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We consider the little-known one parameter Lindley distribution. This distribution may be of interest as it appears to be more flexible than the exponential distribution, the Lindley fitting more data than the exponential. We give smooth tests of fit for this distribution. The smooth test for the Lindley has power comparable with the Anderson-Darling test. Advantages of the smooth test are discussed. Examples that illustrate the flexibility of this distributions is given.

The Cochran-Mantel-Haenszel (CMH) methodology is a suite of tests applicable to tables of count data. The traditional CMH tests assess association, mean, and correlation effects. Here, testing for mean effects is extended to testing for higher order moment effects. Of especial interest are dispersion effects, reflecting, for example, market segmentation in certain scenarios. Correlation testing is extended to testing for generalized correlations. Of especial interest are correlations of order (1, 2) and (2, 1), which may reveal umbrella effects.

Data consisting of ranks within blocks are considered for randomized block designs when there are missing values. Tied ranks are possible. Such data can be analysed using the Skillings–Mack test. Here we suggest a new approach based on carrying out an ANOVA on the ranks using the general linear model platform available in many statistical packages. Such a platform allows an ANOVA to be calculated when there are missing values. Indicative sizes and powers show the ANOVA approach performs better than the Skillings–Mack test.

Summary 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.

Sometimes the number of treatments to be compared is so large that a complete blocks experiment is impractical. This happens, for example, in some agronomic variety trials. A balanced incomplete block (BIB) design can be used in such a situation. In sensory evaluation trials loss of sensitivity can occur when the subjects are not be able to compare more than a few products with any certainty. Again BIB designs are useful. Consider a balanced incomplete block design with the data being ranks within blocks. A traditional test for treatment differences for such a scenario is the Durbin (1951) test, based on the statistic D, given by

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.

Summary The rank transform procedure is often used in the analysis of variance when observations are not consistent with normality. The data are ranked and the analysis of variance is applied to the ranked data. Often the rank residuals will be consistent with normality and a valid analysis results. Here we find that the rank transform procedure is equivalent to applying the intended analysis of variance to first order orthonormal polynomials on the rank proportions. Using higher order orthonormal polynomials extends the analysis to higher order effects, roughly detecting dispersion, skewness etc. differences between treatment ranks. Using orthonormal polynomials on the original observations yields the usual analysis of variance for the first order polynomial, and higher order extensions for subsequent polynomials. Again first order reflects location differences, while higher orders roughly detect dispersion, skewness etc. differences between the treatments.

In the analysis of variance (Anova) the use of orthogonal contrasts is quite common and is a traditional topic in many basic Anova courses. Similar ideas apply to rank tests. In this paper we present a simple and general method that allows an orthogonal contrast decomposition of rank test statistics such as the Kruskal-Wallis, Friedman and Durbin statistics. The components of the test statistics are informative, particularly when ordered alternatives are of interest. The method can handle ties, and null distributions are readily available. Most of the methods are not new, but the way we present them is. Moreover, our formulation makes it easier to better understand and interpret the tests when the traditional location-shift assumption does not hold. The methods are illustrated using several data sets.

The generalised Pareto distribution (GPD) is often used to model extreme values. New smooth tests of goodness of fit are proposed for this distribution. Typical problems with the GPD are that not all moments exist and not all classical estimation procedures work well over the whole parameter space. The generalised smooth test has good powers within a subset of the parameter space for which other tests may not be defined or appropriate, and, conversely, the Anderson-Darling test performs well when the test proposed here does not.

The gamma distribution is often used to model data with right skewness. Smooth tests of goodness of fit are proposed for this distribution. Their powers are compared with powers of the Anderson–Darling test and tests based on the empirical Laplace transform, the empirical moment generating function and the independence of the mean and coefficient of variation that characterizes the gamma distribution.

The analysis of partially balanced incomplete block (PBIB) ranked data is discussed. Two examples are given to illustrate two alternative approaches. Analysis of PBIB ranking data is not covered in any of the standard sensory evaluation texts and this expository note is meant to help fill this gap. For some data sets, the calculations for the first approach are simple enough to do by hand. The second approach that we consider assumes that computer software for general analysis of variance is available. Such analysis of variance software should cope with missing values via a regression method. A suggested multiple comparisons algorithm is also illustrated. R code is given to allow easy application of our first approach. PRACTICAL APPLICATIONS Sensory fatigue can be a problem in some sensory evaluation trials. In the taste-test area, this is sometimes called “palate paralysis.” To cope with this fatigue, balanced incomplete block designs can be employed. However, these restrict the sensory scientist to particular combinations of products, subjects and evaluations per subject. Sometimes, such restrictions can be prohibitive, and then partially balanced designs, which allow more freedom in the choice of these parameters, can be used. Here we consider statistical analysis of ranking data from partially balanced incomplete block designs.

In this note two easily applied moment tests of the Rayleigh hypothesis are compared with the Anderson-Darling test, some smooth tests and tests based on the empirical Laplace transform.

Tests based on the Anderson–Darling statistic, a third moment statistic and the classical Pearson–Fisher X 2 statistic, along with its third-order component, are considered. A small critical value and power study are given. Some examples illustrate important applications.

Smooth tests of goodness of fit may be constructed by defining an order k alternative to the hypothesised probability density function and deriving the score test to assess whether or not the data are consistent with the hypothesised probability density function. For many important distributions the form of the score test statistic is the sum of squares of components that are asymptotically independent and asymptotically standard normal. Moreover each component has a moment interpretation that assists with interpreting rejection of the null hypothesis. Here a sufficient condition is given for the score test statistic to have this form and for the components to have this simple and convenient moment interpretation. Alternative approaches, using generalised score tests, are given for when the sufficient condition is not satisfied. This enables the construction of convenient tests of fit for distributions not from exponential families of distributions, such as the logistic and extreme value distribu...

We set the context for this special issue on modern goodness of fit methods, or modern methods of assessing statistical models.

We propose a methodology for informative goodness of fit testing that combines the merits of both hypothesis testing and nonparametric density estimation. In particular, we construct a data-driven smooth test that selects the model using a weighted integrated squared error (WISE) loss function. When the null hypothesis is rejected, we suggest plotting the estimate of the selected model. This estimate is optimal in the sense that it minimises the WISE loss function. This procedure may be particularly helpful when the components of the smooth test are not diagnostic for detecting moment deviations. Although this approach relies mostly on existing theory of (generalised) smooth tests and nonparametric density estimation, there are a few issues that need to be resolved so as to make the procedure applicable to a large class of distributions. In particular, we will need an estimator of the variance of the smooth test components that is consistent in a large class of distributions for which the nuisanc...

The negative binomial model is an important and flexible two parameter distribution that models data from many application areas. Here we re-examine tests of fit for the negative binomial distribution that were introduced by Anscombe (1950); they are based on a dispersion statistic U and a third moment statistic T. Small sample power calculations are given for U and T. We are not aware that such powers have been given previously. We show Anscombe's tests are smooth tests in the sense of Rayner and Best (1989). Comparisons are made with an empirical probability generating function test suggested by Meintanis (2005). We suggest U not be used and that decisions on the fit of data to the negative binomial be made using bootstrap p-values rather than comparison with standard errors as suggested by Anscombe (1950). We show that tests based on a fourth moment component of a smooth test statistic have good power.

We consider testing for an unobservable normal distribution with unspecified mean and variance. It is only possible to observe the counts in groups with boundaries specified before sighting the data. On the basis of a small power study, we recommend the usual X2 test be used as an omnibus test, augmented by informal examination of the first two non-zero components of X2. We also recommend use of maximum likelihood and method of moments estimation.

The zero-truncated Poisson distribution is an important and appropriate model for many applications. Here, we give and assess several tests of fit for this model.

Tests for the Extreme Value distribution based on the sample skewness and kurtosis coefficients are shown to be related to components of smooth tests of goodness of fit and are compared with tests due to Anderson-Darling, Shapiro-Brain and Liao-Shimokawa. Two examples are given.

A power study suggests that a good test of fit analysis for the binomial distribution is provided by a data-dependent Chernoff–Lehmann X 2 test with class expectations greater than unity, and its components. These data-dependent statistics involve arithmetically simple parameter estimation, convenient approximate distributions and provide a comprehensive assessment of how well the data agree with a binomial distribution. We suggest that a well-performed single test of fit statistic is the Anderson–Darling statistic.

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.

The one-sample Wilcoxon signed rank test was originally designed to test for a specified median, under the assumption that the distribution is symmetric, but it can also serve as a test for symmetry if the median is known. In this article we derive the Wilcoxon statistic as the first component of Pearson's X 2 statistic for independence in a particularly constructed contingency table. The second and third components are new test statistics for symmetry. In the second part of the article, the Wilcoxon test is extended so that symmetry around the median and symmetry in the tails can be examined seperately. A trimming proportion is used to split the observations in the tails from those around the median. We further extend the method so that no arbitrary choice for the trimming proportion has to be made. Finally, the new tests are compared to other tests for symmetry in a simulation study. It is concluded that our tests often have substantially greater powers than most other tests.

A power study suggests that a good test of fit analysis for the Poisson distribution is provided by a data dependent Chernoff-Lehmann X2 test with class expectations greater than unity, and its components. These data dependent statistics involve arithmetically simple parameter estimation, convenient approximate distributions, and provide a comprehensive assessment of how well the data agree with a Poisson distribution. We suggest that a well-performed single test of fit statistic is the Anderson-Darling statistic. Three examples are discussed.

In this expository note examples are given to demonstrate that if categories are ordinal rather than nominal then it may be possible to extend the common Stuart (1955) analysis for cross-classified data by using a statistic which takes order into account.

Summary Standard analysis for ranks from two-way layout data with ties or for 'rank transformed' data with ties can be extended to allow market research to make better comparisons between products. In addition to detecting product average ranks effects, the new analysis allows detection of significant nonlinear effects, umbrella effects, linear contrasts and differences in distributions. The paper presents market research results comparing three types of french fries. There are no differences, according to the standard Friedman analysis. However significant nonlinear effects are found using the new analysis, which give the manufacturer important information.

Anderson (Biometrics 15 (1959) 582) proposed a χ2-type statistic for the nonparametric analysis of a randomized blocks design with no ties in the data. In this paper, we propose an Anderson statistic that allows for ties in the data. We show that the asymptotic distribution of the statistic under the null hypothesis of no treatment effect is a χ2 distribution. Under weak assumptions on the tie structure it is shown that the degrees of freedom for the asymptotic distribution is unchanged compared to the untied case. An extended analysis based on a partition of the statistic into independent components is suggested. The first component is shown to equal the Friedman rank statistic corrected for ties. The subsequent components allow for the detection of dispersion effects, higher order effects and differences in distribution. A simulation study is given and the new analysis is applied to a sensory evaluation data set.

Anderson's statistic, which compares distributions of ranks, is generalized to cope with tied rankings. A simple example is given to illustrate the calculations involved. In market research work it is often important to compare distributions as well as mean values. Comparison of distributions can indicate either market segmentation or product variability.

This article gives power comparisons of some tests of fit for the Geometric distribution. These tests include a Chernoff–Lehmann X 2 test, some smooth tests, a Kolmogorov–Smirnov test, and an Anderson–Darling test. This article suggests that a good test of fit analysis is provided by a data dependent Chernoff–Lehmann X 2 test with class expectations greater than unity, and its components. These data dependent statistics involve arithmetically simple parameter estimation, convenient approximate distributions, and provide a fairly complete assessment of how well the data agrees with a Geometric distribution. The power comparisons indicate also that the best performed single statistic is the Anderson–Darling statistic.

The purpose of a one-sample test of fit is to give an objective measure of how well a probability model agrees with observed data. Here we discuss the test of Karl Pearson and derivatives of it, tests based on the empirical distribution function and the construction of the Neyman–Barton smooth tests. In the final section, we then address some modern developments in smooth testing: diagnostics, Cholesky components, data-driven tests and model selection. Other tests of fit, such as correlation tests and Laplace transform tests, are not considered here.

Diverse food choices and preferences are commonly observed between consumers cross-culturally, however, many aspects of these behavioral differences are not yet fully understood. In order to investigate cross-cultural texture preferences, six extruded snack samples, manufactured under different processing conditions to represent a range of textural characteristics were characterized by a trained panel using descriptive sensory analysis. In parallel, thirty-eight naive European-Origin and thirty-seven naive Chinese-Origin Australian consumers (t = 75) rated their preference for the snacks. The snacks were not representative of any snacks on the market and thus represented a “ novel” product in both cultures. Additionally, consumers answered a simple questionnaire and completed the food neophobia scale. Results of descriptive analysis indicated significant differences (p≤0.01) between the samples for 13 out of 15 descriptive attributes but no significant differences in preference were found between European-origin and Chinese-origin consumers. However, overall analysis of mean effects indicated that Chinese-origin consumers rated samples significantly higher than European-origin consumers (p≤0.05), suggesting a cultural bias in the use of line scales. Gender did not influence texture preference, however, age significantly influenced preference for three of the samples (p≤0.05) and analysis of mean effects also indicated differences in consumers’ ratings which were related to age. Food neophobia classification did not influence preference, although many more Chinese-origin consumers (28) were classified as neophobic than were European-origin consumers (11).

Examples are given to illustrate the application of the Stuart test to taste-testing and other sensory evaluation or market research applications. Convenient formulae for a generalised Stuart test, a mean scores test and a linear trend test are given. These statistical tests allow a quite thorough analysis of consumer sensory data and give an objective assessment of whether market segmentation or other nonlinear effects exist.

Sensory evaluation data is sometimes collected using trinary category scales. Nonparametric data analysis for such data is discussed and a homogeneity statistic for trinary data is proposed which is simple enough to calculate with a pocket calculator. A statistic for identifying market segmentation in trinary data is also suggested. Les donnees sensorielles sont quelquefois collectees en utilisant une echelle a 3 categories. L 'analyse non parametrique de ces donnees est etudiee et l'analyse statistique simple est presentee.

The usual Durbin analysis for ranked data obtained from a balanced incomplete block design is extended to allow for (1) dispersion effects, (2) a predetermined ordering of products and (3) more general trends. An application to paired comparison data is given. Calculation of p-values via Monte Carlo simulation is suggested.

This study examined whether the perception of, and preference for, varying levels of sucrose in water, orange juice, custard and biscuit were related to dietary intakes of sugar and sweet foods in 69 Caucasian Australian and 63 Malaysian university students living in Australia. The two ethnic groups were equally able to discriminate the relative differences in sucrose concentration among the test samples, and a similar percentage of each group were sweetness likers or dislikers. Some significant differences between the ethnic groups' sweetness intensity and preference ratings were observed, but the differences were food-specific and were not always apparent for the same sucrose level in the different test foods. Within each ethnic group, more subjects had greater difficulty discriminating different sucrose levels in a solid, high fat biscuit. The Australian group had higher mean liking ratings than the Malaysian group for the orange juice and biscuit samples. However, both groups preferred similar levels of sucrose in the juice, custard and biscuit, which were similar to the amounts of sucrose present in current commercial products. The hedonic response pattern to the sucrose solution was not a good predictor of the hedonic responses to the other test stimuli, which indicates that preferred levels of sweetness are food-specific. Sweetness intensity ratings were not significantly associated with any of the dietary intake variables or bitterness ratings to 6-n-propylthiouracil. However, subjects who preferred higher levels of sucrose in the test stimuli tended to have higher dietary intakes of sugar and sweet foods.

Product maps which give a quick, visual summary of sensory evaluation data are introduced. These maps are associated with a partition of chi-squared non parametric analysis which often indicates the need for a two-dimensional view of consumer categorical data.

Analysis of two-way nominal-ordinal categorical data can be done via the parametric linear-logistic approach of McCullagh (1980) or by the non-parametric partition of chi-squared method given by Nair (1986). Monte Carlo simulations are used to compare test sizes and powers for these two approaches. These comparisons do not favour either approach over the other. The orthogonal analysis is an advantage of the Nair method.

SUMMARY For randomized block and 'balanced incomplete block designs, nonparametric, treatment comparisons are usually made using the usual Friedman and Durbin tests and the multiple-comparisons approach. This permits an assessment of only the mean rankings. Such comparisons are here extended to permit linear (mean) and quadratic assessments simul­ taneously by the use of product maps incorporating confidence circles. If the blocks are judges or consumers then differences in quadratic effects can indicate product differences in consensus or market segmentation. Preference ranking to compare products is well known. Statistical procedures are commonly used to verify that apparent differences between products are due to other than chance effects. Anderson (1959) suggested a statistical procedure for comparing t products which had been ranked by n judges or consumers. This procedure is closely allied to Pearson's classical x2-test of homogeneity for a tXt table. Anderson (1959) noted that his proposed test statistic could be partitioned to give more information about the product comparisons. Best (1993) discussed such a partitioning, ob­ taining Friedman's rank test for comparing treatment means and a new statistic for comparing quadratic or dispersion effects. Schach (1979) discussed the asymptotic distribution theory for Anderson's statistic and extended it to balanced incomplete-block designs as well as random­ ized block designs. In this paper we use the partitioning suggested by Best (1993) to produce product reference maps which gr,oUR or classify the products. Section 2 gives definitions, Section 3 considers the accuracy of the approximate confidence circles ~d Sections 4 and 5 look at typical appli­ cations.