omitted variables in multilevel models statistical methodology for handling omitted variables is presented in a multilevel modeling framework in many nonexperimental studies the analyst may not have access to all requisite variables and this omission may lead to biased estimates of model parameters by exploiting the hierarchical nature of multilevel data a battery of statistical tools are developed to test various forms of model misspecification as well as to obtain estimators that are robust to the presence of omitted variables the methodology allows for tests of omitted effects at single and multiple levels the paper also introduces intermediate level tests these are tests for omitted effects at a single level regardless of the presence of omitted effects at a higher level a simulation study shows not surprisingly that the omission of variables yields bias in both regression coefficients and variance components it also suggests that omitted effects at lower levels may cause more severe bias than at higher levels important factors resulting in bias were found to be the level of an omitted variable its effect size and sample size a real data study illustrates that an omitted variable at one level may yield biased estimators at any level and in this study one cannot obtain reliable estimates for school level variables when omitted child effects exist however robust estimators may provide unbiased estimates for effects of interest even when the efficient estimators fail and the one degree of freedom test helps one to understand where the problem is located it is argued that multilevel data typically contain rich information to deal with omitted variables offering yet another appealing reason for the use of multilevel models in the social sciences

estimating functions of probability distributions from a finite set of samples this paper addresses the problem of estimating a function of a probability distribution from a finite set of samples of that distribution a bayesian analysis of this problem is presented the optimal properties of the bayes estimators are discussed and as an example of the formalism closed form expressions for the bayes estimators for the moments of the shannon entropy function are derived then numerical results are presented that compare the bayes estimator to the frequency counts estimator for the shannon entropy we also present the closed form estimators all derived elsewhere for the mutual information mathrm ensuremath chi 2 covariance and some other statistics c 1995 the american physical society

augmented weighted estimators dealing with practical positivity violation to causal inferences in a random coefficient model the inverse probability of treatment weighted iptw estimator can be used to make causal inferences under two assumptions 1 no unobserved confounders ignorability and 2 positive probability of treatment and of control at every level of the confounders positivity but is vulnerable to bias if by chance the proportion of the sample assigned to treatment or proportion of control is zero at certain levels of the confounders we propose to deal with this sampling zero problem also known as practical violation of the positivity assumption in a setting where the observed confounder is cluster identity i e treatment assignment is ignorable within clusters specifically based on a random coefficient model assumed for the potential outcome we augment the iptw estimating function with the estimated potential outcomes of treatment or of control for clusters that have no observation of treatment or control if the cluster specific potential outcomes are estimated correctly the augmented estimating function can be shown to converge in expectation to zero and therefore yield consistent causal estimates the proposed method can be implemented in the existing software and it performs well in simulated data as well as with real world data from a teacher preparation evaluation study

the jackknife estimate of variance tukey s jackknife estimate of variance for a statistic s x 1 x 2 cdots x n which is a symmetric function of i i d random variables x i is investigated using an anova like decomposition of s it is shown that the jackknife variance estimate tends always to be biased upwards a theorem to this effect being proved for the natural jackknife estimate of operatorname var s x 1 x 2 cdots x n 1 based on x 1 x 2 cdots x n

propensity score weighting with multilevel data propensity score methods are being increasingly used as a less parametric alternative to traditional regression to balance observed differences across groups in both descriptive and causal comparisons data collected in many disciplines often have analytically relevant multilevel or clustered structure the propensity score however was developed and has been used primarily with unstructured data we present and compare several propensity score weighted estimators for clustered data including marginal cluster weighted and doubly robust estimators using both analytical derivations and monte carlo simulations we illustrate bias arising when the usual assumptions of propensity score analysis do not hold for multilevel data we show that exploiting the multilevel structure either parametrically or nonparametrically in at least one stage of the propensity score analysis can greatly reduce these biases these methods are applied to a study of racial disparities in breast cancer screening among beneficiaries in medicare health plans

estimation of the information by an adaptive partitioning of the observation space we demonstrate that it is possible to approximate the mutual information arbitrarily closely in probability by calculating the relative frequencies on appropriate partitions and achieving conditional independence on the rectangles of which the partitions are made empirical results including a comparison with maximum likelihood estimators are presented

asymptotic bias in information estimates and the exponential bell polynomials we present a new derivation of the asymptotic correction for bias in the estimate of information from a finite sample the new derivation reveals a relationship between information estimates and a sequence of polynomials with combinatorial significance the exponential bell polynomials and helps to provide an understanding of the form and behavior of the asymptotic correction for bias

a bayesian tutorial for data assimilation data assimilation is the process by which observational data are fused with scientific information the bayesian paradigm provides a coherent probabilistic approach for combining information and thus is an appropriate framework for data assimilation viewing data assimilation as a problem in bayesian statistics is not new however the field of bayesian statistics is rapidly evolving and new approaches for model construction and sampling have been utilized recently in a wide variety of disciplines to combine information this article includes a brief introduction to bayesian methods paying particular attention to data assimilation we review linkages to optimal interpolation kriging kalman filtering smoothing and variational analysis discussion is provided concerning monte carlo methods for implementing bayesian analysis including importance sampling particle filtering ensemble kalman filtering and markov chain monte carlo sampling finally hierarchical bayesian modeling is reviewed we indicate how this approach can be used to incorporate significant physically based prior information into statistical models thereby accounting for uncertainty the approach is illustrated in a simplified advection diffusion model

geometrizing rates of convergence ii consider estimating a functional t f of an unknown distribution f in mathbf f from data x 1 cdots x n i i d f let omega varepsilon denote the modulus of continuity of the functional t over mathbf f computed with respect to hellinger distance for well behaved loss functions l t we show that inf t n sup mathbf f e fl t n t f is equivalent to l omega n 1 2 to within constants whenever t is linear and mathbf f is convex the same conclusion holds in three nonlinear cases estimating the rate of decay of a density estimating the mode and robust nonparametric regression we study the difficulty of testing between the composite infinite dimensional hypotheses h 0 t f leq t and h 1 t f geq t delta our results hold in the cases studied because the difficulty of the full infinite dimensional composite testing problem is comparable to the difficulty of the hardest simple two point testing subproblem

estimating entropy on m bins given fewer than m samples consider a sequence p sub n of discrete probability measures supported on m sub n points and assume that we observe n independent and identically distributed i i d samples from each p sub n we demonstrate the existence of an estimator of the entropy h p sub n which is consistent even if the ratio n m sub n is bounded and as a corollary even if this ratio tends to zero albeit at a sufficiently slow rate

nonparametric triple collocation triple collocation has found widespread application in the hydrological sciences because it provides information about the errors in our measurements without requiring that we have any direct access to the true value of the variable being measured triple collocation derives variance covariance relationships between three or more independent measurement sources and an indirectly observed truth variable in the case where the measurement operators are additive we generalize that theory to arbitrary observation operators by deriving nonparametric analogues to the total error and total correlation statistics as integrations of divergences from conditional to marginal probability ratios the nonparametric solution to the full measurement problem is under determined and we therefore retrieve conservative bounds on the theoretical total nonparametric error and correlation statistics we examine the application of both linear and nonlinear triple collocation to synthetic examples and to a real data test case related to evaluating space borne soil moisture retrievals using sparse monitoring networks and dynamical process models

binless strategies for estimation of information from neural data we present an approach to estimate information carried by experimentally observed neural spike trains elicited by known stimuli this approach makes use of an embedding of the observed spike trains into a set of vector spaces and entropy estimates based on the nearest neighbor euclidean distances within these vector spaces l f kozachenko and n n leonenko probl peredachi inf 23 9 1987 using numerical examples we show that this approach can be dramatically more efficient than standard bin based approaches such as the direct method s p strong r koberle r r de ruyter van steveninck and w bialek phys rev lett 80 197 1998 for amounts of data typically available from laboratory experiments

estimating the entropy rate of spike trains via lempel ziv complexity normalized lempel ziv complexity which measures the generation rate of new patterns along a digital sequence is closely related to such important source properties as entropy and compression ratio but in contrast to these it is a property of individual sequences in this article we propose to exploit this concept to estimate or at least to bound from below the entropy of neural discharges spike trains the main advantages of this method include fast convergence of the estimator as supported by numerical simulation and the fact that there is no need to know the probability law of the process generating the signal furthermore we present numerical and experimental comparisons of the new method against the standard method based on word frequencies providing evidence that this new approach is an alternative entropy estimator for binned spike trains

analyzing neural responses to natural signals maximally informative dimensions we propose a method that allows for a rigorous statistical analysis of neural responses to natural stimuli that are nongaussian and exhibit strong correlations we have in mind a model in which neurons are selective for a small number of stimulus dimensions out of a high dimensional stimulus space but within this subspace the responses can be arbitrarily nonlinear existing analysis methods are based on correlation functions between stimuli and responses but these methods are guaranteed to work only in the case of gaussian stimulus ensembles as an alternative to correlation functions we maximize the mutual information between the neural responses and projections of the stimulus onto low dimensional subspaces the procedure can be done iteratively by increasing the dimensionality of this subspace those dimensions that allow the recovery of all of the information between spikes and the full unprojected stimuli describe the relevant subspace if the dimensionality of the relevant subspace indeed is small it becomes feasible to map the neuron s input output function even under fully natural stimulus conditions these ideas are illustrated in simulations on model visual and auditory neurons responding to natural scenes and sounds respectively

an efron stein inequality for nonsymmetric statistics on etablit l analogue d un resultat d efron et stein 1981 que l on demontre a l aide d une technique d espace de hilbert introduite par vitale 1984

weighting for unequal selection probabilities in multilevel models when multilevel models are estimated from survey data derived using multistage sampling unequal selection probabilities at any stage of sampling may induce bias in standard estimators unless the sources of the unequal probabilities are fully controlled for in the covariates this paper proposes alternative ways of weighting the estimation of a two level model by using the reciprocals of the selection probabilities at each stage of sampling consistent estimators are obtained when both the sample number of level 2 units and the sample number of level 1 units within sampled level 2 units increase scaling of the weights is proposed to improve the properties of the estimators and to simplify computation variance estimators are also proposed in a limited simulation study the scaled weighted estimators are found to perform well although non negligible bias starts to arise for informative designs when the sample number of level 1 units becomes small the variance estimators perform extremely well the procedures are illustrated using data from the survey of psychiatric morbidity

second order noiseless source coding theorems shannon s celebrated source coding theorem can be viewed as a one sided law of large numbers we formulate second order noiseless source coding theorems for the deviation of the codeword lengths from the entropy for a class of sources that includes markov chains we prove a one sided central limit theorem and a law of the iterated logarithm

dealing with limited overlap in estimation of average treatment effects summary estimation of average treatment effects under unconfounded or ignorable treatment assignment is often hampered by lack of overlap in the covariate distributions between treatment groups this lack of overlap can lead to imprecise estimates and can make commonly used estimators sensitive to the choice of specification in such cases researchers have often used ad hoc methods for trimming the sample we develop a systematic approach to addressing lack of overlap we characterize optimal subsamples for which the average treatment effect can be estimated most precisely under some conditions the optimal selection rules depend solely on the propensity score for a wide range of distributions a good approximation to the optimal rule is provided by the simple rule of thumb to discard all units with estimated propensity scores outside the range 0 1 0 9

turbulence models for wall boundary layers boundary layer equations with boundary conditions corresponding to a wide range of plane two dimensional flows including the 14 mandatory test cases of the 1968 afosr ifd standford conference and those over the sub and supercritical airfoils of nakayama have been solved results were obtained from turbulence models based on an algebraic eddy viscosity formula the equations for turbulent kinetic energy and dissipation rate and equations for the reynolds stresses and dissipation rate they are shown to depend upon the model in an almost random manner the cost penalty associated with the higher order model is quantified and it is concluded that for the flows considered a well founded algebraic eddy viscosity model is preferred

universal entropy estimation via block sorting in this correspondence we present a new universal entropy estimator for stationary ergodic sources prove almost sure convergence and establish an upper bound on the convergence rate for finite alphabet finite memory sources the algorithm is motivated by data compression using the burrows wheeler block sorting transform bwt by exploiting the property that the bwt output sequence is close to a piecewise stationary memoryless source we can segment the output sequence and estimate probabilities in each segment experimental results show that our algorithm outperforms lempel ziv lz string matching based algorithms

bootstrapping clustered data summary various bootstraps have been proposed for bootstrapping clustered data from one way arrays the simulation results in the literature suggest that some of these methods work quite well in practice the theoretical results are limited and more mixed in their conclusions for example mccullagh reached negative conclusions about the use of non parametric bootstraps for one way arrays the purpose of this paper is to extend our understanding of the issues by discussing the effect of different ways of modelling clustered data the criteria for successful bootstraps used in the literature and extending the theory from functions of the sample mean to include functions of the between and within sums of squares and non parametric bootstraps to include model based bootstraps we determine that the consistency of variance estimates for a bootstrap method depends on the choice of model with the residual bootstrap giving consistency under the transformation model whereas the cluster bootstrap gives consistent estimates under both the transformation and the random effect model in addition we note that the criteria based on the distribution of the bootstrap observations are not really useful in assessing consistency

the upward bias in measures of information derived from limited data samples extracting information measures from limited experimental samples such as those normally available when using data recorded in vivo from mammalian cortical neurons is known to be plagued by a systematic error which tends to bias the estimate upward we calculate here the average of the bias under certain conditions as an asymptotic expansion in the inverse of the size of the data sample the result agrees with numerical simulations and is applicable as an additive correction term to measurements obtained under such conditions moreover we discuss the implications for measurements obtained through other usual procedures

estimating epistemic and aleatory uncertainties during hydrologic modeling an information theoretic approach 1 with growing interest in understanding the magnitudes and sources of uncertainty in hydrological modeling the difficult problem of characterizing model structure adequacy is now attracting considerable attention here we examine this problem via a model structureindependent approach based in information theory in particular we a discuss how to assess and compute the information content in multivariate hydrological data b present practical methods for quantifying the uncertainty and shared information in data while accounting for heteroscedasticity c show how these tools can be used to estimate the best achievable predictive performance of a model for a system given the available data and d show how model adequacy can be characterized in terms of the magnitude and nature of its aleatory uncertainty that cannot be diminished and is resolvable only up to specification of its density and its epistemic uncertainty that can in principle be suitably resolved by improving the model an illustrative modeling example is provided using catchment scale data from three river basins the leaf and chunky river basins in the united states and the chuzhou basin in china our analysis shows that the aleatory uncertainty associated with making catchment simulations using this data set is significant 50 further estimated epistemic uncertainties of the hymod sac sma and xinanjiang model hypotheses indicate that considerable room for model structural improvements remain

an approach to quantifying the efficiency of a bayesian filter 1 data assimilation is the bayesian conditioning of uncertain model simulations on observations to reduce uncertainty about model states in practice it is common to make simplifying assumptions about the prior and posterior state distributions and to employ approximations of the likelihood function which can reduce the efficiency of the filter we propose metrics that quantify how much of the uncertainty in a bayesian posterior state distribution is due to i the observation operator ii observation error and iii approximations of bayes law our approach uses discrete shannon entropy to quantify uncertainty and we define the utility of an observation for reducing uncertainty about a model state as the ratio of the mutual information between the state and observation to the entropy of the state prior these metrics make it possible to analyze the efficiency of a proposed observation system and data assimilation strategy and provide a way to examine the propagation of information through the dynamic system model we demonstrate the procedure on the problem of estimating profile soil moisture from observations at the surface top 5 cm the results show that when synthetic observations of 5 cm soil moisture are assimilated into a three layer model of soil hydrology the ensemble kalman filter does not use all of the information available in observations

on the accuracy of impulse methods for fluid flow a lagrangian numerical method based on impulse variables is analyzed the relation between discrete impulse vectors and vortex dipoles with a prescribed dipole moment is presented and used to adapt the high accuracy cutoff functions of vortex methods for use in impulse based methods the long time implementation of the impulse method is shown to contain a source of error whose growth in time leads to a severe loss of accuracy in interior flows applications for which impulse methods are well suited are identified

bias analysis in entropy estimation we consider the problem of finite sample corrections for entropy estimation new estimates of the shannon entropy are proposed and their systematic error the bias is computed analytically we find that our results cover correction formulae of current entropy estimates recently discussed in the literature the trade off between bias reduction and the increase of the corresponding statistical error is analysed

estimation of entropy and mutual information we present some new results on the nonparametric estimation of entropy and mutual information first we use an exact local expansion of the entropy function to prove almost sure consistency and central limit theorems for three of the most commonly used discretized information estimators the setup is related to grenander s method of sieves and places no assumptions on the underlying probability measure generating the data second we prove a converse to these consistency theorems demonstrating that a misapplication of the most common estimation techniques leads to an arbitrarily poor estimate of the true information even given unlimited data this inconsistency theorem leads to an analytical approximation of the bias valid in surprisingly small sample regimes and more accurate than the usual 1 n formula of miller and madow over a large region of parameter space the two most practical implications of these results are negative 1 information estimates in a certain data regime are likely contaminated by bias even if bias corrected estimators are used and 2 confidence intervals calculated by standard techniques drastically underestimate the error of the most common estimation methods finally we note a very useful connection between the bias of entropy estimators and a certain polynomial approximation problem by casting bias calculation problems in this approximation theory framework we obtain the best possible generalization of known asymptotic bias results more interesting this framework leads to an estimator with some nice properties the estimator comes equipped with rigorous bounds on the maximum error over all possible underlying probability distributions and this maximum error turns out to be surprisingly small we demonstrate the application of this new estimator on both real and simulated data