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Paleontological Society

Studies in Quantitative Paleontology II. Multivariate Analysis: A New Analytical Tool forPaleontology and GeologyAuthor(s): Benjamin H. BurmaSource: Journal of Paleontology, Vol. 23, No. 1 (Jan., 1949), pp. 95-103Published by: Paleontological SocietyStable URL: http://www.jstor.org/stable/1299717

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JOURNALOFPALEONTOLOGY,OL.23, NO. 1, PP. 95-103, JANUARY 949STUDIES IN QUANTITATIVE PALEONTOLOGY

II. MULTIVARIATEANALYSIS-A NEW ANALYTICAL TOOLFOR PALEONTOLOGYAND GEOLOGYBENJAMIN H. BURMA

ABSTRACT-Multivariatenalysis is the simultaneouscomparisonof threeor morevariables in order to determinethe significanceof their likenessesand differences.The techniquehasbeenin use forsome thirteenyears,but no simpleaccountof themethod has hitherto been available. An illustrative example from invertebratepaleontology is hereused, but the method is applicableto any problem n geologyinvolving severalquantitativelyknown variables.

In recent years, increasing attention hasbeen paid to quantitative methods in bothgeology and paleontology. One result of thistrend has been an increased interest in sta-tistical analysis, particularly that part ofthe discipline which deals with tests for thesignificance of the likeness and differencesbetween two samples (or collections). Thesetests have, however, been rather slow ingaining acceptance and for a variety ofreasons. Most of these reasons are not ger-mane to the issue, but there is one, ratherpoorly defined, which seems to have beeneffective. There seems to have been an al-most subconscious realization that the usualtests of significance involve only one, or atthe most two, variables, whereas geologicalproblems usually involve several variables.A more extended discussion of this phase ofthe problem will be found in the paper onquantitative methods by Burma (1948). Ashort summary will serve here.A comparison of a single variable betweentwo samples is readily effected by comparingtheir means (Simpson and Roe, 1939; pp.192-197, 210-212). For certain simple prob-lems, such a comparison is adequate. Abetter comparison of two samples may bemade on the basis of two variables, providedthat these variables show a notable amountof correlation as will be found to be the casein the great majority of problems. This canreadily be accomplished by computing theregression line of one variable on the otherfor each sample, and then comparing the tworegression lines (Simpson and Roe, 1939; pp.262-280).Both the above methods ignore the fact

that much geological data, and particularlypaleontological data, involves several vari-ables and that the characteristics of a givenspecimen depend not on characters A, andB, and C, but on characters A, B and C con-sidered together and simultaneously. We areusually interested in comparing objects asa whole, and not characters as such. Sincethe methods discussed above will not makea comparison of more than two characterssimultaneously, it is obvious that they willbe inadequate in such cases. Such a compari-son would have to involve a comparison ofthe mutual regressions of several characterssimultaneously.The problem of handling multiple regres-sions was successfully solved by Hotellingin 1931 (Hotelling, 1931). The first personto use this method in the solution of a prac-tical problem seems to have been M. M.Barnard, who applied it to a study of fourseries of Egyptian skulls (Barnard, 1935).It has been used by others since that time,but still is not widely known. Simpson andRoe's "Quantitative Zoology" which treatsother statistical methods for biological datawith some thoroughness, does not mentionthis method. Multivariate analysis is treatedin a number of more advanced works onstatistics (e.g., Fisher, 1946; pp. 285-289)but in these books the method is presentedin a way which is not usable to a personwithout advanced mathematical back-ground.In essence, the method may be picturedthus: If we are dealing with one variable, wemay use the value of this variable as a co-ordinate and plot the variable onto a line,

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BENJAMIN H. BURMAusing a point for each specimen, and thenstudy the distribution of these points. If wehave two variables, we have two coordinantsfor our points and plot them into a plane intwo dimensions. If we have three variables,our points would be plotted in a "solid" ofthree dimensions. If we have n variables,we use n-dimensional space. We would thenstudy the dispersion of the points in thisn-dimensional cluster, doing this for each ofthe two samples we wish to compare. Nextwe would combine our two samples andstudy the dispersion of the combined sample.The relative dispersions of the original andcombined samples will enable us to decidewhether the two samples differ significantlyor whether they do not.Fortunately, the actual computations in-volved in this method are all very simple.These operations are, however, ratherlengthy and ordinarily it will be found thatlong-hand computation is much too time-consuming except possibly for a comparisonof three variables. A slide rule cannot be usedbecause of the low order of accuracy it al-lows. A calculating machine is thus a neces-sity in the use of this method. It should bean electrically driven model with at leastten rows of keys and preferably with auto-matic division and multiplication. Such amachine is suitable for calculations involv-ing up to eight variables or so. If it is neces-sary to handle more than eight variables itwould probably be well to use an Interna-tional Business Machines calculator, feedingthe data on punched cards. Ordinarily, itwill be found that the number of charactersto be compared can be held to a reasonablefigure.

METHOD OF CALCULATIONTo illustrate the calculations of multi-variate analysis, we may use an exampletaken from the blastoids. These were col-lected from the Paint Creek formation nearFloraville, Illinois. They are of the groupcommonly referred to Pentremites godoni,but it was soon discovered that they weresusceptible to division into two groups, oneof which had a somewhat stellate horizontalsection associated with a rather low deltoid-standard radial ratio and another with amore rounded section and a markedly higherdeltoid-standard radial ratio. The first of

these we shall call P. godoni alpha and thesecond P. godoni beta. Comparisons of re-gression lines of two characters indicatedthat the two did not differ significantly. Inspite of this, it was felt that there was a realdifference between the two, so it was de-cided to try multivariate analysis. For thispurpose, eight characters were measured oneach specimen: the standard radial, lengthof the deltoid, the base of the radial, heightof the azygous basal, length of the ambulac-rum, the total height, the thickness, andthe number of side plates per ambulacrum.(For an explanation of this terminology, seeBurma, 1948.) In all, 20 usable specimensof P. godoni alpha and 20 of P. godoni betawere available. These two samples will here-after be referred to as "alpha" and "beta"respectively. The original measurements areshown in Table 1. All measurements are inmillimeters. In this and subsequent tables,the column headed by (1) will be the datafor the standard radial; (2) the deltoid; (3)the base of the radial; (4) the azygous basal;(5) the ambulacrum; (6) the height; (7) thethickness; and (8) the number of side plates.All measurements in any one horizontal roware of one single specimen. Column (1)should always be the character on which theother characters will regress, usually the"time" character.Since we will be dealing with regressionformulae which are accurate only in straight-line relationships, we should be sure that ourdata fulfills this requirement. If the presentdata is plotted arithmetically, it will befound to show a definitely curvilinear trend.Knowing, however, that growth is com-monly governed by an exponential function,we will find that the logarithms of these datawill yield a plot whose trend approximatesa straight line.For that reason, we will in this case dealwith the logarithms of the data rather thanthe original measurements. In order to dealwith whole numbers and digits as low aspossible, the following procedure has beenadopted. Three place logarithms have beenused as sufficiently accurate and each hasbeen multiplied by 1000 to get rid of thedecimal point. The size of the digits was thenreduced by subtracting a number from thelogarithms for each character, the samenumber being subtracted, for example, from

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STUDIES IN QUANTITATIVE PALEONTOLOGYcharacter (1) in both alpha and beta. Thisis important. From character (1), 700 wassubtracted; from (2), 600; (3), 500; (4) 300;(5) 1000; (6), 1000; (7), 1000; and (8), 1500.This computation is then checked. This step

first step is to obtain for each set of data, foralpha and beta, the sum of each column(2(X)), the sum of the square of each mem-ber of each column (Z(X2)), and the sum oftwice the product of corresponding members

TABLEI.-ORIGINAL DATAP. godoni alpha

(1) (2) (3) (4) (5) (6) (7) (8)6.47.47.57.78.89.09.39.49.59.69.89.510.010.110.110.410.510.712.48.3

4.76.65.45.47.47.58.48.27.98.68.68.09.69.08.98.78.28.59.26.5

4.14.04.14.85.95.66.55.54.65.15.56.06.05.46.15.26.04.76.65.0

3.02.52.53.53.53.44.44.13.53.53.24.4'4.33.74.53.24.43.55.03.5

10.313.011.511.614.515.116.916.616.017.016.916.419.018.518.017.517.216.919.913.9

12.314.614.214.517.217.720.019.418.819.019.719.521.019.819.719.019.519.523.615.3

11.512.011.512.416.116.019.017.515.317.516.417.518.817.619.516.817.516.518.714.5

3341373747435047475352475252545354545242

P. god(1) (2) (3) (4)5.5 4.9 4.0 2.56.2 4.7 4.0 2.66.6 6.5 4.0 2.77.0 5.9 3.9 2.47.4 6.4 5.0 3.67.5 9.0 4.8 3.37.5 7.7 4.6 3.27.9 7.0 4.9 3.38.3 7.4 4.9 3.08.4 7.4 5.4 4.08.4 8.2 5.3 3.68.5 7.8 4.6 3.58.5 8.6 4.4 3.38.5 7.9 5.0 3.68.5 9.6 4.9 3.48.6 8.7 4.5 3.58.6 8.1 5.5 4.08.6 9.5 5.3 3.99.5 8.9 5.3 4.010.8 9.9 5.5 4.1

oni beta(5) (6) (7) (8)

10.0 12.0 10.6 3310.1 12.4 11.1 3412.5 14.4 12.0 4112.2 14.4 11.3 3812.2 15.1 12.9 3816.0 16.5 15.9 5014.9 17.0 14.5 4914.2 16.6 15.1 4314.9 16.4 15.4 4914.0 16.5 15.5 4216.2 19.2 16.2 4814.5 17.6 14.7 4616.8 18.6 15.3 5315.9 17.5 16.8 4817.7 19.0 16.4 5016.3 18.0 15.0 4915.8 18.6 16.2 4618.0 19.5 17.0 5716.5 18.4 17.5 5319.5 21.9 19.4 58

TABLE 2.-ADJUSTED LOGARITHMSOF DATAP. godoni alpha(1) (2) (3) (4) (5) (6) (7) (8)

106169175186244254268273278282291278300304304317321329393219

072220132132269275324314298334334303382354349340314329364213

113102113181271248313240163208240278278232285216278172320199

177098098244244231343313244244205343333268353205343244399244

013114061064161179228220204230228215279267255243236228299143

090164152161236248301288274279294290322297294279290290373185

061079061093207204279243185243215243274246290225243217272161

P. godoni beta(1) (2) (3) (4) (5) (6) (7) (8)019113068068172133199172172224216172216216232224232232216123

gives the data as recorded in Table 2. Fromthis point on, these data will be used en-tirely. If the original data have an essentiallystraight-line trend, such a step as this couldbe omitted.The actual computation now begins. The

040092120145169175175198219224224229229229229234234234278333

090072213171206354286245269269314292334298382340308378349396

102 098102 115102 131091 080199 256181 219163 205190 219190 177232 302224 256163 244143 219199 256190 231153 244240 302224 291224 302240 313

000004097086086204173152173146210161225201248212199255217290

079093158158179217230220215217283246270243279255270290265340

025045079053111201161179188190210167185225215176210230243288

019031113080080199190133190123181163224181199190163256224263

of columns (1) and (2), (1) and (3), (1) and(4), and so on (Z(XY)). To explain this lastpart of the computation further, in alpha,one would multiply 106 by 072, 169 by 220,175 by 132, and so on, and then sum theseproducts. With a calculating machine, 2(X),

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BENJAMIN H. BURMA2(X2), and 2(XY), may be obtained simul-taneously for each pair of columns, greatlyreducing the work involved. The result forthe data in hand is given in Table 3. Do not"round off" any results of computations toless than ten figures on any place exceptwhere it is done in this sample computation.Such random dropping of figures can havea very adverse effect on the accuracy of themethod.This computation is checked by the fol-

TABLP. godoni

(1) (2) (3)

other checks mentioned must be done. Asingle, small, undetected error early in thecomputation will mean hours of work wastedto obtain an erroneous result. The frequentchecks which are recommended will preventthis.The next step is to obtain Z(X), 2(X2),and E(XY) for the combined data, that is,by combining both samples or collectionsinto one. This is very easily done by addingthe results already obtained for 2(X)

E3alpha

(4) (5) (6) (7) (8)5,291 5,652

1,483,209 1,737,0583,187,5806,407,847

4,010 5,566885,022 1,704,2982,426,9085,016,228

4,4501,071,7922,472,9565,027,957

5,1731,462,6032,878,7626,824,574

P. godoni beta3,552 4,460

676,304 1,088,9401,519,658 1,935,9883,080,984 3,909,940

TABLE 4Combined data

(1) (2) (3) (4) (5) (6) (7) (8)9,301 11,2182,368,231 3,441,3562,807,244

8,002 9,6331,748,096 2,551,5431,996,307 2,407,375

7,206 9,614 7,422 6,6211,536,208 2,492,750 1,585,456 1,262,5091,872,507 2,415,014 1,908,918 1,692,38511,424,075 8,108,941 9,734,524 7,649,453 9,691,009 7,771,523 7,015,510

55.548 46.582 51.3554 -42.663 37.2389 -21.6328 -7.397890.967 0.660 0.814 0.958 0.873 0.891 0.7432170 0068 835 231 502 013 674

lowing method: For each pair of columnsused in computing 2(XY), add the corre-sponding figures in each row, square thissum, and add the squares for the entirecolumns (2(X + Y)2). For example in alpha,columns (1) and (2), 106+072 =178,(178)2 =31,684; 169 +220 =389, (389)2=151,321; and so on, these successivesquares being summed. The following rela-tion should then hold true,

Z(X+ Y)2= (X2) + 2(XY) +Z-( Y).The right hand terms have already beencalculated and the individual terms needonly be added. If the two sides of the equa-tion are not equal, the computation for thosetwo columns must be redone. This and all

2(X2), Z(XY) and (2(X+ )2) for alphaand beta. This has been done in Table 4.This addition is checked as before,z(X+ I')2= z(X2) +22(XF)+2z( 2).

The calculation of the regression equationis next. This equation has the formY=a+bX in which we must calculate thecoefficients a and b.z(Y)2(X2)-Z(X)Z(XF)

N2(X2)-[z(X)12Nz(XY) -2(X)2(Y)b= Nz(X2)- [2(X)12

In these equations, X is the value for column(1) and Y for the other columns. N is thetotal number of specimens in the combined

98

2(X)2(X2)2(XY)Check

z(X)2(X2)2(XY)Check

3,867862,7872,231,6764,577,672

3,339673,421

1,513,3383,071,781

5,1071,395,9432,870,0105,749,162

4,5071,096,8071,960,0183,941,847

4,041918,5952,298,6404,700,444

3,381666,861

1,519,1963,071,079

3,419661,5811,955,9564,100,746

3,202600,928

1,428,8142,914,764

2(X)(X2)

2(2XY)Checkab

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STUDIES IN QUANTITATIVE PALEONTOLOGYsample, in this case 40. a and b are calcu-lated for each pair of columns, (1) and (2),(1) and (3), (1) and (4), and so on. The cal-culated value of these coefficients is shownon Table 4.A sample calculation for the coefficient"a" for columns (1) and (2) is as follows:

(11218)2368231)- (9301) 2807244)a= (40) 2368231) (9301) 9301)456638941= 55.548.8220639

These coefficients should be checked by re-

filled in by subtracting a given computedvalue of Table 5 from its corresponding ob-served value of Table 2. For example, incolumn (2), the first item will be 072-158= -82, the second is 220-219=1, and soon. Be sure to keep proper account of plusand minus signs.After Table 6 is completed, the sum ofeach column, and the sum of the squares ofeach item for each column [2(X), 2(X2)] iscomputed as is shown in Table 7. These arecomputed exactly as in Table 3. In obtainingthe sums of the columns, due account mustbe taken of plus and minus signs. To check,

P. godonialpha(1) (2) (3) (4)106169175186244254268273278282291278300304304317321329393219

158219225235292301315320324328337324346350350362366374436267

117158162169208214223227230233239230245247247256258264306191

138189194203250258270274278281288278295298298308312318370230

059119125136191201214219224228236224245249249261265273334167

130185190200250259271276280284291280299303303314318325381229

TABLE53) (1)

73129134144196205217222226230238226246249249261264272329173

071118123131174181192196199202209199216219219228231237285155

040092120145169175175198219224224229229229229234234234278333

P. godo(2) (3) (4)094145172196219225225247267272272277277277277282282282324378

073107126142158162162177191194194198198198198201201201230266

084126149170189194194213230234234238238238238242242242278323

mi beta(5) (6) (7) (8)

-004 072 014 022045 118 060 061072 142 085 082096 164 108 100119 185 129 118125 190 134 123125 190 134 123147 210 155 140167 229 173 155172 233 178 159172 233 178 159177 237 182 163177 237 182 163177 237 182 163177 237 182 163182 242 187 167182 242 187 167182 242 187 167224 280 226 199276 328 275 240

calculation, a matter of a few momentsusing a calculating machine.For the next step (Table 5), column (1)(our X column) is copied directly from Table2 for both alpha and beta. Columns (2) to(8) are filled in with the calculated value(Y) of that particular character whichwould correspond to the observed value ofX, column (1), using the appropriate regres-sion equation. For example, the equationfor calculating the values for column (2) ofalpha would be Y=55.548+0.9672170Xand the first value in column (2) would be55.548+(0.9672170) (160) =158.0730. Thisis rounded off to 158. In this manner, all ofTable 5 is filled in.After this is done, Table 6 is computed.In this Table, column (1) is dropped and isnot used hereafter. The other columns are

add corresponding columns of 2(X) of alphaand beta together. The answer should beclose to zero and never greater than ? N,in this case, ?20.The next step is to obtain the sum of thecross products of each pair of columns asshown in Table 8, that is, 2(XY) for col-umns (2) and (3), (2) and (4), (2) and(5) . .. , (3) and (4), (3) and (5), and so on.In Table 8 these sums of cross products(2XY) are shown in the corresponding rowand column, for example, the item in row(4) and column (6) is the 3(X Y) for columns(4) and (6). These sums of products arecalculated just as the 2(X Y) in Table 3. Thediagonal items are the sum of the squares ofthe corresponding columns. These calcula-tions are checked by calculating 2(X+ y)2for each pair of columns of Table 8 as shown

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(5) (6) (7) (8

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BENJAMIN H. BURMAP. godoni alpha(2) (3) (4) (5) (6) (7) (8)

-86 - 4 39 -46 -40 -12 -521 -56 -91 - 5 -21 -50 - 5-93 -49 -96 -64 -38 -73 -55-103 12 41 - 72 -39 -51 -63-23 63 - 6 -30 -14 11 - 2-26 34 -27 -22 -11 - 1 -489 90 73 14 30 62 7- 6 13 39 1 12 21 -24-26 -67 -34 -20 - 6 -41 -276 -25 -37 2 -5 13 22- 3 1 -83 - 8 3 -23 7-21 48 65 -9 10 17 -2736 33 37 34 23 28 04 -15 -31 38 - 6 - 3- 3- 1 38 54 6 - 9 41 13-22 -40 -105 -18 -35 -36 - 4-52 20 30 -29 -28 -21 1-45 -92 -75 -45 -35 -55 - 5-72 14 27 -35 - 8 -57 -69-54 8 14 -24 -44 -12 -32

2(X)2:(X2)

TABLE 6

TABLEP. godoni

(2) (3) (4)-577 26 -16643,765 39,992 65,674

P. godoni beta(2) (3) (4) (5) (6) (7) (8)- 4 29 14 4 7 11 -3-73 - 5 -11 -41 -25 -15 -3041 -24 -18 25 16 -6 31-25 -51 -90 -10 - 6 -55 -20-13 41 67 -33 - 6 -19 -38129 19 25 79 27 67 7661 1 11 48 40 27 67- 2 13 6 5 10 24 - 72 -1 -52 6 -14 15 35- 3 38 68 -26 -16 12 -3642 30 22 38 50 32 2215 -35 6 -16 9 -15 057 -55 -19 48 33 3 6121 1 18 24 6 43 18105 -8 -7 71 42 33 3658 -48 2 30 13 -11 23

26 39 60 17 28 23 -496 23 49 73 48 43 8925 -6 24 -7 -15 17 2518 -26 -10 14 12 13 23

:7alpha

(5) (6) (7) (8)-352 -261 -242 -36620,058 12,417 28,278 20,652

576 -25 164 349 25959,108 17,981 29,380 29,177 12,979alpha minus beta -1,153

242 36817,084 31,99451 -330 -701 -520 -484 -734

Combined data(X2)(X) 102,873 57,973 95,054 49,235 25,396-1 1 -2 -3 -2 45,362 52,6460 2

TABLE 8P. godoni alpha

(2) (3) (4) (5) (6) (7) (8)43,765 6,219 4,648 28,481 19,24939,992 38,085 6,899 9,44365,674 8,511 11,40820,058 13,24812,417

P. godoni beta59,108 -1,022 11,336 39,198 23,18717,981 16,487 -1,406 48329,380 3,305 4,971.29,177 17,25512,979

23,068 37,9227,690 -4,12811,084 1,21016,411 27,7969,201 15,41417,084 15,93431,994

100

(2)(3)(4)(5)(6)(7)(8)

24,15123,25327,14317,76313,59428,278

(2)(3)(4)(5)(6)(7)(8)

25,884914-91615,9099,16114,08820,652

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STUDIES IN QUANTITATIVE PALEONTOLOGYin Table 9. This sum, Z(X+ y)2, will equal2X2+2Z(XY)Z y2, these last items hav-ing been already been calculated.We are now ready to set up a determinatewhich is shown in Table 10. The diagonalterms (column (2), row (2); column (3), row

(3)96,195

75,045

(4)118,735181,836

111,16080,335

(5)120,78573,848102,754

(3), etc.) in this determinant are the sumsof the squares of the corresponding columnsof the combined data. The other items arethe sums of the corresponding items of alphaand beta in Table 8, e.g., column (3), row(2) of the determinant equals 6219

BLE 9(6)

94,68071,295100,90758,971

P. godonibeta (Check)166,681 118,46144,346 31,92665,167 52,30176,666

TABLE10

(7)120,345114,776148,23883,86267,883

122,32850,44568,63279,08348,465

(8)116,18562,47284,49472,52851,39177,106

166,94641,79163,794116,7635,80180,946

(2) (3) (4) (5) (6) (7) (8) Difference(2) 102,873 5,197 15,984 67,679 42,436 47,219 63,806 57.65(3) 57,973 54,572 5,493 9,926 30,943 -3,214 -2.55(4) 95,054 11,816 16,379 38,227 294 16.50(5) 49,235 30,503 34,174 43,705 35.05(6) 25,396 22,795 24,575 26.00(7) 45,362 30,022 24.20(8) 52,646 36.70

TABLE 11(2) (3) (4) (5) (6) (7) (8) Difference

49.83503524 211.01 132.30 147.2100945 72795 97528 198.9348260240.23 223.80 8.6331 32.394 118.87 -26.7960005 43107 9138 75706 59172 77691

206.11 -3.0655 12.300 20.791 -17.5761862 2440 96980 41076 32891

0.17974160-0.022738198

0.061284574

68.013 34.447 31.561 28.012 -0.036631209 18632 52825 23093 5634574.185 -25.303 -21.92018053 72986 64849 0.046698337

86.534 32.900 0.017420859 38940 0225098.469 0.029381712 38193

1To conserve space, 320.7382110 is written320.7382110

(2) 320.73182110 16.20324563(3)

(4)

(5)(6)(7)

(8)

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BENJAMIN H. BURMA+(-1022) =5197. The rest of the determi-nant is filled in in this manner. The figuresfor the difference column are obtained fromZ(X) of alpha minus 2(X) of beta, as shownon Table 7, and divided by ?N, in this case,20. Thus the first item of the differencecolumn is -1153/20=57.65. The calcula-tions involved should be checked by re-calculation.There are several ways to solve a deter-minant set up in this manner. The onechosen here is probably the simplest andshortest method. First an auxiliary deter-minant is set up as shown in Table 11. This isaccomplished in the following manner. Ingiving these directions, which are rathercomplex, we will refer to the "determinant"with numbered and lettered squares asshown in Table 12. In this table, the diag-

TABLE12A 1 2B 3C456D

78910E

1112131415F

161718192021G

IIIIIIIVVVIVIIonal terms are given letters and other terms,arabic numerals. The terms of the differencecolumn are shown by roman numerals.The diagonal term A of the auxiliary isthe square root of the original term(V/102873=320.7382110). Term "1" of theauxiliary is the original term divided by aux-iliary A (5197/320.7382110 = 16.20324563).Auxiliary B is equal to square root of theoriginal B minus the square of auxiliaryterm 1. Auxiliary C and the rest of the diag-onal terms equal the square root of the orig-inal diagonal terms minus the sum of thesquares of the auxiliary terms in the columnabove it. Terms 2, 4, 7, 11, 16 and I of theauxiliary equal the original term divided byauxiliary A. Other terms are calculated inthe following manner; all terms used inthese calculations use auxiliary terms cal-culated as shown above.

(2)(4)+(3)(5)Auxiliary 6= (2)(4)+(3)(5)Auxiliay19 (4)(16)+(5)(17)+(6)(18)Auxiliary19= D

(7) I) +(8) (II)+(9) (III)+(10) (IV)Auxiliary V= E

Again, all figures in these calculations arethose of the auxiliary determinant. All thevalues of the auxiliary determinant arechecked by recalculation.The solution of this auxiliary determinantfor B values is next accomplished (not to beconfused with square B of Table 12). In thissolution, we begin at the bottom of the de-terminant and work up. Again we will usethe square lettering and numbering systemof Table 12 and referring entirely to theterms of the auxiliary determinant. B8, thefirst B to be calculated, equals

/0.029338193VII/G, 0.029338193 0.000297940973).' 98.46981712VI-(21)(Bs)B7= FV-(20)(B8) - (15)(B7)B6=

The rest of the B's are calculated by thesame system. Note that 21, 20, and 15 in thecalculation of B7 and B6 refer to the termsin squares numbered in that manner in Table12. The values of these B terms for ourproblem are shown in Table 13. These B's

TABLE 13B2= 0.000719136629B3= 0.000402709216B4=0.000253196212B5=0.001081004484B6= 0.000747476560B7=0.000087825110Bs = 0.000297940973N2BD=0.04126785739

are then multiplied by the correspondingvalue in the difference column (D) of theoriginal determinantBD2= (0.000719136629)57.65)

and these successive products are added(2(BD)). This sum is shown in Table 13also.The calculation of R2 is then possible,which equals lNBD.Nl+N2N1 and N2 are the numbers of specimens inthe respective samples. The value of R2turns out to be, in this case, 0.4126785739.F is next calculated according to the formula

n-p+1 R2p 1-R2

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STUDIES IN QUANTITATIVE PALEONTOLOGY(n is the number of degrees of freedom; inthis type of problem it will be N-+N2-2;p is the number of characters originally usedminus 1.In this particular problem n = 20 +-20-2 or 38 and p =8-1 or 7). Thus F equals3.212.The final step in the process is to look upthe corresponding F in a table of varianceratios. That of Fisher and Yates in "Statis-tical Tables for Biological, Agricultural andMedical Research," p. 33, is eminentlysatisfactory. This table has columns num-bered nl and rows numbered n2 (nl =p,n2= n-p + 1, both as in the paragraphabove). The 1% table should be used for al-most all problems. If we look for our par-ticular values in this table (ni =7, n2=38)we find the consequent to be, by interpola-tion, 3.284. If F is greater than the value inthe table, it means that the two sampleswould differ as much as they do by chance,less than once in a hundred times. In thiscase, the F is slightly smaller than the tablevalue, but not greatly. We may in this case,then, conclude that the differences observedin these two samples are probably signifi-cant.Although the example above is taken

from paleontology, the method is applicableto any two series of samples of several varia-bles. For example, the method could beused to tell whether the difference in pro-portions of several different kinds of heavyminerals in two formations was significantor not. Although the method is laborious, itdoes have the advantage of reducing largemasses of data into a usable form.

ACKNOWLEDGEMENTSThe writer is greatly indebted to Dr. K. J.Arnold for his patient help in guiding himthrough the steps of this statistical tech-nique. BIBLIOGRAPHY

BARNARD,. M., 1935,The secularvariationsofskull characters in four series of Egyptianskulls:Annalsof Eugenics,vol. 6, pp. 352-371.BURMA,. H., 1948,Studies in quantitative pa-leontology, I. Some aspects of the theory andpractice of quantitative invertebrate paleon-tology:Jour.Paleontologyvol. 22,pp. 725-761.FISHER,R. A., 1946, Statistical methodsfor re-search workers: Tenth edition, Oliver andBoyd, London, pp. 354.HOTELLING, H., 1931, The generalization ofStudent's ratio: Annals of Math. Statistics,vol. 2, pp. 360-378.SIMPSON, G. G., and ROE, ANNE, 1939,Quantita-tive zoology: McGraw-Hill,New York.

MANUSCRIPT RECEIVEDAUGUST 13, 1948.

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