BioStatistics — 15. Topic 11. Certain Methods Characteristic of Zoology and Ecology
A number of specific data-processing techniques are used in zoology and ecology. Some of them are discussed in this section.
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Topic 11. Certain Methods Characteristic of Zoology and Ecology
Appendix. Syllabus of the Major Practical Section
Biostatistics-14
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Biostatistics-16 Topic 11. Certain Methods Characteristic of Zoology and Ecology 11.1. Analysis of Fluctuating Asymmetry Here we shall examine one of the remarkable methods that can be applied in zoological and ecological research — and, of course, in other fields as well. This method combines simplicity of data collection, potential analytical power… and highly non-trivial circumstances that must be taken into account when interpreting the results obtained. Everyone knows that the twin method proves very effective in determining the heritability of many traits. If we wish to determine the contribution of innate factors to the development of some trait in humans, we can compare the variability of that trait in monozygotic twins (clones) and in ordinary brothers and sisters (sibs). On the other hand, by comparing the variability of a trait of interest in clones that have developed in different environments, we can assess the degree to which environmental influences have promoted or impeded the realization of the genetic program governing the development of that trait. In cases where it is possible to obtain genetically identical offspring, comparing the outcomes of their development proves to be a very interesting method both for assessing the stability of the system controlling ontogenesis (when comparing the development of different traits under the same conditions) and for assessing the quality of the environment in which development occurs (when comparing the development of the same traits under different conditions). The form of variability studied in such research was termed realizational variability by B. L. Astaurov — the differences among the various outcomes of the realization of the same hereditary program. The idea described here is that the body of bilaterally symmetrical organisms can be regarded as two "clones" — the right and left halves. If the developmental program for the right and left halves of the body is identical, the genetic control of the development of the traits under consideration is stable, and the environment is favourable, the developmental outcome will be entirely symmetrical. However, one must bear in mind that asymmetry can be a consequence not only of realizational variability; it may also represent an adaptation to particular factors. For example, in lobsters one claw is stronger and has a blunt edge, being used primarily for crushing shells and carapaces, while the other is less strong but sharp, and is used for cutting off pieces of relatively soft food. The American biologist Leigh Van Valen proposed a classification of forms of bilateral asymmetry, illustrated in the figure. To determine which type of asymmetry one is dealing with, one should construct the distribution of deviations from symmetry to the right (with greater trait development on the right side) or to the left. [IMG_2] Fig. 11.1.1. Three types of asymmetry according to Leigh Van Valen (1962). For humans, an example of directional asymmetry is the position of the heart; of antisymmetry — the sizes of the right and left hands; of FA — the sizes of the right and left auricles. In the case of directional asymmetry, a systematic predominance of the trait on one side is registered. For example, the human heart is laid down symmetrically, but one of its halves turns out to be larger than the other. The left heart pumps blood into the systemic circulation, the right into the pulmonary circulation. The heart is almost always displaced to the left. In the case of antisymmetry, deviations can occur in either direction, and these deviations are more frequent than the symmetrical state of the traits. For example, right-handed people have larger right hands, and left-handed people have larger left hands. The most rare state is equal development of both hands. Lobster claws are also an example of antisymmetry. In the case of fluctuating asymmetry (FA), the most frequent state of a bilateral trait is its symmetry; strong deviations from symmetry are observed less frequently than weak ones. Analysing the sizes of the right and left auricles in humans, one can see that deviations toward a larger right or left ear are approximately of such a character. It is precisely traits with this type of distribution that can be used to assess realizational variability. Thus, FA is defined as random, non-directional deviations from bilaterally symmetrical condition. FA can be used to assess developmental stability and the influence on it of diverse internal and external factors. It is generally accepted that FA is reduced (and developmental stability is accordingly increased) by: – optimal conditions; – a relatively clean environment; – high individual fitness; – optimal genetic distance between the parents; – evolutionary conservatism, well-regulated development of the traits under study. FA is increased and developmental stability is reduced by: – extreme conditions; – a polluted environment; – low individual fitness; – inbreeding or distant hybridization; – the lack of functional importance, evolutionary novelty, and unstable regulation of development of the traits under consideration. The study of FA belongs to those methods in which the investigator attempts to draw conclusions about a whole complex of factors from the value of a single integral trait. Work of this kind demands a high degree of care in interpretation and attention to research design. A sound programme for the study of FA is contained in the article cited below. "In the statistical analysis of FA of bilateral traits, two aspects can be distinguished. The first is associated with the analysis of individual traits and includes: — the study of the directional (or non-directional) nature of trait asymmetry; — the study of the dependence of the magnitude of trait asymmetry (L-R) on the magnitude (size) of the trait on both sides of the body (L+R) or (L+R)/2; — the study of the degree of correlation among the magnitudes of asymmetry for different traits used in the integral assessment of individual FA; — the study of the presence (or absence) of sex (gender) differences in trait asymmetry; — the study of the contribution of measurement error to the final FA assessment (important for plastic traits and irrelevant for meristic ones). The second aspect is associated with the selection and correct application of integral indices that assess FA of a sample of organisms across a complex of individual traits in biomonitoring tasks." Gelashvili D. B., Yakimov V. N., Loginov V. V., Yeplanova G. V. Statistical analysis of fluctuating asymmetry of bilateral traits of the multicolored racerunner Eremias arguta // Current Problems in Herpetology and Toxinology: Collection of Scientific Papers. Issue 7. — Tolyatti, 2004. — Pp. 45–59. Thus, to study FA one should, first of all, select bilateral traits whose variability can be measured quantitatively. The values of these traits must be measured on the right and left sides of the body in a sufficiently large sample of the organisms under study. To make the notation more comprehensible, let us specify that it refers to a trait we shall designate as C (character). The values of this trait on the right and left sides can then be designated C_r and C_l, respectively. An important step in the study of a trait that is planned for use as a measure of FA is the construction of its distribution taking into account its sign. One should calculate the quantity D_C = C_r – C_l and determine whether its distribution corresponds to the character expected for FA. Gelashvili and co-authors in the article cited above recommend assessing the normality/non-normality of the trait designated by us as D_C using the Kolmogorov-Smirnov, Shapiro-Wilk, and Lilliefors tests. These tests serve to verify the null hypothesis of no difference between the observed distribution and the normal distribution. Significant differences indicate non-normality of the distribution. However, following this recommendation will result in the situation that the larger the sample studied, the more probable will be a significant deviation of its distribution from the normal. In our view, the optimal course of action is different. One should construct the distribution of the trait D_C and at least visually assess whether it has a bell-shaped form or not. If the most frequent state of the trait is symmetry and the greater the deviation from symmetry the less frequent it is, this trait can be used to assess FA. In the case of a small sample, the empirical distribution will be "rough"; in such a case statistical tests can be used to assess the significance of its deviation from the normal. If the samples are sufficiently large, the distribution is relatively uniform and bell-shaped, there is probably no pressing need to employ statistical tests. The distribution of asymmetry of bilateral structures can have a non-trivial character. Below is presented the distribution of Eurasian perch by symmetry/asymmetry of the number of scales in the lateral line on the right and left sides (from the work: Vynohradova K. P., Sakun Yu. V., Belousova K. M., Goncharov H. L., Shabanov D. A. Study of fluctuating asymmetry of the Eurasian perch (Perca fluviatilis L., 1758) // Biolohiia ta valeolohiia, 2012. — Issue 14. — Pp. 9–17). The characteristic distribution in the shape of a trident stands out: differences of one scale between the right and left sides of the body are encountered less frequently than differences of two (of 256 perch, a deviation of one scale was observed in 72 individuals, and of two scales in 91 individuals; upon comparison of the difference in proportions by Fisher's test using a one-tailed criterion, the predominance of deviations of two scales proves significant, p=0.04). [IMG_3] Fig. 11.1.2. Distribution of perch by asymmetry of the number of scales along the lateral line on the right and left sides. The x-axis represents the difference in the number of scales on the right and left sides; the y-axis represents the number of observations. (Vynohradova K. P., Sakun Yu. V., Belousova K. M., Goncharov H. L., Shabanov D. A. Study of fluctuating asymmetry of the Eurasian perch (Perca fluviatilis L., 1758) // Biolohiia ta valeolohiia, 2012. — Issue 14. — Pp. 9–17) How can the distributional feature shown in Fig. 11.1.2 be explained? According to the version set out in the cited article, this may be associated with the fact that those individuals whose deviation from symmetry is relatively small may be able to regulate their development, returning it to the norm. If the level of deviations that are the cause of FA (i.e. "ontogenetic noise," spontaneous deviations, and environmental perturbations) is low, the epigenetic mechanisms governing development may ensure symmetrical development. Beyond a certain threshold, the probability of deviations from the symmetrical state increases. [IMG_4] Fig. 11.1.3. The model of the epigenetic landscape proposed by C. H. Waddington. Ontogenesis is compared in this model to the rolling of a ball across a surface of complex shape. More probable states of the developing system correspond to hollows on the surface. Small deviations from the normal trajectory of development, as a result of the self-regulatory capacity of the developmental system, will be corrected; more substantial deviations will lead to a change in the final state of the developing system. Epigenetic mechanisms (whose action, in the metaphor of the epigenetic landscape, is expressed as the deepening of the trajectories of normal development) stabilize development, while environmental perturbations and ontogenetic noise destabilize it. From this perspective, a distribution such as that shown in Fig. 11.1.2 does not contradict the assumption that it is generated by FA. Consequently, asymmetry in the number of scales in perch can be used to assess the stability of their development. The next item in the protocol for processing the results of an FA study, as proposed in the article by Gelashvili and co-authors, is to test whether the asymmetry is directional. To answer this question, it is sufficient to compare the values of the columns C_r and C_l using the Wilcoxon test (a non-parametric method for paired comparisons, Statistics / Nonparametrics / Comparing two dependent samples (variables)). One can hypothesize that in the case of a study on a truly large sample, a significant asymmetry of the distribution may well be registered in the present case too. In our view, in the case of small deviations from symmetrical distribution of deviations, the trait under analysis can still be used to assess FA, but this circumstance should not be forgotten when interpreting the results. In some cases it is possible to calculate the deviation not from the symmetrical state, but from the median value of the distribution of the trait D_C. Another stage in the analysis of FA data is the assessment of size dependence. To do this, one should calculate the correlation coefficient between two quantities: the absolute value of asymmetry, A_C = Abs(C_r – C_l), and the mean value of the trait C_mean = (C_r + C_l)/2. If the value of asymmetry is not linked to the size of the structures under study, the absolute value of asymmetry, A_C, can be used as a measure of FA. If, with an increase in the size of the organism and the structures under study, the absolute value of asymmetry also grows, the normalized asymmetry should be used to assess FA: N_C = 2×Abs(C_r – C_l)/(C_r + C_l). The next stage of the Gelashvili et al. "protocol" is concerned with determining how strongly different measures of asymmetry that can be used in studying the same organism correlate with one another. The authors of the article under discussion recommend using measures that are weakly associated with each other. "It is obvious that the less correlated the magnitudes of trait asymmetry, the more independent and objective an assessment can be obtained using a given set of traits" (Gelashvili et al., 2004). On the other hand, FA measures are of interest not in themselves, but as means of assessing the stability/instability of organismal development. Developmental stability (for example, the "depth" of creodes, the ontogenetic trajectories shown in Fig. 11.1.3) cannot be directly assessed. The traits that we use in the course of research must be linked to the quantity we are using them to assess. From this perspective, developmental stability is optimally assessed from several traits that are correlated with it, and consequently with one another. If the sample consists of sub-samples (for example, individuals of different sexes), it should be tested whether the sub-samples are homogeneous in the asymmetry of the traits under consideration. If the parts of the sample are homogeneous, they can be treated jointly (for example, data on females and males can be combined). When using several bilateral traits to assess FA, various integral measures can be applied. The simplest of these is the straightforward summation of the values (absolute or normalized) of the asymmetry of all the traits employed. 11.2. An Example of Processing Fluctuating Asymmetry Data Let us consider the analysis of FA using an example. The data presented here were obtained by second-year students during a field practical in vertebrate zoology in Haidary and published in the article cited above (Vynohradova K. P., Sakun Yu. V., Belousova K. M., Goncharov H. L., Shabanov D. A. Study of fluctuating asymmetry of the Eurasian perch (Perca fluviatilis L., 1758) // Biolohiia ta valeolohiia, 2012. — Issue 14. — Pp. 9–17). The full dataset is not presented here, but only a portion thereof, edited in such a way that processing it yields the same result as processing the original data. Studies with an analogous design were carried out subsequently as well, on perch and on other fish species; the processing of the data collected in them led to analogous conclusions. A sample of Eurasian perch was studied. For each fish, body length (L_c) was measured and age was determined (from annuli on the scales). Data are presented for first-year fish, designated I (0+), and second-year fish, designated II (1+). The designations are related to the fact that first-year fish do not yet have the annuli on their scales that form during wintering, while second-year fish have one such annulus. Fish of both ages were divided into three groups differing in growth rate. This is possible because perch spawning takes place within a sufficiently narrow timeframe, and differences in fish size are attributable primarily not to differences in age of a few days, but to differences in growth rate. Slow-growing individuals are designated Slow, fish with a moderate growth rate Moderate, and fast-growing fish Fast. Three bilateral traits were determined in the perch studied: the number of scales along the lateral line (S), the number of rays in the pectoral fins (P), and in the pelvic fins (V). The designation "_r" corresponds to the value of the trait on the right side of the body, and "_l" to the left side. Table 11.1. Results of the description of 100 Eurasian perch (after Vynohradova et al., 2012, abbreviated and edited version)
L_c
Age
Growth
S_r
S_l
P_r
P_l
V_r
V_l
L_c
Age
Growth
S_r
S_l
P_r
P_l
V_r
V_l
31
I (0+)
Slow
69
68
10
10
6
5
74
II (1+)
Slow
70
69
10
11
6
6
31
I (0+)
Slow
66
64
10
10
6
6
74
II (1+)
Slow
69
68
9
10
6
6
31
I (0+)
Slow
62
64
10
10
6
6
74
II (1+)
Slow
63
61
11
11
6
6
31
I (0+)
Slow
66
68
10
10
6
6
74
II (1+)
Slow
67
65
12
11
6
6
32
I (0+)
Slow
66
65
11
10
6
6
75
II (1+)
Slow
71
70
11
11
6
6
32
I (0+)
Slow
63
65
10
10
6
6
76
II (1+)
Slow
69
67
11
11
6
6
32
I (0+)
Slow
64
62
10
9
6
5
77
II (1+)
Slow
73
73
11
11
6
6
33
I (0+)
Slow
69
67
10
10
6
6
77
II (1+)
Slow
68
67
11
10
6
6
35
I (0+)
Slow
70
69
10
10
6
6
77
II (1+)
Slow
68
66
10
11
6
6
35
I (0+)
Slow
70
67
9
10
5
7
78
II (1+)
Slow
72
72
11
10
6
6
36
I (0+)
Slow
61
63
9
10
5
6
78
II (1+)
Slow
70
68
11
11
6
6
36
I (0+)
Slow
66
68
10
11
5
6
79
II (1+)
Slow
72
72
10
10
6
6
36
I (0+)
Slow
60
63
11
10
6
6
80
II (1+)
Slow
73
73
11
10
6
6
37
I (0+)
Slow
65
66
10
10
6
6
80
II (1+)
Slow
67
65
10
11
6
5
37
I (0+)
Slow
66
66
11
9
6
L_c Age Growth S_r S_l P_r P_l V_r V_l L_c Age Growth S_r S_l P_r P_l V_r V_l 31 I (0+) Slow 69 68 10 10 6 5 74 II (1+) Slow 70 69 10 11 6 6 31 I (0+) Slow 66 64 10 10 6 6 74 II (1+) Slow 69 68 9 10 6 6 31 I (0+) Slow 62 64 10 10 6 6 74 II (1+) Slow 63 61 11 11 6 6 31 I (0+) Slow 66 68 10 10 6 6 74 II (1+) Slow 67 65 12 11 6 6 32 I (0+) Slow 66 65 11 10 6 6 75 II (1+) Slow 71 70 11 11 6 6 32 I (0+) Slow 63 65 10 10 6 6 76 II (1+) Slow 69 67 11 11 6 6 32 I (0+) Slow 64 62 10 9 6 5 77 II (1+) Slow 73 73 11 11 6 6 33 I (0+) Slow 69 67 10 10 6 6 77 II (1+) Slow 68 67 11 10 6 6 35 I (0+) Slow 70 69 10 10 6 6 77 II (1+) Slow 68 66 10 11 6 6 35 I (0+) Slow 70 67 9 10 5 7 78 II (1+) Slow 72 72 11 10 6 6 36 I (0+) Slow 61 63 9 10 5 6 78 II (1+) Slow 70 68 11 11 6 6 36 I (0+) Slow 66 68 10 11 5 6 79 II (1+) Slow 72 72 10 10 6 6 36 I (0+) Slow 60 63 11 10 6 6 80 II (1+) Slow 73 73 11 10 6 6 37 I (0+) Slow 65 66 10 10 6 6 80 II (1+) Slow 67 65 10 11 6 5 37 I (0+) Slow 66 66 11 9 6 6 81 II (1+) Slow 70 70 11 10 6 6 37 I (0+) Slow 56 57 12 11 6 6 83 II (1+) Moderate 69 68 11 11 6 6 38 I (0+) Moderate 65 66 9 9 6 6 83 II (1+) Moderate 62 65 11 11 6 6 38 I (0+) Moderate 71 70 10 10 6 6 83 II (1+) Moderate 62 60 10 10 6 6 38 I (0+) Moderate 67 69 10 9 6 6 84 II (1+) Moderate 72 72 11 11 6 7 38 I (0+) Moderate 64 62 11 10 5 6 84 II (1+) Moderate 73 73 12 12 6 6 39 I (0+) Moderate 58 58 9 9 5 5 84 II (1+) Moderate 69 68 11 10 6 6 39 I (0+) Moderate 68 68 10 9 6 6 84 II (1+) Moderate 67 65 11 12 6 6 39 I (0+) Moderate 68 68 9 9 5 6 85 II (1+) Moderate 71 70 11 11 6 6 39 I (0+) Moderate 66 64 10 10 6 6 85 II (1+) Moderate 74 73 11 11 6 6 39 I (0+) Moderate 63 65 10 10 6 6 85 II (1+) Moderate 64 65 10 11 6 6 39 I (0+) Moderate 66 68 9 9 5 5 86 II (1+) Moderate 76 76 11 11 7 6 40 I (0+) Moderate 58 57 10 10 6 6 86 II (1+) Moderate 71 72 11 11 6 6 40 I (0+) Moderate 63 63 10 11 5 6 87 II (1+) Moderate 70 70 12 11 6 6 40 I (0+) Moderate 66 68 10 10 6 6 89 II (1+) Moderate 63 63 10 10 5 5 40 I (0+) Moderate 65 67 10 11 6 6 91 II (1+) Moderate 68 69 11 11 6 6 41 I (0+) Moderate 67 66 11 11 6 6 91 II (1+) Moderate 58 58 11 11 5 5 41 I (0+) Moderate 61 60 10 10 6 6 91 II (1+) Moderate 69 70 10 11 6 6 41 I (0+) Moderate 67 66 10 11 6 6 92 II (1+) Moderate 72 72 11 11 6 6 43 I (0+) Fast 67 67 10 11 6 6 95 II (1+) Fast 70 70 10 11 6 6 43 I (0+) Fast 68 66 11 11 6 6 95 II (1+) Fast 65 63 11 11 6 6 43 I (0+) Fast 64 66 11 11 6 7 98 II (1+) Fast 63 64 12 12 5 6 43 I (0+) Fast 68 72 10 10 6 5 99 II (1+) Fast 72 70 11 11 5 5 44 I (0+) Fast 65 65 10 10 6 6 99 II (1+) Fast 69 67 11 11 6 6 44 I (0+) Fast 68 66 10 10 6 6 100 II (1+) Fast 74 74 12 12 6 6 44 I (0+) Fast 66 68 10 11 6 6 101 II (1+) Fast 62 61 12 11 6 6 45 I (0+) Fast 66 66 11 11 6 6 102 II (1+) Fast 58 59 12 11 5 6 45 I (0+) Fast 69 70 10 11 5 6 104 II (1+) Fast 67 67 12 12 5 5 45 I (0+) Fast 70 68 11 10 6 6 105 II (1+) Fast 72 72 10 11 6 6 45 I (0+) Fast 68 71 11 10 6 6 105 II (1+) Fast 68 67 11 11 6 6 46 I (0+) Fast 69 69 11 10 6 6 105 II (1+) Fast 72 74 11 11 6 6 46 I (0+) Fast 68 66 11 10 6 5 107 II (1+) Fast 67 67 11 12 6 6 47 I (0+) Fast 65 65 9 9 5 5 110 II (1+) Fast 69 69 11 10 6 6 49 I (0+) Fast 66 67 10 11 6 6 113 II (1+) Fast 71 71 11 10 7 6 50 I (0+) Fast 68 66 11 10 6 6 115 II (1+) Fast 69 70 11 11 6 6 52 I (0+) Fast 70 71 11 10 6 7 116 II (1+) Fast 72 72 10 11 6 6 The results of the analysis of the distributions of the traits considered here will not be presented, since in this case we are dealing only with a portion of the data. The distribution of the trait D_S for the entire sample studied is shown in Fig. 11.1.2. No significant deviations to the right or left (upon comparison of the right and left values using the Wilcoxon test) were found either. Which traits exactly were calculated in the course of data processing is clear from Fig. 11.2.1. The trait designations in this figure correspond to those used in the article by Gelashvili and co-authors. [IMG_5] Fig. 11.2.1. Measures of directional and non-directional asymmetry for individual traits, as well as the overall measure of fluctuating asymmetry, employed in the study of perch. The third step of the programme set out in the previous section is concerned with the study of size dependence — that is, determining the correlation between the absolute value of asymmetry and the size of the structures whose asymmetry is under consideration. The article by Gelashvili and co-authors discusses the need to normalize asymmetry if the absolute value of asymmetry grows with individual growth. In such a case, the increase in absolute asymmetry may be a consequence of the increase in body size. The case in which the absolute asymmetry decreases with increasing body size is not considered in the article by Gelashvili and co-authors. Strange as it may seem, our case is precisely of this kind (Fig. 11.2.2). [IMG_6] Fig. 11.2.2. Fluctuating asymmetry demonstrates a significant negative (!) correlation with body length. If the level of FA were to increase with body size, it would need to be used in its normalized form (in order to remove the effect of the increase in the asymmetry index that is simply linked to an increase in size). Since it decreases, we are probably dealing with the manifestation of some regularity that requires elucidation. Let us conduct a two-way analysis of variance, in which we shall treat age and growth rate as independent factors, and the overall measure of fluctuating asymmetry (the sum of three measures of asymmetry for individual traits) as the dependent variable. The result is shown in Fig. 11.2.3. [IMG_7] Fig. 11.2.3. Both age and the group identified on the basis of growth rate are significantly associated with the fluctuating asymmetry of perch. Although the interaction of factors in the two-way analysis of variance proved non-significant, the graph illustrating it is convenient for showing the FA values characteristic of each group (Fig. 11.2.4). [IMG_8] Fig. 11.2.4. Younger individuals are more asymmetrical than older ones, and slow- and fast-growing individuals are more asymmetrical than those displaying moderate growth rate. Why? The result shown in Fig. 11.2.4 can probably be explained by several hypotheses. Let us enumerate them, and then establish which of them deserves to be accepted as the best explanation. We shall consider only one aspect: the significantly (p=0.00132) lower asymmetry of perch of the older age group. Hypothesis 1. Conditions have changed. The year preceding the observations was favourable, the development of the fish was stable. The year of observations proved far more stressful, and the development of juvenile fish became dysregulated. Hypothesis 2. As fish grow and mature, their degree of asymmetry decreases. The relatively symmetrical individuals of the older group may have been less symmetrical at a younger age. Hypothesis 3. The death of less symmetrical individuals is more probable; as a result of selective mortality, the proportion of symmetrical individuals in a cohort increases. To refute Hypothesis 1, data from other years must be brought in. This has been done (outside the scope of the study described here). In the case of perch, and in the case of certain other species, based on the results of studies from different years, fish of older age classes prove to be more symmetrical. Hypothesis 2 should be considered extremely improbable. Deviations from the symmetrical state impose asymmetric loads on paired structures, which lead to the intensification of asymmetry. Observations of asymmetric individuals under zoocultural conditions likewise do not allow this option to be considered likely. Hypothesis 3 corresponds to the situation of stabilizing selection. The fact that individuals with a moderate (probably corresponding to the population norm) growth rate prove to be more symmetrical also accords well with this hypothesis. Thus, it can be reasonably assumed that the increase with age in the proportion of symmetrical individuals within a cohort is the result of stabilizing selection: the more probable survival of more symmetrical individuals.