To bring out the relative efficiency of various types of fishing gears, in the analysis

of catch data, a combination of Tukey's test, consequent transformation and graphical

analysis for outlier elimination has been introduced, which can be advantageously used

for applying ANOVA techniques, Application of these procedures to actual sets of data

showed that nonadditivity in the data was caused by either the presence of outliers, or

the absence of a suitable transformation or both

Alagaraja, K

Nair, A K Kesavan

To bring out the relative efficiency of various types of fishing gears, in the analysis of catch data, a combination of Tukey's test, consequent transformation and graphical analysis for outlier elimination has been introduced, which can be advantageously used for applying ANOVA techniques, Application of these procedures to actual sets of data showed that nonadditivity in the data was caused by either the presence of outliers, or the absence of a suitable transformation or both. As a corollary, the concurrent model: X sub(ij) = µ + α sub(i) +  β sub(j) + λ α sub(i) β  sub(j) + E sub(ij) adequately fits the data

Kesavan Nair, A.K.

Alagaraja, K.

Aquatic Commons

n the Proble of Non ad itivity in Two-way 
alysis of Variance of Data for 
Comparing the Efficiency of Fishing Gears 
A. K. KESA VAN NAIR 
Centra! Institute of Fisheries Technology, Cochin - 682 029 
and 
K. ALAGARAJA 
Central Marine Fisheries Research Institute, Cochin- 682 018 
To bring out the relative efficiency of various types of fishing gears, in the analysis 
of catch data, a combination of Tukey's test, consequent transformqtion and graphical 
analysis for outlier elimination has been introduced, which can be advantageously used 
for applying ANOV A techniques, Application of these procedures to actual sets of data 
showed that nonadditivity in the data was caused by either the presence of outliers, or 
the absence of a suitable transformation or both. As a corollary, the concurrent model: 
Xii = fJ. + cc i + ~i +A c. i~ i + Eii adequately fits the data. 
The difficulties in using analysis of variance 
(ANOVA) F-test for comparing the efficiency 
of fishing gears have been discussed by Nair 
(1982) and Nair & Alagaraja (1982). Broadly, 
these problems arose from the lack of satis-
faction of the assumptions underlying analy-
sis of variance. The importance of each 
assumption has been clearly discussed (Eise-
nhart, 1947). Kempthorn.e (1967) has indi-
cated that the main requirements on the use-
fulness of a model are the additivity of treat-
ment effects and homogeneity of errors and 
that of these two, additivity is more important. 
Treatment of nonadditivity in two-way 
classification has received much attention 
(Tukey, 1949; Mandel, 1961; Daniel, 1976; 
Johnson & Graybill, 1972a, b; Krishnaiah & 
Yochmowitz, 1980; Marasinghe & Johnson, 
1981, 1982; Bradu & Gabriel, 1978 and Snee, 
1982). Snedecor & Cochran (1968) describe 
the usefulness of Tukey's (1949) test of 
additivity " (i) to help decide if a transfor-
mation is necessary (ii) to suggest a suitable 
transformation and (iii) to learn if a trans-
formation has been successful in producing 
additivity." Federer (1967) has observed 
that Tukey's sum of squares for nonadditi-
vity is increased when one or more observa-
tions are usually discrepant and when the 
row and column effects are not additive and 
that nonadditivity could arise from more than 
Vol. 21, 1984 
one source. Johnson & Graybill's (1972b) 
and Rao's (1974) methods of derivation and 
interpretation of Tukey's test show that when 
the above type of nonadditivity is present, 
the model is: 
Xii = fJ. +oc.j+ f3i + Aoc:.ifli + Eii 
and that Tukey's test correspond to testing 
A =0. Xij stands for catch on the ith day for 
the jth gear, fJ. is the overall mean catch, OC:.i 
and ~ i are the effects due to the ith day and 
jth gear respectively, f.. a constant and Eii is 
the error team. Mandel, as quoted by 
Krishnaiah & Y ochmowitz (1980), identified 
this model as the concurrent model and the 
concurrent model can be tested effectively by 
using Tukey's test for nonadditivity. Johnson 
& Graybill (1972b) and Hegemann & Johnson 
(1976b) have discussed that when Tukey's 
test shows significant nonadditivity, that is 
when the model given above describes the 
data, then the best way to analyse the data 
may be to find a transformation that will 
1estore additivity. Bartlett (1947) gives a 
number of transformations suitable for 
various forms of relationship between the 
variance in terms of the mean and the dis-
tribution for which those are appropriate. 
He recommended logarithmic transformation 
for certain type of data with considerable 
heterogeneity. Nair (1982) has found that 
76 A. K. KESAVAN NAIR AND K. ALAGARAJA 
for data on fishing experiments with trawl 
nets logarithmic transformation did not 
stabilize the variance. Also application of 
Tukey's test to the data after logarithmic 
transformation showed highly sugnificant 
nonadditivity (p<0.001). Cochran(1947) has 
observed that nonadditivity tends to produce 
heterogeneity of the error variance. Snee 
(1982) discusses procedures to examine 
whether non-additivity is caused due to non-
homogeneous variance or interaction between 
row and column factors. These show the 
relative importance of the assumption of 
additivity and this communication presents 
the results of an investigation on non-
additivity in trawl net-catch data on com-
parative fishing efficiency studies and pro-
cedures to tackle the problem using graphical 
analysis and transformation. 
Materials and Methods 
To decide whether a transformation is 
necessary and if required what would be the 
appropriate one, Tukey's (1949) test of addi-
tivity was applied to the four sets of data 
given in Nair (1982). Graphical analysis 
of nonadditivity (Tukey, 1949) was applied 
to these data to check whether the nonaddi-
tivity was due to analysis in the wrong form 
or due to one or more usually discrepant 
values. Tukey's test of additivity leads to 
transformation of the form Y = XP in which 
X is the original scale. The procedure 
followed in Snedecor & Cochran (1968) was 
applied to determine 'p', to which X, the 
observation must be raised to produce addi-
tivity. 'p' is estimated by (1-BX .. ), where 
B is the regression coefficient in the linear 
1\ 
regression of the residual (X .. -X..) on the 
IJ IJ 
variate (x .. -x .. ) (X.j-X .. ). An estimate of B 
I 
is obtained from B = N, where N = ~Widi, 
-D i 
wi= ~xif dj, di= cxi.-x .. ), dj = cx.j-x .. ) 
J 
and D=( ~d~) ( ~dJ) ;Xi., x.i and)( .. refer to 
the row (block) means, column (treatment) 
means and grand mean respectively. Tests 
for nonadditivity is given by F, where 
F follows Snedecor's F distribution with 1 
and[(r-1) (c-1) -1] degrees of Freedom, rand 
c indicating numbers of rows and columns, 
respectively. Tukey (1949) discusses trans-
formations which are additive for O<P< 1, 
p= 1 and 1 < p and log (x +a) corresponding 
to none of these. Snedecor & Cochran (1968) 
stated that when p=-1, it is a reciprocal trans-
formation analysing 1/X, instead of X. (p=o 
corresponds to logrithmic transformation 
because for p very small XP behaves like 
log X) 
Results and Discussion 
Application of Tukey's test of additivity 
for the four sets of data on trawl catch (Nair, 
1982) showed that there was significant non-
additivity in all the sets (Table 1 ). For sets 
Table 1. Test for nonadditivity of the four 
sets of data 
F for nonad- Degrees of 
ditivity freedom 
Set 1 38.64*** 1,67 
Set 2 63.87*** 1,67 
Set 3 87.70*** 1,67 
Set 4 4.80* 1,18 
1-3 (that is for the actual data), nonadditivity 
was found to be very highly significant with 
p< 0.001. Tukey's (1949) procedure was fol-
lowed to check whether nonadditivity was 
caused by the presence of one or more dis-
crepant observations or due to the need for 
a transformation. His method of graphical 
analysis for detecting the discrepant obser-
vations (outliers) was applied to the four 
sets of data. The method involves in plot-
ting wi against the block means. According 
to Tukey, "a usually discrepant observation 
will tend to be reflected by one point high 
or low and the others distributed at ound a 
nearly horizontal line. An analysis in the 
wrong terms will tend to be reflected by a 
slanting regression line." To determine the 
points high or low Tukey provided a 2s 
limit, namely, 
(average cross p1 oduct) ± 2 { 
(= ~wi/no. ofrows) 
sums of squares } ! {Means square for}! 
of deviations of balance 
column means ( = ~d~) 
. . J 
FISHERY TECHNOLOGY 
NONADDITIVITY IN TWO-WAY ANALYSIS OF VARIANCE 77 
The plots of w i against the row means 
with the 2s limits for sets 1-4 are presented 
in Figs. 1-4. The figures show the presence 
10() 
" 
80 
60 
40 
20 
" 
ilf-
i 
: 
6~ 
4 
357r, ------------~~ 
• a zr • .,. •• 
05~~:_2 __ _ 
"'ow means 
..,~h-'!..#~">1--.,=--,__,2o:r0;-"-~-'--,3b0;-"-_.__._-;;lr40 Fig. 3. Plot of wi on row means with 2 s limits 
-20~----------~-------------------
-40 Row means 
Fig. 1. Plot of w. on row means with the 2 s 
1 
limits for set 1 
lt:JO 
160 
140 
120 
100 
" 
60 
55 5f---------
40 
·20 
. 256 
40 
Fig. 2. 
----------=---------------
Row means 
Plot of wi on row means with the 2 s 
limits for set 2 
of outliers in all the four sets ranging from 
1 to 5 in number. It is clear from the figures 
that the points excluding the outliers are 
distributed on a nearly horizontal line for 
set 1 and on a slanting regression line for 
Vol. 21, 1984 
for set 3 
1400 
, ... 1-------------------------
IZOO 
1000 
BOO 
600 
400 . . . 
200 jJ • •• • 
. . 
. 
200 
<00 
600 
!000 
.1200 
Fig. 4. Plo't of w. on row means with 2 s limits 
1 
for set 4 
sets, 2 to 4. This shows--that no transfor-
mation is required for set 1, after removing 
the outliers while it is required for the other 
sets. This was confirmed by applying Tukey's 
test to the outlier-eliminated data (Table 2). 
Sets 2A showed the presence of nonadditi-
vity indicating the need for a transformation 
for these sets. 
The power transformation Y = XP , 
suggested by Tukey's test of additivity were 
worked out for. sets 2-4. These have been 
78 A. K. KESAVAN NAIR AND K. ALAGARAJA 
Table 2. Test for nonadditivity of the out-
lier-eliminated data 
F for nonad- Degrees of 
ditivity freedom 
Set 1 0.02 
(not significant) 
Set 2 9.90** 
Set 3 34.37*** 
Set 4 15.23** 
*Significant at 5% level, 
**Significant at 1% level, 
***Significant at 0.1% level 
1,59 
1,61 
1,57 
1,17 
presented in Table 3 along with the estimated 
values of B and P. For set 2, the transfor-
mation worked outto Y=X- 0 ' 31 , which is 
Table 3. Tukey's transformation after eli-
minating the outliers 
B p Y=XP 
Set 1 Data additive after exclusion of 
outliers 
set 2 0.1594 -0.31 x-o.s1 
set 3 1.0335 0.0618 xo.0618 
set 4 0.0166 0.1594 xo.1594 
a reciprocal transformation. For set 3, the 
transformation obtained was Y = X 0 · 0 618 
and for set 4, y = X 0 ' 1594 . 
The data were analysed after carrying out 
these transformations. Tukey's test of addi-
tivity now showed, nonadditivity to be insi-
gnificant for all the sets (Table 4). The 
Table 4. Test for nonadditivity of the out-
lier-eliminated and transformed data 
F for nonad-
ditivity 
Degrees of 
freedom 
Set 1 Not applicable as data is addi-
tive after exclusion. of outliers 
Set 2 
Set 3 
Set 4 
2.55 Not significant 
0.05 
0.13 " 
" 
1,57 
1,57 
1,17 
reduction by 4 in the lower d.f. for set 2 is 
due to omission of two rows where one 
observation each was zero. Though p was as 
small as 0.0618 for set 3, logarithmic trans-
formation did not remove nonadditivity, 
F for nonadditivity being 12.97***, which 
is highly significant for 1 and 57 degrees of 
freedom. Thus application of the power 
transformation suggested by Tukey's test 
to the data after eliminating the outliers has 
been found to be effective in making the data 
additive. In case where nonadditivity is 
not accounted for by Tukey's transforma-
tion and outlier elimination by graphical 
analysis or in other words where the con-
current model does not describe the data, 
there are other methods for testing the stru-
cture of interaction and testing the main 
effects, for instance, methods mentioned by 
Marasinghe & Johnson (1982) (for a multi-
plicative interaction structure) and Krish-
naiah & Yochmowitz (1980). The work in 
this line would be considered later. 
Daniel (1976) points out that nonadditi-
vity is often associated with a few rows or 
columns of the two-way table. Snee (1982) 
states that nonadditivity in a two-way classi-
fication with one observation per cell may 
be either due to nonhomogeneous variance 
or interaction and the data may not be suffi-
cient to distinguish between tl)ese two. 
However, ways and means for interpreta-
tion of the observed nonadditivity has been 
discussed by him. Federer (1967) states that 
the sum of squares associated with Tukey's 
one degree of freedom for nonadditivity 
gives the linear row by linear column inter-
action. Nair (1982) reported the dependence 
of standard error per unit on the average 
catch. A look at the model considered in 
this paper will show that when the availa-
bility of fishes changes over period of 
days, the a: /s may change, for different perio-
ds causing this situation. (The dependence 
of variance on the mean also suggests non-
normality). 
Apart from graphical procedure, much 
work has been done on the rejection of out-
liers. Rules for rejection has been discussed 
by Anscombe (1960), Anscombe & Tukey 
(1963) and Snedecor & Cochran (1968). 
Lately, Gaplin & Hawkins (1981) have 
presented bounds for the fractiles of maxi-
mum normed residuals (MNR). The present 
procedure is convenient to apply along with 
FISHERY TECHNOLOGY 
NONADDITIVITY IN TWO-WAY ANALYSIS OF VARIANCE 79 
additivity test because the steps involved 
in testing provide the material for graphical 
analysis. 
The present study shows that elimination 
of the outliers by graphical analysis and 
application of Tukey's test of additivity can 
be adopted to tackle the problem of nonaddi-
tivity in the analysis of catch data. Nair 
& Alagaraja (1982) suggested Wilcoxon's 
m~tched-pairs signed-rank test as an appro-
pnate procedure for comparing the efficiency 
of two fishing gears and illustrated with a 
set of data the superiority of this method 
over ususal ANOVA. (Ordinary ANOVA 
was less sensitive in this case). The same 
set of data was analysed using the above 
procedure (that is outlier-elimination and 
application of Tukey's test of additivity and 
the consequent transformation as introduced 
and discussed in this paper) and the same 
result as that given by Wilcoxon's test was 
obtained. This shows the usefulness of 
this combination procedure in statistical 
comparison of the efficiency of fishing gears. 
We are grateful to Dr. C. C. Panduranga Rao 
Director, Central Institute of Fisheries Technology' 
Cochin for permission to publish this paper and t~ 
Shri M. Rajendranathan Nair, Scientist-in-Charge, 
Extension, Information and Statistics Division for 
encouragement. 
References 
Anscombe, F. J. (1960) Technometrics. 2, 123 
Anscombe, F. J. & Tukey, J. W. (1963) 
Technometrics. 5, 141 
Bartlett, M. S. (1947) Biometrics. 3, 39 
Bradu, D. & Gabriel, K. R. (1978) Tech-
nometrics. 20, 47 
Cochran, W. G. (1947) Biometrics. 3, 22 
Daniel, C. (1976) Application of Statis-
tics to Industrial Experimentation. John 
Wiley, New York 
Eisenhart, C. (1947) Biometrics. 3, 1 
Federer, W. T. (1967) Experimental Design 
Theory and Application. Oxford & 
IBH Publishing Co. New Delhi 
Gaplin, J. S. & Hawkins, D. M. (1981) 
Technometrics. 23, 65 
Vol. 21, 1984 
Hegemann, V. & Johnson, D. E. (1976a) 
Teclmometrics. 18, 273 
Hegemann, V. & Johnson, D. E. (1976b) 
J. Amer. Statist. Assoc. 71, 945 
Johnson, D. E. & Graybill, F. A. (1972a) 
J. Amer. Statist. Assoc. 67, 388 
Johnson, D. E. & Graybill, F. A. (1972b) 
J. Amer. Statist Assoc. 67, 862 
Kempthorne, 0. (1967) The Design and 
Analysis of Experiments. Wiley Eas-
tern University edn., New Delhi 
Krishnaiah, P. R. & Yochmowitz, M. G. 
(1980) Inference on the Structure of 
Interaction in Two-way Classification 
Model in Hand-book of Statistics 
(Krishnaiah, P. R. ed.), Vol. 1 North 
Holland Publishing Company, p. 973 
Mandel, J. (1961) J. Amer. Statist. 
Assoc. 56, 878 
Marasinghe, M.G. & Johnson, D. E. (1981) 
Technometrics. 23, 385 
Marasinghe, M.G. &Johnson, D. E. (1982) 
J. Amer. Statist. Assoc. 77, 869 
Nair, A. K. Kesavan (1982) Fish. Techno!. 
19, 49 
Nair, A. K. Kesavan, & Alagaraja, K. (1982 
Proc, Symp. Harvest and Post-Harvest 
Technology of Fish Organized by Society 
of Fisheries Technologists (India), Nov. 
24-27, 1982 (in press) 
Rao, C. R. (1974) Linear Statistical Infer-
ence and its Applications. 2nd edn., 
Wiley ~astern Private Limited, 
New Delhi 
Snedecor, G. W. & Cochran, W. G. (1968) 
Statistical Methods. 6th edn. Oxford 
and IBH Publishing Company, New 
Delhi 
Snee, R. D. (1982) J. Amer. Statist. 
Assoc. 77, 515 
Tukey, J. W. (1949) Biometrics. 5, 232 


On the problem of nonadditivity in two-way analysis of variance of data for comparing the efficiency of fishing gears

CMFRI Digital Repository

360 On the Problem of Nonadditivity in Two-way Analysis of Variance of Data for Comparing the Efficiency of Fishing Gears A. K. KESAYAN NAIR Centra! Institute of Fisheries Technology, Cochin - 682 029 and K. ALAGARAJA Central Marine Fisheries Research Institute, Cochin - 682 018 To bring out the relative efficiency of various types of fishing gears, in the analysis of catch data, a combination of Tukey's test, consequent transformation and graphical analysis for outlier elimination has been introduced, which can be advantageously used for applying ANOYA. techniques, Application of these procedures to actual sets of data showed that nonadditivity in the data was caused by either the presence of outliers, or the absence of a suitable transformation or both. As a corollary, the concurrent model: Xi; = fL + ex i + ~ ; +). C> ! ~; + El; adequately fits the data. The difficulties in using analysis of variance (ANOYA) F-test for comparing the efficiency of fishing gears have been discussed by Nair (1982) and Nair & Alagaraja (1982). Broadly, these problems arose from the lack of satis-faction of the assumptions underlying analy-sis of variance. The importance of each assumption has been clearly discussed (Eise-nhart, 1947). Kempthorne (1967) has indi-cated that the main requirements on the use-fulness of a model are the additivity of trea t-ment effects and homogeneity of errors and that of these two,additivity is more important. Treatment of nonadditivity in two-way classification has received much attention (Tukey, 1949; Mandel, 1961; Daniel, 1976; Johnson & Graybill, 1972a, b; Krishnaiah & Yochmowitz, 1980 ; Marasinghe & Johnson, 1981, 1982 ; Bradu & Gabriel, 1978 and Snee, 1982). Snedecor & Cochran (1968) describe the usefulness of Tukey's (1949) test of additivity .. (i) to help decide if a Iransfor-mation is necessary (ii) to suggest a suitable transformation and (iii) to learn if a trans-formation has been successful in producing additivity." Federer (1967) has observed that Tukey's sum of squares for nonadditi-vity is increased when one or more observa-tions are usualJy discrepant and when the row and column effects are not additive. and that nonadditivity could arise from more than Vol. 21, 1984 one source. Johnson & Graybill's (1972b) and Rao's (1974) methods of derivation and interpretation of Tukey's test show that when the above type of nonadditivity is present, the model is : 'Xij =11 + oc. i+ fJj + ACCjfJj + Eij and that Tukey's test correspond to testing 1.=0. Xij -stands for catch on the ith day for the jth gear, fL is the overall mean catch, OCi and ~; are the effects due to the ith day and jth gear respectively, ). a constant and El ; is the error team. Mandel, as quoted by Krishnaiah & Yochmowitz (1980), identified this model as the concurrent model and the concurrent model can be tested effectively by using Tukey's test for nonadditivity. Johnson & GraybilJ (I 972b) and Hegemann & Johnson (1976b) have discussed that when Tukey's test shows significant nonadditivity, that is when the model given above describes the data, theo the best way to analyse the data may be to find a transformation that will lestore additivity: Bartlett (1947) gives a number of transformations suitable for various forms of relationship between the variance in terms of the mean and the dis-tribution for which those are appropriate. He recommended logarithmic transformation for certain type of data with considerable heterogeneity. Nair (1982) has found that 76 A. K. KESAYAN NAJR AND K. ALAGARAJA for data on fishing experiments with trawl nets logarithmic transformation did not stabilize the variance. Also application of Tukey's test to the data after logarithmic transformation showed highly sugnificant nonadditivity (p<O.OOI). Cochran (1947) has observed that nonadditivity tends to produce heterogeneity of the error variance. Snee (1982) discusses procedures to examine whether non-additivity is caused due to non-homogeneous variance or interaction between row and column factors. Tbese show the relative importance of the assumption of additivity and this communication presents the results of an investigation on non-additivity in trawl net-catch data on com-parative fishing efficiency studies and pro-cedures to tackle the problem using ·graphical analysis and transformation. Materials and Methods To decide whether a transformation is necessary and if required what would be the appropriate one, Tukey's (1949) test of addi-tivity was applied to the four sets of data given in Nair (1982). Graphical analysis of nonadditivity (Tukey, 1949) was applied \0 these data to check whether the nonaddi-tivity was due to analysis in the wrong form or due to one or morc usually discrepant values. Tukey's test of additivity leads to transformation of the form Y = XP in which X is the original scale. The procedure followed in Snedecor & Cochran (1968) was applied to determine ' p', to which X, the observation must be raised to produce addi-IIvlty. 'p' is estimated by (I-BX .. ), where B is the regression coefficient in the linear 1\ regression of the residual (X .. -X .. ) on the IJ lJ variate (5( .. -5( .. )(5(.j-X .. ). An estimate of B I is obtained from B = N, where N = :Ewidi , -0 j wj= :r:Xjj" dj , d j= (5(\.-5( .. ), dj = (XT X .. ) and D=( ::§;d~) ( ::§;dj) ;X,., X'j and X .. refer to the row (block) means, column (treatment) means and grand mean respectively . . Tests for nonadditivity is given by F, where F follows Snedecor's F distribution with I and [(r-I) (c-I) -I] degrees of Freedom, rand c indicating numbers of rows and columns, respectively. Tukey (1949) discusses trans-formations which are additive for O:O;:P < I, p=1 and I <p and log (x +a) corresponding to none of these. Snedecor & Cochran (1968) stated that whenp=-I , it isa reciprocaltrans-formation analysing I /X , instead of X. (p=o corresponds to logrithmic transformation because for p very small XP behaves like log X) Results and Discussion A pplication of T ukey's test of additivity for tbe four sets of data on trawl catch (Nair, 1982) showed that there was significal;lt non-additivity in all the sets (Table I). For sets .Table 1. Test for nonadditivity of the four sets of data Set I Set 2 Set 3 Set 4 F for nonad- Degrees of ditivity freedom 38.64'" 63.87'** 87.70**' 4.80* 1,67 1,67 1,67 1,18 1-3 (that is for the actual data), nonadditivity was found to be very highly significant with p< 0.001. Tukey's (1949) procedure was fol-lowed to check whether nonadditivity was caused by the presence of one or more dis-crepant observations or due to the need for a transformation. His method of graphical analysis for detecting the discrepant obser-vations (outliers) was applied to the four sets of data. The method involves in plot-ting W; against the block means. According to Tukey. "a usually discrepant observation will tend to be reflected by one point high or low and the others distributed alound a nearly horizontal line. An analysis in the wrong terms will tend to be reflected by a slanting regression line." To determine the points high or low Tukey provided a 2s limit, namely, (average cross plOduct)± 2 { (= ::§;w;/no. of rows) sums of squares }! {Means square for}! of deViations of balance column means (= ::§;d:) FlSHERY TECHNOLOGY NONADDITIVITY IN TWOcWAY ANALYSIS OF VARIANCE 77 The plots of Wi against the row means with the 2s limits for sets 1-4 are presented <J!-in Figs. 1-4. The figures show the presence' i!-! • J • • • •• .. . ' • • ~ow means !oII~~~::-'M ....... ~,,~~--;tr~~""4<!<O Fig. 3. Plot of wi on row means with 2 s limits for set 3 -AO ~ow means Fig. I. Plot of wi on row means with the '- 2 s limits for set 1 ,"0 '00 60 3i • . , bO,, _______ ~ ________________ ___ 555t-•• 20 . i~iJCf ---~-----'---------__ _ 40 Row means Fig. 2. Plot of Wi on row means with the 2 s limits for set 2 of outliers in all the four sets ranging from I to 5 in number. It is clear from the figures that the points excluding the outliers are distributed on a nearly horizontal line for set I and qn a slanting regression line for Vo]. 21, 1984 .~ " • " , .. .. Fig. 4. " --Plot of w. -on row means with 2 s limits , for set 4 sets, 2 to 4. This shows-that no transfor- . mation is required for set' 1, after removing the outliers while it is required for the other sets. This was confirmed by applying Tukey's test to the outlier-eliminated data (Table 2). Sets 2-4 showed the presence of nonadditi-vity indicating the need for a transformation for these sets. The power transformation Y = Xp , suggested by Tilkey's test of additivity were wotked' out for sets 2-4. These have been 78 A. K. KESAVAN NAIR AND K. ALAGARAJA Table 2. Test for nonadditivity of the out-lier-eliminated data F for nonad- Degrees of ditivity freedom Set I 0.02 Set 2 (not significant) 9.90" Set 3 34.37'" Set 4 15.23" 'Significant at 5% level, "Significant at 1 % level, "'Significant at 0.1 % level 1,59 1,61 1,57 1,17 presented in Table 3 along with the estimated values of Band P. For set 2, the transfor-mation worked Qutto y=X_O,S\ which is Table 3. Tukey's transformation after eli-minating the outliers B P Y=XP Set Data additive after exclusion of outliers set 2 0.1594 ~.31 X- O. S I set 3 1.0335 0.0618 XO.0618 set 4 0.0166 0.1594 XO.1594 a reciprocal transformation. For set 3, the transformation obtained was Y = X o.o 618 and for set 4, Y = XO .1594. The data were analysed after carrying out these transformations. Tukey's test of addi-tivity now showed, nonadditivity to be insi-gnificant for all the sets (Table 4). The Table 4. Test for nonadditivity of the out-lier-eliminated and transformed data F for nonad-ditivity Degrees of freedom Set I Not applicable as data is addi-tive after exclusion of outliers Set 2 Set 3 Set 4 2.55 0.05 0.13 Not significant " " 1,57 1,57 1,17 reduction by 4 in the lower d.f. for set 2 is due to omission of two rows where one observation each was zero. Though p was as small as 0.0618 for set 3, logarithmic trans-formation did not remove nonadditivity, F for nonadditivity being 12.97"', which is highly significant for I and 57 degrees of freedom. Thus application of the power transformation suggested by Tukey's test to the data after eliminating the outliers has been found to be effective in making the data additive. In case where nonadditivity is not accounted for by Tukey's transforma-tion and outlier elimination by graphical analysis or in other words where the con-current model does not describe the data, there are other methods for testing the stru-cture of interaction and testing the main effects, for instance, methods mentioned by Marasinghe & Johnson (1982) (for a multi-plicative interaction structure) and Krish-naiah & Yochmowitz (1980). The work in this line would be considered later. Daniel (1976) points out that nonadditi-vity is often associated with a few rows or columns of the two-way table. Snee (1982) states that nonadditivity in a two-way classi-fication with one observation per cell may be either due to nonhomogeneous variance or interaction and the data may not be suffi-cient to disLinguish between these two. However, ways and means for interpreta-tion of the observed nonadditivity has been discussed by him. Federer (1967) states that the sum of squares associated with Tukey's one degree of freedom for nonadditivity gives the linear row by linear column inter-action. Nair (1982) reported the dependence of standard error per unit on the average catch. A look at the model considered in this paper will show that when the availa-bility of fishes changes over period of days, the a: is may change, for different perio-ds causing this situation. (The dependence of variance on the mean also suggests non-normality). Apart from graphical procedure, much work has been done on the rejection of out-liers. Rules for rejection has been discussed by Anscombe (1960), Anscombe & Tukey (1963) and Snedecor & Cochran (1968). Lately, Gaplin & Hawkins (1981) have presented bounds for the fractiles of maxi-mum normed residuals (MNR). The present procedure is convenient to apply along with FISHERY TECHNOLOGY • NONADDITIVITY IN TWO-WAY ANALYSIS OF VARIANCE 79 additivity test because the steps involved in testing provide the material for graphical analysis. The present study shows that elimination of the outliers by graphical analysis and application of Tukey's test of additivity can be adopted to tackle the problem of nonaddi-tivity in the analysis of catch data. Nair & Alagaraja (1982) suggested Wilcoxon's matched-pairs signed-rank test as an appro-priate procedure for comparing the efficiency of two fishing gears and illustrated with a set of data the superiority of this method over ususal ANOV A. (Ordinary ,ANOV A was less sensitive in this case). The same set of data was analysed using the above procedure (that ,is outlier-elimination and application of Tukey's test of additivity and the consequent transformation as introduced and discussed in this paper) and the same result as that given by Wilcoxon's test was obtained. This shows the usefulness of this combination procedure in statistical comparison of the efficiency of fishing gears. We are grateful to Dr. C. C. Panduranga Rao Director, Central Institute of Fisheries Technology: Cochin f or permission to publish this paper and to Shri ~. Rajendranathan Nair, Scientist-in-Charge, ExtenSion, Information and Statistics Division for encouragement. References Anscombe, F. J. (1960) Technometrics. 2, 123 An~combe, F. J, & Tukey, J, W, (1963) Technometrics. 5, 141 Bartlett, M. S, (1947) Biometrics. 3, 39 Bradu, D. & Gabriel, K. R. (1978) Tech-nometries. 20, 47 Cochran, W. G, (1947) Biometrics. 3, 22 Dal'iel, C. (1976) Application of Statis-tics to Industrial Experimentation. John Wiley, New York Eisenhart, C. (1947) Biometrics. 3, I Federer, W. T. (1967) Experimental Design Theory and Application. Oxford & IBH Publishing Co. New Delhi Gaplin, J. S, & Hawkins, D, M, (1981) Technometrics. 23, 65 Vol. 21, 1984 Hegemann, V. & Johnson, D. E. (I 976a) Technometrics. 18, 273 Hegemann, V. & Johnson, D. E. (1976b) J. Amer. Statist. Assoc. 71, 945 Johnson, D. E. & Graybill, F. A. (l972a) J. Amer. Statist. Assoc. 67, 388 Johnson, D. E. & Graybill, F. A. (I 972b) J. Amer. Statist Assoc. 67, 862 Kempthorne, O. (1967) The Design and Analysis of Experiments. Wiley Eas-tern University edn., New Delhi Krishnaiah, P. R. & Yochmowitz, M. G. (1980) Inference on the Structure of Interaction in Two-way Classification Model in Hand-book of Statistics (Krishnaiah, P. R. ed,), VoL I North Holland Publishing Company, p. 973 Mandel, J. (1961) J. Amer. . Statist. Assoc, 56, 878 Marasinghe, M, G. & Johnson, D. E. (1981) Technometrics. 23, 385 Marasinghe, M. G. & Johnson, D. E. (1982) J. Amer, Statist. Assoc. 77, 869 Nair, A: K. Kesavan (1982) Fish. Technol. 19, 49 Nair, A, K. Kesavan, & Alagaraja, K, (1982 Proc, Symp. Harvest and Post-Harvest Technology of Fish Organized by Society of Fisheries Technologists (India), Nov. 24-27, 1982 (in press) Rao, C. R. (1974) Linear·Statistical Infer-ence and its Applications, 2nd edn., Wiley Eastern · Private Limited, New DeIhl Snedecor, G. W. & Cochran, W. G, (1968) Statistical Methods. 6th edn. Oxford and IBH Publishing Company, New Delhi Snee, R. D. (1982) J. Amer. Statist, Assoc, 77, 515 Tukey, J. W. (1949) Biometrics. 5, 232 

Problem of nonadditivity in two-way analysis of variance of data for comparing the efficiency of fishing gears

http://eprints.cmfri.org.in/7354/1/360-_FISHERY_TECHNOLOGY_1976.pdf

Problem of nonadditivity in two-way analysis of variance of data for comparing the efficiency of fishing gears

Abstract

Similar works

Full text

Available Versions

Aquatic Commons

CMFRI Digital Repository