Arunachalam, V.

Murty, B. R.

English

Publications of the IAS Fellows

~ -. STER NEGATIVE BER: 09295.25 
Arunachalam, V. and ~ urty, B. R. 
27 1967: 60-69. 
Record no. D-6 
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COMPUTER PROGRAMMES FOP\. SOME PROBLEMS IN BIOMETRICAL 
GENETICS-I. USE OF MAHALANOBIS' D2 IN CLASSIFICATORY 
PR()BLEMS 
B. R. MURTY and V. ARUNACHALAM 
Division of Genetics, Indian Agricultural Research Institute, Delhi·.} 2 
(Accepted : 16-xii-1966) 
DURING the various types of investigations on population dynamics carried out 
at the Division of Genetics, Indian l~gricultural Research Institute, New Delhi, 
the assessment of genetic divergence between populations by multivariate analyses 
was found to be useful for classificatory problems as well as choice of parents for 
breeding work in crops with diverse breeding systems (Murty and Arunachalam, 
1966) . 
While the initial work was limited to a few populations in each crop, the· 
fundamental inforITlation on the factors influencing genetic divergence means 
scanning a large collection with as many characters related to fitness under 
natural and human selection included in the analysis. Similarly, the extraction 
of characteristic roots from characteristic equations involving several variables 
means several iteratiolls. Since the magnitude of computations increase several. 
fold with an increase in the size of the matrices and the number of populations, 
the assessment of a large collection of germ plasm can be undertaken only with 
the facilities of a computer for accurate and rapid analysis. 
During the past three years, programmes have been developed in the 
Biometrics Unit for problems involving multivariate ·analysis, canonical analysis, 
factor analysis, fractional and full diaIlel analyses for combining ability, serial 
analysis over different" environments for assessment of genotype-environment 
interaction and similar problems encountered frequently in biological investiga- . 
tions. 
Computer programmes which are of special interest, were recently published 
by some workers (Littlewood et al., 1964, Kobetich, 1964). Since the investiga-
tions carried o·ut here are of particular interest to biological workers, in 
general, it is felt desirable to make these programmes available for such 
investigations. 
These programmes are written in FORrrRAN language for an IBM 1620 
(Model II) computer and can easily be modified to suit the-occasion in fields 
other than biology. For most of the plant breeding investigations of these types 
we come across, the programme is general enough to be. used without any 
modification. The first. of these programmes is on an assessment of divergence 
by the use of Mahalanobis' D2-statistic. 
61 Indian Journal of Genetics & Plant Breeding [Vol. 27, No.1 
"DESCRIPTION OF THE PROGRAMME 
This programme is intended to compute D~ values be-tween all possible 
·combinations for a maximum of 80 varieties and 10 characters. Upto 100 
varieties with 10 characters (or any cornbination "such that number of varieties x 
number of characters is less than 1000), the programme will work without any 
.alteration by changing the card No. 2 as For X 53. For investigations involving 
more varieties and characters the programme can easily be modified. The 
methodology used in this programme is completely given by C. R. Rao 
(1952, 1958). 
COMPUTATIONAL METHOD 
The calculation of D2 values involves three major steps : 
(i) A set of uncorrelated linear combinations (y's) is obtained by the 
pivotal condensation of the common ldispersion matrix (Rao, 1952) of a set of 
correlated variables (x's). " 
(ii) Using the relations between y's and x's, the mean values of different 
varieties for different characters are transformed into the mean values of a set 
of uncorrelated linear combinations (1{'s). 
(iii) The D2 between the ith and jth variety for k characters is calculated 
as D2ij=~ (Yit_·Yjt)2. The k component D-squares are calculated separately 
t== 1 
.and added up to give D2ij • . . 
(iv) The k component D-squares for each combination are ranked in des-
cending order of magnitude, equal values, if occur, which are very rare, 
receiving same ranks. . 
(v) The ranks are added up for each component D-square over all 
combinations and the rank totals are got. 
Input Data-The input mediunl for all these programmes is SO-column 
punch cards. The following data are required as input data. 
(i) Title of the experiment not exceeding 80 letters punched in one card 
from columns 1--80. 
(ii) The number of characters and the number of varieties each occupying 
column 1 to 3 and 4 to 6 respectively of one punch card. 
(iii) The" upper half of the common dispersion matrix designated as 
A-matrix i.e., ((aij)), (j~i,i=I,2, .. n) 
( e.g.) a22 • • • • a2n 
3nn 
.Each card contains 7 quantities, each quantity occupying 11 columns 
with four decimal digits, i.e., quantities 1, 2, 3, etc. occupying columns 1-11, 
March, 1967] Computer programmes for D2 ana?Jsis 62 
12-22, 23-33, etc. respectively of a card~·· The decimal point need not be 
punched. For e.g., if a 4 X 4 matrix is used with values 
832 · 2382 -0· 7632 -0 · 0078 6 · 7002 
11576·2000 76·3280 -6·0007 
6766327-1181 -6650·7634 
181·0705 
the input will take the form 
bbbb'83J22382bbbbbb-7632bbbbbb-0078bbbbbb67002b bl15762000bbbbb 
763280bbbbb--60007 
(Card I, the 7 quantities occupying columns 1-77 of the card, b's representing 
blanks) 67663271181bb-66507634bbbb1810705 (Card 2, the remaining 3 quan-
tities occupying columns 1-33).. The computer will read the quantities as if 
the decimal point is placed after the 4th digit from the right of each 
quantity. 
The format i.e., the mode of allotting columns to each quantity, may be 
changed if necessary according to the size of the quantities in individual experi-
ments which requires changing the statement 11 in the programme. 
(iv) The mean values for each character for each variety. The values of 
each of the characters for one variety are punched in one card. 
The mean value for each character occupies 4 columns with 'one decimal 
digit. The characters are arranged in the same order as in the common 
dispersion matrix. The mean values for a variety are punched in one card in 
the case of 10 characters for e.g., from column 1 to 40 as indicated in 
(iii) '!-bove. Tl)us for each variety one card is used for the input of 
mean values. 
Output.-The programme renders the following printed output in a neat 
form giving spaces and underlined sub-titles wherever necessary as in the 
example presented here. 
(i) Title of the experiment. 
(ii) y, the coefficients of x in the uncorrelated linear combinations. 
(iii) Uncorrelated mean values for each variety for each component YI' 
y~, etc. 
(iv) D-square values and. ranks for each combination. All the component 
D-squares and the total D-square are printed out. 
(v) Rank totals. 
In addition, the programme renders the mean values for each of the un-
correlated linear-combinations YI' 'Y2 etc. for each variety as punched output 
in cards. These cards can be used as such for doing the principal component 
analysis to be published as Part II of the paper. The values Yo Y 2 etc. for one 
variety is punched in one card, similar to the input (iv), the change being that 
each quantity occupies 10 columns ,with four decimal digits. 
The programme with the exa:mple of the data collected in this laboratory 
published in Sankhya, Vol. 27 (1965) is presented as an appendix. The input 
and printed output excluding punched output are given. 
63 Indian Journal of Genetics & Plant Breeding [Vol. 27, No. I 
l\CKNOWLEDGEMENT 
Our thanks are due to the officer and memb.ers of the Mechanical Tabula-
tion Unit, Institute of Agricultural Research Statistics, New Delhi for their help 
:and for making the computer available for these investigtions. 
REFERENCES 
Kobetich, E.J. (1964). General Computer plotting subroutine, Trans. Kansas. Acad. Sci., 67: 640-45. 
Littlewood, R.K., Carmer, S.G. and Hittle, C.N. (1964. A computer programme for estimating comb-
ining abilities in relation to diallel crossing systems, Crop Sci., 4: 662-63. 
Majumdar, D.N. and Rao C.R. (1958). Bengal anthropometric survey, 1945; A statistical study, Sankhya, 
19: 201-408. . 
Murty, B. R. and Arunachalam, V. (1966). The nature of divergence in relation to breeding system in 
some crop plants, Indian J. Genet., 26A: 18~98. 
-- Mathur, J.B.L. and Arunachalam, V. (1965). Self-incompatibility and genetic divergence 
in Brassica campestris var. brown sarson, Sankhya B,. 27: 271-78. 
Rao, C.R. (1952). Advanced Statistical Methods i:n Biometrical Research, John Wiley & Sons, New York • 
• 1 
March, 1967] Computer progr~~mmes for D2 analYsis 
~:JOB 5 
-*.FORX54 
APPENDIX 
COMPUTATION OF D-SQUARE VALUES-PROGRAMMED BY V. ARUNACHALAM, I.A.R.I. 
DIMENSION A(lO, 10), B (10, 10), PA (10), PH (10, 10), x (20), Y (80, 10), DS (20) 
1, M (10), KT (to) 
250 READ 251, (DS (I), 1= 1,20) 
251 FORMAT (20A4) 
PRINT 252, (DS (I), 1= 1,20) 
252 FORMAT (lX, 20A4/1X, 80 (lH-)/I) 
-C THE ABOVE CAUSES THE PRINTING OF THE TITLE OF THE EXPERUvIENT 
eN-NO. OF CHARACTERS, IV-NO. OF VARIETIES 
C A(N, N)-COMMON DISPERSION :MATRIX, BeN, N)-UNIT MATRIX 
C Y(IV, N)-UNCORRELATED MEANS MATRIX OF DIMENSION IV X N 
C DS (N)-COMPONENT D-SQUARES 
READ 1, N, IV 
1 FORMAT (213) 
NO=N-I 
IVV=IV-I 
FIV=IV 
READ 11, «A(I,J), J=I, N), 1= 1 · 
11 FORMAT (7F 11'4) 
PA (I)=A(I, 1) 
DO 101 1=1, N 
101 B (I, 1)=1. 
DO 1010 1= I, NO 
IA=I+l 
DO 10 J=IA, N 
A (J, I)=A(I, J) 
B (I, J) =0. 
10 B (J, I)=B(I, J) 
1010 CONTINUE 
DO 1001 J = 1, N 
1001 PB (I,J)=B (I, J) 
DO 21=1, NO 
L=I+l 
DO 12 J=L, N 
12 A(I, J)=A(I, J)/A(I, I) 
D0121J=I,I 
121 Bel, J)=B(I, J)iA(l, I) 
DO 114 K-L, N 
64 
001 
002 
003 
004 
005 
006 
007 
008 
009 
010 
011 
012 
013 
014 
015 
016 
017 
018 
019 
020 
021 
022 
023 
024 
025 
026 
027 
028 
029 
030 
031 
032 
033 
OM 
035 
036 
037 
038 
039 
()4Q 
-
65 Indian Journal of Genetics & Plant Breeding [Vol. 27, No.1 
DO 112 J=L, N Of l' 
112 A(K, J)=A(K, J)-A(K, 1)* A(I, J) 04~ 
DO 113 J = 1, I 043 
113 B(K, J)=B (K, J)-A(K, 1)* Bel, J) 044 
114 CONTINUE 045· 
PA(L)=A(L, L) 046 
DO 1200 J=I, N 04; 
1200 PB (L, J) = BeL, J) 048-
2 CONTINUE 049-
DO 8 1= 1, N 05()< 
8 PA (I)=SQRTF (PACI» 051 
DO 80 1= 1, N 052 
DO SO J = 1, N 053 
80 PB(I, J) =PB(I, J)fPA(I) 054-
PRINT 9 055 
I; 
9 FORMAT (lX, 59HCOEFFICIENTS OF:K: IN THE UNCORRELATED LINEAR FUNCTION 056, 
IS OF Y/IX, IH-, 19(3H*-*), IH-I/) 051' 
PRINT 90, IH 058 
90 FORMAT (lX, 65 (lH-)/29X, 15HCOEFFICIENTS OFI/IOX, 4HX(l), 6X, 4HX(2) , 6X, 059-
14HX(3), 6X, 4HX(4), 6X, 4HX(5), 6X, 4HX(6)JIX, 65(lH-)/) 060 
DO 91 1=1, N 061 
91 PRINT 92, I, (PB(I, J), J = 1, N) 062-
92 FORMAT (lX, 2HY(, 12, IH), 6 (2X,F8A», 063 
PRINT 7000 064· 
7000 FORMAT (f) 065 
PRINT 222 -066 
222 FORMAT (1 x 24HUNCORRELATED MEAN VALUES/IX, 24 (lH-)/!) 061 
PRINT 224 068 
224 FORMAT (lX,74(lH-)/lX,3HVAR, 5X, •• HY(1), 8X, 4HY(2), 8X,4HY(3). ex. 4HY(4) 069-
1,8X, 4HY(5), ax, 4HY(6) IX, 74(1H-)/I) 070 
DO 215 KD= I, IV 071 
READ 212, (X(J), J = 1, N) 072 
212 FORMAT (lOF 4'1) 073 
DO 214 1= I, N 074-
Y(KD, I) =0. 075 
DO 214 J= 1, I 076· 
214 Y (KD, I) = Y (KD, I) + PB (I, J) *X(J) 071 
215 PRINT 225, KD, (Y (KD, I), 1= 1, N) 078 
225 FORMAT (2X, 12, 3X, FS' 3, 5(4X~F8'3)) 079· 
C STATEMENTS 224 AND 225 TO BE CHANGED FOR PROCESSING MORE THAN SIX 080, 
-Computer programmes for Df}. analYsis 66 
·e CHARACTERS UPTO TEN 081 
DO 9991 I= 1, IV 082 
9991 PUNCH 9990, (Y(I, J), J= 1, N), I 083 
9990 FORMAT (6FIO'4,17X,12) 084 
PRINT 7000 085 
PRINT 216 086 
216 FORMAT (53X,15HD-SQUARE VALUES/5~IX,15(1H-)II) 087 
PRINT 217 088 
217 FORMAT (21X, 5HCOMBN, 4X, 6HDI-SQR, 4X, 6HD2-SQR, 4X, 6HD3-SQR, 4X, 6HD4-SQR 089 
C, 4X,6HD5-SQR, 4X, 6HD6-SQR, 4X, 8HD-SQUARE/21X, 77 (IH-)/I> 090 
DO 2213 1= I, N 091 
2213 KT(I)=O 092 
DO 221 KM= 1, IVV 093 
n=~+l ~ 
DO 221 I=KK, IV 095 
DSS=O. 096 
DO 223 J= 1, N 097 
DS(J) = Y(I, J)-Y(KM, J) 098 
DS(J) =DS(J)* DS(J) 099 
223 DSS=DSS+DS(J) 100 
PRINT 2 I 8, KM, I, (DS(l), J = 1, N), DSS 101 
21 FORMAT (21X, I2,IH-, 12, 3X, 6(F9 t 4,lX), 2X,F7·2) 102 
8 DO 2212 LI= 1, N 103 
M(Ln=1 104 
DO 2211 LJ= I, N 105 
IF (DS(LI)-DS(LJ» 2210,2211,2211 106 
2210 M(LI)=M(LI)+1 107 
2211 CONTINUE 108 
2212 KT(LI)=KT(LI) +M(LI) 109 
PRINT 2180, (M(LI), LI= 1, N) 110 
2180 FORMAT (22X,4HRANK, 2X, 6 (5X, II, 4X)/) 111 
221 CONTINUE 112 
PRINT 2215, (KT(!), 1= I, N) 113 
2215 FORMAT (21X, 77(IH-)/16X, 10HRANK TOTAL, 2X,6 (3X, 14, 3X)/21X,77(lH-)//) 114 
'C STATEMENTS 224, 225, 9990, 218:, 2180, AND 2215 TO BE CHANGED FOR 115 
PROCESSING MORE THAN SIX CHARACTERS UPTO TEN 116 
C FORMAT AND DIMENSION STATEMENTS SHOlJLD BE ALTERED FOR PROCESSING MORE 117 
eTHAN 10 CHARACTERS AND 80 VARIETIES 118 
PRINT 2216 119 
67 Indian Journal of Genetics & Plant Breeding [Vol. 27, No 1. 
2216 FORMAT (10(1)) 120-
GO TO 250 121 
END 122 
DATA ON BRASS ICA-SANKHY A, n., VOL. 28, 1966 123-
6 10 124 
97500 -56600 2400 -46000 -31600 1500 423000 125-
37800 162400 147500 24500 13400 31600 8900 126-
8900 340300 98700 30400 490500 5700 43300 121 
472 466 55 240 394 167 128-
485 526 73 242 385 192 129-
579 597 82 275 435 196 130' 
511 426 66 243 342 167 131 
594 414 76. 328 208 151 132-
599 427 78 309 211 141 133-
755 639 77 199 365 217 134· 
544 451 77 260 382 188 ISS. 
724 583 72 188 320 213 ISS 
602 501 76 314 383 171 13i 
DATA ON BRASSICA-SANKHYA, B., VOL. 28, 1966 ~ ~ 
~ ("') 
COEFFICIENrS OF X IN THE UNCORRELATED LINEAR FUNCTIONS OF Y t:r 
"-
"""""" ~
COEFFICIENTS OF 0") 
" X(l) X(2) X(3) X(4) X(5) X(6) L--I 
Y (1) ·3202 0·0000 0·0000 0·0000 0·0000 0·0000 
Y (2) ·0929 ·1600 0·0000 0·0000 0·0000 0·0000 
Y (3) -·0855 -·1035 1·0312 0·0000 0·0000 0·0000 
~ 
Y (4) ·0908 -·0298 --4211 ·2073 0-0000 0·0000 ~ 
, ~ Y (5) ·0025 -·0517 ·1144 -·0299 ·1529 0·0000 ~ 
"""t 
'T le\ 
-·0079 ·0035 --3044 -·0215 ·0022 ·5199 ~ .I. \U} ~ 
~ 
----- ---
.--. .... --
---- -- ~ UNCORRELATED MEAN VALUES ;§ 
------
------
-- ~ 
VAR Y(I) Y(2) Y(3) Y(4) Y(5) Y(6) ~. 
~ 1 15·116 11·847 -3·192 5·560 3·649 6·370 """t 
t1 
2 15·532 12·928 -2·069 4·783 3·405 7 ·127 J.,;) 
~ 
~ 3 18·542 14·939 -2·680 5·731 g·831 6·951 ~ ~ GoO) 
4 16·365 11·569 -1·977 5·633 3·188 5·972 ~. c.., 
5 19·023 12·148 -1-531 7·764 1·082 4·552 
6 19· 183 12·403 -1· 502 7·293 1·141 4-014 
7 24·179 17·247 -5·136 5·840 2·759 8·219 
8 17·421 12·276 -1·384 5·747 3·754 6·684 
9 23-186 16·062 -4 .. 806 5·708 2·328 8·182 
10 19·279 is-615 -2-501 7·287 3-352 5·686 
.... --~-,,-. S!) 
SQ 
• 
D-SQUARE VALVES 
COMBN Dl-SQR D2-SQR D3-SQR D4-SQR 
1-2 -1733 1-1694 1-2621 · 604,1 
RANK 5 2 1 3 
1-3 11·7425 9-5588 -2622 ·0291 
RANK 1 2 4 6 
1-4 1-5600 ·0772 1-4766 -0052 
RANK 1 5 2 6 
........••........................... . etc ................................... 
8-9 33-2307 14·3354 11·7132 ·0015 
RANK 1 2 3 6 
8--10 3·4502 1·7943 1·2479 2·3712 
RANK 1 3 4 2 
9-10 15·2656 5·9861 . 5-3145 2-4954 
RANK 1 3 4 5 
RANK TOTAL 87 132 179 198 
D5-SQR D6-SQR 
·0597 ·5720 
6 4 
·0330 ·3377 
5 3 
·2129 -1584 
3 4 
2-0327 2·2438 
5 4 
-1611 ·9956 
6 5 
1-0491 6·2290 
6 2 
117 -152 
D-SQUARE 
3-84 
21-96 
3·49 
63·55 
10·02 
36-33 
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Computer programmes for some problems in Biometrical Genetics. I. Use of Mahalanobis' D<SUP>2</SUP> in classificatory problems

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Computer programmes for some problems in Biometrical Genetics. I. Use of Mahalanobis' D<SUP>2</SUP> in classificatory problems

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