Joseph W. McKean

Design Specification Issues in Time-Series Intervention Models

It has been recognized that the two-phase version of the interrupted time-series design can be frequently modeled using a four-parameter design matrix. There are differences across writers, however, in the details of the recommended design matrices to be used in the estimation of the four parameters of the model. Various writers imply that different methods of specifying the four-parameter design matrix all lead to the same conclusions; they do not. The tests and estimates for level change are dramatically different under the various seemingly equivalent design specifications. Examples of egregious errors of interpretation are presented and recommendations regarding the correct specification of the design matrix are made. The recommendations hold whether the model is estimated using ordinary least squares (for the case of approximately independent errors) or some more complex time-series approach (for the case of autocorrelated errors).

Autocorrelation Estimation and Inference With Small Samples

The small sample properties of 6 autocorrelation estimators were investigated in an extensive Monte Carlo study. It was demonstrated that conventional estimators yield problems of estimation and inference in the form of(a) inconsistencies between theoretical and empirical expectations, (b) inconsistencies between theoretical and empirical error variances, and (c) dramatic differences between nominal and empirical Type I errors

Tests of Hypotheses Based on Ranks in the General Linear Model

A unified approach is developed for testing hypotheses in the general linear model based on the ranks of the residuals. It complements the nonparametric estimation procedures recently reported in the literature. The testing and estimation procedures together provide a robust alternative to least squares. The methods are similar in spirit to least squares so that results are simple to interpret. Hypotheses concerning a subset of specified parameters can be tested, while the remaining parameters are treated as nuisance parameters. Asymptotically, the test statistic is shown to have a chi-square distribution under the null hypothesis. This result is then extended to cover a sequence of contiguous alternatives from which the Pitman efficacy is derived. The general application of the test requires the consistent estimation of a functional of the underlying distribution and one such estimate is furnished.

High-Breakdown Rank Regression

Abstract A weighted rank estimate is proposed that has 50% breakdown and is asymptotically normal at rate √n. Based on this theory, inferential procedures, including asymptotic confidence and tests, and diagnostic procedures, such as studentized residuals, are developed. The influence function of the estimate is derived and shown to be continuous and bounded everywhere in (x, Y) space. Examples show that robustness against outlying high-leverage clusters may approach that of the least median of squares, while retaining more stability against inliers. The estimator uses weights that correct for both factor and response spaces. A Monte Carlo study shows that the estimate is more efficient than the generalized rank estimates, which are generalized R estimates with weights that only correct for factor space. When weights are constant, the estimate reduces to the regular Wilcoxon rank estimate.

A comparison of methods for studentizing the sample median

Various methods for “Studentizing” the sample median are com-pared on the basis of a Monte Carlo study. Several of the methods do rather poorly while two, the bootstrap and the standardized length of a distribution free confidence interval, behave accept-ably acrors a wide range of sample sizes and several distributions of varying tail length. These two methods seem to agree closely with the distribution free confidence intervals and moreover, un-like these intervals, the methods can be extended to a method of accurate inference for λ1 regreasion.

A Geometric Interpretation of Inferences Based on Ranks in the Linear Model

Abstract Four different approaches, based on ranks, to testing hypotheses are unified through the geometry of the linear model. The various tests are identified with different but algebraically equivalent forms of the classical F test. Small sample differences are investigated via a Monte Carlo study using both rank and signed rank tests.

Regression Diagnostics for Rank-Based Methods

Abstract Residual plots and diagnostic techniques have become important tools in examining the least squares fit of a linear model. In this article we explore the properties of the residuals from a rank-based fit of the model. We present diagnostic techniques that detect outlying cases and cases that have an influential effect on the rank-based fit. We show that the residuals from this fit can be used to detect curvature not accounted for by the fitted model. Furthermore, our diagnostic techniques inherit the excellent efficiency properties of the rank-based fit over a wide class of error distributions, including asymmetric distributions. We illustrate these techniques with several examples.

The use and interpretation of residuals based on robust estimation

Abstract Residual plots and diagnostic techniques are important tools for examining the fit of a regression model. In the case of least squares fits, plots of residuals provide a visual assessment of the adequacy of various aspects of the fitted model. An important question is whether plots of robust residuals can be interpreted in the same manner as their least squares counterparts. This article addresses this problem for two popular classes of robust estimates: M estimates and GM estimates of the Mallows and Schweppe types. First-order properties of the residuals and fitted values are derived under correct and misspecified models. These properties are insightful on the general interpretability of robust residual plots and on their ability to detect curvature in misspecified models. The results of a simulation study consisting of tests for randomness and curvature in residual plots supports these theoretical properties. Standardization of robust residuals is also presented.

Multivariate L-estimation

In one dimension, order statistics and ranks are widely used because they form a basis for distribution free tests and some robust estimation procedures. In more than one dimension, the concept of order statistics and ranks is not clear and several definitions have been proposed in the last years. The proposed definitions are based on different concepts of depth. In this paper, we define a new notion of order statistics and ranks for multivariate data based on density estimation. The resulting ranks are invariant under affinc transformations and asymptotically distribution free. We use the corresponding order statistics to define a class of multivariate estimators of location that can be regarded as multivariate L-estimators. Under mild assumptions on the underlying distribution, we show the asymptotic normality of the estimators. A modification of the proposed estimates results in a high breakdown point procedure that can deal with patches of outliers. The main idea is to order the observations according to their likelihoodf(X1),...,f(Xn). If the densityf happens to be cllipsoidal, the above ranking is similar to the rankings that are derived from the various notions of depth. We propose to define a ranking based on a kernel estimate of the densityf. One advantage of estimating the likelihoods is that the underlying distribution does not need to have a density. In addition, because the approximate likelihoods are only used to rank the observations, they can be derived from a density estimate using a fixed bandwidth. This fixed bandwidth overcomes the curse of dimensionality that typically plagues density estimation in high dimension.

A Simple and Powerful Test for Autocorrelated Errors in OLS Intervention Models

The important assumption of independent errors should be evaluated routinely in the application of interrupted time-series regression models. The two most frequently recommended tests of this assumption [Moods runs test and the Durbin-Watson (D-W) bounds test] have several weaknesses. The former has poor small sample Type I error performance and the latter has the bothersome property that results are often declared to be “inconclusive.” The test proposed in this article is simple to compute (special software is not required), there is no inconclusive region, an exact p-value is provided, and it has good Type I error and power properties relative to competing procedures. It is shown that these desirable properties hold when design matrices of a specified form are used to model the response variable. A Monte Carlo evaluation of the method, including comparisons with other tests (viz., runs, D-W bounds, and D-W beta), and examples of application are provided.