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This study deals with the evaluation of inter-observer reliability (IOR) among three raters in the case of dichotomous and trichotomous individual animal-based welfare indicators. The performance of the most documented agreement indices proposed in the literature was compared, using udder asymmetry (UA) as a dichotomous indicator and body condition score (BCS) as a trichotomous indicator, both obtained from the AWIN Goat protocol. Nine dairy goat farms, exploiting three alpine pastures (AP1 to AP3), were used for data collection. Krippendorff’s α, the agreement indices belonging to the Kappa statistic and their weighted forms were in some cases affected by the paradox behaviour. This phenomenon was observed for both UA and BCS [e.g., P0(BCS-AP2) = 80%; Fleiss’ K = 0.22]. In the case of UA, Gwet’s γ(AC1), followed by BP coefficient and Quatto’s S, gave the best agreement results [e.g., P0(UA-AP1) = 86%; γ(AC1) = 0.84]. In the case of BCS, the best agreement results were obtained with Gwet’s γ(AC2), followed by the weighted forms of BP and S. When the evaluation is performed by three raters, γ(AC1), BP and S are suggested to evaluate IOR in the case of both dichotomous and trichotomous indicators, while the related weighted forms are suitable for trichotomous indicators only.

This study focuses on assessing inter-observer reliability (IOR) between two observers in the case of trichotomous and four-level animal-based welfare indicators assessed at individual level. The Body Condition Score (BCS) and Knee calluses (KNC) were chosen as trichotomous indicators; data were collected in fourteen intensively managed dairy goat farms in Italy (ITF1 to ITF7) and Portugal (PTF1 to PTF7) and in extensively managed dairy goat farms exploiting three alpine pastures (AP1, AP2 and AP3) in Italy. The Ear posture (EP) and Eye white (EW) were chosen as four-level indicators; data were collected in three intensively managed dairy cattle farms (F1, F2 and F3) in Italy. The performance of the most documented agreement indices was compared. In the case of trichotomous indicators, Scott’s π, Cohen’s K, Cohen’s KC, Cohen’s weighted K and Krippendorff’s α were affected by the paradox effect: when the concordance rate (P0) was high, they sometimes gave very low or even negative values (e.g. P0(BCS-ITF3) = 74%; Scott’s π = 0.05; Cohen’s K = 0.09; Krippendorff’s α = 0.06; P0(BCS-AP3) = 74%; Scott’s π = −0.12; Cohen’s K = Krippendorff’s α = −0.11). Bangdiwala’s B, Gwet’s γ(AC1) and Quatto’s weighted S were not affected by this phenomenon and provided values very close to P0 (e.g. P0(KNC-PTF1) = 88%; Bangdiwala’s B = Gwet’s γ(AC1) = 0.85; P0(BCS-AP1) = 82%; Bangdiwala’s B = Gwet’s γ(AC1) = 0.79). In the case of four-level indicators, Cohen’s K and Krippendorff’s α were not affected by the paradox behaviour. However, Cohen’s KC in some cases exceeded the observed P0 (e.g. P0(EP-F3) = 78%; Cohen’s KC = 1). Gwet’s γ(AC1) showed the best results for four-level indicators (e.g. P0(EP-F1) = 88%; Gwet’s γ(AC1) = 0.86), followed by Quatto’s S and Holley and Guilford’s G (e.g. P0(EP-F1) = 88%; Quatto’s S = Holley and Guilford’s G = 0.84). To evaluate IOR between two observers, Bangdiwala’s B, Gwet’s γ(AC1) and Quatto’s weighted S are suggested for trichotomous indicators, while Gwet’s γ(AC1), Quatto’s S and Holley and Guilford’s G are suggested for four-level indicators.

The Fleiss’ kappa statistic is a well-known index for assessing the reliability of agreement between raters. It is used both in the psychological and in the psychiatric field. Unfortunately, the kappa statistic may behave inconsistently in case of strong agreement between raters, since this index assumes lower values than it would have been expected. The aim of this paper is to propose a new method to avoid this paradox through permutation techniques. Furthermore, we study the problem of kappa confidence intervals and, in particular, we suggest to use Bootstrap confidence intervals free of paradoxes.

We describe the long-term (1981–2008) dynamics of several physico-chemical and biological variables and how their changes may have influenced zooplankton structure in Lake Maggiore (Italy). Data was available for the 1981–1992 and 1995–2008 periods. Standardized time-series for temperature and total phosphorus (TP), chlorophyll-a, phytoplankton density (cel m−3), and cell size (µm3), as well as zooplankton structure (Copepoda, Cladocera, and Rotifera density, ind m−3) were smoothed using penalized B-splines and analyzed using Functional Principal Components (FPCs) to assess their dominant modes of variation. The first four FPCs explained 55% of 1981–1992 and 65% of 1995–2008 overall variation. Results showed that temperature fluctuated during the study period, particularly during 1988–1992 with a general tendency to increase. TP showed a declining trend with some reversions in the pattern observed in the years 1992, 1999, and 2000. Phytoplankton estimators and chlorophyll-a concentration showed a variable trend along the study period. Zooplankton groups also had a variable trend along the study period with a general increase in density of large carnivorous (mainly Bythotrephes longimanus) and a decrease of large herbivorous (mainly Daphnia), and a similar increase in the ratio of raptorial to microphagous rotifers. Our results suggest that the lake experienced a strong trophic change associated with oligotrophication, followed by pronounced climate-induced changes during the latter period. TP concentration was strongly associated with changes in abundance of some zooplankton taxa.

This study focuses on the problem of assessing inter-observer reliability (IOR) in the case of dichotomous categorical animal-based welfare indicators and the presence of two observers. Based on observations obtained from Animal Welfare Indicators (AWIN) project surveys conducted on nine dairy goat farms, and using udder asymmetry as an indicator, we compared the performance of the most popular agreement indexes available in the literature: Scott’s π, Cohen’s k, kPABAK, Holsti’s H, Krippendorff’s α, Hubert’s Γ, Janson and Vegelius’ J, Bangdiwala’s B, Andrés and Marzo’s ∆, and Gwet’s γ(AC1). Confidence intervals were calculated using closed formulas of variance estimates for π, k, kPABAK, H, α, Γ, J, ∆, and γ(AC1), while the bootstrap and exact bootstrap methods were used for all the indexes. All the indexes and closed formulas of variance estimates were calculated using Microsoft Excel. The bootstrap method was performed with R software, while the exact bootstrap method was performed with SAS software. k, π, and α exhibited a paradoxical behavior, showing unacceptably low values even in the presence of very high concordance rates. B and γ(AC1) showed values very close to the concordance rate, independently of its value. Both bootstrap and exact bootstrap methods turned out to be simpler compared to the implementation of closed variance formulas and provided effective confidence intervals for all the considered indexes. The best approach for measuring IOR in these cases is the use of B or γ(AC1), with bootstrap or exact bootstrap methods for confidence interval calculation.

In this paper we consider a controversy on the use and interpretation of p-values in applied research. In recent years several applied and theoretical journals have started to discuss on the appropriate use of p-values in research fields such as Psychology, Ecology, and Medicine. First, the notion of p-value has some intrinsic limitations, which have been already highlighted in the statistical literature, but are far from being recognized in applied research. Second, it has emerged the so-called practice of p-hacking, which consists in analyzing and re-analyzing data until obtaining a significant result in terms of a p-value less than 0.05. In the light of these problems, we review two alternative theoretical frameworks, given by the use of Bayes factor and a recent proposal that leads to evaluate statistical hypotheses in terms of a priori and a posteriori odds ratios.

The 2 × 2 table is the simplest of data structures yet it is of immense practical importance. It is also just complex enough to provide a theoretical testing ground for general frequentist methods. Yet after 70 years of debate, its correct analysis is still not settled. Rather than recount the entire history, our review is motivated by contemporary developments in likelihood and testing theory as well as computational advances. We will look at both conditional and unconditional tests. Within the conditional framework, we explain the relationship of Fisher's test with variants such as mid-p and Liebermeister's test, as well as modern developments in likelihood theory, such as p* and approximate conditioning. Within an unconditional framework, we consider four modern methods of correcting approximate tests to properly control size by accounting for the unknown value of the nuisance parameter: maximisation (M), partial maximisation (B), estimation (E) and estimation followed by maximisation (E + M). Under the conditional model, we recommend Fisher's test. For the unconditional model, amongst standard approximate methods, Liebermeister's tests come closest to controlling size. However, our best recommendation is the E procedure applied to the signed root likelihood statistic, as this performs very well in terms of size and power and is easily computed. We support our assertions with a numerical study.

Fuzzy sets represent an extension of the concept of set, used to mathematically model veiled and indefinite concepts, such as those of youth, poverty, customer satisfaction and so on. Fuzzy theory introduces a membership function, expressing the degree of membership of the elements to a set. Intuitionistic fuzzy sets and hesitant fuzzy sets are two extensions of the theory of fuzzy sets, in which non-membership degrees and hesitations expressed by a set of experts are, respectively, introduced. In this paper, we apply intuitionistic fuzzy sets to questionnaire analysis, with a focus on the construction of membership, non-membership and uncertainty functions. We also suggest the possibility of considering intuitionistic hesitant fuzzy sets as a valuable theoretical framework. We apply these models to the evaluation of a Public Administration and we assess our results through a sensitivity analysis.

Line transect sampling is a distance sampling method widely used for estimating wildlife population density. Since the usual approach assumes a model for the detection function, the estimate depends on the shape of such a function. In particular, the estimate is influenced by the so-called shoulder condition, which ensures that detection is nearly certain at small distances from the line transect. For instance, the half-normal model satisfies this condition, whereas the negative exponential model does not. The aim of this paper is to propose the exponential mixture model of the half-normal and the negative exponential in order to estimate the population density in the case where the shoulder condition is not guaranteed. Such a case study on Hooded crow is described in the paper.