fitting regression models to case control data by maximum likelihood summary we consider fitting categorical regression models to data obtained by either stratified or nonstratified case control or response selective sampling from a finite population with known population totals in each response category with certain models such as the logistic with appropriate constant terms a method variously known as conditional maximum likelihood breslow cain 1988 or pseudo conditional likelihood wild 1991 which involves the prospective fitting of a pseudo model results in maximum likelihood estimates of case control data we extend these results by showing the maximum likelihood estimates for any model can be found by iterating this process with a simple updating of offset parameters attention is also paid to estimation of the asymptotic covariance matrix one benefit of the results of this paper is the ability to obtain maximum likelihood estimates of the parameters of logistic models for stratified case control studies compare breslow cain 1988 scott wild 1991 using an ordinary logistic regression program even when the stratum constants are modelled

kardar parisi zhang equation in the weak noise limit pattern formation and upper critical dimension we extend the previously developed nonperturbative weak noise scheme applied to the noisy burgers equation in one dimension to the kardar parisi zhang equation for a growing interface in arbitrary dimensions by means of the cole hopf transformation we show that the growth morphology can be interpreted in terms of dynamically evolving textures of localized growth modes with superimposed diffusive modes in the cole hopf representation the growth modes are static solutions to the diffusion equation and the nonlinear schrodinger equation subsequently boosted to finite velocity by a galilei transformation we discuss the dynamics of the pattern formation and briefly the superimposed linear modes implementing the stochastic interpretation we discuss kinetic transitions and in particular the preliminary scaling properties pertaining to the pair mode or dipole sector in the dipole sector we obtain the hurst exponent h 3 d 4 d or dynamic exponent zdip 4 d 3 d for the random walk of growth modes below d 3 the dipole growth modes show anomalous diffusion above d 3 the dipole growth modes freeze finally applying derrick s theorem based on constrained minimization we show that the upper critical dimension is d 4 in the sense that growth modes cease to exist above this dimension

the elusive singularity in 1934 j leray 1 proved that navier stokes equations the partial differential equations used to describe viscous incompressible fluids admit global weak solutions the term global refers to the fact that there are no restrictions neither in the size of the initial data nor in the length of time these solutions persist the term weak is technical and refers to the smoothness of these solutions they are not very smooth leray s work did not preclude the possibility that starting from smooth data a singularity might form in finite time sixty three years later we still don t know the spontaneous generation and subsequent evolution of singularities in smooth

flexible maximum likelihood methods for assessing joint effects in case control studies with complex sampling case control studies can often be made more efficient by using frequency matching randomized recruitment stratified sampling or two stage sampling these designs share two common features 1 some first stage variables are ascertained for all study subjects while complete variable ascertainment is carried out for only a selected subsample and 2 the subsampling of subjects for second stage variable ascertainment depends jointly on their disease status and their observed first stage variables because first stage variables alter the subsampling fractions standard analyses require a multiplicative specification of any joint effects of a second and a first stage variable we show that by making use of missing data methods maximum likelihood estimates can be obtained for risk parameters of interest even those characterizing interactions between first and second stage variables joint effects can thus be modelled flexibly with allowance for both additive and multiplicative models preliminary data from a case control study of lung cancer as related to age sex and smoking provide an example leading to the suggestion that the combined effect of age and smoking is multiplicative language en

canonical phase space approach to the noisy burgers equation probability distributions we present a canonical phase space approach to stochastic systems described by langevin equations driven by white noise mapping the associated fokker planck equation to a hamilton jacobi equation in the nonperturbative weak noise limit we invoke a principle of least action for the determination of the probability distributions we apply the scheme to the noisy burgers and kardar parisi zhang equations and discuss the time dependent and stationary probability distributions in one dimension we derive the long time skew distribution approaching the symmetric stationary gaussian distribution in the short time region we discuss heuristically the nonlinear soliton contributions and derive an expression for the distribution in accordance with the directed polymer replica and asymmetric exclusion model results we also comment on the distribution in higher dimensions

power laws and stretched exponentials in a noisy finite time singularity model we discuss the influence of white noise on a generic dynamical finite time singularity model for a single degree of freedom we find that the noise effectively resolves the finite time singularity and replaces it by a first passage time or absorbing state distribution with a peak at the singularity and a long time tail exhibiting power law or stretched exponential behavior the study might be of relevance in the context of hydrodynamics on a nanometer scale in material physics and in biophysics

damped finite time singularity driven by noise we consider the combined influence of linear damping and noise on a dynamical finite time singularity model for a single degree of freedom we find that the noise effectively resolves the finite time singularity and replaces it by a first passage time distribution or absorbing state distribution with a peak at the singularity and a long time tail the damping introduces a characteristic cross over time in the early time regime the probability distribution and first passage time distribution show a power law behavior with scaling exponent depending on the ratio of the nonlinear coupling strength to the noise strength in the late time regime the behavior is controlled by the damping the study might be of relevance in the context of hydrodynamics on a nanometer scale in material physics and in biophysics

the bergman kernel function of some reinhardt domains the boundary behavior of the bergman kernel function of some reinhardt domains is studied upper and lower bounds for the bergman kernel function are found at the diagonal points z z let d be the reinhardt domain n d z cnilll 0 j 1 2 n and let k z wi be the bergman kernel function of d then there exist two positive constants m and m and a function f such that mf z z k z z mf z z holds for every z e d here n f z z r z n 1 j7j r z lzj 12 c3 1 o j 1 and r z jjzjje1 is the defining function for d the constants m and m depend only on ce cel cen and n not on z

finite time singularity in the dynamics of the world population economic and financial indices contrary to common belief both the earth s human population and its economic output have grown faster than exponential i e in a super malthusian mode for most of the known history these growth rates are compatible with a spontaneous singularity occurring at the same critical time 2052 10 signaling an abrupt transition to a new regime the degree of abruptness can be infered from the fact that the maximum of the world population growth rate was reached in 1970 i e about 80 years before the predicted singular time corresponding to approximately 4 of the studied time interval over which the acceleration is documented this rounding off of the finite time singularity is probably due to a combination of well known finite size effects and friction and suggests that we have already entered the transition region to a new regime as theoretical support a multivariate analysis coupling population capital r d and technology shows that a dramatic acceleration in the population growth during most of the timespan can occur even though the isolated dynamics do not exhibit it possible scenarios for the cross over and the new regime are discussed

logistic regression methods for retrospective case control studies using complex sampling procedures there are a number of possible designs for case control studies the simplest uses two separate simple random samples but an actual study may use more complex sampling procedures typically stratification is used to control for the effects of one or more risk factors in which we are interested it has been shown anderson 1972 biometrika 59 19 35 prentice and pyke 1979 biometrika 66 403 411 that the unconditional logistic regression estimators apply under stratified sampling so long as the logistic model includes a term for each stratum we consider the case control problem with stratified samples and assume a logistic model that does not include terms for strata i e for fixed covariates the prospective probability of disease does not depend on stratum we assume knowledge of the proportion sampled in each stratum as well as the total number in the stratum we use this knowledge to obtain the maximum likelihood estimators for all parameters in the logistic model including those for variables completely associated with strata the approach may also be applied to obtain estimators under probability sampling

oscillatory finite time singularities in finance population and rupture abstract we present a simple two dimensional dynamical system where two nonlinear terms exerting respectively positive feedback and reversal compete to create a singularity in finite time decorated by accelerating oscillations the power law singularity results from the increasing growth rate the oscillations result from the restoring mechanism as a function of the order of the nonlinearity of the growth rate and of the restoring term a rich variety of behavior is documented analytically and numerically the dynamical behavior is traced back fundamentally to the self similar spiral structure of trajectories in phase space unfolding around an unstable spiral point at the origin the interplay between the restoring mechanism and the nonlinear growth rate leads to approximately log periodic oscillations with remarkable scaling properties three domains of applications are discussed 1 the stock market with a competition between nonlinear trend followers and nonlinear value investors 2 the world human population with a competition between a population dependent growth rate and a nonlinear dependence on a finite carrying capacity 3 the failure of a material subjected to a time varying stress with a competition between positive geometrical feedback on the damage variable and nonlinear healing

canonical phase space approach to the noisy burgers equation presenting a general phase space approach to stochastic processes we analyze in particular the fokker planck equation for the noisy burgers equation and discuss the time dependent and stationary probability distributions in one dimension we derive the long time skew distribution approaching the symmetric stationary gaussian distribution in the short time regime we discuss heuristically the nonlinear soliton contributions and derive an expression for the distribution in accordance with the directed polymer replica model and asymmetric exclusion model results

the design and analysis of case control studies with biased sampling summary a design is proposed for case control studies in which selection of subjects for full variable ascertainment is based jointly on disease status and on easily obtained screening variables that may be related to the disease recruitment of subjects follows an independent bernoulli sampling scheme with recruitment probabilities set by the investigator in advance in particular the sampling can be set up to achieve on average frequency matching provided prior estimates of the disease rates or odds ratios associated with screening variables such as age and sex are available alternatively for example when studying a rare exposure one can enrich the sample with certain categories of subject following such a design there are two valid approaches to logistic regression analysis both of which allow for efficient estimation of effects associated with the screening variables that were allowed to bias the recruitment the statistical properties of the estimators are compared both for large samples based on asymptotics and for small samples based on simulations

instantons and intermittency we describe the method for finding the non gaussian tails of the probability distribution function pdf for solutions of a stochastic differential equation such as the convection equation for a passive scalar the random driven navier stokes equation etc the existence of such tails is generally regarded as a manifestation of the intermittency phenomenon our formalism is based on the wkb approximation in the functional integral for the conditional probability of large fluctuation we argue that the main contribution to the functional integral is given by a coupled field force configuration char22 the instanton as an example we examine the correlation functions of the passive scalar u advected by a large scale velocity field ensuremath delta correlated in time we find the instanton determining the tails of the generating functional and show that it is different from the instanton that determines the probability distribution function of high powers of u we discuss the simplest instantons for the navier stokes equation textcopyright 1996 the american physical society

solitons in the noisy burgers equation we investigate numerically the coupled diffusion advective type field equations originating from the canonical phase space approach to the noisy burgers equation or the equivalent kardar parisi zhang equation in one spatial dimension the equations support stable right hand and left hand solitons and in the low viscosity limit a long lived soliton pair excitation we find that two identical pair excitations scatter transparently subject to a size dependent phase shift and that identical solitons scatter on a static soliton transparently without a phase shift the soliton pair excitation and the scattering configurations are interpreted in terms of growing step and nucleation events in the interface growth profile finally we show that growing steps perform an anomalous random walk with dynamic exponent z 3 2

logistic regression for stratified case control studies fears and brown 1986 developed a procedure for logistic regression analysis of stratified case control data where the sampling fractions for cases and controls and thus their population frequencies were assumed known they fitted the usual prospective model to the case control data treating case control status as a binary outcome variable in order to adjust for the biased sampling they included the logarithm of the odds ratio relating the actual sample sizes and the population frequencies in each stratum as an offset in the regression equation however no adjustments were made to the estimated variances of the regression coefficients of variables associated with the strata to account for the information about them available in the population frequencies furthermore fears and brown incorrectly claimed that their procedure gave restricted maximum likelihood rml estimates aitchison and silvey 1958 based on the likelihood of the retrospectively sampled data breslow and cain 1988 show that the fears and brown procedure does yield consistent and asymptotically normal estimates of the regression parameters in a logistic regression model for the probability of disease development in fact it is equivalent to the conditional maximum likelihood cml estimate developed by manski and mcfadden 1981 for estimation of quantal response functions from stratified data see also hsieh manski and mcfadden 1985 breslow and cain extended the work of manski and mcfadden for use in the more realistic situation where the distribution of cases and controls in each stratum is estimated from a first stage sample rather than being assumed known they developed variance estimators for the regression coefficients that accurately reflect the additional information available in the first stage sample and that are easily modified to accommodate an infinite population at that stage we first present a reanalysis of the fears and brown data that contrasts the correct variances computed under the assumptions that the first stage sample is finite and infinite respectively with the incorrect variances obtained from the standard logistic analysis then using a subset of the data with only three strata we demonstrate the differences between the cml estimates of manski and mcfadden and rml estimates calculated from the retrospective probabilities a small scale simulation study investigates the properties of cml and rml estimators in samples of moderate size

semiparametric efficient estimation of a conditional density with missing or mismeasured covariates summary pepe and fleming and carroll and wand have recently proposed estimators in a parametric model for the density of a random variable y conditional on a vector of covariates x v when data on one of the regressors x is missing for some study subjects we propose a new class of estimators that remains consistent and asymptotically normal even when the probability that x is missing depends on the observed v and y includes an estimator whose asymptotic variance attains the semiparametric variance bound for the model and when the data are missing completely at random includes an estimator that is asymptotically equivalent to the inefficient estimators proposed by pepe and fleming and by carroll and wand the optimal estimator in our class depends on the unknown probability law generating the data when the vector v of non missing regressors has at most two continuous components we propose an adaptive semiparametric efficient estimator and compare the performance of the proposed semiparametric efficient estimator with the estimators proposed by pepe and fleming and carroll and wand in a small simulation study when v has many continuous components we propose an alternative class of adaptive estimators that should have high efficiency

logistic regression for two stage case control data summary samples of diseased cases and nondiseased controls are drawn at random from the population at risk after classification according to the exposure of interest subsamples of cases and controls are selected for purposes of covariable ascertainment a modification of the usual logistic regression analysis yields consistent estimates of covariable adjusted relative risks and their standard errors by balancing the numbers of exposed and nonexposed for whom covariable inforniation is ascertained within case and control samples some efficiency may be gained over the usual single stage design particularly when the exposure is rare and the relative risks associated with the covariables are large the procedure may be useful also when covariable information is missing for a large part of the sample

logistic analysis in case control studies under validation sampling summary this paper proposes an analysis of case control data under a double sampling scheme when covariates are missing or measured with error at the first stage of sampling and are validated at the second stage in a subsample the method combines risk information from both samples it is derived under the assumptions that i the prospective disease incidence model is of logistic form ii the proxy or partial information takes on finitely many values and iii the error is nondifferential the method of prentice pyke 1979 is extended to a two sample design to derive an estimating equation for the odds ratio parameters it provides an alternative estimator to that given by breslow cain 1988 consistency and asymptotic normality of the estimates are derived and a variance formula is presented parameters can be estimated by use of standard packages for quantalresponse data the method can easily be extended to the analysis of stratified designs with large strata

fitting logistic regression models in stratified case control studies summary methods are developed for fitting logistic models to data in which cases and or controls are sampled from the available cases and controls within population strata particular attention is paid to models in which stratum differences are modelled as well as the effects of the different risk factors experienced by individuals within strata maximum likelihood estimation is developed for discrete explanatory variables and is compared with the method of fears and brown 1986 biometrics 42 955 960 and breslow and cain 1988 biometrika 75 11 20 in which the prospective logistic model is fitted with a fixed offset an example is explored in which maximum likelihood estimation proves to be substantially more efficient than the fears brown method and the modelling of the stratum effects leads to much more efficient estimates of regression coefficients for the remaining variables

first passage and first exit times of a bessel like stochastic process we study a stochastic process x t which is a particular case of the rayleigh process and whose square is the bessel process with various applications in physics chemistry biology economics finance and other fields the stochastic differential equation is dx t nd x t dt 2d dw t where w t is the wiener process the drift term can arise from a logarithmic potential or from taking x t as the norm of a multidimensional random walk due to the singularity of the drift term for x t 0 different natures of boundary at the origin arise depending on the real parameter n entrance exit and regular for each of them we calculate analytically and numerically the probability density functions of first passage times or first exit times nontrivial behavior is observed in the case of a regular boundary