‘” “W“ “*FFFFF FF FFFFF I
CONTROLLED FERTILIZER INPUT OUTPUT " "
XPERIMENTS IN MICHIGAN '
Thus. ﬁr thobogrgo aI mp;
MICHIGAN STATEUNIVERSITY
Wuloy Buﬂon Sundquisi" .
. 1957 .
This is to certify that the
thesis entitled
An Economic Analysis of Some Controlled
Fertilizer InputOutput hbcperiments in Michigan
presented by
Wesley B. Sundquist
has been accepted towards fulfillment
of the requirements for
PhD degree in gricultural Economics
1", v4 ,‘ '/
~V'Ltrw‘m‘,"\ \J... VWW
Major professor
/
Date 8/21/57
0169
L [BRA R Y
Michigan State
University
_._ _._< 44
_,_.___—
AN ECONOMIC ANALYSIS OF SOME CONTROLLED FERTILIZER
INPUTOUTPUT EXPERIMENTS IN MICHIGAN
By
‘wesley Burton Sundquist
A THESIS
Submitted to the School of.Advanced Graduate Studies of Michigan
State University of.Agriculture and.Applied Science
in partial fulfillment of the requirements
for the degree of
DOCTOR OF PHILOSOPHY
Department of.Agricultural Economics
1957
1‘ A .
‘ r" I.) t (1
‘1 ca ’;‘1‘ ""
I
I", /“ I ’ y
L'. [I (7/ {1‘ Li.
ACKNOWLEDGMENTS
The author is indebted to several members of the departments of
Agricultural Economics and Soil Science of Michigan State University
who aided in organizing and conducting the research reported in this
thesis. Dr. R. L. Cook, Head of the Department of Soil Science, Dr. J.
F. Davis, L. N. Shepard and J. C. Shickluna of the Department of Soil
Science and C. R. Hoglund of the Department of.Agricultural Economics
aided in numerous ways in conducting the research reported here.
Dr. L. L. Boger, Head of the Department of.Agricultural Economics
at Michigan State University and C. W. Crickman and H. L. Stewart of the
Farm Economics Research Division,.Agricultural Research Service, United
States Department of.Agriculture, provided financial aid without which
the author could not have conducted this study.
Much of the computational work was performed by the girls in the
statistical pool of the Department of.Agricultural Economics under the
direction of Mrs. Iantha Perfect. Beverly Hamilton deserves a vote of
thanks for typing an earlier draft of this thesis.
The author's sincerest appreciation is due to Jack L. Knetsch,
formerly of Michigan State University, now with the Tennessee Valley
Authority with whom much of the experimental work was planned and con
ducted and to Dr. Lynn 8. Robertson Jr. of the Department of Soil
Science who aided in planning the experimental work and who has super
vised the field work. ‘Without a tremendous amount of work on his part,
the data utilized in this thesis would not have been produced.
Above all, the author appreciates the encouragement and guidance
of Dr. Glenn.L. Johnson, not only while writing this thesis but through
out the course of my graduate studies. Association with him has helped
to make graduate work a really enjoyable and profitable experience.
\I \I \I \I \I \I \< V \I \I \I \I \I \l \I
.. —. — . ~ ~ — ~ —   .~ — — —
l\ u'\ I\ I\ a\ I‘ I\ l\ I\ l\ l\ I\ I\ I\ l\
ii
AN ECONOMIC ANALYSIS OF SOME CONTROLLED FERTILIZER
INPUTOUTPUT EXPERDIENTS IN MICHIGAN
By
wesley Burton Sundquist
AN ABSTRACT
Submitted to the School of.Advanced Graduate Studies of Michigan
State University of.Agriculture and Applied Science
in partial fulfillment of the requirements
for the degree of
DOCTOR OF PHILOSOPHY
Department of.Agricultural Economics
Year 1957
Approved wﬁpﬂ'Z/‘W
ABSTRACT
The three primary plant nutrients, nitrogen, phosphoric acid and
potash are major farm resource inputs. In l9Sh, farmers in the United
States paid over a billion dollars for various commercial forms of these
plant nutrients. In order to allocate optimally their resources, farmers
need information as to the productivity of expenditures made for various
production inputs including the three primary plant nutrients.
In the spring of l9Sh the Michigan.Agricultural Experiment Station,
aided in part by resources contributed by other interested agencies,
initiated a series of plant nutrient inputcrop yield output experiments
which has been expanded in each succeeding year. Experimental input
output information analyzed in this thesis included data for the following
crops: (1) a rotation of oats, wheat, alfalfa and corn on a Kalamazoo
sandy loam soil in Calhoun and Kalamazoo counties (2) a rotation of corn,
field beans and wheat on a Simms loam soil in Gratiot county (3) corn
produced in continuous culture on a'Wisner clay loam soil in Tuscola
county and (h) potatoes grown on a Houghton muck soil at the Experiment
Station muck farm near East Lansing. In total, over 1150 individual
experimental plots were contained in these experiments in 1956.
The primary objectives of this thesis are (l) to estimate plant
nutrient inputcrop yield output production surfaces and then (2) to
provide an economic analysis of the physical inputoutput relationships
iv
derived. Continuous function analysis is utilized to estimate the
inputoutput relationships of interest to researchers and farmers.
Two general formulations of the production functions for plant
nutrients are fitted for most crops. These are a polynomial of the type:
I  a + blN + sz2 + b3P + b4P2 + b5K + b6K2 + b7NP + bBNK + bQPK
where N, P and K represent pound per acre inputs of nitrogen, phosphoric
acid and potash. The second production function formulation is an
exponential of the CarterHalter type:
r  allolclNPb202PKb3c3K
Both equations are fitted by least squares techniques, the latter being
first converted to logarithms.
Significant yield reSponse to applied nitrogen was found for corn,
wheat, oats and field beans. Corn, wheat and field beans showed a signifi
cant yield response to applications of phosphoric acid. Only potatoes
showed a significant reSponse to applied potash for crOps produced during
the growing seasons for which experimental data were analyzed.
DeSpite statistically significant response to applied plant nutrients
for several crops, applications of plant nutrients were profitable for
only two crops assuming current crop and fertilizer prices. Nitrogen
applications were profitable for corn produced on a Kalamazoo sandy loam
soil in 1955 and for field beans produced on a Simms loam soil in 1956.
In computing high~profit plant nutrient inputs, however, no credit was
made for residual fertility or benefits derived from seedings in the
small grain crops. Mid and late summer drouths in 1955 and 1956 very
probably reduced the crop yield benefits which might have been derived
from applied plant nutrients particularly on the lighter soils. Further
information on inputoutput relationships over time and with varying
weather conditions is needed to establish a probability distribution of
these relationships.
The experimental results analyzed in this thesis are from a very
limited number of soil types. These soils tend to be either very fertile
or very unproductive. One might expect the largest yield reSponses to
applied plant nutrients on soils with a high production potential but
depleted in fertility; such soils are not included in the experiments
analyzed here. However, as individual lowtreatment plots in the experi
ments become depleted and if treatments are rerandomized, a wide range
of combinations of residual fertility and applied nutrients should be
observed.
The adjusted coefficients of multiple correlation between applied
plant nutrients and crOp yields ranged from .28 to .78 for the various
production function formulations for the different crOps studied.
Further analysis indicated substantial amounts of yield variance not
associated with regression were due to experimental error and inability
to control entirely unstudied variables. Limited analysis to relate
residual fertility, as measured by soil tests, to the deviations of
predicted from observed yields (Yi  Ii), was relatively unsuccessful.
vi
However, further extension of this type is needed in order to provide
conclusive results.
vii
T.’ TIE OF CCI’T'” T
CHAPTER  Page
I TIIE IL: TUI.E PL.“ D 1313. .II\‘4IT[JDE OF FIRTHT .L/"LR. Us?) FTC/73.4.13 o o o o o o o l
H
Current Research Problems in Fertilizer Use............
The Impor*"nce of Fertilizer as an.Agricultural
Production Factor...................................
Reasons for Increased Fertilizer Use...................
LongRun Agricultural Production Needs.................
The Type of Infornation Needed by Farmers.............. l
OCOO\I\)
II I'ETEIODS OF AI':1:LLYSISO.....‘.................‘C............. 13
Methods of Collecting Data............................. 1h
The Concept of Functional Relationships................ 16
Determining Economic Optima............................ 20
Alternative Types of Analysis.......................... 22
Continuous Function unal\sis........................... 23
Analysis of Variance................................... 2o
Discrete Point Analysis................................ 30
III EXPERILELTLL WCZLK CCFDUCTED BY THE KICH_ CAN A"1IL ULTURLL
MILLILILFP SrifxmiLC'I‘looo000000000000...ooooooooooooooooooo
LL)
DJ
Characteristics of minerim,L a1 Dcsi;ns................ 32
The Cats, Hdi3at, Alfalfa and Corn Rotation............. 33
Continuous Corn........................................ 35
Fie d B ans, U7 eat and Corn iotatien................... 38
Potatoes............................................... 39
The lj;h Potato Expe riment............................. h2
Tm Lﬂirot. EM),imam.uu.n..uu.u..u..u..u.. MI
Thefmallgengwma Pm) anun.u.u.n.n.u.u.n. h]
IV HIRLYSI CF Tn} LLLZ'L...................................... hf;
The Cats, wheat, iljalfa and Corn Rotation............. h?
iralysis of the Late Data........................... L?
Interpre to .tion cf the Statistical Results........ 53
High Profit Ccrcinations of Plart Int ients...... F"CF
ﬁnal sis of the wheat Data.......................... 5;
;aximum Yieles and High Profit Plant Lutrient
applications.................................. 75
aralﬁ SLS of tie Spin Data........................... 75
viii
,I .73! .l»
t. ‘ l‘_f"!'T—V~" (V L I .. w
«iﬁLb Or deth n3  corhiunied
CHAPTER Page
1&1f.alfja0 O O O O O 0 O O O O O O O O O O O O O I O O 0 O O O O I O O C O O O O O O O O O O O O O O 79
Analysis of theC ontinuous Corn Data.................,,. 79
ﬁaximum Yield and High Profit Combinations of
Plant Putrients................................ BC
Analysisc of t1e Bean Data Is on the Corn, Beans and
Wheat Rotation....................................... cl
3x1nun Yields and Optimum Ir put sof Plant
Nutrients...................................... 83
Analysis of the Potato Data............................. 90
Laximum Yiel;ls and Iiign Profit Plant Nutrient
npplications................................... 92
V SlUiCES C‘F UNEIPIJI I VIIRLLIV‘“ IN YIELDS IIIID BLIS OF
1 SI
iLOEﬁIﬁICILJ:::‘OOOOOOOOOOOOOOOOOOOOOO0.00.0000... 9L1.
Sources of Unexplained Variance in Yields............... 05
Experimental Error................................... 95
Uncontrolled and Unmeasured Variables................ 97
Affects of Within Treatment Variance on Stauistical
Estimates......................................... 93
Factors Related to the Independent Variables Used in
Regression Analysis.................................. 100
Incidence of beads, Lodging and Plant Disease........ 100
Relationships Between Residual Fertility and Crop
Yields............................................ 102
Affects of Residual Fertility on Wheat Yie lds..... lOU
Affects of Res’dual Fertility on Bean Yields...... 10?
Conclusions............................................. 110
VI EYJILLUA'LTTOhI OF PROCL:JUI)1‘J’S IXIT.) PBZ—fbrnvaTs 0 O O O O O O O O O C O O O O O C O O O O O I 11:]—
Evaluation of Ecperimental Des irns...................... lll
Evaluation of EXperirn ental P'rocedures................... llh
Evaluation of Analytical Procedures..................... 116
The Continuous Function Analysis..................... llé
Utilization of Soil Test Heasures.................... 121
‘Economic Interpretation and.Evaluation of Results....... 12
Concluding Remarks...................................... 12L
BIBLJIOQmPEIYOOCOOOOOOOOOOOOOCOOOOOOQOOOOOOCOOOOOOOOOOOOOOOOOOOOOO. 127
ix
10.
ll.
12.
13.
1h.
15.
16.
17.
LIST OF TABLES
United States Fertilizer Consumption 19101955..............
Michigan Fertilizer Consumption 19391955...................
Experimental Design for the Oats, Wheat, Alfalfa and Corn
Rotation....................................................
Experimental Design for the Continuous Corn Experiment......
Experimental Design for the Beans, 'heat and Corn Rotation..
Experimental Design for the l95h Potato Experiment..........
Experimental Desipn for the I956 Potato Experiment..........
Observed and Estimated Oat Yields, l956.....................
Changes in Oats Yields Resulting from Unit Changes in
Nitrogen fippljcati0nSOOOO0.00.0.0...OOOOOOOOOOOOOO0.00.00...
Changes in Cats Yields Resulting from Unit Changes in
P205 ApplicationSoOOOOOOOOOOOOOOO0..OOOOOOOOOOOOOOOOOOOOOOOO
Changes in Cats Yields Resulting from Unit Changes in K20
ApplicationSOO0.0.0.0....OOOOOOOOOOOIOCCOOOOOOOOOOOOOOOOO...
Observed and Estimated Wheat Yields, 1956...................
Changes in Wheat Yields Resulting from Unit Changes in
Nitrogen Iipplj.cati0n800000..OOOOOOOOOOOOOOOOOOOOOOOOO0......
Changes in Wheat Yields Resulting from Unit Changes in
P205 AppliLZationSOO.OOOOOOOOOOOOOO...0.0.0.0...0.0.0.0000...
Changes in Wheat Yields Resulting from Unit Changes in K20
Applications.OOOOOOOOOOOOO0.00.00.00.000.0.0....0.00.0000...
Comparison of Observed and Predicted Corn Yields on a
Kalamazoo Sandy Loam Soil, l95§.............................
Observed and Estimated Bean Yields, l956....................
61
63
67
‘3
\J.)
C37)
pf
LIST OF TRBLES  continued
TABLE
18.
19.
20.
21.
22.
Changes in Bean Yields Resulting from Unit Changes in
Ap131jed hIitrO£jeHOOOOOOOOOIIOOOOOOOOOOOOOOOOOOOOOIOOOOOOOOOOO
Changes in Bean Yields Resulting from Unit Changes in
Applied F‘IIOSC’IﬁlCI'iC [LeidOOOOOOOOOOCOOOOOOOOOOOOOOOOOOOOOOOOO.
High Profit Fertilizer Inputs for Field Beans with Varying
Bean Pricesoo0000000000000.000.00.00000000000coco0.00.00...
Incidence of weed Infestation and Plant Lodging on Oat Plots
as Related to Nitrogen Applications......................... 1
Comparison of Amounts of Yield Variance Associated with
Alternative Production Function Formulations................ 1
Estimated HighProfit Plant Nutrient Applications for
90
Cl
If?
4.x)
Varj.0us CropSOOOOOOOOOOOOOCOOC...0......OOOOOOOOOOOOIOOOOOOO 1—23
xi
FIGURE
1.
\‘l
o
O\
0
LIST OF FIGURES
Partial derivatives of polynomial and exponential functions
for oats with reSpect to nitrogen...........................
Partial derivatives of the polynomial and exponential
functions for oats with respect to phOSphoric acid..........
Partial derivatives of polynomial and exponential functions
for oats Pfi'tkl reSpeCt t0 pOtaShoooooooo0000.00.00.00.0000000
Partial derivatives
functions for wheat
Partial derivatives
functions for wheat
Partial derivatives
functions for wheat
Partial derivatives
functions for beans
Partial derivatives
functions for beans
of the polynomial and exponential
with respect to nitrogen................
of the polynomial and exponential
with reSpect to phOSphoric acid.........
of the polynomial and exponential
with respect to
of a polynomial
with reSpect to
of a polynomial
with
a 4
espect to
potaShOOOCOOOOOOOOOOOOOO
and two exponential
IlitrOSGHOOOOOOOOOOOOOOOO
and two eXponential
phosphoric acid.........
Pap
(D
70
72
CHAPTER I
THE NATURE AND MAGNITUDE OF FERTILIZER USE PROBLEMS
Current Research Problems in Fertilizer Use
Recently much attention has been devoted to the economics of ferti
lizer use in the United States. Research receiving increased emphasis
includes various attempts to determine the most efficient forms and
carriers of the three primary plant nutrients: nitrogen, phOSphorus
and potassium.
A second important area of research is that of attempting to determine
the relative effectiveness of alternative methods of fertilizer appli
cation. One such alternative is broadcasting the fertilizer and plowing
it down prior to planting the crop. A second alternative is placing
all or a portion of the fertilizer in bands of varying depths and dis
tances from the seed. A third method is that of applying all or a
portion of the fertilizer by tOp dressing the growing crop. Other
alternatives include combinations of the above listed procedures.
A third major area of fertilizer research, which is interrelated
with the previously mentioned two, is that of deriving fertilizer input
crop output ratios and relationships. Current research includes deriving
such inputoutput relationships for the three primary plant nutrients
to
for a number of crops on a variety of soil types and for differing
management practices.
This by no means exhausts the list of fertil
work currently being conducted. iowever, it in‘icates three of the
major areas in which fertilizer research is being conducted and illustrates
N
I
the diversity of current fertilizer research. The latter researca area,
that of der
1%
Vin: inputoutput relationships plus an economic interpreta
1*
tion of these relationships, s the primary concern in ‘his thesis.
Derivation of physical inputoutput ratios or physical production
functions is only the first step in an economic analysis designed to
determine optimal fertilizer use. Once such physical relationships have
been empirically established, profit maximization principles can be
employed to determine Optimal fertilizer use with a given set of crop and
fertilizer prices and given the earning power or marginal value productiv
ity of other farm expenditure or investment categories.
The Importance of Fertilizer as an Aoricultural
Productux1Factor
The eXpanded interest and resources currently being allocated to
obtaining more detailed and reliable information about the economics of
fertilizer use appears to be warranted by (l) the importance of fertilizer
as production factor in United States and Michigan agriculture and (2) the
need for greater production from.American agriculture in the years ahead.
The latter can be obtained only by the use of more production resources
and/or a more efficient combination of production factors.
Fertilizer consumption in the United States has increased rapidly
over the past several decades as indicated by the data shown in Table 1.
Consumption of the primary plant nutrients in 1910 totaled h6,000 tons
of nitrogen, h99,000 tons of P205, the common fertilizer form of phos
phorus, and 211,000 tons of K20, the common fertilizer form of potassium.
By 195h these totals had increased to 1,868,000 tons of nitrogen,
2,228,000 tons of P205 and 1,868,000 tons of KéO. Recently particularly
large increases have occurred in the consumption of nitrogen and potassium
with nitrogen consumption more than doubling from 19h9 to 195h. Preliminary
estimates indicate further substantial increases for 1955 with a slight
decline in 1956. The decline in consumption in 1956 was accompanied by
a decrease in total crOp acreage for the United States as a whole during
that year .
Increases in fertilizer consumption have occurred in Michigan with
even greater relative increases in recent years than for the United StateS‘
as a whole. In contrast to total United States consumption which declined
slightly in 1956, Michigan consumption increased slightly over that of
1955. The annual consumption of the primary plant nutrients for Michigan
during the period 1939 to 1955 is indicated in Table 2. During this
period total nitrogen consumption increased over 11 fold from 3,31h to
37,18h tons while consumption of P205 increased from 18,016 to 88,228
tons. Consumption of K20, which was only 9,97h tons in 1939, increased
to 85,3h3 tons in 1955. Assuming a price of $.15 per pound for elemental
. l .
nitrogen, $.10 per pound for P205 and $.11 per pound for £20, the total
1These are the prices currently being used by fertilizer experts
as being typical of prices paid by Michigan farmers.
TABLE 1
UNITED scares FERTILIZER CONSUNPTION 19101955
:—
Year Prhmry Plant I‘éutrients in Thousands of Tons
N P205 th
1910 us u99 211
1920 228 660 257
1925 279 680 282
1930 377 793 35h
l9h0 h19 912 h35
19hl hSB 993 h67
19h2 399 1,131 5&6
19u3 508 1,238 6&3
l9hh 63S 1,h05 6h9
19h5 6&1 1,h35 7S3
l9u6 759 1,671 85h
19h? 835 1,775 878
19h8 8hl 1,8h2 956
19h9 911 1,88h 1,065
1950 1,126 2,073 1,215
1951 1,265 2,091 1,h13
1952 1,h8h 2,218 1,607
1953 1,6u8 2,209 1,720
195h 1,868 2,228 1,868
1Source: agricultural Statistics 1955, U. S. Department of.Agri
culture (Washington:
U. 3. Government Printing Office, 1956).
TABLE 2
MICHIGAN FERTILIZER CONSUMPTION 19391955
Year Primary Plant Nutrients in Tonsl
N P205 K20
1939 3,318 18,016 9,9'8
1980 3,931 19,672 11,078
1981 8,588 21,328 12,175
1982 8,991 38,730 19,303
1983 8,651 39,967 19,280
1988 7,223 36,687 19,628
1985 7,995 37,078 28,909
1986 9,235 51,291 26,096
1987 9,821 87,823 27,986
1988 9,898 56,361 32,186
1989 12,078 59,923 37,898
1950 18,898 66,786 85,171
1951 16,981 70,002 56,272
1952 21,798 75,937 66,513
1953 23,887 75,117 70,253
1958 30,190 76,27? 78,172
1955 37,188 88,228 85,383
1Source: Michigan Agricultural Statistics, (Michigan Department of
Agriculture, July, 1956). These estimates were made by the Soil Science
Department at Michigan State University.
cash expenditure for Michigan would have been $11,155,200 for nitrogen,
$17,685,600 for phosphorus, and $18,775,860 for potassium in 1955.
The total cost for all three of the primary plant nutrients would have
been $80,630,280 in 1958 and $87,576,260 in 1955.
Although not all fertilizer is used in production of agricultural
crOps, nonagricultural uses in Michigan were estimated1 to be only about
5.3 percent of the total nitrogen, 2.1 percent of the total P205 and
0.9 percent of the total K20 consumed. Estimates made in The 1958 Census
2
of.Agricu1ture indicate the total expenditure for fertilizer for farm
use in Michigan was only'$31,l63,000 in 1958. Consequently, at least a
portion of the plant nutrients were purchased at prices less than those
listed as typical. The estimated cost for the total onfarm consumption
of the three primary plant nutrients for the entire United States was
$1,028,105,000 in 1958.3 Thus, farm expenditures on fertilizer exceeded
a billion dollars in 1958and was still increasing.
Reasons for Increased Fertilizer Use
Several reasons exist for increased use of commercial fertilizer
by farmers. Plant nutrients have become much cheaper relative to most
other farm inputs due primarily to a reduction in bulk and utilization
lEstimates made by'W. H. Heneberry, Department of.Agricu1tura1
Economics, Michigan State University.
2"Use and Expenditures for Fertilizer and Lime," adapted from
The 1958 Census of.Agriculture, (washington: U. S. Government Printing
Office, 1956).
3Ibid.
of more efficient manufacturing processes. Excluding transportation
costs, the 195855 price of a unit of nitrogen was only about one
third of the adjusted 1920 price.1 A unit of K20 was only onefifth of
the adjusted 1920 price in 195855 while the adjusted price of a unit
of P205 decreased about 27 percent during this 35year period.
A second important reason for increased fertilizer use is the
availability of more information concerning the yield benefits realized
by various crops from application of the primary plant nutrients. This
information has been forthcoming in increasing quantities from numerous
sources. Experimental results from Agricultural Experiment Stations and
private fertilizer companies have been.utilized by farmers. .Agencies
such as the Federal Extension Service, the Tennessee Valley.Authority
and others have aided in providing farmers with educational materials
and demonstrations of the affects of fertilizer on crop yields. In ad
dition, farmers personal experiences with plant nutrients together with
those of their neighbors are the basis for increased fertilizer use by
many farmers.
It seems that we can validly conclude that commercial fertilizer
is an important agricultural production factor as indicated by the fact
that the value of the three primary plant nutrients used exceeded a
billion dollars in 1958. It is a productive input used by a great number
1T. P. Hignett, "Our Changing Technology," Methodological Procedures
in the Economic Analyses of Fertilizer Data, Edited by E. L. Baum,
Earl O. Heady and John Blackmore7fhmes: Iowa State College Press, 1956)
p. 205.
of farmers producing a variety of crops. Farmers have greatly expanded
fertilizer use in the past decade. They need additional information as
to what expenditures for fertilizer are yielding in dollar returns.
Such information is ne essary if farmers are to allocate optimally their
capital resources between alternative farm investments and expenditures.
Lonngun Agricultural Production Needs
One of the major problems currently facing.American.Agriculture is
that of surpluses for some of the major farm crops. In view of this
problem, a question arises as to the logic of engaging in research which
could result in recommendations indicating greater use of commercial
fertilizer, larger crOp yields and greater total production.
Several studies have been made in which attempts have been made to
forecast future needs for farm products in the United States. Predictions
of future potential demands for agricultural products are all considerably
higher than quantities supplied by current production. Two factors
seem to be of primary importance in these higher predictions. First,
large population increases have been predicted. Using the period 195153
1
as a base, predictions made by the Bureau of the Census in 1955 are for
a pOpulation increase of 11 percent by 960 and an increase of more than
onethird of the base period population by 1975.
U)
ca
0
O
E
p:
(*1
U
“s
(5
[—1 o
ncreases in consumer income accompanying an xpand
l
' nave been predicoed. Estimates made by the United States
4
ing econom
1Bureau of the Census, Current POpulation Reports, Series P25
No. 123 (washington: U. S. Government Printing Office, October 20, 1955).
'Department of Agriculture indicate an increase of real per capital con
sumer income of almost twothirds greater than the 195153 base period
by'1975.l Estimates of total crop production needs for 1975 are about
25 percent above actual 195153 production.2 This overall increase is
not uniformly distributed over all crops, however. For example, more
than preportional increases are predicted for pasture and feed grain
crOps since a needed increase of h5 percent in livestock production is
forecast. The needed average yearly increase in production of feed
grains from the 195153 base period to 1975 is 5 1/2 times the historical
average annual longterm increase.
No attempt will be made here to provide a comprehensive analysis of
future agricultural production needs. Rather, the point being made here
is that agricultural production needs will be much higher in the years
ahead. This greater production must come from use of more resources,
3
more productive resources and/or a more productive combination of resources.
1H. H. Wboten and J. R..Anderson, “Agricultural.Land Resources in
the United States~with Specia1.Reference to Present and Potential Crop
land and.Pasture," Agricultural Information Bulletin IhO (washington:
U. S. Department of.Agriculture, June, 1955).
2G. T. Barton and R. 0. Rogers, "Farm Output, Projected Changes
and Projected Needs," Agricultural Information Bulletin No. 162
(washington: ‘Agricultural Research Service, August, 1956).
3A particularly critical problem currently faced by farmers and by
‘farm management researchersis that of finding combinations and quantities
of other resources which will increase the marginal value productivity of
labor. Numerous farm management studies have indicated an extremely low
marginal value product for this extremely important farm resource.
A discussion of the low'marginal value productivity of labor as
well as a bibliography of other work on this subject may be found in,
E. I. Fuller, "Michigan Dairy Farm Organizations Designed to Use Labor
‘Efficiently," Unpublished Masters Thesis, Department of.Agricultural
Economics, Michigan State University, 1957.
10
Research workers in the Agricultural Research Service1 have made pro
jections of probable increases in pasture and cropland acreage of 25
million acres by 1975 or an increase of about one million acres per
year. If the projected increase in cropland occurs, the necessary annual
increase in crop production per acre will still be about 50 percent
larger than that occurring in the post world war II period.
In view of the longrun needs for farm products it is apparent that
there will be a need for improved or increased use of farm resources in
the next two decades. Improved information about the productivity of
various resources, including fertilizer, will help farmers make the
necessary production adjustments on an economical basis.
The_Iype of Information Needed by Farmers
In order to make economically sound decisions regarding how much
and what analysis of fertilizer to use, farmers need rather Specialized
information. First, they need information on yield reSponse to the three
primary plant nutrients of the various crOps which they produce. They
need information about the affects of different forms of fertilizer and
different application methods. In addition, this information must be
applicable to their particular type of soil, the soil management practices
which they use or should use, and the weather conditions which they
encounter. Differences in the fertility level of the soil will influence
the yields obtained by various amounts of applied plant nutrients; thus,
ca.
1H. H. Wboten and J. R..Anderson, gp, cit.
11
.effects of residual fertility need to be known. Finally, the price of
fertilizer and the price of the crop produced will influence the high
profit combination of plant nutrients to apply.
Specification of the type of information needed by farmers is a
guide in determining what research is needed and what research procedures
may be followed in obtaining this information. For example, the effects
of different variables such as the effects of the various plant nutrients
on crop yields, the affects of weather on yield reSponses and the signifi
cance of crop and fertilizer prices on Optimal fertilizer use will be
treated quite differently in the analysis. Applied plant nutrients can
be measured and controlled and their effect on crop yields determined by
statistical estimation. weather cannot be controlled but if experi
mentation is carried out over a number of years and a variety of weather
conditions, yield reSponses for several sets of weather conditions and
a probability distribution of responses with respect to weather can be
acquired. In the case of crops and fertilizers, various prices may be
applied to the physical inputoutput relations to correSpond with expected
farm conditions.
Currently, researchers are attempting to devise methods for incorporat
ing information about soil fertility acquired by chemical soil tests into
their predictions of yield reSponses to fertilization. An attempt will
be made in this thesis to reduce unexplained variances in crOp yields by
taking into account soil test data.
l2
Succeeding chapters of the thesis will pertain to alternative
analytical procedures, specification of the experimental work being
carried on at the Michigan Agricultural Experiment Station, analysis of
the experimental data and, finally, evaluation of the results.
CHAPTER II
METHODS OF.ANALYSIS
It is the generally recognized task of scientific endeavor to
establish and verify relationships which are universal to some popula
tion.1 When relating various phenomena in the real world we find two
dimensions of such relationships subject to variance. First, the
relationships may vary with respect to the reliability of the empirical
estimates which we can derive or establish for them, i.e., variance in
the reliability dimension. Secondly, the size of the population to
which such relationships are universal may vary considerably, i.e.,
variance in the application dimension. One would not expect, for example,
to establish relationships between plant nutrients and crop yields as
accurate or as general as those which have been established between the
volume and pressure of gas as Boyle's law. However, if we believe that
there are logical, systematic and describable relationships existing
between plant nutrients and crop yields, it seems to be our task as
scientists to attempt to quantify such relationships to the best of our
ability. This is true particularly in view of the need for such infor
mation indicated in Chapter I. In so doing, an optimum level of
1Most of these relationships will of course be probability state
ments about relationships. Thus the universality referred to here does
not imply absoluteness of the relationships specified, but rather implies
universal applicability to some population of the deductions and infer
ences made.
1h
accuracy of quantitative estimates can be defined by equating the cost
of additional accuracy with its value. Failure to structure and
quantify relationships systematically, when such action is possible, is
likely to result in a failure to make Optimum use of scientific procedure
in deveIOping a body of interpersonal information useful to researchers
working on this and related problems of soil fertility and/or farm
management.
Methods of Collecting Data
Two methods of securing data for use in determining the relation
ships existing between variables are generally recognized as being valid
forms of scientific methodologg. These are (1) controlled experimentation
and measurement of relationships and (2) collection of noncontrolled
observations, as in astronomy, which typify the population being studied
and to which relational inferences are to be made. Both of these two
methods have advantages as well as some disadvantages which vary some
what with the nature of the Specific problem being studied. The discussion
which follows is an attempt to evaluate the two procedures in the context
where determination of plant nutrient input—crOp yield output relation~
ships is the problem being investigated.
The former method, controlled experimentation, has the relative
advantage of lending itself to more precise estimation of rel tionships
between relevant variables. Greater accuracy is usually obtained in
controlled experimentation for two reasons. First, variables can be
measured more accurately. Fertilizer applications and crop yields,
15
for example, can be measured quite accurately on experimental plots.
Secondly, controls can be enforced quite rigorously; for example,
tillage practices, insect infestations, soil characteristics,etc.,can be
controlled better on experimental plots than under farm conditions. .
Such controls, though facilitating accurate estimation of relationships
between studied variables have an accompanying disadvantage. This dis
advantage is that there is a possibility that no pOpulation other than
the experimental one may have exactly the same combination of controlled
and uncontrolled variables interacting in the production processes being
studied. It follows that one may not be able to draw inferences from
the experimental results and apply them validly to any given farm popu
lation. The alternative method, that of collecting noncontrolled
observations by a sample survey procedure, has proven effective in
numerous types of research. It is difficult, however, to utilize this
method when acquiring fertilizer reSponse information because (1) dif
ferences in numerous uncontrollable factors such as insect damage,
weather, tillage and harvesting methods are apt to bias the results or
introduce excessive unexplained variance and (2) studied inputs are
difficult to measure accurately. Another shortcoming of using the sample
survey method in estimating fertilizer reSponse surfaces is the difficulty
of acquiring observations dispersed over the range and combination of
plant nutrients necessary to obtain a statistically reliable estimate
of the yield reSponse surface. These and other problems have been
16
encountered by researchers working with nonexperimental data.
It is the opinion of most soil scientists and other researchers
that the controlled experiment method of obtaining data is not only the
more scientific method but the only one producing reliable estimates of
fertilizer inputoutput functions. As experimental inputoutput data
.1.
become available a logical followup stage of analysis would be CO
V , L J. L.) a
test the applicability of these results under farm conditions. This
procedure should indicate whether or not results obtained from the
experimental sample may be validly inferred to some farm population.
The Concept of Functional Relationsnips
The principles utilized by economists in determining various Optimal
conditions of resource use and production output are stated in numerous
publications by numerous authors. However, it seems desirable to outline
briefly some of the principles of economic theory which can readily be
applied to the production relationships of interest in agronomiceconomic
work. In order to apply effectively the deductive principles of economic
theory, the relevant production relationships need to be specified
rather systematically or formally.
1For a discussion of problems encountered and results obtained using
nonexperimental data in fertilizer inputcrOp output determinations see
E. W} Kehrberg, "Some Problems Involved in Fitting Production Functions
to Data Recorded by SoilTesting Laboratories," Methodological Procedures
in the Economic.Analy§es of Fertilizer Data, Edited by E. L. Baum, Earl
O. Heady and John Blackmore (Ames: Iowa State College Press, 1956) pp.
lBhlhO and H. H. Yeh, "Estimating InputOutput Relationships for Wheat
in Michigan Using Sampling Data, l9525h, Unpublished Masters Thesis,
Michigan State University, 1955.
17
Agronomists have hypothesized for years that plant nutrients and
crOp yields are functionally related. Numerous attempts have been made
to Specify these relationships in equation form for various creps and
plant nutrients. In its simplest form, this functional relationship may
be written
Y  f(X)
where Y is the crop yield and X the plant nutrient, in this example
nitrogen. Recognizing that other factors interact with nitrogen, X1,
and are necessary for crop production, we write:
Y . f(X1,X2,oooo,Xi,oooo,Xn)
where X1 represents nitrogen andX2 to Xn are other factors such as
P205, K20, water, temperature etc. To symbolize that all factors except
nitrogen are fixed at some constant level, we write
Y . f(Xl :2,oooo,Xi’oooo,Xn)o
Furthermore, if all factors affecting crOp yields cannot be isolated
and specified, we say
Y I f(X1/X2,oooo’Xi,oooo,Xn)+ U
A
where U is an error term representing the unexplained variance of Y
1
(predicted yield) from Y (observed yield). If it can be validly
11f unexplained variance is to be validly attributed solely to
components of the error term, U, the specified functional relationship
must be the right one, i.e., it must be the real world functional
relationship.
18
assumed that: (1) factors which contribute to U, i.e., unspecified
factors, are normally and randomly distributed with respect to the
measured variables (in this case X1) and (2) that the expected value of
U is zero, the existence of this unSpecified source of yield variance
does not bias statistical estimates of the influence of the observed
variables on Y.
The specification of the functional relationship between plant
nutrients and crop yields, commonly called a production function, has
taken different forms over a period of years. Justice Von.Liebig's
"Law of the Minimum" was an early attempt to Specify the form of fertili
zer production functions. This formulation postulated that crop yields
increased in direct proportion to additions of the nutrient which was
limiting plant growth. Thus, other production factors were assumed to
be perfect complements of the limiting factor. This formulation of the
fertilizercrOp yield production function has been rejected because
researchers have observed that: (1) production factors are not perfect
complements, i.e., a given crop yield may be produced with varying
quantities and combinations of applied N, P205, K20, water etc. and (2)
additional inputs of a factor limiting crop yields does not typically
result in linear additions to crOp yields but rather it results in
diminishing additions to crOp yields for a time and eventually further
additions of the factor cause an actual decrease in total yield.
Since Von Liebig's early formulation, numerous attempts have been
made to use different forms of production functions to describe these
inputoutput relationships. although numerous types of functions have
been formulated, none has been accepted as "best." These various func
1
tions have received adequate discussion in other literature and will
not be analyzed here. There are, however, several criteria which must
be satisfied by a particular function if it is to provide a re:
fornulation of the inputoutput relationships between fertilizer inputs
and crop outputs. The function should be cap able of reflecting suc
cessively the following yield responses to added inputs of plant nutrients:
(l) vields increasing at a diminisuinw rate and (2) decreasi Lng total
yields. If the soil is relatively low in initial fertility, an earlier
state of inputoutput relationship may be present. This is the stage
C.
"I 1
where yieics increase at an incram sin: rate in response to additional
3
input 3 of plant nutrients. In addition, if interaction between plant
nutrients is expected, Le formulation should incluce equma Wior 1 variables
'3
to Specify this interaction.
If the characteristics of plant growth and crOp yields could be
formulated theoretically to the extent that a prOper equational form
lHistorical descr ription of use of production functions in estimat—
t ilizcrCTOP yield relations may be fornd in the following 7ubli
ing fer ‘
cations: John C. Redman ands at eph 1en Q. allen "Some Int+3 rolationsutns
f acoromnc and.A:ronomic Concepts ," Journal of Farm BC or emits, vol. KIIVI
(nu :ust, l9:~d), pp. L5 2 UGS and T r‘l C. deady, John T. Pese: and William
Br wn, Crop i": orte Surf? '0'}: and Esono C (Ah a 1:1 Ft.ert?'l__.i7;er Use,
Research Bulletin l¢., Agricultural ixperiment Station, Iowa
tate College, lQSS).
2Such interaction may be incorporated into the functional relation~
ship in several ways. It is in a sense automatically included in a
production function of product form such as an exponential. Special
cross product terms may be included in a polynomial type equation. The
point of importance is that it be included so that partial derivatives of
yield with reSpect to individual plant nutrients reflects the level at
which other interacting nutrients are considered.
20
could be deduced, statistical estimation of the production function would
be greatly singlified. The statistical task would then be only that of
estimating parameters for the variables in the functional relationship
and obtaining reliability measures for these parameters. However, lack
ing a precise theory as to the proper functional form, it is necessary
to compare various equations to see which "best" describes the observed
relationships. Various problems of design and alternative analysis
necessitated by lack of knowledge about the apprOpriate functional form
will be developed later in this chapter.
Detrnnhioijvt'ﬂeormuvic (hitira
After obtaining an estima,e of the production function for plant
ﬁ
nutrients, various optimal combinations of phant nutrients may be
determine". If for example, the following equation:
Y  + b " + h ” 3 + b T + b V 9 + b " + b V 2
‘ a 1&1. “2“1 see ar~2 the see
describes the relation of yield to the three plant nutrients Kl, KB, and
X3, then the following proeoiuce is used to find the combination of
plant nutrients producing the naximua yield.
the *artial derivatives of the three nutrients with respec
to yield gives:
0 ’
 1 FA 1.
(l) —+ = 01 + aogal
A
h)
v
n
0"
m
+
2134:: 2
TT'
+ 20043.
G]
t. _
U)
u)
c1
('1‘
r
,4
(I)
as
,0
’t‘
{11
’ S
C 
t...
9)
p4
CL
6)
*3
t...
f
U:
pvacive equal to zero and olving the three
equations simultaneously gives the combiaation of plant nutrients pro
ducing the maximum crop yield. To obtain the economically optir O . u ‘ﬂ  V '7 0 J’ J. . w’ .0 _ Q _. “'1 ~~ > ‘ ‘_¢ '1: 0 1‘ ‘ r _ '§ 3.
have sen or.s1 ied primarily with the OOjective Oi pIOvLLlRH ahuQHdhC
data for estiration of the reSpective crop j/ie ld surfaces icr c..l;1ut
#3 A] ‘1. nsx'» . ‘\r\ . . ,‘A‘. '~“ mmmmm ,. .\ ‘I 1‘,
mvzral int er ested ajenciss LaVC aLuJQ the e: p “ilenta 20" by
.L . ,. ... A.  J.‘  .. , . ,. .L‘. ,.. .' m1, .1. ' , .
contrisuting funds or OUJ’F r; {urCUu, ULCQC include: ;.3 lLa at the l Ilant
ﬂ r a '..‘ m“, ' ‘l‘ 9v. " ‘45 o ‘r (\J“ . h ,1, r“ \ "if“ '3'! 1. '1"
reed COdnCil, The navison Che teal Covworetion and The iw.vess3) Ja'iq
‘ J‘ .3 J
all CEO 1‘4. by .
’3’“
it“.
.1
33
crops. This objective, together with the restriction Of limited funds
for experimentation, provide the main restrictions for the framework
within which the experimental designs were developed. In conforming to
these restrictions the designs have the following general characteris
tics: (1) individual observations cover those portions of the production
surfaces of interest to researchers, (2) the experiments contain a
minimum number of replicated plots since the objective is to estimate
the entire surface over the range in which it is Of economic importance
thus minimizing the need for establishing accurate measurements of
individual surface points (3) the designs involve numerous check plots
(plots to which no fertilizer is applied) to establish the origin of
fitted functions, i.e., the yield value with no plant nutrients applied
and (h) to the extent possible, intercorrelations among the amounts of
nutrients applied have been minimized to facilitate estimation of the
equational parameters with greater reliability than would be the case if
such intercorrelations were high. The designs vary somewhat for different
experiments but may be broadly classified as incomplete factorials.
The Cats, Wheat,gglfalfa and Corn Rotation
In the Spring of 1955 an experiment was initiated for a rotation
of oats, wheat, alfalfa and corn. This experiment is located at two
sites in Kalamazoo and Calhoun counties on a Kalamazoo sandy loam soil.
This is a light upland soil having a tendency to be somewhat drouthy and
of relatively low natural fertility. Each crop of the rotation is
3h
grown each year; thus there are four fields each having the same experi—
mental design. The experiment includes all three Of the primary plant
nutrients, nitrogen, phosphoric acid and potash in varying combinations.
Six treatment levels, including the zero application level, are included
in the experiment for each of the plant nutrients. These treatment
levels measured in pounds per acre are:
N  o 20 to 80 160 2&0
P.o5  o to 80 160 320 hBO
K20  o 20 to 80 160 2ho
Ninetyone individual surface points are sampled, twentyseven of which
are replicated twice in a 3 x 3 x 3 factorial at the 2nd, hth and 6th
treatment levels. There are eleven replications of the check (0,0,0,)
treatment.
There are one hundred and thirty plots in each of the four fields
in the experiment.
Individual plots are SO x 1h feet in size, making a total area per
plot of about l/62.S of an acre. The lh foot width facilitates use of
a 7 foot grain drill for fertilizer and seed application and a 7 foot
self propelled combine for harvesting Operations. Almost all fertilizer
applications are made by broadcasting the fertilizer, either mechanically
or by hand, prior to plowing the ground and preparatory to planting the
crop. Two notable exceptions are: (l) the first level of applied P205
(ho pounds per acre) is applied in the row at planting time as a starter
35
fertilizer and (2) the alfalfa crop is fertilized by topdressing in
the Spring. The design for this experiment is shown in detail in
Table 3.
Continuous Corn
An experiment in which corn is grown in continuous culture was
initiated in Tuscola County in 1956. This experiment is located on a
Wisner clay loam soil which is one of the heavier,more productive soils
occurring in the state. The experiment contains 20h individual plots
representing 139 surface points. Included in the design is a 3 x 3 x 3
factorial replicated three times including observations at the 2nd, hth,
and 6th treatment levels. In addition, there are eight check plots.
Inclusion of the triplicated factorial allows limited study of yields
and other experimental data by analysis of variance techniques. The
seven treatment levels in pounds per acre for the three plant nutrients
in this experiment are as follows:
N  0 20 &0 80 160 2&0 320
19.05  0 &0 80 160 320 &80 6&0
6
K20  o 20 &0 80 160 2&0 320
Individual plots are 55 x lh feet in size allowing h rows Of corn Spaced
h2 inches apart to be grown on each plot. The design for this experiment
is shown in detail in Table h.
36
TABLE 3
EXPERIMENTAL DESIGN FOR THE OATS, WHEAT, ALFALFA AND CORN ROTATION
Plant Nutrients NO. Plant Nutrients NO.
(Pounds Per.Acre) of (Pounds Per.Acre) of
N P205 K20 Plots N P205 K20 Plots
0 O O l 80 160 80
O O O 80 160 2&0
O hO 20 80 320 ‘ hO
0 160 O 80 320 160
O 160 80 80 320 ZhO
O 160 2h0 80 NBC 0
O hBO 80 80 h80 20
O hBO 2hO 8O hBO 80
2O 0 20 80 b80 160
20 &0 O 80 hBO 2hO
20 hO 20 160 hO to
20 DO 80 160 to 160
20 hO ZhO 160 80 20
20 80 hO 160 80 DO
20 80 160 160 80 80
20 160 20 160 80 2hO
160 160 hO
160 160 160
160 160 2&0
20 160 80
20 160 ZhO
20 320 hO
n>FJADRJFJFJFJF’FJRDRJRDFJRJRJRJFJF4FJFJFJFJFJRDFJFJFJFJP‘F‘F4FJFJPJFJRJFJAJRDFJF‘F‘FJAJAJ
20 320 160 160 320 20
20 &80 20 160 320 80
20 &80 80 160 320 160
20 &80 2&0 160 320 2&0
&0 &0 &0 160 320 &0
&0 &0 160 160 &80 80
&0 80 20 160 h80 160
&0 80 &0 160 &80 2&0
ho 80 80 QhO O 80
&0 80 2&0 2&0 0 2A0
&0 160 &0 2&0 &0 20
&0 160 160 2b0 &0 80
&0 320 20 2&0 ho 2h0
&0 320 80 2&0 80 160
&0 320 2&0 2&0 160 20
&0 &80 he 2&0 160 80
&0 &60 160 2&0 160 2&0
8O 0 O ZhO 320 O
80 0 80 2&0 320 &0
80 0 2&0 2&0 320 160
80 &0 20 2h0 320 2h0
80 &0 80 2&0 &80 0
80 &0 2&0 2&0 use 20
80 80 hO 2h0 hBO BO
80 80 160 2&0 &80 160
80 160 0 2&0 &80 2&0
nDF‘FJFJAJRDRDFJFJFJFJFJF‘F’F’F’F‘F’F’RDFJFJFJnDRJAJFJF’AJAJRJFJF4AJADAJFJPJF‘F‘F‘F’F‘F‘F‘P‘
80 160 20
‘1
1
J
313 h
tn: _
Li
EKPeRILEKIAL ssszen FOR THE
csnumxtsccmzssustuir
37
Plant I‘éutrients I~.(:. Plant I‘Iutrients I20.
(Pounds Per Acre) of (Pounds Per Acre) of
N P205 K20 Plots H P20r K20 Plots
0 0 c a hC 320 to 1
C C 11.0 1 LC! 22C 1 :11) l
0 &3 2&0 1 &o 32‘ 320 1
0 80 0 1 &0 &80 &0 1
0 80 &0 1 &0 &80 80 1
0 160 320 1 &0 &80 2&0 1
0 320 160 1 &0 6&0 20 1
0 &80 20 1 &0 6&0 160 1
0 6&0 80 1 &0 6&0 320 1
0 6&0 320 1 80 0 0 1
20 &0 20 3 80 0 160 1
20 &o 80 3 80 &0 20 3
20 &0 160 1 80 &0 80 3
20 &0 2&0 3 80 &0 2&0 3
20 8O 20 1 80 80 &0 1
20 80 80 1 80 80 160 1
20 80 2&0 1 80 80 2&0 1
20 160 20 3 80 160 20 3
20 160 &0 1 80 160 80 3
20 160 80 3 80 160 2&0 3
20 160 2&0 3 80 160 320 1
20 320 20 1 80 320 0 1
20 320 160 1 80 320 &0 1
20 320 320 1 80 320 160 1
20 &80 20 3 80 &80 20 3
20 &80 &0 1 80 &80 &0 1
20 &80 80 3 80 &80 80 3
20 &80 2&0 3 80 &80 2&0 3
20 6&0 160 1 80 6&0 80 1
20 6&0 320 1 80 6&0 320 1
&0 o 0 1 160 0 20 1
&0 0 &0 1 160 0 80 1
&0 &0 20 1 160 &0 0 1
&0 &0 &0 1 160 &0 80 1
&0 &0 80 1 160 &0 2&0 1
&0 &0 160 1 160 80 &0 1
&0 &o 320 1 160 80 160 1
&o 80 0 1 160 160 0 1
&0 80 &0 2 160 160 20 1
&0 80 2&0 1 160 160 80 1
to 160 20 1 160 160 160 1
&0 160 80 1 160 160 2&0 1
no 160 160 1 160 160 320 1
&0 160 2&0 1 160 320 &0 1
&0 320 20 1 160 320 80 1
&0 320 &0 1 160 320 160 2
Continued
38
Table h concluded
Plant Nutrients NO. Plant Nutrients No.
(Pounds Per Acre) of (Pounds Per.Acre) of
N P205 K20 Plots N ’ P205 K20 Plots
160 h80 20
160 hBO 80
160 hBO 320
160 6hO 80
160 6hO 2hO
2&0 6h0 160
2&0 6&0 320
320 0 80
320 &0 20
320 to 2&0
2&0 0 0 320 80 to
2&0 0 &0 320 80 80
2&0 &0 20 320 80 160
2&0 &0 80 320 80 2&0
2&0 &0 2&0 320 80 320
2&0 80 0 320 160 0
2hO 80 &0
2&0 80 160
2hO 80 320
2&0 160 20
2hO 160 80
2h0 160 2h0
2h0 320 to
2h0 320 160
2h0 320 320
2h0 h80 20
2hO hBO 8O
2hO h8O 2h0
2hO OhO &0
320 160 &0
320 160 320
320 320 &0
320 '320 80
320 320 160
320 320 2&0
320 &80 20
320 &80 160
320 &80 320
320 6&0 &0
320 6&0 80
320 6&0 2&0
320 6&0 320
FJbJUJU)F‘FJFJuJquJFJFJF‘FJUJbJUJF‘F‘F’F’F’F’F’
nJF‘F’F‘F‘F’FJFJFJFJF‘F‘F’F‘F‘F’F‘k’k‘k‘k‘k’k‘F‘
Field Beans,_Wheat and Corn Rotation
An intensive rotation of field beans, wheat and corn was initiated
in Gratiot county in 1955. Corn was produced on these plots in 1955 and
field beans in 1956. The experiment is located on a Simms loam soil, a
heavy productive soil which can be crOpped.quite intensively without
hazard of erosion damage. The seven treatment levels for the three
plant nutrients are identical to those in the continuous corn experiment.
39
The treatments in pounds per acre of applied plant nutrients are:
N  O 20 hO 80 160 2hO 320
PnO5  0 ho 80 160 320 hBO 6hO
~ K20  O 20 &0 80 160 2hO 320
The experiment is an incomplete factorial consisting of 193 individual
surface points of which twentyseven are replicated twice in a 3 x 3 x 3
factorial at the lst, hth, and 6th treatment levels. There are ll check
plots in the basic experimental design which contains a total of 233
individual plots. Extra plots were included in the experiment for purposes
of other analyses bringing the total number of plots to 258. Individual
plots in this experiment are SO x lh feet in size.
The design for this experiment includes a more complete Specification
of the production surface than any of the other experiments with the total
of 193 different surface points exceeding that of any other experiment
currently being conducted. The experimental design for this experiment
is shown in Table 5.
Potatoes
Two experiments have been established to measure the response of
potatoes to variable quantities of applied P205 and K20. The first of
these experiments was initiated in l95h and the second in 1956. Both
experiments are being conducted on a Houghton muck soil on the Experiment
Station muck farm near East Lansing. Only two plant nutrients, P205 and
K20 are treated as variables in this experiment. The muck soil is high
in organic matter content and consequently high in nitrogen. Available
TABLE 5
EXPERIMENTAL DESIGN FOR THE BEANS, WHEAT AND CCBN ROTATION
hO
I
Plant Nutrients NO. Plant Nutrients No.
(Pounds Per.Acre) of (Pounds Per Acre) of
N P305 K20 Plots N P205 K20 Plots
0 O 0 ll 20 320 160 2
O 0 2O 1 20 320 320 2
O 0 ho l 20 hBO 2O 1
O O 80 1 2O hBO hO l
O O 160 l 20 htO 160 l
O O 2hO 1 2O hBO 320 l
O O 320 l 20 OhO 2O 2
0 ho O l 20 GhO hO l
O &0 2hO l 20 ého to l
0 to 0 1 20 6&0 160 2
O to &0 l 20 6hO 2hO l
O 160 O l 20 OhO 320 2
O 160 320 l &0 O O l
0 320 0 1 &0 0 &0 1
O 320 160 l &0 hO 2O 1
O hSO O 1 ho hO to l
O th 2O 1 hO hO 160 l
0 6&0 0 1 &0 &0 320 1
O 6hO 60 1 ho to O l
0 6&0 320 1 &0 80 &0 2
2O 0 O l hO BO 2hO 1
2O &0 2O 2 ho to 320 l
20 hO hO 1 ho 160 20 l
20 &0 60 1 &0 160 60 1
2O hO 160 2 hO 160 160 1
2O hO 2hO 1 ho 160 320 l
20 hO 320 2 hO 320 20 l
20 8 2O 1 hO 320 80 l
20 SO 80 l hO 320 160 l
20 80 160 l hO 32 320 l
20 to 2hO l &0 hCO LO 1
2O . 60 32C 1 hO hOO to l
20 ISO 20 l &0 th 2hO 1
20 ISO &o 1 ho 6&0 2O 1
20 160 150 l hO ehO 60 1
2C 160 2&0 l hO éhO 160 l
20 160 32 l &o 6&0 320 l
20 32C 20 2 OC O O l
20 320 LO 1 CO 0 320 l
20 320 tO I SO no 20 l
_‘ .  I
Continued
Table 5 continued
Plant Nutrients NO . Plant Nutrients NO .
(Pounds Per.Acre) of (Pounds Per Acre) of
N P 205 K 20 Plots N P 205 K 20 Plots
80 to 80
8O &0 160
80 &0 320
80 80 &o
80 80 2&0
80 80 320
160 320 20
160 320 &0
160 320 80
160 320 160
160 320 2&0
160 320 320
80 160 20 160 &80 20
80 160 80 160 &80 &0
80 160 160 160 &80 160
80 320 20 160 &80 320
80 320 80
80 320 160
80 320 320
80 &80 &0
80 &80 2&0
80 &80 320
80 6&0 o
80 6u0 20
80 6&0 &0
80 6&0 160
80 6&0 2h0
80 6&0 320
160 6&0 20
160 6&0 &0
160 6&0 80
160 6&0 160
160 6&0 2&0
160 6&0 320
2&0 0 0
2&0 &0 20
2&0 hO 80
2&0 &0 160
2&0 &0 320
2h0 80 NO
160 O O 2h0 80 320
160 O 160 2hO 160 20
160 hO 20 2hO 160 80
2h0 l60 160
2h0 160 ZhO
2h0 320 20
2ND 320 80
2ND hBO 160
2&0 320 6&0
2hO hBO hO
2h0 hBO 80
2hO hBO 2h0
2h0 6h0 20
2110 611,0 80
2&0 6ND 160
2h0 6&0 320
320 O 0
320 O 80
160 &0 &0
160 &0 80
160 &0 160
160 &0 2&0
160 &0 320
160 80 20
160 80 80
160 80 160
160 80 2&0
160 80 320
160 160 20
160 160 &0
160 160 160
160 160 320
160 320 0
I—‘l—‘l—‘i—JHl‘Hl‘HHmHmHHml‘l—‘HI‘HHHI—‘I—‘HHHHHHHNHl—‘Hl‘l—‘HH
HHHHHHNHHI—JHHHHHHHHHHHHHHNHNHHml—‘HHHNHNHHM
Continued
h2
Table 5 concluded
Plant Nutrients No. Plant Nutrients NO.
(Pounds Per Acre) of (Pounds Per Acre) of
N P205 K20 Plots N P205 K30 Plots
320 0 320 1 320 320 &0 1
320 &0 20 2 320 320 80 1
320 &0 &0 1 320 320 160 2
320 &0 80 1 320 320 2&0 1
320 &0 160 2 320 320 320 2
320 &0 2&0 1 320 &80 20 1
320 &0 320 2 320 &80 &0 1
320 80 20 1 320 &80 160 1
320 80 80 1 320 &80 320 1
320 80 160 1 320 6&0 0 1
320 80 2&0 1 320 6&0 20 2
320 160 0 1 320 6&0 &0 1
320 160 20 1 320 6&0 80 1
320 160 &0 1 320 6&0 160 2
320 160 160 1 320 6&0 2&0 1
320 160 320 1 320 6&0 320 2
320 320 20 2
nitrogen may be in short supply at a given time but largely because Of
weather conditions not conducive to sufficiently rapid nitrification.
Thus, it usually does not pay to apply commercial nitrogen fertilizers
unless temporary nitrogen deficiencies are evident. Furthermore, if
nitrogen applications are made, the amount of applied nitrogen is, at
best, a poor indicator of the amount of nitrogen available for plant use.
The l95h Potato Experiment
1
The experiment initiated in l95h consists of an incomplete factorial
1The experimental design and treatment rates used in this experiment
were established by Professors G. L. Johnson and J. F. Davis of the
Michigan.Agricultural Experiment Station.
b3
of seven levels of P205 and nine levels of K20. The treatment rates in
pounds per acre are as follows:
P205  l 25 50 100 200 300 hSO
K20  l 25 50 100 200 350 550 750 900
The design includes fortyseven surface points, twentynine of which are
replicated twice for a total of seventysix plots in the experiment.
Individual plots are h9 by ll feet in size.
Application of a single pound of a plant nutrient to some plots
constituted a substitution for the zero treatment level. Utilizing a
one pound treatment alleviated the problem of having to use negative
logarithms when fitting exponential functions to zero treatments. The
design for this eXperiment is presented in detail in Table 6.
After the first potato crop was produced on these plots, the plots
were Split and half of the plot continued to receive the original
fertilizer treatment while the other half of the plot received no ferti
lizer in subsequent years. This procedure was practiced because of
complications due to the high fertility level of the land at the time
the experiment was initiated. This original high fertility level resulted
in negligible yield changes with additional applications of plant
nutrients. Continuous soil testing Of these experimental plots in
succeeding years should provide valuable information with reSpect to
residual fertility values Since over time a wide range of fertility levels
should deveIOp on these plots.
*J
1.4) '._l 6
assumes. Deena»: FOR "we 19% POTATO Hemmer?
1
P205 Treatment (pounds per acre)
1 25 50 100 200 300 hSO
F"
N
N
>4
(R
p;
“i.
j
:3
{>4
s per acre)
g.
N
ii
1
'C‘;
X XX
£301 71
H
O
O
E: ii
(~.
+
U
200 X XX XX X
:3;
he
use
>4
£2
350
550
750 "5
K20 Treatmen
ii ii
MN
:23
{>4
ii
>5
:2”
g;
XX XX
1
1 r~ : 1’,
AA A
p4
900 X Y
lEach X represents one experimental plot.
m1 A r 3 1,.:. 1..... .' M. .1.
nr3135e Iotam3.iperinenu
A new potato experiment was inititted in l956. This eXperiment was
established on a newly cleared muck soil which had not been previously
farmed and which had not received previous applications of plant nutrients.
The experimental design includes at different surface points. neplications
and check plots bring the total number of plots to 114. All surface points
are replicated twice with the exception of a h x h triplicated factorial
amw.h.check plots.
The design utilized in this experiment is basically an incomplete
factorial but it also includes several additional features. Included
in the design are:
(l) A h x h triplicated factorial. The treatment levels included
in this factorial are the following in pounds of plant nutrients per
acre:
P205  100 200 300 hOO
K20  200 &00 600 800
Inclusion of this triplicated factorial allows limited study by analysis
of variance procedures. The experiment produces a large amount of useful
agronomic data as a byprOduct of the basic inputoutput study. Such
data includes information on the quality and chemical compositions of
the product, plant characteristics, residual fertility,etc. Analysis of
these data by continuous function analysis may not be feasible or
appropriate.1 By including a triplicated factorial in the experimental
design, analysis of variance treatment Of such data is facilitated at a
small additional cost.
2
A 3 x 3 composite design is included in the experiment for the
1For example, protein content Of wheat may increase linearly or
curvilinearly with additional nitrogen inputs up to some maximum value
and then remain unchanged with additional nitrogen inputs. In such an
event, continuous function analysis might not be the appropriate means
of analyzing data to acquire determinations of quality differences.
2This design is described by R. L. Anderson in "A Comparison of
‘Discrete and Continuous Models in.Agricultural Production.Analysis,“
Methodological Procedures in the Economic Analyses of Fertilizer Data,
Edited by E. L. Baum, Earl O. Heady and John Blackmore (Ames: Iowa
State College Press, 1956) p. h9.
A6
purpose of comparing it with a 3 x 3 factorial design as to its effective
ness as a basis for estimating the reSponse surface by least squares
techniques. The treatments used for this comparison are shown in Table 7.
TABLE 7
EXPERIMENTAL DESIGN FOR THE 1956 POTATO EXPERIMENT
r ‘1
Plant Nutrients No. Plant Nutrients NO.
(Pounds Per Acre) of (Pounds Per Acre) of
13,305 K20 Plots 13205 K20 Plots
0 O h 200 600 3
O 200 2 200 700 2
25 25 2 200 800 3
25 75 2 250 250 2
50 50 2 250 hOO 2
50 100 2 250 500 2
50 150 2 250 600 2
50 hOO 2 250 750 2
75 225 2 300 200 3
100 O 2 300 300 2
100 100 2 300 hOO 3
100 200 3 300 600 3
100 300 2 300 700 2
100 hOO 3 300 800 3
100 600 3 300 900 2
100 800 3 350 350 2
150 150 2 350 hOO 2
150 300 2 350 500 2
150 hSO 2 350 700 2
ISO 600 2 hOO 200 3
200 100 2 hOO hOO 3
200 200 3 hOO 600 3
200 hOO 3 hOO 800 3
200 500 2 hOO 900 2
1:7
(3) Points on three constant~proportion_PgOsKQO diagonals are
sufficiently sampled to permit estimation of these diagonals individually.
These are the diagonals in the experimental design in which P205 and
K20 are applied in 1:3, 1:2 and 1:1 preportions reSpectively. Estimation
of yield reSponse along these constant proportion of P205 and K20
diagonals allows comparison of these estimates with those derived from
estimates Of the entire surface.
The Total Experimental Prggram
Exclusive of the sugar beet experiment initiated in 1957, the
nutrient level experiments described in this chapter contain about 1150
individual plots.l Relative to experimental work undertaken elsewhere,
this is an elaborate project. Over 20 acres of land are required for
the experimental work. Acquiring soil samples, plant tissue samples,
crop yields and quality determinations for the various crOps are tasks
entailing large labor inputs. In addition, chemical analysis as well as
tabulation and computational analysis of these data are time consuming
undertakings.
In addition to the basic inputoutput determinations, the experi
Inents produce a large amount of byproduct data of interest to agronomists
éxnd economists. For example, data acquired from these experiments are
1Deingutilized to compare alternative methods of testing soil for
1Numerous other experiments are conducted by the Michigan.Agri
Ctﬁltural Experiment Station, many Of which also provide data for
fertilizer inputoutput determinations.
AB
residual quantities Of the three plant nutrients. The influence of
various plant nutrient treatments on the quality of crops produced is
being studied utilizing data from these experiments. A comparison of
experimental results from field and greenhouse experiments is being
conducted in conjunction with the basic inputoutput studies. In brief,
the experiments described in this chapter produce a wealth of data which
is being used for a diversity Of research projects.
Data produced from the experiments described in this chapter from
l95h56 were used for the analysis conducted in Chapter IV.
CHAPTER IV
ANALYSIS OF THE DATA
The Oats,_Wheat,_Alfalfa and Corn Rotation
The oats, wheat, alfalfa and corn rotation experiment was initiated
in 1955 and data have been collected for two years. Only two harvested
crOpS were produced in 1955 as alfalfa and wheat stands could not be
established in time for harvest during the 1955 crop year. field data
were acquired for both corn and cats in 1955 and all four crops were
produced in 1956. Due to a heterogeneous stand of alfalfa, no data were
acquired for that crop in 1956.
Analysis of the Oats Data
Oats were produced on two of the experimental sites in Calhoun and
Kalamazoo counties in 1955. Preliminary graphic analysis of these data
indicated that the variance present in the yield data was not associated
with different quantities of applied plant nutrients. This hypothesis
was further substantiated by fitting a polynomial equation to the data.
The equation fitted was of the type
I = a + bl N + ‘02 N2 + b:3 P + b4 P:3 + be Ii + by, K2 + b7 'J‘TP +
b8 1;}; + 6, P1:.
The variables N, P and H represent per acre applications of N, P305
Le
O‘
and K20, respectively. None of the variables in this equation had
estimated parameters significantly different from zero. Apparently
weather conditions were the main determinants limiting crOp yields
during the 1955 crop growing season. Unfavorable weather conditions,
largely the result of a late summer drouth, prevented crop yield increases
which might have occurred with increased applications of plant nutrients
under more favorable weather conditions.
Yield data for cats were acquired again in 1956. Preliminary
graphic analysis of these data indicated that positive relationships
existed between oat yields and applied N and P205. Furthermore, these
relationships appeared to be curvilinear, reflecting diminishing returns
to plant nutrient inputs.
The first formulation of the fUnctional relationship which was
attempted for the 1956 data was a polynominal equation identical to the
one fitted to the 1955 data. This polynomial contains first and second
degree terms for N, P205 and K20 and first degree, crossproduct terms
for all nutrients taken two at a time. This formulation containing the
estimated parameters is shown in equation I. Values listed below the
estimated parameters and included in parentheses are standard errors of
the respective parameters. N, P and K represent per acre applications
of N, P205 and K20 reSpectively as is the case in all equations unless
otherwise indicated.
Equation (I): i0  h3.326378 + .u0112190 N  .00130761 N2  .00650205 P
(.05115313) (.0019075) (.02579697)
+ .0000053h P2 + .06186818 K  .00010387 K2 +.00000068 NP  .COOlC905 NK
(.oooou775) (.051965u8) (.000191h8) (.ooooéh95) (.00013020)
+ .OOOO75h2 PK
(.00006h30)
51
The adjusted coefficient of multiple correlation for this equation
was .690. The coefficient of multiple determination indicated that
about h8 per cent of the variance in crop yields was associated with
variance explained by the regression equation. Estimated coefficients
for the nitrogen variables were significant at the one per cent prob
ability level. None of the coefficients for other variables were sig
nificant at the ten per cent level of probability.
Because of the large amount of variance not associated with re
gression and the nonsignificant coefficients which were estimated for
several variables, a second formulation of the production function
relationship was attempted. This formulation was an exponential equation
of the CarterHalter1 type. This exponential equation is quite flexible
depending on the magnitude of parameters estimated for the variables.
In addition to retaining the curvilinear properties postulated to exist
in fertilizer inputcrop output relationships, use of this equation
facilitates estimation of inputoutput relationships ranging over all
three stages of production, i.e., returns to additional plant nutrients
which (1) increase at an increasing rate (2) increase at a decreasing
rate and (3) become negative. This formulation in equational form is:
N bgci K
Y = aNbl c1 P b3c§
By taking the logarithm of this equation, we can acquire an equational
form of this relationship for which the parameters can be estimated by
1The usefulness of this equation as a production function formu
lation was first noted by H. 0. Carter and A. N. Halter.
52
the technique of least squares. The form in which this equation is
fitted statistically is:
LogY== loga+bl logNIN 10g Cl+b2 logP+P log c24
b3 log K +‘K leg cs.
The equation with estimated parameters is shown in Equation II.
Egation £11): Log 1?, = 157315152 + .16L175’028 log N  .00057687 N 
(.02022815) (.000150h6)
.ozuu1092 log P + .000096095838 P + .ooé3u3hh87 log K + .000217567362 K
(.016u801o) (.0000669u) (.02017332) (.OOClHYlh)
The coefficient of multiple correlation for this equation was .760.
The coefficient of multiple determination indicated that about 58 per
cent of the variance in oat yields was associated with regression.
In this equation, coefficients for nitrogen variables were signifi
cant at the five per cent probability level as was the coefficient for
the first phOSphorus variable.
Testing the significance of coefficients for individual variables
in an equation which contains more than one variable for a given plant
nutrient is a practice of limited usefulness. The related variables in
an equation such as N, N2, log N, etc., are obviously highly correlated.
Estimates of individual parameters may be subject to large standard
errors reflecting these high intercorrelations. One might conclude that
since individual parameters are not statistically significant, no sig
nificant effects are present. This conclusion might well be fallacious.
If the aggregate effect of all variables representing a particular
plant nutrient could be tested for significance, the test might indicate.
a significant aggregate effect. This situation illustrates an inadequacy
of current statistical testing procedures. In cases where (1) two or
more independent variables in a production function occur in product
form or (2) more than one variable is used to measure the effects of a
particular plant nutrient, it would be desirable to obtain a reliability
measure on the derivative of crep yield with respect to individual plant
nutrients. Such derivatives are necessarily utilized in determining
marginal nutrient effects and consequently Optimal applications of plant
nutrients. A satisfactory procedure for computing reliability measures
for such derivatives has not yet been develOped but is a critical need
in much analytical production economics work.
Interpretation of the Sta istical Results
It appears desirable to investigate several aSpects of the two
alternative production function formulations presented here. A comparison
of the production surfaces generated ey the two functions is of particular
J
interest. In addition, it is interesting to compare the combinations of
plant nutrients which (1) maximize yields and (2) maximize profits under
various plant nutrient and crop prices. A comparison of predicted eat
«.1
. the two functions an‘
ye
‘
yields usin; selected combinations of applied plant
L
nutrients is shown in Table 8. Observations from twentyeight combinations
n a o ‘ .‘ .' 7_ ﬂ ; T, , A .o ' . q p. _ f5 ‘ “va r _,',  ._‘
01 plant nutrients are included in Taole o. fuss: ineiuce UUSei‘aolOLS
from all 2" ‘lots in the l x 7 x 3 replicated factorial in addition to
.3 2 .J ._
a w
yield from all check plots.
(I)
the averag«
TABLE 8
OBSERVED AND ESTIMATED OAT YIELDS, 1956
Treatment Observed Residual2
(Pounds per Acre) Predicted Yield Yieldl v §
LBu. per Acre) Lgu._per.Acre) (‘1’ i)
N P205 K20 Exp.3 Poly. Exp. Poly
0 O O 37 01$ )4303 387 103 1416
20 80 20 56.7 51.8 67.5 10.8 15.7
20 80 80 58.9 58.9 55.1 “.8 0.2
20 80 280 68.3 59.6 63.5 0.8 3.9
20 160 20 56.3 51.3 51.0 ~5.3 ~0.3
20 160 80 58.5 58.9 70.8 11.9 15.5
20 160 280 63.8 61.2 57.9 5.9 3.3
20 880 20 58.8 50.8 56.8 2.8 5.6
20 880 80 61.1 55.9 60.0 1.1 0.1
20 880 280 6617 66.0 60.6 6.1 5.8
80 80 20 65.7 67.9 75.8 10.1 7.9
80 80 80 68.8 70.6 72.1 3.7 1.5
80 80 280 78.6 78.3 88.2 9.6 9.9
80 160 20 65.3 67.8 76.9 11.6 9.5
80 160 80 67.9 70.7 89.8 ~18.5 ~21.3
80 160 280 78.0 75.8 71.0 1.8 “.0
80 880 20 68.2 66.9 61.2 7.0 5.7
80 880 80 70.9 71.7 72.3 1.8 0.6
80 880 280 77.8 80.6 88.3 6.9 3.7
280 80 20 63.7 68.8 71.7 8.0 6.9
280 80 80 66.2 66.5 66.6 0.8 0.1
280 80 280 72.3 67.3 61.7 10.6 55.6
280 160 20 63.2 68.3 57.2 6.0 7.1
280 160 80 65.8 66.6 66.2 0.8 0.8
280 160 280 71.7 68.9 69.2 2.5 0.3
280 880 20 66.1 63.9 76.2 10.1 12.3
280 880 80 68.7 67.6 72.3 3.6 8.7
280 880 280 75.0 73.7 80.6 5.6 6.9
1The observed yield for the 000 treatment is an average of yields
from 11 plots, all other observed yields are averages of two plots.
2Residuals are deviations of predicted yields from average observed
yields.
3In computing Ii for zero treatments of plant nutrients using the
exponential equation, inputs of a single pound of N, P205 and K20 were
used. This introduces a slight upward bias in the predicted yield but
overcomes the problem of having Yi = 0 when any of the treatments is
zero. This procedure is utilized throughout this chapter when computing
Yi from exponential equations.
Statistical measures derived for the equations, including the co
efficient of multiple correlation and standard errors of the regression
coefficients, indicate that the exponential is a slightly, but not
significantly, more appropriate formulation than the polynomial.
Measures such as correlation coefficients and standard errors of regres
sion coefficients and equations are not without some limitations in
comparing these two functions. The observations, and hence the variances,
of the variables are not readily comparable since in one instance they
are in real numbers and in the other in logarithms. The real numbers
and logarithms, although bearing a consistent monotonic relationship to
each other, do not maintain a relationship of equivalence or of constant
ratios, Hence, the listed statistical measures should not be given an
absolute interpretation for comparative purposes, i.e., they should,
instead, serve as a basis for a rough comparison. InSpection of the
residual values, (Ii  Ii) for both functions provides little basis for
choice between functions since the individual residual values about the
two functions are about equally dispersed with reSpect to magnitude and
direction.
Some additional insight into the appropriateness of the two altern
ative functions may be gained by comparing the derivatives of these
functions with reSpect to their correSpendence to inputoutput relation
ships postulated to exist in accordance with currently held theory.
In addition, the derivatives are used to calculate plant nutrient combi
nations which preduce (1) maximum yields and (2) maximum profits.
lThese residuals are shown in columns 7 and 8 in Table 8.
56
Maximum yields occur where the first order partial derivatives of the
functions are equal to zero. Maximum profits occur where the partial
derivatives with reSpect to individual plant nutrients are equal to the
plant nutrientcrOp price ratios.
Since most of the variance explained by regression is associated
with the nitrogen variable, the derivatives of the functions with reSpect
to nitrogen are of particular interest. The partial derivatives of the
two functions with reSpect to nitrogen,’:%%ﬁ, are represented by the
following equations III and IV. All derivatives are taken for a unit
(one pound) change in plant nutrients.
E9
Eggation ETD  polynomial: 5% a 81  282 N + 1»7 P + be K
Substituting in the estimated parameters from equation (I) gives
21:1 = .80112190  2(.00130761)N + .00000068 P  .00010905 K
49 N
, , . ey Nbl Nlnc N b1_1
Equation (IV)  exponential. éfifl a R(I cl ,1 + 01 b1 N )
Where R a antileg (a + b2 log P + + log 02 + b3 log K + K log Cs).
The expression of the partial derivative of the exponential may be
simplified by factoring out YO which leaves
ﬂD'Y b
A: 4K
,2 N YO (ln cl«+ N ).
Substituting in the estimated parameters from equation (II) gives
2 Y , 16875028
—._—Q 2: — , 1” —'———.—'.i———
The partial derivatives of the two functions with reSpect to N are shown
in Table 9 with P205 and K30 fixed at three different levels, 20hO,
80160 and 2h0880 pounds per acre respectively. These derivatives are
also shown in Figure I. Derivatives of the exponential function are
larger at small nitrogen inputs than is the case for the polynomial
function. It is the Opinion of the author that the exponential generates
a production surface rising too rapidly with small nitrogen inputs. If
this is true, the derivatives are probably too reSponsive to input
changes, i.e., they probably are too large with small inputs and change
rapidly to become too small with larger inputs. This phenomenon is due
in part to the fact that when Xi = O, Y a O. The function may still be
quite reliable ever the range of moderate inputs.
The derivative of the exponential function is l.h56 bushels per
pound of nitrogen with a nitrogen input of 5 pounds and decreases to
.77? bushels when 10 pounds are applied. These values of the derivative
seem excessively high from a viewpoint of plant physiology, i.e., it is
difficult to visualize how one pound of additional nitrogen could result
in the production of l.h56 bushels of additional wheat. However, the
derivatives of the exponential type function are not restricted to a
linear function of plant nutrient inputs as is the case with a polynomial
with only first and second degree terms. This linear restriction on the
derivatives of a polynomial can be overcome by modifying the formulation
to include variables raised to fractional powers, e.g., powers such as
3/2, l/2 etc. and/or by adding variables involving powers hig
TABLE 9
CHANGES IN OATS YIELDS RESULTING FROM UNIT CHANGES
IN NITROGEN APPLICATIONS
Treatment Level Nitrogen Treatment Derivative of Derivative of
of P205 &zK20l Level Polynomial2 Exponential2
(pounds per acre) (bu. per acre) (bu. per acre)
1 20 .387 .393
l 80 .298 .17h
l 80 .190 .050
1 120 .085 .005
l 160 .019 .018
1 200 .128 .031
1 2hO .229 .O39
2 2O .3hl .h06
2 80 .288 .180
2 80 .188 .051
2 120 .079 .005
2 160 .013 .018
2 200 .130 .032
2 280 .235 .080
3 20 .323 .h62
3 hO .270 .205
3 80 .166 .059
3 120 .061 .006
3 160 .Oh3 .021
3 200 .188 .037
3 280 .253 .086
1Nitrogen is varied with P305 and K20 fixed at three levels:
(1) 8020, (2) 16080 and (3) 880280 pounds per acre reSpectively.
2The derivatives are those resulting from an additional pound of
nitrogen.
The statistical fit might not be improved by such a modification but
derivatives would be allowed to become a curvilinear function of
additional plant nutrients. Further experimentation with the use of
fractional powered and more complex polynomials and additional
derivative of exponential
derivative of polynomial
‘
“L
di
1.0
0.75
13
$4
(J
CO
I.
Q)
8* 0.5
3
U)
,4
c0
0
3 0.25
'0
r"
(1)
H
>~.
.5
ﬁg 0
[1
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0.25
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— ——*... .. .
"‘ «X
hO 80 1’0 160
‘6
Nitrogen Applied (pounds per acre)
200
Applications of P205 and K30 are fixed at 160
and 80 pounds per acre respectively.
Fig. 1. Partial derivatives of polynomial and exponentiai
functions for cats with respect to nitrogen.
O\
O
inSpection of the derivatives of these f1nct ions is neeee
The partial derivatives of yield with respect to P205 and K20 are
y.‘
shown in Tables 10 and 11 and Figures 2 and 3 respectively. As in tle
.1
case for the derivatives witn reSpect to nitrogen, the de1vatiV.es f
O
the expenentials take more extreme values than those of the polynomial.
However, the derivatives of be oh functions are small and may be non
significant and the absolute value of the difference between the two
derivatives is not large.
High Profit Cem'inatidns of Pl_ant lutrients
The Optimal amount of plant nutrients to apply, as has been previously
stated, is a function not only of the productivity of applied nutrients
but also of plant nutrient and crep prices. To solve for the combination
of applied plant nutrients which will maximize yields, the partial
derivatives of yield with respect to all plant nutrients are set equal
to zero and solved simultaneously. For the polynomial equation, the maxi
mum estimated yield is obtained with 153 pounds of N, a slightljf neg;at1ve
quantity of Png and .l p aid of NBC. The estimated amounts of P205 and
K20 resulting in maximum yields are neither statistically nor economically
significant, 1. e., they are not significantly different from zero. It
becomes profitable to a13ply plant nutrients to oats only when the price
of cats is in excess of 1 CO per bushel and even then only nitreg en
1
appl lications are profitable.
¥Plant nutrient prices used in computing the high profit inputs
were $0.15 per pound for nitrogen, $0.10 for phOSphoric acid and $0.11
for potash.
61
TABLE 10
CHANGES IN OATS 111133 RBSULTBJG FROM UNIT Gimmes
IN P205 APPLICATIONS
Treatment Level
of N and K201
Derivative of
Exponential3
(bu. per acre)
P205 Treatment Derivative of
Level Polynomial2
(pounds per acre) (bu. per acre)
1 20 .005 .022
1 ho .00u .00h
1 80 .003 .003
1 160 .002 .007
1 2&0 .002 .008
1 320 .001 .009
1 u80 .000 .010
2 20 .000 .027
2 no .000 .006
2 80 .001 .005
2 160 .002 .008
2 2u0 .003 .010
2 320 .oou .011
2 use .005 .012
3 20 .012 .028
3 to .013 .006
3 80 .013 .005
3 160 .011 .009
3 ZhO .OlS .011
3 320 .016 .012
3 uso .017 .013
IPhosphoric acid is varied with N and K20 fixed at three levels:
(1) 2020 (2) 8080 and (3) 2hO2h0 pounds
2Regression coefficients for phOSphoric acid variables were not
significant.
per acre reSpectively.
3The regression coefficient for only one phosphoric acid variable
was significant.
. per acre)
(bu
Change in yield of oats
O
O.?S
70.75
l.25
1.S
oz
A‘t__‘_‘ ”””””” =—’"’P’”’——'.'
."7"' ‘t__
f derivative of polynomial
E derivative of exponential
I
I
I
a
I
I
I
I
I
I
’I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
l
I
80 160 2DO 320 L00 &80
PhOSphoric Acid Applied (pounds per acre)
Applications of N and K30 are fixed at 80 pounds
per acre. ‘
Fig. 2. Partial derivatives of the polynomial and exponential
functions for oats with respect to phosphoric acid.
TABLE 11
CHANGES IN OATS YIELDS RESULTING FROM UNIT CHANGES
IN K20 APPLICATIONS
Treatment Level K20 Treatment Derivative of Derivative of
of N and P2051 Level Polynomial Exponential
(pounds per acre) (bu. per acre) (bu. per acre)
1 20 .059 .0h6
1 b0 .Oéh .038
l 00 .0h6 .03h
1 120 .037 .033
1 160 .029 .033
1 200 .021 .031
1 2u0 .012 .03A
2 20 .061 .05
2 hO .057 .Ohh
2 80 .0L9 .039
2 120 .oto .038
2 100 .032 .030
2 200 .024 .039
2 2&0 .015 .C39
3 20 .C07 .tjh
3 MO .06h .Uhh
3 00 .CjS .OHO
3 120 .0h? .039
3 180 .039 .ce9
3 200 .030 .039
3 2h0 .022 .OhO
1Potash is varied with N and P205 fixed at three levels: (1) 20hO
(2) 80160 and (3) 2hOh80.
0.33
0.3
Ci:
cl—u‘.
'
. derivative of exponential
\ derivative of polynomial
h0 80 120 160 200 iin
L A:(/
Potash Applied (pounds per acre)
Apptications of N and P205 are fixed at 80 and 100 pounds per acre
respectively.
Fig. 3. Pirtial derivatives of polynomial and exponential funCLitns
for oats with respect to potash.
On the basis of yield reSponse measured for 1955 and 1956, icrtiliz
ing of oats was not a profitable practice. The only possible justifi
cations for application of plant nutrients to the cats crop appear to
be when a seeding is being established with the cats and/or the benefits
derived from residual plant nutrients by other crops in the rotation.
Analysis of the Wheat Data
Wheat was produced on the Kalamazoo County experimental site in
1956. The yield data produced in this experiment were analyzed in the
same manner as the oats data. The original function fitted to the wheat
data was a nine variable polynomial. This formulation with estimated
parameters is shown in Equation V.
Equation (v): §¥,= 28.538730321 + .08598h69h1 N  .0002208h56 N2 +
(.016959906h) (.0000632u18)
'0163750688 P ' 000035115h P3 + .0085708071 K + .000021323n K2 +
(.0085530282) (.0000158325) (.0172292t2u) (.000063h867)
.00001902h6 P  .0000799h31 NK + .0000151210 PK
(.0000215352) (.0000hh5667) (.709260110)
The adjusted coefficient of multiple correlation for this equation was
.66 and the coefficient of multiple determination indicated that about
hh.per cent of the variance in yield was associated with variance in
applied plant nutrients. As was the case for cats, only the estimated
parameters for the nitrogen variables were statistically Significant at
the one per cent probability level. However, the phosphoric acid vari
ables, P and P%.were significant at the five per cent probability level.
66
A CarterHalter type exponential function was also fitted to the
wheat data. The results of this fit are shown in Equation VI.
Equation (VI): Log §w = l.hh50h7179 + .0226358023 log N + .0001726882h8 N +
(.0117u51331) (.0000873600h8)
.0165763652 log P  .0000173738hl.P4.0012796889 log K + .000108232188 K
(.0095688935) (.000038867869) (.0117132969) (.000085h36998)
The adjusted coefficient of multiple correlation for this equation
was .65. The adjusted coefficient of multiple determination indicated
‘that about h3 per cent of the variance in crOp yields was associated with
variance in the amounts of applied plant nutrients. The first three
estimated coefficients in this equation were significant at the l0 per
cent probability level and were almost significant at the five per cent
probability level. The last three coefficients in the equation were not
statistically significant. A comparison of observed yields with yields
estimated by using the two functions is shown in Table 12. As was the
case for the cats data, the tabular comparison includes observations and
predictions for 28 combinations of applied N, P205 and K20. The observed
yield values are averages of two replications for all treatments except
the check (0,0,0) treatment which is an average of eleven replications.
The coefficients of multiple correlation and determination indicated
that the two functions were about equally effective in explaining variance
in wheat yields. Inspection of the residuals for the two functions,
(Yi  Ti), further substantiates the conclusion that the two functions
produce about equally good fits. These residuals are shown in columns
7 and 8 of Table 12.
67
TABLE 12
OBSERVED AND ESTIMATED WHEAT YIELDS, 1956
‘ _._‘—_——: _
I H r *
Treatment Predicted Yield IObserved Yield1 Residual2
(pounds per acre) (bu. per acre) (bu. per acre) (Yi ~ Ti)
N P205 K20 Exp. Poly. Exp. Poly.
0 o o 27.9 28.5 28.2 0.3 o.3
20 to 20 32.2 30.9 29.5 2.7 1.u
20 to 80 32.7 31.5 29.3 3.h 2.2
20 to 2t0 3h.l 33.8 3u.6 0.5 0.8
20 160 20 32.8 32.1 31.2 1.6 {L9
20 160 80 33.3 32.8 3o.t 2.9 2.t
20 160 2uo 32.7 35.u 35.1 o.u o.3
2o t8o 20 33.0 3o.t 32.1 o.9 1.7
20 ubo 80 33.5 31.u 31.9 1.6 0.5
20 t80 2u0 3h.9 3h.8 3t.9 0.0 0.1
80 to 20 3h.0 3h.7 37.5 3.5 2.8
80 to 80 3u.6 35.0 3u.9 0.3 o.1
80 to 2to 36.1 36.6 37.2 1.1 0.6
80 160 20 3u.6 36.1 no.9 6.3 h.8
80 160 80 35.2 36.5 36.7 1.5 0.2
80 160 2&0 36.7 38.3 36.5 o.2 1.8
80 u80 2o 3u.8 3u.7 28.7 ~6.1 6.0
80 hbo 80 35.b 3S.h 39.0 3.6 3.6
80 u80 2uo 36.9 38.0 38.6 1.7 0.6
2uo to 20 37.2 37.0 36.6 o.6 o.h
2t0 to 80 37.8 36.6 35.1 2.7 1.5
2to to 2to 39.1 36.1 35.1 “.3 ~1.o
2to 160 20 37.8 38.8 t2.1 t.3 3.3
2uo 160 80 38.5 38.h 39.3 0.8 0.9
2to 160 2to no.1 38.2 38.h 1.7 0.2
2u0 A60 20 38.1 38.h t2.9 t.8 h.5
2uo t8o 80 38.7 38.3 38.2 0.5 0.1
2&0 hBO 2&0 no.3 38.8 38.8 1.5 0.0
1The observed yield is the average of two replications except for
the check (0,0,0) treatment which is the average of 11 replications.
estimated yields.
gﬁesiduals are the difference between average observed yields and
68
Derivatives of the two functions with respect to all three of the
plant nutrients are presented in Tables 1315 and in Figures h6. The
derivatives of the two functions produce different estimates of the
productivity of the various plant nutrients. For example, the derivative
of the polynomial indicates that the marginal productivity of nitrogen
over the range of 30 to 100 pounds, which is a common range of application,
is almost double the marginal productivity schedule generated by the
derivative of the exponential. Derivatives of the two functions with
reSpect to P205 also exhibit substantial differences over the range of
usual applications. However, the marginal productivity of phOSphorus
is low and the absolute value of the differences between the two deriva
tives is small as is shown in Table lb and Figure 5.
Estimated derivatives of the two functions with respect to K20 also
differ widely as indicated in Table 15 and Figure 6. The derivative of
the polynomial with reSpect to K20 exhibits increasing returns to addi
tional applications of K20 which is not a logical phenomenon. The deriva
tive of the exponential exhibits only slightly diminishing returns. As
previously indicated, however, the K20 variables in both equations lack
statistical significance at any acceptable probability level.
Inferences made about the productivity of all three plant nutrients
will vary considerably depending on which function is chosen as "best".
InSpection of residuals of the two functions does not provide any
satisfactory basis for choosing between the two functions.
MP‘W. 7“
Diana 13
69
CHANG'S IN WHEAT YIELDS RESULTING FROM UICIT CHANGES
.LN
TTT 110/” 71".
APTTICATICIS
UULM
Treatme
nt Level
of
‘v
01
IIitrogen Tr atment
Level
(pounds per
Derivative of
Polynomial
acre)
(bu. per acre)
Derivative of
Exponential
(bu. per acre)
lwmit
(l) hC '20
F’FJFJFJFJFJFJ
TORDRJNNMM
\nwwwwwxn
en is varied with P305
PO
C," C:
:uﬁ'
.‘\I\)'\
J C) CL"
V\
I
r“;
[UNI—1H
\
J
4_
)
C.“ 1:" f\
C. O C)
120
150
200
ZQC
F)“
(.U
bu
120
160
200
2210
.076
(2) L301’0 and (3) hEG2SIC pounds per a01e
.0h9
.032
. c2 3
. L 21
. 019
.019
o L]8
. 0;1
.033
.62'
and K 0 iixed at three level
'33
reSpectively.
Change in yield of wheat (bu. per acre),
‘ya— deriVative of exponential
derivative of polynomial
.05 \
.035
to 80 120 160
T\J
O
C)
r
E:
Nitrogen Applied (pounis per acre)
Applications of P205 and K20 are fixed at 160 and
80 pounds per acre respectively.
Fig. h. Partial derivatives of the polynomial and exponential
functions for wheat with respect to nitrogen.
TLBLE 1h
012151150513 I“? L'JIZBLIT 11:11:13 :‘tESLlTl’i‘lCI E“. 101'. UL ITL " ._.3.I=.7<_}.ZS
Treatment Level P205 Treatment Derivative of Derivative of
of 1 Level Polynomial Exponential
N and K20 (pounds per acre) (bu per acre) (bu. per acre)
1 ho .Olh .012
1 to .011 .006
1 130 .006 .002
l 2h0 .000 .001
1 320 .005 .001
1 800 .011 .000
l h80 .017 .000
2 hO .016 .013
2 80 .013 .006
2 160 .008 .003
2 2h0 .002 .001
2 320 .003 .001
2 hOO .009 .000
2 h80 .015 .000
3 hO .022 .015
3 80 .019 .007
3 160 .013 .003
3 2h0 .008 .002
3 320 .002 .001
3 800 .008 .000
3 h80 .009 .000
1P2 05 is varied with N and K30 fixed at three levels: (1) 2020
(2) 8080 and (3) 2h02h0 pounds per acre respectively.
00 ’15.
.05

I
I
I
‘
u
I
I

I

I

./ darivat ive of exponential
‘
derivative of poiynomiai
80 160 ELO 3‘0
Phosphoric Acid Applied (pounds per acre)
N and Ké0 are fixed at 80 pounds per acre
Fig. 5.
Partill derivatives of the ptlynonial anl exponential
functions for wheat with respect to phosphoric acid.
73
TABLE 15
CHANGES IN WHEAT YIELDS RESULTING FROM UNIT CHANGES
IN K20.APPLICATIONS
—_.
T _._‘_
Treatment Level Potash Treatment Derivative of Derivative of
of Level Polynomial Exponential
N and P2051 (pounds per acre) (bu. per acre) (bu. per acre)
1 20 .008 .010
1 h0 .009 .009
l 80 .011 .009
1 120 .013 .009
1 160 .018 .009
1 200 .016 .009
1 2h0 .018 .009
2 20 .005 .010
2 hO .006 .009
2 80 .008 .009
2 120 .010 .009
2 160 .011 .009
2 200 .013 .009
2 2h0 .015 .009
3 20 —.003 .011
3 80 .002 .011
3 80 .000 .010
3 120 .002 .010
3 160 .003 .010
3 200 .005 .010
3 2h0 .007 .010
tPotash is varied with N and P205 fixed at three levels:
(1) 20h0 (2) 80160 and (3) 2h0h80 pounds per acre reSpectively.
‘F
ii ulv loll}.
Change in yield of wheat (bu. per acre)
C......—.
C'
CD' ‘
derIVative of exponential
b0 80 120 160 300 Ad?
PotaSh applied (pounds per acre)
Applications of N and P205 are fixed at 80 and I60 pounds per acre
respectively.
Fig. 6. Partial derivatives of the polynomial and exponential
functions for wheat with respect to potash.
75
Maximum Yields and High Profit Plant Nutrient Applications
Maximum yields of about 39 bushels per acre are predicted using
the polynomialequation. This yield occurs with plant nutrient appli
cations of about 196 pounds of N, 300 pounds of P205 and 61 pounds of
K20. The maximum yield predicted using the eXponential is in excess
of any yield observed in the experiment and requires plant nutrient
applications in excess of any quantities applied in the experiment.
Since the predicted maximum yield and the plant nutrient inputs producing
this yield lie beyond the range of observed values, they are probably
invalid inferences.
Both functions generated reSponse surfaces which illustrated sub
stantial positive yield reSponse to N and P205. However, because of
the moderate slopes of the reSponse surfaces, the value of additional
production was less than the cost of plant nutrients necessary to obtain
the increases in yields.1
Applications of nitrogen and phOSphoric acid would.have been profit
able only at wheat prices of about $3.00 per bushel. Such wheat prices
appear extremely unlikely. It is important to note that although the
derivatives of the two functions differ considerably, the same conclusion,
that no fertilizer applications were profitable at typical prices, would
be reached using either function as a basis for computing high~profit
fertilizer inputs.
1No credit was given for possible residual fertility. With large
fertilizer applications some carr"over fertility would be expected,
however, the magnitude and value of this carryover can only be determined
over time. .
76
Analysis of the Corn Data
Two corn crOps have been produced and harvested in the rotation
experiment. The corn plots were located at the Calhoun county site in
1955. A severe summer drouth reduced corn yields in this area,
particularly on the lighter upland soils. An extensive analysis of the
1955 corn data was conducted by Jack Knetsch and has previously been
reported.1 Knetsch found that a CarterHalter type exponential provided
the best statistical fit to the data. Significant reSponse was found to
exist only for applied nitrogen. The fitted function was
.18627 (N + 0.1)
YC  39.?1(N + .01) .96230 ,
where N was measured in 20pound units. The addition of .1 of a unit
alleviated the problem of forcing the function to have a value of zero
when any one of the plant nutrient inputs was zero. The coefficient of
multiple correlation for this equation was .69. The high profit nitrogen
application varied from 29 to Sh pounds per acre as the price of corn
was varied from $.80 to $2.00 per bushel with nitrogen priced at $.15
per pound. A comparison of observed and predicted yields is shown in
Table 16.
1The results of this analysis are contained in: Jack L. Knetsch,
"Methodological Procedures and Applications for Incorporating Economic
Considerations into Fertilizer Recommendations," Unpublished Master's
Thesis, Michigan State University, 1956, and Jack L. Knetsch, L. S.
Robertson, Jr., and w. B. Sundquist, "Economic Considerations in Soil
Fertility Research," Michigan Agricultural Experiment Station, Quarterly
Bulletin, August, 1956, pp. 1016.
TABLE 16
COMPARISON OF OBSERVED AND PREDICTED CORN YIELDS
ON A KALAMAZOO SANDY LOAM SOIL, 1955
Average Predicted Marginal Product
Number of Actual Yield of of 20pound
N Per Acre of Yields, Corn Units of N
(pounds) Plots (bu. per acre) (bu. per acre) (bu. per acre)
0 18 26.3 25.8 0
20 2h 80.6 38.2 12.8
to 18 83.5 81.8 3.6
80 29 h3oh hh.l 1.15:L
160 18 82.8 83.0 0.501
2h0 27 h0.7 39.8 O.8Sl
1Average marginal product of 20pound units of nitrogen for the
hOpound incremental intervals shown in column 1.
Corn was produced on the Kalamazoo county site in 1956. Once again
the crop was damaged by a severe late summer drouth. Check plot yields
were not significantly dif erent from those receiving applied plant
nutrients. Preliminary tabulations indicated very little association
of yield variance with variance in any of the three applied nutrients.
This lack of relationship was further substantiated by functional analysis.
A.nine—termgpolynomial was fitted to the data with the estimated para
meters shown in Equation VII.
Equation (VII): Y0 = 58.8ouc809  .0085761891 N + .000113767680 N2 
(.0391267523 (.000185895858)
.0181889261 P + .0000093102508 P2 + .0269173892 K  .000092531u50 K2 +
(.019683161) (.0000365288729) (.039750uu01) (.00186857332)
.0000669778119 NP + .0000357228950 Nx;¢ .000056027392 PK
(.0000u96773588) (.0000995h8385) (.0000891638629)
‘ l
None of the {ma ineters in thi seqLation are sL _nificaatly diiferent from
zero. This lack of significance is not surpri31ng since the adjusted
coefficient of multiple correlation for the ecuation is only .23 and
the adjusted coefficient of mul‘iple det ruina ion is only .Oj a value
which is not signifi Lcantlv cil' rent from zero.
A CarterD alter trLe equation which was fitted to the oata is snown
in Equation VIII.
S“
A
~\“e+“on (Vlll)= L02 Y0 = 1.719957583 + .012753199 103 N  (0((l43. I
+ CO :7 W977
31L~ P  .080110811 P ~ .005317996 103 K + .00c18587o K
(. 0131:;326) (.0Loc55c33) (.t16278 , 3
203) (.LL8118L;6)
Only the fourth tern in this equation, F, is Statistically Sivniica 8.
Since the phOSphoric ac (1 variable is represx rted by two terms, orie of
which is not significant, no very valid inierenccs can be made about
the aggregate influence of phi 'Lhoric acid. None of as terms repie
r‘ v “I ’L r “. ‘ r‘ o r“ ‘ '9‘. rs r‘ “L 7' : O ' ' “ﬁr/I rv “ ‘1
Senting nitrogen 01 pOCaSJ are signiiioantly oiiierent 110m zero. about
the same amount of total variance in yield is associated with regression
for t11e polunom al. The adgusted coeiliciro nts of multiple
U]
c l"
P a
h—J
C}
O
9.}
U]
U)
a S 1'58.
‘ y a n K" J...
 .‘ "~‘~ ~~ ‘a' ~'~ ' t‘i 3 ~.». ,\ 1"!ch '. *F
multiple determ1haoion are .24 ano .co SQyUCULVClJo
O
*5
*3
L.J
h)
CE
‘rJo
O
:5
E
E.
n "
.L_° . _. 1, °,,._..‘  .1. rs .  _. .. 9  ,  ,3 i .. ‘.. ..'.
oisticalif SILUIIICano. ihe only inierence alien Sdphn o)
0 O !  _:’_ _f‘
Till11.081113 IC‘LLI‘C 01 V8.1" 1611103
rieid w‘s associa tad w th applied plan t nutiiints.
' \
\—‘
O‘I’IC‘ \“n TY» I‘T‘( "“ﬁ’l '3".,_ 1".C)“1)1(‘.u
 LA.) (1.L 1.... J'. )1 .141, JL. L” La. wax)
(I A. 
 it. ,
In sunrary, onlv moo2rLte apuiica
per acre, were profitable on corn on Kalamazoo sa.id/ loan In 1933 and no
.‘1
\VL.)
7w 1.3.3 .r' . x . ° n,  r" J.  A, D 1. . 2 : 3,4. ,
no aliaiia was frosn in tdd iirsc year 01 tJG experiment o
v. .' n... 4. 1 11 ' 1. r . ‘. . » 2. .21 a  ' a; c
the inaullloy to esca31153 any harvestaole grouti 1n CA5 first year 01
1 . ~.~ fr. , .L —.. ' d’” .L' J... 1 n , h, ‘ f‘r. , 7 _ ..  ~ .1 ._ i .i
tile 1:}C}2231"l3:1:;l‘1u. in .L‘," O om? 83.3arlo Ol d11dlhd. ILLS JCI‘J 1123 oOI‘O_’~IIJI’J_';Ollb.
w «— ’ —'— . .. A ‘ ' o '_ 1. . ‘o .. 3 r‘ .~‘ "a. f " J . . .4. t \  u ..
LarLe contiguois areas in use i11ld had hO‘uTao31y good SoanS while
7
other areas had almost no alfalia growing on them. to attempt was made
U
to collect and analyze yield data because yield difchGnCVS were obviously
(
”I
Q
a function of difiercnces in stand not associated with applied plant
J._, ‘ , . J
Analgsis o” the Continuous Corn Data
The initial corn crOp in a continuous corn rotation was produced on
a Wismer clay—loam soil in Tuscola county in 1956. Preliminary inspection
of the data indicated small and heterOgeneous yield reSponses to applied
plant nutrients. The eight check plots in this experiment had an average
yield of 100.6 bushels per acre, while the average of all 210 plots in
the experiment was 109.7 bushels per acre.
A nine variable polynomial was fitted to the data and the results
of this formulation are shown in Equation IX.
Equation gig); it . 10h.565510278 + .06991h3u6 N + .05075haso p 
(.03u70b916) (.017279507)
.001629512 K  .000356932 N2  .000068956 P2  .000053579 K2 
(.OBhBSSSB? (.000108282) (.0000287h3) (.000116055)
.000039695 NP + .000112058 NK + .000060759 PK
(.000039782) (.000079363) (.oooou383o)
80
Coefficients of four of the variables N, N2, P, and P2 were sig
nificant at the one per cent probability level, whereas none of the
potash variables were statistically significant. Only a small portion
of yield variance was associated with applied plant nutrients as the
coefficients of multiple correlation and multiple determination were
only .hO and .16 respectively;
Since none of the independent variables containing a potash term
were statistically significant, the polynomial was reformulated, drOpping
the variables containing a potash term. The shortened polynominal is
shown in Equation.X.
A
Egpation X: Y + 10h.082698823 + .07370h5h6 N + .050022736 P 
° (.O3h298683) (.017116212)
.000331599 N2  .000056021 P2  .OOOOZShéO NP
(.000107266) (.00002733h) (.000038963)
In equation X the first four coefficients are significant at the
one per cent probability level. The fifth term, a cross product, was
not significant at any acceptable significance level. The coefficients
of multiple correlation and multiple determination for the shortened
polynomial were .39 and .16 reSpectively.
Due to the small portion of yield variance associated with applied
plant nutrients as indicated by inspection and the fitted polynomials,
no attempt was made to fit an exponential type equation to the data.
Maximum Yield and High Profit Combinations of Plant Nutrients
Coefficients for the nitrOgen and phOSphoric acid variables were
similar for the two polynomials fitted to the data. Since the potash
coefficients were not significant, the plant nutrient combination
providing maximum yields was restricted to N and P205 and was calculated
from Equation X. The maximum predicted yield, 123.h bushels per acre,
was obtained using 95 pounds of N and h25 pounds of P205. The cost of
using any amount of applied plant nutrients exceeded the returns unless
corn prices exceeded $2.00 per bushel. The latter corn price situation
is, of course, an unlikely phenomenon.
The high check plot yields, in excess of 100 bushels per acre,
probably indicates the soil was quite fertile prior to additional appli
cations of plant nutrients although soil tests indicate only a moderate
fertility level. Other possible sources of yield variance were present
in the experimental field, including differences in previous crepping
history; Although yields from the plot areas with different crOpping
histories were not statistically different, this factor of heterogenity
may have contributed some variance to crop yields.
Analysis of the Bean Data from the Corn,
Beans and Wheat_ﬁotation
Field beans were produced on a Simms loam soil in Gratiot county
in 1956. The bean crOp is part of an intensive cash crop rotation of
corn, beans and wheat. Experimental plots had received plant nutrient
treatments in 1955 identical to the 1956 treatments. Thus, some residual
fertility might have been expected to be present in 1956, particularly
on plots receiving heavy fertilizer applications the previous year.
Preliminary tabulation of the data indicated a substantial response to
82
nitrogen applications, a smaller reSponse to phOSphoric acid, and no
appreciable response to applied potash.
Three functions were fitted to the bean data. The first two func
tions are exponential type formulations and the third a five variable
polynomial. The original production function formulation is a six
variable exponential of the CarterHalter type. Although preliminary
analysis had indicated no reSponse to potash, variables containing
potash terms were included in this original exponential which is shown
in Equation XI.
Equation 131); Log Y5 = 1.203h797 + .032812261 10g N + .000398971 N +
(.017529035) (.001c69u)
.019527h3h log P + .000062271 P + .001880612 log K + .000050911 K
(.015569387) (.oooougsoh) (.018591118) (.000068525)
The adjusted coefficient of multiple correlation for this equation
was .605 and the coefficient of multiple determination was .366. This
indicates that about 37 per cent of the variance in bean yields was
associated with regression. Because of the large standard errors for ﬁne
potash coefficients, a second formulation of the exponential was made
dropping the potash terms. This exponential is shown in Equation XII.
Equation (X11): Log Y5 = 1.207h135791 + .O3h7393520 leg N + .000396596h N
(.016076657) (.00010650195)
+ .021h607700 log P + .0000597327 P
(.01u609617) (.0000L8352231)
The adjusted coefficient of multiple correlation for the shortened
exponential was .607 and the coefficient of multiple determination was
b.)
.369. Coefficients of the nitrogen and phOSphoric acid variables were
not changed appreciably by omitting the non—significant potash terms.
Phosphoric acid terms were not significant at the 10 per cent probability
level as the size of the estimated coefficients for these terms exceeded
their reSpective standard errors. Finally, a five variable polynomial
was fitted to the bean data. The results of this fit are shown in
Equation XIII.
.1:
Equation (XIII): ﬁt = 17.60231tu + .0636878985 N  .OOC10708hh1 r2 +
(.011h222) (.000035h159)
.0127h99698 P  .0000105617 P2 + .OOOOO63h92 NP
(.00580265) (.00000373620) (.000013095?)
The adjusted coefficients of multiple correlation and determination
for this equation were .6h6 and .hl7, respectively.
A comparison of observed and predicted yields using the three
functions fitted to the data are presented in Table 17. As in previous
cases, inSpection of the residual quantities, i.e., differences between
predicted and observed values, of the three functions provides little
basis for choosing any one function over the others. This is true because
of the relative uniformity of the magnitude and direction of the residuals.
Partial derivatives of the three functions with reSpect to nitrogen are
shown in Table 18 and Figure 7. Partial derivatives with reSpect to
phOSphoric acid are presented in Table 19 and Figure 8.
Kaximum Yields and Optimum Inputs of Plant Nutrients
Derivatives of the two exponential equations with respect to nitrogen
are characterized by preperties which are unusual for marginal product
OBSERVED AND ESTIMATED BEAN YIELDS, 1956
h
Treatment Observed Residual3
gpounds per acr§)_ Predicted Yieldl Yieldg (r1  ii)»
N P205 K20 Poly 325(1) EJMZ) Poly mm m7?)
0 o 0 17.6 16.1 16.0 17.h 0.2 1.3 1.8
20 no 20 19.3 19.8 19.6 25.h 6.1 5.6 5.8
20 no 160 19.3 19.8 20.0 25.9 6.1 6.1 5.9
20 no 320 19.3 19.8 20.h 19.6 0.3 0.2 0.8
20 320 20 21.9 21.5 21.2 15.h r6.5 6.1 5.8
20 320 160 21.9 21.5 21.6 2h.5 2.6 3.0 2.9
20 320 320 21.9 21.5 22.1 21.h 0.5 o.1 0.7
20 6&0 20 22.7 22.9 22.5 25.8 3.1 2.9 3.3
20 6ho 160 22.7 22.9 23.0 21.0 1.7 1.9 2.0
20 6u0 320 22.7 22.9 23.u 18.8 7.9 8.1 8.6
no 80 to 21.0 21.1 20.9 21.9 0.9 0.8 1.0
80 160 80 23.9 23.0 22.8 31.1 7.2 8.1 8.3
160 no 20 25.6 28.2 23.8 23.8 1.8 O.h 0.0
160 no 160 25.6 2u.2 2h.3 26.9 1.3 2.7 2.6
160 to 320 25.6 2h.2 2h.8 27.8 2.2 3.6 3.0
160 320 20 28.8 26.3 25.8 31.6 3.2 5.3 5.8
160 320 160 28.8 26.3 26.h 33.8 5.h 7.5 7.h
160 320 320 28.11 26.3 26.9 25.6 2.8 0.7 1.3
160 6h0 20 29.5 27.9 27.h 2u.7 h.8 3.2 2.7
160 6h0 160 29.5 27.9 28.0 33.5 h.0 5.6 5.5
160 6h0 320 29.5 27.9 28.5 29.3 0.2 1.h 0.8
2uo hBO 2u0 31.1 29.6 29.9 29.5 1.6 0.1 0.u
320 no 20 27.6 28.7 28.2 32.8 u.8 3.7 8.2
320 no 160 27.6 28.7 28.8 27.5 0.1 1.2 1.3
320 to 320 27.6 28.7 29.h 26.u 1.2 2.3 3.0
320 320 20 30.7 31.2 30.6 311.1 3.1; 2.9 3.5
320 320 160 30.7 31.2 31.2 29.7 1.0 1.5 1.5
320 320 320 30.7 31.2 31.9 27.8 2.9 3.8 ~h.1
320 6h0 20 32.2 33.1 32.5 3u.8 2.6 1.7 2.3
320 6u0 160 32.2 33.1 33.1 33.7 1.5 0.6 0.6
320 6u0 320 32.2 33.1 33.8 30.6 1.6 2.5 3.2
l
1Exp (1) is the four term exponential and Exp (2) is the six term
exponential.
2The observed yield for the 000 treatment is an average of yields
from ll plots, all other observed yields are averages of two plots.
3Residuals are deviations of predicted yields from average observed
yields.
85
TABLE 18
CHANGES IN BEAN YIELDS RESULTING FROM UNIT CHANGES
IN APPLIED NITROGEN
Treatment Nitrogen Treatment Derivativecﬂ? DerivativecﬂTDerivative of
Level of Level Polynomial Exp. (1) Exp. (2)
P205 and K201 (pounds per acre) (bu.per acre) (bu.per acre) (bu.per acre)
1 20 .060 .052 .050
1 to .055 .037 _ .035
1 80 .0h7 .030 .029
l 120 .038 .028 .027
1 160 .030 .027 .027
l 200 .021 .027 .027
l 2&0 .013 .027 .027
l 320 .005 .029 .029
2 20 .060 .055 .052
2 hO .056 .039 .037
2 80 .0h8 .031 .030
2 120 .039 .029 .029
2 160 .030 .029 .028
2 200 .022 .029 .028
2 2h0 .013 .029 .029
2 320 .OOh .031 .031
3 20 .062 .057 .055
3 hO .058 .0h0 .039
3 80 .050 .032 .032
3 120 .0h1 .030 .030
3 160 .032 .030 .030
3 200 .02h .030 .030
3 2uo .015 .030 .030
3 320 .002 .032 .032
h 20 .063 .060 .060
u to .059 .0t2 .0h2
h 80 .051 .03h .03h
h 120 .0h2 .032 .033
h 160 .033 .031 .032
h 200 .025 .032 .032
h 2h0 .016 .032 .033
u 320 .001 .033 .035
lNitrogen is varied with P205 and K20 fixed at: (1) h020 (2) 160
80 (3) 320160 and (h) 6h0320 reSpectively. Derivatives of the poly—
nomial and Exp. (1) are independent of applied K20 since there were no
K20 variables in the functions for which these derivatives were taken.
Y‘""
54..
Z’VII LV 5 131 6) v::r;;xf i? ‘3Y7N=f‘7:1?..;i
O
r
K
._


K‘g‘ianttt
V 9 . 
dau'lvat :vr'cwl li‘Jar
I e/ 131519. ‘,'):‘;)._,r[ﬁ_ nfjiul
I
I
II
I
'K
'(
10 “
O ‘
X
.I
I
'1
‘l
\
.1
l'\
0.18 "
“
‘K
.‘
“
derivative of po.ynom.al
k _
“ ,“~‘.‘¢‘*.‘. .‘‘& !_x<_x Xx’xx'!
SO TOO 150 ECO
(X;
3'.
n
‘_ v)
A4
Nitrogen applied {pounds per acre)
PpOS and KdO are fixed at 160 and &0 pounds per acre
respectively.
Fig. /. Partial derivatives of a polynomial and (we expunentiui
Tenetions for beans with respect to nitrogen.
("h
\,
—4
TABLE 19
CHAN ES IND KIN YILLDS IﬂNULIILa FROh UIIT CEaIGES
IN APPLIED PIL‘SPH IC ACID
Treatment P205 Treatment Derivative of Derivative of Derivative of
Level of Level Polynomial Exp. (1) Exp. (2)
N and K20 (pounds per acre) (bu.per acre) (bu.per acre) (bu.per acre)
to .012 .013 .012
80 .011 .008 .008
160 .009 .006 .005
2ho .C08 .005 .C05
320 .006 .00h .00h
hOO .OOh .CCh. .OCh
u80 .003 .00h .00h
6u0 .001 .00h .00h
FJFJFJF‘F‘FJFJF‘
2 L0 .012 .015  .015
2 80 .012 .CC9 .009
2 130 .010 .tts .006
2 2nc .008 .kk .005
2 32‘ .006 .005 .005
2 hIO .C05 .C05 .005
2 h80 .003 .00; .oou
2 6h0 .000 .eoh .005
3 hC .013 .01” .015
3 80 .c13 .tio .010
3 160 .011 .CC? .C;7
3 233, .CC9 .CChS .(X,6
3 320 .000 .C05 .CC5
3 LOO .006 .005 .CC5
3 . L80 .CCa .005 .C
3 6LO .OCd .005 .CL:
9‘11“
C‘C’
PC
0 0
CC)
l#}4
‘JJL"
o o
C‘C
I_‘l—
[0N
m
.JL—J
J:
h 160 .011 .00) .uLu
h 230 .010 .0’7 ..L(
N 30‘ .CCS .LL) .LCS
h nCC .CC3 .ct3 .cL7
h hLC .CC5 .Cxﬁi .Lcé .
1 04C .CCI .CCG .LCG
ngOr is varied with N and K30 fixed at (1)2 20 (2) tCJO (f) 170
10v and (V) 320 320 reSpectively. Jen'vaulv7r of the polvnom a1 and
h.;p. (l) are Independent of a7plied p70 since there wece no ‘70 variaoles
Change in yield of beans (bu. per acre)
.09
Kﬂﬁlx
0)
L0
¥
O
0
U1
statueﬁn eff##1##?
f.
.03 ' *
. \
00..) \
derivative of pcivnomiai
*¥1‘°*——*—
100 200 300 hOO 500 600
PhOSphoric Acid Applied (pounis per acre)
N and K30 are fixed at 80 pounds per acre
Fig. 8. Partial derivatives of a polynomial and two exponentia;
functions for beans with respect to phOSphoric acid.
d)
v A 1 1 _ ’ 3 ,_ _o_ J_ o y, 1 ___ _0 7'1, 1. g r) r. , j n: _, r; if  !,
seneoules. Tnese derivauives, shown in ladle lo and Figure ;, fired
(‘3‘,"(\ I.“ '. J‘ n ‘V‘f‘, V] .—() n " ': rv': 1'] .1 r". . I I" c '7'“ u \l" '\ v" ‘ 3“ “‘ﬁ ’) ‘3 ‘ " 7 ': ‘1 "3 ’ '_‘ I" ' (I
suing/it; d. lanky OI L ........ inL_L_.'x;J tawtl‘eg “LU. 0.511 “CULHL; ‘9 Jaiuio Oi
. r . ’ ’ 7.1. r 1 I  r H
l‘ncrUa‘S'L‘Ilc; r ~( 4. Ultra 39 .4‘ XJCIUJ) v .2 U _ 1 '9. U‘ ./ 1,1) U1. LI'I ) Y 'K).l(\.41‘1{J IGNLL “ 1” .. U
.r‘ ' ,. 1.1. _"\" ._'., ,._ _ .. .1 n 13,: 1' 33".,” .1. .r‘ " . 1 ‘ ,, ' ,
this rapier 1110,1ca1 UfOquuJ oi Qiﬂiniohldw returns lolioued 0v increts—
ing returns to successive nitrogen inputs, the polynomial equation is
probably a more appropriate approximation to the fertilizer response
surface.
Because of the phenomenon of increasing returns to nitrogen inputs
exhibited by the exponential functions, maximum yields and high profit
plant nutrient inputs lie beyond the range of experimental inputs. The
maximum yield as calculated from the polynomial equation is 32.2 bushels
per acre. This maximum is achieved using slightly less than 318 pounds
of nitrOgen and about 629 pounds of P205. The quantities of N and P205
producing the maximum bean yield are almost identical with those of the
highest treatment level in the experiment.
Despite the large phosphoric acid inputs which produced maximum
yields, applications of phosphoric acid were not profitable at typical
crop and fertilizer prices. Assuming a price of $0.10 per pound for
P205, use of P205 became profitable only with bean prices in excess of
$7.00 per bushel. Nitrogen inputs, on the other hand, were profitable
over a wide range of been and nitrogen prices. Assuming a price of $0.15
per pound for nitrogen, the high profit quantity of nitrOgen ranged from
about 76 pounds with bean prices at $3.50 per bushel, to 205 pounds at
$7.50 per bushel. Estimated high profit nutrient inputs for various
bean prices are shown in Table 20.
90
TABLE 20
HIGH PROFIT FERTILIZER INPUTS FOR FIELD BEANS
wITH VARYING BEAN PRICES
High Profit Plant
Bean Price Nutrient Inputl Predicted Yield
(per bushel) N P205 (bu.per acre)
$3.00 35 o 19.70~
3.50 76 0 21.82
h.00 106 0 23.15
8.50 130 0 2h.07
5.00 lh8 0 2h.68
5.50 16h 0 25.16
6.00 177 0 25.52
6.50 188 O 25.79
7.00 197 O 25.99
7.50 205 35 26.60
1N and P205 were priced at 20.15 and 0.10 per pound reSpectively.
Analysis of the Potato Data
V
M
The original potato experiment was initiated in 195h. Data have
been collected for three successive years. Only'P20s and K20 were varied
in this experiment. The response surface estimated for the 195h data
was one of diminishing absolute yields with additional inputs of P205 and
K20. The pretreatment fertility level of the plots was such that,
given the weather conditions existing in 195b, the portion of the reSponse
91
surface characterized by the experiment was that of stage three in
the inputoutput dimension, i.e., negative marginal returns to additional
plant nutrient inputs.
Preliminary analysis of data collected from the two succeeding
years, 1955 and 1956, indicated no significant change in potato yields
associated with applied plant nutrients. Further analysis of the soil
test data and associated changes in yields over time may provide useful
information as to depletion rates and residual fertility as well as
yield response to applied plant nutrients. However, data collected to
date from the original potato experiment do not indicate P205 and K20
reSponses of economic consequence.
Data were also collected in 1956 for the lit plot potato experiment
initiated on a previously unfarmed parcel of muck soil. Preliminary
analysis of these data indicated that much of the variance in potato
yields was not associated with variance in applied plant nutrients.
However, as some discernible relationships were evident in the data, a
functional analysis was conducted and is presented in Equations XIV and XV.
The first formulation attempted was a fivevariable polynomial
which is shown in Equation XIV.
A
l
Equation (XIV) : Yb = 38.u8031h37  .06692653 P + .Izobosoo K +
(.OL313US9) (.019792e?)
.00003556 P2  .00013183 K2 + .00022502 PK
(.00009859) (.00002250) (.00003975)
1Yields expressed in Equations XIV and.XV are pounds per plot.
IMultiplying pounds per plot by a conversion factor of 6.8062 gives the
potato yield in bushels per acre.
\O
I’D
The adjusted coefficients of mul iple correlation and multiple de termin
ation for this equation are .501 and .251 re ectively. The second,
fourth and fifth variables of this equation have coefficients which are
statistically Sig ii" Icant at the one per cent probability level.
The second formulation of the functional relationship was an
emponenti ial type equation as shown in Equation XV.
quatlcn (XV)? L08 Y = 1h213095;  .OJSGPPYC log P  .00013065 P +
p (03h07L68) (.0001h319)
.20502059 log K  .ooo21112 K
(.0289h7h2) (.oooeo317)
The adjusted coefficients of multiple correlation and multiple determin
ation for Equation XV are .585 and .3h2 respectively. In the latter
ormulaiion , coefficients estimated for both K20 variables are sitnifican+
H)
at the one percent probability level.
r—o
0 f" ' ro o _ \_ _ ~ L _o r v r. ‘ _o c_ 0
.axrr 3m YiCl as and Tjrh Profit Plant huurient applI aLlOﬁS'
Because of the complex nature of the PROSK20 yield relationship,
it is extremely difficult to determine the amounts of plant nutrient inputs
(1) whi h maximize yields or (2) which maximize profits. The complexity
of these relations hips is further exemplified by the derivatives of the
functions. For example, he part al derivative of yield with reSpect
to phosphoric acid for the polynomial is neca tive for almost any quantity
of P205 unless K20 is fixed at a level of at leas 250 pounds per acre.
In the case of the exponential, the partial ee Ivative ofy wlo with
ream) ct to phos sphoric acid is always negative. Because of these unusual
93
phenomena, at ordinary potato prices the calculated highdprofit quantity
of P205 is negative and consequently outside of the range of observed
values.
It is the expressed Opinion of soil scientists that the interaction
between P205 and K20 is an important complementary relationship for
potato production on muck soils. This interaction effect may exceed in
importance the individual effects of either plant nutrient. There is,
in particular, a commonly held belief that applied P205 will cause
significant increases in potato yields only if adequate amounts of K20
are concurrently present in the soil. In view of this, it may not be
illogical to assume that P205 applications had a nonsignificant effect
on yields at lOW'KQO treatment levels. Both equations contain at least
one nonsignificant P205 coefficient which may bias the estimate of the
production surface and consequently the derivatives of the function.
Further detailed analysis of these data is needed; however, it
appears that, given the weather conditions of l956,ru>substantial appli
cations of plant nutrients were profitable. If any plant nutrient
applications were profitable at all in 1956 they were only moderate
applications of potash.
CHAPTER V
SOURCES OF UNEXPLAINED VARIANCE IN YIELDS.AND
BIAS OF REGRESSION COEFFICIENTS
The analysis presented in Chapter IV was designed prinarily to
explain variance in crop yields with logically formulated functional
relationships between quantities of applied plant nutrients and crop
yields. Statistical estimates of the parameters of the plant nutrient
variables were made using alternative production function formulations.
On the basis of these estimates, inferences were made as to the shape
of the plant nutrientcrop yield production surface, plant nutrient
combinations producing maximum crop yields, plant nutrient combinations
producing maximum dollar profits, etc.
Variance in crop yields was not solely a function of variance in
the quantities of applied plant nutrients. The adjusted coefficients
of multiple determination ranged from a high of .58 to a low of .05
for the crops analyzed. Lacking knowledge of the exact form of the
functional relationship between applied plant nutrients and crop yields
and, furthermore, lacking completely effective control over unstudied
variables, one should not expect 100 per cent of the variance in crop
yields to be associated with regression. One might, however, expect a
greater proportion of yield variance to be associated with regression
than was found to be the case in the analysis of the preceding chapter.
9h
Failure to characterize the major portion of yield variance by
functional analysis raises questions as to whether or not experimental
controls were rigidly enforced.
This chapter will be directed first towards an explanation of
variance in crOp yields not explained by the regression of applied plant
nutrients. An additional problem deals with whether or not unspecified
variables were randomly and normally distributed with respect to the
independent variables studied.
Sources of Unexplained.Variance in Yields
Sources of unexplained variance in yield can be broadly classified
as being due to (1) experimental error with reSpect to variables Specified
and measured and (2) inadequate control over unspecified and unmeasured
variables. Since these two sources of yield variance should be normally
and randomly distributed with reSpect to treatment variables, they may
be viewed as being sources of within treatment yield variance.
Experimental Error
Some portion of the unexplained variance in crop yields is undoubt—
edly due to experimental error. Such errors are made by not applying
the Specified amounts of plant nutrients on individual plots or errors
made in acquiring yield measurements from the plots. ther sources of
experimental error are uneven seed and fertilizer distribution on plots
to mention only a few. In general, however, these errors are expected
to be somewhat normally and randomly distributed with reSpect to
96
treatments and should be averaged out in the statistical estimating
process. Researchers should recognize that this component of variance
is present even in rigorously controlled experiments. Competent
researchers should attempt to minimize such errors subject to the con
dition that the cost of reducing the errors is not in excess of the
value of the gain in accuracy resulting from their reduction. For
example, mechanization of controlled experiments may introduce experi
mental error in excess of that occurring with the use of hand—labor
methods. However, minor increases in experimental error may be more
than offset by the acquisition of additional information and better
functional analysis resulting from additional plots and/or larger plots.
Thus, reduction of experimental error should not be established as an
absolute goal but rather one subject to economic considerations.
It is the opinion of the author that the data analyzed in the
preceding chapter did not, in general, have excessive experimental error.
Some experimental error, however, was present. In particular, the con
tinuous corn experiment was characterized by a considerable amount of
such error. Due to unfavorable weather conditions it was necessary to
harvest the continuous corn plots by hand. Only a subsample from each
plot was harvested; consequently, due to the smaller harvested sample a
larger experimental error would be expected. Furthermore, the previous
crOpping history varied for some of the plots in this experiment.
Although corn yields from plots on the two areas with different cropping
histories were not significantly different statistically, this hetero~
geneity of previous land use probably contributed to a minor amount of
97
variance in yields. Since the total yield variance was small originally,
the existence of experimental error made it difficult to isolate the
effects on yield variance due to variance in the quantity of applied
plant nutrients.
Uncontrolled and Unmeasured Variables
Numerous factors such as weather, insects, bacterial action in the
soil, etc.,are possible sources of variance in crOp yields not explained
by the functional analysis in Chapter IV. The field bean input—output
experiment was duplicated in the greenhouse. Results of the greenhouse
experimentation are presented here to substantiate the hypothesis that
yield variance could be explained by functional analysis given adequate
control of unmeasured variables affecting yield and/or specification and
measurement of these variables.
A nine variable polynomial was fitted to data produced in the green
house. The soil contained in individual greenhouse pots was acquired
from the correSponding field plots. The same number of observations were
acquired using the same treatment levels as in the field experiment.
Yields acquired in the greenhouse were for bean numbers per pot since the
1
beans could not be allowed to mature under greenhouse conditions. The
results of th's regression analysis are presented in the following equation:
1Bean count and bean yields are not perfectly correlated, however,
the two measures should be sufficiently correlated to allow valid infer
ences to be made from one to the other quantity(yield) wise. Bean count
might, however, be a considerably less valid measure of the quality of
the crop.
98
Igh = 9.22679560 + .hShooioB N + .0203075 P + .07h73918 K  .00081362 n2 
(.o378h008) (.01919871) (.03779oho) (.00011092)
.00001986 P2  .000258338 K2 + .000197867 NP + .00012933 HK  .00003616 PK
(.000019855) (.00011181) (.00003765) (.00007785) (.00OO3762)
i‘ = .91
P? = .828
These results indicate that about 83 per cent of the variance in
bean count for the greenhouse pots was associated with regression. In
the functional analysis of the field data, however, only h2 per cent or
about onehalf as much of the variance in bean yields was associated.
with regression. The inference suggested by this comparison of analyses
is that explanation of more of the variance in yield under field condi
tions would be possible if variables affecting yield could be better
controlled and/or measured and Specified in the functional relationship.
Effects of Within Treatment Variance
on Statistical Estimates
The presence of within treatment variance should be noted when evalu
ating the relative success of particular functional forms in characterizing
inputoutput relationships. If there is a difference in the yields from
plots receiving the same plant nutrient applications, any function fitted
to these data by least squares techniques, or any valid estimating
procedure, will miss one or both yield observations. The greater the
difference in yields between replicated plots, the greater will be the
variance which cannot be explained by the function. Failure to explain
this within treatment variance is not therefore a valid criticism of a
particular functional form.
99
The effects of within treatment variance on the amount of total
variance explained by regression may be exemplified by use of data from
the 1956 wheat experiment. Data from all 130 plots were used in acquir
ing the statistical estimates made for the nine variable polynomial
function presented in Equation V. A second polymial equation was fitted
to the average yields of the plots which had a minimum of two replications
for a given treatment. These observations include yields from the
3 x 3 x 3 factorial which was replicated twice and the ll check plots
for a total of 65 plots averaged into 28 observations. Statistical
results for the function fitted to the average yields from replicated
treatments is shown in Equation XVII.
A
Eiuation XVII: Yﬁ a 27.87287 + .11222667 N  .ooo28877 N2 + .02207660 P 
(.0292398h) (.00010520) (.01h61992)
.oochoos P2  .01h07013 K + .OOOllOll K2 + .00003065 NP  .00013398 NK +
(.oooo2o3o) (.29239Ch) (.0001c32o) (.oooo273o (.ooooSM72)
.00001889 PK
(.oooo2736)
The adjusted coefficients of multiple correlation and multiple determin
ation for this equation were .79 and .62 respectively as compared to .66
and .hh for the function fitted to all 130 individual observations. This
sizeable increase in the amount of yield variance explained by regression
illustrates that within treatment variance was an important component of
total yield variance. The parameters of a function fitted to the average
value of replicated plots, if all plots are replicated an equal number of
100
times, should be the same as those for a function fitted to the non
averaged observations, however, avera jng values for replicate Wb
ations discards part of the information prov: ded oy the experime ental data.
rectors R.lated to the Independent Variables
O
[Jsﬁxi IIII;:1*:s;,IoIIIInasﬂgsIIs
LIL
Visual obse ervation oft he experimental plots indicates that there
were yield variance creating comg>onents which were associated with plant
nutrients and which therefore were either (1) sources of biases in the
estimat ed mf etc of plant nutrients on yields or (2) sources of yield
riance which should be considered when evaluating the aggreg'te ef ects
of applied plant nutrients.
Incidence of Needs, LOdging and Plant Disease
Observational data collected for oats in 1956 ind cated mignificant
differences in weed growth and plant lodging which were as wsociat d With
nitrogen applications. Prior to harves gthe oats crop, individual
plots were ranked as to the degree of weed infesta oion and plant lodging
and then these ranks were tabulated against nitrogen applications. The
results of this classification are shown in Table 21. The incidence of
weeds in plots was ranked from O to 3 with an increase in number rank
indicating an increase in weed infestation. Lodging was ranked similarly
from 0 to 8. ‘Weeds and lodging not only affected the absolute crop
yields produced on some plots but the harvestability of the crop as well.
The ratio of the amount of grain produced to the amount of grain
lOl
TMEE21
INCIDENCE OF hﬁBD INFESTATIUN AND PLANT LODGIKG ON CAT PLOTS
AS RELATED TO NITROGEN APPLICATIONS
it
Standard Standard
Nitrogen Number ‘Average Deviation Average Deviation
Application of Lodging of Lodging Weed2 of Need
(pounds per acre) Plots Scorel Score Score Score
0 18 .889 .7hl O O
20 2h 2.250 1.561 .167 .3hO
ho 1h 2.lh3 1.187 .lh} .32h
80 29 b.690 l.Sll .828 .hS?
160 18 6.hhh 1.257 1.556 .889
iPlots were ranked from O to 8 according to the extent of lodging
present.
2Weed incidence was ranked from O to 3.
harvested was probably significantly different for badly lodged plots as
compared to nonlodged plots. No statistical measures of these differences
were made, however.
Incidence of plant disease as well as weed growth varied with plant
nutrient applications on the field bean plots. Particularly, quack
grass infestations were more pronounced on high nitrogen plots than on
plots receiving smaller applications of nitrOgen. Plots with a large
amount of plant foliage tended to have more shading of lower leaves and
bean pods and consequently more disease infestation. The quantity of
foliage on plots was, in turn, associated with the quantity of applied
nitrogen.
102
These and other variance generating factors which are not inde
pendent of quantities of applied plant nutrients, but which may influ—
ence yields in a manner other than that Specified in the production
function formulation are sources of bias, i.e., they distort the absolute
crop yield producing effects of plant nutrients. If such distortion
or bias is a necessary consequence of applying plant nutrients it should
be measured and considered when evaluating the effects of plant nutrient
applications. In some instances, however, utilization of improved crOp
management practices may eliminate such effects. For example, if weeds
could be adequately controlled and if a sufficiently strong strawed
variety of wheat were available for planting, the potential effects of
n
eercts
(D
applied plant nutrients on crOp yields might be realized. Th
of factors which are sour es of bias in plant nutrient inputcrOp yield
output estimates as well as a discussion of other factors interacting
with plant nutrients in the production of crOps are discussed adequately
i
in other literature and will not be enlarged upon here.
Relationships Between Residual Fertility and Crop Yields
L
All plots in the two rotation experiments received the same plant
a.
nutrient applications in 1955 and in 1956. It seemed logical to expect
1For a discussion of these factors see L. S. Robertson Jr., G. L.
Johnson and J. F. Davis, "Problems Involved in the Integration of Agrono
mic and Economic Methodologies in Economic Optima Experiments," Fertiliser
Innovations and Resource Use, Edited by E. L. Baum, E. O. Heady, J. T.
Peach and C. G. ﬁildreth (Ames: Iowa State College Press, 1956) pp. 226—
2h2, and L. S. Robertson Jr., W} B. Sundquist and L. N. Shepherd "A Frog
ress Report of the Studies on the Economics of Fertilizer Use on Beans
and Potatoes," himeographed Report presented at a T.V.A. sponsored sym
posium on the economics of fertilizer use at Knoxville, Tennessee, Harch
1957.
103
some carryover or residual effects in 1956 from plant nutrient appli
cations made in 1955. This was particularly true for the plots receiving
heavy plant nutrient applications in the preceding year. Soil tests for
P205 and K20 were taken preceding and following every crOp produced.
Consequently, differences in fertility between plots prior to plant
nutrient applications made in 1956 would be expected to be related to
these soil test measures. The method utilized in attempting to relate
variance in crop yields to soil fertility as measured by soil tests will
be summarized briefly. Soil test measures were first correlated with
the applied amounts of the same nutrient for the individual plots. If
this correlation was very high it would indicate that (l) variance in
yield could probably be explained as well by the original functional
analysis using only applied plant nutrients and/or (2) it would be diffi
cult to include both soil test and applied plant nutrient measures in a
1
functional analysis since the presence of high intercorrelations would
reduce the reliability of estimated parameters of a function containing
both measures as variables. If, on the other hand, the correlation
between quantities of applied plant nutrients and soil tests was low,
indicating some independence of the two measures, soil tests might
successfully be used to explain a portion of the variance not associated
with regression. The procedure used in relating soil test data to un
explained variances was to correlate the soil test data with the
1The distinction between high and low correlations is quite arbi
trary, however, as intercorrelations approach .70 the reliability of
estimated parameters probably begins to decrease quite rapidly.
th
residuals computed from the original functional analysis. The results
obtained by this method of analysis are shown for one crop from each of
the rotation experiments, namely, the wheat and bean crOps produced in
1956.
,Effects of Residual Fertility;on wheat Yields
Soil test data were collected for P205 and K20 preceding the wheat
crop grown in 1956. No analysis of residual nitrogen has been completed
to date.1 ‘The first analysis conducted was that of correlating prel956
crop soil test measures2 with the applications of P205 and K20 made in
1955. In the following discussion, soil test values of P205 and K20 are
designated as Pst and.KSt reSpectivelyy Applied P205 and K20 are
designated Pa and Ka'
A regression analysis was conducted using Pst as the dependent
variable and Pa as the independent variable. The resulting regression
equation is as follows:
13340 a memo + .166659 Pa
(.016565)
The coefficient of correlation for this equation was .662 and the co
efficient of determination .h38. A similar regression analysis was
1Several nitrogen tests determinations have been made for soil
samples from these plots. No nitrogen soil tests have as yet been
generally accepted as satisfactory. A statistical comparison of the
effectiveness of alternative nitrogen soil tests for residual nitrogen
is currently in process using soil samples from this experiment.
zThese soil samples were actually acquired in September of 1955
immediately preceding seeding of the wheat crop which was harvested in
1956.
105
conducted relating soil test measures and applied quantities of K20.
The resulting regression equation is as follows:
A
Kst = 75.108u + .276689 Ka
(.039190)
The coefficients of correlation and determination for this equation were
.526 and .277 reSpectively. A preliminary inspection of the residuals
of these functions indicates that little, if any, improvement could be
made by changing the formulation, i.e., by fitting a curvilinear form
such as Pst = a + Pa + P: to the P205 variables.
As evidenced by the preceding analysis, applied and residual plant
nutrients Show a moderate amount of interdependence or correlation.
A correlation as high as .66, as was found between Pst and Pa, might
indicate that the effects of residual and applied P205 could not be
easily separated. The correlation between Kst and Ka, .53, does not,
however, appear to be prohibitively high.
On the basis of the preceding exploratory analysis it was decided
that some reduction in unexplained yield variance might be accomplished
by incorporating the residual fertility measures into the analysis.
Simple correlation analysis was conducted using Pst and Kst as separate
independent variables and the residuals from the nine variable polynomial
(Equation V) as the dependent variable. Designating the residuals or
deviations from the polynomial (Ii  ii) as D, the results of these
simple correlation analyses are as follows:
1C6
U)
= .l§9696 + .OO’ZZ‘D'E) Psi)
(.006331)
”I
r2 I: .OCDCS
D = 1.L58§83 + .01h232 Kst
(.ooohto)
F” = .190
Eb = .0362
Phosphoric acid soil test measures appear to bear no relation to the
unexplained residuals of the original functional analysis, whereas,
potash soil tests are slightly, but not significantly, related to these
residuals. The inference suggested by this analysis appears to be that
soil test measures do not provide an aid in reducing unexplained yield
variance in the case of wheat, at least not in the simple relational
form analyzed here. It should be remembered, however, that soil test
measures were correlated with quantities of applied plant nutrients and
that most of their effects on yields are probably incorporated in the
1
original functional analysis. Additional work is currently in progress
evaluating soil test procedures and relating these measures to quanti
ties of applied plant nutrients.
lGordonhnderson of the Department of Agricultural Economics and
Arthur welcott of the Department of Soil Science of the hichigan.Agri
cultural Experiment Station are COOperating on this phase of research
work.
107
Effects of Residual Fertility n Dean Yielcs
As in the case of wheat, soil test data were collected prior to
fertilization and planting of the 1956 bean crop. P205 soil test
observations ranged in values from h8 to 50h, however, only one observ
ation was in excess of hOO. K20 soil tests ranged from a low of 8h to a
high of hOO. Application rates in the bean experiments ranged from O
to 320 for K20 and from O to 6hO for P205. A regression analysis was
conducted using Pst as the dependent variable and Pa as the independent
variable. The results of this regression were as follows:
97.9670078 + 2.253587 P
(.11616h)
"U >
U)
(+
II
a
F2 = .593
The same analysis was conducted using potash soil test measures and
treatment rates as variables. The results of this regression are shown
in the following equation:
Kst = ~7h.970613 + 1.2h59oo Ka
(.091887)
E7 a .6h5
F2 = .hlé
As in the case of the wheat experiment, a greater portion of the
variance in P205 soil tests was associated with variance in P205 appli
cations of the preceding ear than was the case for K20. Correlations
as large as these, .770 and .6h5 reSpectively, indicate that the effects
108
of residual fertility on bean yields migit well have been explained at
least in part by the original functional analysis in which yield vari
ance was formulated as a function of variance in applied plant nu.trients
alone.
Despite the high correlation between soil test measures and quanti
ties of applied P205 and K20, a multiple regression analysis was conducted
. A
using Pet and Kst as dependent variables and the residuals, (Ti  Ti),
from Equation XI, the six variable enponen al equation, as the dependent
variable. The results of this multiple regression analysis are as
follows:
E = 2.0157  .07973 Pst  .020h2 Hot
(.C'OET9) (.Oﬂhhg) o
The parameters of this equat ion are highly SLfniiicant however, the
adjusted coefficient of mul_tiple correlation is only .068. This value
of §.is not significantly different from zero. As in the wheat experiment,
no significant amount of the variance in yields not explained by the
original regression analysis with applied plant nutrients as independent
variables can be attribute d to residual fertility as measured by K20
and P205 soil tests.
(D
p)
‘1
O
m
0
<1
0
*1
p
H
'1
C.)
:1,
£0
0
E
Ther why failure to relate unexplained yield
variance to soil test measures should not be interpreted as meaning that
crOp yields are not a function of residual fertility. P305 and K20 soil
tests were found to be significantly related to amounts of applied
nutrients, hence, a portion of their effect on yield variance would be
expected to be characterized by the original functional analysis.
In addition, soil test measures for nitrogen, which in almost all
experiments had the predominant effect on crop yields, were not included
in the analysis. Soil test measures are themselves subject to consider
able variance beCause of errors in sampling and in testing the samples.
Additional research needs to be undertaken in calculating sampling
and testing variances for soil test procedures. Such research would
provide an aid in evaluating the accuracy and adequacy of Soil testing
procedures currently being used. Another possible explanation of the low
correlation between soil tests and residuals is that a more complex
formulation of the relationship between soil est measures and unex
plained residuals would have been more appropriate, i.e., the linear
relationship assumed in simple correlation analysis may be an over
simplification of the relationship between these variables.
Because of the importance of soil test data in making current
fertilizer recommendations, additional work needs to be done relating
alternative soil test measures to: (l) variance in crOp yields (2) quanti
ties of applied plant nutrients to establish substitution ratios between
applied and resi‘ual plant nutrients and (3) other soil testing methods
to determine the most effective soil test procedures available.
The plant nutrient input—output experiments described earlier should
provide data well adapted to an analysis of soil esting procedures.
The experiments contain extremely high and extremely low levels of plant
nutrient applications and consequently a wide range of residual fertility
values is develOping on the plots. As individual plots become extremely
depleted of plant nutrients or extremely fertile they will provide a
110
wide range of soil test observations. If plant nutrient applications
were to be rerandomized on the plots, a wide range of residual and
applied nutrient combinations could be observed. Thus the effects of
residual fertility might be studied without the complicating influence
of highly correlated plant nutrient applications. ,Furthermore, substi
tution ratios between residual and applied nutrients could be estimated
from a wide range in the combinations of the two.
Conclusions
Several subinferences may be drawn from the analysis presented in
this chapter. The important conclusion, however, seems to be simply
this: Given (1) adequate control over Specified factors affecting crOp
yields, and (2) a random and normal distribution of other factors affect
ing crop yields, functional analysis should provide an adequate repre
sentation of plant nutrient inputcrop yield output relationships. The
relatively small amount of total variance in crop yields explained by
functional analysis is not an inherent characteristic of the analysis
and/or the functional forms used but rather is largely a function of the
uncontrolled factors enumerated in this chapter.
EVALUATION OF PROCEDURES AND RESU TS
Evaluation of Experimental Desirns
The experimental designs used in the several experiments described
in this work were formulated with several restrictions and objectives in
view. Prior to designing the experiments, it was decided that continuous
function analysis of the experimental data would provide a better basis
for (l) estimating plant nutrient inputcrop yield output coefficients
and (2) facilitating an economic analysis to determine optimal plant
nutrient applications, than would alternative methods of analysis. Thus
the experiments were designed to provide data suitable for continuous
function analysis. Restrictions on funds, labor and equipment limited
the number and/or size of the experimental plots. Individual treatments
or cells in the eXperimental designs were selected to: (1) describe the
economically relevant portion of the production surface sufficiently to
obtain reliable estimates of parameters of the production functions
(2) establish with adequacy inputoutput measures for critical points on
the production surfaces, e.g., origin of the functions and their inflec
tion points and (3) minimize intercorrelations among treatment variables.
It is the Opinion of the author that the experimental designs were
quite satisfactory as a basis for providing data for continuous functions
lll
H
}J
[‘0
analysis. The experimental designs utilized in the two original rota
tion experiments are not highly efficient in providing data which
1
reatnent yield variance
cf
readily facilitates estimation of (1) within
(2) crop quality differences associated with treatments, (3) differences
in plant nutrient content of plant tissue and (h) differences in other
plant and soil characteristics associated with plant nutrients but not
in the manner prescribed for the basic inputoutput relationship.
Once committed to an incomplete factorial design with a minimum
number of replications, analysis of such factors as those listed above
2
may be quite difficult. However, he designs which were used are adequate
for these determinations if (1) the determinations can be made by corre—
lation analysis or (2) if the determinations for one plant nutrient can
be assumed to be independent of the treatment level of other plant
nutrients. In the latter case this means that all observations for
which the treatment level of the studied variable are constant can be
considered as replications of that treatment.
A modification of the incomplete factorialminimum replication
design used in the rotation experiments was incorporated into the con
tinuous corn, the 1956 potato and the new sugar beet experiments. These
designs include a triplicated factorial in addition to other treatments
which were replicated twice. This modification was incorporated into
1Inability to Specify within treatment variance is not considered
to be an important criticism of the experimental design.
2The inference being made here is that some of the determinations
listed above can best be acquired by analysis of variance techniques.
113
the designs to facilitate analysis of byproduct data produced in the
experiment. The experimental designs, as modified, still provide numerous
nonreplicated treatments in order to Specify the production surface
adequately for continuous function analysis. Inclusion of a factorial
into the experimental design facilitaoes utilization of analysis of
variance techniques on a limited basis at little additional cost.
A possible criticism of the experimental designs which were used
might be the large Spacing between treatment levels of the various plant
nutrients. Obviously, it would be desirable to have observations at
treatment levels intermediate to those contained in the experiment,
however, the experiments already were large and required a considerable
amount of land, labor, machinery, equipment and supervision. Larger
experiments would have created problems in conducting experimental work,
such as seeding, harvesting etc., with apprOpriate timeliness. The primary
consideration in not enlarging the experiments by including intermediate
treatment levels was that of the additional time and cost which would be
necessitated by such an expansion.
The correlation between applied and residual plant nutrients is
relatively high in these exgeriments since individual plots receive the
same treatment in successive years. A more comprehensive analysis of
residual and applied plant nutrient relationships would be facilitated
by rerandomizing treatments on the experimental fields. Such a modifi
cation of the experimental design would provide observations over a much
wider range of combinations of residual and applied nutrients. This is a
modification of the experimental design currently being contemplated.
11h
Evaluation of Experimental Procedures
To the extent feasible, the experimental work was conducted utiliz
ing mechanized procedures. When soil conditions allowed, plant nutrient
applications were made with a 7 ft. ‘ractor drawn drill. Small grain
seedings were also made with a 7 ft. drill which required one round on
the plots which were 1h feet wide. Wheat and oats crOps were harvested
with a 7 ft. self propelled combine. A portion of the corn crop was
harvested by using an eSpecially constructed singlerow corn picker.
In instances where weather conditions prevented fertilizer application
and corn harvest by machine, this work was accomplished by use of hand
labor.
Some amount of additional experimental error undoubtedly occurs due
to use of machinery as compared to hand labor; for example, plant
nutrient applications are not precisely weighed out and delivered in exact
amounts to individual plots. Small amounts of grain remain in the combine
from one plot to another when harvesting etc. and introduce some small
experimental error. These errors should, for the most part, however,
average out and not bias the plant nutrient inputcrOp yield output esti—
mates made.
Mechanization of experimental work provides some interesting and
important implications particularly with reSpect to the number and size
of individual plots which can be satisfactorily included in an experiment.
Two objectives of plant nutrient inputcrop yield output research appear
to be of relevance here. First, we want research results to be validly
inferrable to some farm population. Farmers typically operate as units
fields of a minimum of several acres in size. The larger the experi
mental plots, the more nearly they represent the conditions actually
existing on farms. Farmers, and consequently researchers whose objective
is to make inputoutput estimates applicable to farm conditions, are not
particularly interested in measuring within treatment yield variance.
Rather, they are interested in determining the variance in yield which
can be attributed to variance in plant nutrient applications under farm
conditions e.g., the change in yield resulting from application of an
additional 20 pounds of nitrogen etc. Researchers are interested,
however, in having some assurance that within treatment yield variance
is not prohibitively large so as to constitute a large portion of total
yield variance. Within treatment variance is reduced by increasing the
size of individual experimental plots and the harvested portion of these
plots. Increases in plot size are facilitated by mechanizing the experi
mental procedures used. Errors of inference due to excessive within
treatment yield variance can be eliminated alternatively by replicating
a given treatment several times and averaging the yields of the several
replications. Additional replications of a treatment require more labor
and have a higher cost than is true for a comparable enlargement of a
given plot. It is the opinion of the author that when the main objective
of experimentation is to estimate plant nutrient reSponse surfaces,
increasing plot size is a more efficient alternative.
A second objective of our research, that of estimating input—output
coefficients to which we can attach acceptable reliability measures,
116
is aided by increasing the number of individual plots in an experiment.
.9 standard error of estimate for parameters in a functional equation
diminishes as the number of observations increases. .Attainment of both
accurate and applicable research results is, therefore, enhanced by
increasing the size and number of experimental plots.
It is the opinion of the author that within treatment variarce in
the experiments was probably higher than was necessary. Use of larger
plots and/or harvesting a larger portion of individual plots would
probably have provided results the additional accuracy of which would
have been worth the cost of obtaining this accuracy.
Y"! 0 v o ' 
evallation of Analytical Procedures
The Continuous Function Analysis
A brief justification for utilizing continuous function analysis was
presented in Chapter II and will not be repeated or expands here.
Rather, a brief a posteriori evaluation of the effectiveness of the con
tinuous function analysis used will be attempted here. Both polynomial
and exponential type formulations of the reSpective production functions
were fitted for all creps for which preliminary analysis indicated that
an appreciable amount of variance in yield was associated with variance
in applied plant nutrients. No criteria are available which provide a
basis for saying one formulation is "absolutely" more appropriate than
the other; however, some measures which provide somewhat of a quantitative
basis for comparis n are available. rthermore, logic and theory provide
ll?
a basis for selecting one formulation in preference to the other in at
least two instances.
As previously mentioned, comparison of the coefficients of multiple
correlation for the two functions provides a guide as to the relative
amount of yield variance associated with regression. This comparison is
somewhat subjective, however, since: (1) in the case of the eXponentials,
variance is measured in logarithms and in the polynomials it is measured
in real numerical values. Although the logarithms and real numbers
bear a consistent monotonic relationship to each other over the range of
the values which they take in the data, they do not retain a relationship
of constant ratios. (2) The two formulations differ as to the number of
variables in the respective equations, hence there is a small difference
in the number of degrees of freedom used in the two analyses. The latter
difficulty is not an important one, however, because of the large number
of observations and, hence, degrees of freedom, present in the analysis.
A comparison of the coefficients of multiple correlation and determin
ation for the functions fitted is shown in Table 22. In three of the
six comparisons a larger amount of yield variance is explained by
regression for the exponential equations than for the polynomials. In
one case, that of the field beans, the polynomial equation has larger
values of h and ﬁg, whereas, in the remaining two comparisons values
of E and i2 for the two equations are almost identical. This comparison
provides no very conclusive indication as to the superiority of either
type of formulation.
TM1322
COMPARISON OF AMOUNTS OF YIELD VARIANCE ASSOCIATED WITH
.ALTERHATIVE PRODUCTION FUECTICN FCRLULATIOKS
CrOp
Function
Number of
Variables R R3
Oats, I956 Polynomial 9 .69 .h8
Exponential 6 .76 .58
Wheat, 1956 Polynomial 9 .66 .bb
Exponential 6 .65 .h2
Corn, 1955 Exponential 6 .70 .h?
Polynomiall 9 .6h .hl
Corn, 1956 Polynomial 9 .23 .CS
Exponential 6 .2h .06
Cont. Corn, I956 Polynomial 9 .hO .16
Polynomial S .39 .16
Beans, I956 Polynomial S .65 .h2
Exponential 6 .61 .37
Exponential h .61 .3
Potatoes, I956 Polynomial S .50 .25
Exponential h .59 .)h
P7
1The polynomial used on the
polynomial of the form Y = a + b1
+ b9
[EF—
1955
N +
Corn data was a square root
2
b /ﬁ. + b3P + b4 /P" + b5K + b6 /i— +
119
A second comparison of the two types of functions was included in
A
the analysis. Residual measures, (Ii  Ii), were computed for both
types of functions. These residuals are measures of the deviation of
predicted yields from observed yields. The residuals are almost identi
cal for both t; es of functions for all crops. This is true for the
magnitude of residuals as well as for their sign or direction. In
summary, inSpection and measurement of the residuals provides no discern—
able basis for choosing one function in preference to the other.
A third comparison of the polynomial and exponential functions
which might provide some basis for choosing the more apprOpriate one is
an inspection of the derivatives of these functions. InSpection of the
partial derivatives of the exponential functions with respect to individual
plant nutrients shows that the derivatives are usually of extreme
magnitude (negative or positive) for small inputs of the plant nutrients
1
and then become extremely small quite rapidly. Extremely large deriva
Y O 0 ﬂ 9
., With small inputs or the Xi are a consequence of he yield
tives ~—:~
’ Z>Al
being zero when any of the Xi = O. Derivatives of the polynomials in
comparison usually take less extreme values.
It is the opinion of the author that over moderate plant nutrient
input ranges for most crOps, generally in the range of 20 lbs. to 200 lbs.,
the exponential is probably a satisfactory formulation of most of the
1There are exceptions to this statement. For example, the partial
derivatives of bean yields with reSpect to nitrogen decreases at firs
and then increase with additional nitrogen inputs. here are other
exceptions to this statement as well.
inputoutput relationships. Derivatives of the exponentials for field
beans and potatoes, however, are contradictory to the usually accepted
concept of diminishing returns. Maximum yields predicted using the
exponential functions were outside of the range of observed inputs for
1
the bean and potato crops. However, the maximum potato yield predicted
was secured using quantities of plant nutrients outside of the range of
observed inputs using the polynomial as well.
Calculation of the quant it es of plant nutrients which result in
maximum profits is a much more complex procedure using the exponential
type formulation than using a polynomial. Solving the exponential for
optimal inputs requires use of a series of successive approximations
2
known as Newton‘s method. This method requL was in part a graphic
approximation refined by solving a series of equations. Statistical
estimates of the parameters of both types of equations are rather easily
acquired by methods of least squares.
The primary advantap e of the exponential type formulation as com
pared to the particular polynomial used is that it permits derivatives,
Q
X.)
2’ l
to takecninonlinear forms. Deeratives of a polynomial containi ng
" 1
range of observed inputs is a criticism 01 the inn
reality the maximum yield does occur within t1:e ranfe of ooserved inputs
1The phenomena of maximum predicted yields being outside of the
c
and is fallaciously pr mt d to be outside. I1, indeed, the true maxi
mum yield exists in ziond the range of observed inputs it is the experi
mental desi;.ii, not the function, v1111c1i s1‘lould oe critici ed.
3a complete ex,1anation of t 3 method of solving a Carter—halter
t pe e: :ponent ial is explained in a forthcoming article in the Journal
of Farm Economics by A. h. haltei E. C. circa“ and J. G. hocking.
u.)
.. . ‘ < O “’._1“ V. a A)". ,'.‘._' r‘ nr‘n“ur n
Tne authors also discuss in some cecail the cronnroic) of this ianiiy oi
"unctions.
._ D. L 1 r an “ “‘1 — a. >7 a P 1. 
only llfSu ant Szcuvu 1333:; c»rm3 xiii necessarily be restrictc to
g «f V. 5 "q _. ﬂ 3‘.> _ 5c , .. _ . f‘ ‘. ' P '.  s ' O _ 0 ~.. .
linear iorm. Iv .Lb L113 opinion Oi {1:123 author brat until a 31:1“ 001111.111
._ , ,.1 1H  '1 I,” , ,1 , ,., 7__'1I... ‘ I "I I‘l. '
113 proceeuies for soiVan a Carter—naiter ty] e .ponkn iai icr OptiMal
plant nutrient inputs are available that modifications of the polynomial
3
"““j '7‘ I, ~r"'—.l ' '1‘ ' 1.3 '13 :ﬁru'."vr~"‘  ~— ‘.4.,.. 3}.  _ ’7‘
Byte: iOl I1 Lei 1.; on 1111,3311; Du {7.011, C in: 11; :A'Jl *3 . .LIlCUl pol a biﬂf5‘ \fal'lLJ 1.13:? ()l
v. '1 ’3 WW) . pa . 7~........ . A . .—‘ . .7’1 ....'I  9
degree i/e or nxa or oi a CULECB 2< c dLU not equal to l will resait in
non—linear derivatives.
uliiuaercn of Soil Test Measures
No significant amount of variance not explained by regression was
explained by use of P205 and K20 soil test measures. Soil tests for
residual quantities of P205 and K20 were not significantly correlated
A
with residuals (Vi  Yi) from the functions fitted for wheat and beans.
The analysis presented here does not provide a very comprehensive
exploitation of the possibilities of using soil test measures in supple
menting functional analysis of applied plant nutrients. The Federal
Extension Service, as well as Agricultural Experiment Stations and private
fertilizer companies rely heavily on soil test data as a basis for making
fertilizer recommendations. Because of the wideSpread use of these soil
test procedures, any additional information relating variance in soil
test measures to crOp yields would be a very valuable contribution.
Economic Interpretation and Ev alu a ion of Results
The most profitable amounts of plant nutrients to apply we*e co puted
for all crops except alfalfa. The analysis presented in Chapter IV
122
indicated a significant reSponse to nitrogen for corn produced on a
Kalamazoo sandy loam soil in 1955. Significant yield responses to
applied nitrogen were recorded for oats, wheat, field beans and corn
produced on a Wismer clay loam soil in 1956. The only crop not showing
a significant reSponse to nitrogen in 1956 was the corn produced on a
Kalamazoo sandy loam soil.
Statistically significant reSponse to applied phosphoric acid was
recorded for wheat, field beans and the corn produced on a Wismer clay
loam soil in 1956. Oats and corn produced on a Kalamazoo sandy loam
soil in 1956 did not Show significant yield response to applied phosphoric
acid. The only crOp showing significant yield reSponse to applied
potash was the potato crop produced in 1956 on a Houghton muck soil.
DeSpite the several significant reSponses recorded, only small
amounts of plant nutrient applications were indicated to be profitable.
Predicted high~profit plant nutrient inputs for the various crOps are
shown in Table 23. No applications of P205 and K20 were indicated to be
profitable for any of the crOps produced at typical crop and fertilizer
prices. Nitrogen applications were profitable for five of the crOps
produced if crOp prices were sufficiently high. Assuming typical prices,
however, nitrogen applications were profitable only for corn produced
in 1955 and field beans produced in 1956.
Some qualification of these results seems to be warranted. First,
the 1955 and 1956 growing seasons were characterized by severe summer
drouths. Thus the reSponses recorded.may not typify the longrun
expected responses to applied plant nutrients. Additional data collected
TABL 3 23
ESTIMATED HIGHéPROFIT PLANT NUTRIENT APPLICATIONS
FOR VaRIOUS CROPS
Estimated High Profit Plant Nutrient
Tintitsl
Crop V e i A v 5
N PROS 320
Oats, 1955 None None None
Oats, 1956 Only if the
price of oats None None
> €211.00
Wheat, 1956 Only if the
price of wheat None None
> 253 .00
Corn, 1955 About hO lbs.
(on Kalamazoo Sandy loam soil) at Corn None None
prices of
Corn, 1956
(on Kalamazoo Sandy loam soil) None None None
Corn, 1956 Only if the
(on Wiener Clay loam soil) price of corn None None
> 2.00
Field Beans, 1956 75 lbs. with Only if None
beans 33.50, bean prices
150 lbs. with are >iﬁ7.00
beans at $5.CO
and 200 lbs.
with beans at
$7.00
Potatoes, 1956 Not varied in None None at
experiment ordinary
prices
1Computed with N at $0.15 per 1b., P205 at $0.10 per lb. and K20
at $0.11 per 1b.
over time are needed to obtain a probability distribution of yield
reSponses over the range of existing weather conditions.
As a further qualification, it should be noted that the experimental
results reported in the preceding analysis were obtained from soils
either (1) relatively unproductive, as in the case of the Kalaaazoo
sandy loam soil or (2) relatively heavy and productive in the cases of
the Simms loam and Wiener clay loam soils. One might expect, a priori,
to obtain the greatest yield reSponse to applied plant nutrients from
soils with a high productive potential but with low fertility levels.
Greater yield reSponse may be noted in future years on low nutrient level
‘
0
plots as residual fertility is depleted.
Concludinv Remarks
The analysis of experimental work presented here is rather limited
in sc0pe with respect to nunber of soils, crops and growing seasons.
Additional work is needed before the optimal plant nutrient treatments
estimated here can be substantiated or invalidated as longrun Optimal
applications. The distribution of yield responses over time is likely
to be characterized by wide diSpersions, particularly in the case of the
lighter soils which are frequently subject to damaging drouth periods.
However, some interesting questions and implications are posed by the
results of the analyses presented here.
J
P
w
No significant response was obtained from applied potash for t
several crops grown on mineral soils durin* a twoyear period of
I I
L.)
experimentation. This lack of reSponse poses a question as to the
validity of recommending a program of "ba lanced” plant nutrient appli
cations. Rather, the general reSponses recorded from thes experiments
indicate that nitrogen was the rite ary source of crOp yi as r Sponse.
On the basis of these results it appea‘s that a plant nutrient combination
weighted more heavily with nitr05:en relative to potash might be Optimal
at least until residual potash is depleted somewhat.
A second general implication posed by the experimental results is,
"despite statistically significant yield responses, in most cases the
cost of pplying additional plant nutrients exceeded the value of the
additional crOp produced." This general result would indicate that
analysis which only detects significant yield differences which are
associated with plant nutrient applications is not an adequate procedure
for determinin~ the most profi able application rates. This result in
itself would seem to validate or at least vindicate the general type of
anal uiS used in this dissertation, i.e., that of continuous function
analysis to which economizing principles may be applied.
In conclusion, at the farm management application level of fertili
zation practices, these practices cannot be considered independent of
other alternative farm business expenditures nor can they be considered
independent of the numerous factors with which they interact. For examp_e ,
a livestock farmer may find it profitable to fertilize oats, not for the
oaty ield benefits, but in ordeu to establish a clever or grass seeding
which is essential to his livestock ente rplise However, if a farm
manager is to intelligently and economically Synth ethe costs and
benefits of the numerous components of his farm business he needs infor
mation as to the productivity of expenditures made for plant nutrients
for the various crops he produces. Additional plant nutrient inputcrOp
yield output estimates will help to provide this information.
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