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Numerical consequences of computing structural formulas
formulas JOHN C. BUTLER
I.ITHOS
Buffer, J. C. 1980: Numerical confluences of compuffng structural formulas. Lithos 1J, 55-59. Oslo. ISSN 0024-4937. The six steps involved in transforming oxide weight percentages to coefficients of a structural fi:¢mula are a series of column type transformations and a row transformation that have the effect of reclosmg the oxide weight percentage d~ta. Statistical properties of the coefficients of the structural formula (such as means, coeffici©nts of variation and correlation coefficients) are controlled by the properties of the oxide weight percentages. Care is necessary when comparing the properties of the oxide weight percemages with those of the coefficients of the structural formulas. In no way does tha conversion to structural formulas eliminate the effects that closure has on the properties of the original weigh~ percentages and, in f~ct, the transformation actually compounds the effect.
John C. Butler, Department o f Geology, University o f Houston, Houston, Texas 77004, U.S.A.
In an earlier paper (Butler 1979) the numerical consequences of converting oxide weight percer~tages to units of grams per 100 cc were exami~md. This type of transformation was classifted as a row transformation in the sense that each complete chemical analysis is multiplied by a property of the sample itself- the bulk density. Interest in transformations used by petrologists continues and attention in this note will be focused on the effects of computing the coefficients of the structural formula of a given mineral species.
Computation ,of the structural formulas of pyroxenes Although the generalizations that follow can be extended to all mineral groups for which a structural formula can be written, it will be useful to focus attention on a specific example. The structural formula for the pyroxene mineral l;roup can be written as: (W),_~(X, Y)~,Z2Oe
(1)
where W equals Na or Ca; X equals Fe 2., Mg, Mn, Li; Y equals Fe a~ Al, Cr, Ti; and Z equals Si, AI (Hess 1949). To compute the coefficients of each cation from the oxide weight percentage analysis one performs the following calculations in stepwise fashion: (1) Each oxide weight percentage is divided by
the formula weight of the oxide to yield the molecular proportion. (2) Each molecular proportion is mu]itiplied by the number of cations in the oxide formula to yield the number of cations contributed to the analysis. (3) The number of cations contributed by each component is multiplied by the number of positive charges on the cation to yield the number of positive charges contributed by each component. (4) The sum of the positive charges contributed by all of the components is divided into the number of negative charges in the structural formula (12.0 for the pyroxenes) to yield a factor F. (5) The number of positive charges contributed by each component (ste~ 3 above) is multiplied by F (step 4 above) to yield the number of charges of each component adjusted so that the sum of the charges equals the number of negative charges in the s~ructura~ formula. (6) The number of positive charge,~ contributed by each cation is divided by the charge on the cation to yield the proportionate share of the total positive charge contributed by the cation. Values obtained following completion of step 6 are the subscripts f~r each cation in the structural formula. In the following discussion it wiU be assumed that the investigator ha~ assembled a set of
56 John C. Butler
LITHOS 13 (1980)
chemical analyses of pyroxenes in matrix form. Each row represents an analyzed pyroxene and values of each measured component are stored in the columns of the raw data matrix. The six steps outlined above are applied to each row of the raw data matrix with the end result being a matrix of coefficients of the structural formulas. It is instructive to examine the interrelations between and among the columns of the matrices following each step in the procedure. Steps 1 through 3 are column transformations (Butler 1979) in the sense that the~e steps involve multiplication or division by a factor that is the same for each column. For example, the weight percentage SiO2 is always divided by 60.028 (step 1), multiplied by 1.0 (step 2) and divided by 4.0 (step 3). Column transformations (Koch & Link 1970; Butler 1979) result in changes in the iaean (,i') and variance (s z) for each column but not for the coefficient of variation (C = s/X). Following step 3 the variables have means (AO that are related to the oxide weight, percentage means (~~) by: = 2,
(2)
and variances (~') that are related to the oxide weight percentage variances (s~~) by: /
= s~2 (ArfZ~/W~O
(3)
where N . Z~ and W~ are the nuraber of cations in the oxide formula, the charge on the cation and the oxide formula weight respectively of the ith variable. The correlation (r) between a pair of variables (which gives a measure of the strength of the linear association between the pair of variables) is unchanged followir~g a column type of transformation (Butler 1979). That is, the correlation between the number of positive charges contributed by Fe z+ anJ Mgz~- is ident~.cal ~o the correlation between ~he oxide weight percentages of FeO and MgO. Step 5, however, is a ro~ transformation (Buder 1979) in the sense that each value in the row of the matrix is multiplied by a function of the sum of all values in that row. Thus, step 5 creates a set of closed data in that each row sum equals a constant (12.0 for the pyroxenes, for example). Thus, the effect of the row sum transformation is analogous to percentage formatim,~. Chayes (summarized in Chayes 1971) has been a leader in demonstrating the effect that percentage formatiGn has on the correlation between a pair of variables. In general, the correlation belween the row transformed variables
(Step 5) are rather complex functions of the properties of the initial oxide weight percentages and of the row sum variable (the sum of the number of positive charges contributed by each component - Step 3). Step 6, a column transformation, results in changes in mean and variance of the order of z~ and ~ (where z~ is the charge on the cation) but the correlations are identical to those from Step 5. To complicate interpretation, mos,t pyroxene analyses requite that some AP + be distributed into both the X and Z sites to bring the number of cations in the Z site up to 2.0. When Ithis is done a
correlation
of
-1.0
between
Si 4+
and
tetrahedral AP ÷ is induced as their sum is constrained to be 2.0. Structural formulas have been computed for v variety of purposes. Chief among these appears to be analyses of petrogenetic relations and substitutional patterns. For example, Nieva (1974) examined variations in major, minor and trace elements in schorlites from granites, aplites and pegmatites by analysis of binary plots of structural formula coefficients. Reasonably strong linearity of such plots led Neiva (1974) to conclude that observed composition~d variation resulted from magmatic fractionation. Schweitzer et at. (p. 502, 1979) argued that 'the number of cations per formula unit was used in the statistical programs because it gives more insight into the crystal chemistry of the pyroxenos than does similar treatment of the oxides, and provides information equally .,mitable to discriminate among the basalt groups'. Central to such interpretations is the realization that the correlation coefficients, and thus the linearity, may be considerably modified as a result of forming a closed data set. Chayes (1971) has shown that formation of a closed array will often induce a negative correlation between pairs of variables. The magnitude of the induced correlation appears to be a function of the magnitude of the variances of the original variables - the larger the variances the greater the induced negative correlation. Induced positive correlation between pairs of variables is also possible. As developed by t.hayes (1971) chemica! anal':/ses :~re closed by observation and thus the correlatioras among oxide weight percentages contain intractable amounts of induced correlation. Step 5 potentially results in induced correlation; this suggests that the correlation between
Computing structural formulas
LITHOS 13 (1980)
57
T a b l e 1. Summary statistics for the clino and orthopyroxene data set.
Oxide weight percentages SiOa Mean 54.80 Coef. variation ~ 0.037 Skewness* 0.218 Kurtosis a 8.60
TiO, 0.28 1.260 4.280 28.51
AhO~ 4.95 0.280 0.089 1.90
FeO 4.80 0.433 0.248 1.45
MgO 24.69 0.330 0.012 1.08
CaO 10.47 0.932 0.021 1.00
FeZ+ 0.140 0.424 0.343 1.59
Mga + 1.385 0.313 0.007 1.08
Ca~ ÷ 0.401 0.935 0.026 1.01
Row Sum* 5.69 0.021 -0.286 2.52
* The sum of the number of positive ch~ges(Step3) Structural formula coefficients Si4+ Mean 1.925 Coef. variation t 0.025 Skewness* 0.677 Kurtosis s 9.123
Ti 4+ 0.008 1.307 4A85 30.42
Ap+ 0.21{) 0.350 0.320 2.45
~The coefficient of variation is given by the standard deviation divided by the mean and ~s a measure of relative variability. *Skewness is a measure of the symmetzy of the distribution. A no~qally distributed se~-of valaes has zero skewness. aKurtosis is a measure of the peakedness of the distribution. A normally distributed ~,et of values has a kurtosis of 3.0. The larger the kurtosis the greater the concentration of values close to the mean.
coefficients of structural formulas have a rather complex origin: (I)
The initial variables (open and unobservable, Chayes 1971) may have non-zero correlations, which is a measure of their strength of linear association. (II) Correlations between the observed weight percentage variables contain a potential contribution from the closure effect. (lid Correlations between the coefficients of the structural formulas may contain an effect of reclosing the initial data array. V¢hile it is apparently difficult, if not impossible, to assess the effects of percentage formation in going from I to lI such an assessment can be m:'~de for the step II to step III transformation. Chayes' equation 4.10 (1971) enables estimation of the correlation between closed varia,bles/f the means and variances of the initial variables are known and ~f the initial variables are uncorrelated. An extension to Chayes" (1971) first order expression which allows for correlated initial variables leads to :he following expression for the expected correlation between percentage variables (/,~j) (see Appendix I for the derivation of the following expression): ~o =
roC~Cj + ~ - r~tC~Ct - rjtCjCt (C~ +'C~: - 2C~Ctru)v*(C]+ C~t- 2CjC~rjt) v*
(4)
where ro is the initial correlation between the oxide weight percentage variables, C. is the
coefficient of variation of the nth variable and r,t is the part-whole correlation between the row sum variable (t) and a part of tile sum 0t). In the ce.se under consideration the row sum is the sum of the positive charges contributed (step 3) and the part-whole correlation is the correlation between this sum and any one of the ,~omponents. As is the case with closed data in general, the lack of knowledge as to an appropriate null value against which to test an observed correlation hampers a detailed interpretation of the matrix of correlation coefficients for the coefficients of structural formulas. Tlais poses an especially difficult problem if one uses visual appearance and intuition to interpret correlations on a binary scatter diagram. From equation (4) it is clear that correlations between the closed array (step 5) are fimctions of the properties of the oxide weight percentage array. One should not expect that computing the coefficients of a structural formula will reve',d information which is not already present in the original data array.
Analysis of the compositions of coexisting clino and ortho pyroxenes A set of 72 coexisting clino pyroxenes (36 sam-. pies) and orthopyroxenes (36 samples) from six spinel lherzolites from the Xalapasco de la Joya
58 John C, Butler
LITHOS 13 (1980)
Table 2. Correlati,on <:aefficients.* Pyroxene data sets. SiO= Si4+ Ti4+ AP + Fe =+ Mg~+ Ca =*
TiO=
AI=Oa
FeO
MgO
CaO
I.O00 --0.&i~7 -.0.531 0.234 0.456 -.0.541 --0.372 1.0~J0 0.362 --0.257 -0.597 0.601 -0.318 0.394 i.000 -0.502 -0.674 0.625 -0.178 .196 -0.508 1 ° 0 0 0 0.787 -0.838 -0.074 -0.578 -0.708 0.769 l.O00 -0.981 -0.052 0 . ~ 0 0.662 -0.807 -0.977 1.000
* Correlation coefficwnts between the oxide weight per,::enrages given in the upper h~df of the matrix at~d correlation coefficients between the coefficients of the structu: J form~das given in the lower half of ~:he matrix. Correlations betw:~en coefficients that differ by mo~e than 0.250 from those betwee~ ~he oxide weight percentages are underlined.
structure near San Luis Potosi, Mexico (Greene 1974; Greene & Butler I979) was selected solely to illustrate the nature of changes that can accompany calculation of coefficients of structural formulas. A~,:alyses of SiOz, TiO2, AlzOa, total iron as FeO, MgO, and CaO were obtained using the Rice University microprobe and analytical procedures described in detail in Greene (1974). Only analyses which had a row sum between 98.0 and 1020 were included in the initial data matrix•
Summary statistics for the oxide weight percentages and the coe/ficients of the structural formulas are given in Table 1. Correlations between the oxide weight ~ r c e n t a g e variables and between the c0effici,:T ~s are given in the upper and rower t dve~ resT.ectiveid of the matrix in Table 2. There h ~ be ~=n no distribution of AF + between the Y an~ Z sites so as to be able to compare the oxide v~eight percentage and *he structural formula correlation coefficients. Correlations between the coefficients are underline .i if they differ from the oxide weight percentage correlations by more than 0.25 (Table 2). Only the correlations involving SP + exhibit marked changes following the transformation. Those between Si 4+ and TP + and Ca ~+ change in a positive sense whereas those between SP + and Fe z+ (total iron as Fe ~+) and Mg 2+ change in a negative sense. U s e of zero as the appropriate null value or an intuitive assessment of strensth of linear association on a binary scatter diagram could suggest an apparent discrepancy when comparing the properties of the oxide weight percentages with those of the coefficients of the structural formulas. F o r example, there is a moderate positive correlation (0.456) between weight percent SiO2 and MgO (Fig. 1) whereas there is a very weak negative correlation
40. 1.90 •
q,
o
.:v
~z:
°
30.
•:
,
• %,
•
•
150
LU
had n
I--
¢,q
'4"m,,~ "=
~Z
20.
1.10 i
•..;...:: . . . . ,.~ • •
~0.
=~
•
•
.70
50.
54. 58. SiO2 WEIGHTPERCENT
62.
Fig. /. Binary scatter diagram showing the variation of MgO weig3t percent versus SiO= weight percent for the 72 Xalapasco de la Joya pyroxenes.
1.7o
t.04
Si +4
t.92
2,o0
Fig. 2. ~;inary scarer diagram showing ghe variation of Mgt÷ versus Si4+ for the 72 Xal,~pasco de la ~oya pyrox~nes.
LITHOS 13 (1980) (-0.074) between Si 4+ and Mg~+ (Fig. 2). Unwise use e f zero as a null value would indicate that weight percentages SiO~ and MgO were significantly correlated at the 95% confidence level (assuming, of course, that the investigator had reason to believe that this type o f test was wdid for the data) whe,'eas Si '~+ and Mg ~+ were t~acorrelate~. The fact is, however, that the I reperties of the matrix of coefficients is conl o l l e d by the properties of the weight ! ercentages and any such differences between ':~e two to:ms o f the data are due to the transformation itse|f. Figs. 1 and 2 also illustrate the effects of mixing two discrete groups of samples into a stogie array. In both f~,ures the two groups (clino and ortho pyroxenes) are quite distinct. In Fig. 1, however, a crudely defined elliptical distribution of points with its positive correlation for the overall data set (in spite of the fact that the correlation within eac ~ group is close to 0.0) is apparent whereas in F~g. 2 the points are crudely distributed on the pe~'imeter of a circle which results in an overall correlation very close to 0.0. Regardless of the nature of this data array, the fact remains that the ~:onversion to coefficients of a structural formu~:a has considerably modified the measure of linear assoeiatic,n. In general, correlations which do not involve Si 4+ rarely exhibit changes exceeding 0.1 (based on an examination of 12 data sets ot' pyroxenes, garnets, olivines and feldspars). That is, r'e is approximately equal t(~ r,j (equation ,l).
Summary and conclusion Care should be exercised in comparmg the properties of the initial data matrix of oxide weight percentages with those, of the matrix of coefficients of the structural formulas. In no way does the conversion to structural formula ~:oeffic~ents ehminate the effects ~that closure has on the properties of the oxide weight percentage data and, in fact, such a tnmsformation compounds the closure problem.
Computing stn~cturalformula~ 59 units in which chemical analyses of igneous . - ~ ~xe analyzed. Lithoz 12, 33-39. Chayes, F. 1977:Ratio Correlation. U niv. Chicago Press. Greene, G. M. 1974: The gcocbemist,'y of spinel lherzo[ites from Xedapasco de la Joya, San Luis Potosi, Mexico. MS thesis, University of Houston. 63 pp. Greene, G. M. & Buffer, J. C. 1979: Geochemistry of spinel lherzolites from the Xalapasco de la Joya structure, San Luis Potosi, Mexico, S.L.P. Mineral. Mug. in press. Hess, H. H. 1949: Chemical composition and optical proparties of common clinopyroxenes. Part I. Am. Mineral 34 621. Koch, G. S, & Link, R. F. 1~70 ~tatist!ca! Analysis ~,[ Geological Data. John Wiley & Sons, New Yozk. Neiva, A. M. R. 1974: Geochemistry of t o u ~ i n e (:~chorl :~:) from ~'anites, ap;ites and pegmatites from northern Portugal. Geochim. et Cosmochim. Acts 38, 1307-~317. Schweitzer, E. L., Patpike, J. J. & Ben,.e, A. E. ]979: Statistical analysis of cfinopyroxenes from deepsea basalts. Am. Mineral. 64, 501-513. Accepted for publication August 1979 Printed January 1980
Appendix I - Derivation of text equation (4) Equation (4) can be derived by following and expanding the derivations given by Chayes (1971:37-39) and Buffer (1979:75gl). The wuiance of a proportion Y, .~sgiven by:
v ~ r, = (X~I~,~ct+ ~ - 2r,,c,c,) (Buffer 1979, Eq. 8)
where X is the mean of the variable indicated by the sub,.icript, C is the coeff,cient of the indicated variable, the subscript t refers to the row sum variable (the sum of the values that are being convened to proportions), and ru is the part whole correlation - the correlation between variable i and the. row sum t). A similar expression can be written for variance of the proportion Yj by replacing appropriate subscripts in (5). From Chayes' (1971:38) equation (4.9) the product of the deviations of Y~and Yj can be written as fol[ows: where ~. is the deviation oftho value ofn rio,,, IL~,scan value (in Chayes" exF~ression he uses p,=,fd~',, otherwise the expression above is identical to Chayes" (197i:38) equation 4.9). Taking expectations of I~oth sides of the above and not requiring that the expectation of the product of two deviations is zero (that is, E(SmS~)=r,ts~,s, where r is the correlation between m and 1 and s is the standard deviation) yields the following expression for the covariance between Y,Yj: COVf, = ( ! 1 ~ ) (r.s,sj + ~ . , - (.t~,s ~,~.t,))
References Buder, J. C. 1979: The effects of closure on the moments of a distribution. Math. Geol. 12, 75.-g4. Buffer, J. C. 19~9: Numerical consequences of ehaf~glngthe
(5)
- (~us,s,l.f~)(6)
Rearranging and remembering that C, ---s,I](, and dividing by the product of the square roots of the variances for Yeand Yj (equation (5)) yields the expression for the correlation between two proportions i~ terms of th~ p~openies of the initial values (text equation 4).
I.ITHOS
Buffer, J. C. 1980: Numerical confluences of compuffng structural formulas. Lithos 1J, 55-59. Oslo. ISSN 0024-4937. The six steps involved in transforming oxide weight percentages to coefficients of a structural fi:¢mula are a series of column type transformations and a row transformation that have the effect of reclosmg the oxide weight percentage d~ta. Statistical properties of the coefficients of the structural formula (such as means, coeffici©nts of variation and correlation coefficients) are controlled by the properties of the oxide weight percentages. Care is necessary when comparing the properties of the oxide weight percemages with those of the coefficients of the structural formulas. In no way does tha conversion to structural formulas eliminate the effects that closure has on the properties of the original weigh~ percentages and, in f~ct, the transformation actually compounds the effect.
John C. Butler, Department o f Geology, University o f Houston, Houston, Texas 77004, U.S.A.
In an earlier paper (Butler 1979) the numerical consequences of converting oxide weight percer~tages to units of grams per 100 cc were exami~md. This type of transformation was classifted as a row transformation in the sense that each complete chemical analysis is multiplied by a property of the sample itself- the bulk density. Interest in transformations used by petrologists continues and attention in this note will be focused on the effects of computing the coefficients of the structural formula of a given mineral species.
Computation ,of the structural formulas of pyroxenes Although the generalizations that follow can be extended to all mineral groups for which a structural formula can be written, it will be useful to focus attention on a specific example. The structural formula for the pyroxene mineral l;roup can be written as: (W),_~(X, Y)~,Z2Oe
(1)
where W equals Na or Ca; X equals Fe 2., Mg, Mn, Li; Y equals Fe a~ Al, Cr, Ti; and Z equals Si, AI (Hess 1949). To compute the coefficients of each cation from the oxide weight percentage analysis one performs the following calculations in stepwise fashion: (1) Each oxide weight percentage is divided by
the formula weight of the oxide to yield the molecular proportion. (2) Each molecular proportion is mu]itiplied by the number of cations in the oxide formula to yield the number of cations contributed to the analysis. (3) The number of cations contributed by each component is multiplied by the number of positive charges on the cation to yield the number of positive charges contributed by each component. (4) The sum of the positive charges contributed by all of the components is divided into the number of negative charges in the structural formula (12.0 for the pyroxenes) to yield a factor F. (5) The number of positive charges contributed by each component (ste~ 3 above) is multiplied by F (step 4 above) to yield the number of charges of each component adjusted so that the sum of the charges equals the number of negative charges in the s~ructura~ formula. (6) The number of positive charge,~ contributed by each cation is divided by the charge on the cation to yield the proportionate share of the total positive charge contributed by the cation. Values obtained following completion of step 6 are the subscripts f~r each cation in the structural formula. In the following discussion it wiU be assumed that the investigator ha~ assembled a set of
56 John C. Butler
LITHOS 13 (1980)
chemical analyses of pyroxenes in matrix form. Each row represents an analyzed pyroxene and values of each measured component are stored in the columns of the raw data matrix. The six steps outlined above are applied to each row of the raw data matrix with the end result being a matrix of coefficients of the structural formulas. It is instructive to examine the interrelations between and among the columns of the matrices following each step in the procedure. Steps 1 through 3 are column transformations (Butler 1979) in the sense that the~e steps involve multiplication or division by a factor that is the same for each column. For example, the weight percentage SiO2 is always divided by 60.028 (step 1), multiplied by 1.0 (step 2) and divided by 4.0 (step 3). Column transformations (Koch & Link 1970; Butler 1979) result in changes in the iaean (,i') and variance (s z) for each column but not for the coefficient of variation (C = s/X). Following step 3 the variables have means (AO that are related to the oxide weight, percentage means (~~) by: = 2,
(2)
and variances (~') that are related to the oxide weight percentage variances (s~~) by: /
= s~2 (ArfZ~/W~O
(3)
where N . Z~ and W~ are the nuraber of cations in the oxide formula, the charge on the cation and the oxide formula weight respectively of the ith variable. The correlation (r) between a pair of variables (which gives a measure of the strength of the linear association between the pair of variables) is unchanged followir~g a column type of transformation (Butler 1979). That is, the correlation between the number of positive charges contributed by Fe z+ anJ Mgz~- is ident~.cal ~o the correlation between ~he oxide weight percentages of FeO and MgO. Step 5, however, is a ro~ transformation (Buder 1979) in the sense that each value in the row of the matrix is multiplied by a function of the sum of all values in that row. Thus, step 5 creates a set of closed data in that each row sum equals a constant (12.0 for the pyroxenes, for example). Thus, the effect of the row sum transformation is analogous to percentage formatim,~. Chayes (summarized in Chayes 1971) has been a leader in demonstrating the effect that percentage formatiGn has on the correlation between a pair of variables. In general, the correlation belween the row transformed variables
(Step 5) are rather complex functions of the properties of the initial oxide weight percentages and of the row sum variable (the sum of the number of positive charges contributed by each component - Step 3). Step 6, a column transformation, results in changes in mean and variance of the order of z~ and ~ (where z~ is the charge on the cation) but the correlations are identical to those from Step 5. To complicate interpretation, mos,t pyroxene analyses requite that some AP + be distributed into both the X and Z sites to bring the number of cations in the Z site up to 2.0. When Ithis is done a
correlation
of
-1.0
between
Si 4+
and
tetrahedral AP ÷ is induced as their sum is constrained to be 2.0. Structural formulas have been computed for v variety of purposes. Chief among these appears to be analyses of petrogenetic relations and substitutional patterns. For example, Nieva (1974) examined variations in major, minor and trace elements in schorlites from granites, aplites and pegmatites by analysis of binary plots of structural formula coefficients. Reasonably strong linearity of such plots led Neiva (1974) to conclude that observed composition~d variation resulted from magmatic fractionation. Schweitzer et at. (p. 502, 1979) argued that 'the number of cations per formula unit was used in the statistical programs because it gives more insight into the crystal chemistry of the pyroxenos than does similar treatment of the oxides, and provides information equally .,mitable to discriminate among the basalt groups'. Central to such interpretations is the realization that the correlation coefficients, and thus the linearity, may be considerably modified as a result of forming a closed data set. Chayes (1971) has shown that formation of a closed array will often induce a negative correlation between pairs of variables. The magnitude of the induced correlation appears to be a function of the magnitude of the variances of the original variables - the larger the variances the greater the induced negative correlation. Induced positive correlation between pairs of variables is also possible. As developed by t.hayes (1971) chemica! anal':/ses :~re closed by observation and thus the correlatioras among oxide weight percentages contain intractable amounts of induced correlation. Step 5 potentially results in induced correlation; this suggests that the correlation between
Computing structural formulas
LITHOS 13 (1980)
57
T a b l e 1. Summary statistics for the clino and orthopyroxene data set.
Oxide weight percentages SiOa Mean 54.80 Coef. variation ~ 0.037 Skewness* 0.218 Kurtosis a 8.60
TiO, 0.28 1.260 4.280 28.51
AhO~ 4.95 0.280 0.089 1.90
FeO 4.80 0.433 0.248 1.45
MgO 24.69 0.330 0.012 1.08
CaO 10.47 0.932 0.021 1.00
FeZ+ 0.140 0.424 0.343 1.59
Mga + 1.385 0.313 0.007 1.08
Ca~ ÷ 0.401 0.935 0.026 1.01
Row Sum* 5.69 0.021 -0.286 2.52
* The sum of the number of positive ch~ges(Step3) Structural formula coefficients Si4+ Mean 1.925 Coef. variation t 0.025 Skewness* 0.677 Kurtosis s 9.123
Ti 4+ 0.008 1.307 4A85 30.42
Ap+ 0.21{) 0.350 0.320 2.45
~The coefficient of variation is given by the standard deviation divided by the mean and ~s a measure of relative variability. *Skewness is a measure of the symmetzy of the distribution. A no~qally distributed se~-of valaes has zero skewness. aKurtosis is a measure of the peakedness of the distribution. A normally distributed ~,et of values has a kurtosis of 3.0. The larger the kurtosis the greater the concentration of values close to the mean.
coefficients of structural formulas have a rather complex origin: (I)
The initial variables (open and unobservable, Chayes 1971) may have non-zero correlations, which is a measure of their strength of linear association. (II) Correlations between the observed weight percentage variables contain a potential contribution from the closure effect. (lid Correlations between the coefficients of the structural formulas may contain an effect of reclosing the initial data array. V¢hile it is apparently difficult, if not impossible, to assess the effects of percentage formation in going from I to lI such an assessment can be m:'~de for the step II to step III transformation. Chayes' equation 4.10 (1971) enables estimation of the correlation between closed varia,bles/f the means and variances of the initial variables are known and ~f the initial variables are uncorrelated. An extension to Chayes" (1971) first order expression which allows for correlated initial variables leads to :he following expression for the expected correlation between percentage variables (/,~j) (see Appendix I for the derivation of the following expression): ~o =
roC~Cj + ~ - r~tC~Ct - rjtCjCt (C~ +'C~: - 2C~Ctru)v*(C]+ C~t- 2CjC~rjt) v*
(4)
where ro is the initial correlation between the oxide weight percentage variables, C. is the
coefficient of variation of the nth variable and r,t is the part-whole correlation between the row sum variable (t) and a part of tile sum 0t). In the ce.se under consideration the row sum is the sum of the positive charges contributed (step 3) and the part-whole correlation is the correlation between this sum and any one of the ,~omponents. As is the case with closed data in general, the lack of knowledge as to an appropriate null value against which to test an observed correlation hampers a detailed interpretation of the matrix of correlation coefficients for the coefficients of structural formulas. Tlais poses an especially difficult problem if one uses visual appearance and intuition to interpret correlations on a binary scatter diagram. From equation (4) it is clear that correlations between the closed array (step 5) are fimctions of the properties of the oxide weight percentage array. One should not expect that computing the coefficients of a structural formula will reve',d information which is not already present in the original data array.
Analysis of the compositions of coexisting clino and ortho pyroxenes A set of 72 coexisting clino pyroxenes (36 sam-. pies) and orthopyroxenes (36 samples) from six spinel lherzolites from the Xalapasco de la Joya
58 John C, Butler
LITHOS 13 (1980)
Table 2. Correlati,on <:aefficients.* Pyroxene data sets. SiO= Si4+ Ti4+ AP + Fe =+ Mg~+ Ca =*
TiO=
AI=Oa
FeO
MgO
CaO
I.O00 --0.&i~7 -.0.531 0.234 0.456 -.0.541 --0.372 1.0~J0 0.362 --0.257 -0.597 0.601 -0.318 0.394 i.000 -0.502 -0.674 0.625 -0.178 .196 -0.508 1 ° 0 0 0 0.787 -0.838 -0.074 -0.578 -0.708 0.769 l.O00 -0.981 -0.052 0 . ~ 0 0.662 -0.807 -0.977 1.000
* Correlation coefficwnts between the oxide weight per,::enrages given in the upper h~df of the matrix at~d correlation coefficients between the coefficients of the structu: J form~das given in the lower half of ~:he matrix. Correlations betw:~en coefficients that differ by mo~e than 0.250 from those betwee~ ~he oxide weight percentages are underlined.
structure near San Luis Potosi, Mexico (Greene 1974; Greene & Butler I979) was selected solely to illustrate the nature of changes that can accompany calculation of coefficients of structural formulas. A~,:alyses of SiOz, TiO2, AlzOa, total iron as FeO, MgO, and CaO were obtained using the Rice University microprobe and analytical procedures described in detail in Greene (1974). Only analyses which had a row sum between 98.0 and 1020 were included in the initial data matrix•
Summary statistics for the oxide weight percentages and the coe/ficients of the structural formulas are given in Table 1. Correlations between the oxide weight ~ r c e n t a g e variables and between the c0effici,:T ~s are given in the upper and rower t dve~ resT.ectiveid of the matrix in Table 2. There h ~ be ~=n no distribution of AF + between the Y an~ Z sites so as to be able to compare the oxide v~eight percentage and *he structural formula correlation coefficients. Correlations between the coefficients are underline .i if they differ from the oxide weight percentage correlations by more than 0.25 (Table 2). Only the correlations involving SP + exhibit marked changes following the transformation. Those between Si 4+ and TP + and Ca ~+ change in a positive sense whereas those between SP + and Fe z+ (total iron as Fe ~+) and Mg 2+ change in a negative sense. U s e of zero as the appropriate null value or an intuitive assessment of strensth of linear association on a binary scatter diagram could suggest an apparent discrepancy when comparing the properties of the oxide weight percentages with those of the coefficients of the structural formulas. F o r example, there is a moderate positive correlation (0.456) between weight percent SiO2 and MgO (Fig. 1) whereas there is a very weak negative correlation
40. 1.90 •
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o
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,
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54. 58. SiO2 WEIGHTPERCENT
62.
Fig. /. Binary scatter diagram showing the variation of MgO weig3t percent versus SiO= weight percent for the 72 Xalapasco de la Joya pyroxenes.
1.7o
t.04
Si +4
t.92
2,o0
Fig. 2. ~;inary scarer diagram showing ghe variation of Mgt÷ versus Si4+ for the 72 Xal,~pasco de la ~oya pyrox~nes.
LITHOS 13 (1980) (-0.074) between Si 4+ and Mg~+ (Fig. 2). Unwise use e f zero as a null value would indicate that weight percentages SiO~ and MgO were significantly correlated at the 95% confidence level (assuming, of course, that the investigator had reason to believe that this type o f test was wdid for the data) whe,'eas Si '~+ and Mg ~+ were t~acorrelate~. The fact is, however, that the I reperties of the matrix of coefficients is conl o l l e d by the properties of the weight ! ercentages and any such differences between ':~e two to:ms o f the data are due to the transformation itse|f. Figs. 1 and 2 also illustrate the effects of mixing two discrete groups of samples into a stogie array. In both f~,ures the two groups (clino and ortho pyroxenes) are quite distinct. In Fig. 1, however, a crudely defined elliptical distribution of points with its positive correlation for the overall data set (in spite of the fact that the correlation within eac ~ group is close to 0.0) is apparent whereas in F~g. 2 the points are crudely distributed on the pe~'imeter of a circle which results in an overall correlation very close to 0.0. Regardless of the nature of this data array, the fact remains that the ~:onversion to coefficients of a structural formu~:a has considerably modified the measure of linear assoeiatic,n. In general, correlations which do not involve Si 4+ rarely exhibit changes exceeding 0.1 (based on an examination of 12 data sets ot' pyroxenes, garnets, olivines and feldspars). That is, r'e is approximately equal t(~ r,j (equation ,l).
Summary and conclusion Care should be exercised in comparmg the properties of the initial data matrix of oxide weight percentages with those, of the matrix of coefficients of the structural formulas. In no way does the conversion to structural formula ~:oeffic~ents ehminate the effects ~that closure has on the properties of the oxide weight percentage data and, in fact, such a tnmsformation compounds the closure problem.
Computing stn~cturalformula~ 59 units in which chemical analyses of igneous . - ~ ~xe analyzed. Lithoz 12, 33-39. Chayes, F. 1977:Ratio Correlation. U niv. Chicago Press. Greene, G. M. 1974: The gcocbemist,'y of spinel lherzo[ites from Xedapasco de la Joya, San Luis Potosi, Mexico. MS thesis, University of Houston. 63 pp. Greene, G. M. & Buffer, J. C. 1979: Geochemistry of spinel lherzolites from the Xalapasco de la Joya structure, San Luis Potosi, Mexico, S.L.P. Mineral. Mug. in press. Hess, H. H. 1949: Chemical composition and optical proparties of common clinopyroxenes. Part I. Am. Mineral 34 621. Koch, G. S, & Link, R. F. 1~70 ~tatist!ca! Analysis ~,[ Geological Data. John Wiley & Sons, New Yozk. Neiva, A. M. R. 1974: Geochemistry of t o u ~ i n e (:~chorl :~:) from ~'anites, ap;ites and pegmatites from northern Portugal. Geochim. et Cosmochim. Acts 38, 1307-~317. Schweitzer, E. L., Patpike, J. J. & Ben,.e, A. E. ]979: Statistical analysis of cfinopyroxenes from deepsea basalts. Am. Mineral. 64, 501-513. Accepted for publication August 1979 Printed January 1980
Appendix I - Derivation of text equation (4) Equation (4) can be derived by following and expanding the derivations given by Chayes (1971:37-39) and Buffer (1979:75gl). The wuiance of a proportion Y, .~sgiven by:
v ~ r, = (X~I~,~ct+ ~ - 2r,,c,c,) (Buffer 1979, Eq. 8)
where X is the mean of the variable indicated by the sub,.icript, C is the coeff,cient of the indicated variable, the subscript t refers to the row sum variable (the sum of the values that are being convened to proportions), and ru is the part whole correlation - the correlation between variable i and the. row sum t). A similar expression can be written for variance of the proportion Yj by replacing appropriate subscripts in (5). From Chayes' (1971:38) equation (4.9) the product of the deviations of Y~and Yj can be written as fol[ows: where ~. is the deviation oftho value ofn rio,,, IL~,scan value (in Chayes" exF~ression he uses p,=,fd~',, otherwise the expression above is identical to Chayes" (197i:38) equation 4.9). Taking expectations of I~oth sides of the above and not requiring that the expectation of the product of two deviations is zero (that is, E(SmS~)=r,ts~,s, where r is the correlation between m and 1 and s is the standard deviation) yields the following expression for the covariance between Y,Yj: COVf, = ( ! 1 ~ ) (r.s,sj + ~ . , - (.t~,s ~,~.t,))
References Buder, J. C. 1979: The effects of closure on the moments of a distribution. Math. Geol. 12, 75.-g4. Buffer, J. C. 19~9: Numerical consequences of ehaf~glngthe
(5)
- (~us,s,l.f~)(6)
Rearranging and remembering that C, ---s,I](, and dividing by the product of the square roots of the variances for Yeand Yj (equation (5)) yields the expression for the correlation between two proportions i~ terms of th~ p~openies of the initial values (text equation 4).