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

a numerical approach for solving an extended fisher kolomogrov petrovskii piskunov equation in the present paper a numerical method based on finite differences and spline collocation is presented for the numerical solution of a generalized fisher integro differential equation a composite weighted trapezoidal rule is manipulated to handle the numerical integrations which results in a closed form difference scheme a number of test examples are solved to assess the accuracy of the method the numerical solutions obtained indicate that the approach is reliable and yields results compatible with the exact solutions and consistent with other existing numerical methods convergence and stability of the scheme have also been discussed

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

adjustments to the mantel haenszel chi square statistic and odds ratio variance estimator when the data are clustered dependence between observations on a dichotomous variable renders invalid the usual chi square tests of independence and inflates the variances of parameter estimates such a situation occurs for example when subjects consist of members of the same family or with repeated observations on the same person in this paper we describe methods of measuring the extent of correlation or clustering we also develop methods of adjusting the mantel haenszel chi square test statistic and the variance of the mantel haenszel estimate of a common odds ratio in sets of 2 2 contingency tables

comparison of the proportions of affected relatives of cases and controls analysis and minimum sample size formula the problem of familial aggregation has frequently been approached by comparing the proportion of affected relatives of index cases with the proportion of affected relatives of index controls this type of study has an analogy with the case control design and is frequently analysed as if it was one it is however an essentially different design which may involve dependent observations we show that conventional tests for comparing proportions of affected subjects are still valid with this design we give a formula for computing the minimum sample size required to detect a given degree of aggregation measured by the intracluster correlation coefficient which can be estimated very simply by the difference in proportions of affected relatives in the two groups

non inferiority tests for clustered matched pair data non inferiority tests for matched pair data where pairs are mutually independent may not be appropriate when pairs are clustered the tests may require an adjustment to account for the correlation within a cluster we consider the adjusted score and wald type tests and a modification of obuchowski s method for non inferiority and compare them with the non inferiority test based on a method of moments estimate in terms of type 1 error rate and power by simulations for a small cluster size under various correlation structures in general the score test adjusted by an inflation factor and the modified obuchowski s method perform as good as the test based on moments estimate in the accuracy of type 1 error rates the latter does not provide reasonably close type 1 error rates to the nominal level when the number of clusters is 25 or smaller and a positive response rate for the standard procedure is 20 per cent or lower the adjusted score test the method based on moments estimate and the modified test are comparable in power the adjusted wald type test is too anti conservative and we should caution use of the test since number of clusters is strongly related to the accuracy of empirical type 1 error rate and power it is very important to have a sufficiently large number of clusters in designing a clustered matched pair study for non inferiority

sample size requirements for stratified cluster randomization designs sample size requirements are provided for designs of studies in which clusters are randomized within each of several strata where cluster size itself may be a stratifying factor the approach generalizes a formula derived by woolson et al which provides sample size requirements for the cochran mantel haenszel statistic issues of data analysis are also discussed

interval estimation of the common odds ratio from k 2 x 2 tables under cluster sampling we propose a simple correction factor for the variance of the logarithm of the common odds ratio estimated by the method of mantel and haenszel from a series of 2 2 tables when data are cluster correlated the adjustment is applied to the variance estimators proposed by hauck and by robins breslow and greenland for the log of the mantel haenszel common odds ratio and its performance is evaluated in a simulation study the key features of the proposed adjustment are i it has closed form ii it can accommodate covariates defined at the cluster specific level the site specific level or both and iii it does not require the user to specify a particular correlation structure for the response data the correction derives from liang and zeger s generalized estimating equations gee technique for logistic regression modelling via simulation we examine empirical versus nominal coverage probabilities for interval estimation of the common odds ratio using adjusted and unadjusted variance estimates and we present ratios of observed to estimated variances results are compared to those obtained from the fully iterated gee analysis the characteristics of the simulation study mimic scenarios common in the periodontal research setting with small numbers of subjects n 25 50 moderate numbers of sites per cluster m 4 16 32 and modest intracluster correlation levels 0 0 0 1 0 2 0 3 results show that adjusted confidence intervals applied to the hauck or the robins breslow greenland variance estimate provide coverage probabilities close to the nominal level for the mantel haenszel common odds ratio over a variety of cluster sizes and levels of correlation copyright 1999 john wiley sons ltd

a brief biography and appreciation of calyampudi radhakrishna rao with a bibliography of his books and papers we present a brief biography and appreciation of calyampudi radhakrishna rao to whom this fifth special issue on linear algebra and statistics of linear algebra and its applications is dedicated in celebration of his 75th birthday a bibliography of his books and papers is also included

correlated binomial variates properties of estimator of intraclass correlation and its effect on sample size calculation in group randomized studies the sample size calculations are complicated by within group worksite community etc correlation we compare by simulation the moment method and the more standard anova method of estimating the intraclass correlation we find the former is less biased for a small to moderate number of clusters but the difference disappears when the appropriate degree of freedom is used for the anova estimator we propose a simulation approach for sample size determination and illustrate it with an example

an algorithm for minimizing the norm of state feedback controllers in eigenvalue assignment abstract minimization of the norm of parametric feedback controllers which assign desired eigenvalues to the closed loop system is achieved by implementation of vector companion forms described in an earlier paper the simple algorithm which emerges gives insight into the mechanism by which parametric eigenvalue assignment is affected a couple of numerical examples are presented to illustrate advantages of this algorithm which easily computes the controller gain matrix with minimum norm

controller design with regional pole constraints a design procedure is developed that combines linear quadratic optimal control with regional pole placement specifically a static and dynamic output feedback control problem is addressed in which the poles of the closed loop system are constrained to lie in specified regions of the complex plane these regional pole constraints are embedded within the optimization process by replacing the covariance lyapunov equation by a modified lyapunov equation whose solution in certain cases leads to an upper bound on the quadratic cost functional the results include necessary and sufficient conditions for characterizing static output feedback controllers with bounded performance and regional pole constraints sufficient conditions are also presented for the fixed order i e full and reduced order dynamic output feedback problem with regional pole constraints circular elliptical vertical strip parabolic and section regions are considered

numerical solution of a non classical parabolic problem an integro differential approach a numerical method based on an integro differential formulation and approximation by local interpolating functions is proposed for solving a one dimensional parabolic partial differential equation subject to non classical conditions some specific test problems are solved using the proposed method numerical results obtained indicate that it can give accurate solutions and that it is an interesting and viable alternative to existing numerical methods for solving the class of problems under consideration

sequential design of linear quadratic state regulators with prescribed eigenvalues and specified relative stability this paper considers the problem of optimal regulator design of linear multivariable systems with prescribed pole locations and or poles corresponding to specified relative stability a sequential method based on the frequency domain optimality condition is proposed for achieving the desired pole assignment and determination of the corresponding quadratic performance index this design method enables the retention of some stable open loop poles and the associated eigenvectors in the closed loop system an illustrative example is provided to demonstrate the effectiveness of the proposed method

sample size considerations for stratified cluster randomization design with binary outcomes and varying cluster size stratified cluster randomization trials crts have been frequently employed in clinical and healthcare research comparing with simple randomized crts stratified crts reduce the imbalance of baseline prognostic factors among different intervention groups due to the popularity there has been a growing interest in methodological development on sample size estimation and power analysis for stratified crts however existing work mostly assumes equal cluster size within each stratum and uses multilevel models clusters are often naturally formed with random sizes in crts with varying cluster size commonly used ad hoc approaches ignore the variability in cluster size which may underestimate overestimate the required number of clusters for each group per stratum and lead to underpowered overpowered clinical trials we propose closed form sample size formulas for estimating the required total number of subjects and for estimating the number of clusters for each group per stratum based on cochran mantel haenszel statistic for stratified cluster randomization design with binary outcomes accounting for both clustering and varying cluster size we investigate the impact of various design parameters on the relative change in the required number of clusters for each group per stratum due to varying cluster size simulation studies are conducted to evaluate the finite sample performance of the proposed sample size method a real application example of a pragmatic stratified crt of a triad of chronic kidney disease diabetes and hypertension is presented for illustration

estimation of a common odds ratio in paired cluster randomization designs we develop two estimators of a common odds ratio for designs in which the investigator randomly assigns each of two clusters to interventions within strata the estimators rely on an empirical adjustment for clustering to provide improved estimators of relative to the standard woolf and mantel haenszel estimators respectively the results of a simulation study show that the suggested adjustment improves the accuracy of both of these well known estimators under conditions likely to arise in practice we find the clustered woolf estimator as particularly effective in terms of mean squared error reduction we also discuss interval estimation

a gametheoretic approach for non uniform pole shifting and pole homothety this paper investigates the problems of non uniform pole shifting or pole homothety in multivariable linear systems by using a gametheoretic approach and more specifically a particular nash strategy with an open loop information structure for a game including a time preference rate in the quadratic criteria such a result is possible due to properties of non symmetric algebraic riccati equations associated with an open loop nash strategy numerical examples allow to illustrate the efficiency of the proposed approach

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

optimal pole shifting for continuous multivariable linear systems a method for shifting the real parts of the open loop poles to any desired positions while preserving the imaginary parts is presented the method is based on the mirror image property which has been reported by molinari in other words molinari s results are extended and then a recursive approach is developed in each step of this approach it is required to solve a first order or a second order linear matrix lyapunov equation for shifting one real pole or two complex conjugate poles respectively the presented method yields a solution which is optimal with respect to a quadratic performance index the attractive feature of this method is that it enables solutions to complex problems to be easily found without solving any non linear algebraic riccati equation

numerical solution of the heat equation with nonlocal boundary conditions abstract the method for the heat equation with nonlocal boundary conditions is discussed in this paper the unconditional stability is proved for 1 2 subject to a condition that is much weaker than the one assumed in a paper by ekolin due to the nonlocal boundary conditions the systems of linear equations generated by the method have a coefficient matrix that is tridiagonal except its first and last rows three efficient algorithms for solving this kind of linear systems are presented a simple numerical example is given to compare their efficiency

a simple solution to the optimal eigenvalue assignment problem the problem of the optimal eigenvalue assignment for multi input linear systems is considered it is proven that for an n order system with m independent inputs the problem is split into the following two sequential stages initially the n m eigenvalues are assigned using an n m order system this assignment is not constrained to satisfy optimality criteria next an m order system is used to assign the remaining m eigenvalues in such a way that linear quadratic optimal criteria are simultaneously satisfied therefore the original n order optimal eigenvalue assignment problem is reduced to an m order optimal assignment problem moreover the structure of the equivalent m order system permits further simplifications which lead to solutions obtained by inspection

odds ratio inference with dependent data summary when data can be presented as a series of k 2 x 2 tables with cell counts xi ni xi yi mi yi it is often assumed that xi and yi are binomially distributed this paper deals with inference for the common odds ratio i l when the binomial assumption is invalid when k increases the consistency and asymptotic normality of the mantel haenszel 1959 estimator is derived the conditional maximum likelihood estimator is shown to be inconsistent and the asymptotic bias is computed when either the first order markov chain or beta binomial model is assumed the mantel haenszel test for testing f 1 is also shown to be inappropriate through some simulation studies two consistent test statistics are proposed and shown to be comparable to each other in terms of size and efficiency some possible further work is described

bayesian methods of analysis for cluster randomized trials with binary outcome data we explore the potential of bayesian hierarchical modelling for the analysis of cluster randomized trials with binary outcome data and apply the methods to a trial randomized by general practice an approximate relationship is derived between the intracluster correlation coefficient icc and the between cluster variance used in a hierarchical logistic regression model by constructing an informative prior for the icc on the basis of available information we are thus able implicitly to specify an informative prior for the between cluster variance the approach also provides us with a credible interval for the icc for binary outcome data several approaches to constructing informative priors from empirical icc values are described we investigate the sensitivity of results to the prior specified and find that the estimate of intervention effect changes very little in this data set while its interval estimate is more sensitive the bayesian approach allows us to assume distributions other than normality for the random effects used to model the clustering this enables us to gain insight into the robustness of our parameter estimates to the classical normality assumption in a model with a more complex variance structure bayesian methods can provide credible intervals for a difference between two variance components in order for example to investigate whether the effect of intervention varies across clusters we compare our results with those obtained from classical estimation discuss the relative merits of the bayesian framework and conclude that the flexibility of the bayesian approach offers some substantial advantages although selection of prior distributions is not straightforward

sample size and power for mcnemar s test with clustered data mcnemar s test is used to compare the distribution of two paired binary random variables when the data are clustered adjustment is needed to ensure that it is still a valid test this article presents two approximations for calculating the power and sample size for the adjusted mcnemar s test for clustered data working with a particular adjustment a simulation study is conducted to demonstrate the accuracy of these approximations the method is also applied to the design of a study involving positron emission tomography in detecting metastatic colorectal cancer and sensitivity of sample size computations to the design parameters are explored in this context

linear quadratic regulators with eigenvalue placement in a specified region a linear optimal quadratic regulator is developed for optimally placing the closed loop poles of multivariable continuous time systems within the common region of an open sector bounded by lines inclined at 2k k 2 or 3 from the negative real axis with a sector angle 2 and the left hand side of a line parallel to the imaginary axis in the complex s plane also a shifted sector method is presented to optimally place the closed loop poles of a system in any general sector having a sector angle between 2 and the optimal pole placement is achieved without explicitly utilizing the eigenvalues of the open loop system the design method is mainly based on the solution of a linear matrix lyapunov equation and the resultant closed loop system with its eigenvalues in the desired region is optimal with respect to a quadratic performance index

random effects models for serial observations with binary response this paper presents a general mixed model for the analysis of serial dichotomous responses provided by a panel of study participants each subject s serial responses are assumed to arise from a logistic model but with regression coefficients that vary between subjects the logistic regression parameters are assumed to be normally distributed in the population inference is based upon maximum likelihood estimation of fixed effects and variance components and empirical bayes estimation of random effects exact solutions are analytically and computationally infeasible but an approximation based on the mode of the posterior distribution of the random parameters is proposed and is implemented by means of the em algorithm this approximate method is compared with a simpler two step method proposed by korn and whittemore using data from a panel study of asthmatics originally described in that paper one advantage of the estimation strategy described here is the ability to use all of the data including that from subjects with insufficient data to permit fitting of a separate logistic regression model as required by the korn and whittemore method however the new method is computationally intensive

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

analyzing k 2 x 2 tables under cluster sampling summary in many data analytic applications such as ophthalmologic longitudinal or periodontal studies multiple observations are recorded over several sites or timepoints within the same subject bringing about dependence between measurements this correlation in turn precludes the use of standard statistical methods that assume independence between outcome measurements for example the mantel haenszel statistic used to assess association between a binary outcome and a binary exposure while adjusting for a categorical covariate does not follow the usual chi squared distribution under the null hypothesis when there is correlation between observations a modified mantel haenszel procedure which makes adjustment for dependence is proposed no particular correlation structure is assumed for responses within a cluster this closed form adjustment stems from liang and zeger s 1986 biometrika73 13 22 generalized estimating equations approach for clustered data the difference between this tabular i e noniterative technique and many earlier tabular methods is that the current method allows for consideration of site specific exposure and covariate information an example from a periodontal research study illustrates application of the method

a spline collocation approach for a generalized parabolic problem subject to non classical conditions abstract a numerical method based on finite differences and spline collocation is proposed for solving a generalized parabolic problem subject to non classical conditions it is shown numerically that this scheme has second order convergence the stability conditions relating the time step size to mesh sizes were examined through numerical examples it turns out that the stability of the scheme in certain cases is subjected to a stability condition and depends on the nature of the problem a number of specific test problems are solved to assess the efficiency and performance of the method the numerical results obtained indicate that the approach is viable and yields accurate solutions when compared with the existing closed form solutions and generates results consistent with recent numerical approaches

pole placement by performance criterion modification a procedure is presented that modifies the state weighting matrix q and introduces a degree of relative stability into the original performance criterion to shape the resulting optimal dynamics by positioning the closed loop eigenvalues along the real axis of the optimal system in the linear quadratic regulator problem it is based on the results of the algebraic riccati equation are which establish the invariance of certain eigenspaces of the associated hamiltonian matrix with respect to certain perturbations of q and the degree of relative stability

numerical solution of a parabolic equation with non local boundary specifications the parabolic partial differential equations with non local boundary specifications model various physical problems numerical schemes are developed for obtaining approximate solutions to the initial boundary value problem for one dimensional second order linear parabolic partial differential equation with non local boundary specifications replacing boundary conditions the method of lines semi discretization approach will be used to transform the model partial differential equation into a system of first order linear ordinary differential equations odes the spatial derivative in the pde is approximated by a finite difference approximation the solution of the resulting system of first order odes satisfies a recurrence relation which involves a matrix exponential function numerical techniques are developed by approximating the exponential matrix function in this recurrence relation the new algorithms are tested on two problems from the literature the central processor unit times needed are also considered

beta binomial model for meta analysis of odds ratios in meta analysis of odds ratios ors heterogeneity between the studies is usually modelled via the additive random effects model rem an alternative multiplicative rem for ors uses overdispersion the multiplicative factor in this overdispersion model odm can be interpreted as an intra class correlation icc parameter this model naturally arises when the probabilities of an event in one or both arms of a comparative study are themselves beta distributed resulting in beta binomial distributions we propose two new estimators of the icc for meta analysis in this setting one is based on the inverted breslow day test and the other on the improved gamma approximation by kulinskaya and dollinger 2015 p 26 to the distribution of cochran s q the performance of these and several other estimators of icc on bias and coverage is studied by simulation additionally the mantel haenszel approach to estimation of ors is extended to the beta binomial model and we study performance of various icc estimators when used in the mantel haenszel or the inverse variance method to combine ors in meta analysis the results of the simulations show that the improved gamma based estimator of icc is superior for small sample sizes and the breslow day based estimator is the best for n 100 the mantel haenszel based estimator of or is very biased and is not recommended the inverse variance approach is also somewhat biased for ors 1 but this bias is not very large in practical settings developed methods and r programs provided in the web appendix make the beta binomial model a feasible alternative to the standard rem for meta analysis of ors 2017 the authors statistics in medicine published by john wiley sons ltd

estimation of a common odds ratio under binary cluster sampling we present results from a simulation study for the estimation of a common odds ratio in multiple 2 x 2 tables when the data are correlated within clusters we model the correlation of the data by the beta binomial distribution through a simulation study we compare the mantel haenszel estimator with rao and scott s estimator in terms of their biases observed variances relative efficiencies of their variances and 95 per cent coverage proportions we limit the simulation study to the case where there are the same number of subjects in each cluster and the same number of observations in each row of each stratum when rho 0 we recommend use of the mantel haenszel estimator gamma mh with an unadjusted variance and rao and scott s estimator gamma rsp with a pooled design effect in general when rho 0 we recommend the mantel haenszel estimator gamma mh with an adjusted variance and rao and scott s estimator gamma rsp with a pooled design effect

correlated binary regression with covariates specific to each binary observation regression methods are considered for the analysis of correlated binary data when each binary observation may have its own covariates it is argued that binary response models that condition on some or all binary responses in a given block are useful for studying certain types of dependencies but not for the estimation of marginal response probabilities or pairwise correlations fully parametric approaches to these latter problems appear to be unduly complicated except in such special cases as the analysis of paired binary data hence a generalized estimating equation approach is advocated for inference on response probabilities and correlations illustrations involving both small and large block sizes are provided

robustness results in linear quadratic gaussian based multivariable control designs the robustness of control systems with respect to model uncertainty is considered using simple frequency domain criteria available and new results are derived under a common framework in which the minimum singular value of the return differences transfer matrix is the key quantity in particular robustness results associated with multivariable control systems designed on the basis of linear quadratic lq and the linear quadratic gaussian lqg design methodologies are presented

an evaluation of methods for the stratified analysis of clustered binary data in community intervention trials a simulation study is conducted in a community intervention setting several methods of stratified analysis of clustered binary data are compared in terms of empirical significance and empirical power levels they are the mantel haenszel test statistic chi 2 mh the adjusted mantel haenszel test statistic of donald donner chi 2 dd rao scott chi 2 rsn and chi 2 rsp and zhang boos chi 2 zbn and chi 2 zbp wald chi 2 w robust wald chi 2 rw score chi 2 s robust score chi 2 rs and the test statistic based on generalized linear mixed model glmm chi 2 glmm when rho not equal 0 chi 2 mh has inflated type i error and it should not be used when observations are correlated the results also warn of the use of chi 2 rsn and chi 2 rw due to their poor performance in terms of empirical significance level chi 2 zbp and chi 2 glmm have better empirical significance levels as compared to other statistics however chi 2 zbp tends to have lower empirical powers than other statistics when the number of clusters n is less than 24 chi 2 rsp provides the highest empirical powers when rho or 0 1 and n or 12 when rho or 0 01 we recommend the use of chi 2 rs and chi 2 glmm since they have better overall performance in terms of empirical significance levels and empirical power levels

a locally most powerful test for homogeneity with many strata summary a locally most powerful test is proposed for homogeneity of nuisance parameters from many strata in the mixed model setting the test is based on the difference between the square of the score and the observed information about the nuisance parameters all evaluated at the null hypothesis several examples including constant normal mean with possibly different variances and matched case control studies are presented in detail the relative efficiencies of the unstratified estimator of parameter of interest versus the stratified one are calculated two data sets on matched case control studies are used for illustration

optimal linear regulator pole placement by weight selection a recursive algorithm for selecting the state weighting matrix of a linear quadratic regulator lqr problem to result in a set of desired closed loop eigenvalues is presented aggregation is used so that at each step of the recursive process where either a real pole or a complex conjugate pair is being placed one has to deal with two or four dimensional matrix equations regardless of the system s dimensionality in addition the algorithm is capable of placing a complex conjugate pair at a new complex conjugate location or at two distinct real locations

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

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

intraclass correlation coefficients and bootstrap methods of hierarchical binary outcomes intraclass correlation coefficients are designed to assess consistency or conformity between two or more quantitative measurements when multistage cluster sampling is implemented no methods are readily available to estimate intraclass correlations of binomial distributed outcomes within a cluster because statistical distribution of the intraclass correlation coefficients could be complicated or unspecified we propose using a bootstrap method to estimate the standard error and confidence interval within the framework of a multilevel generalized linear model we compared the results derived from a parametric bootstrap method with those from a non parametric bootstrap method and found that the non parametric method is more robust for non parametric bootstrap sampling we showed that the effectiveness of sampling on the highest level is greater than that on lower levels to illustrate the effectiveness we analyse survey data in china and do simulation studies

a spline collocation approach for the numerical solution of a generalized nonlinear klein gordon equation in this paper a finite element collocation approach using cubic b splines is employed for the numerical solution of a generalized form of the nonlinear klein gordon equation the efficiency of the method is tested on a number of examples that represent special cases of the extended equation including the sine gordon equation the numerical results are compared with existing numerical and analytic solutions and the outcomes confirm that the scheme yields accurate and reliable results even when few nodes are used at the time levels

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

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

estimating the number of clusters for the analysis of correlated binary response variables from unbalanced data dominating the literature on sample size determination are papers concerned with observations collected from independent responses we discuss in this paper under the common correlation model for binary responses the issue of estimating the number of groups when the sampling units are clusters of variable sizes

a parametric lyapunov equation approach to the design of low gain feedback low gain feedback has found several applications in constrained control systems robust control and nonlinear control low gain feedback refers to a family of stabilizing state feedback gains that are parameterized in a scalar and go to zero as the scalar decreases to zero such feedback gains can be constructed either by an eigenstructure assignment algorithm or through the solution of a parametric algebraic riccati equation are the eigenstructure assignment approach leads to feedback gains in the form of a matrix polynomial in the parameter while the are approach requires the solution of an are for each value of the parameter this note proposes an alternative approach to low gain feedback design based on the solution of a parametric lyapunov equation such an approach possesses the advantages of both the eigenstructure assignment approach and the are based approach it also avoids the possible numerical stiffness in solving a parametric are and the structural decomposition of the open loop system that is required by the eigenstructure assignment approach

effects of cluster sampling on epidemiologic analysis in population based case control studies we consider population based case control designs in which controls are selected by one of three cluster sampling plans from the entire population at risk the effects of cluster sampling on classical epidemiologic procedures are investigated and appropriately modified procedures are developed in particular modified procedures for testing the homogeneity of odds ratios across strata and for estimating and testing a common odds ratio are presented simulations that use the data from the 1970 health interview survey as a population suggest that classical procedures may be fairly robust in the presence of cluster sampling a more extreme example based on a mixed multinomial model clearly demonstrates that the classical mantel haenszel 1959 journal of the national cancer institute 22 719 748 and woolf haldane tests of no exposure effect may have sizes exceeding nominal levels and confidence intervals with less than nominal coverage under an alternative hypothesis classical estimates of odds ratios may also be biased with non self weighting cluster samples the modified procedures we propose remedy these defects

numerical solution of a parabolic equation subject to specification of energy in this paper special attention is devoted to one particular problem the diffusion equation subject to the specification of mass in a portion of the domain which has been studied quite extensively both analytically and numerically in recent years parabolic partial differential equations with a non local constraint in place of one of the standard boundary specifications feature in the mathematical modeling of many phenomena in this paper the application of the method of lines mol to such problems is considered the mol semi discretization approach will be used to transform the model partial differential equation into a system of first order linear ordinary differential equations odes the mol is a method of solving pdes by discretizing the equation with respect to all but one variable usually time the spatial partial derivative is approximated by a finite difference method the solution of the resulting system of first order odes satisfies a recurrence relation which involves a matrix exponential function numerical techniques are developed by approximating the exponential matrix function in this recurrence relation the matrix exponential function is approximated by a rational approximation consisting of four parameters new parallel algorithms are developed using the resulting approximation numerical experiments on two challenging examples are presented to illustrate the performance of the algorithms

application of the mcnemar test to non independent matched pair data mcnemar s one degree of freedom chi square test for the equality of proportions appears frequently in the analysis of pairs of matched binary outcome data y1i y2i an assumption underlying this test is that the responses from pair to pair are mutually independent in certain applications however the pairs may represent repeated measurements on the same experimental unit and hence this assumption is violated in this paper we suggest an adjustment to the mcnemar test to account for the repeated measures clustering effect and we report on a monte carlo simulation that evaluates the effectiveness of this approach

adjustments to the mantel haenszel test for data from stratified multistage surveys the usual form of the mantel haenszel test statistic assumes independent observations this is inappropriate for data from a stratified multistage survey two alternative adjustments to the test statistic are developed to deal with this a a modification of the effective sample size for each row of each table using the design effects extending a method proposed by donald and donner for familial aggregation studies b a taylor series approximation to the variance of the square root of the numerator of the mantel haenszel statistic both methods are evaluated by application to both simulated and real data the two methods perform equally well and offer a considerable improvement over the unadjusted test statistic when observations from the same cluster are highly correlated a simplified adjustment is also considered as is the need for correction to the variance of the odds ratio estimator

partial pole placement by lq regulators an inverse problem approach this paper gives a necessary and sufficient condition under which a state feedback control law places part of the closed loop poles exactly at specified points and at the same time is linear quadratic optimal for some quadratic weightings this is made possible by means of a solution to the inverse problem of optimal control a design example is given to illustrate the result

a simulation study of the analysis of sets of 2 2 contingency tables under cluster sampling estimation of a common odds ratio abstract this article presents the results of a simulation study of the estimation of a common odds ratio in multiple contingency tables when the data are correlated within clusters the correlation is modeled by the beta binomial distribution the three objectives of the study were to determine a the variances biases and relative efficiencies of the beta binomial maximum likelihood estimator mle the binomial mle and the mantel haenszel estimator of the common odds ratio b the variances and biases of two estimators of the intracluster correlation coefficient and c the effectiveness of adjusting variance estimates of the binomial and mantel haenszel estimators using the two estimators of the simulation shows that increasing the size of the intracluster correlation coefficient increases the variance and the positive bias of all three estimators of the odds ratio the mantel haenszel estimator displays less variance and bias than the mle except for large values of and an unexpected b

score test for familial aggregation in probands studies application to alzheimer s disease when studying familial aggregation of a disease the following two stage design is often used first select index subjects cases and controls then record data on their relatives the likelihood corresponding to this design is derived and a score test of homogeneity is proposed for testing the hypothesis of no aggregation this test takes into account the selection procedure and allows adjustment to be made for explanatory variables it appears as the sum of three terms a pure test of homogeneity a test of comparison of observed minus expected cases in the two groups and a term which adjusts for the possible unequal probabilities of disease of the index subjects asymptotic efficiency and a simulation study show that the proposed test is superior to either the pure homogeneity test or tests based on the comparison of numbers of affected in the two groups the test statistic which has an asymptotically standard normal distribution is applied to a study of familial aggregation of early onset alzheimer s disease for which a highly significant value 9 46 is obtained this is the highest value among the three tests compared in agreement with the simulation study a logistic normal model is fitted to the data taking account of the selection procedure it allows to estimate the regression parameters and the variance of the random effect the likelihood ratio test for familial aggregation seems less powerful than the score test

longitudinal data analysis using generalized linear models summary this paper proposes an extension of generalized linear models to the analysis of longitudinal data we introduce a class of estimating equations that give consistent estimates of the regression parameters and of their variance under mild assumptions about the time dependence the estimating equations are derived without specifying the joint distribution of a subject s observations yet they reduce to the score equations for multivariate gaussian outcomes asymptotic theory is presented for the general class of estimators specific cases in which we assume independence m dependence and exchangeable correlation structures from each subject are discussed efficiency of the proposed estimators in two simple situations is considered the approach is closely related to quasi likelih ood some key ironh estimating equation generalized linear model longitudinal data quasi likelihood repeated measures

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