Statistical comparisons of heavy metal pollutants between seven regions of the Polish exclusive economic zone

Abstract

This paper addresses three intractable difficulties associated with the statistical analysis of compositional data, such as percentages or ppm. These are: (1) that such data do not follow multivariate normal distributions thus rendering inappropriate, standard parametric statistical tests and estimation procedures, (2) the covariance/correlation coefficients between specific pairs of components are determined in whole or in part by the presence or absence of other components, and, (3) the negative bias property. That is, at least one covariance and therefore at least one correlation, must be negative, hence the remaining correlations are prevented from ranging freely between −1 and +1. It follows that correlation coefficients formed from compositional data are not only not absolute, but also frequently spurious. Standard multivariate procedures based on them are unreliable, and intrinsic associations between components inferred from strong positive correlations in particular, are potentially false. In a recent 2009 paper, it was reported that 59 surface sediment samples from 7 regions in the Polish exclusive economic zone had been chemically analyzed for 16 elements. Enrichment factors together with crude correlation coefficients between selected elements were presented. All these quantities were computed from the initial raw compositional data resulting from the chemical analyses In this paper, a statistical procedure is presented which is distinctly different to the enrichment factor computations based on the same raw compositional data. The procedure generates a log-ratio measure of the abundance of each element in each of the seven regions, thus enabling comparisons of relative levels of pollution between the regions. Although the two techniques are quite unrelated, it is shown that in general, extremely high or low measures of the relative abundances in the regions are associated with correspondingly high or low values of the enrichment factors in the same regions that were reported in the 2009 paper. That is, the statistical analysis confirms the results of the enrichment factor data in the identification of the most to the least polluted regions. In an additional analysis, the residue term was excluded from each sediment sample by rescaling the 16 element concentrations to sum to 100%, thus forming 59 residue-free sub-compositions. Crude correlation coefficients were computed for pairs of elements of this sub-compositional data. These revealed that certain correlations based on the initial raw data that were reported in the 2009 paper for the same pairs of elements, were not only inconsistent, but sometimes also contradictory. Such contradictions imply that intrinsic geochemical element associations inferred in that paper from such correlations were false.

Appendix

Perturbations

Let $x_{0} + y_{0} = 1$ be a two-part composition where x ₀ is the initial proportion of some element X in a region (e.g., Fe), and y ₀ is the proportion of Y, the entire remainder. Subsequently, due to either natural attrition or augmentation, the initial proportions x ₀ and y ₀ are perturbed by factors a ₁ and b ₁, respectively. The new proportions of X and Y, are related to x ₀ and y ₀ since by definition, x ₁ ∝ a ₁ x ₀, and y ₁ ∝ b ₁ y ₀. However, $x_{ 1} + y_{ 1} = 1$, necessarily, hence,

$$ x_{ 1} = a_{ 1} x_{0} /\left( {a_{ 1} x_{0} + b_{ 1} y_{0} } \right) \, \;{\text{and}}\;\;y_{ 1} = b_{ 1} y_{0} /\left( {a_{ 1} x_{0} + b_{ 1} y_{0} } \right), $$

Forming a log-ratio of x ₁ and y ₁, the denominator (a ₁ x ₀ + b ₁ y ₀) cancels and,

$$ { \log }\left( {x_{ 1} /y_{ 1} } \right) = { \log }\left( {a_{ 1} x_{0} /b_{ 1} y_{0} } \right), $$

$$ { \log }\left( {x_{ 1} /y_{ 1} } \right) = { \log }\left( {x_{0} /y_{0} } \right) + { \log }\left( {a_{ 1} /b_{ 1} } \right) $$

If over time there were n sequential perturbations $\left( {a_{ 1} ,b_{ 1} } \right), \, \left( {a_{ 2} ,b_{ 2} } \right), \ldots , \, \left( {a_{\text{n}} ,b_{\text{n}} } \right),$ then by the same algebra, the second log-ratio is given by,

$$ { \log }\left( {x_{ 2} /y_{ 2} } \right) = { \log }\left( {x_{ 1} /y_{ 1} } \right) + { \log }\left( {a_{ 2} /b_{ 2} } \right) $$

Substituting in this for log(x ₁/y ₁) from above,

$$ { \log }\left( {x_{ 2} /y_{ 2} } \right) = { \log }\left( {x_{0} /y_{0} } \right) + { \log }\left( {a_{ 1} /b_{ 1} } \right) + { \log }\left( {a_{ 2} /b_{ 2} } \right) $$

Similarly for log(x ₃/y ₃), log(x ₄/y ₄), and so on to the n-th log-ratio which is given by,

$$ { \log }\left( {x_{\text{n}} /y_{\text{n}} } \right) = { \log }\left( {x_{0} /y_{0} } \right) + { \log }\left( {a_{ 1} /b_{ 1} } \right) + { \log }\left( {a_{ 2} /b_{ 2} } \right) + \cdots + { \log }\left( {a_{\text{n}} /b_{\text{n}} } \right) $$

That is, log(x _n /y _n ) is formed from the initial log(x ₀/y ₀) plus the sum of these n effects. Assuming n is sufficiently large, and given certain mild regularity conditions, the Central Limit Theorem of probability theory predicts that the distribution of a family of observations like log(x _n /y _n ) will follow the bell-shaped curve of the normal distribution. Denoting a real observation of X by x and the remainder Y, by y, we have y = 1 − x. The log-ratio value, u, for x is u = log[x/(1 − x)]. For a percentage, v = 100x, we have u = log[v/(100 − v)], which defines the transformation of all element concentrations in the statistical analysis. Assuming all such values x are the results of a continuous history of natural perturbations, we treat the log-ratios as normally distributed, or approximately so. In any event, the t statistic based on that assumption is robust.

The linear model and the t statistic

The linear model adopted for the seven regional means of the log-ratios of each element is described as follows. Denoting the mean log-ratio of an element in region i by $\mu_{i} ,\,i = { 1},{ 2}, \ldots , 7,$ the pooled mean μ for the element is defined to be $\mu = \, \left( {\mu_{ 1} + \mu_{ 2} + \cdots + \mu_{ 7} } \right)/ 7$. So that by setting a regional mean $\mu_{i} = \mu + \alpha_{i} ,$ each α _i measures the deviation of μ _i from the pooled mean μ. Adding these seven equations it follows that $\alpha_{1} + \alpha_{2} + \cdots + \alpha_{7} = 0$ necessarily. This is a sum-to-zero constraint. One further assumption is that the variability in all the measurements of an individual element is assumed to be accounted for by a constant variance, σ ² (the homoscedastic assumption), the parameters $\mu ,\alpha_{1} ,\alpha_{2} , \cdots \alpha_{7}$ are estimated by ordinary least squares regression, such that the estimates all obey the equations above. Consequently, unless they are all zero, the estimates for the deviations must include positive and negative values, indicating abundances above and below the pooled mean. If â _i denotes the estimate for each α _i then a t statistic, $T_{i} = \left( {\hat{a}_{i} - \alpha_{i} } \right)/s_{i}$ is defined for any given α _i (s _i is the so-called standard error of the estimate). Assuming an underlying normal distribution of log-ratios, the expected value of T _i is zero, but its observed value, which varies with â _i will be assessed by whether or not it falls within or outside a range of probable values for the t statistic. In this study, it was assumed that each α _i  = 0 implying that for each element, the regional averages μ _i were all equal to the pooled average μ. When α _i  = 0, the expression for the observed t statistic becomes, $T_{i} = \left( {\hat{a}_{i} - 0} \right)/s_{i}$ or $T_{i} = \hat{a}_{i} /s_{i} ,\;i = { 1},{ 2}, \cdots ,{ 7}$. So it follows that an extremely large positive or negative value of â _i will result in a large positive or negative value of T _i . In that case, the assumption that the true deviation, α _i  = 0, may be improbable. Representative two-tailed probabilities for the t statistic with 52 degrees of freedom are set out in Table 2. For example, the second column of Table 2 defines the 5% “significance level” for this study. A frequentist interpretation of the Prob(|T| > 2) < 5.0% as indicated, would be that 95% of values of the t statistic lie between −2 and +2, so an observed value outside that range is “unlikely”, with only a 1 in 20 chance. Note, a 5% significance level is conventional in many fields and hence included here. Tail-end probabilities are very close to zero for |T| > 4.5. That is, 100.00% of the values of T are in the range −4.5 to 4.5. Observed values outside that range are extremely rare unless |α _i | ≫ 0, indicating a large deviation from the pooled mean.

Statistical notes

In the equations below, y is the 59 × 1 column vector of the log-ratios of the element concentrations, X is the 59 × 7 incidence (or design) matrix, b is the 7 × 1 vector of estimates for μ and six of the α _i (the sum to zero constraint determines the 7th), ${\mathbf{b}} = \left( {{\mathbf{X}}^{\text{T}} {\mathbf{X}}} \right)^{ - 1} {\mathbf{X}}^{\text{T}} {\mathbf{y}}$, and $\left( {{\mathbf{y}} - {\mathbf{Xb}}} \right)$ is the 59 × 1 vector of residuals.

The estimate for σ ² is given by $s^{2} = \frac{1}{n - p}\left[ {\left( {{\mathbf{y}} - {\mathbf{Xb}}} \right)^{\text{T}} \left( {{\mathbf{y}} - {\mathbf{Xb}}} \right)} \right]$ where n = 59 and p = 7. The i-th diagonal element of the 7 × 7 matrix $\left( {{\mathbf{X}}^{\text{T}} {\mathbf{X}}} \right)^{ - 1}$ is denoted by k _i and so the Standard Error of The Estimate associated with region i is given by $s_{i} = s\sqrt {k_{i} }$.