Wayne A. Woodward

Global Warming and the Problem of Testing for Trend in Time Series Data

Abstract In recent years a number of statistical tests have been proposed for testing the hypothesis that global warming is occurring. The standard approach is to examine one or two of the more prominent global temperature datasets by letting Yt = a + bt + Et, where Yt represents the temperature at time t and Et represents error from the trend line, and to test the hypothesis that b = 0. Several authors have applied these tests for trend to determine whether or not a significant long-term or deterministic trend exists, and have generally concluded that there is a significant deterministic trend in the data. However, we show that certain autoregressive-moving average (ARMA) models may also be very reasonable models for these data due to the random trends present in their realizations. In this paper, we provide simulation evidence to show that the tests for trend detect a deterministic trend in a relatively high percentage of realizations from a wide range of ARMA models, including those obtained for the te...

Statistical Analysis Quick Reference Guidebook

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A Comparison of Minimum Distance and Maximum Likelihood Estimation of a Mixture Proportion

Abstract The estimation of mixing proportions in the mixture model is discussed, with emphasis on the mixture of two normal components with all five parameters unknown. Simulations are presented that compare minimum distance (MD) and maximum likelihood (ML) estimation of the parameters of this mixture-of-normals model. Some practical issues of implementation of these results are also discussed. Simulation results indicate that ML techniques are superior to MD when component distributions actually are normal, but MD techniques provide better estimates than ML under symmetric departures from component normality. Interestingly, an ad hoc starting value for the iterative procedures occasionally outperformed both the ML and MD techniques. Results are presented that establish strong consistency and asymptotic normality of the MD estimator under conditions that include the mixture-of-normals model. Asymptotic variances and relative efficiencies are obtained for further comparison of the MD and ML estimators.

On the Relationship between the S Array and the Box-Jenkins Method of ARMA Model Identification

Abstract This paper investigates an extension of the partial autocorrelation function, which we call the generalized partial autocorrelation function. These generalized partial autocorrelations, which are not true correlations except when p = 0, are useful in examining the relationship between the S array method of Gray, Kelley, and McIntire (1978) and the Box-Jenkins approach to ARMA model identification. Also, the generalized partial autocorrelation is shown to be a useful model identification tool to be used along with the S array. Also discussed is a reformating of the S array into a shifted S array that the authors believe is easier to use in practice than the S array. The methods of this paper are illustrated by means of examples, including an analysis of the Makridakis (1978) metals series data.

Selecting a Model for Detecting the Presence of a Trend

Abstract The authors consider the problem of determining whether the upward trending behavior in the global temperature anomaly series should be forecast to continue. To address this question, the generic problem of determining whether an observed trend in a time series realization is a random (i.e., short-term) trend or a deterministic (i.e., permanent) trend is considered. The importance of making this determination is that forecasts based on these two scenarios are dramatically different. Forecasts based on a series with random trends will not predict the observed trend to continue, while forecasts based on a model with deterministic trend will forecast the trend to continue into the future. In this paper, the authors consider an autoregressive integrated moving average (ARIMA) model and a “deterministic forcing function + autoregressive (AR) noise” model as possible random trend and deterministic trend models, respectively, for realizations displaying trending behavior. A bootstrap-based classificatio...

Epidemiological similarities between appendicitis and diverticulitis suggesting a common underlying pathogenesis

BACKGROUND Nonperforating appendicitis is primarily a disease of children, and nonperforating diverticulitis affects mostly older adults. Apart from these age differences, the diseases share many epidemiological features, such as association with better hygiene and low-fiber diets. HYPOTHESIS Nonperforating appendicitis and nonperforating diverticulitis are different manifestations of the same underlying colonic process and, if so, should be temporally related. DESIGN Data from the National Hospital Discharge Survey were analyzed to investigate the incidence of admissions for appendicitis in children and diverticulitis in adults between 1979 and 2006. SETTING Statistical sampling of all US hospitals. PATIENTS Children admitted for appendicitis and adults with diverticulitis. MAIN OUTCOME MEASURES Time trends were assessed for stationarity using unit root analysis, and similarities between time trends were tested using cointegration analysis. RESULTS The incidence rates of nonperforating appendicitis and nonperforating diverticulitis exhibited U-shaped secular trends. The rates of perforating appendicitis and perforating diverticulitis rose slowly across all the study years. Cointegration analysis demonstrated that the rates of nonperforating and perforating diverticulitis did not cointegrate significantly over time. The rates of nonperforating and perforating appendicitis did not vary together. Nonperforating appendicitis and nonperforating diverticulitis rates were significantly cointegrated over time. CONCLUSIONS Childhood appendicitis and adult diverticulitis seem to be similar diseases, suggesting a common underlying pathogenesis. Secular trends for their nonperforating and perforating forms are strikingly different. At least for appendicitis, perforating disease may not be an inevitable outcome from delayed treatment of nonperforating disease. If appendicitis represents the same pathophysiologic process as diverticulitis, it may be amenable to antibiotic rather than surgical treatment.

Association of viral infection and appendicitis.

HYPOTHESIS What causes appendicitis is not known; however, studies have suggested a relationship between viral diseases and appendicitis. Building on evidence of cyclic patterns of appendicitis with apparent outbreaks consistent with an infectious etiology, we hypothesized that there is a relationship between population rates of appendicitis and several infectious diseases. DESIGN Epidemiologic study. SETTING The National Hospital Discharge Survey PATIENTS Estimated US hospitalized population. MAIN OUTCOME MEASURES International Classification of Diseases, Ninth Revision, Clinical Modification discharge diagnosis codes of the National Hospital Discharge Survey were queried from 1970 to 2006 to identify admissions for appendicitis, influenza, rotavirus, and enteric infections. Cointegration analysis of time series data was used to determine if the disease incidence trends for these various disease entities varied over time together. RESULTS Rates of influenza and nonperforating appendicitis declined progressively from the late 1970s to 1995 and rose thereafter, but influenza rates exhibited more distinct seasonal variation than appendicitis rates. Rotavirus infection showed no association with the incidence of nonperforating appendicitis. Perforating appendicitis showed a dissimilar trend to both nonperforating appendicitis and viral infection. Hospital admissions for enteric infections substantially increased over the years but were not related to appendicitis cases. CONCLUSIONS Neither influenza nor rotavirus are likely proximate causes of appendicitis given the lack of a seasonal relationship between these disease entities. However, because of significant cointegration between the annual incidence rates of influenza and nonperforated appendicitis, it is possible that these diseases share common etiologic determinates, pathogenetic mechanisms, or environmental factors that similarly affect their incidence.

Minimum Hellinger distance estimation of mixture proportions

Beran (1977) showed that, under certain restrictive conditions, the minimum distance estimator based on the Hellinger distance (MHDE) between a projection model density and a nonparametric sample density is an exception to the usual perception that a robust estimator cannot achieve full efficiency under the true model. We examine the MHDE in the case of estimation of the mixing proportion in the mixture of two normals. We discuss the practical feasibility of employing the MHDE in this setting and examine empirically its robustness properties. Our results indicate that the MHDE obtains full efficiency at the true model while performing comparably with the minimum distance estimator based on Cramer-von Mises distance under the symmetric departures from component normality considered.

Improved agreement between Talairach and MNI coordinate spaces in deep brain regions.

Disagreement between the Talairach atlas and the stereotaxic space commonly used in software like SPM is a widely recognized problem. Others have proposed affine transformations to improve agreement in surface areas such as Brodmanns areas. This article proposes a similar transformation with the goal of improving agreement specifically in the deep brain region. The task is accomplished by finding an affine transformation that minimizes the mean distance between the surface coordinates of the lateral ventricles in the Talairach atlas and the MNI templates. The result is a transformation that improves deep brain agreement over both the untransformed Talairach coordinates and the surface-oriented transformation. While the transformation improves deep brain agreement, surface agreement is generally made worse. For areas near the lateral ventricle, the transformation presented herein is valuable for applications such as region of interest (ROI) modeling.

A New Test for Outlier Detection from a Multivariate Mixture Distribution

Abstract The problem of testing an outlier from a multivariate mixture distribution of several populations has many important applications in practice. One particular example is in monitoring worldwide nuclear testing, where we wish to detect whether an observed seismic event is possibly a nuclear explosion (an outlier) by comparing it with the training samples from mining blasts and earthquakes. The combined population of seismic events from mining blasts and earthquakes can be viewed as a mixture distribution. The classical likelihood ratio test appears to not be applicable in our problem, and in spite of the importance of this problem, little progress has been made in the literature. This article proposes a simple modified likelihood ratio test that overcomes the difficulties in the current problem. Bootstrap techniques are used to approximate the distribution of the test statistic. The advantages of the new test are demonstrated via simulation studies. Some new computational findings are also reported.