use of a discriminant function for differentiating soils with different azotobacter populations

8/13/2019 USE OF A DISCRIMINANT FUNCTION FOR DIFFERENTIATING SOILS WITH DIFFERENT AZOTOBACTER POPULATIONS

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. I . . ~ ~ -USE OF A DISCRIMINANT FUNCTION FOR DIFFERENTIATING

SOILS WITH DIFFERENT AZOTOBACTER POPULATIONS'GERTRUDE M. Cox A:SD '\V ILLIAM P. MARTIN

rom th Statistical S ~ c t l o n and Soils Subsection. Iowa Agricultural Experiment ~ t i o nAccepted for publication April 7, 1937

.. After measuring several characteristics of each member of two or

1ore groups, the investigator may wish to know i the groups differ significantly. The usual method is to test the significance of the differencebetween the group means, taking each character separately. But unfor-

tunately, there is no way to combine the knowledge gained. The value ofthe information furnished by the several varieties may be different. Fur.thermore, correlation among the variates will make it inappropriate totreat the differences as independent. Another method used is the Coefficient of Racial Likeness (6), whichgives a single numerical measure of the whole system of differences. TheCoefficient of Racial Likeness is made up of the sums of squares of the


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324 GERTRUDE M COX AND WILLIAM P MARTIN

able phosphates from 16 to 520 pounds per acre, and the total nitrogencontent from 7 to 78 milligrams per 10 grams of soil. The mean value foreach variate, for the samples n group I and group II, is given in table 3The samples which contained Azotobacter have, on an average, a higherpH, more available phosphate and a larger total nitrogen content thanthe samples which did not contain the organisms. .The mean differences,1.408, 82.007, and 8.260, are given at the bottom of the table. The variatesused:

X1=PHX 0 the amount of readily available phosphate,

/ Xa = the total nitrogen content,. I

\ are quite different in numerical size with correspondingly large differ-Ll _2ences in variance. . .r What weighted compound of the three variates will afford the maxi.mum differentiation between the two groups of soil samples? Or, whatcoefficients of the linear function of the three variates,

TABLE L pH Xt)1 available phosphate content (XJ) and total nitrogen. content X1


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; GR l1\UDE l\1. coX AND WILL AM P ..MMTINX1 . X2. X3f1 X L 1 L,----=:::=. L3 --;::::::::::::fi y ~ x . y ~ x , y ~ x s

[J will maximize the ratio or the difference between the means of the groups

IJ to. the standard deviation within the groups? X is the weighted compound.f. of the measurements of pH, available phosphate and total nitrogen. Thequantities L,. L and La are the coefficients by which the respective[' measurements (each divided by the square root of the sum of squaresfi within groups of any individual soil samples should be multiplied in order to form its compound measurementX.The difference betwee n the means of X in the two groups is:

d, d. dD = L -:=:=: + L . ...::_. +La....,.---- ,-..;-sx, v VJ.xswhere d d and d3 are the mean differences between the three variates inthe two groups. The problem is then to find the values of L L and Lasuch that D is a maximum. The method involves the solution of a set ofnormal equations similar to those leading to multiple regression. Sincethere are t\vo groups of observations, some of the calculations are likethose which have become familiar in analysis of variance. The newfeatures are to be described in some detaiLTABLE 3. Number of samples, sums, means and mean differences for pH, availablephosp tate content nd total nitrogen content

pH I hosphate. \ NitrogenGroup x. X X. Number 100 100 100L With Azotobacter Sum

742.3 13312 2940Mean 7.423

133}20 29.400

Number 186 186186

lL Without Azoto Sum. 1118.7 9507 3932

bacter Mean6.015 51.113 21.140

Mean differenCe1.408 . 82.007 8.260

In table 4 are recorded the computations leading to the pooled sumof squares and products within the two groups of soil measurements.the line of totals, the entries are the sums of squares and products of tentire 286 observations in tables 1 and 2 no distinction being made asgroup. In the lines for groups are put down the sums of squares aproducts of the group sums in table 3 c l c u l t ~ d in the manner charteristic of analysis of variance. As examples, the entry for column Xrow X 1 of table 4 is,(742.3) (1,118.7) = 12 238.5321100 186


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DISCRL INANT FUNCTION FOR .1

.tl- >OTOBACTER

-1111,:;1 - 1 II>

327

328 GERTRUDE M COX AND WILLlt\lV P :MARTIN. 1(742.3) (13,312) (1;118.7) (9507).;__ _;___ + = 155,994.9808.100 186

The differences in the third line are the sums of squares and products ofdeviations from means \vithin the groups.The calculation. of the standard deviations and the correlation coeffi-cients now proceeds in the usual manner (8, table 7a, page 32). As eamples

S x l=y111.0879 10.5398= = 0.625422,084 y2s4

2,292.7192 = 0.213019.(10.5398) (1,02.1.1754)The degrees of freedom used, 284, are those. within the two groups,(100 - 1 ) + (186 - 1 ) .t may be observed that tl1e pooled standard deviations of these variates are very different, and that there is little correlation between thevariates within the two groups. t is usually of interest. to observe thesestatistics, and it takes little extra time to compute them. In addition, ithas been found convenient to use the correlation coefficients in the solu-tion of the normal equations which follow.The correlation coefficients from- table 4 are carried into table 5where they are used to solve the linear function, :X, which best discriminates the two groups of soils. The coeffiCients L, Lz and L 3) requiredare proportional to the solutions of the equations,

r 11Lt + 12L2+ 13La = 1, o 0,r12L1 + zoL2 + zaLa = 0, 1, 0'1.r13L1+ zaL2+ aaLa = 0, 0,

Each expression, in turn, is set equal to 1 with tl1e other expressions equalTable 5 is worked in a manner similar to tahle .8 (page 36 in Wal-o 0.lace and Snedecor (8) except for the three back solutions. The k valuesobtained constitute the matrix in table 6. 'I11ey will be used below to

calculate the desired L coefficients.Going back, now, to the rnean differences given at the bottom oftable 3, d ~ 1 . 4 0 8

d ,= 82.007,d,=3.260,

cnrh cliffrlrnrr is cliviclrrllw tho s ~ u n r r rnot nf its snm nf SCI11nl'OS w\tbin


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... l ..1:::o.MN' ......>.."' ...00

M......,JI.,_;; ........' .;.; ..... ~ ; i : 0- '' e,

..

. DISCRIMINANT FUNCTION FOR AZOTOBACTER

/

0 o ..0 OM0 o n0 o ....0 0r ".-irl...

-00 e>e>mco rlrl ..co OOrl00 ~ ~ H 30

: : i ~ C " CI. e>.1- I I+ ,c, :-' ' Mt t l : : I O.... Nt-"


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DISCRL\UNANT FUNCTION FOR AZOTOBACTER 331This value is entered in table 7. The sum of squares witldn. roups isthe value of D, 0.021777. In determining lhe degrees of freedom betweengroups, note that in addition to the specific mean difference, two adjustableratios have been used, making the thre e degrees of freedom. The analysisof variance is completed in table 7. The difference between the groupmeans is highly significant, showing that the pH, available phosphate andtotal nitrogen content of the soils tested give significant information aboutthe presence of Azotobacter.

jTABLE 7. mrlysis of va.n ance of the cntde compound X b e ~ w e e n and wit1t.in groupsDegrees of Sum ofSource of variation freedom squares Mean square

Be tween groups J 3 1/ .030842 -lilt 01028Within groups ,, 282 .021777 :i f .00007722Total 7 285The relative value of these variates for discriminating between thegroups is apparently indicated by the values of the coefficients,

IL 1 = .11837 4,L 2 = .054118,L 3 = .033599.

t may be concluded from these results, therefore, that the pH, thecontent of available phosphate and of total nitrogen serve to significantlydistinguish the samples of Iowa soils which contained Azotobacter fromthose which contained none of the bacteria. In addition, the results indicate that the presence of Azotobacter in Iowa soils may be most closelyassociated with tl1e pH, closely associated with the available phosphatecontent of the soil and least associated with its total nitrogen content.LITERATURE CITED

1. BARNARD, M. M.1935. The secular variations of skull characters in four series of Egyptian. skulls. Ann. Eugen., 6:352-371..2. FISHER, R. A.1936. The use of mulliple meosurements in taxonomic problems. Ann. Eugen.,7:179-188.3, 1934. Statistical methods (or research workers. Ed. 5 Edinburgh, Olivernnd Boyd.4. MARTrN' E. A.1936. A study of nn Egyptian series of mnndibles, with speci;tl reference tomathematical methods of sexing. Biomelrika, 28:149-178.i. M.l\nTN. W. P_ R T T. W M f ~ t r t Aim P. R Br.oWN

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t

332 GERTRUDE M COX AND WILLIAMP. MARTIN6. PEAJISON, K. \ .1926 On the coefficient of racial likeness. Biomelrika, 18:105-117.7. Sr-.nrn H. F.1936. A discriminant unction or plant selection. Ann. Eugen. 7:240-250.8. 'VALLACE,H A. AND G. w SNEDECOR1931. Correlation and machine calculation. Iowa State CoUelie Off. Pub., 30No. 4 Revised edition.

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use of a discriminant function for differentiating soils with different azotobacter populations

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