Ordinal Quantity Research Articles

The Issue of Using Ordinal Quantities to Estimate the Vulnerability of Seabirds to Wind Farms

The article is a follow-up study to research on analysing methodological approaches to estimate anthropogenic impacts on marine biota. The work examines relevant publications about the vulnerability of seabirds to the effects of offshore wind farms, relying upon the provisions of the measurement theory and the median theorem for ordinal quantities. It has been shown that the final ranking of sensitivity of certain bird species to wind farms, or indices of a risk of collisions with turbine blades and indices of a risk of birds’ displacement by wind farms, can vary at permissible monotonic transformations of the values of initial factors since all of these are estimated on ordinal scales. We conclude such estimates are incorrect. The summation of the two indices (exposure to collisions with turbine blades and to habitat change) of the birds’ species vulnerability in the context of the proposed models is incorrect. It has been demonstrated that the model based on dividing factors into primary and aggravation, when the latter are incorporated into the formula of computation through the exponent of primary factors, is incorrect: when primary factors have maximum values (equal to 1), the effect of aggravation factors is no longer taken into account; with some values of factors in the model, infinite vulnerability values can be obtained. The models are to operate within the entire presented range of initial data, but that is not fulfilled. Thus, none of the analysed models are correct, and an approach is needed, based on the use of metric values. Considering that the bird flight altitude primarily determines the impact of wind farms on birds, the proposals have been formulated to take into account the impact of various factors on the flight altitude.

What we count dictates how we count: A tale of two encodings

We argue that what we count has a crucial impact on how we count, to the extent that even adults may have difficulty using elementary mathematical notions in concrete situations. Specifically, we investigate how the use of certain types of quantities (durations, heights, number of floors) may emphasize the ordinality of the numbers featured in a problem, whereas other quantities (collections, weights, prices) may emphasize the cardinality of the depicted numerical situations. We suggest that this distinction leads to the construction of one of two possible encodings, either a cardinal or an ordinal representation. This difference should, in turn, constrain the way we approach problems, influencing our mathematical reasoning in multiple activities. This hypothesis is tested in six experiments (N = 916), using different versions of multiple-strategy arithmetic word problems. We show that the distinction between cardinal and ordinal quantities predicts problem sorting (Experiment 1), perception of similarity between problems (Experiment 2), direct problem comparison (Experiment 3), choice of a solving algorithm (Experiment 4), problem solvability estimation (Experiment 5) and solution validity assessment (Experiment 6). The results provide converging clues shedding light into the fundamental importance of the cardinal versus ordinal distinction on adults' reasoning about numerical situations. Overall, we report multiple evidence that general, non-mathematical knowledge associated with the use of different quantities shapes adults' encoding, recoding and solving of mathematical word problems. The implications regarding mathematical cognition and theories of arithmetic problem solving are discussed.

The Issue of Using Ordinal Quantities to Estimate the Vulnerability of Seabirds to Oil Spills

Oil spills can have a serious negative effect on seabirds. Numerous studies have been carried out for relative vulnerability assessment of seabirds to oil, with the majority of such works based on ordinal quantities. This study aims to assess (from the aspect of measurement theory) the methodological approaches used for calculating the vulnerability of seabirds to oil spills, and corresponding conclusions. We assess several well-known works on the vulnerability of seabirds (1979–2004). We consider the effect on derived conclusions of (a) monotonic initial data transformations on an ordinal scale, (b) multiplication operations on the same scale, and (c) the replacement of initial metric data to ordinal. Our results show the following: (a) the conclusions for arithmetic operations may not be saved with permissible monotonic transformations of ordinal quantities; (b) partially uncertain results can be obtained with arithmetic operations on an ordinal scale as compared with metric; (c) the replacement of metric values to scores changes the real relationships among initial data and affects the final result. Thus, conclusions in works which use arithmetic operations with ordinal quantities cannot be considered to be justified and correct, since they are based on unacceptable operations and, quite often, on the distorted original data.

Heritability estimation of osteoarthritis in the pig-tailed macaque (Macaca nemestrina) with a look toward future data collection.

We examine heritability estimation of an ordinal trait for osteoarthritis, using a population of pig-tailed macaques from the Washington National Primate Research Center (WaNPRC). This estimation is non-trivial, as the data consist of ordinal measurements on 16 intervertebral spaces throughout each macaque’s spinal cord, with many missing values. We examine the resulting heritability estimates from different model choices, and also perform a simulation study to compare the performance of heritability estimation with these different models under specific known parameter values. Under both the real data analysis and the simulation study, we find that heritability estimates from an assumption of normality of the trait differ greatly from those of ordered probit regression, which considers the ordinality of the trait. This finding indicates that some caution should be observed regarding model selection when estimating heritability of an ordinal quantity. Furthermore, we find evidence that our real data have little information for valid heritability estimation under ordered probit regression. We thus conclude with an exploration of sample size requirements for heritability estimation under this model. For an ordinal trait, an incorrect assumption of normality can lead to severely biased heritability estimation. Sample size requirements for heritability estimation of an ordinal trait under the threshold model depends on the pedigree structure, trait distribution and the degree of relatedness between each phenotyped individual. Our sample of 173 monkeys did not have enough information from which to estimate heritability, but estimable heritability can be obtained with as few as 180 related individuals under certain scenarios examined here.

Validity Concerns with Multiplying Ordinal Items Defined by Binned Counts: An Application to a Quantity-Frequency Measure of Alcohol Use.

Social and behavioral scientists often measure constructs that are truly discrete counts by collapsing (or binning) the counts into a smaller number of ordinal responses. While prior quantitative research has identified a series of concerns with similar binning procedures, there has been a lack of study on the consequences of multiplying these ordinal items to create a desired index. This measurement strategy is incorporated in many research applications, but it is particularly salient in the study of substance use where the product of ordinal quantity (number of drinks) and frequency (number of days) items is used to create an index of total consumption. In the current study, we demonstrate both analytically and empirically that this multiplicative procedure can introduce serious threats to construct validity. These threats, in turn, directly impact the ability to accurately measure alcohol consumption.

Concept system on ‘quantity’: formation and terminology

The third edition of the International Vocabulary of Metrology occasions continued useful discussion. A Commentator, in four recent papers, presents viewpoints on the generic division of ‘property’, ‘quantity’, and ‘ordinal quantity’, as well as a perceived problem of inheritance. Alternative terms for ‘kind-of-quantity’, ‘kind-of-quantity with chosen sort of component’, and ‘dedicated kind-of-quantity’ are suggested to be formed by prefacing “quantity” with the respective adjectival modifiers “generic”, “subgeneric”, and “specific”. Furthermore, a neoterm for ‘quantity expressed by a measurement unit’ and a redefinition of ‘property’ are offered. The Commentator’s proposals and arguments are discussed with emphasis on ways of constructing concept systems to ensure proper inheritance. Some formal and semantic problems of the proposed alternative terms and definitions are advanced. Terminological concepts are defined in an Appendix.

Ordinal quantities in clinical laboratory sciences

From an ontological point of view, the concept ‘property’ may be defined as “that which when possessed by an object contributes to it being as it is”. Ordinal quantity is one of the types into which properties may be divided. Ordinal quantity is understood internationally as a “quantity defined by a conventional measurement procedure, for which a total ordering relation can be established, according to magnitude, with other quantities of the same kind, but for which no algebraic operations among those quantities exist”. The term ordinal quantity may be preceded by an adjective (generic, subgeneric, specific, or individual) to clarify the relation object–property. Ordinal quantities may have values belonging to binary ordinal scales and polinay ordinal scales.

Generic division of ‘quantity’ and related terms

In the slow evolution of the International vocabulary of metrology (VIM), the first concept of ‘quantity’ has now been divided generically into ‘ordinal quantity’ and a coordinate primitive without definition and term. An analysis of the concepts by their characteristics is made, and the nature of inheritance is discussed in response to a recent communication in this journal. A completion of the initiated generic division of ‘quantity’ is suggested, and a neoterm for the sister of ‘ordinal quantity’ is offered on the basis of two proposals.

Scalar and ordinal quantities

The last edition of the International Vocabulary of Metrology (thereafter VIM3) [1] includes the concept ‘ordinal quantity’ and its definition. In the VIM3, the concepts ‘quantity’ and ‘ordinal quantity’ are in generic relation. As ‘quantity’ is a superordinate concept, all its characteristics would also belong to its subordinate concept ‘ordinal quantity’. However, this is not the reality: there are many characteristics that are consequence of the algebraic operations (e.g. mean and standard deviation calculations) that are not applicable to ordinal quantities. On the contrary, all these peculiarities are applicable to the rest of (non-termed) quantities (i.e. quantities the values of which need to be expressed by way of a measurement unit). On the other hand, but stressing the above comment, in the concept diagram given in Fig. A.1 of the VIM3, the concept ‘ordinal quantity’ (see entry 1.26 in [1]) and the (neither termed nor numbered) concept explained by the sentence ‘quantity expressed by a measurement unit’ are both subordinated to the concept ‘quantity’. One logical derivation from the VIM3 is the necessity of a term to designate the non-defined concept mentioned in the previous paragraph. Among all the different terms I taken into account, I think the best term is ‘scalar quantity’. The VIM3 recognizes the existence of scalar, vector, and tensor quantities (though vector and tensor quantities are out the scope of VIM3) in a note to the definition of ‘quantity’ (see entry 1.1 in [1]) and explicitly uses ‘scalar quantity’ to define ‘measurement unit’ (see entry 1.9 in [1]). In addition, the term ‘scalar quantity’, denoting a quantity which is completely specified by magnitude only, appears in a plethora of scientific and technological books (see http://books.google.com/books at the Internet). The term ‘unitary quantity’ proposed elsewhere [2] is not a good choice, because the general language meaning of the adjective ‘unitary’ can cause confusion. So, let me propose the concept, and the corresponding term scalar quantity as a complement to ordinal quantity, reserving quantity in case a broader concept is needed.

An algorithm for generating color scales for both categorical and ordinal coding

AbstractPrevious research has shown multihue scales to be well‐suited to code categorical features and shown lightness scales to be well‐suited to code ordinal quantities. We introduce an algorithm, Motley, that produces color scales varying in both hue and lightness, intended to be effective for both categorical and ordinal coding, allowing users to determine both absolute and relative quantities efficiently and accurately. The algorithm first determines the lightnesses of scale colors to maximize perceived lightness differences and establish the lightness ordering, generating separate search spaces for each scale position. It then selects hues by heuristic search to maximize the discriminability of the scale. It produces scales that are ordered with respect to lightness but unordered with respect to hue and thus more discriminable than typical multihue lightness scales. In an experimental evaluation on human subjects, Motley's scales enabled accurate judgments of relative quantity, with response times superior to unordered multihue scales and comparable to ordered lightness scales, and enabled accuracy and speed of judgments of absolute quantity superior to lightness scales and comparable to multihue scales. Published 2009 Wiley Periodicals, Inc., Col Res Appl, 2010.

On imputing continuous data when the eventual interest pertains to ordinalized outcomes via threshold concept

Multiple imputation under the multivariate normality assumption has often been considered a workable model-based approach in dealing with incomplete continuous data. A situation where the measurements are taken on a continuous scale with an eventual interest in ordinalized versions via threshold concept is commonly encountered in applied research, especially in medical and social sciences. In practice, researchers ordinarily impute missing values for continuous outcomes under a Gaussian imputation model, and then ordinalize them via pre-specified cutoff points. An alternate strategy is creating multiply imputed data sets after ordinalization under a log-linear imputation model that uses a saturated multinomial structure. In this work, the performances of the two imputation methods were examined on a fairly broad range of simulated incomplete data sets that exhibit varying distributional characteristics such as skewness and multimodality. Behavior of efficiency and accuracy measures was investigated to determine the degree to which the procedures work appositely. The conclusion drawn is that ordinalization before carrying out a log-linear imputation should be the preferred procedure except for a few special cases. It is recommended that researchers use the less common second strategy whenever the interest centers on ordinal quantities that are obtained through underlying continuous measurements. This postulate is probably due to the transformation of non-Gaussian features into better-behaving categorical trends in this particular missing-data environment. This premise preponderates the factual argument that continuous variables intrinsically convey more information, leading to a counter-intuitive, but potentially beneficial result for practitioners.

Modelling neuromotor ratings with floor‐effects

Associated movements (AMs) are a classical diagnostic tool to assess differences between normal children and children with some motor dysfunction. This paper presents a methodology to produce age- and gender-dependent reference-curves for AMs of normal children, for various tasks of a test battery. Data available consist of separate ratings of duration and extent of AMs, which are ordinal quantities with few levels. Other problems are severe age- and gender-dependent floor-effects (as well as some ceiling-effects), leaving little information for analysis at older ages. To get a better scale, we combined the two ordinal ratings into one meaningful and quasi-continuous quantity referred to as intensity of AMs. In order to solve problems due to floor-effects, ceiling-effects and discreteness, we assumed left- , right- and interval-censored values, respectively. We considered a censored regression problem and postulated a truncated normal distribution for the non-censored values (after an appropriate transformation of the data). Using Wei and Tanner's poor man's data augmentation algorithm, together with the technique of linear mixed effects modelling, useful reference-curves could be produced. In contrast to the cumulative probabilities approach for ordinal data, our methodology allows the calculation of individual age- and gender-standardized values, which puts us in a position to investigate numerous scientific questions.

The development of rulebound strategies for manipulating seriated cups: A parallel between action and grammar

Systematic observation of American children from 11 to 36 months of age playing with seriated nesting cups established the existence of a developmental sequence of rulebound or consistent strategies for combining the cups. The three action strategies seem formally homologous to certain grammatical constructions, and the manipulative strategies are acquired in the same developmental order as the corresponding grammatical structures. These strategies might therefore constitute a manifestation of some underlying structural capacities critical for language acquisition. The development of seriation described by Piaget for children from 4 to 8 years of age working with other materials was replicated in this much younger age range using nesting cups. These stages in the growth of a concept of ordinal quantity turn out to be empirically and theoretically related to the rulebound strategies upon which the study focused.