we consider the inference problem of estimating covariate and genetic effects in a family based case control study where families are ascertained on the basis of the number of cases within the family however our interest lies not only in estimating the fixed covariate effects but also in estimating the random effects parameters that account for varying correlations among family members these random effects parameters though weakly identifiable in a strict theoretical sense are often hard to estimate due to the small number of observations per family a hierarchical bayesian paradigm is a very natural route in this context with multiple advantages compared with a classical mixed effects estimation strategy based on the integrated likelihood we propose a fully flexible bayesian approach allowing nonparametric modeling of the random effects distribution using a dirichlet process prior and provide estimation of both fixed effect and random effects parameters using a markov chain monte carlo numerical integration scheme the nonparametric bayesian approach not only provides inference that is less sensitive to parametric specification of the random effects distribution but also allows possible uncertainty around a specific genetic correlation structure the bayesian approach has certain computational advantages over its mixed model counterparts data from the prostate cancer genetics project a family based study at the university of michigan comprehensive cancer center including families having one or more members with prostate cancer are used to illustrate the proposed methods a small scale simulation study is carried out to compare the proposed nonparametric bayes methodology with a parametric bayesian alternative

the asymptotic variance of the interclass correlation coefficient summary in genetics the term interclass correlation has been used to refer to the parent offspring and son daughter correlations within families under the assumption of variable family sizes and normality we derive the asymptotic variance of the maximum likelihood estimator of the sondaughter interclass correlation

every 2n by 2n complex matrix is a sum of three symplectic matrices abstract let j 2 n 0 i n i n 0 an a m 2 n c is called symplectic if a t j 2 n a j 2 n if n 1 then we show that every matrix in m 2 n c is a sum of two symplectic matrices if n 1 then we show that every matrix in m 2 n c is a sum of three symplectic matrices moreover we show that some matrices cannot be written with less than three symplectic matrices we also show that for every a m 2 n c there exist symplectic p q m 2 n c and b c d m n c such that p a q b c 0 d if a is skew hamiltonian j 2 n 1 a t j 2 n a then we show that a is a sum of two symplectic matrices

properties of alternative estimators of familial correlations under variable sibship size selected distributional properties of a variety of estimators of familial correlations spouse parent child and sibling were investigated numerically focusing particularly on sibling correlations under variable sibship size maximum likelihood estimators were evaluated for each of the three familial relationships an additional parent offspring estimator was studied using a parent midoffspring estimate pooled over variable sibship size for sibling intraclass correlations three estimators based on analysis of variance and three pairwise estimators using different weighting functions for variable sibship sizes were investigated correlations were estimated from data simulated under eight sampling conditions each replicated 1 000 times using two sets of true parameter values moderate and large sample sizes and normal versus highly non normal sample distributions of data the estimators are nearly unbiased and efficient but none of the sibling correlation estimators are normally distributed in small samples estimates from highly non normal data are nearly unbiased but are less efficient than those from normal data

a note on statistical methods adjusting for intraclass correlation on utilise un resultat sur les distributions de robbins et pitman 1949 pour obtenir une procedure de calcul exacte des vraies p valeurs des tests de signification

confidence intervals for intraclass correlation coefficients in a nonlinear dose response meta analysis this work is motivated by a meta analysis case study on antipsychotic medications the michaelis menten curve is employed to model the nonlinear relationship between the dose and d2 receptor occupancy across multiple studies an intraclass correlation coefficient icc is used to quantify the heterogeneity across studies to interpret the size of heterogeneity an accurate estimate of icc and its confidence interval is required the goal is to apply a recently proposed generic beta approach for construction the confidence intervals on iccs for linear mixed effects models to nonlinear mixed effects models using four estimation methods these estimation methods are the maximum likelihood second order generalized estimating equations and two two step procedures the beta approach is compared with a large sample normal approximation delta method and bootstrapping the confidence intervals based on the delta method and the nonparametric percentile bootstrap with various resampling strategies failed in our settings the beta approach demonstrates good coverages with both two step estimation methods and consequently it is recommended for the computation of confidence interval for iccs in nonlinear mixed effects models for small studies

robust asymptotic sampling theory for correlations in pedigrees methods to unravel the genetic determinants of non mendelian diseases lie at the next frontier of statistical approaches for human genetics it is generally agreed that before proceeding with segregation or linkage analysis the trait under study ought to be shown to exhibit familial correlation by coding dichotomous traits as binary variables a single robust approach in the estimation of pedigree correlations rather than two distinct approaches can be used to assess the potential heritability of a trait and latterly to examine the mode of inheritance the asymptotic theory to conduct hypothesis tests and confidence intervals for correlations among different members of nuclear families is well established but is applicable only if the nuclear families are independent as a further contribution to the literature we derive the asymptotic sampling distribution of correlations between random variables among arbitrary pairs of members in extended families for the pearson product moment estimator with generalized weights this derivation is done without assuming normality of the traits the sampling distribution is shown to be asymptotically normal to first order and hence large sample hypothesis tests and confidence intervals with estimates of the variances and correlation coefficients are proposed discussion concludes with an example and a suggestion for future research

the sum of two ϕs orthogonal matrices when s ts is normal and 1 σ s ts abstract let a nonsingular s m n c be given for a m n c set s a s 1 a t s we say that a is s symmetric if s a a we say that a is s orthogonal if a gl n and s a a 1 we say that a has a s polar decomposition if a u p for some s orthogonal u and s symmetric p suppose that s t s is normal and 1 s t s we determine conditions on a m n c so that a can be written as a sum of two s orthogonal matrices

on confidence intervals of a common intraclass correlation coefficient this study presents a novel approach for the confidence interval estimation of a common intraclass correlation coefficient derived from several samples under unequal family sizes this approach is developed using the concept of generalized pivots comparisons are made with a large sample procedure on the coverage probabilities

the length problem for a sum of idempotents abstract it was shown by r e hartwig and m s putcha and independently by p y wu that the complex square matrix t is a sum of finitely many idempotent matrices if and only if tr t is an integer and tr t rank t we determine the minimal number of required idempotents for such matrices in terms of their traces and sizes and we completely solve the problem for matrices t with size 2 3 4 or 5 also we give some sufficient necessary conditions for a complex matrix t to be the sum of three or more idempotents

an integer programming problem and rank decomposition of block upper triangular matrices a necessary and sufficient condition is given for a block upper triangular matrix a to be the sum of block upper rectangular matrices satisfying certain rank constraints the condition is formulated in terms of the ranks of certain submatrices of a the proof goes by reduction to an integer programming problem this integer programming problem has a totally unimodular constraint matrix which makes it possible to utilize farkas lemma

on the properties of maximum likelihood estimators of familial correlations under variable sibship size selected distributional properties of the maximum likelihood estimator and its z transformation of three familial correlations parental parent offspring filial were investigated numerically for the case of nuclear families with variable sibship size this investigation was based on six different sets of the three correlations and four different sample sizes defining 24 sampling conditions which were replicated 1 000 times each it was found that the distributional properties of the correlation estimator are affected by the magnitude of the correlations even in large samples although approximate normality is achieved locally fisher s z transformation here used only in its interclass form achieves reduction of skewness stabilization of variance and approach to normality already in small samples except for the filial correlation where it may be deemed inappropriate in smaller samples for both the correlation estimator and its z transformation the estimated relative efficiency was shown to be high better than 90 in most sampling conditions suggesting that the estimated minimum variance bound is a satisfactory estimator of the sampling variance it is concluded that the maximum likelihood estimation of familial correlations under variable sibship size is feasible and when prudently applied especially in the form of their z transformations provides an appropriate method in analyses of family studies

a comparison of parent offspring correlation estimators in terms of large sample mean squared error and bias in the analysis of familial data there has been some confusion regarding the best estimator of the parent offspring correlation and or regression rather than rely on simulation evidence we examine explicit formulae for the bias and mean squared error of four asymptotically unbiased estimators the pairwise correlation and regression coefficients the sib mean regression coefficient and the ensemble estimator in terms of bias it is shown that the ensemble estimator is a better estimator of the interclass correlation than the pairwise in terms of mean squared error the pairwise correlation is on balance the best estimator certainly when the intraclass correlation is less than about 3 this watershed value has been observed by other workers in simulation studies and varies from about 3 for families with a low mean number of progeny to about 1 for those with a high mean number an explicit expression for the watershed value is obtained in terms of the arithmetic mean the harmonic mean and the variance of the family size distribution

estimating intraclass correlation for binary data summary this paper reviews many different estimators of intraclass correlation that have been proposed for binary data and compares them in an extensive simulation study some of the estimators are very specific while others result from general methods such as pseudo likelihood and extended quasi likelihood estimation the simulation study identifies several useful estimators one of which does not seem to have been considered previously for binary data estimators based on extended quasi likelihood are found to have a substantial bias in some circumstances

estimation of interclass and intraclass correlations in multivariate familial data asymptotically normal estimators of interclass and intraclass correlations are derived for more than two quantitative characteristics of parent and siblings in a simple random sampling of families that have different numbers of offspring these estimators are proposed as an alternative to the maximum likelihood estimators which can be found only by iterative methods requiring prohibitively large amounts of computation the asymptotic variances of the proposed estimators are also given in an illustrative example these easily computable estimators are seen to be comparable to the corresponding maximum likelihood estimators

analysis of familial data linear model approach summary a linear model approach is proposed for deriving the maximum likelihood estimates of the mother sib interclass correlation and other parameters from familial data when the families have unequal numbers of offspring it leads to an algorithm which involves the maximization of an explicit function of a variable for estimating one parameter and direct substitutions for other parameters the algorithm is believed to be much simpler and more practical than that proposed by rosner 1979 some other advantages of the linearmodel approach are also discussed

non parametric bootstrap confidence intervals for the intraclass correlation coefficient the intraclass correlation coefficient rho plays a key role in the design of cluster randomized trials estimates of rho obtained from previous cluster trials and used to inform sample size calculation in planned trials may be imprecise due to the typically small numbers of clusters in such studies it may be useful to quantify this imprecision this study used simulation to compare different methods for assigning bootstrap confidence intervals to rho for continuous outcomes from a balanced design data were simulated for combinations of numbers of clusters 10 30 50 intraclass correlation coefficients 0 001 0 01 0 05 0 3 and outcome distributions normal non normal continuous the basic bootstrap t percentile bias corrected and bias corrected accelerated bootstrap intervals were compared with new methods using the basic and bootstrap t intervals applied to a variance stabilizing transformation of rho the standard bootstrap methods provided coverage levels for 95 per cent intervals that were markedly lower than the nominal level for data sets with only 10 clusters and only provided close to 95 per cent coverage when there were 50 clusters application of the bootstrap t method to the variance stabilizing transformation of rho improved upon the performance of the standard bootstrap methods providing close to nominal coverage

product of invertible operators of quadratic type abstract suppose all invertible quadratic operators t are assumed to satisfy t 2 bt i 0 we show that a complex matrix t is the product of two invertible quadratic matrices if and only if t is similar to a matrix of the form d d 1 i n i m i 1 j i 2 i 2 i 2 i xy 2 i yx 1 where b b 2 4 2 0 1 and gb 2 are not eigenvalues of d n xy and yx are nilpotent and each j i is a nilpotent jordan block of even size also we show that an n n matrix t is the product of finitely many invertible quadratic matrices if and only if det t m where m is an integer for n odd and is an even integer for n even on the other hand for operators on an infinite dimensional hilbert space we characterize the products of two and four invertible quadratic operators among normal operators and show that every invertible operator is the product of six invertible quadratic operators

a test of hypothesis on familial correlations when familial data are analysed the model usually employed assumes independence of family observations and constancy of interclass and intraclass correlations a statistic for testing the validity of the assumptions of family independence and constant interclass and intraclass correlations is developed the test statistic is for constant family size it has an asymptotic chi square distribution an example to illustrate the theory is given using frets s data on head lengths a recommendation is made on how to apply the test to the general case of varying sizes another recommendation is made on models to be tried once the null hypothesis is rejected

decomposition of threshold functions into bounded fan in threshold functions this paper obtains explicit decomposition of threshold functions into bounded fan in threshold functions a small fan in is important to satisfy technology constraints for large scale integration by employing the concept of error in the threshold function we are able to decompose functions in email protected 1 into a network of size o n c m 2 and depth o log 2n logm where n is the number of inputs of the function and m is the fan in bound the proposed construction enables one to trade off the size and depth of the decomposition with the fan in bound combined with the work on small weight threshold functions this implies polynomial size log 2 depth bounded fan in decompositions for arbitrary threshold functions in lt d these results compare favorably with the classical decomposition which has a size o 2 n m and depth o n m we also show that the decomposition size and depth can be significantly reduced by exploiting the relationships between the input weights as examples of this strategy we demonstrate an o n 2 m size decomposition of the majority function and o n m size decompositions of an error tolerant pattern matching function and the comparison function in all these examples except for the first level all other levels use only majority functions

a note on the analysis of familial data because families have varying numbers of offspring the maximum likelihood estimators of the interclass or parent offspring and intraclass or offspring offspring correlations are difficult to compute srivastava 1984 for this reason alternative estimators have been proposed which are not necessarily asymptotically efficient however maximum likelihood estimators of the inter and intraclass correlations in families of constant size can be obtained in closed form

estimation of the interclass correlation coefficient summary a noniterative method for estimating the interclass correlation coefficient is derived from the technique of weighted sums of squares the asymptotic variance of this estimator is derived under the assumption of normality through extensive monte carlo simulations the sample variance of this estimator is found not to differ greatly from that of the maximum likelihood estimator an important problem in genetics is the estimation of the degree of resemblance between a mother and her children as measured by the interclass correlation coefficient because parents produce differing numbers of offspring during their reproductive period the maximum likelihood estimation of the interclass correlation presents computational difficulties requiring an iterative solution therefore a number of noniterative estimators have been proposed in the literature these estimators will be shown to be associated with special cases of the generalized estimator derived from the weighted sums of squares of measurements on parents and offspring because our purpose is to provide a comprehensive approach to the noniterative point and interval estimation problems for the interclass correlation coefficient the asymptotic variance of the proposed estimator is derived under the assumption of normality moreover the sample variances of two special cases of the proposed estimator are compared to that of the maximum likelihood estimator through monte carlo simulations by a theoretical argument it is shown that the estimator proposed by srivastava 1984 has a uniformly smaller asymptotic variance than the ensemble estimator proposed by rosner donner hennekens 1977 smith 1957 used the weighted sums of squares approach for the estimation problem concerning the intraclass correlation coefficient which measures the degree to which siblings resemble one another our results together with those of smith 1957 provide a unified approach to estimation problems concerning familial correlations among a parent and offspring

bounded independence fools halfspaces we show that any distribution on 1 1 n that is k wise independent fools any halfspace or linear threshold function h 1 1 n to 1 1 i e any function of the form h x operatorname sign sum i 1 n w i x i theta where the w 1 dots w n and theta are arbitrary real numbers with error epsilon for k o epsilon 2 log 2 1 epsilon our result is tight up to log 1 epsilon factors using standard constructions of k wise independent distributions we obtain the first explicit pseudorandom generators g 1 1 s to 1 1 n that fool halfspaces specifically we fool halfspaces with error epsilon and seed length s k cdot log n o log n cdot epsilon 2 log 2 1 epsilon our approach combines classical tools from real approximation theory with structural results on halfspaces by servedio comput complexity 16 2007 pp 180 209

sample size calculation for cluster randomized cross over trials the cluster randomized cross over design has been proposed in particular because it prevents an imbalance that may bring into question the internal validity of parallel group cluster trials we derived a sample size formula for continuous outcomes that takes into account both the intraclass correlation coefficient representing the clustering effect and the interperiod correlation induced by the cross over design

multiple population models for multivariate random length data with applications in clinical trials this paper focuses on the development and study of multiple population models for multivariate random length data of the type often encountered in clinical trials if experimental outcomes per subject consist of multiple measurements of a quantitative variable and the number of these measurements then a multivariate random length vector is observed for this type of data the experimental treatment is likely to affect both the quantitative measurements and the number of these measurements one example of such data is from the national heart lung and blood institute type ii coronary intervention study brensike et al 1982 controlled clinical trials 3 91 111 1984 circulation 69 313 324 the outcome data consist of vectors of lesion sizes with lengths determined by the number of underlying lesions assessed from the patients angiograms where both the numbers and the lesion sizes depend on patients overall disease status we propose models which can realistically describe the relationships between the quantitative variables and the number of responses the asymptotic covariance of the maximum likelihood estimators is obtained data from the type ii study are analyzed using this multiple population model

testing hypotheses about interclass correlations from familial data testing problems concerning interclass correlations from familial data are considered in the case where the number of siblings varies among families under the assumption of multivariate normality two test procedures are proposed for testing the hypothesis that an interclass correlation is equal to a specified value to compare the properties of the tests including a likelihood ratio test monte carlo experiments are performed several test statistics are derived for testing whether two variables about a parent and child are uncorrelated the proposed tests are compared with previous test procedures using monte carlo simulation a general procedure for finding confidence intervals for interclass correlations is also derived

principal component analysis for multivariate familial data summary the use of a principal component analysis is considered for multivariate data on families with different numbers of siblings the coefficients in principal components are given as the eigenvectors of the weighted sums of squares and products matrix from the sibling data asymptotic distributions of the eigenvalues and eigenvectors of the estimated covariance matrix are obtained for an elliptical population asymptotic distributions of statistics associated with reduction of dimensionality are also derived the results can be used to construct approximate confidence intervals for the coefficients and variances of principal components

sibling and parent offspring correlation estimation with variable family size abstract a method for estimating intrafamilial correlations under variable family sizes involving weightings of paired data points is introduced classical methods of intrafamilial correlation estimation and those in current use are outlined and critically analyzed extensions of the proposed estimation method to more general data structures are delineated

sums of projections necessary and sufficient conditions are given for an n n complex matrix to be the sum of projection matrices the result is applied to show when a matrix is the sum of involutions

quantum matchgate computations and linear threshold gates the theory of matchgates is of interest in various areas in physics and computer science matchgates occur for example in the study of fermions and spin chains in the theory of holographic algorithms and in several recent works in quantum computation in this paper we completely characterize the class of boolean functions computable by unitary two qubit matchgate circuits with some probability of success we show that this class precisely coincides with that of the linear threshold gates the latter is a fundamental family that appears in several fields such as the study of neural networks using the above characterization we further show that the power of matchgate circuits is surprisingly trivial in those cases where the computation is to succeed with high probability in particular the only functions that are matchgate computable with success probability greater than 3 4 are functions depending on only a single bit of the input

solving non linear estimation equations summary in this paper we consider the numerical computation of a vector parameter estimate 0 which is a root of a system of unbiased non linear estimation equations a sequence 0 r is constructed which converges with a probability approaching 1 to 0 from any starting value 0 0 unlike newton type algorithms the 0 r do not involve matrices of partial derivatives furthermore it can be shown that 0 r 0 is of the order op n 2 so that great accuracy is achieved in very few iterations when the 0 r do not have closed forms an approximation is suggested an alternative and usually simpler way is also found for deriving analytically the expected value of the matrix of partial derivatives in the asymptotic covariance matrix of 0 the efficiency of the proposed algorithm is demonstrated numerically and compared with that of the em and the newton raphson algorithms applications to some well known statistical problems including missing data analysis variance component models familial correlations and errors in variables models are also briefly discussed

rank equalities for idempotent matrices with applications we show by elementary block matrix operations a variety of rank equalities related to sums of two idempotent matrices and give various consequences and applications

decomposition of a scalar matrix into a sum of orthogonal projections we describe the set of all n for which the scalar complex matrix in is a sum of k idempotent hermitian matrices and get the minimal number of summands for each in

log depth circuits for division and related problems we present optimal depth boolean circuits depth o log n for integer division powering and multiple products we also show that these three problems are of equivalent uniform depth and space complexity in addition we describe an algorithm for testing divisibility that is optimal for both depth and space

every matrix is a linear combination of three idempotents abstract it is shown that every n n matrix over a field of characteristic zero is a linear combination of three idempotent matrices it is proved that both 2 2 matrices and complex 3 3 matrices are linear combinations of two idempotents also we present 3 3 and 4 4 matrices that are not linear combinations of any two idempotents

polynomial threshold functions ac 0 functions and spectral norms this paper examines the class of polynomial threshold functions using harmonic analysis and applies the results to derive lower bounds related to ac 0 functions a boolean function is polynomial threshold if it can be represented as the sign of a sparse polynomial one that consists of a polynomial number of terms the main result of this paper is that the class of polynomial threshold functions can be characterized using their spectral representation in particular it is proved that an n variable boolean function whose l 1 spectral norm is bounded by a polynomial in n is a polynomial threshold function while boolean function whose l infty 1 spectral norm is not bounded by a polynomial in n is not a polynomial threshold function j bruck siam j discrete math 3 1990 pp 168 177 the motivation is that the characterization of polynomial threshold functions can be applied to obtain upper and lower bounds on the complexity of computing with networks of linear threshold elements in

interval estimation and hypothesis testing of intraclass correlation coefficients the generalized variable approach in this paper we propose a novel approach using the concept of generalized variable gv for the confidence interval estimation of the difference of two intraclass correlation coefficients under unequal family sizes this approach can also easily provide p values for hypothesis testing simulation results show that the gv approach can provide confidence intervals with good coverage properties and perform hypothesis testing with satisfactory type i error control furthermore the confidence intervals and p values by gv approach can be easily obtained by simulation therefore the gv approach is a suitable candidate for making inference concerning two intraclass correlation coefficients

a taxonomy of problems with fast parallel algorithms the class nc consists of problems solvable very fast in time polynomial in log n in parallel with a feasible polynomial number of processors many natural problems in nc are known in this paper an attempt is made to identify important subclasses of nc and give interesting examples in each subclass the notion of nc 1 reducibility is introduced and used throughout problem r is nc 1 reducible to problem s if r can be solved with uniform log depth circuits using oracles for s problems complete with respect to this reducibility are given for many of the subclasses of nc a general technique the parallel greedy algorithm is identified and used to show that finding a minimum spanning forest of a graph is reducible to the graph accessibility problem and hence is in nc 2 solvable by uniform boolean circuits of depth o log 2 n and polynomial size the class logcfl is given a new characterization in terms of circuit families the class det of problems reducible to integer determinants is defined and many examples given a new problem complete for deterministic polynomial time is given namely finding the lexicographically first maximal clique in a graph this paper is a revised version of s a cook 1983 in proceedings 1983 intl found comut sci conf lecture notes in computer science vol 158 pp 78 93 springer verlag berlin new york

sums of idempotents and logarithmic residues in zero pattern matrix algebras abstract a zero pattern algebra is a matrix subalgebra of c n n determined by a pattern of zeros the issue in this paper is under what conditions is a matrix in a zero pattern algebra a a sum of rank one idempotents in a or a logarithmic residue in a here logarithmic residues are contour integrals of logarithmic derivatives of analytic a valued functions it turns out that there is a necessary condition involving certain rank trace requirements although these requirements are generally not sufficient there are several important cases where they are

unions of rank trace complete preorders abstract a zero pattern algebra is a matrix algebra determined by a pattern of zeros corresponding to a preorder i e a reflexive transitive digraph the issue studied in 6 is under what conditions is a matrix in a zero pattern algebra a a sum of rank one idempotents in a or a logarithmic residue in a here logarithmic residues are contour integrals of logarithmic derivatives of analytic a valued functions as has been established in 6 there is a necessary condition involving certain rank trace requirements algebras for which this necessary condition is also sufficient are said to be rank trace complete several classes of rank trace complete algebras are identified in 6 also operations are described there for producing rank trace complete algebras out of given ones a basic operation of that type is taking a disjoint union in the present article this simple operation is generalized to a more involved one which preserves rank trace completeness too the new operation turns out to be a useful tool for establishing rank trace completeness in situations that up to now could not be handled one of the positive results is concerned with bouquets a special type of rooted trees it provides a partial answer to an issue raised in 7

sums of idempotent matrices abstract we show that any complex square matrix t is a sum of finitely many idempotent matrices if and only if trt is an integer and trt rank t moreover in this case the idempotents may be chosen such that each has rank one and has range contained in that of t we also consider the problem of the minimum number of idempotents needed to sum to t and obtain some partial results

efficient bayesian joint models for group randomized trials with multiple observation times and multiple outcomes in this paper we propose a bayesian method for group randomized trials grts with multiple observation times and multiple outcomes of different types we jointly model these outcomes using latent multivariate normal linear regression which allows treatment effects to change with time and accounts for 1 intra class correlation icc within groups 2 the correlation between different outcomes measured on the same subject and 3 the over time correlation otc of each outcome moreover we develop a set of innovative priors for the variance components which yield direct inference on the correlations avoid undesirable constraints and allow utilization of information from previous studies we illustrate through simulations that our model can improve estimation efficiency lower posterior standard deviations of iccs and treatment effects relative to single outcome models and models with diffuse priors on the variance components we also demonstrate the methodology using body composition data collected in the trial of activity in adolescent girls taag

skew coninvolutory matrices we study the properties of skew coninvolutory ee i matrices and derive canonical forms and a singular value decomposition we study the matrix function s a sa 1s 1 defined on nonsingular matrices and with s satisfying ss i or ss i we show that every square nonsingular a may be written as a xy with s x x and s y y 1 we also give necessary and sufficient conditions on when a nonsingular matrix may be written as a product of a coninvolutory matrix and a skew coninvolutory matrix or a product of two skew coninvolutory matrices moreover when a is similar to a 1 or when a is similar to a 1 or when a is similar to a or when a is similar to a we determine the possible jordan canonical forms of a for which the similarity matrix may be taken to be skew coninvolutory

score tests for interclass correlation in familial data we consider a generalized interclass correlation model which can be considered as a sister brother interclass correlation model and which incorporates the parent child interclass correlation model as a special case our main interest is in testing the significance of the interclass correlation for this we develop a score test an adjusted score test corrected for small samples is also obtained maximum likelihood estimates of the parameters under the null hypothesis required for the score test are obtained using an estimating equation approach of paul 1990 biometrilka 77 549 555 the score test and the adjusted score test for the parent child interclass correlation are then compared through simulation with the likelihood ratio test and a test statistic zp based on the pairwise estimator of the parent child interclass correlation and its asymptotic standard error based on size and power we recommend the score test for large k number of families and the adjusted score test when k is small

combined maximum likelihood estimates for the equicorrelation coefficient combined maximum likelihood estimates for equicorrelation covariance matrices are considered the case of a common equicorrelation rho and possibly different standard deviations sigma 1 sigma k among k experimental groups is examined first and the estimation of rho sigma 1 sigma k is discussed second under the assumption of a common standard deviation and possibly different equicorrelations the estimation of rho 1 rho k sigma is considered in each case maximum likelihood solutions and corresponding large sample variances are presented

the intraclass correlation coefficient distribution free definition and test a definition of the intraclass correlation coefficient is given on the basis of a general class of random effect model the conventional intraclass correlation coefficient and the intracluster correlation coefficient for binary data are both particular cases of the generalized coefficient we derive the score test for the hypothesis of null intraclass correlation in the exponential family the statistic does not depend on the particular distribution in this family and is related to the pairwise correlation coefficient the test can be adjusted for explanatory variables