Michael D. Conerly

Alarm rates for quality control charts

There is a direct relationship between a single alarm probability and the average run length only for basic Shewhart charts such as the X-chart. Alarm rates are defined in this paper that can be applied with charts such as the cumulative sum (CUSUM) chart and the exponentially weighted moving average (EWMA) chart that base decisions on several observations, not just the most recent one. Methods for determining EWMA chart limits are compared on the basis of their false alarm rates. It is shown how control charts can be more flexibly and carefully defined by considering a desired pattern of in-control false alarm rates in conjunction with a desired in-control average run length.

Exact Properties of Demerit Control Charts

A demerit rating system is used to simultaneously monitor counts of several types of defects in a complex product. The demerit statistic is a linear combination of the counts of these types of defects. The traditional recommendation is to plot the dem..

An Approximate Test for Comparing Heteroscedastic Regression Models

Abstract This article addresses the problem of testing whether the vectors of regression coefficients are equal for two independent regression models when the error variances are unequal. The usual Chow statistic, appropriate when equality of variances can be assumed, is modified by replacing the pooled residual variance in the denominator with a weighted average of the residual variances from each data set. The weights are functions of the mean of the eigenvalues of W = X′ 1 X 1(X′ 1 X 1 + X′ 2 X 2)-1. Both numerator and denominator are then approximated by scalar multiples of chi-squared distributions. The parameters of these approximated distributions are chosen to equate their first two moments to those of the exact distribution. The resulting approximation for the modified Chow statistic, C*, is an F distribution with degrees of freedom that depend on the two sample sizes, the number of regressor variables, the average eigenvalue of W, and the true ratio of error variances. Since the latter is unknow...

A SIMULATION STUDY AND EVALUATION OF MULTIVARIATE FORECAST BASED CONTROL CHARTS APPLIED TO ARMA PROCESSES

Much research had been performed in the area of control charting techniques for monitoring autocorrelated processes, especially regarding forecast based monitoring schemes. Forecast based monitoring schemes involve fitting an appropriate time-series model to the process, generating one step ahead forecast errors, and monitoring the forecast errors with traditional control charts. Another method introduced into the literature involves using multivariate control charts to monitor the ARMA derived one-step-ahead (OSA) and two-step-ahead (TSA) forecast errors. This article provides a broad simulation study and evaluation of the suggested multivariate approaches in regards to various ARMA(1,1) and AR(1) processes, and a comparison to their univariate counterparts.

The reverse moving average control chart for monitoring autocorrelated processes

Forecast-based monitoring schemes have been researched extensively in regards to applying traditional control charts to forecast errors arising from various autocorrelated processes. The dynamic response and behavior of forecast errors after experiencing a shift in the process mean make it difficult to choose a suitable control chart. In this paper we propose the reverse moving average control chart as a new forecast-based monitoring scheme, compare the new control chart to traditional methods applied to various ARMA(1,1), AR(1), and MA(1) processes, and make recommendations concerning the most appropriate control chart to use in a variety of situations when charting autocorrelated processes.

Diagnostic Value of Residual and Partial Residual Plots

Abstract This article illustrates why partial residual plots in addition to the usual residual plots are useful in a multiple regression analysis. The expected values of the vector of residuals and the vector of partial residuals are presented and examined for the situation when a regressor variable is misspecified in the model. If curvature exists in a predictor variable, the plot of residuals versus the variable displays the points scattered around a line that is a linear transformation of the correct functional form of the variable. Hence a nonrandom pattern may appear in the plot, but the appropriate transformation may not be evident. For a partial residual plot, the underlying signal displays the correct functional form of the predictor variables across the relevant range of interest, except in instances when severe collinearity exists.

An approximate test for comparing independent regression models with unequal error variances

Abstract The usual F statistic for comparing two independent regression equations is commonly used by practitioners. This test, however, presumes the equality of the error variance of the two populations. For applications where this assumption is not valid, an approximate test based on the same statistic is proposed that improves the Toyoda (1974) approximation by using Satterthwaites (1946) approximation not just for the numerator but also for the denominator of the usual F statistic. The unconditional significance level of this approximate test is computed for a variety of design configurations. The power of the approximate test relative to an upper bound is also considered.

Sex difference in muscle cross‐sectional area of athletes and non‐athletes

The purpose of the present study was to determine whether there is a sex difference in limb muscle cross-sectional area by comparing upper- and lower-body limb fat-free cross-sectional areas (FFCSAs) adjusted for differences in fat-free weight (FFW), in male and female athletes with similar histories of upper-body physical conditioning and in non-athletes. Limb FFCSAs were calculated from circumferences corrected for subcutaneous fat thickness and FFW was estimated from body density measured by underwater weighing in 24 male and 25 female swimmers and 23 male and 25 female non-athletes, 15 to 28 years of age. The male swimmers had 32% larger FFWs and 49% larger upper-arm, similar forearm and 23% larger thigh FFCSAs compared to the female swimmers. The male non-athletes had 34% larger FFWs, 61% larger upper-arms, 54% larger forearms and 35% larger thighs than female non-athletes. To adjust for differences in body size, analysis of covariance was performed on the FFCSAs using FFW as the covariate. For the swimmers there were no significant differences (P greater than 0.05) in the adjusted FFCSAs. For the non-athletes, males had significantly larger adjusted upper-arm and forearm FFCSAs than the females but thigh FFCSAs were not significantly different (P greater than 0.05). These results suggest that sex differences in muscle area of the arms, may be partially attributed to long-term activity differences between sexes. Possible long-term differences in activity between sexes should be considered in comparisons of functional or performance measures between sexes.

A comparison of biased regression estimators using a pitman nearness criterion

Biased regression estimators have traditionally benn studied using the Mean Square Error (MSE) criterion. Usually these comparisons have been based on the sum of the MSEs of each of the individual parameters, i.e., a scaler valued measure that is the trace of the MSE matrix. However, since this summed MSE does not consider the covariance structure of the estimators, we propose the use of a Pitman Measure of Closeness (PMC) criterion (Keating and Gupta, 1984; Keating and Mason, 1985). In this paper we consider two versions of PMC. One of these compares the estimates and the other compares the resultant predicted values for 12 different regression estimators. These estimators represent three classes of estimators, namely, ridge, shrunken, and principal component estimators. The comparisons of these estimators using the PMC criteria are contrasted with the usual MSE criteria as well as the prediction mean square error. Included in the estimators is a relatively new estimator termed the generalized principal...

Phase I control chart based on a kernel estimator of the quantile function

To measure the statistical performance of a control chart in Phase I applications, the in-control average run length (ARL) is the most frequently used parameter. In typical start up situations, control limits must be computed without knowledge of the underlying distribution of the quality characteristic. Assumptions of an underlying normal distribution can increase the probability of false alarms when the underlying distribution is non-normal, which can lead to unnecessary process adjustments. In this paper, a control chart based on a kernel estimator of the quantile function is proposed. Monte Carlo simulation was used to evaluate the in-control ARL performance of this chart relative to that of the Shewhart individuals control chart. The results indicate that the proposed chart is more robust to deviations in the assumed underlying distribution (with respect to the in-control ARL) and results in an alternative method of designing control charts for individual units. Copyright