Linear and nonlinear Rossby waves in basins both with and without a thin meridional barrier

Submitted in partial fulfillment of the requirements for the degree of Master of Science at the Massachusetts Institute of Technology and the Woods Hole Oceanographic Institution February 2002The linear and nonlinear Rossby wave solutions are examined in homogeneous square basins on the ß-plane both with and without a thin meridional barrier In the presence of the meridional barrier the basin is almost partitioned into two; only two small gaps of equal width, d, to the north and south of the barrier allow communication between the eastern.and western sub-basins. Solutions are forced by a steady periodic wind forcing applied over a meridional strip near the eastern side. Bottom friction is present to allow the solutions to reach equilibrium. The linear solution for the basin containing the barrier is determined analytically and the nonlinear solutions for both basins are found numerically. In the linear solution with the barrier present, particular attention was paid to the resonant solutions. We examined the effects of varying the symmetry of the forcing about the mid-latitude, the frequency of the periodic forcing and the strength of the bottom friction. For each solution we focus on how the no net circulation condition, which is central to any solution in a barrier basin, is satisfied. The nonlinear solutions were studied for both basin configurations. In each case the transition from the weakly nonlinear solution to the turbulent solution was examined, as the forcing frequency and forcing strength were varied. Only integer multiples of the forcing frequency are present in the weakly nonlinear solutions. The turbulent solutions were accompanied by the appearance of many other frequencies whose exact origins are unknown, but are probably the result of instabilities. A hysteresis was found for the turbulent solutions of both the barrier-free and barrier basins. In the weakly nonlinear solutions of the barrier basin it was predicted and confirmed that there is never a steady net flow from sub-basin to sub-basin. It was also shown that with a symmetric forcing all modes oscillating with an odd multiple of the forcing frequency are symmetric and all modes oscillating with even multiples of the forcing frequency are antisymmetric

Testing the assumptions for the analysis of survival data arising from a prevalent cohort study with follow-up

In a prevalent cohort study with follow-up subjects identified as prevalent cases are followed until failure (defined suitably) or censoring. When the dates of the initiating events of these prevalent cases are ascertainable, each observed datum point consists of a backward recurrence time and a possibly censored forward recurrence time. Their sum is well known to be the left truncated lifetime. It is common to term these left truncated lifetimes "length biased" if the initiating event times of all the incident cases (including those not observed through the prevalent sampling scheme) follow a stationary Poisson process. Statistical inference is then said to be carried out under stationarity. Whether or not stationarity holds, a further assumption needed for estimation of the incident survivor function is the independence of the lifetimes and their accompanying truncation times. That is, it must be assumed that survival does not depend on the calendar date of the initiating event. We show how this assumption may be checked under stationarity, even though only the backward recurrence times and their associated (possibly censored) forward recurrence times are\ud observed. We prove that independence of the lifetimes and truncation times is equivalent to equality in distribution of the backward and forward recurrence times, and exploit this equivalence as a means of testing the former hypothesis. A simulation study is conducted to investigate the power and Type 1 error rate of our proposed tests, which include a bootstrap procedure that takes into account the pairwise dependence between the forward and backward recurrence times, as well as the potential censoring of only one of the members of each pair. We illustrate our methods using data from the Canadian Study of Health and Aging. We also point out an equivalence of the\ud problem presented here to a non-standard changepoint problem

Bayesian optimal design for changepoint problems

We consider optimal design for changepoint problems with particular attention paid to situations where the only possible change is in the mean. Optimal design for changepoint problems has only been addressed in an unpublished doctoral thesis, and in only one journal article, which was in a frequentist setting. The simplest situation we consider is that of a stochastic process that may undergo a, change at an unknown instant in some interval. The experimenter can take n measurements and is faced with one or more of the following optimal design problems: Where should these n observations be taken in order to best test for a change somewhere in the interval? Where should the observations be taken in order to best test for a change in a specified subinterval? Assuming that a change will take place, where should the observations be taken so that that one may best estimate the before-change mean as well as the after-change mean? We take a Bayesian approach, with a risk based on squared error loss, as a design criterion function for estimation, and a risk based on generalized 0-1 loss, for testing. We also use the Spezzaferri design criterion function for model discrimination, as an alternative criterion function for testing. By insisting that all observations are at least a minimum distance apart in order to ensure rough independence, we find the optimal design for all three problems. We ascertain the optimal designs by writing the design criterion functions as functions of the design measure, rather than of the designs themselves. We then use the geometric form of the design measure space and the concavity of the criterion function to find the optimal design measure. There is a straightforward correspondence between the set of design measures and the set of designs. Our approach is similar in spirit, although rather different in detail, from that introduced by Kiefer. In addition, we consider design for estimation of the changepoint itself, and optimal designs for the multipath changepoint problem. We demonstrate why the former problem most likely has a prior-dependent solution while the latter problems, in their most general settings, are complicated by the lack of concavity of the design criterion function.Nous considérons, dans cette dissertation, les plans d'expérience bayésiens optimauxpour les problèmes de point de rupture avec changement d'espérance. Un cas de pointde rupture avec changement d'espérance à une seule trajectoire se présente lorsqu'uneséquence de données est prélevée le long d'un axe temporelle (ou son équivalent) etque leur espérance change de valeur. Ce changement, s'il survient, se produit à unendroit sur l'axe inconnu de l'expérimentateur. Cet endroit est appelé "point derupture". Le fait que la position du point de rupture soit inconnue rend les tests etl'inférence difficiles dans les situations de point de rupture à une seule trajectoire

A graphical perspective of marginal structural models : an application for the estimation of the effect of physical activity on blood pressure

Estimating causal effects requires important prior subject-matter knowledge and, sometimes, sophisticated statistical tools. The latter is especially true when targeting the causal effect of a time-varying exposure in a longitudinal study. Marginal structural models (MSMs) are a relatively new class of causal models which effectively deal with the estimation of the effects of time-varying exposures. MSMs have traditionally been embedded in the counterfactual framework to causal inference. In this paper, we use the causal graph framework to enhance the implementation of MSMs. We illustrate our approach using data from a prospective cohort study, the Honolulu Heart Program. These data consist of 8006 men at baseline. To illustrate our approach, we focused on the estimation of the causal effect of physical activity on blood pressure, which were measured at three time-points. First, a causal graph is built to encompass prior knowledge. This graph is then validated and improved utilizing structural equation models. We estimated the aforementioned causal effect using MSMs for repeated measures and guided the implementation of the models with the causal graph. Employing the causal graph framework, we also show the validity of fitting conditional MSMs for repeated measures in the context implied by our data

A model for sequential evolution of ligands by exponential enrichment (SELEX) data

A Systematic Evolution of Ligands by EXponential enrichment (SELEX) experiment begins in round one with a random pool of oligonucleotides in equilibrium solution with a target. Over a few rounds, oligonucleotides having a high affinity for the target are selected. Data from a high throughput SELEX experiment consists of lists of thousands of oligonucleotides sampled after each round. Thus far, SELEX experiments have been very good at suggesting the highest affinity oligonucleotide, but modeling lower affinity recognition site variants has been difficult. Furthermore, an alignment step has always been used prior to analyzing SELEX data. We present a novel model, based on a biochemical parametrization of SELEX, which allows us to use data from all rounds to estimate the affinities of the oligonucleotides. Most notably, our model also aligns the oligonucleotides. We use our model to analyze a SELEX experiment containing double stranded DNA oligonucleotides and the transcription factor Bicoid as the target. Our SELEX model outperformed other published methods for predicting putative binding sites for Bicoid as indicated by the results of an in-vivo ChIP-chip experiment.Comment: Published in at http://dx.doi.org/10.1214/12-AOAS537 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org

A new simple interpretation of an optimal design criterion

We show that an often-used Bayesian optimal design criterion function fits into a standard decision theoretic framework. This suggests a sensible two-stage decision theoretic approach whereby this criterion function is also used for inference once it is used for design.

Generalized Linear Mixed Models for Binary Data: Are Matching Results from Penalized Quasi-Likelihood and Numerical Integration Less Biased?

<div><p>Background</p><p>Over time, adaptive Gaussian Hermite quadrature (QUAD) has become the preferred method for estimating generalized linear mixed models with binary outcomes. However, penalized quasi-likelihood (PQL) is still used frequently. In this work, we systematically evaluated whether matching results from PQL and QUAD indicate less bias in estimated regression coefficients and variance parameters via simulation.</p><p>Methods</p><p>We performed a simulation study in which we varied the size of the data set, probability of the outcome, variance of the random effect, number of clusters and number of subjects per cluster, etc. We estimated bias in the regression coefficients, odds ratios and variance parameters as estimated via PQL and QUAD. We ascertained if similarity of estimated regression coefficients, odds ratios and variance parameters predicted less bias.</p><p>Results</p><p>Overall, we found that the absolute percent bias of the odds ratio estimated via PQL or QUAD increased as the PQL- and QUAD-estimated odds ratios became more discrepant, though results varied markedly depending on the characteristics of the dataset</p><p>Conclusions</p><p>Given how markedly results varied depending on data set characteristics, specifying a rule above which indicated biased results proved impossible.</p><p>This work suggests that comparing results from generalized linear mixed models estimated via PQL and QUAD is a worthwhile exercise for regression coefficients and variance components obtained via QUAD, in situations where PQL is known to give reasonable results.</p></div

Parameters used for data generation.

<p>Parameters used for data generation.</p

Results from a linear mixed quantile regression model with absolute percent bias in σ²_u estimated via PQL or QUAD as the dependent variable and absolute percent different in σ²_u as estimated via PQL and QUAD as the independent variable, adjusted for data set characteristics (β₁, σ²_u, proportion with the outcome (p), total number of observations in the data set and data set composition).

<p>^a : This is the variance of the random slope.</p