Conjoint Professor John Rayner

Sensory evaluation is the perception science of the food industry. Sensory data can be costly to obtain and so gleaning the most information possible from the data is key. Increasingly, value is added to sensory evaluation by the use of statistics, especially to improve the quality of product development and to make the most of market research. Nonparametrics for Sensory Science is written to complement existing parametric methodology. Nonparametric methods are appropriate when facts are only available in nominal or ordinal form, and when the model assumptions necessary for parametric procedures do not hold. Author Rayner and his colleagues consider problems including the most commonly occurring and important experimental designs: the one-sample, k-sample, blocked samples, samples with factorial structure and samples with correlation structure. Innovative new techniques are outlined and complemented with real examples. Techniques described may be applied to data where the traditional, most frequently applied nonparametric tests, such as the Kruskal-Wallis, the Friedman and the Spearman tests, are applied. Those familiar with traditional nonparametric testing will be able to update their knowledge, acquiring powerful new methods. Those without prior knowledge of nonparametric testing will be able to acquire that knowledge through this book. Aimed at sensory scientists and statisticians interested in nonparametrics, the techniques of Nonparametrics for Sensory Science are of broad general interest, but are of particular interest in sensory evaluation applications.

Most texts on nonparametric techniques concentrate on location and linear-linear (correlation) tests, with less emphasis on dispersion effects and linear-quadratic tests. Tests for higher moment effects are virtually ignored. Using a fresh approach, A Contingency Table Approach to Nonparametric Testing unifies and extends the popular, standard tests by linking them to tests based on models for data that can be presented in contingency tables. This approach unifies popular nonparametric statistical inference and makes the traditional, most commonly performed nonparametric analyses much more complete and informative. It also makes tied data easily handled, and almost exact Monte Carlo p-values can be obtained. With data in contingency tables, one can then calculate a Pearson-type, chi-squared statistic and its components. For univariate data, the initial tests based on these components detect mean differences between treatments. For bivariate data, they detect correlations. This approach leads to tests that detect variance, skewness, and higher moment differences between treatments with univariate data, and higher bivariate moment differences with bivariate data. Although the methods advanced in this book have their genesis in traditional nonparametrics, incorporating the power of modern computers makes the approach more complete and more valid than previously possible. The authors' unified treatment and readable style make the subject easy to follow and the techniques easily implemented, whether you are a fledgling or a seasoned researcher.

The Kruskal¿Wallis tests are appropriate tests for the completely randomised design, both for when the data are untied ranks, and, with adjustment, for when there are ties and mid-ranks are used. Both these tests are shown to be Cochran¿Mantel¿Haenszel mean score tests. The relationship between the Kruskal¿Wallis test statistic and the ANOVA F test statistic when there are no ties generalises to the same relationship between the Cochran¿Mantel¿Haenszel mean score test statistic and the ANOVA F test statistic. It thus also relates both Kruskal¿Wallis test statistics to the ANOVA F test statistic. A small simulation study finds that p-values may be more accurately found using the F test.

A number of nonparametric tests are compared empirically for a randomized block layout. We assess tests appropriate for when the data are not consistent with normality or when outliers invalidate traditional analysis of variance (ANOVA) tests. The objective is to assess, within this setting, tests that use ranks within blocks, the rank transform procedure that ranks the complete sample and continuous analogs of the Cochran-Mantel-Haenszel tests. The usual linear model is assumed, and our primary foci are tests of equality of means and component tests that assess linear and quadratic trends in the means. These tests include the traditional Page and Friedman tests. We conclude that the rank transform tests have competitive power and warrant greater use than is currently apparent.

For multifactor experimental designs in which the levels of at least one of the factors are ordered we show how to construct components that provide a deep nonparametric scrutiny of the data. The components assess generalised correlations and the resulting tests include and extend the Page and umbrella tests. Application of the tests described is straightforward. Orthonormal polynomials on the ANOVA responses and the factors are required and the formulae needed are given subsequently. These depend on the moments of the responses and of each factor and are easily calculated. Products of at least two of these orthonormal polynomials are then used as inputs into standard ANOVA routines. For example, using the first order orthonormal polynomial on factor A and the second order orthonormal polynomial on the ANOVA response will assess if, with increasing levels of factor A there is an umbrella response with either an increase and then a decrease or a decrease and then an increase.

© Grace Scientific Publishing, LLC. Data consisting of ranks within blocks are considered for randomized complete block layouts where treatment effects are expected to be ordered. Ranks with and without ties are considered, as well as missing values. A small indicative test size study indicates both a new test and the Page test, modified by Thas et al. (2012) to easily permit ties, perform well. An advantage of the new test is that it can be easily applied to data with missing values. Three real examples are given. One of these illustrates a quadratic trend test. Comparisons between the new test and the Alvo and Cabilio (1995) extended Page test for missing values are given.

Data consisting of ranks within blocks are considered for balanced incomplete block designs. An F test statistic from ANOVA is better approximated by an F distribution than the Durbin statistic is approximated by a chi-squared distribution. Indicative powers demonstrate that the F test is generally superior to Durbin's test.

Cochran-Mantel-Haenszel (CMH) test statistics can be used to compare products rated on JAR 'just about right' scales and for similar categorical data. In this note we consider the case of paired or related data. The simplified statistics we give allow 'by hand' calculation when the data set is small. We also give an R package for larger data sets or for when the user is not interested in 'by hand' calculation. © 2013 Elsevier Ltd.

This paper compares and investigates the generalised Neyman smooth test, its components, and the classical chi-squared test with a variety of equiprobable classes. Each test is evaluated in terms of its power to reject a wavelike alternative to the uniform distribution, chosen to quantify the complexity of the alternative. Results indicate that if broadly focused tests (rather than strongly directional or weakly omnibus) are sought, then smooth tests of order about four, or the chi-squared test with between five and ten classes, will perform well. Copyright© 2001, Lawrence Erlbaum Associates, Inc.

There are few techniques available for testing if modes take specified values. We show that standard tests of location such as the t-test and the Wilcoxon test, which test for the mean and median respectively, can perform poorly as tests for modes when the data is other than unimodal and symmetric. Carolan and Rayner [1] proposed a score test of location for symmetric nonnormal data. We consider a family of distributions similar to those considered by Carolan and Rayner [1] and propose a test for the mode or modes of data from multimodal or skewed distributions and demonstrate by way of simulations that it is reasonably effective. Copyright© 2001, Lawrence Erlbaum Associates, Inc.

For randomized block designs, nonparametric treatment comparisons are usually made using the Friedman test for complete designs, and the Durbin test for incomplete designs; see, for example, Conover (1998). This permits assessment of only the mean rankings. Such comparisons are here extended to permit assessments of bivariate effects such as the linear by linear effect and the quadratic by linear, or umbrella effect. © Journal of Applied Mathematics & Decision Sciences.

Pearson's statistic is investigated for nominal-ordinal two-way contingency tables in which we wish to test for identical rows. The statistic is expressed as a sum, the first summand of which is a statistic given by Yates (1948), and examines location effects for the nominal category. The second and subsequent summands reflect the corresponding moments: for example, dispersion, skewness, kurtosis etc. The summands are shown to be weakly optimal in that they are score statistics. © Journal of Applied Mathematics & Decision Sciences.

The sample correlation coefficient R is almost universally used to estimate the population correlation coefficient p. If the pair (X, Y) has a bivariate normal distribution, this would not cause any trouble. However, if the marginals are nonnormal, particularly if they have high skewness and kurtosis, the estimated value from a sample may be quite different from the population correlation coefficient ¿. The bivariate lognormal is chosen as our case study for this robustness study. Two approaches are used: (i) by simulation and (ii) numerical computations. Our simulation analysis indicates that for the bivariate lognormal, the bias in estimating p¿ can be very large if ¿ ¿0, and it can be substantially reduced only after a large number (three to four million) of observations. This phenomenon, though unexpected at first, was found to be consistent to our findings by our numerical analysis. © Journal of Applied Mathematics & Decision Sciences.