Boris Iglewicz

Fine-Tuning Some Resistant Rules for Outlier Labeling

Abstract A previous study examined the performance of a standard rule from Exploratory Data Analysis, which uses the sample fourths, FL and FU , and labels as “outside” any observations below FL – k(FU – FL ) or above FU + k(FU – FL ), customarily with k = 1.5. In terms of the order statistics X (1) ≤ X (2) ≤ X (n) the standard definition of the fourths is FL = X(f) and FU = X (n + 1 − f), where f = ½[(n + 3)/2] and [·] denotes the greatest-integer function. The results of that study suggest that finer interpolation for the fourths might yield smoother behavior in the face of varying sample size. In this article we show that using f i = n/4 + (5/12) to define the fourths produces the desired smoothness. Corresponding to a common definition of quartiles, fQ = n/4 + (1/4) leads to similar results. Instead of allowing the some-outside rate per sample (the probability that a sample contains one or more outside observations, analogous to the experimentwise error rate in simultaneous inference) to vary, some us...

Performance of Some Resistant Rules for Outlier Labeling

Abstract The techniques of exploratory data analysis include a resistant rule for identifying possible outliers in univariate data. Using the lower and upper fourths, FL and FU (approximate quartiles), it labels as “outside” any observations below FL − 1.5(FU — FL ) or above FU + 1.5(FU — FL ). For example, in the ordered sample −5, −2, 0, 1, 8, FL = −2 and FU = 1, so any observation below −6.5 or above 5.5 is outside. Thus the rule labels 8 as outside. Some related rules also use cutoffs of the form FL — k(FU — FL ) and FU + k(FU — FL ). This approach avoids the need to specify the number of possible outliers in advance; as long as they are not too numerous, any outliers do not affect the location of the cutoffs. To describe the performance of these rules, we define the some-outside rate per sample as the probability that a sample will contain one or more outside observations. Its complement is the all-inside rate per sample. We also define the outside rate per observation as the average fraction of outs...

Some Implementations of the Boxplot

Abstract An increasing number of statistical software packages offer exploratory data displays and summaries. For one of these, the graphical technique known as the boxplot, a selective survey of popular software packages revealed several definitions. These alternative constructions arise from different choices in computing quartiles and the fences that determine whether an observation is “outside” and thus plotted individually. We examine these alternatives and their consequences, discuss related background for boxplots (such as the probability that a sample contains one or more outside observations and the average proportion of outside observations in a sample), and offer recommendations that lead to a single standard form of the boxplot.

A treatment allocation procedure for sequential clinical trials.

A dynamic treatment allocation procedure is proposed for clinical trials which require balancing across several prognostic factors. The treatment allocation decision is based on the minimization of a function which is an easily evaluated approximation to the variance of the treatment effect in a linear model relating the outcome variable to the chosen prognostic factors and selected interactions. By use of simulations, the procedure is shown to be superior to ad hoc procedures proposed by Pocock and Simon (1975, Biometrics 31, 103-115), over a variety of reasonable experimental situations. It is shown that it is feasible to evaluate the procedure by hand calculations and that it is extremely easy if a small programmable calculator is available. Practical problems relating to implementation of the procedure are discussed with special reference to multi-institutional clinical trials.

Illustrating the Impact of a Time-Varying Covariate With an Extended Kaplan-Meier Estimator

In clinical endpoint trials, the association between a baseline covariate and the risk of an endpoint is often measured by the hazard ratio as calculated by a Cox regression model, and illustrated by Kaplan-Meier curves comparing cohorts defined by levels of the covariate. The Cox regression model is easily extended to the case of time-varying covariates; however, there is no clear approach for similarly extending the standard Kaplan-Meier estimator. Various ad hoc procedures that have been used in the medical literature are seriously flawed. This article discusses an extended Kaplan-Meier estimator that can be used with time-varying covariates and illustrates this method using data from a long-term clinical trial.

Bivariate extensions of the boxplot

The boxplot has proven to be a very useful tool for summarizing univariate data. Several options of bivariate boxplot-type constructions are discussed. These include both elliptic and asymmetric plots. An inner region contains 50% of the data, and a fence identifies potential outliers. Such a robust plot shows location, scale, correlation, and a resistant regression line. Alternative constructions are compared in terms of efficiency of the relevant parameters. Additional properties are given and recommendations made. Emphasis is given to the bivariate biweight M estimator. Several practical examples illustrate that standard least squares ellipsoids can give graphically misleading summaries.

The influence of uninterpretability on the assessment of diagnostic tests.

A frequent problem faced by physicians utilizing diagnostic tests is the occurrence of uninterpretable test results. Such results, if they occur commonly, can seriously impair the diagnostic performance of the test. Moreover, in assessing the characteristics of the test, i.e. sensitivity, specificity, etc. failure to consider the impact of uninterpretability will artificially inflate the test characteristics. In this paper we explore the implications of this issue. We observe that a relevant factor is the potential repeatability of the test, i.e. whether the cause of uninterpretability is a transient phenomenon or an inherent property of the subject. We distinguish uninterpretable results, in which no result is obtained, from indeterminate results, in which the result is equivocal, or for which predisposing concomitant factors limit the interpretability of the result. Our results demonstrate that the naive approach of ignoring uninterpretable results in test assessments may indeed be unbiased in certain circumstances. However, if the cause of uninterpretability is related to disease status or to the potentially observable test result, then this approach will lead to bias. In either case, the frequency of uninterpretability is an important consideration in the cost-effectiveness of the test.

Comparisons of Approximations to the Percentage Points of the Sample Coeffcient of Variation

It may be of some interest in certain statistical problems to have available percentage points of the sample coefficient of variation (S.C.V.). If the parent population is assumed to be normal, the obtaining of such percentage points from tables of the non-central t distribution might be thought to be simple. However, one does encounter difficulty in using these tables to calculate the percentage points of the S.C.V. One can, as an alternative, use the exact table of Iglewicz (1967). A number of approximations for the percentage points of the S.C.V. exists in the literature. These approximations may be useful as alternatives to either of the tables mentioned above. The applicability of several of these approximations is further enhanced by their simplicity. In this paper we compare these approximate results with the exact ones in order to determine the quality of the approximations.

Trimmed Mean X̄ and R Charts

A modified approach to the computation of control limits for X and R charts is introduced. This procedure consists of replacing X with the trimmed mean of the subgroup averages, and R with the trimmed mean of the subgroup ranges. Standard tables of control limits may continue to be used, requiring only adjustment by a constant multiplier, which is tabulated. The proposed control chart limits are shown to be less influenced by extreme observations than their classical counterparts, and to lead to tighter limits in the presence of out-of-control observations. These concepts are illustrated by several examples.

Use of Boxplots for Process Evaluation

The use of boxplots in place of single points in a quality control chart can provide an effective display of the information usually given in X-bar and R charts, show the degree of compliance with specifications and identify outliers. An example from a ..