Paul L. Speckman

Bayesian t tests for accepting and rejecting the null hypothesis

Progress in science often comes from discovering invariances in relationships among variables; these invariances often correspond to null hypotheses. As is commonly known, it is not possible to state evidence for the null hypothesis in conventional significance testing. Here we highlight a Bayes factor alternative to the conventional t test that will allow researchers to express preference for either the null hypothesis or the alternative. The Bayes factor has a natural and straightforward interpretation, is based on reasonable assumptions, and has better properties than other methods of inference that have been advocated in the psychological literature. To facilitate use of the Bayes factor, we provide an easy-to-use, Web-based program that performs the necessary calculations.

A hierarchical model for estimating response time distributions

We present a statistical model for inference with response time (RT) distributions. The model has the following features. First, it provides a means of estimating the shape, scale, and location (shift) of RT distributions. Second, it is hierarchical and models between-subjects and within-subjects variability simultaneously. Third, inference with the model is Bayesian and provides a principled and efficient means of pooling information across disparate data from different individuals. Because the model efficiently pools information across individuals, it is particularly well suited for those common cases in which the researcher collects a limited number of observations from several participants. Monte Carlo simulations reveal that the hierarchical Bayesian model provides more accurate estimates than several popular competitors do. We illustrate the model by providing an analysis of the symbolic distance effect in which participants can more quickly ascertain the relationship between nonadjacent digits than that between adjacent digits.

Confidence bands in nonparametric regression

Abstract New bias-corrected confidence bands are proposed for nonparametric kernal regression. These bands are constructed using only a kernel estimator of the regression curve and its data-selected bandwidth. They are shown to have asymptotically correct coverage properties and to behave well in a small-sample study. One consequence of the large-sample developments is that Bonferroni-type bands for the regression curve at the design points also have conservative asymptotic coverage behavior with no bias correction.

Proteomic analysis of nipple aspirate fluid using SELDI‐TOF‐MS

Proteomic analysis of body fluids, including breast nipple aspirate fluid (NAF), holds promise to aid in early cancer detection. We conducted a prospective trial that collected NAF from women scheduled for diagnostic breast surgery to determine 1) the consistency of proteomic results, 2) protein masses associated with breast cancer, 3) subsets of women with a unique proteomic profile and 4) a breast cancer predictive model. NAF was collected preoperatively in 114 women and analyzed by SELDI‐TOF mass spectrometry over a 3–50 kDa range using H4, NP and SAX ProteinChips. For all 3 chips, the same protein peaks were detected over 90% of the time in duplicate samples. The overall coefficient of variation was ≤ 0.17% for each chip for the internal standard and ≤ 0.29% for the unknown proteins. Seven candidate protein ion masses frequently expressed in NAF were identified. Three (5,200‐H4, p=.04, 11,880‐H4, p=.07 and 13,880 Da‐SAX, p=.03) were differentially expressed in women with/without breast cancer. Protein expression differed between women with/without pathologic nipple discharge (PND), but the 5,200, 11,880 and 13,880 proteins remained associated with breast cancer even if PND samples were excluded. Subset analysis identified differences in expression between benign disease and DCIS and between DCIS and invasive cancer for the 5,200 and 33,400 Da proteins. The best cancer detection model included age, parity and the 11,880 Da protein, and excluded women with PND. 1) NAF proteomic analysis using SELDI‐TOF is reproducible with the same sample set across different platforms, 2) differential proteomic expression exists between women/without breast cancer and 3) combining proteomic and clinical information that are available before surgery optimizes the prediction of which women have breast cancer.

A hierarchical bayesian statistical framework for response time distributions

This paper provides a statistical framework for estimating higher-order characteristics of the response time distribution, such as the scale (variability) and shape. Consideration of these higher order characteristics often provides for more rigorous theory development in cognitive and perceptual psychology (e.g., Luce, 1986). RT distribution for a single participant depends on certain participant characteristics, which in turn can be thought of as arising from a distribution of latent variables. The present work focuses on the three-parameter Weibull distribution, with parameters for shape, scale, and shift (initial value). Bayesian estimation in a hierarchical framework is conceptually straightforward. Parameter estimates, both for participant quantities and population parameters, are obtained through Markov Chain Monte Carlo methods. The methods are illustrated with an application to response time data in an absolute identification task. The behavior of the Bayes estimates are compared to maximum likelihood (ML) estimates through Monte Carlo simulations. For small sample size, there is an occasional tendency for the ML estimates to be unreasonably extreme. In contrast, by borrowing strength across participants, Bayes estimation “shrinks” extreme estimates. The results are that the Bayes estimators are more accurate than the corresponding ML estimators.

A hierarchical process-dissociation model.

In fitting the process-dissociation model (L. L. Jacoby, 1991) to observed data, researchers aggregate outcomes across participant, items, or both. T. Curran and D. L. Hintzman (1995) demonstrated how biases from aggregation may lead to artifactual support for the model. The authors develop a hierarchical process-dissociation model that does not require aggregation for analysis. Most importantly, the Curran and Hintzman critique does not hold for this model. Model analysis provides for support of process dissociation--selective influence holds, and there is a dissociation in correlation patterns among participants and items. Items that are better recollected also elicit higher automatic activation. There is no correlation, however, across participants; that is, participants with higher recollection have no increased tendency toward automatic activation. The critique of aggregation is not limited to process dissociation. Aggregation distorts analysis in many nonlinear models, including signal detection, multinomial processing tree models, and strength models. Hierarchical modeling serves as a general solution for accurately fitting these psychological-processing models to data.

Regression models for air pollution and daily mortality : analysis of data from Birmingham, Alabama

In recent years, a very large literature has built up on the human health effects of air pollution. Many studies have been based on time series analyses in which daily mortality counts, or some other measure such as hospital admissions, have been decomposed through regression analysis into contributions based on long-term trend and seasonality, meteorological effects, and air pollution. There has been a particular focus on particulate air pollution represented by PM10 (particulate matter of aerodynamic diameter 10 µm or less), though in recent years more attention has been given to very small particles of diameter 2.5 µm or less. Most of the existing data studies, however, are based on PM10 because of the wide availability of monitoring data for this variable. The persistence of the resulting effects across many different studies is widely cited as evidence that this is not mere statistical association, but indeed establishes a causal relationship. These studies have been cited by the United States Environmental Protection Agency (USEPA) as justification for a tightening on particulate matter standards in the 1997 revision of the National Ambient Air Quality Standard (NAAQS), which is the basis for air pollution regulation in the United States. The purpose of the present paper is to propose a systematic approach to the regression analyses that are central to this kind of research. We argue that the results may depend on a number of ad hoc features of the analysis, including which meteorological variables to adjust for, and the manner in which different lagged values of particulate matter are combined into a single ‘exposure measure’. We also examine the question of whether the effects are linear or nonlinear, with particular attention to the possibility of a ‘threshold effect’, i.e. that significant effects occur only above some threshold. These points are illustrated with a data set from Birmingham, Alabama, first cited by Schwartz (1993, American Journal of Epidemiology137: 1136 – 1147) and since extensively re-analyzed. For this data set, we find that the results are sensitive to whether humidity is included along with temperature as a meteorological variable, and to the definition of the exposure measure. We also find evidence of a threshold effect, with the greatest increase in mortality occurring above 50 µg/m3, which is the long-term average level permitted by the current NAAQS. Thus, on the basis of this data set, the need for a tighter NAAQS is not established. Although this particular analysis is focussed just on one data set, the issues it raises are typical in this area of research. We do not dispute that there is a reasonable level of evidence linking atmospheric particulate matter with adverse health outcomes even within the levels permitted by current regulations. However, the impression has been created by some of the published literature that such associations are overwhelmingly supported by epidemiological research. Our viewpoint is that the statistical analyses allow different interpretations, and that the case for tighter regulations cannot be based solely on studies of this nature. Copyright

Statistical Models for Limiting Nutrient Relations in Inland Waters

Abstract The ecological theory of limiting factors holds that the observed level of response in a biological process will be governed by the input factor in least supply—the limiting factor. This theory has formed the basis for numerous attempts by aquatic ecologists to describe the relation between the biological productivity of inland waters and the availability of plant nutrients required for algal growth. Regression analysis has been the primary statistical tool used in the development of such relations, yet any statistical model that represents the limiting effect of some explanatory factor as an expectation contradicts the substantive theory of limiting factors. Limnological data not resulting in an adequate regression of chlorophyll on phosphorus have been viewed as failing to support the limiting effect of this nutrient on algal biomass in lakes. But when represented by a more appropriate model, such data may be seen to provide similar evidence for the relation of chlorophyll to phosphorus as does...

Detecting chance: a solution to the null sensitivity problem in subliminal priming.

In many paradigms, the persuasiveness of subliminal priming relies on establishing that stimuli are undetectable. The standard significance test approach is ill-suited as null results may reflect either truly undetectable stimuli or a lack of power to resolve weakly detectable stimuli. We present a novel statistical model as an alternative. The model provides for estimates of the probability that each individual is truly at chance. Researchers may select individuals for whom there are sufficiently high probabilities of true undetectability. The model is hierarchical, and estimation is done within the Bayesian framework.

A comment on Heathcote, Brown, and Mewhort's QMLE method for response time distributions

Heathcote, Brown, and Mewhort (2002) have introduced a new, robust method of estimating response time distributions. Their method may have practical advantages over conventional maximum likelihood estimation. The basic idea is that the likelihood of parameters is maximized given a few quantiles from the data. We show that Heathcote et al.’s likelihood function is not correct and provide the appropriate correction. However, although our correction stands on firmer theoretical ground than Heathcote et al.’s, it appears to yield worse parameter estimates. This result further indicates that, at least for some distributions and situations, quantile maximum likelihood estimation may have better nonasymptotic properties than a more theoretically justified approach.