A method is proposed to make decisions jointly on a number of dependent character variables. A score was allotted to each entry for each character. The scores were added across characters to provide a final score for each entry. Based on the final scores, the entries were ranked on their performance over a set of characters

Arunachalam, V.

Bandyopadhyay, A.

English

Publications of the IAS Fellows

~,..--.STER ~GATIVE 
A etho to 
44 1984: 419-424. 
Record no. D-55 
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Indian J. Genet., 44(3): 419 - 424 {19,84) 
A METHOD TO MAKE DECISIONS·JOINTLY ON A NUMBER 
OF DEPENDENT CHARACTERS 
V.' ARUNACHALAM AND A. ,BANDYOPADHYAY 
National Research Centre, Indian Agricultural Research lnstitute l 
Regional Station, Hyderabad-500030 
(Accepted : Decem~er 5, 1983) 
ABSTRACT' 
A method is proposed to make decisions jointly ona number of dependent character 
variables. A score was allotted to each entry for each character.. The scores ,were added 
across characters to provide a final score for each entr,y. Based on the final scores, the entries 
were ranked on their performance over a set of characters.' 
Experiments in diverse research disciplines, in general and, agriculture, in 
particular deal inevitably with dependent characters and warr·ant, quite, often, 
decisions taken jointly on them. While methods of an~lysis would be available 
to arrive at conclusions on each .. of ?the characters individually, it remains a 
problem to collate them over the characters and arrive at a final decision. When 
made, such decisions become subjective ,and would reflect the variation in the 
decision making p~ocesses adopted by individual investi_gators. The need is 
increasingly felt in this context to provide a method adequate to draw practical 
decisions. This paper is an attempt in this direction. 
MATERIALS AND METHODS 
A number of F2 plants from selected single, crosses was ady;a~c~d, to .. Fa on a plant- . 
to-progeny row basis in groundnut. Selection of upto 4 grades was made in each Fa family 
based oil the improvement of pod number over a national check.,·The unselected pods were 
.·pooled to form, a bulk in eac,hFs ,.fam.ily .. _Based on kernel weight, kernel size and kernel 
, colour, seIec.tions. from· differ~ntF3 .families were pooled to form populations. t. Each popUlation 
,thus consisted, of selections from vario~~ ~~ .. famiiies derived from one or more single· crosses. 
. . . .'
The term 'population' is used here to define such a pool of Fa selections. The populations were 
advanced to F 4 in plots of 50 m 2 containing 1500 plants and their kernel yield measured. 
The ANOVA. of F5.populations "'as ,based on a cOIl1:plet~ly .. ran<;loDlised design. For 
, . . ." " . .. ' ," : ~, '-'P .. - # 
illustration, the four direct yieldc~mponents, pod weight (PW)., kern~l weight (K~)', tOO-kernel 
weigh t (TW) all measured in g/plant and shelling per cent (SP) were only considered. 
1 
420 V. Arunachalam and A. Bandyopadhyay [Vol. 44. No.3 
. . 
The differences in the mean values of populations were tested by t-test. For example, 
the difference, in the.m,eaQ.~ of popUlations i and j was tested by 
. ~ ;-
where Xi was the mean of the pop ulation i, 
Xj that of j, ;J 
Di was the number of components (~elections) making up p'opulation i, 
Dj that in j, 
e was the error m.s. in the ANOVA 
The 't' would follow a t-distribution with error d.f. 
The population means were arranged --in groups based on t-test (and l.s.d.). The 
topmost group containing population with the highest mean was given a score 1, the next best 
a score 2 and so on. If 'k' ,was the number of groups for a particular character, the popula- ' 
tions in group 1 were given a score= 11k, those in group 2, a score=2/k and so on· to obtain 
standardised scores across the characters later on. - When overlapping of groups occur, it is 
possible that a pop~lation was found in group 1 and also in group 2. The score for that 
population was taken to be the average which would thus be equal to (1 + 2)/2k = 3/2k. 
Populations occurring in more than 2 gr oups would be treated in a like manner for allotme'nt 
of scores. ~he above points would becom'e clear by a study of Ta.bles 2 and 3. 
, The individual scores for each character were added up to provide a total score, for 
each population. The populations were then ranked in descending order of the numerical 
values of total scores,. 
RESULTS 
The differences among the 8 Fa populations were highly significant 
(Table 1) indicating it would be useful to assess their relative order of impor-
tance. While only 2 groups, which .were also non-overlapping, were obtained 
for PW and TW, there were 3 and 5 groups for KWand SP, some ofwbjch were 
overlapping (Tab"Ie 2). The individual scores obtained by each population for 
TABLE 1 
ANOVAfor 4 yield components in 8 F3 populations of groundnut 
Mean squares 
Source d.f. PW KW TW SP 
Between pops 7 1983.98* 4103.18* 464.74* 731.66* 
Within po ps 147 428.13 165.91 69.94 46.9S 
PW=Pod weight; KW=Kernel weight; TW=100-Kernel weight; SP = Shelling per ceni. 
·Significant at 5% level. . 
,. 
TABLE 2 
I-test of signiJi~ance between means of populations for 4 yield component} 
Pop· D PW r Pop KW r Pop TW r Pop 
III 9 82.87 . J 59.36 1 I 53.00 I I 
I S 81.10 . III 54.72 III 52.35 1 III 
V· 9 78.41 1 V 50.69 VI 4S.59 I V 
IV 7 . 77.92 IV 2 'X 40.09 I IV 
VI 26 74.17 Vl 45.83 XII 38.84 VI 
'IX 7 40." I IX . 23.91 IV 38.13 t ·2 lX X 36 37.62 2. X 21.79 I 3' IX 37.72 I X XII IS 32.60 ·XII· 17.56 . V 36.89 XlI 
D -= no. of ~omponent lines in the population; r = rank 
SP r 
73.03 1 1 67.26 r % 
64.38 . 
64.16 I I 3 
62.09 
60~O6 
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·58.37 S 
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422 .v. A'Tunachalam and A.. Bandyopadh~ay [Vol. 44, No 3 
each component character and their total ,scores across the characters are presen~ 
ted in Table 3. 'It ,was 'found that populations I'and III were at the top while 
X and XII were at the :bottom. Further the populations IV and V reqeived 
. . 
identical scores for individual characters and hence their total scores were also 
equal, implying they were of equal merit (See also 'Discussion'). 
The relative order' of importance 'of Fa populations 'was compared with the 
ranking of these- populationsbasee on their '.kernel yield in F4 (Table 4). 
Populations I and 'VI had identical ranks in Fa and F4• Populations III, IV, 
IX and XII.had adjace~franks i.n.F; and F,.. Papulations V and X had quite 
. different r'anks in Fa and F4 •. 
TABLE 3 . 
Scores allott611.10 Fa :populatlonl /or.4 yield components 
. 
Pop PW KW TW SP Total 
score 
. 
I ' 1/2 1/3 t/2 lIS 1.533 
III 1/2 3/6 1/2 3/10 1.800 
VI 1/2 2/3 1/2 S/10 2.167 
IV 1/2 3/6 2J2 S/10 2.S00 
V . 1/2 3/6 2/2 S/10 2.S00 
·IX 2/2 3/S ,2/.2 7/10 3 .. 700 
X 2/2 3/3 .. 2/2 9/10 3.900 
XII 2/2 3/3 212 5/S 4.000 
. . ' : 
Comparison of the ranked performance olFs populations with their performance 
in F""ranked .. on kernelyield 
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Rank in 
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Pop Fs F, , 
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I 11 1 
III !2 4 
VI 13 3 
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IV l4 6 
V i4 8 
IX 6 ~ 
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XII 18 , 
n '"It •... 
November, 1984] Method to Decide Jointly on Characters L~23 
DISCUSSION 
Various approaches are adopted to tackle the complex problem considered 
in this paper. For example, breeders assign a floor value to each character and 
aIlot a p~pulation a score + ( or -1 when it exceeds or falls short of the floor 
value. The aggregate scores across the characters are used for ranking the popu-
lations. Alternatively, an arbitrary weight is associated with each character and 
a discriminant function is St~t up. USIng the values of discriminant scores, the 
populations are arranged in their order of Inerit. 
, 
.The method suggested in this paper has its own merits. For examlJle, 
the differences are tested statistically for. its significance. The number of groups 
that is obtained by t-test is taken into account in assigning the score for each 
population and each character. If there are no significant differences among 
populations for a character (which, incidentally, will also be reflected in the 
F-test in the ANOVA), an the populations will occur ill one group, consequently 
each getting a score 1. It·can be easily see~l that sucl1. a character does not add 
to the differentiation between populations (as uniform addition of 1 to the score 
of every population will not alter the order of tTIcrit). But the scoring process 
takes c.are of the varying potential of characters in different,iatiol1. For example, . 
if 10 groups are obtained for a character X, the population ill. gr.oup 1 gets a 
score 1/10; if only 3 groups are obtained for a character y, the population in 
group 1 gets a score 1/3. Since lower tqtal score will mean higher rank (as the 
topmost bnOUP is allotted score 1), this inlplies that charac~er X is weighted more 
compared to Y for'population 1 and siluilarly for other's. Further, the t-test of 
, . 
significance being based on the error .m.s. of ANOV A, if the character Y has a 
high error variance associated with it, its differentiating potential is propor-
tionately lower compared to another 'character X with low erro~ variance. For, 
the critical difference in the former would be high resulting in a comparatively 
fewer significant differences for Y compared to X. Following ear)ier arguments 
this would imply that X is automatically weighted higher than Y. 
When a number of characters are considered, there could be a sequential 
, ,. . 
relationship between theIn, in general. For example, poor germination affects 
seedling vigour that affects in turn initial leaf area and hence photosynthesis.· 
Hence decisions based on several important characters spanning the entire 
growth phase will be fair and precise due to an auto~atic weighting in the 
expression of the various characters measured sequentially over the growth 
period. 
Thus the three mechanislns~scoring process,. t-test of differenc'es and 
decision based on a large number of characters-ensure the superiority of the 
proposed method over the traditional, ones. 'In the latter .. only the numerical 
-
424 v. Arunachalam and A. Bandyopadhyay [VoI..44,No. 3 
superiority over the floor value of a character is considered. Such superiority 
may not be upheld by statistical test leading to erroneous decisions. Though 
apparently no weight is associated with each character, itis evident the method 
proposed here takes into adequate, account the· relative importance of the 
characters. 
On the other hand, discriminant· fun\Ction techniquesuifers essentially 
from arbitrary assignm.ent of weights to 'characters. The problem of character 
weighting has been dealt with in great detail by Sneath and Sokal (1973) who 
. ' 
defended equal weighting on several independent grounds~ Moreover, exact 
rules for estimating ,,]eights cannot be formulated. 'When m:any charact~rs are 
employed, . the statistical analysis of s..i~ilarity is only slightly affected by 
weighting (unless this weighting is· extreine).' 
The efficiency of the method is also borne out by the comparative perfor-
mance of popUlations in F,' (Table 4). Though there is no a priori reason to 
expect identical ranking of Fa populations in F" it is also difficult to conceive of 
top-or bottom-ranking Fa populations to perform otherwise in F4• The results 
, ' 
have upheld this point in brin'ging out the superior performance of populations I. 
III and VI and the inferiority of XII. The performance of Fa population X 
sugg~sts that the constitutent lines of the 'populations in Fa must have been 
affected to a great extent by environment, diseases a'nd pests while those of V 
could have had a higher contribution by environment. In addition, genetic 
differences in the constituent lines sbould also have' been pronounced in those 
two populations to explain their differential performance in Fa andF4• But the 
results are, by and large, consistent in. F3 and F,. ' 
, , 
,Sometimes two or more popUlations may receive the same total ,score as 
has happened for populations IV and V in Fa. The scores may, be equal for 
every component character as in these populations in which case it can be 
concluded that they were of equal potential. Alternatively, there' may be internal 
compensa~ion in characters leading to ·such a result. In ,that case, those 'other-
wise equal' populations can be differentiated on the scores computed on a subset 
of characters important for the objectives of the study. 
ACKNOWLEDGEMENT 
The authors are grateful to' Dr. S.' Ramanujam for critical comments that helped to 
improve an earlier version. 
REFERENCE 
Sneath, P.H.A. and R.R.· SokaI. ,(t973).Numerica/ Taxonomy. W.R .. Freeman and Co. SaD 
Francisco., pp. 96-113 


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