j?*^
ff OF THE
(( UNIVERSITY
OF
Statistical Laboratory
Dept. of Mathematics
University of California
! F _ CCUPLPTE I £ B £
of
; A R T K U R H A R R I
Volume III
191? - 1921
Property of
Statistical Laboratory
University of California
MATH-STAT.
Volume III
MATH..
1 q STAT.
— LIBRARY
1. Further Illustrations of the Applicability of a
Coefficient Measuring the; Correlation Between a Variable
and the Deviation of a Dependent Variable Frorr; Its Pro
bable Value.
Genetics 31 32° -3^2, July, 191?.
2. Practical Universality of Field Heterogeneity as
a Factor Influencing Plot Yields.
Journal of Agricultural Research. Vol. XIX, No. 7,
July, 192C.
3. Permanence of Differences in the Plots of an Experi
mental Field. (J. Arthur Harris and C. f. Scofield) .
Journal of Agricultural Research. Vol. XX, No. 5,
Dec.. 1020.
4. Tissue '"eight and Water Content in a Tetracotyle-
donou? Llutant of Phaseolus vulgaris.
Proceedings of the Society for Experimental Biology and
Medicine. 1921, XVIII, pp. 2C-7-2C9.
5. Leaf-Tussue Production arid VJater Content in a
ivlutant Rac<~ of F^ llif_Ji_ li.-
Botanical Gazette. Vol.LXXII, No. 3, S€jt. , 1921.
6. Correlations Between Anatomical Characters in the
Seedling of Phase plus vulgaris. (*J . Arthur Harris,
Ecjmuncl W. Gii.r.ott, John Y. Pennyp. . and G. B. Durham)
American Journal of Botany, °: 339-36:;, July, 1921.
7. The Vascular Anatomy of H^m.i trimerous Seedlings of
Phase clus vi \ ^_s_. (J. Arthur Harris, Edmund VT. Sinnctt
. Y. Pennypacki >•. anr1 G . 3. Durham).
Journal of Botany, °: 375-3^1, ::t.. 1921.
°. "nterrelati 1 Lp of "
-.s of Vascular Bundles ' ' Trar: ' " n Zone of the
'^ Phaseolus Vulgaris. (J. Arthur Harr ' ' ".
lot, John Y. Pennypac'-er, an^ '" . B. Di c*ha ).
Journal of Botany, °: - 2, fov. , 1921.
M779976
Contents, Volume III
. e 2.
•9. The Vascular Anatomy of Dimerous and Trimerous
Jllngs of Pha£_ _ luj ___ l__is.. (J. Arthur Harris,
Edmui." ''. -'Lii'iott, John Y. Pennypacker, and G. 3. Durham)
American Journal of Botany, P: 63-102, Feb., 1921.
10. The Vascular Anatomy of Normal and Variant Seed
lings of Pliaseolus vulgaris.
Edmund T"r. "innott) .
Procer . ' of the National Acadeny el rciences, Vol. 7,
No. 1, Jan., 19-"1 .
11. Correlation of Morphological Variations in the
Seedling of ?„_. ___ 1 ___ __L___l£- (3 • Arthur Harris and
B. T. A very" .
Bulletin of the Torrey Botanical Club 4$: 109-119.
12. .-ther ftudies on the Interrelationship of Morphc-
ical and Physiological Characters in Seedlings of
Phase olus .
Bro: lyn Botanic Garden Memoirs, 1: 167-174, June, b.
13. Note on the Relation of Blood Fat to ex, and on
the Correlation Between Blood Fat and ^gg Production
in the Domestic Fowl. (Oscar Biddle and J. Arthur Karris
urnal of Biological Chemistry, Vol. XX7IV, I'o. 1,
ril, 19GL&3
14. The Fgg Record of Limited Periods as Criteria for
^dieting the Egg-Production of the V/hite Leghorn Fowl.
(J. Arthur Harris, ' kpatrick, A. F. lakeslee,
.
6: 26:1-309, May, 1921.
15. The Prediction of Annual Egg Production From the
cords of Limited Periods. (J. Arthur Harris, IV. F.
"' -kpatr-' F. Blake s lee ).
oceed"' J ' onal Academy of Scirnces, Vol. 7,
T-i-iT-ii ] -.pi
c - _ . j. , C- 4- .
16. A Bicmetric ,ctudy of Human Basal Metabolism.
(J ' and Francis G. Benedict).
Proceedings of the National Academy of rc-' &; Vol. 4,
17. Biometric Standards for Energy Requirement 5-
triti: (J. Arthur Iferri.
' '0.
cientific Monthly, May,
Contents, Volume III
Page 3.
18. Charles Buckman Goring.
Science, N. f: . , Vol. LI., Ho. 1310, Feb., 1920.
19. The Variation and the Statistical Constants of
Basal J.Ietabolism in Hen. (J. Arthur Harris and F. G.
Benedict) .
The Journal of Biological Chemistry, Vol. XLVI, No. 1,
March, 1921.
20. On the Osmotic Concentration of the Tissue Fluids
of Phanerogamic Epiphytes.
American Journal of Botany, ';:, November, 19lB.
21. On the' Relationship Between Freezing Po^'nt Lowering,
A . and Specific Electrical Conductivity, k, of Plant
Tissue Fluids . (J.A.Harris , R.A. Gortner, J.V. Lawrence).
Science, N. S., Vol. LII., No. 13;1, Nov., 1920.
22. The Specific Electrical Conductivity of the Tissue
Fluids of Desert Loranthaceae . (J. Arthur Harris and
A. T. Valentine).
Proceedings of the "ociety for Experimental Biology and
Medicine, "1920, xviii.
23. On the Differentiation of the Leaf Tissue Fluids
of Ligneous and Herbaceous Plants V/ith Respect to
Osmotic Concentration and Electrical Conductivity.
Journal of General Physiology, Jan., 1921., Vol.iii,
No. 3. (J. Arthur Harris, Ross Aiken Gortner, and John
V. Lawrence) .
24. The Osmotic Concentration and Electrical Conducti
vity of the Tissue Fluids of Ligneous and Herbaceous
Plants. (J. Arthur Harris, Ross Aiken Gortner. and John
V. Lawrence) .
Journal of Physical Chemistry, Vol. 25, Feb., 1921.
25. Maximum values of Osmotic Concentration in Plant
Tissue Fluids. (J. Arthur Harris, P. A. Gortner, W.F.
Hofmann, and A. T. Valentine).
Proceedings of the ^ociety for T xperimental Biology and
Medicine, 1921, xviii.
26. The Transformation of the Plant Ovule into an Ovary.
Proceedings of the Society for T xperimental Biology
and Medicine, 1919, xvi .
27. rormulae for the Determination of the Correlations
of cize and of Growth Increments in the Developing Organ
ism.
Proceedings of the Society for Experimental Biology and
... dicine, 1921, xviii.
Contents, Volume III
Page 4.
28. Inter -Periodic Correlation in the Analysis of
Growth. (J. Arthur Harris and H. S. Reed).
Biological Bulletin, Vol. XL., No. 5, Hay, 1921.
29. Notes on the Occurrence of Gammer us llrnnaeus Fmith
in a Saline Habitat. (Ross Aiken Gortner, and J. Arthur
Harris )
Science* N. S., Vol. LIII. , No. 1376, May, 1921.
30. Secondarv Parsitisra in Phoradendron.
The Botanical Gazette, Vol. IXVI, No. 3, Sept., 1918.
31. The Interrelationship of the Number of Stamens and
Pistils in the Flowers of Ficaria.
Biological Bulletin, Vol. XXXIV, No. 1, Jan., 1918.
32. The Second -Year Record of Birds Y/hich did and. did not
Lay During Individual Months of the Pullet Year. (J. Arthur
Harris and Earry R. L^wis).
Science, N. S., Vol. LIV. . No. 1393, Sept. 9, 1921.
FURTHER ILLUSTRATIONS OF THE APPLICABILITY OF A
COEFFICIENT MEASURING THE CORRELATION BE
TWEEN A VARIABLE AND THE DEVIATION OF
A DEPENDENT VARIABLE FROM ITS
PROBABLE VALUE
J. ARTHUR HARRIS
Station for Experimental. Evolution, Cold Spring Harbor, New York
Reprinted from GENETICS 3 : 328-352, July 1918
GENETICS
A Periodical Record of Investigations Bearing on
Heredity and Variation
EDITORIAL BOARD
GEORGE H. SHULL, Managing Editor
Princeton University
WILLIAM E. CASTLE EDWARD M. EAST
Harvard University Harvard University
EDWIN G. CONKLIN ROLLINS A. EMERSON
Princeton University Cornell University
CHARLES B. DAVENPORT HERBERT S. JENNINGS
Carnegie Institution of Washington Johns Hopkins University
BRADLEY M. DAVIS THOMAS H. MORGAN
University of Pennsylvania Columbia University
RAYMOND PEARL
Johns Hopkins University
GENETICS is a bi-monthly journal issued in annual volumes of about
600 pages each. It will be sent to subscribers in the United States, the
Philippines, Porto Rico, etc., for $6 per annum for the current volume,
and $7 per volume for completed volumes until the edition is exhausted.
Canadian subscribers should add 25 cents for postage. To all other
countries 50 cents should be added for postage. Single copies will be
sent to any address postpaid for $1.25 each.
All subscriptions, notices of change of address, and business corre
spondence should be sent to the Princeton University Press, Princeton,
New Jersey, and all remittances should be made payable to the Princeton
University Press.
Entered as second-class matter February 23, 1916, at the post office at
Princeton, N. J., under the act of March 3, 1879.
FURTHER ILLUSTRATIONS OF THE APPLICABILITY OF A
COEFFICIENT MEASURING THE CORRELATION BE
TWEEN A VARIABLE AND THE DEVIATION OF
A DEPENDENT VARIABLE FROM ITS
PROBABLE VALUE
J. ARTHUR HARRIS
Station for Experimental Evolution, Cold Spring Harbor, New York
Reprinted from GENETICS 3 : 328-352, July 1918
FURTHER ILLUSTRATIONS OF THE APPLICABILITY OF A
COEFFICIENT MEASURING THE CORRELATION
BETWEEN A VARIABLE AND THE DEVIA
TION OF A DEPENDENT VARIABLE
FROM ITS PROBABLE
VALUE
J. ARTHUR HARRIS
Station for Experimental Evolution, Cold Spring Harbor, New York
[Received November 1, 1917]
TABLE OF CONTENTS
PACK
INTRODUCTORY 328
Earlier applications 3 3 j
Further illustrations 332
Illustration I. Proportionality of parts in Paramecium 332
Illustration 2. Absence of relationship between size of litter and sex in swine 336
Illustration 3. Proportion of pistillate and hermaphrodite flowers in the
inflorescence of the composite Homogyne 336
Illustration 4. Fertility of capsules and viability of seed in carnation crosses 337
Illustration 5. Relationship between the total number of pedicels and the
number of abnormal pedicels in Spiraea Vanhouttci 337
Illustration 6. Interrelationship of cotyledons and primordial leaves in a race
of Phaseolus vulgaris highly variable in seedling characters 340
Illustration 7. Change in proportion of parts in developing trout 344
Illustration 8. Relationship between total solids and sucrose content in the
j uice of sugar beets 346
Illustration 9. Relationship between total number of spikelets* and number
of sterile spikelets in wheat 348
Illustration 10. Viability of dominants and recessives in F0 generation of
Mendelian hybrids " 3^
RECAPITULATION --
LITERATURE CITED .
3o i
INTRODUCTORY
Eight years ago I pointed out (HARRIS 1909 a) that wheels some
fraction of x the correlation between them, rxv, while of descriptive value,
GENETICS 3: 328 Jl 1918
APPLICABILITY OF A COEFFICIENT OF CORRELATION 329
does not give all of the information which is required concerning the
interrelationship of these two variables, and that a coefficient showing
whether the value of y becomes relatively larger or smaller with increas
ing values of x} would have considerable analytical value.
I showed1 then that if s = px, where p = y/lc, the bars denoting pop
ulation means,
Vx— TI
The purpose of this paper is to illustrate the range of usefulness
of this coefficient by noting biological progress which has been made by
its use, and by actually applying it to series of data which have not been
heretofore fully analyzed by the higher statistical methods.
Before passing to actual illustrations of applicability of the coefficient,
some questions of method should be considered.
In cases in which the coefficient rxz is desired the correlation table
has usually been formed to determine rvu, where v -\- y = x. The mean
and standard deviation for x made up of any number of components
(HARRIS 1917 c) are well known (PEARL 1909, HARRIS 1918 a) ;
when x = v + y
•x = v + y, vx = V o-v2 + a-y + 2 rvy.r] is the first moment about zero as
origin of z for any array, the product moment for the population is
where S denotes summation of the values of the 3' arrays of x.
Practically it is more convenient to determine the product moment
from
where S(xy) and S(S) are the product moments of x and y and the
second moment of x for the population.
APPLICABILITY OF A COEFFICIENT OF CORRELATION 331
EARLIER APPLICATIONS
The method has been most extensively applied to problems of fertility
and fecundity. Thus the relationship between the number of ovaries
formed and the number of ovaries developing into fruits has been investi
gated in the inflorescence of Staphylea (HARRIS 1909), Celastrus (HAR
RIS 1909 b) and Crinum (HARRIS 1912). In Staphylea and Crinum
inflorescences which produce larger numbers of flowers mature relatively
fewer fruits. In Celastrus there is apparently no relationship between
the number of flowers formed and the capacity of the inflorescence for
maturing the ovaries into fruits.
In the fruit, the relationship between the total number of ovules
laid down and the deviation of the number of seeds matured from their
probable number has been investigated in Sanguinaria (HARRIS 1910 a).
For Phase ohts rulgaris a first study (HARRIS 1913) of 53 series
comprising 166,130 pods and a supplemental investigation of 16 series
comprising- 56,698 pods (HARRIS 1917 a) leave no doubt that the
pods with the larger number of ovules mature relatively fewer of their
ovules into seeds. The same relationship holds in the arborescent legume,
Ccrcis canadcnsis, as is shown by studies based on massed data (HARRIS
1914 a) and on series from individual trees (HARRIS 1914 b).
The relationship found in Cercis and Phaseolus is not universal for the
Leguminosae. In a series of 1427 pods of Robinia (HARRIS 1909 a),
the pods with larger numbers of ovules mature a relatively higher propor
tion of their ovules into seeds. The correlation between the actual num
ber of ovules formed and the actual number of seeds developing is
rog •== .693 ± .009, while that between the number of ovules formed and
the deviation of the number of seeds matured from their probable value
is roz = .365 ± .015.
That this result represents a real biological relationship is indicated
by the correlations, hitherto unpublished, for the individual trees. Only
three of the twelve constants in table i are negative in sign. No one of
these can be regarded as statistically significant when the probable error
is taken into consideration, while seven of the nine positive coefficients
must be looked upon as statistically trustworthy.
The formula has also been advantageously applied to the problem of
the interrelationship of the number of male and female flowers in the
inflorescence of the aroid Arisarum (HARRIS 1916 a) and that of the
interdependence of numbers of stamens and pistils in the ranuncula-
ceous genus, Ficaria (HARRIS 1918). In Arisarum the relative number
GENETICS 3: Tl 1918
332 J. ARTHUR HARRIS
TABLE i
Relationship between seed and ovule number in Robinia.
Number of
Number of
r
r
r IE
OS
oz
OZ/ T
tree pods
OZ
I
122
.524±x>44 — .O55±.o6i
(— )o.oo
2
64
.802 ±.030
468±.o66
7.09
3
in
.478 ±.049
.074 ±.064
1.16
4
102
.430 ±.054
. 105 ±. 066
1-59
5
122
.671 ±.034
.338±.054
6.26
6
I2O
-533±.044
.2-2±.057
477
7
120
•259±-057
— .ii5±.o6i
(-)i-89
8
159
.6i4±x>33
.2i6±.o5i
4.24
9
128
•5oo±.o40
•35o±.052
6-73
10
78
.507 ±.057
.225±.o73
3-o8
ii
105
.714^.032
— .044 ±. 066
(-)o.67
12
196
•797±.oi8
48o±.037
13-22
All trees
1427
.693±.oo9
.365 ±.015
24-33
of pistillate flowers increases as the total number of flowers per inflo
rescence increases. In Ficaria the relative number of pistils increases
as the total number of sporophylls becomes larger.
Dr. BLAKESLEE and I (1918) have applied this coefficient to the deter
mination of the relationship between the total annual egg production
and the monthly egg production of White Leghorn fowl. We have there
shown by means of this coefficient that the winter months, November,
December, January and February, and the following autumn months!
August, September and October, show an increase over their theoretical
quota of eggs when the annual total egg production rises above the
normal. That is, reg, the correlation between total annual egg produc
tion and the deviation of the monthly production from its probable value,
is on the whole significantly and substantially positive. The spring and
summer months, April, May, June and July, show negative values of
rn, that is, they make a lower relative contribution to the annual total
than might be expected when the total varies in the direction of an in
crease above the normal egg production of the flock as a whole.
FURTHER ILLUSTRATIONS
Illustration i. Proportionality of parts in Paramccinm
JENNINGS (1911) in his masterly investigation of assortative mating
in Paramecium, has given data for determining the relationships between
APPLICABILITY OF A COEFFICIENT OF CORRELATION 333
(a) distance from the anterior end of the organism to the posterior mar
gin of the mouth, (fe) distance from the posterior margin of the mouth
to the posterior end of the organism, and (c) the total length in series of
conjugant and non-conjugant Paramecia.
He has calculated and discussed for a purpose which does not con
cern us here the correlations between certain of these dimensions. All
the correlations between the absolute measurements, calculated from his
data, are given in table 2.
TABLE 2
Relationship between total length, I, and anterior length, a, and between anterior length
and posterior length, p, in Paramecium.
Series*
rap
ria
ri.
ri,/Er
MP
Lot 7, table 40
.246 ±.O5O
.735 ±.025
—.733 ±.025
3-0
Lot 7, table 41
.57o±.047
.893 ±.014
— .427±.o57
7-5
Lot 19, table 51
.382 ±.044
.832 ±.o 1 6
-434±.042
10.4
Lot 19, table 52
.67 1 ±.034
.939 ±.074
— .I96±.o6o
3-3
Lot 22, table 55
— .403 ±.026
.485^.049
— .282±.o6o
4-7
Lot 22, table 56
.62O±.O25
.906 ±.073
—.405 ±.034
I2.O
Lot 24, table 64
.277 ±.040
.82o±.oi4
— .282 ±.040
7-1
Lot 24, table 65
.488±.03i
.885 ±.089
— .288±.o38
7-6
* Lot /, table 40, conjugates of wild cultures. 2. Lot 7, table 41, non-conjugants
of wild cultures. 3. Lot 19, table 51, conjugants of race g. 4. Lot 19, table 52,
non-conjugants of race g. 5. Lot 22, table 55, wild culture conjugants not yet sep
arated. 6. Lot 22, table 56, wild culture conjugants about twelve hours after separa
tion. 7. Lot 24, table 64, unseparated conjugants of race k. 8. Lot 24, table 65, con
jugants of race k about twelve hours after separation.
In this table the constants for anterior and posterior length are ar
ranged in pairs of conjugants and non-conjugants or ex-conjugants. In
every instance the correlation between the anterior and posterior por
tions of conjugants is lower than that between the same dimensions in
non-conjugants or ex-conjugants.2
All of these values are low, as JENNINGS has noted. In the case of
lot 22 the coefficient for the conjugants is actually negative in sign.
The correlation between total length and the length of the section
anterior to the mouth is high. In every case the value of ria is higher
in non-conjugants or ex-conjugants than in conjugants. The additional
relationship to be brought out by the formulae here under discussion is
2 See in this connection the discussion by JENNINGS (1911, pp. 65-66, 71-73)-
GENETICS 3: Jl 1918
334
J. ARTHUR HARRIS
that between the total length of the organism and the relative length of
either anterior or posterior element.
JENNINGS (1911, p. 63) has emphasized the high variability of the
post-oral dimension. Table 3, in which all the coefficients of variation
TABLE 3
Coefficients of variation for anterior and posterior fractions of length
in Paramccium.
Series
Total
length
Anterior
length
Posterior
length
v»-va
Lot 7, table 40
6.90
5-79
13-86
4-8.07
Lot 7, table 41
9.68
8.76
14-53
~r 5-77
Lot 19, table 51
8.55
7v8
14-31
+ 6.53
Lot 19, table 52
12.18
12.08
15.13
+ 3-05
Lot 22, table 55
548
7.38
14.29
+ 6.91
Lot 22, table 56
7.96
7.28
11.2O
+ 3-92
Lot 24, table 64
6-34
6.42
9.98
+ 3o6
Lot 24, table 65
6.62
6-45
9-35
+ 2.90
are laid side by side, fully confirms his conclusion in this regard. Utiliz
ing these coefficients of variation we obtain the values for the correlation
between total length and the deviation of the anterior length from its
probable value, given in the fourth column of table 2.
These constants are negative in sign throughout, and while variable
in magnitude all may reasonably be considered statistically significant
in comparison with their probable errors.
Thus when Paramecium varies in length both anterior and posterior
fractions of the body contribute to this variation, but as length in
creases the anterior portion becomes relatively shorter.
For one series, the unseparated conjugants of race k. I have deter
mined the regression of the anterior length on total length and the re
gression of the deviation of the anterior length from its probable value
on the total length of the organism. The equations are
a == 3-0359 + -4977 I
^==.7616, sa = 3.3743 --.1135 /
The equations and empirical means are represented graphically in
diagrams i and 2. In both cases the relationships are sensibly linear.
APPLICABILITY OF A COEFFICIENT OF CORRELATION
335
-2J
-20
•/s
2$
Tote/
26 27
28
29
30
31
32
33 34-
3$ 36
DIAGRAM i. — Relationship of anterior length to total length in Paramecium, Com
pare diagram 2.
DIAGRAM 2. — Relationship of deviation of anterior length from its probable value
to total length in Paramecium. Compare diagram i.
GENETICS 3: Jl 1918
336 J. ARTHUR HARRIS
Illustration 2. Absence of relationship between size of litter
and sex in swine
PARKER and BULLARD (1913) have discussed the possible relationship
between the size of the litter and sex in the contents of 1000 uteri of
swine. From a simple percentage table they state that the relative num
bers of males and females are "even in the extreme cases so nearly uni
form that we may conclude with reasonable assurance that there is no
intimate relation between sex and the size of the litters."
The correlations between the total numbers of pigs in the litter, /,
and the number of males, m, and females, /, may be deduced from their
data. They are
rlm-= .6833 ± .0114
rtf = .6875 ± .0112
From these and the three coefficients of variation one may deduce
For males rte = — -0177 ± -O2I3
For females rlz = + .0177 ± .0213
The correlation is sensibly zero, with regard to its probable error.
This method of analysis therefore fully confirms the conclusion drawn
by PARKER and BULLARD.
Illustration 3. Proportion of pistillate and hermaphrodite flowers in
the inflorescence of the composite Homogyne
LUDWIG (1901) has given data for the correlation between the num
ber of pistillate and the number of hermaphrodite flowers in the inflo
rescence of Homogyne. From his data we deduce
For hermaphrodite flowers, h,
h = 31.8333- °» == 7-398i, Vh = 23.240
For pistillate flowers, p,
P= I0-537o, °P = = 2.6460, Vp = 25.112
For total flowers, f,
/"= = 42.3704, *f == 8.7749, V, = 20.7099
For hermaphrodite and pistillate flowers,
r*P = -3899 ± -0449
For total flowers and hermaphrodite flowers,
rfh = .9607 ± .0049, rfih = .2429 ± .0499
For total flowers and pistillate flowers,
rfp == .6303 ± .0339, rfSf = — .2429 ± .0499
APPLICABILITY OF A COEFFICIENT OF CORRELATION 337
It follows, therefore, that in the larger heads the purely pistillate flow
ers are relatively less, and the hermaphrodite flowers relatively more,
numerous.
Illustration 4. Fertility of capsules and viability of seed in
carnation crosses
STUART (1912) has recorded the number of seeds obtained and the
number germinated, planted into the field, and producing flowers in
various carnation crosses. Our problem is to determine whether the
seeds which come from capsules producing a large number of seeds are
relatively more (or less) viable than those from capsules producing
small numbers. .
Using his two larger tables of data, tables 3 and 6,3 and confining at
tention to the relationship between number of seeds per capsule, and
the number which germinated, I find
For commercial X commercial, STUART'S table 3, N = 23,
rsg = -775 ± -056, rsa = -.072 ± .141
For single flower X double flower, STUART'S table 6, N = 32,
rg(J = .649 ± .069, rs~ = — .118 ± .118.
The signs are both negative, indicating a relatively higher failure to
germinate among the seeds which are produced many in a capsule. With
regard to their probable errors, the constants are untrustworthy. Be
cause so few observations are available, no biological significance is at
tached to these two series, which serve merely as another illustration
of the kind of problems to which the method may be applied.
Illustration 5. Relationship between the total number of pedicels and
the number of abnormal pedicels in Spiraea Vanhonttei
In Spiraea Vanhouttei the pedicels of the umbel-like raceme normally
produce but a single flower each. An abnormal condition in which one
or more pedicels may bear a relatively large number of flowers is fre
quently observed (HARRIS 1917 d).
Let x be the total pedicels in an inflorescence and a the number which
are abnormal. Then if abnormality be distributed purely at random
among the pedicels one would expect material values of rxa. The cor
relation rxs meets our requirements since it shows whether inflorescences
with a large number of rays have relatively more or fewer of their rays
abnormal than those with a small number.
3 In table 6 the cases in which the seeds are not normally developed are omitted.
GENETICS 3: Jl 1918
33§ J. ARTHUR HARRIS
During the last fifty years a great deal has been said about the in
fluence of nutrition, vegetative vigor, etc., upon the development of
anomalies. If a larger number of rays indicates greater vigor or better
nutrition one might a priori expect larger inflorescences to have a pro
portionately higher number of branched rays, providing of course, that
the classic theories are true.
The constants for a short series of data collected in 1906 were pub
lished in 1909. Since then a large number of determinations have been
made on a general sample of inflorescences from a number of shrubs in
1909 and from three large individual shrubs in 1913.
In the latter series the data have been analyzed in two ways. First, the
inflorescences which contain at least a single abnormal pedicel have been
used as the basis of the correlations. These are designated as the ab
normal inflorescences. Second, the normal inflorescences from the
same plants have been included and counted as zero in the distribution
of number of abnormal rays.
The results are:
For 1906* rm = • -f .121 ± .034
*W = — .071 ± .034
For 1909. Massed statistics. Inflorescences producing some abnor
mal pedicels (N = 785),
rM = + -1542 ± .0235
r** = — -0915 ± .0239
For 1909. Massed statistics. All 2040 inflorescences,
rxa = + -1584 ± .0146
r« = + -0370 ± .0149.
For 1913. Individual plants. Inflorescences producing some abnor
mal pedicels,
Plant i. N = 747 inflorescences.
rXa = 4- .0880 ± .0244
TXZ = ~ .2846 ± .O227
Plant 2. N == 641 inflorescences.
Tm = + .1148 ± .0263
« = - -3855 .± .022
* Since in the 1906 series only synanthies were observed, the total pedicels rinclud
APPLICABILITY OF A COEFFICIENT OF CORRELATION 339
Plant 3. Ar = 548 inflorescences.
rxa = + -0941 ± -0285
r*z = — -2849 ± .0265
For 1913. Individual plants. All inflorescences,
Plant i. Ar = 1135 inflorescences.
rxa = — .0067 ± .0200
rxz = — .2125 ± .0191
Plant 2. Ar = 975 inflorescences.
rxa = — .0821 ± .0214
rxx = — .3681 ± .0187
Plant 3. Ar = 912 inflorescences.
rxa = —.0360 ± .0223
rxg = - .2342 ± .0211
For all the samples of inflorescences in which there is at least one
abnormal pedicel the correlations between the total number of pedicels
and the number of normal pedicels is positive in sign and perhaps sta
tistically significant, but low in actual magnitude. Thus the number of
abnormal pedicels increases on the average as the total number of pedi
cels per inflorescence becomes larger. The relationships are, however,
very slight indeed.
For these five series the correlation between the total number of pedi
cels and the deviation of the abnormal pedicels from their probable
value, is negative in sign. Thus the larger inflorescences have a rela
tively smaller proportion of abnormal pedicels than do those with a
smaller total number of pedicels.
In the four series in which the wholly normal inflorescences are in
cluded, the correlations between total number of pedicels and number of
abnormal pedicels is positive in 1909 but negative throughout and insig
nificant in magnitude in 1913. The three series from individual shrubs
studied in 1913 show low but significantly negative correlations between
the total number of pedicels per inflorescence and the deviation of the
number of abnormal inflorescences from their probable value. The con
stant for the heterogeneous data of 1909 is positive but insignificant.
Taking the data altogether, there can be no reasonable doubt that the
relative number of abnormal pedicels decreases as the total number of
abnormal pedicels increases.
This is shown in diagram 3, which represents the regression of the
deviation of the number of abnormal rays from their probable value on
GENETICS 3: Jl 1918
340
J. ARTHUR HARRIS
the total number of rays in the series showing the lowest, and in one of
these showing the highest, correlation.
23 25 27 29 31 33 35 36
DIAGRAM 3.— Regression of the deviation of the number of abnormal pedicels from
their probable value on the total number of pedicels in Spiraea.
The standard deviations are :
For 1909, -i "-> rdered when the corrected galley-proofs are returned.
Manuscripts and all editorial correspondence should be addressed to the Managing
Editor, Dr. Georjje H. Shull, 60 Jefferson Road, Princeton, N. J.
GENETICS, JULY 1918
TABLE OF CONTENTS
PAGj
SAX, KARL, The behavior of the chromosomes in fertilization. . . . 305
HARRIS, J. ARTHUR, Further illustrations of the applicability of
a coefficient measuring the correlation between a variable
and the deviation of a dependent variable from its prob
able value 32!
EAST, E. M., and PARK, J. B., Studies on self -sterility. II. Pollen-
tube growth 35;
WRIGHT, SEWALL, On the nature of size factors 36;
ROBBINS, RAINARD B., Some applications of mathematics to breed
ing problems. Ill 37;
ROBBINS, RAINARD B., Random mating with the exception of sister
by brother mating 39
G-196
PRACTICAL UNIVERSALITY OF FIELD HETEROGENEITY
AS A FACTOR INFLUENCING PLOT YIELDS
BY
J. ARTHUR HARRIS
Reprinted from JOURNAL OF AGRICULTURAL RESEARCH
Vol. XIX, No. 7 : : : : Washington, D. C., July 1, 1920
PUBLISHED BY AUTHORITY OF THE SECRETARY OF AGRICULTURE. WITH
THE COOPERATION OF THE ASSOCIATION OF LAND-GRANT COLLEGES
WASHINGTON : GOVERNMENT PRINTING OFFICE : 1920
PRACTICAL UNIVERSALITY OF FIELD HETEROGENEITY
AS A FACTOR INFLUENCING PLOT YIELDS
By J. ARTHUR HARRIS
Collaborator, Office of Western Irrigation Agriculture, Bureau of Plant Industry,
United States Department of Agriculture
INTRODUCTION
With the development of a more intensive agriculture there must be
a wider use and a progressive refinement of the method of plot tests in
agronomic experimentation. Betterment of the method of plot tests
must be sought along two lines, (i) the perfection of biological technic
and (2) the more extensive use of the modern higher statistical methods
in the analysis of the results.
In 1918 Mr. C. S. Scofield, in charge of the Office of Western Irrigation
Agriculture, and Prof. E. C. Chilcott, in charge of the Office of Dry-Land
Agriculture, asked the writer to undertake an investigation of the
statistical phases of the problem of the accuracy of plot tests. The
present paper deals with one aspect only of the general problem, that of
the lack of uniformity of the experimental field. This is both the most
potent cause of variation in plot yields and the chief difficulty in their
interpretation.
Many of the careful writers on field experimentation have noted the
existence of soil heterogeneity. Few have, however, sufficiently recog
nized and none have adequately emphasized the importance of this
factor.
The problem of field heterogeneity is twofold. First, some measure
of the amount of its influence upon crop yields must be obtained. Sec
ond, some means of avoiding or of correcting for its influence must, if
possible, be secured.
An exact measure of the influence of field heterogeneity, and not
merely a vague notion that it may influence experimental results, is the
first and most fundamental step in the closer analysis of the factors
determining the variability of plot yields. If the application of such a
criterion to results obtained by practised agriculturalists from fields
selected for their uniformity shows no evidence of heterogeneity, plot
tests may be carried out along conventional lines with confidence that
Journal of Agricultural Research, Vel. XIX, No. 7
Washington, D. C. July i, 1920
utn Key No. G-i$>6
(279)
280 Journal of Agricultural Research vol. XIX.NO. 7
with reasonable precautions reliable results will be obtained. If, on the
other hand, the application of such a criterion shows a high degree of
irregularity in fields selected for their uniformity by experienced agri
culturalists, it is evident that very special precautions must be taken
to obtain trustworthy results. Some quantitative measure, and some
probable error of this measure, of the amount of irregularity of the soil
of a field, as shown by actual capacity for crop production, and not
merely a demonstration of its existence is, therefore, required.
The purpose of this paper is to show by the analysis of the actual
yields of test plots reported by agricultural experts that the securing of
fields suitable for a direct comparison of yields is, practically speaking,
an impossibility. The results show that unless special precautions are
taken irregularities in the field may have greater influence upon the
numerical results of an experiment than the factors in crop production
which the investigator is seeking to compare.
The results of this study may seem to be altogether negative — destruc
tive rather than constructive. The unbiased student must, however,
admit that a full evaluation of all the sources of error is an essential
prerequisite to constructive work. Furthermore, large expenditures of
public funds are being devoted to fertilizer tests, variety tests, and rota
tion experiments. It is preeminently worth while to ascertain to what
extent results derived from methods now in use may be considered
reliable.
Subsequent papers will treat other phases of the problem.
FORMULAE
A criterion of field homogeneity (or heterogeneity) to be of the greatest
value should be universally applicable, be comparable from species to
species, character to character, or experiment to experiment, and be
easy to calculate.
In 1915 the suggestion was made (5)* that we may proceed as follows:
Suppose a field divided into N small plots, all sown to the same variety of
plants. Let p be the yield of an individual plot. The variability of p
may be due purely and simply to chance, since the individuals of any
variety are variable and the size of the plots is small, or it. may be due in
part to the diversity of conditions of the soil. If irregularities in the
experimental field are so large as to influence the yield of areas larger
than single plots,2 they will tend to bring about a similarity of adjoining
plots, some groups tending to yield higher than the average, others lower.
Now let the yields of these units be grouped into m larger plots, Cn,
each of n continguous ultimate units, p. The correlation between the
1 Reference is made by number (italic) to " Literature cited, " p. 313-314.
2 Irregularities of soil influencing the plants of only a single small plot may in most work be left out of
account, since they are of the kind to which differences between individuals are to a considerable extent
due and are common to all the plots of a field.
July i, i93o Universality of Field Heterogeneity 281
p's of the same combination plot, Cn, will furnish a measure (on the scale
of o to ± i) of the heterogeneity of the field as expressed in capacity
for crop production. If this correlation be sensibly o (under conditions
such that spurious correlation is not introduced), the irregularities of
the field are not so great as to influence in the same direction the yields
of neighboring small plots. As heterogeneity becomes greater the cor
relation will also increase. The value of the coefficient obtained will
depend somewhat upon the nature of the characters measured, some
what upon the species grown, somewhat upon the size of the ultimate
and combination plots, and to some degree upon the form of the combina
tion plots.
Knowledge of the values of the correlations to be expected must be
obtained empirically.
Let 5 indicate summation for all the ultimate or combination plots of
the field under consideration, as may be indicated by Cn or p. Let
p be the average yield of the ultimate plots and