Illustration of steps to compute GCAI based on multiple structures in different species having muscles as example, according to Aversi-Ferreira et al.
Rules, formulas, and statistical tests have been widely used in studies that analyze continuous variables with the normal (Gaussian) distribution or defined parameters. Nevertheless, in some studies such as those in gross anatomy, only statistics with discrete or nominal variables are available. In fact, the existence or absence of an anatomical structure, its features and internal aspects, innervation, arterial and vein supplies, etc. can be analyzed as discrete and/or nominal variables. However, there have been no adequate methods, which allow transformation of data with qualitative/nominal variables in gross anatomy to those with quantitative variables. To resolve the issue, we have purposed a new method that allows, in order, descriptions based on numerical analyses, the statistical method for comparative anatomy (SMCA), and proposed the formula for comparison of groups of anatomical structures among different species that allows to infer evolutionary perspective. The important features of this method are as follows: (1) to allow to analyze numerical data, which are converted from discrete or nominal variables in morphological areas and (2) to quantitatively compare identical structures within the same species and across different species. The SMCA fills the lack of a specific method for statistical works in comparative anatomy, morphology, in general, and evolutional correlations.
- statistic nonparametric
- nominal variables
The statistical analysis is widely used in studies in almost all scientific fields to lead to discussions and conclusions of the data . In most of the cases, the variables analyzed are continuous ones that can be analyzed by calculating an average, standard deviation, i.e., the data can be fitted approximately to Gaussian distribution (normal distribution) of probability. Gaussian distribution means that most population, in studied variables, is concentrated around the average, i.e., the data are grouped around the average symmetrically ; these data, when fitted in Cartesian plane, display a geometrical shape similar to an inverted bell, called Gaussian curve showing a central tendency. In a perfect Gaussian distribution, the average is located at the center of the curve and the frequency of the data, in a studied variable, decreases quantitatively toward lateral extremes. This type of data distribution, similar distribution around the average, is called parametric statistic.
It is important that data characteristics regarding their distribution must be analyzed before any statistical calculation. However, variables of data that do not follow Gaussian distribution are sometimes submitted to statistical calculation under assumption of normal distribution of probability. In these cases, application to mathematical tools under assumption of normal distribution would induce errors, to be more specific, acceptance or rejection of statistical hypotheses should be incorrect . These kinds of errors are becoming common mainly because of indiscriminate and incorrect use of statistical software. The statistical programs are very important to allow fast, concise, and reliable analyses of variables and include a test of normality of data. However, sometimes, data that do not display normal distribution are analyzed using these programs, and the results usually indicate no statistical significance.
For understanding the importance of correct statistics, imagine that although average atmospheric temperature around a man is 24°C, his feet are put in a refrigerator and the head is put in a stove. Then, he should consider the temperatures as very uncomfortable . In this case, the average hardly reflects correct interpretation of the data. Indeed, these extreme values could provide an acceptable average. However, mostly these values are not fitted into Gaussian distribution. Therefore, they cannot be analyzed by parametric statistics [1–3].
Data that cannot be submitted to parametric designs should be analyzed using nonparametric statistics. Indeed, other types of averages can be calculated based on nonnormal distributions, as in the binomial or chi-square (
In gross anatomy, there are many cases in which numerical data are not available for analyses. Anatomical studies must analyze absence or presence of a structure or organ and characteristics associated with these organs; for example, presence or absence of specific nerves and vessels in muscles, and distribution of these structures if these structures are present. They are qualitative variables, but not numerical ones. This means that numbers cannot provide this type of information.
It is well-known that anatomical texts include vast descriptions of structures, relationships between the structures, axes, and positions of the body. These findings indicate that specific statistical methods are required especially in comparative anatomical studies. However, any previous statistical methods do not allow accurate analyses of anatomical data, but only could assist discussion on them. Some anatomical studies tried to analyze qualitative variables more objectively, using nonparametric statistics such as chi-square (
It is reasonable that when the central tendency measures cannot be used, nonparametric distributions must be chosen , mainly due to small sample sizes . The nonparametric distributions are also used when it is difficult to set up quantitative variables. Indeed, the percentage of a given structure based on the frequency of the structure in the samples is one of the key measures in nonparametric statistics used in gross anatomy. In gross anatomy, the highest percentage of occurrence of a given structure is called
Gross anatomy has no specific statistical method for analyses of noncontinuous variables regarding anatomical structures until a few years ago. Here, we show a new statistical method based on nonparametric statistics, more consistent with anatomical descriptions. We also compare this new method with cladistics used for evolutionary analyses to indicate usefulness of this new method in this discipline.
2. Concepts of the statistical methods for gross anatomy
In this section, we will show that the new statistical method is based on the anatomical concept of normality, and appropriate weight is provided with each variable (parameter for a specific feature) of structures based on the importance of the variable, and that conclusions can be drawn based on the values integrated across multiple variables. The results by this new method have been reported in our previous papers, in which this method was designed to compare muscles not only within the same species but also across different species in comparative gross anatomy [1, 7, 9–11].
2.1. Anatomical concept of normality and variation
The initial step in the statistical method for comparative anatomy (
|Species (||Specie 1 (control specie) (||Specie 2 (||…||Specie s (|
|Investigated structures (||M1||M2||…||Mm||M1||M2||…||Mm||…||M1||M2||…|
|Weighted averages for single muscle (PAF = ||…||…||…||…|
|Weighted averages for multiple muscles (mean of ||…|
Good examples of structures in animals that should be applied to SMCA are muscles, because muscles require different variables to describe their characteristics: shapes, innervation, vascularization, origin, insertion, and number. Different individuals in the same species or individuals in different species could display different numbers, as in the contrahentes muscles in primates.
The formula indicating the relationship among a total number of studied structures and numbers of normal and variation in the structures is shown below:
In case of muscles, the parameters should include at least the following four: (1) innervation, (2) origin, (3) insertion, and (4) vascularization. For instance, in a case of the biceps in a specific species,
The next step is calculation of the relative frequency (
When the structure is pair organs,
It is noted that qualitative features are transformed into quantitative data after the initial data are expressed as percentages. Thus, the method allows numerical description of anatomical structures, which increases preciseness in description of characteristics of anatomical structures. Another usefulness of this method is that the value of
Normal structure in each parameter means 0.5 <
For accurate and detailed analyses, it is required to calculate
By the introduction of this parameter (
2.2. Definition of pondered average of frequency (PAF)
In the next step of the SMCS in which multiple features (
After designation of pondered values as weighted coefficients, the pondered average of frequencies (
The idea of weighted coefficients (
Zero cannot be accepted as weighted coefficient (
2.3. Definition of comparative anatomy index (CAI) for comparison among different species
In normal structures,
It is noted that
For example, there are two types of origin in the coracobrachialis (
Although any species can be defined as control species, the species studied in the first time or the species with abundant known data should be chosen as control species. To compare any single structure (e.g., muscle) between two different species (
It is noted that the
2.4. Definition of group comparative anatomy index (GCAI) for comparison of a group of structures among different species
Based on the above inferences, using
In Table 2, we show that different structures (muscles) among different species can be compared in reference to a control species. The 8 specimens of the
|Superficial dorsal group of the forearm|
|Extensor digitorum communis||Origin||Lateral epicondyle of the humerus||Lesser variation regarding the distribution of tendons to fingers.|
Somewhat similar to BC
CAI = 0.22
|Highly similar to BC|
CAI = 0.00
|Highly similar to BC|
CAI = 0.00
|Insertion||Dorsal aponeurosis in the second to fifth proximal phalanges|
|Extensor digiti quinti proprius||Origin||Lateral epicondyle of the humerus||Only one insertion tendon to little finger.|
Somewhat similar to BC
CAI = 0.22
|Fleshy portion is well detached.|
Somewhat similar to BC
CAI = 0.22
|Highly similar to BC|
CAI = 0.0
|GCAI = 0.22; somewhat similar to BC||GCAI = 0.11; similar to BC||GCAI = 0.00; highly similar to BC|
For humans, variation of the structures is very well informed, and too for great apes. However, others animals, except for domestic ones, are scarcely studied, and if any the number of the specimens or species could be small. However, calculation of the SMCA based on the previous studies, except the values of the
2.5. Comparison of the SMCA with other nonparametric statistics
Another possibility to study nominal variables is to use the cladistics method that is commonly used in evolutionary studies. This method supposes binary characteristic of data, and any other possibility could be an error, because these features are mutually exclusives [15, 16]. This method is useful to obtain objective and/or precise information of evolution-related structures regarding the absence or presence of such structure across different species. However, this characteristic limits its application to morphological analyses of structures, since it considers just two parameters; 0 for absent characteristic, and 1 for its presence. Nevertheless, this method is important in evolutionary studies, since this method might provide evolutionary information. Cladistics analyses prioritize the primitive and derivative features [15, 16], while the morphological analyses studied here (SMCA) prioritize utmost characters observed in a given structure.
We previously compared SMCA with other nonparametric methods including cladistics [7, 10]. In fact, the SMCA accept more variables to be analyzed for each structure than the cladistics method. For a more detailed comparison, see Aversi-Ferreira et al. .
It is desirable to quantitatively assess any kinds of data, even in gross anatomy [7, 10], which is important for more precise discussions and more reliable conclusions . Indeed, according to Lord Kelvin, “When you can measure what you are speaking about, and express it in numbers, you know something about it.” Our objective is to provide a statistical test for gross anatomy to numerically compare structures of different subjects within the same species and those across different species, which should be useful to analyse more precisely and objectively the data in comparative anatomy.
The SMCA is a new statistical method and requires further verification using many data. We reported SMCA analyses previously [7, 9–11] and the SMCA could satisfactorily incorporate many qualitative data numerically. In conclusion, the main features of SMCA are as follows: (1) to allow numerical description of the data shown by discrete or nominal variables in comparative anatomy or in other areas of morphology and (2) to provide a, at least, more precise (numerical) method for comparison of samples of structures from the same species and from different species. Thus, the SMCA fills the lack of an appropriate method for statistical works in comparative anatomy, and in other areas of morphology and other disciplines such as taxonomy, phylogenetic, and evolution.
T.A. Aversi-Ferreira is a recipient of Scholarship Research Productivity from National Council of Technology and Development (CNPq/Brazil). This work was supported partly by the Japan Society for the Promotion of Science (JSPS), Grant-in-Aid for Scientific Research (B) (16H04652). The authors declare no competing financial interests.
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