The constant sum problem

At first sight there appears to be some similarity between the method of Aitchison quoted with approval here and the ratio correlation technique heavily criticized in an earlier section of this chapter. The two are the same inasmuch as both require the taking of ratios, but there the similarity ceases. At the heart of Aitchispn s argument is the observation that you cannot correlate ratios, and yet this is an essential part of ratio correlation. Rather, Aitchison s method involves not the correlation of log-ratios themselves but the structure of a log-ratio covariance matrix. 

In his 1986 text Aitchison proves (for the mathematically literate reader) that die covariance structure of log-ratios is superior to the covariance structure of a percentage array (the crude covariance structure, as it is termed in his text). The covariance structure of log-ratios is free from the problems of the negative bias and of subcompositions which bedevil percentage data. In detail he shows that there are three ways in which the compositional covariance structure can be specified. Each is illustrated in Table 2.5. Firsdy, it can be presented as a variation matrix in which the log-ratio variances are plotted for every variable ratioed to every other variable. This matrix provides a measure of the relative variation of every pair of variables and can be used in a descriptive sense to identify relationships within the data array and in a comparative mode between data arrays. 

A second approach is to ratio every variable against a common divisor. The covariances of these log-ratios are presented as a log-ratio covariance matrix. (The correlation coefficient between two variables x andy is the covariance nonnalized to the square root of the product of their variances — Eqn [2.1]). The choice of variable as the divisor is immaterial because it is the structure of the matrix which is of importance rather than the individual values of the covariances. Nevertheless, this does give rise to a large number of different solutions. This form of covariance structure is used by Aitchison (1986 — Chapter 7) in a wide variety of statistical tests on log-ratio data which include tests of normality and regression. An example of its use in testing the independence of subcompositipns in pollen analysis is given by Woronow and Butler (1986). 

The final form of compositional covariance structure is centred log-ratio covariance matrix. In this case the single divisor of the log-ratio covariance matrix is replaced by the geometric mean of all the components. This form is used with 

Si02/Ti02 AijOj/TiOj Fe203/Ti02 MnO/TiO MeO/TiOi CaO/TiOz-..  

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This approach is not described in this text for there is a still more fundamental problem, and one which is not resolved directly by the application of multivariate techitiques. This is the problem caused by the summation of major element analyses to 100% — the constant sum problem. The statistical difficulties resulting from this feature of geochemical dam are foraiidable and are discussed below in Section 2.6. 

Meisch A.T., 1969, The constant sum problem in geochemistry. In Merriam D.F. (ed.), Computer applicatiom in the earth sciertces. Plenum Press, New York, pp.. 161-176. Mensing T.M., Faure G., Jones L.M., Bowman J.R. and Hoefs J., 1984, Petrogenesis of the Kirkpatrick basalt, Solo Nunatak, northern Victoria Land Antarctica, based upon isotopic compositions of strontium, oxygen and sulfur. Contrib. Mineral. Petrol., 87, 101-108. 

Rollinson H.R., 1992, Another look at the constant sum problem in geochemistry. Mineral. Mflg. 56, 469-475. 

Skala W., 1979, Some effects of the constant sum problem in geochemistry. Chem. GeoL, 27,... 

The induction period is measured experimentally at the constant sum of concentrations of two antioxidants, namely, Co = [S]o + [InH]0 = const. Theoretically this problem was analyzed in [9] for different mechanisms of chain termination by the peroxyl radical acceptor InH (see Chapter 14). It was supposed that antioxidant S breaks ROOH catalytically and, hence, is not consumed. The induction period was defined as t = (/[InH /v, where vV2 is the rate of InH consumption at its concentration equal to 0.5[InH]o. The results of calculations are presented in Table 18.1. 

In the panel data models estimated in Example 21.5.1, neither the logit nor the probit model provides a framework for applying a Hausman test to determine whether fixed or random effects is preferred. Explain. (Hint Unlike our application in the linear model, the incidental parameters problem persists here.) Look at the two cases. Neither case has an estimator which is consistent in both cases. In both cases, the unconditional fixed effects effects estimator is inconsistent, so the rest of the analysis falls apart. This is the incidental parameters problem at work. Note that the fixed effects estimator is inconsistent because in both models, the estimator of the constant terms is a function of 1/T. Certainly in both cases, if the fixed effects model is appropriate, then the random effects estimator is inconsistent, whereas if the random effects model is appropriate, the maximum likelihood random effects estimator is both consistent and efficient. Thus, in this instance, the random effects satisfies the requirements of the test. In fact, there does exist a consistent estimator for the logit model with fixed effects - see the text. However, this estimator must be based on a restricted sample observations with the sum of the ys equal to zero or T muust be discarded, so the mechanics of the Hausman test are problematic. This does not fall into the template of computations for the Hausman test. 

Summing over all pairs of segments in the solution we strictly speaking rim into problems with the tf-function interaction, which could be avoided by using a potential of finite range (cf. Sect. 2.2). Furthermore the constant Uq in principle differs from the definition (5.25). We here ignore all such complications, which do not affect the essential features of the derivation. We rather argue on a purely formal level. 

The various coefficients A , Afcp9.. . . a002, gp+> gpu- . up i are constants and involve quantities of two types parameters such as those in Equations 1 and 2, which are characteristic of the ionic species involved and a number of sums associated with the particular lattice structure. Numerical methods for evaluating the latter have been described (14, 15). Formulas for these coefficients are not given in detail here, but those for the coefficients in Equation 5 are similar to expressions given by Benson, Balk, and White (2) for the single layer problem and the forms of the coefficients in Equation 6 follow an analogous pattern. 

For H2/ at temperatures below 1000 K, the rotational constant B = 59.4 cm 1 is so large that integration is invalid, and one must sum directly the leading terms of the two sums. The same holds for D2, where B = 29.9 cm-1. For the heavier homonuclear diatomics, like 02 (B = 1.437 cm ) or N2, B is so small that the high-temperature integration of Problem 5.3.11 works well, and the difference between ortho and para states becomes experimentally indistinguishable. 

Here the indices /, j run over electrons, while A, B run over nuclei. The. second sum must now account for the attraction of each electron to multiple nuclei. The last sum represents repulsion between each pair of nuclei. Tt acts as a constant term added to the electronic energy. The Born-Oppenheimer approximation is assumed, so that the positions of the nuclei R.4 are fixed with no nuclear kinetic-energy contributions. The Schrbdinger equation for the jV-electron problem is written symbolically as... 

This tabular representation is a convenient way of doing the bookkeeping for equilibrium problems. Under each substance in the balanced equation an entry is made on three lines (1) the amount of starting material (2) the change (plus or minus) in the number of moles due to the attainment of equilibrium and (3) the equilibrium amount, which is the algebraic sum of entries (1) and (2). The entries in line (2) must be in the same ratio to each other as the coefficients in the balanced chemical equation. The equilibrium constant can be found from the entries in line (3). 

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