ILS: An R package for statistical analysis in Interlaboratory Studies

Abstract

In this paper we present an R package with routines to perform Interlaboratory Studies (ILS). The aim of the ILS package is to detect laboratories that provide not consistent results, working simultaneously with different test materials, from the perspective of the Univariate Data Analysis and the Functional Data Analysis (FDA).

The ILS package estimates the Mandel's h and k scalar statistics, based on the ASTM E691 and ISO 5725-2 standards, to identify laboratories that provide significantly different results. Cochran and Grubbs tests to evaluate the presence of outliers are also available. In addition, Analysis of Variance (ANOVA) techniques are provided, both for the cases of fixed and random effects, including confidence intervals for the parameters.

One of the novelties of this package is the incorporation of tools to perform an ILS from a functional data analysis approach. Accordingly, the functional nature of the data obtained by experimental techniques corresponding to analytical chemistry, applied physics and engineering applications (spectra, thermograms, and sensor signals, among others) is taking into account by implementing the functional extensions of Mandel's h and k statistics. For this purpose, the ILS package also estimates the functional statistics $H\left(t\right)$ and $K\left(t\right)$, as well as the $d^H$ and $d^K$ test statistic, which are used to evaluate the repeatability and reproducibility hypotheses where the critical $c_h$ and $c_k$ values are estimated by using a bootstrap algorithm.

Introduction

An Interlaboratory Study (ILS) can be defined as a control procedure to evaluate the performance of a group of laboratories through a collaborative trial [1,2]. In an Interlaboratory Study, an adequate number of laboratories are chosen to participate in the experiment with the aim of analysing the samples and obtain results.

Participating laboratories receive samples (previously homogenized or to be homogenized by the laboratories) for analysis, then, the measurements results of the laboratories are evaluated according to the degree of data variability. Some of the most common factors that may be a cause of variability are: the equipment of laboratories, operators, materials, temperature and humidity, among others.

Several scalar statistical techniques are frequently applied to study the consistency of test results from the different laboratories that participate in an ILS. Standard ASTM E-691 (Standard Practice for Conducting an Interlaboratory Study to Determine the Precision of a Test Method) recommends applying only one graphical technique from Mandel's k and h statistics [2], while ISO 5725-2 (Accuracy – trueness and precision – of measurement methods and results) recommends, in addition to the graphic technique, to use the Cochran and Grubbs tests [1].

Additionally, through Analysis of Variance (ANOVA), the effect of the laboratory factor over the response can be studied. The variance of repeatability and reproducibility can be also estimated when an ANOVA random effects model is considered, as shown in ISO 5725-2 [1]. On the other hand, if a fixed effect model is fitted, in addition to the F test, multiple comparisons of means can be performed with the Tukey Honest Significant Difference (HSD) method.

To perform consistency tests for the repeatability and reproducibility hypotheses, as well as for the detection of outliers, the values of the statistics should be compared with their corresponding critical values. If these are greater, inconsistency is detected in the results of laboratories. ISO 5725-2 provides some critical values depending on the number of laboratories p, number of measurements n and level of significance α.

At present, both ISO 5725 and ASTM-E691 do not provide a methodology for performing an ILS when data are functional, this is, in the case where the test results are curves (functional data). Functional Data Analysis (FDA) is a relatively new branch of statistics that takes curves as unit of analysis, also surfaces, and volumes defined in a continuum (such as time, or frequency's domain). Considering the recent advances in computing science, and the increasing amount of data generated by experimental techniques and sensors, the FDA has had a great development in recent years. In fact, we have many statistical methodologies that have been developed and extended to the functional case, such as exploratory analysis, regression, classification, analysis of variance, and time series [[3], [4], [5]]. In the specific case of ILS, FDA extensions for Mandels's h and k have been proposed and described by Flores et al. [6], in addition to other works where the FDA descriptive analysis had been introduced for ILS studies [7].

The aim of the ILS package [8] is, on the one hand, to facilitate the use of new tools in the FDA context and, on the other hand, to provide a comprehensive set of the more used univariate outlier tests for ILS with scalar response. It is important to note that FDA techniques for Interlaboratory Studies are based on the proposals of Naya et al. [7], Flores et al. [9] and, above all, Flores et al. [6], whereby new functional extensions of h and k statistics are introduced for identifying non consistent laboratories. The functions that have been implemented in the ILS package can determine the Mandel's h and k statistics both in a graphical and analytical way, using a functional approach. These test statistics have also been implemented to facilitate the implementation of Repeatability and Reproducibility studies (r&R) when the data are functional. In addition, the ILS package apply the methods suggested by the norms ASTM E-691 and ISO 5725-2 for the scalar case. The ILS package is available on the Comprehensive R Archive Network at http://CRAN.R-project.org/package=ILS.

The present ILS library implements and calls some of their routines in order to perform outlier detection in the framework of the Interlaboratory Studies. Thus, regarding to ILS with scalar response, there are some interesting and useful computational tools in R software. Namely, the metRology package estimates the uncertainty of the measurement, and performs the required statistical calculations for Interlaboratory Studies [10], whereas multcomp performs analysis of variance (ANOVA) through F and Tuckey tests [11]. On the other hand, due to the exponential increasing of FDA available techniques, there are also a continuously increasing number of R libraries devoted to this branch of statistics. Among all of them, the most important and used packages (on which the present proposal is based) are fda.usc [12], that implements outlier detection techniques and functional ANOVA, among other tools for FDA, and the fda [13]. The present ILS package uses the applications of the before mentioned multcomp and fda.usc packages.

This work is organized as follows. In Section 2, we describe examples of Interlaboratory Studies defined by four sets of experimental data. Then, the ILS package is used to summarize two of these sets. In Section 3, the functionality of the ILS package is illustrated through a standard ILS procedure using the Glucose dataset. Further, in Section 4, the TG and DSC datasets (composed by thermogravimetric and colorimetric curves of calcium oxalate, respectively) are used to show the ILS package utilities when experimental data are curves (functional). Finally, the principal remarks of this study are summarized in the conclusion section.