The distribution of fish caught by experimental gill nets has been found to be in the Poisson or Negative binomial form. Using this information, application of Chi-square test as suggested by Mood et al. (1974) has been illustrated, for comparing the efficiencies of gill nets. This test provides an alternative to Anova F-test especially in the context of significance of non-additivity for the two-way model. Based on the present work and the findings by Nair (1982) and Nair & Alagaraja (1982, 1984) an outline approach for statistical comparison of the efficiencies of fishing gear is presented

Kesavan Nair, A.K.

Aquatic Commons

Statistical C 
 
p riso 	 f Efficieracrn7 (7)f 	 Gea it 
 
A. K. KESAVAN NAIR 
Central Institute of Fisheries Technology, Cochin--682 029 
The distribution of fish caught by experimental gill nets has been found to be in 
the Poisson or Negative binomial form. Using this information, application of 
Chi-square test as suggested by Mood et al. (1974) has been illustrated, for comparing 
the efficiencies of gill nets. This test provides an alternative to Anova F-test especially 
in the context of significance of nonadditivity for the two-way model. Based on the 
present work and the findings by Nair (1982) and Nair & Alagaraja (1982, 1984) 
an outline approach for statistical comparison of the efficiencies of fishing gear is 
presented. 
A study of the distribution of the data is 
important in view of developing test pro-
cedures. If the form of the distribution is 
known, that information could be used to 
construct a test to compare the location. 
With this in view the gill–net catch data 
were examined. This assumes importance 
because of the fact that non-additivity in the 
two-way model was found to be present 
when the experimental data for comparing 
the efficiencies of gill-nets were examined. 
Further, Nair (1982) and Nair & Alagaraja 
(1982, 1984) have investigated the applica-
bility of some tests to compare the efficien-
cies of trawl nets. The difficulties caused 
by lack of satisfaction of relevant assump-
tions for applying parametric tests and some 
approaches to obviate some of these diffi-
culties are discussed by them. It is also the 
purpose of this communication to use these 
findings along with the present work to 
suggest an outline for a practical approach 
for the statistical comparison of the effi-
ciencies within trawl nets and within gill-
nets. 
Materials and Method3 
Data on catches of different types of gill-
nets, for instance (Kunjipalu et al., 1984) 
obtained under comparable conditions for 
different days were used to compare the 
efficiencies. Frequency distribution of the 
numbers of fish caught according to the 
frequency (in terms of number of hauls) of 
occurrence of 0, 1, 2 etc fish in the catch 
was made for different types of gill–nets. 
As the largest frequencies corresponded to  
occurrence of 0 or 1 fish and the frequencies 
decrease for increasing numbers of fish, the 
Poisson, Negative binomial and Geometric 
distributions were considered for the data. 
The theoretical frequencies were calculated 
using the densities, 
f(x) =e ?`Tx I (0, 	 (x) (Poisson) 	 ( 1 ) 
xi 
f(x)= {r x-1} 	 ,r P 	 (0, 1, 	 (x) ; x 	
0 < p < 1, r > 0 (2) 
(q 	 1 — p) (Negative binomial) 
and f (x) = pqX I (0, 1, ..) (x); 0<pl; 
(q= 1—p) (Geometric) 
	 (3) 
as given by (Mood et al., 1974). a, p, q 
( =1—p) and r are parameters of the dist-
ributions. The goodness of fit was tested 
by chi-square. Further, the chi-square 
test (Mood et al., 1974), 
2 k +1 
Q2k = 
i=1 j=-. 1 
with degrees of freedom equal to (2k-the 
number of parameters estimated) was used 
to test whether two gi 1, en samples are 
drawn from the same population such as 
the Poisson, the Negative binomial or the 
Gamma Here k 1 refers to the number 
of classes and i=1 and 2 for two samples. 
On the basis of this and the results ob-
tained in the studies mentioned above, an 
approach for the statistical comparison of 
the efficiencies within trawl nets and within 
(Nij –  nipj)2 	 (4) 
nipj 
Vol. 23, 1986 
8 	 A. K. KESAVAN NAIR 
gill—nets is listed. Information obtained 
by applying Quad.e (1979) test and rank 
transform test (Lemmer & Stocker, 1967; 
Conover & Iman, 1976; Hora & Conover, 
1984) as presented in Iman et al. 1984, to 
data on trawl and gill—net catches has also 
been utilized to indicate this approach. 
Table L Distribution of .fish caught by 4 
gill—nets A, B, C and D 
A 
No. of fish 	 Fre- 
caught 	 quency 
No. of 
fish caught 
Fre- 
quency 
0 27 0 13 
26 1 18 
2 14 2 24 
3 4 3 9 
4 3 4 5 
5 5 2 
6 1 6 3 
7 7 1 
8 8 
9 1 9 
10 10 1 
11 11 1 
12 1 
Total 77 Total 77 
C D 
0 9 0 6 
1 9 1 12 
2 6 2 11 
3 3 3 4 
4 5 4 3 
5 2 5 1 
6 — 6 - 
7 -- 7 1 
8 3 8 2 
9 1 9 1 
10 — 10 — 
11 1 11 
12 1 12 
13 1 13 
Total 41 Total 41 
frequencies correspond to the occurrences 
of 0 or 1 fish in the catch and the frequen-
cies decrease with occurrences of increasing 
numbers of fish, as already mentioned. The 
comparisons made were between A and B 
and between C and D. The Poisson, Nega-
tive binomial and Geometric distributions 
fitted to these data alongwith the observed 
frequencies are presented in Table 2. 
The chi-square values with the respective 
degrees of freedom for the goodness of tests 
are also presented in Table 2. It can be 
seen from Table 2 (A and B) without any 
test itself that the geometric distribution 
does not fit any set of the data. Therefore, 
this distribution was not fitted for sets C 
and D. The chi-square goodness of fit for 
Poisson and Negative binomial distributions 
as presented by Mood et al. (1974) showed 
Poisson and negative binomial to be a good 
fit for the sets A and B and Negative bino-
mial for sets C and D (Table 2). Poisson 
distribution was however found to be satis-
factory for set D, though not for set C. 
From the mean and variance presented in 
Table 2, it can be seen that they are not 
widely different for sets A, B and D, so 
that Poisson distribution too fitted these 
data. But for set C, variance is very much 
larger than the mean, which made the 
Poisson distribution, a poor fit. Negative 
binomial distribution fitted all the four sets 
of data. However, for any of these distri-
butions, the chi-square test as given by (4) 
can be used to test whether the samples 
came from the same Poisson or Negative 
binomial populations (Mood et al., 1974). 
The application of this test for the two 
distributions, that is, to test whether sets 
A and B came from the same Poisson 
distribution and sets C and D came from 
the same Negative binomial distribution is 
illustrated below. 
(1) Comparison of gear A and B: 
Results and Discussion 
Frequency distribution of the 
fish caught by the two nets is 
number of 
The frequency distribution of the numbers 
of fish caught by gill—nets A, B, C and D 
(per equal area) are presented in Table 1. 
(The frequencies are the number of opera-
tions of equal duration). The maximum 
No. of fish 0 1 2 3 4 or Total 
more 
Net A 27 26 14 4 5 	 76 
Net B 13 18 24 9 12 	 76 
Total 40 44 38 13 17 	 152 
FISHERY TECHNOLOGY 
ON STATISTICAL COMPARISON OF EFFICIENCIES OF FISHING GEAR 	 9 
Table 2. Fit of Poisson, negative binomial 
and geometric distributions to the 
data given in Table 1 
A 
No. of Obser- Poisson Negative Geo- 
fish 	 ved fre- 	 bino- 	 metric 
caught quency 	 mial 
0 27 22 31 41 
1 26 27 21 19 
2 14 17 12 9 
3 4 7 6 4 
4 3 2 3 2 
5 
6 1 
7 
8 1 3 
9 1 
10 
Total 76 76 76 76 
Mean 1.22, Variance = 2.25 
Test for goodness of fit (chi-square) 
1.80 N.S 2.87 N.S. 
d.f. 	 2 	 2 
(Frequenoies in classes 3 and above were 
pooled for Poisson and 4 and above for 
Negative binomial, to computer chi-square) 
B 
0 13 10 15 46 
1 18 20 19 18 
2 24 20 16 7 
3 9 14 11 3 
4 5 7 7 1 
5 2 3 4 
6 3 2 
7 1 
8 
9 2 2 1 
10 1 
Total 76 76 76 76 
Mean e  2,08., Variance ._--. 3.38 
Test for goodness of fit chi-square 
5.06 N.S 5.38 N.S 
d.f 	 4 	 3 
(Frequencies in classes 5 and above were 
pooled to compute chi-square) 
No. of fish 
caught 
C 
Observed 
frequency 
Poisson Negative 
binomial 
0 9 1.76 9.37 
1 9 5.55 7.55 
2 6 8.73 5.85 
3 3 9.15 4.47 
4 5 7.20 3.40 
5 
6 
) 7 8 
9 8.61 10.36 
10 
11 1 
12 1 
13 
Total 41 41 41 
Mean ® 3.15, Variance = 12.28 
Goodness of fit (chi-square) 
37.92** (p<0.005) 1.71 N.S 
d.f 	 4 	 3 
D 
0 6 8.91 4.04 
1 12 9.43 9.36 
2 11 7.55 10.85 
3 4 5A0 8.38 
4 3 3.63 4.85 
5 1 
6 
7 1 
8 
9 
2 
1 6.08 3.52 
10 
11 
12 
13 
Total 41 41 41 
Mean = 2.32, Variance 5.07 
Goodness of fit (chi-square) 5.31N.S 3.89N.S 
d.f 	 4 	 3 
N.S -Not significant; ** -Highly significant; 
d.f - degrees of freedom 
Vol. 23, 1986 
10 	 A. K. KESAVAN NAIR 
Now, the parameter, namely, the mean of 
	
The maximum likelihood estimate of the 
the Poisson population is to be estimated. 	 sample mean is 
0(40) + 1(44) -I- 2(38) + 3(13) + 4(8) + 5(2) + 6(4) + 7(1) + 9(1) -I- 10(1)  = 1.6513 
152 
From (1), the expected number in each group 
of the population is given by 
No. of fish 0 1 2 3 4 or 
more 
Expected 
number 14.58 24.07 19.95 10.97 6.43 
Thus the chi-square given by Mood, Gray-
bill and Boes (1974) is 
2 k + 1 
Q2k= 
i=1 j=1 
(Nij-nipj) 2 	 (27-14.58) 2 + 	 (12-6.43) 2 
nipj 
	 14.58 	 6.43 
= 24.96** with 2k-1 = 8-1 
	 7 degrees 
of freedom (as I parameter is estimated). 
The significance of the chi-square shows that 
the two samples are not from the same 
population which means that the catches 
by the two gear are not equal. This can 
be generalized to several samples, that is, 
to catches by more than two gear also. 
(2) Comparison of gear C and D: 
Assuming that the catches by nets C and 
D are distributed in the Negative binomial 
form (as found already) whether they came 
from the same Negative binomial distribu-
tion is tested by the chi-square test discussed 
and applied above. Frequency distributions 
of the number of fish caught by the two 
nets are as under. 
No. of 
fish 0 1 2 3 4 5 and Total 
above 
Net C 9 9 6 3 5 	 9 	 4 i 
Net D 6 12 11 4 3 	 5 	 41 
Total 15 21 17 7 8 	 14 	 82 
Here two parameters r and p are to be esti- 
mated from the combined data. Estima- 
tion of these parameters by the method of 
moments (p —  mean r= mean X Ft), 
variance 
gave P =0.3125, q =0.6875 and 7. =1.2427. 
Thus from (2), the expected number in each 
group of the population is given by 
No. of fish 
	 0 	 1 	 2 3 and Total 
above 
Expected 
number 	 9.67 8.26 6.36 16.71 41 
(Frequencies in the last three classes were 
pooled to form a single class '3 and above' 
to make all the expected values greater 
than 5, for computing chi-square) 
Q2k = (9-9.67) 2 	 (17-16.71) 2 + 
9.67 	 16.71 
(6-9.67) 2 + 	 + (12-16.71) 2 
9.67 	 16.71 
= 7.94 (N.S.) with 2k-2, that is, 4 
d.f., as two parameters are estimated. 
Thus the hypothesis that the two catches 
come from the same population is not 
rejected. 
The above illustrations show that a test 
based on the distribution of fish catch data 
(for gill-nets) can be constructed. The 
distribution has been found to be either 
Poisson or negative binomial. Negative 
binomial fitted three sets out of the four 
when all the observations were considered 
and the same distribution fitted all the four 
sets when one observation in the extreme 
class after some discontinuity was omitted. 
Poisson distribution fitted 3 sets with and 
without the omission of the observation in 
the extreme class. Geographical and species 
difference may attribute to the difference 
in the distribution. However fitting Poisson 
or negative binomial is easy and can be tried 
for any set of gill-net data. Depending on 
the adequacy of the fit either of these distri-
butions may be assumed and the difference 
between the samples tested by employing 
chi-square test. But it is important to test 
FISHERY TECHNOLOGY 
ON STATISTICAL COMPARISON OF EFFICIENCIES OF FISHING GEAR 
	 11 
the goodness of fit because, when the fit is 
not adequate, that itself will contribute to 
the significance of chisquare, vitiating the 
results of the test for difference of the two 
samples. 
An outline of approach for statistical 
comparison: 
(i) When the efficiency of two trawl nets 
or two gill-nets are to be compared, Wilco-
xon matched-pairs signed-rank test (WSR 
test) may be used, as this has been found 
to be more efficient for the data. Also, 
its application is simple. For normal distri-
butions this test is 95.5 percent as efficient 
as the parameteric F or t-test (Siegel, 1956) 
but, for other types of distribution (for 
instance, some long tailed ones) this test 
may be more than 100% efficient compared 
to the F or t-test (Snedecor & Cochran, 
1968). The superiority of WSR test over 
F-test for trawl catches has been demonstra-
ted by Nair & Alagaraja, (1982) and the test 
has been applied in Narayanappa et al. 
(1982). Moreover, the nonnormality of the 
data has been indicated by Nair (1982) as 
revealed by the dependence of the mean on 
the variance. Lack of satisfaction of other 
assumptions like nonadditivity for ANOVA, 
has also been established by applying 
Tukey's test and the presence of outliers 
have been observed by Nair & Alagaraja 
(1984). Finally among the nonparamteric 
methods for paired comparisons, except 
for randomization test, only Wilcoxon test 
seems to use interblock information. But 
randomization test is unwieldy for even 
moderately large samples (say, when the 
number of pairs exceeds 12) and as Siegel 
(1956) has observed, Wilcoxon test (WSR 
test) is a very efficient alternative to the 
randomization test because it is a randomi-
zation test on ranks. 
(ii) When the efficiency of more than two 
trawl nets are to be compared, Friedman 
test and ordinary ANOVA F-test may be 
tried first. Applications showed Friedman 
test to be as sensitive as F-test, though no 
higher sensitivity was observed in any case. 
As Friedman test depends on fewer assump-
tions than does F-test, as a practical proce-
dure, if either of these tests brings out the 
difference in the efficiency, there is no need 
to test further. If both the tests are not  
found to be sensitive and if the probability 
for an observed difference is close to the 
significance level, the Quade (1979) test and 
if still inconclusive the combination pro-
cedure as demonstrated in Nair & Alagaraja 
(1984) may be applied. The latter, though 
not simple, may bring out the real diffe-
rence, if any, in this case. Recently, Iman 
et al. (1984), while maxing a comparison 
of Friedman test, Quade (1979) test and 
rank transform test (Lemmer & Stoker, 1967; 
Conover & Iman, 1976; Hora & Conover, 
1984) found Quade test to be a better choice 
than Friedman test for normal data for 
the number of treatments, k < 6 and vice 
versa for k >6. For the nonnormal settings 
the result favoured the Quade test for uni-
form case and lognormal case (when k= 3), 
while Friedman test showed more power 
than the Quade test in the remaining 11 of 
the 16 nonnormal cases, they examined. 
They found Quad.e test to be favourable 
for light tailed uniform distributions while 
the Friedman test and the rank transform 
test for heavy tailed distributions. Appli-
cation of Quade test and rank transform 
test to trawl catches showed the same result 
as when Friedman test was applied. How-
ever, Friedman and Quade tests showed 
more or less the same sensitivity but rank 
transform test showed a little less sensitivity. 
(iii) When the efficiency of more than two 
gill-nets are to be compared, Friedman test 
and ordinary ANOVA F-test may be used. 
Friedman test helps to confirm the result as 
its applicability for the data is more valid 
and as applications (Kunjipalu et al., 1984) 
have shown Friedman test to be as sensitive 
as F-test. The performance of Quade test, 
Friedman test and rank transform test were 
compared for gill-net catches too. All the 
tests showed the same result. However, 
Quade test and rank transform test showed 
a little more sensitivtiy than Friedman 
test. Therefore it is advisable to apply 
Quade test and rank transform test when 
the probability for an observed difference 
is close to the significance level. Another 
alternative to confirm the results would be 
the test illustrated in this paper. Fitting of 
the Poisson or negative binomial for this 
purpose is simple. So also the application 
of chi-square test for goodness of fit and 
then for testing equality of samples from the 
" same Poisson or same negative binomial 
Vol. 23, 1986 
12 	 A. K. KESAVAN NAIR 
populations. In fact this test can be applied 
to compare the efficiencies of two or more 
gill-nets. 
The author is grateful to Shri M. Rajendranathan 
Nair, Director, Central Institute of Fisheries Tech-
nology, Cochin for permission to publish this paper, 
to Shri M.K. Kandoran, Scientist-in-Charge of 
Extension, Information and Statistics Division for 
encouragement and Dr. K. Alagaraja of Central 
Marine Fisheries Research Institute and S/Shri 
H. Krishna Iyer and K. Krishna Rao of Central 
Institute of Fisheries Technology for critical com-
ments on the manuscript. 
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FISHERY TECHNOLOGY 


On statistical comparison of efficiencies of fishing gear

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On statistical comparison of efficiencies of fishing gear

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