John Mandel

A New Analysis of Variance Model for Non-additive Data

where nil is referred to as the interaction between rows and columns, and where 7ii , unlike eii, is no longer considered as just random error. However, a part of 77i may be just random error. Because of the presence of r,i , which, apart from its random error component, is a function of two variables, all advantage of the additive model will be lost, unless one can again partition the non-random portion of n7.i into functions of only one variable each. Obviously, an additive partitioning of 7ii is impossible since all additive parts have already

Non-Additivity in Two-Way Analysis of Variance

Abstract In two-way classification analysis of variance situations there often exists a systematic type of row column interaction. A model is proposed in which the interaction is of the type Q i γ i where Q i is a parameter of the ith row, not necessarily associated with the main effect for rows, and γ j is the main effect for column j. The analysis of data according to this model is given, including estimation and tests of significance. The model is more general than that involved in Tukeys “one degree of freedom for non-additivity” and includes the latter as a special case. The relationship between the two methods is discussed. Applications of the method to different types of problems are mentioned and a numerical example is included.

Use of the Singular Value Decomposition in Regression Analysis

Abstract Principal component analysis, particularly in the form of singular value decomposition, is a useful technique for a number of applications, including the analysis of two-way tables, evaluation of experimental design, empirical fitting of functions, and regression. This paper is a discussion in expository form of the use of singular value decomposition in multiple linear regression, with special reference to the problems of collinearity and near collinearity.

Fitting Straight Lines When Both Variables are Subject to Error

(This paper was presented at the Journal of Quality Technology Session of the 27th Annual Fall Technical Conference of the Chemical and Process Industries Division of the American Society for Quality Control and the Section on Physical and Engineering S..

The validation of measurement through interlaboratory studies

Abstract Mandel, J., 1991. The validation of measurement through interlaboratory studies. Chemometrics and Intelligent Laboratory Systems , 11: 109–119 The problem of interlaboratory tests has been investigated with regard to its principal aspects: planning of experiments, analysis of numerical data, treatment of questionable data, and possible models. The discussion is illustrated by means of a real numerical example, comprising 18 laboratories, 5 materials, and duplicate measurements in each of the 90 cells.

Fitting a Straight Line to Certain Types of Cumulative Data

Abstract Many situations giving rise to linear data involve measurements made at progressive stages of a physical or chemical process carried out on the same subject of experimentation. In such cases, the experimental errors include cumulative components related to the process, and the errors corresponding to different points on the line are not independent. Failure to take such cumulation of errors into account results in serious underestimation of the standard error of the estimated rate of the process. The standard errors and correlations of the residuals from regression are derived both for situations involving cumulative and independent errors. The differences between the two situations are striking and may be used as a basis for judging which of the two types of error is predominant in a given case. The procedure is illustrated by means of the data obtained in an experiment in physical optics.

The Measuring Process

This paper deals with the theory of a proposed method for the statistical study of measuring processes. The practical aspects of the method, including computational details, are discussed in a companion paper published in the ASTM Bulletin. In the present article a theoretical framework is proposed for the mathematical expression of the sources of variation in measuring methods and a suitable method of statistical analysis is described. Particular attention is given, both here and in the companion paper, to interlaboratory studies of test methods. An illustration based on data taken from the chemical literature is appended.

A Method for Fitting Empirical Surfaces To Physical or Chemical Data

The central problem of data analysis is the achievement of just the right balance between prior assumptions on the one hand and the findings based on observation of the data on the other hand. This paper is concerned with a special case of this problem. To analyze a set a data without any prior assumptions about an underlying model is impossible. To adopt too rigid a set of assumptions, on the other hand, results in forcing the data into a mold in which they may not fit at all well. There exist cases in which this problem is of secondary importance: those are the relatively rare instances in which a completely specified mathematical model is available prior to analysis. In contrast, the problem is most pertinent when all the information available to the data analyst is that contained in the data themselves. Between those extremes, which we may refer to respectively as the purely theoretical case and the purely empirical case, lie a multitude of situations in which prior information is available to some extent, subject however to confirmation by the experiment, which also serves to supply the missing parts of the assumed model.

Estimation of Weighting Factors in Linear Regression and Analysis of Variance

It frequently happens that data to be treated by regression methods or analysis of variance are subject to marked heterogeneity in the error variance. Weighting of the observations, in the least squares sense, is then appropriate but requires that values of the relative weighting factors be available. A method is presented for deriving weight estimates from the residuals of an analysis carried out with any preliminary set of weights. The method is iterative and may therefore be applied with an initial set of unit weights for all observations. Specifically, the method is concerned with column weighting in two-way tables, treated in accordance with an additive model or a more general linear regression model. The effectiveness of the method has been tested by Monte-Carlo techniques. An application of the method to an interlaboratory study of a test method is discussed.

Standard Reference Materials: A reference method for the determination of potassium in serum

Guided by a committee of experts in clinical chemistry, a reference method was established for the determination of serum potassium based on flame atomic emission spectroscopy (FAES) . Its accuracy was evaluated by comparing the values obtained by use of the method in 12 laboratories against the results obtained by a definitive analytical method based on isotope dilution-mass spectrometry (IDMS) . Seven serum pools with potassium concentrations in the range 1.319 to 7.326 mmol/L were analyzed. Manual and semiautomated pipetting alternatives were tested using sample sizes of 5.0 and 0.25 mL, respectively. The laboratories used several different FAES instruments. The results showed that the standard error for a single laboratorys performance of the procedure ranged from 0.049 to 0.063 mmol/L with a maximum bias of 0.065 mmol/L over the range of concentrations studied. These values were within the accuracy and precision goals that had been set by the committee. The results from the two pipetting techniques were similar. The calibration curve data showed excellent linearity over the total concentration range, with 20 of 22 curves having standard deviations of fit of 0.075 mmol/L or less . With appropriate experimental design, the reference method may be used to establish the accuracy of field methods as well as to determine reference potassium values for pooled sera .