Estimating consumer acceptance limits

A methodology is developed for estimating consumer acceptance limits on a sensory attribute of a manufactured product. In concept these limits are analogous to engineering tolerances. The method is based on a generalization of Stevens' Power Law. This generalized law is expressed as a nonlinear statistical model. Instead of restricting the analysis to this particular case, a strategy is discussed for evaluating nonlinear models in general since scientific models are frequently of nonlinear form. The strategy focuses on understanding the geometrical contrasts between linear and nonlinear model estimation and assessing the bias in estimation and the departures from a Gaussian sampling distribution. Computer simulation is employed to examine the behavior of nonlinear least squares estimation. In addition to the usual Gaussian assumption, a bootstrap sample reuse procedure and a general triangular distribution are introduced for evaluating the effects of a non-Gaussian or asymmetrical error structure. Recommendations are given for further model analysis based on the simulation results. In the case of a model for which estimation bias is not a serious issue, estimating functions of the model are considered. Application of these functions to the generalization of Stevens’ Power Law leads to a means for defining and estimating consumer acceptance limits, The statistical form of the law and the model evaluation strategy are applied to consumer research data. Estimation of consumer acceptance limits is illustrated and discussed.     

Reverse Regression and Employment Discrimination

The importance of interval forecasts is reviewed. Several general approaches to calculating such forecasts are described and compared. They include the use of theoretical formulas based on a fitted probability model (with or without a correction for parameter uncertainty), various “approximate” formulas (which should be avoided), and empirically based, simulation, and resampling procedures. The latter are useful when theoretical formulas are not available or there are doubts about some model assumptions. The distinction between a forecasting method and a forecasting model is expounded. For large groups of series, a forecasting method may be chosen in a fairly ad hoc way. With appropriate checks, it may be possible to base interval forecasts on the model for which the method is optimal. It is certainly unsound to use a model for which the method is not optimal, but, strangely, this is sometimes done. Some general comments are made as to why prediction intervals tend to be too narrow in practice to encompass the required proportion of future observations. An example demonstrates the overriding importance of careful model specification. In particular, when data are “nearly nonstationary,” the difference between fitting a stationary and a nonstationary model is critical.     

Simulation-Based Tests that Can Use Any Number of Simulations

Conventional procedures for Monte Carlo and bootstrap tests require that B, the number of simulations, satisfy a specific relationship with the level of the test. Otherwise, a test that would instead be exact will either overreject or underreject for finite B. We present expressions for the rejection frequencies associated with existing procedures and propose a new procedure that yields exact Monte Carlo tests for any positive value of B. This procedure, which can also be used for bootstrap tests, is likely to be most useful when simulation is expensive.     

A BAYESIAN ANALYSIS FOR INTER-RATER AGREEMENT

An analysis of inter-rater agreement is presented. We study the problem with several raters using a Bayesian model based on the Dirichlet distribution. Inter-rater agreement, including global and partial agreement, is studied by determining the joint posterior distribution of the raters. Posterior distributions are computed with a direct resampling technique. Our method is illustrated with an example involving four residents, who are diagnosing 12 psychiatric patients suspected of having a thought disorder. Initially employing analytical and resampling methods, total agreement between the four is examined with a Bayesian testing technique. Later, partial agreement is examined by determining the posterior probability of certain orderings among the rater means. The power of resampling is revealed by its ability to compute complex multiple integrals that represent various posterior probabilities of agreement and disagreement between several raters.     

Testing hypotheses about covariance matrices using bootstrap methods

The usual chi-squared approximation to test statistics based on normal theory for testing covariance structures of multivariate populations is very sensitive to the normality assumption. Two general bootstrap procedures are developed in this paper to obtain approximately valid critical values for these test statistics when the data are not normally distributed. The first is based on separate sampling from individual samples, and the second is based on sampling from pooled samples. Although the second method requires more assumptions, its small sample properties are better.     

Statistical estimation for a failure model with damage accumulation in a case of small samples

A failure model with damage accumulation is considered. Damages occur according to a Poisson process and they degenerate into failures in a random time. The rate of the Poisson process and the degeneration time distribution are unknown. Two sample populations are available: a sample of intervals between damages and a sample of degeneration times. The case of small samples is considered. The purpose is to estimate the expectation and the distribution of the number of damages and failures at time t. We consider the plug-in and resampling estimators of the above mentioned characteristics. The expectations and variances of the suggested estimators are investigated. The numerical examples show that the resampling estimator has some advantages.     

Understanding the role of facial asymmetry in human face identification

Face recognition has important applications in forensics (criminal identification) and security (biometric authentication). The problem of face recognition has been extensively studied in the computer vision community, from a variety of perspectives. A relatively new development is the use of facial asymmetry in face recognition, and we present here the results of a statistical investigation of this biometric. We first show how facial asymmetry information can be used to perform three different face recognition tasks—human identification (in the presence of expression variations), classification of faces by expression, and classification of individuals according to sex. Initially, we use a simple classification method, and conduct a feature analysis which shows the particular facial regions that play the dominant role in achieving these three entirely different classification goals. We then pursue human identification under expression changes in greater depth, since this is the most important task from a practical point of view. Two different ways of improving the performance of the simple classifier are then discussed: (i) feature combinations and (ii) the use of resampling techniques (bagging and random subspaces). With these modifications, we succeed in obtaining near perfect classification results on a database of 55 individuals, a statistically significant improvement over the initial results as seen by hypothesis tests of proportions.     

Prediction Intervals for Time Series: A Modified Sieve Bootstrap Approach

Traditional Box–Jenkins prediction intervals perform poorly when the innovations are not Gaussian. Nonparametric bootstrap procedures overcome this handicap, but most existing methods assume that the AR and MA orders of the process are known. The sieve bootstrap approach requires no such assumption but produces liberal coverage due to the use of residuals that underestimate the actual variance of the innovations and the failure of the methods to capture variations due to sampling error of the mean. A modified approach, that corrects these deficiencies, is implemented. Monte Carlo simulations results show that the modified version achieves nominal or near nominal coverage.     

Permutation methods in relative risk regression models

In this paper, we develop a weighted permutation (WP) method to construct confidence intervals for regression parameters in relative risk regression models. The WP method is a generalized permutation approach. It constructs a resampled history which mimics the observed history for individuals under study. Inference procedures are based on studentized score statistics that are insensitive to the forms of the relative risk function. This makes the WP method appealing in the general framework of the relative risk regression model. First-order accuracy of the WP method is established using counting process approach with a partial likelihood filtration. A simulation study indicates that the method typically improves accuracy over asymptotic confidence intervals.     

Building multivariable prognostic and diagnostic models: transformation of the predictors by using fractional polynomials

To be useful to clinicians, prognostic and diagnostic indices must be derived from accurate models developed by using appropriate data sets. We show that fractional polynomials, which extend ordinary polynomials by including non-positive and fractional powers, may be used as the basis of such models. We describe how to fit fractional polynomials in several continuous covariates simultaneously, and we propose ways of ensuring that the resulting models are parsimonious and consistent with basic medical knowledge. The methods are applied to two breast cancer data sets, one from a prognostic factors study in patients with positive lymph nodes and the other from a study to diagnose malignant or benign tumours by using colour Doppler blood flow mapping. We investigate the problems of biased parameter estimates in the final model and overfitting using cross-validation calibration to estimate shrinkage factors. We adopt bootstrap resampling to assess model stability. We compare our new approach with conventional modelling methods which apply stepwise variables selection to categorized covariates. We conclude that fractional polynomial methodology can be very successful in generating simple and appropriate models.