32
INDUSTRIAL AND ENGINEERING CHEMISTRY
ever, a few preliminary experiments indicate that galacturonic acid is affected but little by either organism. ACKNOWLEDGMENT
Grateful acknowledgment is made to L. J. Wickerham of the Northern Regional Research Laboratory for his cooperation in supplying the organisms and to Eda Nihlen of The Institute of Paper Chemistry for drawing Figures 1 and 2 of the text. LlTERATURE CITED (1) Anderson, Ernest, Cniversity of Arizona, unpublished data. (2) Assoc. Official Agr. Chem., Official and Tentative Methods of Analyses, 5th ed., p. 500, 1940. (3) Fisher, E . , and Thierfelder, H., Ber., 27, 2031 (1894). (4) Haar, A. W. van der, “Anleitung zum Nachweis, zur Trennung
und Bestimmung der Monosaccharide und Aldehydsaiiren”, p. 123, Berlin, Gebruder Borntraeger. 1920.
Vol. 16, No. 1
(5) Harding, V. J., Nicholson, T. F., and Grant, G. A, J . Biol. Chem., 99, 625 (1933). (6) Hopkins, E . W., Peterson, W. H., and Fred, E. B., J . A m . Chem. SOC.,52, 3659 (1930).
(7) Kluyver, A. J., dissertation, “Biochemische Suikerbepalingen”, Technische Hoogeschool, Delft, 1914. (8) Kurth, E. F., unpublished data, The Institute of Paper Chemistry. (9) Kurth, E . F., and Ritter, G. J., J . Am. Chem. SOC.,56, 2720 (1934). (10) Menzinsky, G., Svensk Papperstidn., 45, 421 (1942). (11) Schmidt, E., Trefz, F., and Schnegg,H., Ber., 59, 2635 (1926). (12) Scott, M., and West, E . S., PTOC. Soc. Ezptl. Biol. Med., 34, 52 (1936). (13) Sherrard, E. C., and Blanco, G. W., IND. ENQ. CHEM.,15, 611 (1923). (14) Wise, L. E., and Peterson, F. C., Ibid., 22, 362 (1930). before the Division of Sugar Chemistry at the 106th Meeting of the AMERICAN CHEMICAL SOCIETY, Pittsburgh, Penna.
PREBENTED
Use of the Discriminant Function in the Comparison of Proximate Coal Analyses W. D. BATEN AND C. C. DEWITT Michigan State College, East Lansing, M i c h .
Comparing a number of analyses of a material from one source with analyses of simila, material from another source has heretofore presented a very real p;oblem. The present paper applies statistical methods to the comparison of nroximate analyses and B.t.u. per pound of coal from two mines. This statistical analysis b y random sampling shows the probability that two coals come from the same source. The discriminant function developed b y Fisher is used. By means of this function a comparison may b e obtained between analytical data based on material from one source and data on similar material from a different source. In addition, the order of significance of the several analytical constituents in a single series of samples may be determined. The method is capable of general application.
F
ISHER (6), in 1936, developed the discriminant function for the comparison of multiple measurements obtained in taxonomic problems. Since then three papers ( 3 , 4 , 5 )have appeared which make use of Fisher’s technique. The present paper 1s concerned with the application of the discriminant function to thedifferentiation of two series of proximate coal analyses and thc B.t.u. per pound of coal. Each series of analytical data is from a different mine. Each series consists of 100 samples of coal. The proximate analysis, covering the volatile matter, fixed carbon, per cent of ash, as well as the B.t.u. per pound of coal, is used in making the comparison of the coal from these mines. The samples were taken from cars of coal sent to this college over a period of several years, and the analyses are reported on samples dried a t 105’ C. The analytical data, while accurate, indicate that the mrthods of sampling may be questioned. The value of the present approach is, however, not in the data reported but in the application of statistical procedures to comparison of similar chemical data. The discriminant function enables one to obtain a numerical comparison of the coals by the use of two linear compounds or equations in which the effects, in the present instance, of all four of these measurements are combined, Further, the application of this function to these analytical data permits a test for signifi-
cance between the constituents of which the compounds are formed. The compound for the first mine is:
X
=
bixi
+
+
b 2 ~ 2
+
b 3 ~ 3
b4x4
where 21,2 2 , xz, and x, represent, respectively, B.t.u., per cent of volatile material, per cent of fixed carbon, and per cent of ash, and bl, b2, bs, and bc are constants to be found. The variables XI, x2, x3, and z4 may be correlated. The compound for the second mine is
X’ = biz:
+
+ b3xi + bax:
b2~:
where xi, 4, x;, and a$ represent, respectively, the above similar measurements; the coefficients are the same as in compound X. The difference between the means of the above two compounds made up of the four measurements is
D where - d, = 2, -
=
bid1
z,
+ bzdz + -
d, = x,
b3d3
+ bdr _
-
(1)
-
- x13.I = x, - xi,andd,
-
= x,
-
z,E , E, and 2, represent the arithmetic means of the respective measurements- made of the coal from the first mine
xi, and
.
~
---
.
2
and x,’,xl, xi, and represent the means of similar measurements made on the coal from the second mine.
Table I. Measurements of B.t.u., Per Cent of Volatile Material Per Cent of Fixed Carbon, and Per Cent of Ash for M i n e s A and Mine B Mine A Per VolaPer Volatile Fixed cent tile Fixed cent matter, carbon, ash, B.t;u., mat;er, oarbpn, as), B.t.u.,
d
21
13,000 13,700 12,800 12,300 14,100
... ...
li,‘900 Av. 13,110
za 25.7 25.0 23.0 22.8 33.5
...
...
2:
64.3 64.2 66.2 59.9 60.2
... ... ...
24
7.9 8.4 10.0 12.9 6.9
... ... ...
$1
13,600 14,300 13,000 14,000 13,700
... ...
22
23
33.0 36.9 35.5 34.9 30.1
59.6 56.3 50.9 58.7 59.3
... ... ...
...
...
...
=4
6.5 6.2 12.9 5.8 10.0
... ... ,..
2i:3
59.4
8.3
13,800
35.2
54.8
9.2
28.33
61.16
8.71
13,596
34.00
56.28
8.65
ANALYTICAL EDITION
ilawav IS, i944
Table I contains a few measurements from mines A and B, taken at random from the sets of observations. The means of the 100 measurements on each of four properties are given in the last line of that table.
(C) bl
-
33
0.049277bz
+
0.089418bs
-
0.041091b4 = -0.000236
Let 8 = b:sn
+ b:Szz + b:sss +
+ 2bib2~1z+ 2bibsSis + + 2bzbs~zs+ 2bzb4sz4 + 2bsbhsst
b:~44
2bibasic
( B ) - (C)
(C)
where
+
0.06231082
100 s11
=
(Z1.k
- Z)’
k-1
+
100 (ZL
k = l
-
a’
In our case this quantity is (using values in Table I) found as follows:
+ . .. + + . +-
+-
= (13 (900 - 13,110)” (13,700 13,110)’ (13,900 13,110)2’+ (13,600 13,596)’ ‘14,300 13,596)’ ., 13,596)’ = 20,110,000 -r 17,718,4,, = 37,828,400 (13,800
811
-
-
0.000282
+ 0.083387ba - 0.030309b4 =
(0) -0.053330bz
-
+ +
+ ... +
+--
-
01
+
Dividing each equation by the absolute value of the coefficient of br in it gives:
(H) -ba
+ +
1.012215b4 = 0.001711 = 0.029157 -0.027446;
(I) -bt 4.950058br (H)- (I) -3.937843b4
From (E) From (F) From (G)
bf = 0.008820 h: = 0.008821 h2 = 0.008821; b~ = 0.008821 (average)
(A) (B) (C) (D) Therefore
bl = 0.000007 bl = 0.000007 bl = 0.000007 bi = 0.000007; bi = 0.000007 (average)
=
-
+
-
-
+
There are several ways of solving these equations. One way (.a), using the computing machines, is as follows: Dividing each equation by the absolute value of the coefficient of bl in it gives:
+ 0.003275b2 + 0.000547ba - 0.003725b4 0.000013 = bl + 0.013033h - 0.008226bs - 0.004610br = 0.000046
b4
= 0.006970
+ 0.008821.~~1+ 0.00534528 + 0.006970~4
The mean compound pertaining to mine A is =
123,906.00b1+ 1,614.87bz - 1,019.31b8- 571.21b4 = 5.67 20,685.00bi 1,019.31bz 1,849.62bs 849.97br = -4.88 -140,927.OObl - 571.21bz 849.97bs 1,519.42b4 = -0.06
0.005345,
0.000007~1
X = 0.000007%
+ 123,906.00bz + 20,685.2Obr 140,927.OOb4 = 486.00
=:
This method of solving simultaneous equations is easy to follow and easy to explain to the average computer. The linear compound which enables one to detect the greatest difference between the mines in relation to these four memurements is
-
37,828,400.OObl
0.006970
br = 0.005344 b3 = 0.005345; bs = 0.005345 (average)
hi = 0.000007, bz = 0.008821, bs
X
From these equations the values of the b’s, of the coefficients in compound X , can be found. The solution of these equations gives, of all linear com ounds in zl,2 2 , z3, and x, the one compound, X , which most cfearly discriminates one mine from the other. These equations are:
=
b4
From (H) From (I)
From From From From
+ - - + + . . .+ -
The quantity .cI is equal t o the sum of squares within compounds. By maximizing the quantity D z / S the following equations arise. The& e uations are actually proportionalities, but for the p ose evaluating constants, bl, 62, ba, and b4 t h y may be u s 3
+
+
By using the valuea in Table I this is 82s = (25.7 - 28.33) (64.3 61.16) (25.0 28.33) (64.2 61.16) (27.3 28.33) (59.4 61.16) (33.0 34 .OO)(59.6 - 56.28) (36.9 34 .OO) (56.3 56.28) (35.2 - 34.00, (54.8 56.28) = - 645.82 373.49 = -1,019.31
+
0.899057b8 0.090695br = -0.003382 ( E ) -b2 (F) be - 1.567068bs 0.585476b4 = 0.004526 1.563604ba - 0.568329b4 = -0.004425 (G) -bs (E) (F) - 0.668011ba 0.676171b4 = 0.001143 (F) (G) - 0.003464bs 0.01714bd 0.000101
The value of s i j where i = 2 and j = 3 is
(B)
=
0.000236 Dividing each equation by the absolute value of the coefficient of bz in it gives:
where k is the variable of summation. The value of s i j when i = 1 m d j = lis
(A) bl
- 0.097644b3 + 0.036481bd
+ +
+
++
0.0058212~ 0.00534528 0.006970Z, 0.000007 (13,100) 0.008821 (28.33) 0.005345 (61.16) 0.006970 (8.71)
or
+
-
X = 0.7293
-
The mean compound pertaining to mine B is
+
+
X’ = 0.000007 (13,596) 0.008821 (34.00) 0.005345 (56.28) 0.006970 (8.65) = 0,7562 The difference D between these two means is 0.7562 - 0.7293 = 0.0269. This can be found directly from Equation 1 as follows:
D
=
0.000007 (486)
+
+ 0.008821 (5.67) + 0.005345 (-4.88)
+ 0.006970 ( -0.06)
or
D = 0.003402 f0.050015
- 0.026083 0.000418 = 0.0269
(2)
as before. The question arises as to whether or not the means of these compounds are statistically significantly different. Table I1 contains an analysis of variance of these compounds and enables
I N D U S T R I A L A N D E N G I N E E R I N G CHEMISTRY
34
one to test for a significant difference between these two means. The two asterisks in the last column indicate that these compounds, .are\highly significantly different. This means that the probability of getting by chance such a large value of D is less than 0.01. This is found from a table of F values (1, 7). The odds in favor of getting by chance such a large difference between the compounds pertaining to the two mines are less than 1 to 99. This indicates that the coal from mine A is definitely different from the coal from mine B, as far as these four measurements are concerned. By examining the four terms in Equation 2 one can determine which characteristic of the coal (B.t.u., per cent of volatile material, per cent of fixed carbon, or per cent of ash) is most important for differentiating the two coals, which is of next importance, etc. The second term, 0.050015, in this equationis greater than the absolute value of each of the other terms; hence the per cent of volatile material is the most important factor, in this case, for determining whether or not the coal from one mine is different from the coal from the other mine. The absolute value of the third term in this equation, 0.026083, is second in size; hence the per cent of k e d carbon is next in importance for revealing a difference between the coal from the two mines. The factor B.t.u. is third in importance for differentiating these two coals, Per cent of ash is of the least importance for these mines. The order of importance of these factors may change for other coals. Table I11 gives the ranks of importance of these characteristics when various combinations are used to calculate the compounds. Table 111gives the ranks of the characteristics of the coals for various combinations of the sets of measurements. I n column 2 it is seen that per cent of k e d carbon is the most important characteristic for differentiating one mine from the other as far aa per cent of volatile material, per cent of k e d carbon, and per cent of ash are concerned.
Vol. 16, No. 1
Table 1V. Arithmetic Averages of B.t.u., Per Cent of Volatile Material, Per Cent of Fixed Carbon, and Per Cent of Ash Measurements from M i n e s C and D Per Cent of Per Cent of Volatile Fixed Per Cent Mines B.t.u. Matter Carbon of Ash C 13,700 35.09 57.18 6.56 D 13,205 29.99 59.81 8.72 Difference 495 5.10 -2.63 -2.16 ~~
Table V. Averages of B.t.u. Per Cent of Volatile Material, Per Cent of Fixed Carbon, and Per Cent of Ash for M i n e s E and F Per Cent of Per Cent of Volatile Fixed Per Cent Mines B.t.u. Matter Carbon of Ash E 14.012 34.15 60.31 4.21 12,134 F 35.78 48.14 13.25 Difference 1,878 -1.63 12.17 -9.04
most important, per cent of fixed carbon is second, per cent of volatile material is third, and per cent of ash is last in importance in testing whether or not the mines differ as far as a compound of these four sets of measurements is concerned. Table V contains averages of measurements pertaining to B.t.u., volatile material, k e d carbon, and per cent of ash for mines E and F. The difference between the means of the two discriminant functions pertaining to the mines is
++
D = 0.000083 (1878) - 0.001356 (- 1.63) 0.006124 (12.17) + 0.009580 (-9.04) = 0.155874 0.002210 0.074529 - 0.086603 = 0.146010
+
In this case B.t.u. is first, per cent of ash is second, per cent of fixed carbon is third, and per cent of volatile material is fourth in importance for differentiating between the coal from these mines. DISCUSSION
II.
Analysis of Variance of the Compounds Degrees of Source of Variation Freedom Sum of Squares Mean Square 199 Total Between compound mean8 4 500% 0.0362 0.00905** Within 195 D 0.0269 0.00014 Table
--
....
Table 111.
Rank of Characteristics for Various Compounds Pertaining to Coals A and B Compound Composed of
..
B.t.u. Volatile matter Fixed carbon Per cent ash
Per cent B.t.u. of volatile matter Per cent Per cent of volatile of volatile Per cent matter of fixed B.t.u. Per cent matter carbon Per cent of carbon Per cent Per cent of volatile Per cent of fixed Per cent of fixed carbon matter of ash of ash carbon .. 2 .. .. 3 3 1 .. 2 1 1 .. 1 1 2 3 2 .. ..
..
_-
The discriminant function enables one to test for a statistical difference between two linear compounds made up of several variables or measurements. I t has many advantages because it furnishes one measurement pertaining to a combination of several measurements. In the present instance the statistical analysis of the data shows a difference between the coal from the two mines. I t may be useful in the future to compare analytical data from other mines. Such an effort would involve the selection of a standard coal, and it seems apparent that other analytical data, such as per cent of sulfur, per cent of moisture, and the fusing temperature of the ash, should be included in the more accurate statistical comparison. CONCLUSION
Fisher’s discriminant function has been applied to the differentiation of analytical data obtained from one hundred coal samples taken from each of two mines. The intercomparison of the relative significance of the elements of the analytical data relating to coal has been accomplished. The probability that two coals come from the same source by random sampling is given.
Table I V contains the means of measurements pertaining to B.t.u., volatile material, fixed carbon, and per cent of ash for two other minee, C and D. The discriminant function pertaiaing to these mines for the value D is 1) = 0.000020 (495)
- 0.006360 (5.10)
-
0.001659(-2.63) - 0.000468 ( - 2 16) OI‘
D D
0.00990
- 0.00350
LITERATURE CITED
+ 0.00410 + 0.00101
0.01151
An analysis of variance table (not given) shows that the two coals are significantly different. The characteristic B.t.u. is
(4) . .
(5) (6) (7)
Baten, W. D., “Elementary Mathematical Statistics”, pp. 24962, New York, John Wiley & Sons, 1938. Baten, W. D., J . Agr. Research, 61,237-40 (1940). Brier, G. W., Schott, R. G., and Simmons, V. L., Proc. Am. Soc. Animal Prod~Ction,1, 153-60 (1940). Cox, G. M., and Martin, W. P., Iowa State College J . Sci., 11, NO. 3, 323-31 (1937). Day, B. D., Sandomire, M.M., J . A n . Stat. Assoc., 37,461-72 (1942). Fisher, R. A., Ann. Eugen., 7, 179-88 (1936). Snedecor, G. W.,“Statistical Methods Applied to Experiments in Agriculture and Biology”, pp. 184-7, Ames, Iowa, Collegiate Press, 1940.
