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

Abstract

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.

1. Introduction

To protect the marine environment, it is extremely important to assess the possible impact of various anthropogenic factors on the marine biota, and oil spills are perhaps the most important aspect here. Various scientific journals offer numerous publications on the issue; the vital components are the development of vulnerability maps for coastal areas for environmental purposes, oil spill response (OSR) plans, and OSR operations [1,2,3,4], as well as vulnerability maps for other liquid chemicals [5,6]. In this study, we consider the vulnerability of marine avifauna (which is an important and one of the most vulnerable components of the marine ecosystem) to oil spills.

In addition to oil, wind farms can have a serious impact on marine biota, because their construction on the shelf may become the most extensive marine engineering project in Europe. Such projects have raised widespread discussion in connection with possible harmful impacts on the environment and, in particular, on seabirds. There are various publications on this issue, as well as on the effect of oil on avifauna [7,8,9,10,11,12], and the approaches to assessing the impact of oil and wind farms on avifauna are largely similar. In [13], the methodology for calculating vulnerability from wind farms was applied with some inevitable changes to assess the impact of oil spills on birds.

When calculating the estimates of the impact of oil spills or offshore wind turbines on avifauna, various specialists often use both values measured on metric (quantitative) scales and those measured on other scales, including ordinal (rank) quantities. When constructing vulnerability maps of coastal-marine zones (taking into account the impact of oil on the avifauna), ordinal quantities (ranks, scores) are used in numerous works [5,6,14,15,16,17,18]. Publications devoted to the effect of wind farms on birds, have followed a similar rank-based approach [7,8,9,10,11,12,19]. The key questions here are whether ordinal quantities are admissible in such calculations, and whether the results and conclusions are fully reliable.

Any calculations should take into account the scale on which the initial data were obtained, and whether it is possible to use this data to calculate the impact according to the accepted mathematical models (formulas). However, it is not always possible to use strictly quantitative measurements of individual parameters in order to perform the necessary observations of the environmental consequences of anthropogenic impacts or, in general, in order to assess the environmental impact (EIA). Sociology, psychology, and other social sciences face the same problem, since measurements in the generally accepted sense are impossible, and a qualitative approach is often used, i.e., the necessary parameters are estimated on conditionally quantitative scales, that is, the initial data are expertly evaluated and ranked. Thereby, measurements are obtained on an ordinal scale, and these values are usually denoted by natural numbers (1, 2, 3, …). Various arithmetic operations are carried out afterwards, i.e., the necessary estimates and statistical criteria are calculated. Nevertheless, whether any operations with data measured on a particular scale are permissible depends on the type of scale, on the calculated values, and on the applied algorithms, taking into account the permissible operations and transformations on the corresponding scale [20,21,22,23,24,25,26,27]. In this study, we are more interested in arithmetic operations with values measured on two types of scales (ordinal and ratio), however, to draw a more complete picture, we also provide a brief description of other types of scales.

In this study, we aim to assess (from the metrological point of view, including taking into account the statements of measurement theory) the methodological approaches used for calculating the vulnerability of seabirds to oil spills, the conclusions obtained, and, very briefly, the corresponding vulnerability maps calculated in some studies, and we aim to formulate possible recommendations for further research.

Briefly, we describe measurement theory on various scales (classification of scales, permissible transformations, operations on them, etc.); more details are provided in Appendix A, Appendix B and Appendix C. Earlier we have used basic provisions of this theory to analyze application of ordinal quantities in calculations of vulnerability of biota and sea-coastal zones to oil [28,29,30,31,32]. Briefly, general approaches to operations on different scales have been previously described in [29]. In [30], vulnerability map development of sea-coastal zones to oil were assessed considering only unacceptability of arithmetic operations with ordinal quantities. The method of vulnerability assessment of some biota groups and development of vulnerability maps to oil based on metric approach are given in [28,31,32]. Now, we focus, based on the provisions of measurement theory, on the assessment of earlier studies of birds’ vulnerability to oil [33,34,35,36,37], and briefly assess the development of vulnerability maps [14]. These publications [33,34,35,36,37] were chosen since they are frequently referred to in various summarizing works [7,9,13]. Additionally, for the aims of our work, it was also important to select publications which provided the initial data in detail (this allowed replicated calculations) and to assess the corresponding approaches to avifauna vulnerability for the development of oil vulnerability maps. While the main focus of our article is the actual calculations with ordinal values, we briefly explore how the application of resultant vulnerability scores could lead to erroneous applications in oil vulnerability mapping.

It becomes clear that in calculations of biota vulnerability to oil (as in any other calculations) it is necessary to use metric values instead ordinal quantities. Furthermore, there is also a need to develop alternative approaches such as the use of ordinal quantities on the ratio scale [38,39]. In this case, it is necessary to take into account the type of data scale, since, even on some metric scales, not all arithmetic operations are permissible. One of the possible options for such an approach for calculating oil vulnerability maps, based strictly on metric values, is described in [32]. All issues raised in our article relate to the calculations of seabirds’ vulnerability, which has implications for the development of oil vulnerability maps in any sea region.

2. Arithmetic Operations with Values on Different Scales: Elements of Measurement Theory

2.1. Arithmetic Operations with Values of Various Kinds

In practice, arithmetic operations with certain numbers do not always make sense Their mere relevance to socially relevant applications does not imply that these numbers can always be added and multiplied, or that other arithmetic operations can be performed with them.

The usual arithmetic relations are not always adequate. The opinions of experts are often expressed in an ordinal scale, that is, an expert can say (and justify) that the first influence factor is more dangerous than the second, etc., yet they are not able to say how many times more it is dangerous. Although in arithmetic 1 + 2 = 3, it cannot be argued that, for an object occupying the third place in the ordering, the intensity of the studied characteristic is equal to the sum of the intensities of objects with ranks 1 and 2 (see also Appendix B and Appendix C). A rank is a “number” in an ordered series of characteristic values for various objects; in statistics, such series are called variational. Formally, the ranks are expressed by the numbers 1, 2, 3, …, but one cannot perform the usual arithmetic with them.

In addition, it cannot be said that a body with the temperature of 40 °C is twice as warm as one with the temperature of 20 °C, although this is already a strictly quantitative (metric), and not an ordinal scale. Not all arithmetic operations are possible with all the quantities represented by numbers; thus, to analyze this kind of data, we need another theory which provides the basis for the development, study, and application of specific calculation methods, i.e., measurement theory. Initially, it is necessary to take into account the type of scale used to obtain these data [23].

2.2. Measuring the Quantities: Scales of Measurement

The assessment of anthropogenic impact on birds and the subsequent calculation of vulnerability maps mean, to a certain extent, constructing a reality model based on the results of various measurements (direct measurements, estimates based on observations, and expert estimates). Moreover, the measurement results, as a rule, are obtained on different scales.

In the early 1940s, Harvard psychologist S.S. Stevens introduced the terms nominal scale, ordinal scale, interval scale, and ratio scale to describe the hierarchy of measuring scales used in psychology. In his fundamental work “On the Theory of Scales of Measurement” [20], he presented a hierarchy of data scales based on the invariance of their values for various classes of transformations. Measurement in the broadest sense is the attribution of numerical forms to objects or events in accordance with certain rules, and the fact that numerical forms can be attributed to objects in accordance with different rules leads to the use of different scales and different types of measurements [21].

Measurement theory is the subject of different publications worldwide, for instance [20,21,23,24,25,40,41,42]; these works describe the general theory of measurements on different types of scales in sufficient detail.

Notably, there has been criticism of Stevens’s typology of scales. Velleman, Wilkinson [41] wrote that Stevens in his article “Mathematics, Measurement and Psychophysics” [21] went beyond the limits of his elementary typology and classified simple operations, as well as statistical procedures, from the point of view of their ”admissibility” for one or another scale. In their opinion, the application of measurement theory “when choosing or for recommending certain methods of statistical analysis is inappropriate and often leads to errors”; a detailed analysis of their arguments and approach is presented in [43]. The discussion on this issue continues, but the typology of the scales as a whole is not questioned and is currently universally recognized. L. Finkelstein, Professor at the University of London, in his opening speech at the congress of the International Measurement Confederation in 1973 called the scale theory ”a solid logical basis” for constructing the measurement theory [44]. Classification of scales and permissible operations on them were included in the Russian six-volume Physical Encyclopedia [45], in metrology textbooks, and in various regulatory documents (see below).

The first classification of scales by Stevens does not fundamentally differ from the modern generally accepted classification, although there are several similar classifications. Let us very briefly describe the main measurement scales.

Currently, scales are usually grouped into nominal (for quality measurements); ordinal (to reflect the relationship of order (bigger, better, more important, etc.)); and quantitative (based on the usual arithmetic operations, for example, 10 is two times more than five). Sometimes all the measurement scales are divided into the following two classes: scales of qualitative attributes or non-metric scales (ordinal scale and nominal scale) and scales of quantitative attributes or metric (quantitative) scales.

The definitions of certain basic concepts of measurement theory (value, measurement scale, and measurement) from the International Vocabulary of Metrology [26] are given in Appendix A. The concept of measurement is the most important for the present study, and in [46] 23 different definitions are given (see also [20,21,23,24,25,38,40,41,42]). One can proceed from the following definition of this concept, which does not contradict the one given in [26], i.e., measurement is the construction of scales by isomorphic mapping of an empirical system with ratios into a numerical system with ratios. Formally, a scale is a triple that consists of a set of x_i elements, a binary operation ”O” on x_i elements, and a transformation of φ (for x_i) into real numbers [38].

2.3. Methods for Obtaining Measurement Information

Obtaining measurement information is possible in one way, i.e., by comparing the properties (values) of the measured objects. Leonhard Euler, renowned Swiss, German, and Russian mathematician and mechanic, wrote: “It is impossible to determine or measure one value otherwise than by accepting another value of the same kind as a known and indicating the relation between them.” All the cases of comparison of two values of Q₁ and Q₂ are reduced to three options [47]:

Q_i > Q_j, Q_i < Q_j,

(1)

Q_i − Q_j = ΔQ_ij = q_ij[Q],

(2)

Q_i/Q_j = q_ij.

(3)

The first option, Option (1), is the simplest and least informative. An experimental solution to the inequality answers the question of which of the two is larger than the other, however, it does not state how much exactly one is larger than the other, or how many times. A more informative comparison is Option (2), since it does answer the question ”how much”, however, it is still impossible to answer ”how many times”, as in Option (1). To answer this question, Option (3) is needed, since it can determine how many times Size Q_j fits into Size Q_i. This means that Q_j acts as a unit of measure, and certain requirements are imposed on units of measure. Thus, Option (3) is the most informative.

2.4. Types of Measurement Scales and Their Main Characteristics

The scale is best represented in terms of a class of transformations that preserve the information contained in it [23,38]. The type of scale defines a class of permissible transformations of the scale that do not change the objectively existing ratios between measured objects [23]. The opposite is also true, i.e., permissible transformations determine the type of scale. Let us give the classification, features and characteristics of the main types of scales that currently exist, using the data from several publications [21,22,23,27,38,45,46]. Appendix B provides additional information on the subject.

Nominal scales reflect quality properties. Their elements are characterized only by the relations of equivalence (equality), differences, and similarities of specific qualitative manifestations of properties. All mutually unique transformations are admissible on them. These scales do not allow introducing the concepts of the unit of measurement, and therefore of dimension, and they lack the zero element. However, some statistical operations are possible when processing the measurement results in these scales, for example, one can find the modality class, or the most numerous equivalence class, based on the measurement results.

Ordinal scales describe properties for which the equivalence and also the relations in increasing or decreasing of the quantitative manifestation of the property are meaningful in accordance with Option (1) of comparing the values. These scales also do not allow introducing units of measurement, since they are fundamentally nonlinear, i.e., it is logically impossible to establish the equality of intervals in different parts of the scale. The measurement results in such scales are expressed in numbers, scores, degrees, levels, and not in units of measurements. Although the results of measurements on such scales are often indicated by continuous sets of real arithmetic numbers, it is impossible to imply the proportionality of these values (it is logically impossible to determine how many times one implementation of a property is more or less than another). Measurement results in scores, degrees, and levels are often expressed by discrete rows of natural numbers. Ordinal scales allow monotonic transformations; zero of the scale can be present in them [27].

In the international dictionary of metrology, the definition of ordinal quantity is stated as followed: ”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. Ordinal quantities can enter into empirical relations only and do not have measurement units or quantity dimensions. Differences and ratios of ordinal quantities have no physical meaning” [26] (item 1.26) (full definition is given in Appendix A).

Difference scales (interval scales) differ from ordinal scales, i.e., for the properties they describe the equivalence and order relations make sense, as well as the equality and summation of the intervals (differences) between different quantitative manifestations of the properties. Interval scales with size Q are described by the following equation:

Q_i = Q_j + q_ij [Q],

(4)

where Q_i is the value of the physical quantity, Q_j is the origin, q_ij is the numerical value of the interval of the physical quantity, and [Q] is the unit of measurement of the physical quantity in question. As follows from Option (4), the interval scale is completely determined by setting the origin Q_j and unit of measurement [Q].

A typical example here is the scale of time intervals. Time intervals (for example, work periods or study periods) can be added and subtracted, but it is pointless to add up the dates of any events. Scales of this type also include practical temperature scales with a conditional zero (Celsius, Fahrenheit, Réaumur). Interval scales have conditional (accepted by agreement) units of measure and conditional zeros based on any benchmarks; in these scales, linear transformations are permissible.

Equivalence and order relations, as well as subtraction and multiplication operations (for ratio scales of the first and second type, see Appendix B) are applicable to the set of quantitative manifestations in ratio scales. From a formal point of view, ratio scales are interval scales with a natural origin, and are the most advanced measuring scales. Their equation is as follows:

Q = q[Q],

(5)

where Q is the value of the physical quantity, q is the numerical value of the physical quantity, [Q] is the unit of measurement of the physical quantity. Ratio scales have no natural unit of measure, yet they have conditional (accepted by agreement) units and natural zeros. These scales are widely used in physics and technology; all the arithmetic operations are allowed in them, except for summation in scales of the first type.

Permissible transformations, here, are similar transformations (changing only the scale), in other words, linear increasing transformations without a constant term [23]. A ratio scale can be converted to another by being multiplied by a positive constant (for example, kilometers into nautical miles by multiplying by a factor of 1.852, or y = a × x). Moreover, the ratio of two ordered observations is preserved, however, in an interval scale it changes with an allowable transformation (the Celsius scale has one ratio of two intervals, the Fahrenheit scale has another, although these are differences of two identical temperature states of matter); for more details, see Section Appendix B.3 of Appendix B.

Absolute scales have all the attributes of a relation and have a natural unambiguous definition of a unit of measure. Such scales are used to measure the following relative values (the relation of the same values): bird numbers, amplification, attenuation, reflection, and absorption coefficients, etc.

Thus, (taking into account the provisions of the modern measurement theory) it can be argued that it is not permissible to perform any arithmetic operations with all types of data, even on metric scales. Emphasizing again that no algebraic operations among ordinal quantities exist and differences and ratios of ordinal quantities have no physical meaning. Not all actions are permissible and on differences and intervals scales. In mathematical modeling of a real phenomenon or process, it is necessary to establish on which type of scale certain variables are measured. The type of scale defines a group of permissible scale transformations. The opposite is also true, i.e., the group of permissible transformations determines the type of scale.

We further consider several publications on assessing the vulnerability of birds to oil [33,34,35,36,37]. The methodology for calculating vulnerability maps for the Norwegian coastal-marine zones is also considered [14]. According to these works, we estimate the following:

• Whether the obtained conclusions remain invariant with one or another choice of values for ordinal quantities (otherwise, with permissible transformations of these quantities), if arithmetic operations are carried out;
• Whether or not the order is maintained during the multiplication of ordinal quantities, as compared with that for corresponding them metric values (actually often unknown);
• Whether or not replacement of real metric values of quantities (having different ranges of variability) with ordinal quantities with an equal range of variation affects the end result.

3. With Permissible Monotonic Transformations of Ordinal Quantities, Conclusions for Arithmetic Operations May Change

3.1. Calculation of Oil Vulnerability Index for Marine Oriented Birds

As noted in [7,9], the first publication that systematically addressed the vulnerability of seabirds to oil pollution was the work of King and Sanger [33]. To assess the impact of the oil and gas industry on seabirds, they classified 176 species of birds living in the northeastern Pacific Ocean, based on 20 factors that influenced their survival. For each bird species according to each of these factors, a score of 0, 1, 3, or 5 was assigned, representing, respectively, the absence, low, medium, or high significance of the corresponding species in its biology or habitat with respect to the oil industry. Moreover, the choice of four values of each factor is due to the simplification of work with the source data, due to the availability and accessibility of only low-quality information. Scores 0, 1, 3, 5 instead of 0, 1, 2, 3 allowed using the convenient 100 points instead of 60 as the maximum potential total number for any type of birds. In fact, it does not matter which values to use for the points in this case. The key thing is that the increasing number series correspond to the significance of the impact. The transition from the scale 0, 1, 3, 5 to the scale 0, 1, 2, 3 and vice versa corresponds to permissible (decreasing/increasing) transformations on the ordinal scale.

The scores for each of the 20 factors were summarized to obtain a common oil vulnerability index for each of 176 species and the arithmetic mean for families. The sum of scores for 22 families, including 128 species, ranged from 19 for the Marsh hawk (Circus cyaneus) to 88 (out of the possible 100 points) for the Kittlitz’s murrelet (Brachyramphus brevirostris) and the whiskered auklet (Aethia pygmaea), not counting the rare avian or most at-risk bird species. The authors suggest that if a species with a high vulnerability index is found in the area of the proposed oil development, then additional funding will probably be required for research, for modification of the project, for emergency plans for natural disasters, as well as for other environmental measures. In areas where species with low vulnerability scores are present, minimal precautions will be required. For the list of all 20 exposure factors, see [33] or [7].

For each bird species, the authors calculate the vulnerability (OVI, total points) for the family, the arithmetic mean (OVI Average), and the range (OVI Range) of the values of this average if the family has more than one species. On the basis of the data thus obtained, a comparison is made for two large regions (Alaska and the Aleutian Islands).

Let us evaluate whether the conclusions obtained by the authors are stable to admissible transformations on the ordinal scale used in the study, i.e., do these conclusions depend on whether, for example, scores 0, 1, 2, 3 or 0, 1, 3, 5 are selected? Given the characteristics of such a scale, conclusions should not depend on such monotonic transformations of the values of the quantities used. In the present study, we do not regard the issue of calculating the average for ordinal quantities, considering this a topic for a separate research. Let us note that, according to the median theorem, the average of the ordinal quantity is the median or another term of the variation series [48]. Below, is a conditional example similar to the calculations in [33] as follows:

Example 1.

Let the following expert opinions be obtained on the vulnerability of two bird species from 20 types of exposure, with each impact factor, as in the publication under discussion, rated from 0 to 5 (0, 1, 3, 5).

For conditional species A, the impact factors are estimated as follows: vulnerability score 0 is 13 factors, 1 is 2, 3 is 1, and 5 is 4 factors. Then, the sum of the scores is 0 × 13 + 1 × 2 + 3 × 1 + 5 × 4 = 25. For conditional species B, 0 is 0 factor, 1 is 19, 3 is 1, and 5 is 0; the total score will be 0 × 0 + 1 × 19 + 3 × 1 + 5 × 0 = 22. Species A is more vulnerable than B (total 25 > 22).

For the ordinal scale, all strictly monotonic (increasing or decreasing) transformations (see Section 2.3) are permissible, and the conclusions should not be changed after such transformations. With this in mind, let us move from the values 0, 1, 3, 5 to the values 0, 1, 2, 3. This is a strictly reducing transformation of the original values; then, for species A, 0 is 13 factors, 1 is 2, 2 is 1, and 3 is 4 factors. The sum of the scores is 0 × 13 + 1 × 2 + 2 × 1 + 3 × 4 = 16. For species B, 0 is 0 factor, 1 is 19, 2 is 1, and 3 is 0; the total score is 21. Species A is less vulnerable than B (16 < 21); this contradicts the conclusion obtained for scores 0, 1, 3, 5.

Let us compare the average values (OVI average) obtained by the authors for the Pandionidae family (one species in the family), i.e., OVI average = 37, and for the Corvidae family (two species), i.e., OVI average = 34. When changing from ordinal quantities 0, 1, 3, 5 to the values 0, 1, 2, 3, the values are no longer 37 and 34 but 27 and 27, that is, the two families become equally vulnerable on this new scale (it should be noted that although the authors do not make any comparisons for families, however, following their method, the values 37 and 34 also correspond to the equal vulnerability, i.e., they come in a range of 21 to 40 points). The performed OVI average recalculation for all the initial data presented in the article showed nine cases of changes in the relationships between vulnerability of all possible pairs of families, that is, the data showed instability of the sum of 20 factors of individual bird species and arithmetic mean values (OVI average) for families to monotonic data conversion.

When transforming the scale 0, 1, 3, 5 (range OVI 1–100, each subrange consists of 20 points) into the scale 0, 1, 2, 3 (range OVI 1–60, each subrange consists of 12 points), of the 128 species with vulnerability index from 19 to 88 listed in Table 1 from [33], the OVI vulnerability range (classes 1–5) changed in 45 species (35% of all species) in the direction of an increase in 1 class as compared with what is indicated in Tables 4 and 5 of these authors [33]. Among them, 15 species changed their vulnerability class from third (OVI 41–60) to fourth (OVI 37–48) and 26 species, from second (OVI 21–40) to third (OVI 25–36).

Thus, the permissible transformation (monotonic decrease of ordinal quantities used for calculations) changes the ratio of the order of the sum values of vulnerability factors for individual bird species (OVI) and the arithmetic mean values for families (OVI average), that is, the comparison operation for these quantities (the sum of the factors or their arithmetic mean) is unstable to an allowable transformation on an ordinal scale. This means that the calculated OVI does not fully reflect the real situation.

Appendix C additionally provides two examples with respect to summing without a permissible monotonic conversion of the source data on an ordinal scale. They confirm the conclusion formulated above. Even a simple summation for ordinal quantities is unacceptable, since “it is logically impossible to establish the equality of intervals in different parts of the scale” [27].

3.2. The Bird/Habitat Oil Index: A Habitat Vulnerability Index Based on Avian Utilization

To some extent, starting from [33] and modifying the approach adopted there, the authors [35] evaluated the potential effect of oil pollution on seabirds in each habitat in Puget Sound, Washington State [49,50]. They developed and described a relatively simpler method of calculating the bird oil index (BOI), which quantitatively took into account various aspects of the behavior, biology, distribution, and abundance of seabirds, as follows:

$$BOI=({{\displaystyle \sum }}_{i=1}^5\frac{X_i}{2}/2.5)({{\displaystyle \sum }}_{i=1}^5\frac{Y_i}{2}/2.5)({{\displaystyle \sum }}_{i=1}^4\frac{Z_i}{2}/2.0),$$

(6)

where $X_i$ is the first component (X_i, i = 1–5) which is comprised of five elements that describe the behavior of individuals of a species. Species with the highest vulnerability would have the following characteristics of X: roost on water, dive from danger, form large flocks on water, nest in large colonies, and have a highly specialized feeding niche.

$Y_i$ is the second component (Y_i, i = 1–5) which has five elements that describe the characteristics of the total population of a species. The most vulnerable species would have the following characteristics of Y: small population size, low annual reproductive potential, localized breeding distribution, concentrated winter distribution, and always in marine habitats.

$Z_i$ is the third component (Z_i, i = 1–5) has four elements that describe the species’ seasonal abundance in a region. The most vulnerable species would have a large proportion of its North American population in the region each season; the least vulnerable species would be one not present (0).

Here, as in [33], the assessment (ordering) of vulnerability was made using scores 0, 1, 3, 5 (the appearance of scores 2 and 4 in Table 3 from [35] is not entirely clear from the text of the article). All the operations, i.e., summation and multiplication, are carried out with ordinal quantities. As noted above (see Section 2.3), the conclusion based on the results of certain data operations should not change if a permissible transformation of the source data is carried out.

Example 2.

There are such values of X_i, Y_i, and Z_i that the transition from 0, 1, 3, 5 to 0, 1, 2, 3, during an operation, according to Formula (6) radically changes the picture, i.e., the sequence of BOI values on the numerical axis changes (see

Table 1. An example of calculating the bird oil index (BOI) of two conditional types AAA and BBB at different values of the source data (differing by a monotonic transformation). The upper two lines correspond to the values of the components 0, 1, 3, 5, the lower two lines correspond to the values of 0, 1, 2, 3.

Species	X1	X2	X3	X4	X5	Y1	Y2	Y3	Y4	Y5	Z1	Z2	Z3	Z4	BOI	Rank
AAA	5	0	3	5	5	1	0	0	0	5	3	3	0	1	60.5	2
BBB	1	0	1	1	1	1	5	1	1	1	3	1	5	5	40.3	1
AAA	3	0	2	3	3	1	0	0	0	3	2	2	0	1	17.6	1
BBB	1	0	1	1	1	1	3	1	1	1	2	1	3	3	20.2	2

). Species AAA, more vulnerable than Species BBB in the first case, becomes less vulnerable in the second case, that is, after a permissible transformation of the original ordinal variables (when switching to another value to indicate scores), their ranks change.

Therefore, a situation similar to [33] is quite possible. Among the entire array of BOI values calculated in [35], some of these vulnerability indices can change their values under the indicated permissible transformation, and also their order or rank (position on the ordinal scale) with respect to other BOI values, which can change the conclusions. Yet, such virtually inevitable change is hidden from the researcher and is not controlled by them.

Thus, a monotonic transformation permissible on an ordinal scale (that is, a decrease or increase in the initial data, i.e., ordinal quantities, when summing data groups with their subsequent multiplication) can also change the conclusions (in this case, BOI), i.e., a part of the final calculated values can change position on the ordinal scale relative to other values. This means that some calculated final values are incorrect (moreover, which BOI values change rank is unknown), and that the general conclusions are also incorrect, since it cannot be unambiguously said which of the calculated rank values should be used in the end and which should not.

3.3. General Remarks

A comparison of OVI values for individual bird species and their families made in [33] and a comparison of BOI values for birds in [35] cannot be considered strictly correct, since a not entirely correct methodological basis of calculations was used, i.e., vulnerability of the comparison of birds to the general anthropogenic impact of oil is made with summation (and arithmetic mean [33]) and addition, followed by multiplication of ordinal quantities in [35]. However, all these actions are unacceptable with such quantities, thus, conclusions on individual totals based on such methodology depend on the ordinal quantities used (0, 1, 2, 3–0, 1, 3, 5, …), differing only in the permissible transformation. For such different values obtained by the permissible operation of their transformation on the scale, we find that the order of the total vulnerability between individual bird species/families changes on it, that is, the conclusions obtained are unstable to monotonic transformations. On the basis of the analysis, the conclusions presented in the two considered papers are not strictly justified. A similar conclusion applies to other works where summation or multiplication of ordinal quantities is carried out.

4. Arithmetic Operations on an Ordinal Scale Can Yield Partially Uncertain Results

4.1. Multiplication of Ordinal Quantities

Let us separately consider multiplication of ordinal quantities from a slightly different angle, i.e., not from the point of view of the influence of permissible transformations on the final result but of the order of the values obtained in this way. First, we consider a conditional example of the multiplication of ordinal quantities as follows:

Let two datasets (a, b, c) and (f, g, h) be given and their values on the scale of relations X_m and Y_m are known (

Table 2. The initial data of the values (a, b, c) and (f, g, h). X_m and Y_m are the metric values of the initial data; Xr and Yr are the corresponding ranks.

	Xm	Xr			Ym	Yr
a	5	1		f	3	1
b	10	2		g	20	2
c	50	3		h	40	3

). Additionally, the table shows their ordinal quantities X_r and Y_r.

Table 3. Results of the multiplication of both metric and rank values of the source data.

	Xm × Ym	Xr × Yr
af	15	1
bf	30	2
ag	100	2
cf	150	3
ah	200	3
bg	200	4
bh	400	6
cg	1000	6
ch	2000	9

and Figure 1 show that the results obtained by the product of ordinal quantities do not strictly correspond to the products of metric quantities. The points ag and bf, cf and ah, as well as bh and cg must have different ordinal quantities, since their metric products are different, but X_r × Y_r for these pairs coincide. The points ah and bg are different for the product of ranks, yet the products of true values are equal for them. When multiplying ordinal quantities, such uncertainty can lead to the following: In the calculation of vulnerability indices, individual species or species groups are assigned to different categories of vulnerability, or vice versa, species with different vulnerability are assigned to the same category. There is a similar situation for vulnerability maps, i.e., areas with different vulnerabilities may actually have the same vulnerability, or vice versa. In fact, arithmetic operations on an ordinal scale are uncertain for some results. Moreover, in practice, it is not even known which results are distorted, since there is no data on the metric values of the quantities.

4.2. Methods for Impact Assessments Oil/Seabirds

To assess the vulnerability to oil of each bird species within a specific region, Anker-Nilssen [34] developed a set of vulnerability criteria, i.e., nine criteria at the individual level and eight at the population level. According to the knowledge about the biology of the species in the area, for each of these criteria, values are set from 1 (small value criterion) to 3 (very important criterion). They are grouped by five vulnerability factors. The final values are the values of individual vulnerability (IV) and population vulnerability (PV), calculated by the following formulas:

$$IV=Ta\times Ts\times (\frac{2Au+Bs+2Sa}{5})\times (\frac{Pr+4Fc}{5})\times (\frac{Pf+Rc}{2}),$$

(7)

$$PV=IV\times De\times (\frac{Ps+2Ft}{3})\times (\frac{2Fi+4Rp+2Pt+4Vp+Pi}{13}),$$

(8)

These values are converted to a linear index scale (with values from 0 to 1) based on the cumulative probability distributions of all possible values of IV and PV for calculating the final indicators [51]. If we consider only the PV values, this distribution means that a random set of values of the criterion can yield a PV value that is less than or equal to the actual estimated value for the population. This probability value is the final vulnerability index for the population. This is not an absolute value for the vulnerability of a population but a relative index that can be used to compare vulnerability of different populations. According to [52], since all the estimates in the analysis are carried out on the scale of (1, 2, 3), it is reasonable to represent the final vulnerability of a population on the same scale, i.e., 1, low; 2, moderate; and 3, high vulnerability.

Anker-Nilssen’s approach [34] was fully used in the research on the vulnerability of seabirds to oil pollution along the Northern Sea Route [37], in the report of the Norwegian Polar Institute on the assessment of the potential impact of oil production on seabirds and mammals in the North of the Barents Sea [52], in [51], and in other studies. The results of the vulnerability of groups of birds, calculated on the basis of Anker-Nilssen [34] with additions and refinements based on several other works, were also used in the methodology of calculating vulnerability maps for the Norwegian coastal-marine zones adopted at the state level [14]. Next, we consider the publications [14,37] in more detail.

4.3. Vulnerability of Seabirds to Oil Pollution Along the Northern Sea Route

The authors of [37] made some minor changes to the descriptions of individual factors presented by Anker-Nilssen, which did not affect the essence of the methodology and calculation formulas. On the basis of the available data and expert opinions, for each of the considered bird populations living in four areas of the Northern Sea Route (Novaya Zemlya, Western Siberia; Taymyr, Severnaya Zemlya; Yakutia; and Chukotka), an assessment of all 17 indicators on the scale (1, 2, 3) for four seasons (winter, spring, nesting period, and autumn) was given. The PV values were calculated, which were translated into rank values 1, 2, 3), corresponding to the cumulative probability ranges of all the possible PV values (0–1/3, 1/3–2/3, and 2/3–1). The ranges of PV values translated into ranks are not indicated in the report, but they correspond to the ranges <100 (Rank 1), 100–250 (Rank 2), >250 (Rank 3), as in [52]. Of the 311 examined seasonal avifauna populations in the indicated areas of the Northern Sea Route, 92 populations are vulnerable to oil (the population vulnerability index is 2 or 3 in spring and summer, and 3 in the autumn-winter period). Another 28 populations could also potentially fall into this category after some features of their biology and distribution on the NSR route are specified. The remaining populations (191) are less vulnerable to oil.

Notably, Formulas (7) and (8) have a large number of factors. In addition, some multipliers are conditionally “averaged” sums of several criteria (from 2 to 5, and in the latter case, “averaging” is for 13 terms). The scoring for each criterion corresponds to an unknown interval on the metric scale (see examples Figure A1 and Figure A2 in Appendix C). Given the abovementioned uncertainty due to the multiplication of ordinal quantities, the final values for IV and PV are unlikely to reflect the real situation with regard to the vulnerability of various populations and to their ranking, that is, the conclusions, at least for a certain group of species, are incorrect.

4.4. Norwegian Method for Classification of Environmental Resources Priority for Oil Spill

Norway has adopted a unified method for constructing maps of the vulnerability of coastal zones from oil [14], which includes, inter alia, presenting on the final maps of vulnerable areas where certain types of birds that are sensitive to oil may be present. All bird species are classified into six ecological groups, and vulnerability is assessed for each of them for possible areas of summer (four areas) or winter (one area) distribution.

The classification method [14] is based on the following four main factors (V_x, x = I, II, III, IV), which are independent and interconnected:

I.

Naturalness (Is the resource natural?), i.e., assessment of factor V_I = 2 if “yes” and V_I = 1 if “no”;

II.

Compensability (Is it possible to economically compensate the resource?), i.e., assessment of factor V_II = 1 if “yes” and V_II = 2 if “no”;

III.

Conservational value (What conservation value does the resource have?), i.e., V_III = 3 for a national or international resource, V_III = 2 for a regional resource, V_III = 1 for a local resource, V_III = 0 for the absence of protective value;

IV.

General oil vulnerability (How sensitive is the resource when exposed to oil?), i.e., the degree of vulnerability to oil pollution, V_IV = 3, 2, 1, 0, respectively, high, medium, low, zero; the assessment of the vulnerability of birds is based on several works, including [34].

For each resource, including for birds, the factor estimates V_x (independent) are indicated, and the priority of the resource is calculated as P = V_I·V_II·V_III·V_IV. The whole P scale is from 0 to 36 (there are 12 different results in total. P = (1–2) · (1–2) · (0–3) · (0–3)). To operate with fewer categories, the results are grouped [24] in the priority category P (A, B, C, D, E) as follows: A–P = 36, B–P = 24 (18); C–P = 12 (9), D–P = 8, 4 (6); E–P = 2, 1 (3). As a result, five protection priority categories are obtained, which are applied to OSR preparedness maps. Zero-priority resources are excluded so as not to overload the maps. In calculating the priority category for biota groups (marine mammals, fish, and benthic organisms), as well as for seabirds, rank values of factor estimates are used.

In detail, this method, along with other approaches to constructing vulnerability maps was considered in [31] and briefly in [30].

Here, as in the case of the calculations suggested by Anker-Nilssen [34], several (four) factors are multiplied, two of which (V_III and V_IV) are ordinal quantities. The sensitivity factor of bird groups to oil (V_IV) is estimated directly by the method [34] (as already noted, with some refinement). With this in mind, some areas of bird vulnerability in the coastal areas of Norway will be assigned a priority category that does not fully reflect the real situation. Moreover, the developers do not know which specific areas these are.

4.5. General Remarks

The product of ordinal quantities, as well as addition with subsequent multiplication, is unacceptable operations on the ordinal scale. They lead to partially uncertain results, as compared to real metric values, that is, if arithmetic operations were carried out with known metric values on the ratio scale and further with the ranks of these values, then, the order of the product of ordinal quantities (ranks) would be violated with respect to the order of the products of only metric values. Nevertheless, most importantly, it is unknown whether such inconsistencies (permutations) exist, and what values the initial data have. This means that the conclusions obtained on this basis on the vulnerability of seabird populations and the resulting vulnerability maps are not fully correct (see also Section 2.3).

5. Transition from Metric Values to Ordinal (Scores) Alters Actual Ratios between Values of Initial Data

In all the previous reviewed publications, the actual metric values used to calculate the vulnerability of birds to human impact were not known. In this case, the range of variability of ordinal quantities was assumed 1–3 or 1–5 (if necessary, zero is used if there is no effect). These quantities, as a rule, are real integers, that is, natural numbers, in some cases real numbers. Thus, as it were supposed, the maximum and minimum values of such parameters differ no more than three or five times.

Using the publication [36] as an example, let us consider in more detail the effect on the final result of the transition from the range of variability of the source data to the range of score values. To assess the vulnerability of seabird species to surface pollution, the authors of this work developed a quantitative oil vulnerability index (OVI) based on four easily assessed factors. The index was used to calculate the vulnerability of the water area in the North Sea by combining species vulnerability points with seabird density information.

The OVI index includes the following four factors: (Factor a) the proportion of oil-contaminated birds of each species from the total number found dead (or dying) on the coastline, and the ratio of the time that birds of this species spend on the sea surface to the time spent in flight; (Factor b) the size of the biogeographic population of the species; (Factor c) the potential rate of recovery after population decline for each species; (Factor d) the dependence on the marine environment of each species.

Factor (a) was based on percent ranges of polluted birds (from 0 to 100% in increments of 20%), rated from 0.5 to 2.5 (step 0.5) and the ratio of the time that birds of the same species spend on the sea surface to the time they are in flight (five ranges 0–1, 1–3, 3–5, 5–7, >7, also rated from 0.5 to 2.5). The total factor (a) is the sum of these two values and varies from 1 to 5 in increments of 0.5.

Factor (b) varies from 1 to 5 in increments of 1 depending on the size of the population in the North Sea (pairs), i.e., score 1 (1,000,000+); 2 (400,000–1,000,000); 3 (150,000–400,000); 4 (50,000–150,000); and 5 (1–50,000).

Factor (c) consists of the following three components: (i) average clutch size (1, 2, 3, 4–5, 6+ eggs, which corresponds to integer scores from 5 to 1); (ii) maximum clutch size (1, 2–3, 4–5, 6–7, 8+ eggs, which corresponds to scores from 5 to 1); (iii) age of the first breeding (values 6+, 5, 4, 3, 1–2 pass over into scores 5–1, respectively). Furthermore, the sum i + ii + iii is replaced by scores (12–15, 9–11, 7–8, 5–6 and ≤4 by scores, respectively, from 5 to 1).

For Factor (d), it is accepted that bird species spending most of their time at sea are at greater risk than those that also use shore. The percentage of birds using the marine environment (in percent), i.e., 1–40, 41–60, 61–80, 81–90, 91–100 was replaced, respectively, by scores from 1 to 5.

Thus, for each bird species, each of the four described factors was evaluated from one (low vulnerability) to five scores (high vulnerability), and the following formula was used to calculate the OVI [36]:

OVI = 2a + 2b + c + d,

(9)

In this formula, Factors (a) and (b) have a coefficient of 2 as compared with Factors (c) and (d), considering Factors (a) and (b) most important. Such doubling (as the authors of the article write) is arbitrary but explicit and can be changed for future applications of the methodology or if a more correct quantitative method for assigning weights is developed. Estimates for each species were obtained by the authors of the article based on the best available information. They also noted that the quality and quantity of information currently available varied by type, and it would be possible to revise estimates in the future, or if the methodology was applied in other areas. At the time the article was written, the maximum OVI score was 30.

To assess coastal vulnerability, density values were combined with OVI estimates for species using Formula (10) as follows:

$$Area vulnerability score(AVS) = {{\displaystyle \sum }}_{species}Ln \left(\rho +1\right)\times OVI,$$

(10)

where AVS is the water area vulnerability score, ρ is the bird density calculated for the species in the area, OVI is the oil vulnerability index of this bird species. Further, the AVS value for each region was placed in one of four vulnerability categories (very high, high, medium, and low) after dividing the entire range of vulnerability values into four subranges of the same size. On the basis of these results, a vulnerability map was compiled for each month of the year.

In fact, the approach proposed in [36] boils down to the fact that quantitative values or ranges of various metric quantities (factors) are ordered, and they are assigned values from 0.5 to 2.5 in increments of 0.5 (Factor a) or from 1 to 5 in increments of 1 (Factors b and d), and for the components of Factor (c), the values of the three quantities i, ii, iii are added, and the ordered sum is also assigned values from 1 to 5 in increment of 1. Using values from 0.5 to 2.5 in increment of 0.5 does not change the essence. Multiplying all the values of all the factors by 2 produces the same result (the increment for several points will be not 1 but 2).

Replacing real metric values with scores (for the possibility of comparing dissimilar values, including those having different units of measurement) significantly distorts the relationship between the data (between ranges or arithmetic mean values for them) which are initially presented in the original units of measurement. This means that the result of the calculations is also distorted, since arithmetic operations are performed further with this data presented already in scores (ordinal quantities instead of values on metric scales). Thus, for the percentage of oil-stained birds of the total number of dead birds found on shore (component of Factor a), the ratio of the average values for the extreme ranges should be about 9 (≈90/10), and for score estimates it is assumed to be 5 (2.5/0.5 = 5). For Factor (b), the ratio of scores 5 and 2 is 2.5 (the opposite is 0.4), although the real ratio of arithmetic means for these ranges (the penultimate and first) is 700,000/25,000 = 28. In the calculations, this significantly affects the result. For Factor (d), the ratios between the average values of the initial data (ranges) are approximately the same as between the scores and have little effect on the result.

Thus, the transition from real values of factors to scores with other ranges of variability distorts the relationships between factors and the results of the calculation by Formula (9). This conclusion applies to all the previous articles, since everywhere metric quantities with one or the other (usually different and often unknown) ranges of variability are replaced by the equal variability ranges of ordinal quantities.

One more point should be noted in the calculation of the AVS of each bird species based on the product of the vulnerability index a given bird species from oil (OVI) and the logarithm of the density of birds of this species (Ln (ρ + 1)). In fact, for each species $OVI\times Ln \left(\rho +1\right)$=$Ln [{\left(\rho +1\right)}^{OVI}]$, i.e., the vulnerability parameter OVI of each species is included in the final function of each bird species as an indicator of the degree from which the natural logarithm is calculated. A more rigorous justification for such approach, which is absent in [36], is necessary. It may be more correct to not calculate for each bird species the logarithm of their density in the area (which is further multiplied by the species vulnerability index), but rather calculate the logarithm of the product of bird density and their vulnerability coefficient.

6. Conclusions and Recommendations

Numerous publications on the evaluation of the vulnerability of seabirds to oil pollution (including all the five articles examined in detail) and of the methods for assessing vulnerability for marine areas have used ordinal quantities. But ordinal quantity is 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. Ordinal quantities can enter into empirical relations only and do not have measurement units or quantity dimensions. Differences and ratios of ordinal quantities have no physical meaning. Ordinal quantities are arranged according to ordinal quantity-value scales [26].

Summation of ordinal quantities in calculating vulnerability of seabirds sometimes leads to ambiguous correlations between the results and, accordingly, to ambiguous conclusions. The ratio of the final calculated estimates of the vulnerability indices and obtained vulnerability maps becomes dependent on the choice of certain ordinal quantities when these selected values are connected by permissible transformations on the ordinal scale, that is, individual conclusions (for instance, which indices for species of birds or their families are larger/smaller, or how many times they are larger/smaller with permissible scale transformations) often change with such permissible transformations. The conclusions based on values (OVI, BOI, IV, PV) which include arithmetic means of ordinal quantities are not invariant to permissible transformations.

Multiplication of ordinal quantities leads to a change in the order of the final results (products), as if one could proceed from the metric (but unknown) values of the initial parameters. In fact, the multiplication of ordinal quantities leads to partially uncertain results. Moreover, the researcher does not even know which results are incorrect. Values (BOI, IV, PV, V_x) calculated using ordinal quantities have hidden uncertainties.

The transition from the known initial metric values to ranks (ordinal quantities), i.e., replacing the former with the latter distorts real relations between the initial data, which significantly affects the final result of the calculation of the vulnerability indices and vulnerability maps of seabirds (calculation of BOI [36]). Calculations with ordinal quantities in a limited range of variability (0–3 or 0–5), most likely, also do not correspond to the real relations of appropriate metric values and lead to incorrect results (calculations of OVI, BOI, IV, PV, V_x, and others).

In general, the conclusions obtained in the considered works [14,33,34,35,36,37], as in many others which use arithmetic operations with rank (ordinal) quantities, cannot be considered strictly justified in full. To correctly assess seabirds’ vulnerability to oil, it is necessary to develop models based on metric values on a ratio scale. This could include different approaches, for example, (1) the use of species biodiversity [3,53], while taking into account their abundance (more detail see in [30]); (2) parameters assessment based on expert assessments which provides results on ratio scale [38]; (3) the use of several anthropogenic factors strictly on a ratio scale with calculation of vulnerability coefficients based on them [32].

Methods based on arithmetic operations with ordinal quantities lead, in general cases, to uncertain and not always correct results. Such methods are probably useful in a relatively crude approximation (the examples of calculations in Section 3 and Section 4 show this). Thus, one should be careful about using the results of calculations on such a basis for making management decisions. However, utility and significance of all previous assessments of seabirds’ vulnerability to oil and the resulting vulnerability maps which were calculated using ordinal values should be recognized. Such works, on the one hand, have made it possible to accurately identify the most and the least vulnerable species of seabirds and marine areas (taking into account, for example, Figure 1), although for the other species (average in vulnerability values) there is uncertainty of such estimates shown above. On the other hand, a list of factors identifying birds’ vulnerability to oil was determined during such studies [14,33,34,35,36,37]. Specific metric values of a range of parameters, which are necessary for further researches and estimation of vulnerability were identified (Factors a, b, c and, d, see [36]).

Ordinal quantities in arithmetic operations should be abandoned when calculating the vulnerability of individual species or their groups. The development of appropriate models (calculation formulas) of vulnerability is required which would include only metric values on ratio scales. This can significantly complicate the models (formulas) used, since it would no longer be possible to simply sum or average several heterogeneous parameters of different nature and different units of measurement. Such an approach, based on the use of metric data in calculations, is proposed [30,31,32] for vulnerability assessments of different ecological groups of biota to oil and for development, on those bases, of vulnerability maps of sea-coastal zones to oil.

At the same time, one should also take into account the possibility of strictly correct expert assessment of parameters characterizing anthropogenic factors affecting seabirds, while receiving data on a ratio scale. This could include multicriteria decision analysis techniques such as the analytic hierarchy process (AHP). In fact, the method of pairwise comparison, as described in [38], allows making this. The obtained estimates can be considered to be quantities on a ratio scale. The correctness of the obtained values is controlled by the corresponding criteria [38]. For example, AHP was used to evaluate and integrate expert opinion on the risks posed to 14 species of seabird in NW Atlantic [39]. AHP is an effective tool to rank and prioritize conservation concerns, and it can help to guide conservation planning and management decisions. If used correctly, this method could overcome, to some extent, issues associated with calculations based on ordinal data.