UDC 614.7:66]:574.3:579.26

K. V. Geniatulin

Application of Graphical Methods of Mathematical Statistics to the Analysis of the Effects of Chemical Environmental Pollutants on Microorganisms

All-Union Scientific Research Center for the Study of Surface and Vacuum Properties, USSR State Committee for Standards, Moscow

A number of reports have described in detail the possibilities for monitoring various environmental pollution factors by measuring the response of sensitive microorganisms to their effects (Bagdasaryan G. A., Geniatulin K. V., 1981, 1982). It was noted that such measurements of the reaction (response) of microorganisms must be supported by appropriate mathematical and metrological methods.

The present paper uses a number of examples to examine the potential of certain graphical methods of mathematical statistics for describing the effects on microorganisms of chemical substances, which are among the most characteristic environmental pollutants. Graphical methods of statistical data analysis are not yet used in the everyday practice of researchers working in biology and medicine. Only a few specialists, particularly those concerned with the application of mathematical experimental design methods in biology (Maksimov V. N., 1980) and industrial experimentation (Daniel C., 1979; Johnson N., Leone F., 1981), are making increasing efforts to draw them to the attention of a broad range of researchers.

The statistician’s principal task is to help the experimenter obtain reliable—that is, better-substantiated—research results. This is generally achieved by carrying out complex statistical analyses, such as a full analysis of variance. Although the latter is one of the most sophisticated and effective methods of statistical data analysis, it nevertheless involves extremely cumbersome calculations. Wherever and whenever possible, simpler—but no less reliable—methods should be used, including graphical methods.

We examined the potential of two graphical methods for determining the significance of the effects of a number of chemical substances (detergents, petroleum, and heavy-metal salts) on microorganisms (E. coli bacteria, viruses, and phages). The survival of these microorganisms under the action of individual chemical substances and their combinations was used as the response variable (Nedachin A. E., 1976).

The idea of the first graphical method (Daniel C., 1959) is that, if the measured response of the process under study (the survival time of microorganisms) follows a normal distribution, then on half-normal probability paper all insignificant effects (of the factors under study) should lie on a straight line passing through the origin, while significant effects will lie away from it. All effects plotted on the graph (calculated, for example, using Yates’s method) are ranked by absolute magnitude. Instead of the effects themselves, the corresponding regression coefficients or mean effects may be ranked. Table 1 gives detailed experimental conditions, the results of processing the data using Yates’s algorithm, the values of the regression coefficients obtained for the equation of the process under study, the mean effects, and their ranking for a \(2^4\) full factorial experiment. The ranked values of the mean effects (the penultimate column of Table 1) are plotted on half-normal probability paper, whose vertical scale differs from the corresponding scale of normal probability paper as follows: \(P'=2P-100\), where \(P\) is the normal distribution function and \(P'\) is the distribution function used on half-normal probability paper. The absolute values of the mean effects are plotted on the abscissa, and their rank numbers on the ordinate; these are related to \(P\) as follows:

\[ P'=\frac{i-0.5}{n},\qquad i=1,2,\ldots,n, \]

where \(n\) is the total number of effects.

The graph (Fig. 1) shows that some effects deviate noticeably from the straight line passing through the origin and the remaining, larger group of insignificant effects (more precisely, through the points corresponding to those effects). Thus,

Table 1

Effect of sodium alkyl sulfate (AS) and other factors on virus survival

Note. For factor $x_1$, the lower level (−) is 0.5 mg/L and the upper level (+) is 20 mg/L; for $x_2$, river water and seawater; for $x_3$, isolation and presence in association; and for $x_4$, concentrations of $10^5$ and $10^6$. The algorithm step is indicated in parentheses.

…Thus, in the example considered, graphical analysis of the effects of the detergent (AS), the composition of the aqueous medium (river water or seawater), the conditions of virus isolation from other microorganisms (E. coli bacteria and phage T₁), and virus concentration showed that the effects of aqueous medium ($x_2$) and virus concentration ($x_4$), as well as the interaction effect ($x_1x_3$), were significant. This indicates that the absence of a significant effect of the chemical substance (AS) does not necessarily mean that this factor has no influence on the process under study. This is clearly seen from the equation representing a mathematical model of the process that includes only significant effects:

$$ y=48.2-4.7x_2+2.2x_1x_3+8.1x_4. $$

It can be seen that the effect of AS ($x_1$) is determined by the relationship between $x_1$ and $x_3$: when both are at the upper (+1) or lower (−1) level, their interaction increases the process response; otherwise, it decreases it.

To make the differences in effect magnitudes even clearer, some authors (Johnson N., Lyon F., 1981) give their mean squares. These are shown in the corresponding column of Table 1. Graphs showing the confidence limits of the effects also serve this purpose (Fig. 2). For this, the data are normalized: each value of the mean…

Fig. 1. Half-normal distribution plot (based on the data in Table 1).

The horizontal axis shows the effect (in relative units); the vertical axis shows the rank. $x_1x_3$ is the interaction effect of the factors; $x_2$ and $x_4$ are factor effects.

Fig. 2. Standard plot for determining the significance limits of the effects shown in Fig. 1.

The horizontal axis shows normalized effect values (in relative units); the vertical axis shows the rank; $\alpha$ is the significance level.

…of the effect must be divided by an estimate of the variance. As an approximation to the variance, the number whose rank is closest to \(0.683n+0.5\) is selected (Johnson N., Lyon F., 1981). In our case, a full factorial experiment \(2^4\), where the number of effects \(n\) is 15, this rank is 11. The mean effect at this rank is 2.5. Dividing each mean effect by 2.5, we use the resulting data (the last column of Table 1) to construct a normalized half-normal probability plot (see Fig. 2).

On the standard plot, the lines were positioned as follows: the central line passes through the points \((0,0)\) and \((\tau,n)\), while the boundary of the significance interval passes through the point \((\tau+d,n)\) and the intersection of the central line with the horizontal line whose ordinate is rank 11. As the plot shows (see Fig. 2), all three points corresponding to the largest normalized effects lie beyond the line at the significance level \(\alpha=0.05\). We can therefore conclude that the effects of factors \(x_2\) and \(x_4\), as well as the interaction of factors \(x_1\) and \(x_3\), are significant.

The graphical method not only identifies significant effects but also makes it possible to assess the reliability of the experiment as a whole: whether it is free of systematic errors and whether the selected process response follows a normal distribution. The absence of errors is indicated by the straight line passing through the origin. A departure from the normal distribution is indicated by a curved rather than a straight line on the plot.

Another advantage of the graphical method is that no preliminary determination of the reproducibility variance is needed to establish the significance of the regression coefficients in the mathematical model of the process. The method therefore permits the evaluation of regression coefficients even in the not uncommon cases where a factorial experiment is performed without replication. This may occur for a variety of reasons: the experiment may have been conducted under full-scale production conditions, or it may have been performed comparatively long ago, so that the results of intermediate calculations, including data from replicate trials, may have been lost, leaving, as a rule, only averaged data in the article or report, and so on.

It should be borne in mind, however, that the graphical method works best for experiments with four or more factors—that is, when there are at least 15 effects. Another advantage of large experimental designs is that they permit a more thorough examination of the data.

Figure 3 presents a plot for a five-factor experiment \((2^5)\), in which the factors affecting the microorganisms (viruses) included a combination of three chemical substances (AS, oil, and copper salt). The experimental design matrix and Yates algorithm do not differ in principle from those presented in Table 1, except that the number of runs is 32 and the algorithm has one additional step. The reasoning leading to the results shown in Figs. 3 and 4 therefore remains the same as for the plots in Figs. 1 and 2. In Fig. 4, seven points corresponding to the seven largest (significant) effects are clearly separated. On the standard plot, all of them lie beyond the line at the 0.05 significance level. A mathematical model of the process is presented on the basis of the significant effects alone: \(y=47.9+4.2x_1+4.8x_2+2.3x_1x_2-2.5x_3+2.6x_4-2.1x_3x_4+2.8x_5\). We, like a number of researchers (Maksimov V. N., 1980, and others), hold that a model based only on significant effects is closer to the actual response function. The model gives a complete picture of the influence of all five factors studied on the process response: because of the positive interaction effect, the positive effect of either factor—\(x_1\) (AS) or \(x_2\) (oil)—is strengthened as the level of the other increases; because of the negative interaction effect between \(x_3\) (copper salt) and \(x_4\) (the aqueous-medium factor), the effect of the negative factor \(x_3\) is strengthened as the level of \(x_4\) increases, while the effect of the positive factor \(x_4\) diminishes as the level of \(x_3\) increases. The effect of the positive factor \(x_5\) (the virus-isolation factor) increases the process response.

Only through the use of mathematical research methods have we been able to fit all the information on the effects of chemical substances on microorganisms into a few lines. The difficulty lies solely in interpreting this information. This creates a need for continuous contact between specialists in biomedical research and specialists in technical and mathematical fields.

Of particular interest in determining the significance of regression coefficients in equations relating the process response to the effects of four or more factors in unreplicated experiments is a graphical solution based on a somewhat different approach: the use of normal probability paper. A set of residuals—the differences between the experimental results and the calculated values, \(y_i-\hat y_i\)—is plotted on this paper. The values \(\hat y_i\) are calculated using the inverse Yates algorithm for the large (significant) effects. When such effects are present, the plotted residuals should, on average, fall along a straight line, thereby…

Fig. 3. Half-normal distribution plot (based on data from a \(2^5\) design).

The abscissa shows the effect (in relative units); the ordinate shows the rank. \(x_1\)–\(x_5\) are factor effects; \(x_1x_2\), \(x_3x_4\) are factor interaction effects.

Fig. 4. Standard plot for determining the significance limits of the effects shown in Fig. 3.

The abscissa shows the normalized effect value (in relative units); the ordinate shows the rank; \(\alpha\) is the significance level.

…would constitute a set of normally distributed independent variables. This method is also called the method of cumulative distributions constructed from independent standardized normal deviates (Daniel, C., 1979). The complete computational algorithm comprises the following stages: 1) Yates’s algorithm for obtaining the effects; 2) the reverse Yates algorithm, including the large effects, to obtain \(\hat y_i\); 3) calculation of the residuals \((y_i-\hat y_i)\); 4) a test of the normality of the residual distribution \((y_i-\hat y_i)\); 5) plotting the residuals against \(\hat y_i\).

According to C. Daniel (1979), the study of the behavior of sets of residuals should be based not on an examination of the variances and covariances of ordinary order statistics, but on direct examination of substantial sets of empirical cumulative distributions constructed from samples of a given size. Conventional analysis of variance, widely used in discussing the results of multilevel factorial designs, appears less effective for experiments of the \(2^n\) type (much of the information is lost when the results are reduced to analysis-of-variance tables).

Table 2 presents the results of calculations for the first three steps of the computational algorithm described above. In the plot shown in Fig. 5, the residuals obtained are plotted on the vertical lines of normal probability paper. On average, a straight line is obtained, which should mean that the cumulative distributions of the residuals for the \(2^4\) design we selected are consistent with choosing the five largest effects as significant. The process model can therefore be written as \(y=53+2.6x_1+1.2x_2-6x_3-1.1x_4-1.5x_1x_4\). This equation shows how the various factors affect the process response (virus survival): the effects of factors \(x_2\) (AC) and \(x_3\) (the aqueous medium) can be characterized by their regression coefficients or main effects (the numerical value indicates each factor’s contribution to the process response, and the sign indicates the direction of its effect). The effects of factors \(x_1\) (oil) and \(x_4\) (the isolation of viruses from other microorganisms), however, can be characterized only by their partial effects, since there is an interaction between them, characterized by the regression coefficient of \(x_1x_4\). Since the interaction effect is negative,

Table 2

Results of standard calculations for a \(2^4\) design (effects of AS, oil, aquatic environment, and isolation on the survival of polioviruses)

Note. Column (0) contains the largest regression coefficients from column (4)/16; all insignificant coefficients are set to 0. The factors (AS, oil, aquatic environment, isolation) had lower levels of 0.5, 0.3 mg/L, “river water,” and “in isolation,” respectively, and upper levels of 10, 3 mg/L, “seawater,” and “in association with other microorganisms,” respectively.

The effect of the positive factor \(x_1\) weakens, while the effect of the negative factor \(x_4\) intensifies.

It should be borne in mind that all the process equations presented above used coded variables \(x_i\) (equal to \(+1\) or \(-1\)). These can readily be converted to natural variables using the transformation formula \(x_i'=x_{i0}+\lambda_i x_i\), where \(x_i'\) is the natural variable, \(x_{i0}\) is its value at the “center of the experiment,” \(\lambda_i\) is the unit of variation, and \(x_i\) is the coded variable in our equation. As a rule, the resulting equation for the process under study is more conveniently analyzed in coded variables.

Fig. 5. Empirical cumulative distribution of residuals for the \(2^4\) design. The horizontal axis shows the ordinal number; the vertical axis shows residuals (in relative units).

Fig. 6. Residuals plotted against \(\hat y\) for the \(2^4\) design. The horizontal axis shows predicted values \(\hat y\) (in conventional units); the vertical axis shows residuals (in relative units).

Fig. 7. Empirical cumulative distribution of residuals for the \(2^5\) design.
The horizontal axis shows the ordinal number; the vertical axis shows the residuals (in relative units).

Fig. 8. Residuals versus \(\hat{y}\) for the \(2^5\) design.
The horizontal axis shows the values of \(\hat{y}\) (in relative units); the vertical axis shows the residuals (in relative units).

Figure 6 shows the residuals plotted against \(\hat{y}\), in accordance with the fifth step of the algorithm. Ideally, there should be no evident relationship between them (for example, an increase in the residuals as \(\hat{y}\) increases). This requirement is satisfied in the present example.

Figure 7 shows the distribution of residuals for the \(2^5\) design, whose analysis (using the first graphical method) and mathematical model are presented in Fig. 3. The effects selected as large (significant) were those represented in Fig. 3 by points that do not lie on the straight line. As for the \(2^4\) design, calculations were performed using the forward and reverse Yates algorithms (the latter included the significant effects identified in the plot in Fig. 3), and the graph in Fig. 7 was constructed from the calculated residuals. Figure 8 shows the residuals plotted against \(\hat{y}\). A comparison of the results obtained by the two graphical methods for analyzing the effects of the same factors on the microorganism parameter under study—in our case, the survival time of type I polioviruses—showed reasonably good agreement. These graphical methods, which possess the requisite mathematical rigor, are readily interpretable, and shorten the time required for an investigation, can be used either as a primary, independent method or as an auxiliary method for preliminary assessment, after which additional experiments and mathematical and statistical analysis can be carried out.

In addressing the problem of using microorganisms to monitor environmental pollution, which we considered in a series of studies (Bagdasaryan G. A., Geniatulin K. V., 1981, 1982), the graphical methods described here may play an important role in developing a quantitative scale for assessing the degree of environmental pollution (Geniatulin K. V., Bagdasaryan G. A., 1981). These methods are intended to identify significant effects of various environmental pollution factors on test microorganisms, as measured by various parameters of their vital activity.

References

Bagdasaryan G. A., Geniatulin K. V. Gigiena i sanitariya, 1981, no. 11, pp. 11–13.

Bagdasaryan G. A., Geniatulin K. V. Ibid., no. 12, pp. 55–57.

Bagdasaryan G. A., Geniatulin K. V. Gigiena okruzhayushchei sredy (Ekspress-informatsiya), 1982, no. 9, pp. 1–29.

Geniatulin K. V., Bagdasaryan G. A. In: Measurements in Medicine and Their Metrological Support. Moscow, 1981, pp. 202–204.

Maksimov V. N. Multifactorial Experimentation in Biology. Moscow, 1980, p. 61.

Nedachin A. E. In: Hygienic Aspects of Environmental Protection. Moscow, 1976, issue 4, pp. 90–92.

Daniel C. Applications of Statistics to Industrial Experimentation. Moscow, 1979, pp. 81–139.

Johnson N., Leone F. Statistics and Experimental Design in Engineering and the Physical Sciences. Moscow, 1981, p. 228.

Received March 4, 1983.