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This report addresses the characterization of measurements that include epistemic uncertainties in the form of intervals. It reviews the application of basic descriptive statistics to data sets which contain intervals rather than exclusively point estimates. It describes algorithms to compute various means, the median and other percentiles, variance, interquartile range, moments, confidence limits, and other important statistics and summarizes the computability of these statistics as a function of sample size and characteristics of the intervals in the data (degree of overlap, size and regularity of widths, etc.). It also reviews the prospects for analyzing such data sets with the methods of inferential statistics such as outlier detection and regressions. The report explores the tradeoff between measurement precision and sample size in statistical results that are sensitive to both. It also argues that an approach based on interval statistics could be a reasonable alternative to current standard methods for evaluating, expressing and propagating measurement uncertainties.

this paper, we study the use of these intervals as CIs for the partially identified parameter f(P,#). Our most basic finding is Lemma 2.1: Lemma 2.1 Let CN0 0, CN1 0, # #, and P #P

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We propose a two-stage instrumental variable estimator that is consistent if there is selective compliance in the treatment group of a randomized experiment and the outcome variable is a censored duration. The estimator assumes full compliance in the control group. We use the estimator to reanalyze data from the Illinois re-employment bonus experiment. # Faculteit der Economische Wetenschappen en Econometrie, Vrije Universiteit Amsterdam and Department of Economics, Johns Hopkins University,Baltimore, MD 21218-2685, fax. 4105167600, E-mail: gbijwaard@econ.vu.nl, gridder@jhu.edu 1 1 Introduction In theory, data from a randomized experiment produce an unbiased estimate of the e#ect of an intervention or program on an outcome variable. The di#erence of the average outcomes of the treatment and control samples estimates the average treatment e#ect. In practice, a randomized experiment may su#er from the same problems that a#ect behavioral studies. In particular, the random assignme...

u-driven stand-alone package. Users of the first type should focus their attention on the Bounds Elementary Routines and Core Procedures. Users of the second type will employ the Bounds Shell Program. In what follows, the names of the elementary routines, subroutines, and core procedures recognized by Bounds are in italics. Elementary Routines The elementary operations used by Bounds are nonparametric estimation of regressions and bootstrap estimation of sampling distributions. The Elementary Routines are Matlab commands performing these operations. They are kern -- a routine performing kernel estimation of regressions silverman -- a routine computing Silverman's "rule of thumb" bandwidth for use in kern. empirical -- a routine using the empirical distribution of the data to generate bootstrap pseudo-samples and the resulting bootstrap sampling distribution

Inference for causal effects is a critical activity in many branches of science and public policy. The field of statistics is the one field most suited to address such problems, whether from designed experiments or observational studies. Consequently, it is arguably essential that departments of statistics teach courses in causal inference to both graduate and undergraduate students. This presentation will discuss some aspects of such courses based on: a graduate level course taught at Harvard for a half dozen years, sometimes jointly with the Department of Economics (with Professor Guido Imbens, now at UCLA), and current plans for an undergraduate core course at Harvard University. An expanded version of this brief document will outline the courses ' contents more completely. Moreover, a textbook by Imbens and Rubin, due to appear in 2000, will cover the basic material needed in both courses. The current course at Harvard begins with the definition of causal effects through potential outcomes. Causal estimands are comparisons of the outcomes that would have been observed under different exposures of units to treatments. This approach is commonly referred to as 'Rubin's Causal Model- RCM " (Holland, 1986), but the formal notation in the context of randomization-based inference in randomized experiments goes back to Neyman (1923), and the intuitive idea goes back centuries in various literatures; see also Fisher (1918), Tinbergen (1930) and Haavelmo (1944). The label "RCM " arises because of extensions (e.g., Rubin, 1974, 1977, 1978) that

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This paper introduces an empirical strategy to estimate dynamic treatment effects in randomized trials that provide treatment in multiple stages and in which various noncompliance problems including attrition and selective transitions between treatment and control groups arise. Our approach is applied to the highly influential four year randomized class size study, Project STAR. We find benefits from small class attendance initially in all cognitive subject areas in kindergarten and the first grade. We do not find any statistically significant dynamic benefits from continuous treatment versus never attending small classes in either the second or third grade. Finally, statistical tests confirm that one should account for both selective attrition and noncompliance with treatment assignment.

Within the literature on causal statistical inference, an important goal is to examine the causal mechanisms or channels through which the treatment affects the outcome of interest. Net (or direct) causal effects measure the effect of the treatment on the outcome while blocking the effect of the treatment on the variable that represents the mechanism. Hence, net effects are useful in learning about the ways in which the treatment causally affects the outcome. This paper provides sufficient conditions under which net average effects can be partially and point identified without functional form, distributional, or constant-effects assumptions. First, we show that the data usually available to researchers contains information on the relevant potential outcome used in the definition of causal net effects only for a particular subpopulation: those individuals for which the treatment does not affect the mechanism variable. An implication of this result is that estimation of net effects for other subpopulations can only be based on extrapolations involving typically strong assumptions. Second, we show that by imposing a monotonicity condition on the effect of the treatment on the mechanism variable—which is also imposed on methods currently used for estimation