The F Test
2.10 Receptor Models
26 Chapter 2
and later to apply CMB to quantify the contribution of these sources. In addition and prior to FA and CMB, several other statistical and mathematical approaches, such as linear regressions, trend analysis, trace element ratios together with the various enrichment factor calculations are applied statistical techniques employed to get a feeling about the nature of the full total diet data set.
2.10.1 Factor Analysis
Factor analysis is a most commonly used example of “multivariate analysis” which itself is a complex statistical technique for examining the large data sets to observe multiple correlation among the groups, which fluctuate together. In many research fields, a huge number of variables in different conditions are measured. The FA, a powerful statistical technique, is used to identify a relatively small number of factors that can be utilized to represent relationships among sets of many interrelated vari- ables. The basic assumption of factor analysis is that underlying dimensions, or fac- tors, can be used to explain complex phenomena, such as the sources of trace elements in total diet and daily intake of trace elements. One of the main steps of FA is to compute the linear correlation matrix for all normalized variables. Variables that do not appear to be related to other variables can be identified from the matrix and associated statistics.
Since FA is a purely statistical analysis technique, it requires no preliminary assumptions on the source profiles. During the calculations of the factor extraction step, the concentrations of elements in each sample are transformed into the nor- malized standard form given by
zik (2.38)
where zikis the standardized value of the ith element for the kth sample i1,2,…,nis the total number of parameters measured
k1,2,…,mis the total number of observations x¯imean value of the ith element
sistandard deviation of the distribution of concentrations of the ith element xikconcentration of ith element in the kth sample.
This kind of normalization takes equal weighting for each element, regardless of its average concentration. In factor analysis, the aim is to determine the minimum number of factors that can explain most of the common variance of the system.
Owing to this normalization step, the quantitative information on each element is lost and factor analysis ends up with only the qualitative information.7
It is important to note that, for multivariate analysis, sufficient degrees of freedom should be available in the model. Thus, the data set employed must have many more observations than variables if stable results are to be derived.8
The primary objective of applying factor analysis is to derive a small number of components, which explain a maximum of the variance in the data. Initially the fac- tor analysis results in as many components as there are original variables n. Usually, however, only a limited number of these uncorrelated components (i.e. 4–6) are
xik xi si
28 Chapter 2 required to explain virtually all of the variance in data set of original intercorrelated variables. In order for this reduction in the size to be useful, the new variables (com- ponents) must have simple substantive interpretations. Empirically, it has been found that unrotated components are often not readily interpretable, since they attempt to explain all remaining variance in the data set.9This calculation results in a number of sources of variance being grouped together. For this reason, a limited number of components, which explain at least 80% of the total variance are usually subjected to rotation using a criteria such as VARIMAX.10After the VARIMAX rotation, the resulting components have been often found to be more representative of individual underlying sources of variation. This, in turn, results in more interpretable and use- ful components.
In the usual application of FA to any data set, it is desirable to have as many parameters measured in greater number of cases. However, in most of the experi- ments, almost every element measured could be missed in one sample or another, which made the factor score extraction an extremely restricted set. If one omits the whole sample in which an element used in FA was missing, half of the full data set could be wasted that makes no sense at all. A possible solution to this missing data problem is to fill the absent data with a suitable number. There are a number of methods to fill the missing data by using one of the following, such as:
(i) the most frequently observed value, (ii) the mean value of the measured species and
(iii) a number found by the multi-linear correlation to the other elements (extracted by stepwise regression for all elements).
The last approach was found to be more suitable,i.e.to use Statgraphics software package to run stepwise regression for each missing element and find an equation as a result of correlation of each test element being dependent variable to other ele- ments being the independent variables. The equation obtained for the test element Y is as follows:
YConst.a[X1]b[X2]… (2.39) where the constant number is the intercept and corresponds to the residue which is not explained by [X1], [X2] and [Xn]. The independent parameters X1,X2and Xnwere the measured concentrations in the samples in which the element Ywas missing.
One of the major problems of factor analysis is the decision as to the number of factors to be retained. There are no simple rules applicable to each situation that has been advised to aid this choice. Each proposed rule has also generated criticism and counter-examples. It will often fall to the judgement of the investigator as to how many factors to keep. The purpose of factor analysis is to group the elements with the same variation into the same factor in a more understandable framework. The cri- teria that can then be applied is whether or not the factors can be interpreted in a way that aids in understanding of the system being studied. The factor analysis can be conducted using commercial statistical programs such as SPSS package or Statgraphics.
2.10.2 Chemical Mass Balance Method
Factor analysis and enrichment factors calculations provide primarily qualitative infor- mation about the possible sources whose contribution can only be determined by apply- ing another chemical calculation method. In the CBM method, the other basic approach of receptor models, the concentration of an element i in the receptor site is assumed a linear combination of sources contributing to that concentration, and it is given as
cimjfijaij (2.40)
where iis the type of the chemical species,jthe source type,cithe concentration of chemical species measured at a receptor site,mjthe mass concentration contributed dur- ing the sampling period,fijthe mass fractionation term of element iemitted from the jth source and aijthe fractionation term which accounts any loss or any gain, during any step of the cooking processes in the concentration of element icoming from jth source.
Since we assume no change for the concentration of any element in between source and the receptor site,aijvalues are usually set to 1 at the very beginning.
These equations have a unique solution only when the number of species is equal to or greater than the number of sources. Receptor modelling evaluation studies show that the greater the number of chemical species, the more precise the appor- tionment. Keeping these criteria, if one knows the composition of the total diet (i.e.
civalues) and source profiles of the dominant staple foods (i.e. mjvalues), source contribution terms (i.e. fij values) can be determined by a least-squares fit to the observed concentrations of a set of elements called “marker elements”. Marker ele- ments are chosen among the strong indicators of staple foods and they are usually non-volatile elements, which can be measured, reliably in both total diet and the source samples.2This method also requires precision estimates for the ci and fijval- ues as model inputs.
The CMB model generally has the following assumptions:
(i) Compositions of sources are constant over the period of source sampling.
(ii) Chemical species do not react with each other,i.e.they add linearly.
(iii) All sources with a potential for significantly contributing to the receptor site have been identified.
(iv) The source compositions are linearly independent of each other.
(v) The number of sources or source categories is less than or equal to the num- ber of chemical species measured at receptor site.
(vi) Measurement uncertainties are random, uncorrelated and normally distributed.
Although CMB studies provide valuable insight into the relative performances of sev- eral receptor models, certain limitations prevented a demonstration of the models’ full capabilities. The major limitation is the lack of source profiles especially specific to the area where total diet is collected. They obtained most of the profiles from other places.
Unfortunately, the true composition of a source is not usually measured but is approxi- mated by employing a similar source type signature from the reference literature.
Up to this point the preliminary steps of source apportionment in total diet have been discussed. Some potential sources such as wheat and wheat products, vegetables, meat,
30 Chapter 2 etc. have been identified as the contributors to the observed concentrations of the ele- ments in total diet. Because of the limited number of the trace element results on var- ious staple foods in the literature, much more work is needed for the receptor model-based source apportionment, which are the FA and CMB.
2.10.3 Enrichment Factors of the Elements
The qualitative approaches such as enrichment factors and elemental ratios are the preliminary steps of factor analysis and CMB based receptor modelling. The enrich- ment factor and the inter-element ratios are also the necessary steps to interpret the results obtained from the FA and CMB. A direct application of FA and CMB to a total diet data set may end up with uninterpretable results without these preliminary steps.
The enrichment factor model, is a double normalization technique, in which the elemental composition of the local total diet is compared with the elemental com- position of the possible sources. That is,
(EF)source (2.41)
where Cnis the concentration of normalizing element assumed to be uniquely charac- teristics of the source and Cithe concentration of an element whose enrichment is to be determined. In total diet studies, the source may be soil, flour, earth crust, vegetable, etc. According to above definition, if EF is around unity for the test element, one can assume that this element is entirely from the source used in calculations. Elements with enrichment factors greater than 2 are assumed to be due to the other sources rather than the source in question. The enrichment factor approach has been most useful when there has been a limited amount of information available. Enrichment factor approach cannot quantify a source’s contribution, relies heavily on the assumed source compo- sition and is not applicable to complex source mixtures where multiple sources, such as spices are contributing to the same element. Usually, in receptor modelling studies, the enrichment factor calculations and inter-element ratios are preliminary steps giv- ing an insight about the characteristic of the trace element data set.
Aluminium, Sc or Fe are the common reference elements for the soil in (EF)soil calculations. Elements, which have other sources as well as crustal dust, have enrichment factors higher than unity. The other sources, such as wheat products, vegetables and dairy products, contribute to the total diet.
Similarly, the enrichment of the elements with respect to wheat was examined in Turkey.2In this case the reference element was Mg and the source composition was taken from 7-year wheat averages.4
References
1. L.A. Currie, in The Importance of Chemometrics in Biomedical Measurements, K.S. Subramamanian, K. Okamoto and G.V. Iyengar (eds), Biomedical Trace Element Research, American College of Surgeons Symposium Series, 1991, 445.
2. N.K. Aras and I. Olmez,Nutrition Suppl., 1995,11, 506.
(Ci/Cn)sample (Ci/Cn)source
3. K.A. Wolnik, F.L. Fricke, S.G. Capar, M.W. Meyer, R.D. Satzger, E. Bonnin and C.M. Gaston,J. Agric. Food. Chem., 1985,33, 807.
4. N.K. Aras and J. Kumpulainen,Proceedings of the Technical Workshop on Trace Elements, Natural Antioxidants and Contaminants, REU Technical Series, 1996, 49.
5. P.R. Bevington,Data Reduction and Error Analysis for the Physical Sciences, McGraw-Hill, New York, 1969.
6. G.E. Gordon,Environ. Sci. Technol., 1988,22, 1132.
7. R.L. Gorsuch,Factor Analysis, W.B. Saunders, Philadelphia, PA, 1974.
8. T.G. Dzubay,Environ. Sci. Technol., 1988,22, 46.
9. SPSS (Statistical Package for Social Sciences) Manual, SPSS Inc., Chicago, IL, 1990.
10. P. Koutrakis and J.D. Spengler,Atmos. Environ., 1987,21, 519.