Biometrika Trust
The Transformation of Data from Entomological Field Experiments so that the Analysis of
Variance Becomes Applicable
Author(s): Geoffrey Beall
Source: Biometrika, Vol. 32, No. 3/4 (Apr., 1942), pp. 243-262
Published by: Biometrika Trust
Stable URL: http://www.jstor.org/stable/2332128 .
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THE TRANSFORMATION OF DATA FROM ENTOMOLOGICAL
FIELD EXPERIMENTS SO THAT THE ANALYSIS
OF VARIANCE BECOMES APPLICABLEt
By GEOFFREY BEALL
Dominion EntomologicalLaboratory,Chatham,Ontario,Canada
1. INTRODUCTORY
THE presentpaper deals with experimentson the controlof insects in the field.
In such experimentalworkthe problemto be investigatedis whethermoreinsects
survive on plots which have been subjected to one treatment than on plots
subjected to another. It will be shown in the presentpaper that the numbersof
insectsfoundper plot must vary in such a way that one cannot, strictly,subject
the results to the analysis of variance, and it is proposed to findhow the data
may be transformedso that analysis of variance becomes applicable. Such
transformationhas been discussed by Bartlett (1936a,b) in connexion with
entomologicalexperiments,and by Tippett (1934) in connexionwith industrial
experiments.
2.
EXPERIMENTAL
RESULTS CONSIDERED
The data used in the followingwork are results from seven insecticidal
experimentsarrangedby the author at Chatham, Ontario. The workwas carried
out with replicatedblocks containingplots subjected to treatmentsof which the
assignmentwas random. This procedure,normalin agronomicwork,was supplemented by one repetitionof each treatmentwithina block. The assignmentof
the repetitionof a treatmentwas independentof the firstfor that treatment,
except that, of course, the same plot could not be chosen twice. This repetition
was carriedout to obtain estimates of variabilitywithinblocks. In these experiments complete counts were not made but random sampling was employed.
Experiments on Pyrausta nubilalis Hubn., reportedby Beall et al. (1939), for
whichresultsare shownin Tables 1 and 2, weremade on one area at two different
periods,whereas experimentson LeptinotarsadecemlineataSay, forwhichresults
are indicated in Tables 3 and 4, were carriedout on contiguousareas at the same
time. Three,similar experimentswere carried out in one place on the tobacco
hornworm,Phlegethontius
quinquemaculataHaw., forwhich the data are shown
in Tables 5-7. Reference is also made to the data from a uniformitytrial on
insects of Beall (1939).
ScienceService,DepartmentofAgriculture,
t PublicationNo. 2101,DivisionofEntomology,
Ottawa,Canada.
244 Transformation
ofdatafromentomological
fieldexperiments
Table 1. Numbersofan insect,Pyraustanubilalis,perplot. Experiment
I
Block
T rea t-
ment
1
1
2
2
3
3
4
4
5
5
6
6
7
7
_
_
_
_ - _
_
_
_
_
_ - _
_
_
_ - _
_
_
_
_
_ - _
_
_
_ - _
_
_
_
_
_ - _
_
_
1
2
3
4
5
6
7
8
9
10
15
27
19
11
16
19
14
34
16
23
12
15
43
47
23
20
12
28
15
16
23
21
16
12
14
16
28
81
21
23
34
37
22
18
10
9
19
12
17
15
35
30
31
33
16
16
25
21
19
34
26
10
10
28
36
69
22
34
20
26
13
19
17
19
15
12
14
13
50
35
14
27
10
18
21
24
18
9
11
17
24
22
69
29
18
17
24
19
18
21
7
15
18
13
24
11
62
71
11
13
23
13
38
18
18
16
23
21
17
7
63
47
21
20
14
10
27
12
8
12
27
12
7
5
42
50
34
26
13
9
10
18
17
12
6
9
3
4
40
43
Table 2. Numbersofan insect,Pyraustanubilalis,perplot. Experiment
II
Block
T reat--
ment
1
1
2
2
3
3
4
4
5
5
6
6
7
7
_
_
_
_-
_
_
_
--
_
_ - _
_
1
2
3
4
5
6
7
8
9
10
32
18
6
9
10
4
2
24
13
17
13
17
37
44
38
40
23
14
21
21
17
13
2
22
10
9
58
71
27
39
8
20
25
26
11
13
5
23
21
29
28
55
7
12
4
13
10
4
3
10
0
8
4
5
11
20
13
19
3
15
13
9
10
6
18
14
10
18
24
26
14
26
18
14
20
14
10
14
10
16
8
15
44
27
26
30
26
15
33
30
26
28
33
26
17
19
30
43
25
19
27
19
48
18
13
11
23
22
15
16
44
52
22
18
17
19
28
27
22
34
20
15
13
27
56
39
30
28
19
10
27
18
17
7
34
34
16
23
45
58
_
245
GEOFFREY BEALL
Table 3. Numbersof an insect,Leptinotarsa decemlineata,
per plot. ExperimentIII
Block
Treatment
1
1
2
2
3
3
4
4
1
2
3
4
5
6
7
305
207
97
93
270
153
7
12
391
364
49
51
105
190
42
2
420
639
21
25
341
348
34
22
355
527
12
37
469
212
8
4
287
293
3
4
82
100
1
1
175
248
10
12
57
285
10
3
454
397
10
1
221
309
4
3
Table 4. Numbersof an insect,Leptinotarsa decemlineata,
per plot. ExperimentIV
Block
Treatment
1
1
2
2
3
3
4
4
1
2
3
4
5
6
253
239
16
95
18
40
2
2
145
265
13
54
130
137
0
1
309
166
74
159
165
118
22
31
665
230
110
108
137
142
6
8
99
302
14
14
153
239
129
9
93
237
5
13
78
63
3
8
1
Table 5. Alumbersof an insect,Phlegethontiusquinquemaculata,
per plot. ExperimentV
Block
Treatment
1
1
2
2
3
3
1
2
3
4
5
6
6
4
0
2
15
12
5
15
1
2
17
22
6
13
1
1
22
16
13
6
0
4
28
11
6
10
1
1
8
13
11
15
1
1
16
25
246
Transformationof data from entomologicalfieldexperiments
Table 6. Numbersofan insect,Phlegethontiusquinquemaculata,
perplot. Experiment
VI
Block
Treatment
>
3
1
2
3
4
5
6
1
1
2
2
3
12
13
13
20
13
9
6
9
9
5
8
9
5
4
7
1
4
4
11
5
5
12
8
4
10
4
4
5
5
6
6
1
2
13
11
1
2
0
2
12
4
8
2
1
1
3
1
9
4
4
4
6
9
6
4
3
5
11
8
5
12
3
1
7
7
7
8
_
9
9
7
5
6
10
_
4
4
|
_
3
7
7
7
9
2
Table 7. Numbersofan insect,Phlegethontiusquinquemaculata,
perplot. Experiment
VII
Block
Treatment
1
1
2
2
3
3
4
4
0
5
6
6
3.
THE RELATIONSHIP
1
2
3
10
7
11
17
0
1
3
5
3
5
11
9
20
14
21
11
7
2
12
6
3
5
15
22
14
12
16
14
3
1
4
3
3
6
15
16
4
I
10
23
17
17
2
1
5
5
1
1
13
1 0
I
5
6
17
20
19
21
3
0
5
5
3
2
26
26
14
13
7
13
1
4
2
4
6
4
24
13
BETWEEN THE STANDARD DEVIATION AND THE MEAN
IN THE EXPERIMENTAL
DATA
If x is the numberof insects on one of a group of small contiguousareas, say
plots, within a larger area, say a block, let the expectation of x over all these
plots be M and the standard deviation be o; then over a number of the larger
GEOFFREY BEALL
247
areas, when the insectsare distributedin a completelyrandom fashion,fromthe
(1)
Poisson distribution,
0.2 = M.
As is discussed by 'Student' (1919) one cannot, however,anticipate that (1) will
be satisfiedwhen organisms occur in groups, as, say, when insects come from
masses of eggs, or when thereis a change in expectation fromplot to plot within
a block. Generally,0-2 will tend to be greaterthan M and we can only say
02
= f(M).
(2)
The formoff(M), in (2), must be consideredcarefully,since it bears on the form
of the transformationwhich may be developed to make the standard deviation
independentof the mean.
In dealingwith (2), Bartlett (1936a) startedbysupposingthat,approximately,
0.2=
KM,
(3)
whereK is a constant. Generally,in fielddata, however,the relationshipbetween
0.2 and M, or of theirrespectiveestimates,S2 and x, does not, as in Fig. 1, appear
to be linear; rather,the departureof S2 fromx3becomes disproportionatelygreat
as x increases. This relationshipbetween departures and the magnitude of the
mean has been discussed by Clapham (1936) in connexion with data on the
frominsects as much as floweringplants, and
distributionof organismsdiffering
with very low mean have the squared
distributions
he showed that only those
standard deviation close to the mean.
Our discussionabove on the shortcomingsof (3) suggeststhe conclusionthat
0,2- Moc M
(4)
is generallyuntrue. We propose to considerthe possibilitythat the curvilinearity
of (2) mightbe bettermet by supposing that
o,2-
Equation (5) leads to
2=
Moc M2.
(5)
M + kM2,
(6)
wherek is a constant. It will be noticed that
k = (o2 - M) M-2
(7)
is the Charlier coefficientof disturbance from a Poisson distribution. This
was employed by Beall (1935).
coefficient
It is possible to considerthe suitabilityof (3), as comparedwith (6), by finding
and
how,respectively,they fitobservationson S2 and x. To fitexactly is difficult,
it was foundnecessaryto fall back on an empiricaldeterminationof K and of k;
thus, if there are a number of pairs of estimates, x and s2, from(3) and (6) we
K = Zs2/L'x,
(8)
estimate
k = (L's2-2X2)/2?;2,
whereL representsthe summationover all pairs.
(9)
248
Transformationof data from entomologicalfield experiments
Since in the work presentedin ? 2, x and 82, being based on only two observations,are highlyvariable, these experimentsdo not show clearlythe suitability
of (3) and (6). Accordingly,referenceis made instead to the data fromthe uniformitytrial on LeptinotarsadecemlineataSay of Beall (1939). When the mean
and standard deviation of 144 samplingunits withineach of 16 areas were considered, the estimates from (8) and (9) were K = 2*405 and k = 0-2548. For
these values from(1), (3) and (6), curves,describedas lines 1, 2 and 3 respectively,
are plottedin Fig. 1, the observedvalues ofmean and squared standard deviation
3
25-
2
20I
15-
z
u
O
4
X6
I10
Fig. 1. The squaredstandarddeviationplottedagainstthe meanfor144 smallareas withineach
of16 largeareas; line 1 is fromequation(1), line2 from(3) and line3 from(6). The countshad
decemlineatc
been made on Leptinotarsa
Say.
are also shown. In the cases where the mean is near unity the departure of the
squared standard deviation fromthe mean, i.e. fromline 1, appears to be trivial,
but as the mean increases the departure becomes more marked. It can be seen
that the observations lie more snugly about line 3 from(6) than about line 2
from (3). Generally, for the data from field studies the same effecthas been
observed. Such resultssuggestthat (6) may be generallya betterapproximation
to the formoff(M) than (3) and make it preferableto proceed with the analysis
of data fromthe assumption (6).
4.
THE TRANSFORMATIONS OF FIELD
DATA
Fig. 1 shows clearly how, within an area, the variability of the numbers of
insects on sub-areas is related to the mean numberof insectsper sub-area. This
relationshipwill make invalid the use of the analysis ofvariance on experimental
GEOFFREY
BEALL
249
results involving counts on insects, since the expectation of the variance
should be the same for all plots. To overcome this invalidity,Bartlett (1936a)
suggestedtransforming
the observations,x, fromthe basis of (3). The transformation found was xi, which Bartlett modifiedto (x + J)k. From ? 3 it was seen,
however, that for field data the relationship between standard deviation and
mean may be representedbetterby equation (6) than by (3), and, since the form
of the transformation
depends on the formoff(M), a freshtransformation
must
be sought.A transformation,
as is developed in the Appendix to the presentpaper,
is suggestedby the method of Tippett (1934), i.e.
x' = k-i sinh-1(kx)i.
(10)
An advance note of this transformationwas published by Beall (1940). The
adequacy of this transformationmust be judged from the extent to which it
stabilizes variability. In (10), if we express sinh-1(kx)k,when kx < 1, as a welknown series,we have
xi=xi-= k2xl
(11)
kxi+ 430- 5X2kx+...,
where it is obvious that for k = O, x' = xi. Of course, for large values of kx,
x' varies almost as log x, or as the log (x+ 1) used by Williams (1937), and so our
proposed expansion may be regarded,for practical purposes, as embracing the
root and logarithmictransformations.
Table 8 gives the transformation,(10), fora probable range of observations,
x, and fork at intervalswhichwillprobablybe close enoughforpracticalpurposes.
This table was computedin part by inverseinterpolationfromthe table ofhyperbolic functionsof the SmithsonianMathematicalTables (Becker & Van Orstrand,
1931), and in part from (12). Should values of x' be required outside those of
Table 8, these can convenientlybe calculated from
x
=
k-i loge
{(kx)i + (1+ kx)i}.
(12)
In preparingTable 8 the question arose of whether,instead of dealing with
k-i sinh-1(kx)k, one should not use k-i sinh-l {kk(x+ J)k} in the same way as
Bartlett (1936a) dealt with the transformation,(x+ 1)i, instead of xi. This
modificationwas rejected on the basis of results of the transformation,as disand
cussed in ? 5, since it was foundthat the addition of 2 made little difference
did not give, consistently,an improvement.
For fielddata, in making the transformation(10), it is necessaryto estimate
the value of k empiricallyby (9) forwhich estimates x, of the mean and s, of the
standarddeviation,mustbe found.The mostobviousmethodin practiceofmaking
these estimatesseems to be to put more than one plot subjected to a given treatmentin a block and so to estimatethe chance variation of resultsfora plot within
a block. In the presentwork,as is discussed in ? 2, two plots were subjected to a
given treatmentin each block and this is probably good practice.
Transformationof data from entomologicalfield experiments
250
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BEALL
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Biommetrika
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252
Transformationof data from entom,ological
field experimbents
In the special case wherethereare two plots forthe ith treatment(i =
,n)
in thejth block (j = 1, ..., N), and so two observations,xjl and Xij2 the estimate
of the mean will be writtenxij and of the squared standard deviation
(13)
S9) = 2(Xijl-Xij2
Then from(9), we estimate k from
k= 2
(in
N
EE(xijl
n
-xij2)2-
N
j(xijl
i(
+ xij2)j
n
N
Xjl 1+
A-1
Xi2)j
(14)
and the calculation is very light.
5.
RESULTS
SHOWING
ON THE
THE
EFFECT
VARIABILITY
OF THE
TRANSFORMATION
OF DATA
Trheadequacy of our proposed transformationmay be judged in two ways:
first,with respectto its effect,which we shall considerin the presentsection,on
the differences
between repetitionsof a treatmentwithina block, and secondly,
with respect to its effect,which we shall considerin ? 6, on the behaviour of the
quantities submittedto the analysis of variance.
It is a fundamentalassumption in the analysis of variance that the chance
variabilityforeach plot shall be, when the effectof block and of treatmentare
removed, normallydistributedwith a standard deviation common to all plots,
in which situation of course the standard deviation of the chance variabilityfor
a given plot is independentof the expectation forthat plot. In the data of the
presentwork, where each treatmentis repeated in each block, it is possible to
examine the estimates of this standard deviation,8ij, and of the expectation,x,j..
For a clear graphical illustrationofthe situationconsiderFig. 2, as obtained from
the original data of Experiment III on Leptinotarsadecemlineata,where sij is
plotted against xij, and contrast this situation with that obtaining for the
correspondingquantities s'j and x4j, obtained after transformation(k = 008)
in Fig. 3.
In Fig. 2 the points are widely scattered as is natural froma sample of two;
nevertheless,it is apparent that forthe smallest values of xij the values of s8
are correspondingly
small and fall in a close group. In Fig. 3 the clusterof observations in the lower left-handcornerof the previous diagram has disappeared,
and generallythe scatter appears to be independentof x4j,so that apparently
the transformation
gave satisfactoryresults.The nature of the material involved
that
it
is such
does not seem possible to examine the relationshipunder considerationmore exactly, nor to summarizeexactly the correspondingresultsfor
the other treatments;it can only be said that the same type of result appeared
although the magnitudeof the relationshipbeforetransformationdepended on
the magnitudeof the differences
between the effectsof treatments.
The resultsshown in Figs. 2 and 3 suggestthat the proposed transformation
has tendedto make thestandarddeviationiindependentofthe mean,in accordance
GEOFFREY
253
BEALL
theanalysisofvariance.In usingthisprocedure
underlying
withtheassumptions
one actuallyassumes,morebroadly,thata commonstandarddeviationexists,so
should
beforeand aftertransformation
ofobservations
thatthehomoscedasticity
be tested.Thus it is assumedthatxijl and xij2 are observationsfroma normal
200-
0
50
100-
*
50-
0
00
0
200
100
Xij
400
300
500
Fig. 2. The standarddeviationand meanas estimatedfromplotsby pairs,
data on Leptinotarsa
decemlineata.
withuntransformed
2-0
.
1*5(J*
1.0~~~~
0
1*0-
0
*~~~~~~
0.5-~~~~
05
*
0
2
0
*
4
*
/
Xijj
6
*
@0
8
Fig. 3. The standarddeviationand meanas estimatedfromplotsbypairs,withthe transformed
data on Leptinotarsa
decemlineata
Say, i.e. usingx' = k-i sinh-' (kx)1,(k = 0 08).
ofi and j. Then
populationwitha standarddeviation,o, whichis independent
_(x
(Zzk-
)2}-2
is
as x2 withonedegreeoffreedom.tAccordingly,
distributed
t In theL1 test,discussedby Nayer(1936),thiscase ofestimatesofstandarddeviationwith
sincezerovaluestendto arisewhendealingwithgroupedor
is troublesome
one degreeoffreedom
to a geometric
integralobservations.Whenthisis the case L1, whichis theratioofan arithmetic
have a wider
maytherefore
meanof sumsofsquares,cannotbe calculated.The presenttreatment
atrnlication.
I7-2
254
Transformationof data from entomologicalfield experiments
yi=
(x6ijl- xj2)/V2o- should be distributednormallywith unit standard deviation forall i and j. In orderto test the hypothesisofnormalitywithunit standard
deviation it is only necessaryto test forleptokurtosis;forthe distributionmust
and thereforeof yij, is a matter of
be symmetricalsince the sign of differences,
chance. Since the numberof items involved will almost certainlybe < 100, and
since the population,mean is zero, the wncriterionofGeary (1935) will providean
appropriate test. In using this criterionwe must find the ratio of the mean
deviation to the standard deviation,i.e.
Wn
nnN
=
n
(
I xjl -Xij2
i=lj=l
N
_
EE Xj2)
(Xijlf nNi=lj=1
(15)
Of course,values of w,,may be calculated fortransformeddata by substitution
of XZjk for Xijk.
Table 9. The w. teston thehomoscedaisticity
ofcountsbyplots
a
six
within blockfor fieldexperiments
Untransformed
|
{Experi-
mxent anN
I
II
III
IV
V
VI
70
70
28
24
18
36
Upper
Lower
lImit
limit
0-841
0 841
0 866
5%
0 872
0 881
0 857
5%
Transformed
(Beall)
(Bartlett)
-
_
_
-
_
_
_-
_
Wn
Departure
by S.D.
wn
0 757
0 757
0 737
0 6659
0 7885
0 5973
- 5 33
- 0 49
- 5 33
0 7525
0 7807
0 6431
0 728
0 745
0 7554
0 7959
-1 16
- 0 22
0 8155
0 8166
0 732
V
Transformed
0 5692
- 5 67
0 6948
-_
_
-
-
_
-
Departure
by S.D.
Estimated
Employed
n
Departure
by S.D.
191
079
-415
0 078
0 046
0 084
0 08
0 04
0 08
0-7846
0 7499
0 6838
- 0-64
- 2 01
- 3 11
0 08
0 02
0-7823
0 8130
-
- 267
0 285
0 082
+0 13
+0 38 1 0 019
0 30
0 7370
- 166
-0 58
+0 28
Values of w.,fromthe untransformedobservationsand fromthe transformed
observations, both following Bartlett (i.e. the transformation(x + )i) and
followingthe line suggested in the present paper, are shown in Table 9 for the
thevalues ofk as calculated
fielddata ofTables 1-6. For the secondtransformation
from(14) are shown as well as the nearest value of k entered in Table 8. For
each experimentthe value of nN and also the 0 05 limits of probability,from
Geary (1935), are shown. There are also shown the departuresof observed Wn
from the expected value in termsof the standard deviation, a useful criterion
since the distributionof wn is almost normal. From Table 9 it can be seen that
out of the three experimentsin which Wn fell beyond the lower 5 % limit of
probability for the untransformeddata and the data transformedas (x+ 1)i,
in only one experimentdid Wn fall so with the finaltransformation.The results
forExperimentII, in which Wn is decreased by the transformation,
are peculiar.
Considerationof the departuresfromthe mean in termsof the standard deviation indicates more clearly the improvementeffectedby each transformation
GEOFFREY
BEALL
255
and how the transformationsuggestedin the presentwork secures an improvement of the same, but more marked, character than that secured from the
of Bartlett. The resultssuggestthat whilehomoscedasticitymay
transformation
not be attained always, it will be approached by means of the proposed transformation.
6. THE EFFECT OF THE TRANSFORMATION ON THE ANALYSIS
OF VARIANCE
besides
As was indicatedat the beginningof ? 5, our proposedtransformation
makingthe variabilitywithina block fora repeated treatmentthe same for all
treatmentsand blocks, should also provide quantities satisfyingthe assumptionsunderlyingthe analysis of variance. Since it is not quite clear how,in so far
as the transformation
is satisfactoryin the firstway, it will necessarilybe satisfactoryin the second, it will be well to considerdirectlythe suitabilityof our
transformedvalues forthe analysis ofvariance.
In the application of the analysis of variance one would deal with xij rather
than withXijk and suppose that
(16)
xii. = A + Bi + Cj + Dij,
whereA is a contributionfromthe general level of population on the experimental area, Bi the contributionof the ith treatmentand Cj the contributionof
thejth block. The remainderterm,Dij, is called the interactionof treatmentsand
blocks. Of course, the present discussion on the untransformedvalues, xij,
holds for the transformedvalues, xj = (xt1 + x*j2) when the appropriate
symbols,A', B*, CQand Dj are used.
In material satisfyingthe conditionsunderlyingthe analysis of variance, for
the observationsunder each treatment,the calculated squared standard devia1 N
tion is
)2.
(17)
s N-1 jl(x,j.
Following the argumentof the analysis of variance, xj - xi, of which the mean
is 0, is an estimateof Cj + Dij, in whichthe two termsare independent;hence the
expectationofS2is
C2 =
0-2
(18)
where ocj and oDj are the standard deviations of the parameters,Ci and Dij,
respectively,and are independent of treatment.Accordingly,si should be independentoftreatmentand distributedas an estimateofoi, havingN - 1 degrees
of freedom. Conversely,if xij. cannot be built of the independenttermsof (16),
then the various values of si will not be distributedas estimates of a single
standard deviation. The hypothesisthat the values of si in any one experiment
are estimates of one quantity may be tested.t
withDr R. W. B. Jackson,thewriterhas learnedthathe had arrived
t Fromcorrespondence
at the same test.
independently
256
Transformationof data from entomologicalfieldexperiments
The results of the tests on the homogeneityof the values, si, withinthe six
experimentstreated in the present paper are presentedin Table 10, where the
value of the L1 criterionis shown forthe original data and forthe transformed
values togetherwith the appropriate0*05and 0.01 levels of probability(Nayer,
1936). From Table 10 it can be seen that of the values of L1 obtained fromthe
original data, all but one are near or beyond the 0 05 level of significance,but
that aftertransformationall are moved in to less significantvalues. Accordingly,
the values ofsi, whencalculated fromthe originaldata, appear heterogeneousbut
the correspondingvalues obtained afterthe transformation
appear homogeneous.
Thus it is more probable that the analysis of variance is applicable to the transformeddata than to the untransformed.
Table 10. The homogeneity,
as measuredby thecriterionL1, of theestimates
si for various values of i beforeand aftertransformation
Experiment
1
LI beforetransformation
LI aftertransformation
1 % limit
5 % limit
0 867
0 864
0 757
03812
2
0
0
0
0
833
941
757
812
3
4
0.325
0 766
0 604
0 707
0
0
0
0
657
688
542
656
[5
0 344
0 813
0 514
0 648
6
0
0
0
0
680
730
583
673
In Table 10 we have tested the homogeneityof the estimates,si, as in ? 5 we
tested the homogeneityof sij, that is without referenceto the values of the
associated means. In view ofour originalassumptionswe are, however,interested
in the possibilitythat the standard deviations, as calculated, might show every
sign of being estimates of a common standard deviation and yet be dependent
on the associated means. Accordingly,we have investigated such dependence
roughlyby fittingby least squares a firstorderregressionof si on xi.. From this
fittingwe recordthe sign of the regressionas follows:
Experiment
Before transformation
Aftertransformation
I
1
2
31
+*
+
+*
-
+
-
I
4
5
6
+ **
+
+
+
+
-*
GEOFFREY
BEALL
257
By a single asteriskwe have indicated cases where the reductionin variability
effectedby the regressionpassed the 5 % probability limit and by a double
asteriskwhereit passed the 1 % limit. Several points may be noted. (1) In two
cases (Experiments 2 and 6) after transformationthe residual sun1 of squares
about the regressionwas greaterthan the reductionin squares due to the regression, whereas it was consistentlyless beforetransformation.(2) As can be seen
above, the regressiongenerallydid not effecta significantreductionin variability
aftertransformation
but did before(the small numberofdegreesoffreedommade
high significancedifficultof attainment). (3) Aftertransformationthe sign of
the regressionseemed to be a chance matter,whereas beforetransformationit
was consistentlypositive. These results suggest that the transformationproposed did tend to make the variabilitywithina given treatmentindependentof
the mean forthat treatment.
7.
THE EFFECT OF TRANSFORMATION UPON THE CONCLUTSIONS
FROM THE ANALYSIS OF VARIANCE
It has been shown in ?? 5 and 6 that the analysis of variance can be made on
entomologicaldata when a suitable transformationhas been effected.It is of
willhave upon
practicalinterestto see what numericaleffectsuch transformation
tests on the significanceof, say, the effectof treatmentand the significanceof
differences
fortreatments.
First, consider the numerical results to be obtained from the analysis of
variance (1) withoutand (2) with transformation.Thus the mean square ascribable to blocks, treatmentsand their interactionis shown in Table 11, for six
experimentsof which the data are given in ? 2; parallel resultsare presentedfor
untransformedobservationsand forobservationstransformedby (10) with the
values of k fromTable 9. To facilitatethe comparisonof the results,the mean
square for blocks and for treatmentsis expressed in terms of the estimate for
interaction,as the F of Snedecor (1934), and presentedin each case. The transformationof the data has modifiedthe conclusionsto be drawnfromthe analysis
ofvariance in Table 11, in that thereare considerablechangesin the criterion,F,
fortreatmentsor forblocks. In the examples shownthe effectof treatmentswas
highlysignificantin all cases and so the changes introducedby transformation
did not alter the conclusions,as would have been the case forless definiteeffects.
Consider next the effectof transformationon the significanceof differences
between the means fortreatmentsas tested by the criterion,t, calculated with
such estimates of mean square as the interactionof Table 11. For illustration,
values of t, fromthe data on Leptinotarsadecemlineata(Experiment III), are
shownin Table 12 foreach possible comparisonoftreamentswhenuntransformed
data are used, when the transformation,
(x + 1)i, as suggestedby Bartlett (1936a)
is used, and when the transformation,
k-i sinh-1(kx)i, as suggestedin the present
paper is used. In orderthat the influenceof the level of population under each
Table 11. The analysis of variance of untransformed
and
transformed
data in six experiments
D egrees
Variation
Untransformed
data
of
freedom
__
_
_
_
__
Transformed
data
_ _ _ _ _ _ _ _
Mean
Mean
square
square
_ _ _
F
Betweenblocks
Betweentreatments
Interaction
ExperimentI. P. nubilalis
9
92-8
1 21
6
2,839-0
36*95
54
76-8
0-565
7-51
0-341
1 66
22-03
1
Betweenblocks
Betweentreatments 1
Interaction
ExperimentII. P. nubilalis
9
577-0
6-72
6
1,721-0
20-04
54
85-9
537
8-69
0 509
10-55
17-07
Betweenblocks
Betweentreatments
Interaction
ExperimentIII. L. decemlineata
6
20,172-0
2-26
3
390,932-0
43-77
18
8,931-0
4-67
111-5
1 54
3 03
72-16
Betweenblocks
Betweentreatmients
Interaction
ExperimentIV. L. decemlineata
5
12,960.0
2-06
3
124,054.0
20-40
15
6,7270
2-06
20-40
1-05
1-96
19-41
Betweenblocks
Betweentreatments
Interaction
ExperimentV. P. quinquemaculata
5
27.4
2*74
0*349
2
752-0
75.33
19.5
10
9-98
0-083
Betweenblocks
Betweentreatments
Interaction
ExperimentVI. P. quinquemaculata
5
41-7
4*13
5
66*2
6*55
25
10*1
-
1.50
3*33
0*365
4-19
233 77
-
412
9*14
-
Table 12. The values of t, in the comparisonof means, as calculatedfrom the
and thetransformed
data of ExperimentIII on Leptinotarsa
untransformed
decemlineata
urMeansfor
data
Comparison
IXl. -x4..
x2 -x3 .
IX2..-x4. .
X3__-X4
1
[
f
362 30
362 224
362 11
30 224
f30 11
224 11
t without
transformation
+9.27**
+3.84**
+ 9.82**
- 5.43**
+0 54
+5.98**
t from
(x+ -1)i
t from
k-I sinh-I(kx)i
+11-33**
+ 3.52**
+ 12.89**
+ 9.92**
+ 2.11*
+ 12.46**
- 7.81**
+ 1*56
+ 9.37**
- 7.81**
+ 2.54*
+10.35**
259
GEOFFREY BEALL
treatmentmay be judged, there are shown, also in Table 12, the means forthe
untransformeddata. The values of t fallingbeyond the 0.01 level of significance
have been marked with two asterisksand the values beyond the 0 05 level with
one. It can be seen that the transformationresultedin a profoundalteration in
the conclusions.Apparentlyon account of the dependence of variance on mean
in untransformeddata, the pooled estimate of variance was originallytoo low
forthe treatmentswhich resultedin high populations and too high forthe treatments which resulted in low populations. Thus, in the comparison of the first
and thirdtreatments,which appeared to have the two highestsurvivingpopulavalues was high. In the other
tions,the value of t calculated fromuntransformed
extreme case, the comparison between the second and fourthtreatments,the
values was very low. It can be seen
value of t,as calculated fromuntransformed
further,that the firsttransformationonly secured in part the modificationin
the value of t that was secured by the second transformation.
OF TRANSFORMATION IN PRACTICE
THE PROCEDURE
8.
The methods which were found applicable in the preceding discussion will
ofthe data shownin Table 7 (Experinient
now be illustratedin the transformation
quinquemaculata,of the same type as the experiments
VII) on Phlegethontiu8
previouslydiscussed in the presentpaper. The steps in the analysis will be set out
withthe purpose of providinga model forprocedurein estimatingthe constant,
k, which will be used to effecta transformationof the data so that the analysis
of variance may be made.
Supposing that the experimenthas been laid out with a repetitionof each
treatmentin each block, the procedureof estimatingk makes it firstnecessaryto
findthe sum and the absolute difference
of each pair of plots subjected to a given
n
treatmentin a givenplot andthen to sum the sums, z
squared,
nN
N
i=l j
z (xijl
+ xij2), and the sums
n
+ xiJ2)2, and also the differencessquared, E
X(jl
(
.N
(Xijl
-
Xi2)2,
over all such pairs and by substitutingthe resultsin (14) to findk. In the case
being used for an illustrationthe two plots subjected to the firsttreatmentin
each block gave respectively10 and 7, 20 and 14, 14 and 12, 10 and 23, 17 and 20,
14 and 13, so that
N
Similarly,
(Xljl + Xlj2) = (IO + 7) + (20 + 14) + (14 + 12) +
N
z
j=1
(Xljl + Xlj2)2
=
(10 + 7)2 + (20
+ 14)2+...
= 5308
and similarly,
N
(Xljl-Xlj2
)2 =
(10
-
7)2 + (20-14)2+
= 228.
174.
260
Tran8formationof data from entomologicalfield experiment8
Of course, in estimatingk the summationsare not limitedto one treatmentbut
must be extended over all in the experiments.If this is done we find
n
N
i=l j=l
n N
(X1jl+ xij2) = 684,
E
i=l j=l
j(xjl +xj2)2 = 19,656,
k = 2(708 -684)
19,656
From (14) we estimate
n
N
E
E
i=l j=l
(x j
-
Xij2)2 = -708.
0002,
and referring
to Table 8, p. 250, use k = 0*00as the nearestvalue occurringthere.
Of course,in this case, the transformationis simplyxi.
Now fromthe above resultit will be possible to replace the observedvalues of
Table 7 with the correspondingtransformedvalues from the firstcolumn of
Table 8. Thus in Table 7 replace in the firstrow: 10, 20, 14, 10, 17 and 14, by
3'16, 4.47, 3*74,3.16, 4412and 3-74. With such transformedvalues we can now
proceed to carry out a routineanalysis of variance which will be facilitatedby
workingwiththe sum foreach pair ofplots in a givenblockwitha giventreatment.
For example, the finalanalysis of variance forExperimentVII would be carried
out with the values of Table 13.
Table 13. Transformedand summedvalues to be used in theanalysis of
variancefor ExperimentVII on P. quinquemaculata
Block
Treatment
1
2
3
4
5
6
5-81
7.44
1 00
3*97
3*97
6*32
9.
2
3
4
8*21
7.90
4-06
5-91
3 97
8-56
7-20
7*74
2*73
3.73
4-18
7*87
7*96
8-24
2*41
4-48
2-0)
6 77
5
8-59
8-94
1-73
4-48
3 14
i 10-20
6
7.35
6-26
3 00
3-41
4 45
8-51
SUTMMARY AND CONCLUSIONS
The foregoingworkis a study of experimentalresultsfromseven fieldexperiments on the control of insects. In such data, the standard deviation of the
number of insects per plot varies with the mean. By the transformation,
x' = k-I sinh-1(kx)1,where k is a constant and x an observation,the data were
put in a formforwhichthe standard deviationapproached a constantindependent
of the mean. The estimationof the one constant,k, necessaryforthe transformation was made possible by the design of the experimentswith repetitionof treatments withinblocks. In practice, the transformationgave good results so that
GEOFFREY BEALL
261
analysis of variance could be made. From the analysis of the transformeddata,
the results were found to differmarkedly fromthose which would have been
obtained fromthe untransformeddata.
ACKNOWLEDGEMENTS
The presentworkwas suggestedto the writerby Prof. E. S. Pearson to whom
the writeris furtherindebted for advice as the work progressed.The writeris
furtherbeholden to Dr R. W. B. Jackson, to Dr B. L. Welch and to Dr M. S.
Bartlett for criticismand advice. Dr G. M. Stirrettand Mr F. A. Stinson very
kindlypermittedthe use of unpublisheddata fromtheirexperimentscarriedout
at Delhi, Ontario,on Phlegethontiu8
quinquemaculataHaw.
APPENDIX
As has been said, the transformation
of (10) was suggestedby the methodused
by Tippett (1934, p. 61). The procedureis as follows.
It is required to find x' = f(x), such that the standard deviation, yxof x',
shall be approximatelyconstant. Let us write
x' = f(M)+f'(M)
(x-M)+.*.,
(19)
whereM is the expectation of x and whence, approximately,
(x'-M') = f '(M) (x-M),
whereM' is the expectation of x'. Hence
o02, = {f
'(M)}2
o2,
(20)
(21)
where o-is the standard deviation of the observations,x. Replacing ox in (21)
by a constant,c, as is the purpose of our operation,and suibstituting
forC from
equation (6), p. 247, we have
(22)
f '(M) = c(M + kM2)-A,
where k is, as has been previously discussed, a constant peculiar to our data.
Integratingin (22),
f(M) = 2ck-isinh-1(kM)i.
(23)
From (23) the formof the functionsuggestedis sinh-1(kx)i, but it is wise instead
to use k-I sinh-1(kx)1,since the transformation
thenbecomes identical,as shown
in (11), with the established transformation,
xi, when k = 0.
As Tippett (1934) says: 'This derivation is not mathematicallysound, and
the result is only justifiedif on application it is found to be satisfactory.' The
writerwould have hesitated to have used it had it not already led to useful
transformationsin cases analogous to the present,namely to xi where x comes
froma Poisson distribution,to sin-lpi wherep comes froma binomial distribution and, accordingto Tippett, to tanh-1r. wherer is the correlationcoefficient.
262 Transformation
ofdatafromentomological
fieldexperiments
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