Browsing by Author "Williams, James D."

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    On the Distribution of Hotelling's T² Statistic Based on the Successive Differences Covariance Matrix Estimator
    Williams, James D.; Woodall, William H.; Birch, Jeffrey B.; Sullivan, Joe H. (Virginia Tech, 2004-09-30)
    In the historical (or retrospective or Phase I) multivariate data analysis, the choice of the estimator for the variance-covariance matrix is crucial to successfully detecting the presence of special causes of variation. For the case of individual multivariate observations, the choice is compounded by the lack of rational subgroups of observations with the same distribution. Other research has shown that the use of the sample covariance matrix, with all of the individual observations pooled, impairs the detection of a sustained step shift in the mean vector. For example, research has shown that, with the use of the sample covariance matrix, the probability of a signal actually decreases below the false alarm probability with a sustained step shift near the middle of the data and that the signal probability decreases with the size of the shift. An alternative estimator, based on the successive differences of the individual observations, leads to an increasing signal probability as the size of the step shift increases and has been recommended for use in Phase I analysis. However, the exact distribution for the resulting T² chart statistics has not been determined when the successive differences estimator is used. Three approximate distributions have been proposed in the literature. In this paper we demonstrate several useful properties of the T² statistics based on the successive differences estimator and give a more accu- rate approximate distribution for calculating the upper control limit for individual observations in a Phase I analysis.
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    Outlier Robust Nonlinear Mixed Model Estimation
    Williams, James D.; Birch, Jeffrey B.; Abdel-Salam, Abdel-Salam Gomaa (Virginia Tech, 2014)
    In standard analyses of data well-modeled by a nonlinear mixed model (NLMM), an aberrant observation, either within a cluster, or an entire cluster itself, can greatly distort parameter estimates and subsequent standard errors. Consequently, inferences about the parameters are misleading. This paper proposes an outlier robust method based on linearization to estimate fixed effects parameters and variance components in the NLMM. An example is given using the 4-parameter logistic model and bioassay data, comparing the robust parameter estimates to the nonrobust estimates given by SASR®.
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    Statistical Monitoring of Nonlinear Product and Process Quality Profiles
    Williams, James D.; Woodall, William H.; Birch, Jeffrey B. (Virginia Tech, 2007)
    In many quality control applications, use of a single (or several distinct) quality characteristic(s) is insufficient to characterize the quality of a produced item. In an increasing number of cases, a response curve (profile), is required. Such profiles can frequently be modeled using linear or nonlinear regression models. In recent research others have developed multivariate T² control charts and other methods for monitoring the coefficients in a simple linear regression model of a profile. However, little work has been done to address the monitoring of profiles that can be represented by a parametric nonlinear regression model. Here we extend the use of the T² control chart to monitor the coefficients resulting from a parametric nonlinear regression model fit to profile data. We give three general approaches to the formulation of the T² statistics and determination of the associated upper control limits for Phase I applications. We also consider the use of nonparametric regression methods and the use of metrics to measure deviations from a baseline profile. These approaches are illustrated using the vertical board density profile data presented in Walker and Wright[1].