A constrained spline estimator of a hazard function

Bloxom, Bruce

doi:10.1007/bf02294107

Cited by 27 publications

(20 citation statements)

References 16 publications

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“…Psychologists might begin with some recent papers of Bloxom (1983Bloxom ( , 1984Bloxom ( , 1985. Bloxom provided his own approach both to densities and hazard functions but also aided the reader in accessing the general literature on these topics.…”

Section: Statistical Analysesmentioning

confidence: 99%

Truth and consequences of ordinal differences in statistical distributions: Toward a theory of hierarchical inference.

Townsend

1990

Psychological Bulletin

164

131

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A theory is presented that establishes a dominance hierarchy of potential distinctions (order relations) between two distributions. It is proposed that it is worthwhile for researchers to ascertain the strongest possible distinction, because all weaker distinctions are logically implied. Implications of the theory for hypothesis testing, theory construction, and scales of measurement are considered. Open problems for future research are outlined. There are many occasions in psychological research when experimenters are concerned with two or more treatments and their effects on experimental groups. In most instances, the measurements assessing the treatments are assumed to be on at least an ordinal scale. The measurements typically form a sample probability distribution. The investigator may then ask the question as to whether the two distributions differ from one another in some way. The way chosen is often through the testing of summary statistics, particularly sample means, but there are techniques that test entire distributions against one another. In this article I develop a theory that shows that some differences between the distributions are stronger than others, in that the stronger ones imply the weaker ones but not vice versa. In fact, a dominance hierarchy of distinctions between two arbitrary distributions is developed: A distinction may imply another, be implied by another, be equivalent to another, or none of these. The dominance hierarchy is put into the form of an implication graph. The power of the results is that they are, except for special cases to be specified later, general across distributions. That is, most of the relations do not depend on a particular type of distribution such as the normal distribution. Furthermore, as long as the measurements are on at least an ordinal scale, again with the exception of the special cases, they

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Section: Statistical Analysesmentioning

confidence: 99%

Truth and consequences of ordinal differences in statistical distributions: Toward a theory of hierarchical inference.

Townsend

1990

Psychological Bulletin

164

131

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“…If it is believed that these requirements can be fulfilled, then the issue of statistical mimicking is irrelevant. However, while we agree that it might be theoretically possible to avoid the problem of statistical mimicking by simply collecting enough data points, in practice it may not be feasible.To attempt to circumvent the necessity of collecting extraordinarily large numbers of observations, many researchers have advocated the use of the hazard function (the ratio of the density to one minus the distribution function) to discriminate between different distributions (e.g., Balakrishnan & Ashby, 1992;Bloxom, 1984Bloxom, , 1985Burbeck & Luce, 1982;Luce, 1986). The hazard function gives the likelihood that, for a particular point in…”

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confidence: 99%

“…Therefore, the tail of an empirical hazard function is generally quite unstable. The number of observations required to make the estimates ofthese tails stable is another question that we will not address here (but see Bloxom, 1984Bloxom, , 1985. …”

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confidence: 99%

Statistical mimicking of reaction time data: Single-process models, parameter variability, and mixtures

Zandt

Ratcliff

1995

Psychonomic Bulletin & Review

117

126

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Statistical mimicking issues involving reaction time measures are introduced and discussed in this article. Often, discussions of mimicking have concerned the question of the serial versus parallel processing of inputs to the cognitive system. Wewill demonstrate that there are several alternative structures that mimic various existing models in the literature. In particular, single-process models have been neglected in this area. When parameter variability is incorporated into single-process models, resulting in discrete or continuous mixtures of reaction time distributions, the observed reaction time distribution alone is no longer as useful in allowing inferences to be made about the architecture of the process that produced it. Many of the issues are raised explicitly in examination of four different case studies of mimicking. Rather than casting a shadow over the use of quantitative methods in testing models of cognitive processes, these examples emphasize the importance of examiningreaction time data armed with the tools of quantitative analysis, the importance of collecting data from the context of specific process models, and the importance of expanding the database to include other dependent measures.Since the publication ofDonders's (1868/1969) essay, "On the speed of mental processes," psychologists have measured the time required by experimental subjects to perform various tasks. These reaction times (RTs) and the changes in RT under different experimental manipulations have been used as evidence for or against models of mental architecture-the arrangement of the mental processes underlying the subject's performance (Sternberg, 1969;Townsend & Ashby, 1983; Woodworth, 1938, chapter 14). RT data have played an important role in distinguishing between models and in testing hypotheses about processes and structures. Consequently, considerable effort has been devoted to the refinement of RT measures, from techniques to optimize the accuracy of subsequent statistical analyses of RT summary statistics (Ratcliff, 1993; Townsend, 1990b;Ulrich & Miller, 1994) to the estimation of RT distributions and RT hazard functions (Burbeck & Luce, 1982;Luce, 1986; Ratcliff & Murdock, 1976). The concentration on the RT This project was supported by NIMH Grants HD MH44640 and MH00871 to R.R. and was completed while the first author was a postdoctoral fellow at Northwestern University. The manuscript was greatly improved by helpful comments from Barbara Dosher, Ehtibar Dzhafarov, Thomas Fikes, Rich Schweickert, Saul Sternberg, and Jim Townsend. Correspondence may be addressed to T. Van Zandt, Department of Psychology, Ames Hall, Johns Hopkins University, 3400 N. Charles St., Baltimore, MD 21218-2686 (e-mail: trish@maigret. psy.jhu.edu). distributions can be seen as an advance from the use of less informative summary statistics such as the mean or median. In the distributional approach, the density or distribution functions predicted by various models are fit to the usually unimodal, positively skewed RT densities or dis...

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“…THE MODEL Spline functions have been applied in many areas of statistics, including density estimation, time series, and hazard function estimation (Wegman and Wright, 1983). Bloxom (1985) and Whittemore and Keller (1986) propose the use of spline functions to estimate hazard functions in a homogeneous setting ignoring covariables. Bloxom approximates the hazard function using a quadratic spline with knots placed a t each decile of the observed failure times.…”

Section: Restricted Cubic Spline Hazard Functionmentioning

confidence: 99%

The restricted cubic spline hazard model

Herndon

Harrell

1990

Communications in Statistics - Theory and Methods

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A constrained spline estimator of a hazard function

Abstract: hazard function, quadratic spline, maximum penalized likelihood, constrained estimation, response time distribution,

Cited by 27 publications

References 16 publications

Truth and consequences of ordinal differences in statistical distributions: Toward a theory of hierarchical inference.

Truth and consequences of ordinal differences in statistical distributions: Toward a theory of hierarchical inference.

Statistical mimicking of reaction time data: Single-process models, parameter variability, and mixtures

The restricted cubic spline hazard model

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