Statistical analysis of long-term agricultural experiments

156 Pages·07.499 MB·English

by  Dutton, Arthur M

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STATISTICAL AHfcLTSIS Of LQSG-TiUM
AGRICTFITUHAL IXP11IMEHTS
by
Arthur M. Dutton
A Dissertation Submitted to the
Graduate faculty in Partial fulfiHuent of
The lefuiremente for the Degree of
DOCTOR Of PHILOSOPHT
Major Subject: Statist ice
Approved:
S n K i i f g 3^;',i,iii‘'11
Bean of Graduate College
Iowa State College
1951
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UMI Number: DP11946
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fwm of ooswifs
page
x. mmmmmm................................... i
ii. scvxnr of mt rmmm ufmmrn................ k
A. long-Tera Agricultural Experiments......... k
1. Other Pertinent literature.........................10
in. a&owioui© of m nmum......................... 2k
IF. BSTIKATIOH II fBI CASE OF IXIM1 HIGHESSIOI Wifi
SIMHjT AUTOCOBWLAB® XBHBBS..................... 33
A, .linear legression With A Single Series...........35
1. linear ingresslea With fee Correlated Series. . . 5k
S. linear Begreseion With k Correlated Series.......93
?. mtwmim nr m oasi of sxpohhtiai isobissioi wits
s s m r AOfocoBBHAfu w a r n ................ 116
A. Exponential Begreesion With A Single Series . . . . 117
1. Exponential Begreseion With fm Correlated Series . 122
C. Exponential Begreseion With k Correlated Series . . 130
FI. AFPlICAflOI OF 1SS SSfJXATIOI PBOGIDTJISS TO ACfffAl
TIHD SHIES.................................132
A. linear Begreseion ©a fiae......................... 13k
B. Exponential Begreseion...........................139
FII. summary . . .................................. ik>
Fill. 1ITEMTOBS CITED................................ lk3h
H. ACMOWISMSMMT................................ ikhu
X. APFSHDIX....................................... lkkh
T19J-3
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1
x. sarsoBOGtios
The general problem of land use may be briefly stated as that of
maximising the present value ©f all future earnings of a piece ©f land.
Such term# a* value mi earning* mey ho defined differently under dif­
ferent economic system*. la the ease of land suited to agricultural
purpose*. where the «et of possible ueeo to which a piece of laud may
he pit 1* finite, though possibly large la number, the problem oaa he
attacked experimentally. It is possible to subject plot* of land to
various cropping systems ami to measure the economic return* la any
given period. Any measure of present value of the future earning* would
hare to he determined ®a the basis of a (theoretically) Infinitely long
cropping system. If instead of actual economic return* the change* la
yield potential or fertility could he measured and described accurately
ia term* of a simple mathematical function thea the properties of girea
cropping systems might he inferred from long-term experiment*.
It ha* beta pointed oat many times ia the agronomic literature1
that th« yield of a particular crop 1* probably not the ideal measure
of crop producing ability. Xevertheleee it is the yield of test crops
that has been takes almost universally a* the measure of soil fertility,
ffc# statistical problem treated la this dissertation is that of
describing simply uni. accurately ©a the basis of a theoretical agronomic
model the variation* la yields under various cropping systems.
eepliwe»ew<*>e*tiwg»w»»»<li'»i‘ 1'iiiiiiniiuiwswwwqm' "s-om—e
$** for example 1. f. Shaw (19^^).
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2
1
the published work* m the analysis of long-term agricultural
experiment* have taken the view that yield variations with tine e«>
host he described by fitting m linear, quadratic, or, at most, a low
order tine polynomial to the data by the method of least squares. This
approach leads to moderately simple methods of analysis. It has not
boon particularly successful from the agronomic point of visw because
of the fact that the coefficients in the polynomials so fitted hare
little practical meaning, furthermore if the errors from such trend
lines are not Independent in time the method of least squares may lead
to inefficient estimates of the coefficients Involved.
lifter consideration of certain data as well as the remarks of
»
earlens agronomists and: statisticians in print or private conversa­
tion, an alternative model .has been considered which seems to he mere
justified, this model aoomnos that the yield is roprsssatabls by an
exponential trend line and an nnteeorrelated error, the problems of
estimation Involved; in using such a model are similar to these encountered
la tlme-sorlss analysis, the essential difference should be noted, how­
ever, that in this'approach a bio-mathemaileal reaction law is postulated
with theoretical Justification whereas la the ease of time-eeries
analysis there is aa attempt to derive *tru»“ relations from observed
data, the estimation of the parameter# Involved in the model is muds
more involved than.the estimation of the coefficients of polynomials
for instance.' the pnrmietors so estimated, however, have a physical
Hem fisher lift©) and Cochran (1939)
%*♦, for example fisher (1920) footnote 1, p. 109.
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3
mourning whereas 4a the classical polynomial approach ao sack meaning
ia apparent.
»a*4*«i likelihood estimators of the parameters larolred ia a
linear regression *#*#1 with autoeorrelsted error terns will he giren.
fhis analysis alii ho extended to iaeXmi© iJae exponential tread model,
fh# smximmn likelihood estimators are quite imrelved for the case of
•oreral correlated yield eorlea under either model. lay simpler
approach to the estimation problem would oeoa to otea froa emsmiaatiom
of thess rigorous eat last ioa procedures.
the position has h«o» taken In the coarse of this work that the
first step towards the analysis of long*4.era yield data aast he the
examination of data under a staple continuous cropping system. A
model which does not work satisfactorily ia this situation cannot ho
expected to he of mime la the more complex si font lea s.
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H
11. ISri'W Of fill PlBflllSf MflMfS®!
Ia the past 10 year* there hs.«s been an extensive development of
the related subject© ©f timswserles analysis and the analysis of auto*
regressive system*. This development lias coma as Inly la the economic
field with applications to economic data. Sine® It Is felt that the*©
development * are distinct fro* these of the classical analysis of long*,
term agricultural experiments, the literature review will consist of
two mala parts, The first will he devoted to the subject of long-term
agricultural experiments, Th® second will ©over the other statistical
(mainly econometrlc^l) developments which are related to the method of
analysis' advanced ia this dissertation.
A. loag»f«r* Agricultural Ixperlmeats
The most fundamental study of yield variation fro® the statistleal
standpoint as well as one of tho earliest fls that of fisher (1920).
from a long series of observations of wheat yields on the plots of
Broafbalk field (ftotfcamsted) he distinguishes three types of variation,
these arof, (1) annual variation, (2) steady diminution due to soil
deterioration, and (3) slow changes other than diminution. The extent
of slow changes is described by the fittlag of a fifth degree polynomial
in time by the mss of orthogonal polynomials and least squares.
Deterioration Is described by the mean annual diminution over the years
investigated,. The deviations from the fifth degree polynomial are
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5
attributed to random annual variation. In later studies (Fisher (192*) )
the random annual variations were correlated with, reeldmale similarly
obtained from weather data. Fisher states, "The errors involved in the
correlation of residual* of series changing In an unknown manner may
he minimised hy the method of polynomial fitting; such errors are prohahly
insignificant when this procedure is applied to rain data and wheat
yields*. (Fisher (192*0 pg. 1*1).
This sort of technique was repeatedly applied to ether data hy
Fisher as well as hy ether workers at lethansted. (See for example
Cashen (19*?) ).
fariatlons and extensions of the methods of Fisher are set forth
hy other investigators, notably, Cochran (1939). sad Crowther and
Cochran (19*2). The earlier paper is a very exhaustive summary of the
types of long-term agricultural experiments. The use of orthogonal
polynomials Is extended to include replicated experiments and experiments
ia which crop rotations are involved. Cochran (1939) suggests the fol­
lowing preliminary analysis for the case of a long-term experiment laid
out in the ordinary randomized block design with t treatments, r rep.
lleates, and n yearns
Source Degrees of Freedom
Treatments t-1
Replicates r-1
Irrer (l) (t-1) (r-i)
Tear# n-1
Tears x Treatments (n-1) (t-1)
Error (2) (n-1)(r-l)t
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