The most important requirements for analytical methods (a frequently used term is validation), traditionally include such a characteristic as the precision of measurement results subdivided into their repeatability and reproducibility in addition to accuracy, robustness, etc. The evaluation criterion is the minimum standard deviations of the obtained values (or coefficients of variation). Two-dimensional data arrays (first of all, calibration dependences) are characterized by linearity, the criterion of which is the maximum values of correlation coefficients (R) or the minimum values of general variances (S0) [1]. Many practice guidelines specify particular accuracy performance requirements. This is often referred to bioanalytical methods and drug control [2–4]. For example, in accordance with rules [4], the permissible deviations of the results for at least 67% samples should be within ± 15% of their nominal values; otherwise, the reasons for the deviations should be clarified.

However, in real analytical practice, there are a large number of examples of data arrays that are characterized by low reproducibility or poor correlation of variables (in other words, a high degree of uncertainty). This may be due to either a large scatter of direct measurement results or variability because of differences in the nature of objects. Spark source mass spectrometry [5, 6] is an example of an objectively large spread of signal intensities; the same features are often inherent in determinations at a trace level. The general approach to eliminating such uncertainties is to improve the technique of determinations. Variability of the second type sometimes is fundamentally irremovable, but it depends on the accepted formulations of analytical problems (in other words, on accepted analytical hypotheses). As an example of the data scatter caused by the data sample, we can cite the intensities of weak signals of so-called mass spectra of ion series [7]Footnote 1. Their standard deviations may exceed the mean values.

Another example of a weak correlation of values is the assessment of retention indices (RIs) in reversed phase HPLC based on the hydrophobicity factors (log P) of analytes [8, 9]. As a rule, the correlation coefficients of the RI−log P relationships are at a level of 0.9, and S0 (the average accuracy of the obtained estimates) is at least 50–70 index units. Nevertheless, this approach has been used to date due to the relative availability of (predominantly calculated) values of log P. Such correlations are often regarded as having only limited application or even as unsatisfactory. However, the numerous examples of a large scatter of data not only do not allow them to be neglected but also deserve special consideration of the interpretation of such information, which is the goal of this work.

To illustrate the diversity of such examples, we analyzed in detail four different problems. All of them refer to variability due to differences in the nature (properties, characteristics) of objects, and a strong dependence of the features of data interpretation on the formulation of problems under consideration is manifested in all of them. As the simplest example (one-dimensional data array), variations in the composition of developers for black-and-white negative photographic materials are considered. More complex problems include variations in the temperature regimes of gas-chromatographic analysis because the problem becomes two-dimensional when data for different analytes are combined. The toxicity of homologues is another example of large scatter. Finally, the analytically important problem of choosing sample preparation operations for determining drug traces in blood plasma based on the physicochemical characteristics of the test compounds is considered in more detail.

CHOICE OF INITIAL DATA AND SAMPLE PREPARATION FOR ANALYSIS (EXPERIMENTAL)

General characteristics of the four considered examples. To exclude the search for and comparison of information from insufficiently systematized sources of different times of publication, information on the composition of developers for black-and-white negative photographic materials (example 1) was taken from a manual [10], which was published in a period of a great variety of both negative photographic materials themselves and recipes of processing solutions for them (1964). The database of the National Institute of Standards and Technology (NIST, the United States) (http://webbook.nist.gov) [11] was chosen as a source of information on the conditions of gas-chromatographic analysis with temperature programming for five analytes (nitrobenzene, benzonitrile, 2-chlorophenol, 1,4-dimethoxybenzene, and 1,3,5-trichlorobenzene); this database contains detailed information on all compounds whose retention indices are included in it (example 2). The selected parameters include the initial temperature (Т0, °С), the duration of an initial isothermal section (t(T0), min), and the heating rate (r, K/min). Toxicity data (the values of LD50, example 3) for С3–С12 1-alkanols were taken from various literature sources [12–18]. Many of them contain duplicate values of LD50, which were not detected and rejected. Finally, the descriptions of sample preparation procedures for drugs (51 compounds of various chemical nature) are summarized on the basis of original publications and data obtained from JSC Biokad (see below) (example 4). Tables 1–4, the numbers of which are consistent with the numbers of examples, summarize experimental data corresponding to each of the examples and necessary references. The values of hydrophobicity factors (log P) and the degrees of drug binding to proteins are indicated on the Drugbank website [62].

Table 1. Concentrations of components in metol–hydroquinone developers for negative black-and-white photographic materials (according to Yashtold-Govorko [10])
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Table 2. Comparison of the parameters of temperature regimes for gas-chromatographic analysis of some organic compounds (according to NIST Standard Reference Database [11])
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Table 3.   Data on the toxicity of C3–C12 1-alkanols (LD50, mg/kg, p.o.; no indication of the animal corresponds to the use of rats)
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Table 4. Main characteristics of sample preparation operations for drug preparations depending on their hydrophobicity factors (log P) (compounds are arranged in a descending order of the calculated values of log P (ChemAxon))
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Preparation of samples for analysis. Data in Table 4, which corresponds to the last considered example and contains information on the operations of blood plasma sample preparation for HPLC determination of drugs, is supplemented with information for several drugs the analysis methods for which were developed at ZAO Biokad (St. Petersburg). Their standard solutions with a concentration of 1 mg/mL were prepared by dissolving weighed portions of the analyzed substances in a mobile phase (the composition of the solvents corresponded to the initial compositions of eluants in gradient elution modes). The blood plasma of healthy volunteers stored frozen at a temperature of no higher than –70°C was used as a matrix for preparing model solutions.

Samples for HPLC analysis were prepared according to one of the following procedures: (1) Liquid–liquid extraction (LLE) was carried out by adding an extractant to aliquot portions of the samples (1 : 3); the resulting solutions were mixed and centrifuged; the organic extracts were evaporated in a flow of nitrogen, and the dry residues were repeatedly dissolved in the mobile phase. (2) For the precipitation of blood plasma proteins, acetonitrile was used as a precipitant in a ratio of 1 : 3; the resulting solutions were mixed and centrifuged, and supernatant layers were transferred into vials for chromatographic analysis. (3) Solid phase extraction (SPE) was performed using cartridges (Oasis, Waters, the United States); samples were applied to them and various solvents were passed through the cartridges to elute impurities and target analytes separately. (4) Amicon® Ultra 3K centrifuge ultrafilters (Millipore, the United States) were used to filter plasma proteins. Specified volumes of the samples were added to test tubes with filters and centrifuged; the amounts of solutions passed through the membranes were taken for chromatographic analysis.

In all the above examples, the initial information was highly variable, and the scatter of numerical data was large so that any correction of single values in such arrays (identification of outliers) [63] was impossible. The ORIGIN software (versions 4.1 and 8.1) was used for statistical processing of the initial data and plotting. The significant variability of the initial data limits the use of more complex methods of their interpretation (factor or cluster analysis). Moreover, in all of the examples, the number of available values is insufficient for the correct application of these methods.

RESULTS AND DISCUSSION

Component composition of metol–hydroquinone developers for negative black-and-white photographic materials. It is advisable to start considering data sets with a large scatter with one-dimensional arrays to which conventional statistical processing is applicable. According to modern ideas, the best results of processing photographic materials are provided by strict reproduction of the conditions, including the composition of solutions, the duration of processing, and the temperature regime (maximal standardization). However, the situation was fundamentally different 50–60 years ago: for numerous negative photographic materials (with different spectral sensitivity), a large number of developers were used for various purposes, including normal (standard), fine-grained, contrast developers and those to reduce contrast, to increase light sensitivity, for overexposed photographic materials, for high processing temperatures, etc. All of these developers differed in combinations and concentrations of components. If we confine ourselves only to metol–hydroquinone developersFootnote 2 for negative photographic materials, Table 1 summarizes the concentrations of components in 19 of them according to a manual [10].

The purpose of such a variety of compositions is to choose the best option for processing depending on the characteristics of a particular photographic material or shooting conditions. However, another real situation can be imagined: the type of photographic material, the features of its exposure, and, consequently, the a priori requirements for processing are unknown. In other words, here we are dealing with a different formulation of the original hypothesis. In such cases, the choice of a specific developer formulation from among the previously known ones is clearly irrational; therefore, a change in the logical scheme of actions is required. From the uncertainty of the formulation of the problem follows the preferred use of a formulation the composition of which corresponds to the average amounts of all components (arithmetic averages are given in the last row of Table 1).

Variations in the concentrations of individual components (ranges) are very large: 0–14 g/L (metol), 0–45 g/L (hydroquinone), 40–250 g/L (sodium sulfite), 0–90 g/L (sodium carbonate), and 0–20 g/L (potassium bromide). Not surprisingly, the formal calculation of standard deviations leads to values that exceed their arithmetic mean values for some of them (hydroquinone, sodium carbonate, and potassium bromide). For the other two components, the relative standard deviations (coefficients of variation) are 80% (metol) and 58% (sodium sulfite). In the practice of mathematical processing of measurement results [64], if the standard deviation of a value is greater than its mean value, this means that such a value is statistically insignificant, and it can be assumed equal to zero. However, in relation to the problem under consideration, such an interpretation is unacceptable, and it should be changed: this confirms the possibility of the absence of such a component from the developer in some cases.

If the considered data arrays {xi} are obviously asymmetric (as in this case), they can be characterized by an asymmetry factor (A). One way to estimate this parameter is to calculate two standard deviations independently. The first characterizes data that exceed the arithmetic mean, s(+), and the second, accordingly, smaller, s(–). Their ratio A = s(+)/s(–) is a characteristic of asymmetry [65]. In our case, all values of A (indicated in the last row of Table 1) are significantly greater than unity (from 1.5 to 3.5). This is typical for all data arrays bounded from below by zero, while there are no such restrictions from above.

Such a scatter of data on the composition of developers deserves comment. First of all, it corresponds to an average recipe. If we consider the 11 specific compositions listed in Table. 1, the calculated average values correspond best to developers NP-3 and for equalizing image contrast. In addition, note that if we estimate the accuracy of setting the required quantities of ingredients by eye approximately as ±50 rel %, the standard deviations of the obtained mean values exceed this value. This means that it is possible to exclude an operation of taking accurate weights of components from the preparation of these developers; nevertheless, this ensures an acceptable level of quality in the processing of photographic materials.

As noted above, similar examples of the statistical processing of data characterized by a noticeable scatter occur in the calculations of the mass spectra of ion series [7]; this is determined by the nature of the chosen hypothesis (solution of problems of group mass-spectrometric identification of organic compounds).

Comparison of temperature regimes for the gas-chromatographic analysis of various organic compounds. The next most complex example of large-scatter data arrays can be represented as either a one-dimensional or a two-dimensional array. These are the gas-chromatographic separation parameters of various analytes with temperature programming, namely, the initial temperature (Т0, °С), the duration of the initial isothermal step (t(T0), min), and the heating rate (r, K/min). These data from various sources of information are available on the website of the US National Institute of Standards and Technology (NIST) (http://webbook.nist.gov) [11], where they are given for all compounds characterized by retention indices. Any compounds can be chosen to illustrate the features of such information; however, to reduce the volume of discussion, we restrict ourselves to five analytes: nitrobenzene, benzonitrile, 2-chlorophenol, 1,4-dimethoxybenzene, and 1,3,5-trichlorobenzene. Table 2 summarizes some of the values of Т0, t(T0), and r for them.

As in the previous example, noticeable variations in all characteristics of the programming modes attract attention. In this case, the standard deviations of the parameters Т0 and r do not exceed the corresponding average values, which cannot be said about the duration of the initial isothermal section (r). From the general point of view, one could assume that the initial temperatures T0 could be somehow related to the chemical nature of analytes, for example, to the values of their retention indices. However, a check of the RI–T0 relationship shows that these parameters weakly correlate with each other (Fig. 1) and we can only consider a weakly pronounced tendency to increase the initial temperatures of programming regimes for analytes with higher retention indices. This is especially noticeable if the values of Т0, t(T0), and r are not preliminarily averaged for each compound (Fig. 1a), but points corresponding to all values of these parameters are plotted on a graph (Fig. 1b). The correlation coefficient (R) in case (b) is only 0.075, which confirms the absence of a linear relationship between the variables.

Fig. 1.
figure 1

Graph illustrating the absence of a linear dependence of the retention indices (RIs) of analytes on the initial temperatures of the temperature-programming mode (T0) chosen for their gas-chromatographic separation: (a) based on the average values of Т0 and (b) according to all data in Table 2. The parameters of the linear regression RI = aT0 + b are a = 0.18 ± 0.45, b = 1047 ± 28, R = 0.053, and S0 = 73.

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If so, the overall average values and standard deviations of the analytical parameters given in the last row of Table 2 can be calculated and used instead of the average values of Т0, t(T0), and r for each of the analytes. Because approximately 95% of the sample values should fall within ±2sRI intervals, these data correspond to a range of Т0 from 10 to 100°С (no outliers), t(T0) to 8 min (one outlier, 10 min), and r to 11 K/min (no outliers). The coefficients of asymmetry (A) of all these parameters ranged from 1.6 to 3.5 (the last row of Table 2) for the same reason as in the previous case. Hence, an unexpected conclusion follows: when choosing temperature programming conditions for various compounds, the values of Т0, t(T0), and r can be specified in wide ranges, which do not affect the separation of analytes. The most that can be affected by a suboptimal choice of separation conditions (for example, too small values of T0, large t(T0), and low r) is an unjustified increase in the duration of the analysis. Sometimes, the expression development of a procedure (for chromatographic analysis or, more precisely, separation) can be used as applied to the choice of separation conditions. However, taking into account the conclusion on the relatively large variations in the parameters Т0, t(T0), and r, it should be recognized that it would be more correct to speak about the choice of certain values from the allowable ranges.

Characterization of the toxicity (LD50) of homologues based on the example of С3–С12 1-alkanols. The following example can also be considered as both one-dimensional and two-dimensional data sets. For the LD50 values of specific compounds, conventional statistical processing is formally possible (if this is consistent with the accepted hypothesis), and a comparison of such data for several homologues turns the array into a two-dimensional array when the least squares method is necessary to calculate the parameters of regression equations. The toxicity of various chemical compounds is one of their most important characteristics, but its determination (especially in vivo) is a very long, laborious, and expensive operation. Numerous attempts are known to theoretically estimate the toxicity parameters (LC50 or LD50); however, we restrict ourselves to a few references [66–70] because this is not the subject of this work. At the same time, it is important to note that the calculation methods, one way or another, are based on experimental data, but if their scatter is large, the low accuracy of all estimates inevitably follows from this [63].

Table 3 shows data on the toxicity (LD50, mg/kg, oral) of C3–C12 1-alkanols for various warm-blooded animals (rat, rabbit, mouse, hamster, dog, (without breed specification), and bird (the same) and one value for humans (1-propanol)). Here, the significant scatter of the data attracts attention. If we try to average the values of LD50 for С3–С6 alcohols (which were characterized in most detail) separately, we obtain estimates for which the coefficients of variation are 45–80% (shown in Table 3). In addition, it is noteworthy that the data samples for C4 and C5 alcohols are almost symmetrical; the asymmetry factors for them are close to unity: 0.90 and 0.97, respectively. The situation with data scatter did not fundamentally change when we confined ourselves to only one species of animals (rats were represented in most detail). For this reason, we can subsequently consider the entire set of objects without any additional subdivision. Table 3 uses the same principle of citing literary sources as that reported previously [71]: references are given to specific works, but sometimes the SDS pages of companies producing particular chemical reagents available on the Internet are mentioned.

This example illustrates the paradoxes that can result from the interpretation of data characterized by high variability. If we calculate the arithmetic mean values of LD50 for each of the С3–С12 alcohols, they correspond to a smooth sinusoidal curve with two extremums in the graph (Fig. 2a). If we consider the entire set of data without preliminary processing (Fig. 2b), their graphical representation illustrates a significant scatter. The authors cannot fail to note the fact that the opinions of specialists regarding the possibility and impossibility of characterizing the toxicity of homologues by average values are divided approximately in equal proportions. However, it is important to emphasize that this choice is also a consequence of one or another initial hypothesis. Considering that the initial values of LD50 for each of the alcohols were obtained for different animals, their averaging is unacceptable. If we neglect such information, it is acceptable to average the data for not only each of the homologues but also for their entire set. In this case, the entire set of LD50 values can be characterized by the overall mean value and its standard deviation: (2600 ± 1600) mg/kg; on this basis, we can conclude that the available data do not confirm the dependence of the toxicity of C3–C12 1-alkanols on the number of carbon atoms in the molecules. This conclusion is of a general nature: if the standard deviation of the coefficient a for the linear regression y = ax + b exceeds its absolute value, sa > ∣a∣, usual data averaging should be preferred.

Fig. 2.
figure 2

(a) Graphical illustration of the dependence of the arithmetic mean LD50 values of each alcohol on the number of carbon atoms in 1-alkanol molecules; (b) the same for all initial data given in Table 3. The parameters of the linear regression (trend line) LD50 = anC + b are a = 78 ± 73, b = 2200 ± 400, R = 0.10, and S0 = 1600.

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Selection of sample preparation operations in the determination of drugs in blood plasma depending on their hydrophobic–hydrophilic properties. The last considered example of processing and interpreting data characterized by a large scatter is the most complex. First, it refers to two-dimensional arrays, and one of the variables should be chosen artificially (by introducing the ranks of sample preparation operations). The second factor is the relevance of the problem under consideration. The determination of drugs in biological fluids is a very labor-intensive task primarily due to the complexity of a matrix and, as a result, a sample preparation stage. Because reversed phase HPLC is the preferred analytical method for solving such problems, it is necessary to ensure not only the preconcentration of target analytes at this stage but also the removal of interfering components, primarily, proteins. Sample preparation most often includes the following operations: liquid–liquid and solid-phase extraction, precipitation of protein components, ultrafiltration, ion exchange, and, less often, other operations. The selection and optimization of sample preparation operations (or their combinations) is still carried out on the basis of general concepts of the nature of the compounds to be determined. This results in a large amount of time, which could be minimized if it were possible to relate the nature of sample preparation operations to the physicochemical characteristics of the analytes.

Of these characteristics, the hydrophobicity factor log P deserves attention in the first place. However, the experimental values of log P are known not for all characterized compounds. They can be replaced by calculated estimates, in our case, calculated using the ChemAxon software [19]. When such estimates are obtained, the values of log P(ChemAxon) < –2 can be excluded from consideration because there are no examples of the use of LLE for such hydrophilic compounds. Figure 3 illustrates checking the equivalence of calculated (indicated in Table 4 for some drugs) and experimental values (13 pairs of values) and shows that the coefficient a of the linear regression log P(expt) = alog P(ChemAxon) + b is only slightly lower than unity (0.89 ± 0.09), while the coefficient b is statistically insignificant (0.27 ± 0.31). Therefore, the experimental and calculated values of log P can be considered equivalent in some approximation. However, the average accuracy of the resulting estimates is not high because the value of S0 for the linear regression shown in Fig. 3 is 0.69. Moreover, with the use of the calculated values of log P, it is undesirable to use the values of log P calculated in other ways (for example, using the ACD software).

Fig. 3.
figure 3

Correlation between the experimental and calculated (ChemAxon) values of log P. The parameters of the linear regression log P(expt) = alog P(ChemAxon) + b are a = 0.89 ± 0.09, b = 0.28 ± 0.31, R = 0.944, and S0 = 0.69.

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Table 4 compares the most important sample preparation conditions for 51 drug preparations; molecular weight, CAS no., log P(ChemAxon), and a fraction of drug bound to plasma proteins are specified for each particular drug. A reference to the original publication is provided for each version of sample preparation; the absence of a reference (10 preparations) indicates that the procedure was developed and validated at JSC Biokad.

First of all, let us comment on such a characteristic as the fraction of analytes bound to plasma proteins (varies from 2 to 99%), which directly affects their detection limits. Checking a possible relationship of this characteristic with a sample preparation operation such as protein precipitation shows that there is no significant correlation here. The average value of the bound fraction of the analyte with the use of precipitation is (89 ± 16)%, and it is (70 ± 33)% in the absence of this operation (according to the data of Table 4). Thus, the case in point is only a weakly expressed trend, which can be ignored.

For the subsequent interpretation of data, it is necessary to assign tentative codes (ranks) to various sample preparation operations, which will allow us to apply a method somewhat similar to Spearman’s rank correlation. The tentative zero value of a numerical code can be assigned to the protein precipitation operation because it is least of all related to the hydrophobic–hydrophilic properties of the target analytes by its very nature. SPE (code, +2) is used for the preconcentration of the most hydrophobic compounds, and the code can be equated to +1 in the case of less hydrophobic compounds, when the use of LLE is acceptable (the solvents can be varied). Such an operation as ion exchange is applicable only to compounds that exist in the ionic form (the most hydrophilic compounds; code, –2). Then, the ultrafiltration procedure receives the code (–1). Thus, we obtain the following set of variables for rank correlation:

Sample preparation operation

SPE

LLE

Precipitation

Ultrafiltration

Ion exchange

Numerical code

+2

+1

0

−1

−2

As in the previous examples, the results of checking the possible relationship between the values of log P and the proposed numerical codes can be most clearly presented graphically (Fig. 4). From Fig. 4, it follows that, firstly, two versions of extraction (LLE and SPE) are much more popular methods of sample preparation compared to the others; this is mainly due to the hydrophobic properties of the characterized drugs. However, even with such an uneven distribution of points as that in Fig. 4, we can conclude that there are no examples of the use of ultrafiltration and ion exchange at log P > –1 or (alternative wording) only LLE or SPE are used at the stage of sample preparation. The LLE and SPE operations are fundamentally inseparable, and a choice between them is determined primarily by the availability of appropriate equipment, materials, and solvents. Note that LLE at log P < 0 is fundamentally possible but much more difficult to perform, and there are no examples of the use of this extraction in the region of log P < –1 among the drugs under consideration.

Fig. 4.
figure 4

Graphical illustration of a correlation of the numerical codes of the main sample preparation operations in the determination of drug traces in blood plasma with the values of log P (ChemAxon).

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CONCLUSION

Thus, the examples of data arrays with a large levels of variations are widespread. The most difficult are the examples of variability due to differences in the nature of the objects themselves. In these cases, data interpretation is complicated by the influence of the accepted analytical hypotheses. Such data arrays with high degrees of uncertainty are often excluded from consideration, and this is not always justified. The processing of these data has its own characteristics. As a rule, one-dimensional sets {xi} are characterized by high asymmetry. Large values of relative standard deviations δi = sx/\(\left\langle x \right\rangle \) can be interpreted as the absence of the need for precise control of the values of variables x, which can be replaced by a choice from a wide range of their possible values. In contrast to the rules of conventional statistical processing, the condition sx > (\(\left\langle x \right\rangle \)) does not mean that the average value \(\left\langle x \right\rangle \) is zero but that some of the values {xi} of the sample can be equal to zero. If the standard deviation of the coefficient a for the linear regression y = ax + b exceeds its value (sa > ∣a∣) as a result of processing some data arrays by the least squares method, the usual averaging of the data with an estimate of \(\left\langle x \right\rangle \) ± sx should be preferred. The probabilistic nature of the conclusions based on data with high degrees of uncertainty, as a rule, is higher than that for data with normal reproducibility. At the same time, important conclusions can be drawn even from the very facts of high variability of variables. For example, Zenkevich and Deruish [72], who characterized the dependence of the retention indices of analytes in reversed phase HPLC on the methanol content of the eluant (dRI/dc), found that these parameters, in contrast to the retention indices themselves, do not correlate with the values of log P; this fact makes it possible to exclude such a correlation from the subsequent consideration.