•
SOME PROBLEMS IN THE ESTIMATION
OF SINGLE NUTRIENT RESPONSE FUNCTIONS
by
R. L. Anderson and L. A. Nelson
Institute of Statistics
Mimeograph Series No. 737
Raleigh - 1971
•
SOME PROBLEMS ;m THE ESTIl1ATION OF SINGLE NUTRIENT
-e
RESPONSE FUNCTIONS
R. L. Anderson, University of Kentucky
and
L. A. Nelson, North Carolina State University
1.
Introduction.
We will be concerned here with the problem of estimating
the amount of a given nutrient (X) to add to an existing level (8) to
obtain optimal yield, e.g. the amount of nitrogen to add per acre to
achieve maximum profit in the production of a crop.
It is assumed that
all nutrients except the given one (e.g. nitrogen) are already available
in sufficient amounts.
The functional relationship between the average
yield (n) and the total amount of nutrient (N
=X +
8) is usually quite
1
complex ; however, up to the point where added nutrient becomes noxious
a reasonably good function is the exponential-type
(1.1)
This assumes that n
=0
when IT
= O.
Equation (1.1) traditionally has
been called the Mitscherlich equation.
When experiments are conducted to estimate a and y, the variable
nutrient is the amount applied (X) and not the total available (N),
because the amount of usable nutrient in the soil (8) is generally
unknown.
Hence the model is usually changed to
n
1
=a
- Se
-yX
;
a
The actual yield is indicated by Y
residuaL
•
>
S
>
=n+
0;
y
>
O.
(1.2)
e, where e is a random
-2-
•
Equating (1.1) anci (1. 2) for N =_X
8
* 0,
W~
see that_
= ae -yo •
(1. 3)
If the price ratio between the cost of a unit of added nutrient
and the selling price of a unit of product produced is p, the expected
returns due to the .additionof X units of nutrient will be proportional
to
(1. 4)
n - pX
Therefore the optimal yield is attained when dn/dX
= p;
hence, the
optimal X is
Xo
= tn8 +
=
tny
-
tnp
y
tna + tny
y
- tnp
-
(1.5)
0
Exact confidence limits for X cannot be determined from experimental
o
data.
In addition, estimates of 8 and yare often quite poor if few
levels of X are used in the experiment and the
experiment~l
error is
large (e.g. with a coefficient of variation of more than 5%).
For these reasons, many statistical analysts prefer to approximate
(1.1) by a quadratic equation:
(1. 6)
The counterpart to (1.2) would be
(1. 7)
where
•
8~
= 81
-
- 28 20 ;
(1.8)
•
-3__From (l.8) iLis clear that the parameters-, S~ and S~, in the usual
quadratic model are functions of
8,
which may vary considerably from
one field to another.
The optimal amount of X to be added is
(1.9)
One can determine exact confidence limits for X if the experimental
o
yields are normally dis tributed.
appropriate if
n ultimately
Model (1. 6) or (1. 7) may also be
decreases for additional units of X, but
equation (1. 1) needs modification to adapt to this situation.
It should
be indicated that neither the exponential nor the quadratic model is
appropriate if there are increasing returns to scale for small X.
At first glance, it might appear that the problem of determining
the optimal amount of nutrient to add to that already existing can be
obtained from the first part of .(1.9), which does not require a·
knowledge of
function of
6.
o.
This ignores the need to estimate S~, which is a
Hence if an experiment is conducted at an experiment
station and the results are to be applied to a field for which the
8
differs from that at the experimental site, an incorrect recotmnendation
may be made.
Anderson (1956, pages 57-59) discussed this problem.
Some of these COtmnents are expanded here.
Suppose the value of
8 at
the farm is Of and at the experimental
~
si te is 0.
e
Using (1. 9), the recotmnended optimal amount of X to add
would be
•
(1.10)
•
-4whereas the correct amount to add should be
(1.11)
The bias is
(1.12)
In other words, if
site (of <
8e ),
° is smaller at
the farm than at the experimental
the recommended amount of nutrient to add at the farm
~
~
~
should be increased by the difference 0e - of; similarly if of > 0e'
therecominended amount to add at the farm should be decreased by the
~
difference of - 0e·
It is informative to interpret the above result in terms of the
bias in expected yield at the farm where the amount of nutrient
recommended by the experiment station, X ' is added to the soil.
Oe
n
ee
represent the average yield when X
Let
is applied to experiment station
oe
nf e
represent
·
soil with
8e
n
measures the bias in average yield due to application of a
the average
. yield when Xoe is
2
applied to the farmer's soil with ;Sf avai.lable • The difference
fe
- n
ee
available
and
.
non-optimal amount on the farm.
~
~
~
Using (1.6) and (1.10), we obtain
.
-
-
2.
~
-
-
-
2
nfe-nee=[Sl(XOe+of)-S2(XOe+of) ]-[Sl(XOe+oe)-S2(XOe+oe) ]
=(8 f -8 e ) [Sl- 2S 2XOe-S2 (8 f +8 e >]
=(8
Hence ifo
e
>
f
-8 e ) [p+S 2 (6 e -8 f )]·
(1.13)
8f , the average yield to be realized on the farm will
be less· than that based on experiment station results •
•
2
~
In o~her words ni; is the expected yield baseg on the quadratic model
when X is added" to a soil which already has 0i.
Oj
-5-
•
A more informative c:omparisonmigl1tbe basedonexpectedlossm of
returns due to the use of a non-optimal X, where the expected returns
-
are proportional to n - pX, as in (1.4).
In this case we wish to
compare the returns when the correct X is added to the of at the
Of
farm with the returns when the incorrect X is added to the
Oe
8f ,
i.e.
compute the difference
Using (1.6), (1.10), (1.11) and (1.1Z) this difference is
-
-
nff -n fe - p(XOf - XOe )
= Bl(XOf + 6f
=
-
- XOe - Of) - BZ[(XOf +
-
(XOf - XOe)[B l
= BZ(oe
8f )Z
- (X + Of)Z]
Oe
BZ(XOf + XOe + ZOf) - p]
- Z
- 6f ) . ~ O.
(1.14)
The above result (1.14) indicates that if the 0 at the experiment station
differs from that at the farm in either direction and we add n'utrient
based on the experiment station results, the average return will be less
than could have been obtained if
t~e
amount of nutrient added had been
based on farm conditions.
In addition to the above, knowledge of 0 is also useful in
combining the results of experiments conducted on different sites and/or
at different times, especially if the o's may differ widely.
also been discussed by Anderson (1956).
•
This has
For many nutrients, it is
possible to estimate 8- by use of soil measurements.
These measurements
-6-
•
ar~
subject to certain limitations because soiL test determinations are
artificial and involve chemical extractants which are used to simulate plant
uptake of nutrients.
If the simulation is reasonably good
(e.g. present
phosphorus and potassium tests) it is highly desirable that the model
contain terms involving these measurements.
At the present time, good
soil test procedures are not available for some soil nutrients.
Urlfortunately, one of these is available soil nitrogen, which perhaps
is the most important single plant nutrient.
Basic research is needed
for developing more reliable nitrogen soil test procedures.
If soil measurements cannot be obtained for a given nutrient,
efforts should be made to estimate 0 by statistical methods.
L. A. Dean
(1970) has reported that previously statistical methods were used
extensiv~ly
instead of chemical test results to estimate existing levels
of nutrients in Hawaiian soils.
This paper will be concerned with this
statistical problem for a single nutrient, usually nitrogen.
The
problem of simultaneously estimating the o's for several nutrients
is much more difficult [see Hurst (1962)], and will be deferred to
later research.
2.
Estimators Adjusted for Existing Nutrients.
For similar soil, Sl and
~2
should be relatively stable, but
-
because of different past management practices, 0 may vary widely even
from replication to replication on the same site.
vary widely from replication to replication.
Hence So* and Sl* may
To test this we have
analyzed the results of 25 corn fertilization experiments conducted
•
-7-
•
in eastern North Carolina from 1955 through 1957 in which nitrogen was
the only limiting factor. 3
In the analysis of these experiments, the values of X were coded
as deviations from m, the middle level of X:
x
=X-
m.
Hence the
predictor model was
(2.1)
Estimates of the coefficients in (1.7) could be obtained by the
following decoding operations:
Q
I-'
2
= B2'.
(2.2)
In these experiments, X was in units of 62.5 pounds per acre, m = 2,
and the yields were in bushels per acre.
Values ofB ' B and B2 for each of four replications at each
O l
location (three replications for one location in 1955) are presented
in Table 1.
Each replication had 18 treatment combinations of nitrogen,
phosphate and potash in a 15-point central composite design plus
three additional points.
Since nitrogen was the only limiting factor,
we will discuss only that factor here.
there were two plots with x
= ~2,
For each location-replication,
four plots with x
= -1,
five plots
with x =0, four plots with x = 1 and three plots with x = 2.
3See Baird and Fitts (1957) and Hurst and Mason (1957) for a detailed
discussion of these experiments •
•
-8-
e-
fo11owitlg~a1ysis
Thevar:f,.abili ty in the B' s was examined by_ the
of Variance:
Degrees of
Freedom
Source of
Variat'ion
Locations
24
(L)
Bean Squares
B
B
1
O
82.84
1232.57
Expected
Value of MS
8.10
2
0'2 + 0 + 40~
p
r
IV
in L (R)
74
55.45
8.98
2.97
ap2 + 0' r2
Plots in R (P)
1485
10.20
3.40
2.12
0'
Reps.
2
P
The mean squares for locations and replications (in location) are
cotnputed from theB",va1ues given in Table L
The expected mean squares
are based on a model of this type:
(2.3)
where B
is the value of B for the k-th replication of the j-th
ijk
i
location,
e~
is the expected value of B , and
i
!/',
rand p are error
2
2
2
components with zero means and respective variances 0 0 , 0' and 0' •
IV
r
p
A sum of squares for lack of fit was c01l\puted for each locationreplication.
The corresponding mean square with 15 degrees of freedom
is the s2 given in Table 1.
92.6.
The average of the 99 values of s2 is
The estimated variances for the B's for a given location-
replication are:
" 0.11013s 2
Var(B O) =
Var(B )
1
Var(B )
2
"
2
=
0.036711s
(2.4)
= 0.0229448 2 •
A
The average of these estimated variances is obtained by substituting
•
s 2 ::; 92.6 in (2.4).
These are the estimates of the average plot
variance$, 0 2 in the above Analysis of Variance.
p
-9It is evident from the above A,na1)7sis of Variance thatcthere are
large
location-to~loeation differences
differences for
E2~
for B and B and moderate
O
1
large replication differences for EO' modest
differences for B and very small differences for B •
I
2
Since 3
and
0
E differ widely from replication to replication, it was deemed
l
necessary to estimate a separate 0., for each location-replication.
J-K
~
Estimates of 62.5 0jk' designated as D , in pounds of
jk
are presented in Table 1.
n
per acre
Since the 3 -values are relatively stable
2
from replication to replication, an average 3, for each location
"
could have been used in these computations;. however, we decideci to
use a separate B for each locationureplication.
2
From (1. 7) and (2.2), we see that
D
= 62.5
~Bl +J B~ +
t
2B~
4BOB2
(2.5)
L.
For example, consider the first replication for location 551:
BO =
D
6S.9~
= 62.5
HI
t
= 8.2, HZ = 4.8;
[S.2
+J(;~~U~
~.6
J1- 2
= 65.
(2.6)
If the average B of 6.3 instead of 4.8 had been used in (2.6), D
2
would have been 45.
The average value of D for this location would have
been reduced from 52 to 47.
The average values of D for some locations
would have been reduced considerably:
•
_
-10-
•
Usin~
Location
Table 1
552
101
82
554
66
55
652
65
53
658
169
143
660
62
54
752
50
46
B2
The values of D given in Table 1 were much larger than the soil
scientists expected for the co.arse textured soils in eastern North
Carolina.
One purpose of thb paper is to indicate why D as computed
here may be an over-estimate of the true
3.
o.
Bias in Estimation of 0
(a)
Comparison of 0 and
o.
o = (in
~
From (1.3) and (1.8), we see that
- in S)/y ;
o- =
(3.1)
-
To illustrate the problem, let us assume that n(X) and n(X) coincide
for X
=0
and 1 and that n
=n
•
max
max
n(O)
= n(O)
• nO;
To simplify the algebra, let
n(l)
= n(l) = no
+ dl ;
(3.2)
In Qther words, nO is the average yield when no nutrient is added;
d represents .the expected increase in yield due to the first increment
1
of.applied nutrient; d represents the expected additional increase up
2
to the maximum yield. The maximum yield using the exponential model
•
(1.1) is located at X =00.
The. maximum yield using the quadratic
.'
-11model (1.6)_ is_located
~tX=
* 2,
8/28
:Bysuhstituting tbesevalues of
X in (1.2) and (1.7), respectively, we obtain
nmax Substitution of X
and nO + d •
l
ct
=0
and
(3.3)
and X
=1
in (1.2) and (1.7) produce nO
Therefore the equations for the exponential model are
(3.4)
The solutions are
(3.5)
The equation for y is obtained by subtracting the second equation
from the third equation in (3.4) and taking logarithms.
Using (3.3), the equations for the quadratic model are
nO
= 80*; no +
no + dl + d2
dl
= 80* + 81*
= 80* + 81*2 148 2 ,
- 82 ;
(3.6)
By subtracting the first equation in (3.6) from the second equation
and then the first equation from the third equation, we obtain the
following two equations:
(3.7)
Solving for 8 2 in the last equations,
(3.8)
•
-12-
•
UsiIlg (3.1), we.see that
~n(nO+
d1 + d2 ) ~n(d1 + d ) 2
IS
=
=
~n(d1
~n
+ d2 )
(3.9)
d2
-2/8 (d +d ) + 2/8 (d + d ) + n 8
2 1
2 1
2
0 2
2
28 2
1I~0 + d + d 2
1
/d
1 + d2
~
(3.10)
In: (3.9) and (3.10), the results are scale-free, Le. 0 and
-
of 0 and 0 are presented in Table 2, where I =100(0 - 0)/0.
d ~ d ,
1
2
8 often
will be more than 50% in excessdof
o.
6
If.
Directions
of changes in this excess for changes in nO' d and d2 are summarized
1
below:
Fixed
Increase
nO,d 1
n ,d
O 2
d ,d
d
1
-
Change in excess [I]
decrease
2
d
increase
nO
increase
1
2
One explanation for the bias in.
8 is that the slope of n is
much smaller than that of n at X = 0; as a result, the X value for
- =0
n
is much farther from the origin tLQn is t::at for n
-
slopes for nand n at X
=0
The
are:
n t (0) = 8y;
•
= O.
n'(O)
= 81* •
(3.11)
-13-
•
The ratio of the two slopes [using the results in (3.5), -(3-. 7) and (3.9)]
is:
2[1 - Id /(d +d )]
2
l 2
tnT (d +d2 ) 7d ]
l
2
= n' (0)
n' (0)
It is clear that R' is actually a function of the ratio r
= dl /d 2 ,
R'(r) ~ 2[1 - (1+r)-1/2]/tn(1+r).
R' (1)
= 0.84
unity.
i.e.
(3.12)
and R' -+ 1 as r -+ 0 (d -+ co), but R' is always less than
2
Also R' -+ 0 as r-+
co.
Values of R' are presented below for
selected values of r:
r
1/5
1/4
1/3
1/2
1
2
3
4
5
R' (r)
.956
.946
.931
.905
.845
.769
.721
.687
.661
(b)
Estimation Problems for a Quadratic Model when a Yield Plateau is
Reached at an Intermediate Value of X.
6 will
Even if we ignore the fact that
be greater than 6, there is often an added positive bias in the
estimatlon process itself.
We will illustrate the problem with an
experiment involving five equally spaced levels of X:X
4.
Suppose the maximum yield is attained after X
x ..
2; also that the average yields for X
for X = 2.
Therefore,
•
Using the notation of (3.2),
6 is
given by (3.10) •
=3
=1
= 0,
1, 2, 3 and
but on or before
and 4 are the same as
-.
-14It is assumed that the experimenter uses model (2.1) with x
replaced by X and fits the five observations by least squares.
If
we neglect the experimental errors,
B0
= nO
B.
=
(54d +41d )/70;
1
2
B
2
=
(2d +d ) /14;
1 2
1
D
~~~
+ (4d l -5d 2 ) /35;
+4B OB2 -
B~ I~B2'
(3.13)
Some comparisons of D (3.13) with 0 (3.10) and 0 (3.9) are
presented below for selected values of
nO dl d2
D
-
0
0
nO' d1 and d 2 :
no d1 d 2
-
D
0
1
2
1
0.473
0.366
0.262
3
2
1
1.206
0.980
0.631
1
3
1
0.388
0.236
0.161
3
3
1
0.948
0.646
0.404
1
5
2
0.245
0.148
0.107
3
4
2
0.670
0.532
0.369
1
5
3
0.197
0.156
0.120
3
5
2
0.593
0.419
0.285
2
2
1
0.857
0.688
0.465
3 5
3
0,516
0.445
0.325
2
3
1
0.679
0.449
0.292
5
2
1
1.828
1.498
0.893
2
5
2
0.424
0.288
0.201
5
3
1
1.437
1.000
0.585
2
5
3
0.360
0.305
0.228
5
5
2
0.910
0.664
0.430
These results indicate that, for experiments in which a yield plateau is
reached at small values bf X, about 50% of the bias in D is caused by
the attempt to fit a single quadratic function over the entire range.
This tends to move the maximum point to the right (the estimated
X
is too large) and decreases the slope at the origin (producing an
max
•
over-estimate of 0).
The next section presents an estimation procedure
-lS-
•
which might be useful in eliminating much of this bias in D.
It should
be mentioned that the effect of experimental error on the bias in D is
not apparent from a cursory examination of (3.10) and (3.13).
Some
empirical sampling would be useful in assessing this effect, since the
bias cannot be obtained in a closed form when experimental errors are
included in the model.
(c)
An Ad Hoc Procedure to Adjust for Some of the Bias in D.
If there
is evidence from the yields that a yield plateau has been reached at one
of the intermediate levels of X, the following procedure might be used:
(i)
Assume the values of X are Xl' X2 , ... ,Xm, Xm+l"" 'Xn with
the Y-plateau being reached at X
= Xm •
(ii) Replace X = Xm+l' Xm+2'."'Xn by X = Xm•
This implies that
if there is one observation at each X there now will be
n-m+l observations for X
= Xm•
(iii) Determine the B's and D, using these newly designated
levels (Xl' X 2 "" ,Xm) •
Two sets of data taken from Heady, et al. (19SS) are considered below:
X
1
2
3
4
S
6
7
8
Y
23.0
°
88.4
10S.4
128.8
123.0
110.6
127.4
133.1
129.2
Z
22.0
79.1
10S.S
103.8
129.2
127.1
137.9
119.4
123.3
Both Y and Z are bushels of corn per acre.
For Y, each unit of X represents
40 1bs. of N per acre, with each plot also receiving 160 lbs. of P 0 per
2 S
acre.
For Z, each unit of X represents 40 1bs. of P20S per acre with each
plot also receiVing 320 1bs. of N per acre •
•
-16-
•
(i)
'I'~;ese
X represents 40 pound units of applied N.
Y-data have also been analyzed by Anderson (1957).
The
prediction equation using all nine levels of X is as follows: 4
Y = 42.60 + 32.00X - 2.80X
9
2
2
= 38.7(X + 1.20) - 2.80(X + 1.20) ;
hence
09 = 1.20 = 48 lbs. of N per acre.
If we let X=3 for all experimental units receiving 3 or more
units of X, the condensed data are as follows (f
= number
of replica-
tions):
X
f
~Y
0
1
23.0
1
1
88.4
2
1
105.4
3
6
752.1
Total
9
968.9
In this case we obtain the following prediction equation:
2
Y = 25.68 + 65.49X - 10.80X
4
= 73.0(X + 0.347) - 10.80(X + 0.347)2;
04 = 0.347 = 14 lbs. per acre.
The estimate of
° using the exponential model was IS
=
4"
Y or Ok indicates that k levels of X have been used •
k
•
9 lbs. per acre.
-17-
•
One might have preferred 0Illitting all data for X>3.
case,
In this __
5
y' = 25.74 + 64.94X - 10.50X2
4
= 72.8(X + 0.374) - 10.50(X + 0.374)2
A
0'
4
= 0.374=
15 lbs. per acre.
There is very little change in the results by pooling all
observations for
X~3
over those obtained by omitting data for X>3,
except that the .variance of B is reduced by about 20%.
2
There is a
-
decided reduction in the estimate of 0 when one either deletes the
observations for X>3 or pools them with those for X=3.
(ii)
X represents 40 pound units of applied P20S'
Using all plots, Heady, et al. (1955) obtained
2
Z9 = 34.07 + 36.l6X - 3.23X ;
0
9
= 35
lbs. of P 0 per acre.
2 5
Assuming the maximum yield is attained at X=4 and pooling all data for
X~4,
2
Z5 = 27.36 + 48.65X - 6.00X ;
0
5
= 21
lbs. o·f P 0 per acre.
2 5
The result is· almost identical if the data for X>4 are omitted.
The usefulness of the truncation procedure is seen in the
better fit to the yield data for small values of X.
y'
k
5
•
or
5'k
indicate that only k levels of X are used. and that all data
for levels above k have been deleted •
often large increase in yield attained wity an initial application
of fertilizer is revealed even for the truncated data.
In general
Y (or Z) will be too large for the check plots (no added nutrient)
and too small for the plots exhibiting a substantial fertilizer
effect (here those receiving X=l).
This initial jump in yield often
does not occur until X>l; in this case we may experience the typical
logistic growth situation (concave upwards .at the start followed by
concave downwards).
If the quadratic function does not exhibit the
correct large jump in yield for X=IJthe estimated value of 0 will
be too large J and often by a large amount J even if we follow the
truncation procedure proposed in this section.
Because of this, we
have considered the use of other models which might be even more
useful in reducing the bias in
(d)
The Square Root Models.
o.
Many analysts have used the square root
model J
because it appears to more nearly approximate
•
n. If we proceed as
-19-
•
in_the! 1gection above,
(3.15)
(3.16)
Similarlyusing the second and third equations,
(3.17)
Therefore
/1 + 0* -
(11 +
= ;S(l1 +
0* - 15*)
0* - fi*).
Consequently,
0*
~2
=
= _0_-::--
(1 + 5)2 - (&)2
1 + 20
= 6/(2
•
~2
0
+
5-1 )
<
5/2
(3.18)
-20-
•
Since 0 is seldom, if ever, as much as twice o,it appears that
the use of the square root model would
estimate of o.
by
1*
The percentage by which
in Table 2;
0*
al~ost
0*
always lead to an under-
underestimates
is usually less than half as large as
0
is given
0 and
often
much less than this.
Some investigators have used the quadratic-square root model
[see Heady, et a1.(1955) and Anderson (1957) ~
n**
2
= a*0
- a*x
+2
a*1:X 3
- e*x·
1
(3.19)
In order to introduce the amount of soil nutrient into (3.19), it should
be modified to
(3.20)
We have not investigated the use of
a better approximation to
n
n**
than either
in this paper, but it may give
n or
n~
because it isa compro-
mise between two models, one of which overestimates 0 and the other
underesti~ates
o.
A major difficulty with
n**
is that it requires the
estimation of an additional parameter; to date we have had enough
difficulty with
4.
three~parameter models.
Comparison of Numerous Three-Parameter Models for the North Carolina
Data of Table 1.
A general three-parameter polynomial model was studied:
(4.1)
where we considered p
= 0.50
(square root model); 0.75; 0.95; 1.25;
1.50; 1.75; 2.00 (quadratic model).
•
Values of D = D(p) for each model
-21-
•
for each location-replication were compared with a 0 based on the
exponential model.
In addition p, an optimal value of p, was determined
for each location-replication, with the restriction that 0.25
~
p
~
2.00.
The criterion for optimality was minimum residual sum of squares.
For many replications, the optimal value of p was at or near 0.25.
Note that we considered only models for which p
~
0.25; hence,
presumably for some of them p should have been almost zero.
the model would be linear in X.
If p = 0,
The average yields for each X and
estimates of soil nutrient using three models for some of the replications which have p at or near 0.25 are presented below:
LocRep
551...3
552-2
554-1
554-3
556-4
557-1
X=o
X=1
X=2
X=3
X=4
28.2
65.1
61.4
64.3
61.5
50.1
78.4
77.4
80.5
80.9
13.0
40.0
28.6
29.8
35.9
12.9
44.0
42.3
27.6
37.7
40.8
70.2
71.2
70.7
68.6
27.6
51.5
54.2
45.7
40.7
D(2)
D( .5)
.6
78
10
20
134
30
5
129
8
4
69
3
11
102
19
9
75
10
LocRep.
559-2
559-4
651-1
658-1
752-4
754-3
X=o
X=l
X=2
X=3
X=4
26.2
71.9
79.8
77 .1
79.1
12.8
59.6
61.3
63.4
49.3
34.2
78.0
85.4
82.7
80.0
72.2
90.6
88.2
88.4
93.0
36.6
81.0
70.2
66.9
66.0
52.3
81.2
85.1
81.0
84.3
o"
12
51
5
42
29
1
14
59
8
22
294
88
6
97
11
26
137
32
o
D(2)
D( .5)
•
-22For these data, the quadratic model is not flexible enough, whereas the
square root and exponential models produce similar (much smaller) estimates
of
o.
It is unlikely that one should use more than Z
=1
(62.5 lbs. per
acre) for most of these replications; hence, observations are needed
between X
=0
and X
=1
in order to estimate adequately
t~e
parameters in
a three-parameter model.
The values of p varied considerably within a location, implying that
the same polynomial model would not be appropriate for all replications
even at the same location.
There was almost as much variation in p within
locations as between locations.
Also, within locations there were high
positive correlations between p and D(p).
The values of D(p) and 0
, 1) f or t h e rema i
"~cat~ons6 not represente d'~n the a b ove
( exponent~a
ni
ngI
rep
table are presented in Table 3 for p
= 0.50,
0.75, 1.25, 1.50 and 2.00.
From this table we see that D(2) tends to be about double 0
D(o.5) is about one-half of O.
ar::·~_~lat
This correlation between p and D(p) is
in close agreement with results presented in Section 3.
F1:om these comparisons involving numerous three-parameter models
it may be concluded that even when the power is chosen so as to minimize
the residual sum of squares, the effect of the plateau response pattern
on biasing the estimates of 0 cannot be eliminated by choice of polynomial
model.
As
indicated in Section 3, it would be quite useful to compare
6The following location-replications have been omitted, because they had
outliers, which should have been discarded before the analysis was made:
551-2; 552-1; 557-4;753-3. These outliers were responsible, among other
things, for some of the inordinately high values of s2 in Table 1 •
•
-23-
•
these results with those for a_f9ur-parameter model such as the quadraticsquare root model; unfortunately this might not bee too informative here
because there
ar~
only five levels of X, and in many cases there is little
change in Y at the last two levels.
On the whole, it appears that if one has data such as that obtained
in the North Carolina study and he needs a good estimate of 0, he should
use the exponential model.
With modern computers it is relatively easy
to obtain estimates of a, Sand y (and consequently, 0).
Since the
exponential model is non-linear in the parameters, the computations
involve an iterative procedure; unfortunately, the usual computing procedure may not converge for some data sets, especially if there is a
large experimental error.
On other occasions,the estimates of the
parameters may be nonsensical if one insists on minimizing the residual
sum of squares.
This is particularly true if there are one or more
outliers in the data set.
have preferred to use
parameters.
th~
This is one of the reasons that statisticians
quadratic model, which is linear in the
They have also used the square root model as a model linear
in the parameters because they neglect to include 0 in the model, i.e.
the customary square root
mod~l
is
* + YZ*rxX
n* = YO* - YlX
Estimates of the yls are simple least-squares estimates, but they do not
enable us to estimate
o.
One procedure to assure reasonable estimates of
the parameters of the exponential model is to include prior information
on the parameters in the estimation process.
•
problem, see Gregory and Anderson (1970) •
For a recent study of this
-24-
•
As will be pointed out in the next section, there may be good reasons
for desiring to use a quadratic model.
If a quadratic model is used and a
yield plateau is reached for small or intermediate values of X, it is
suggested that some form of truncation procedure be followed.
5.
Comparisons of Estimators of Optimal Amount of Added Nutrient, X '
O
The major effort in this paper has been devoted to a comparison of
various estimators of the amount of nutrient (eS) present in the soil
before a
ferti1i~er
experiment is performed.
It is our belief that even
if the primary objective of an experiment is not to estimate the soil
nutrient level, the model should contain a term to represent this unknown
factor in order to realistically portray the actual response phenomenon.
Fertilizer experiments usually are conducted for the purpose of
obtaining estimates of X ' the optimal amount of nutrient to be applied
O
to the soil for the production of a specified crop.
Although the present
study was not designed to evaluate the effects of the choice of model
upon the estimates of X ' some preliminary information on this aspect was
O
obtained as a by-product of this study.
This preliminary information will
be presented here in hopes that others will expand upon the research
dealing with the effect of choice of model upon the estimates of X '
O
When it was found that choice of model had a decided effect upon
estimates afeS it was also suspected that likewise, estimates of X
o
might also be materially affected by this choice.
This suspicion was
supported by the results for the Iowa data presented in Section 3(c).
For the Y-data, a reasonable value of p is 4.
•
Hence the estimated
optimal value of X using Y , based on (1.9), would be
9
-25-
•
32 - 4
~--:--:......
=
5.6
= 5.0.
However, if we use Y4' we obtain
A
XO,4
=
65.5 - 4
21.6
= 2.8.
For these same data, the estimate of X using the exponential model was
o
3.4.
Clearly there was an effect of choice of model upon the estimates
Paez (1971) found that the estimates (averaged over the 4 rep1ications) of X producing the maximum yield (X
) for the same 25 North
max
Carolina locations described in the present study, using the exponential
model, were considerably higher than those obtained using the quadratic.
The square root model resulted in even somewhat higher estimates of X
max
than those obtained using the exponential modeL
It appears that the value of X using the exponential model may be
o
an overestimate because it has an asymptotic maximum.
Consequently the
exponential may not be as good a standard model for comparison as it was
when we were considering estimates of
o.
In fact, it is not clear what
model should be used as a standard because the XO's obtained using the
square root and quadratic models also are suspected to be too high when
there is a yield plateau,
st~rting
at a low or intermediate value of X.
The upward bias in X ' using the quadratic model, for data with a yield
O
plateau is attributed to estimates of (31* and (32 which are biased downA
ward, with 8
2
(the denominator in the equation for X ) being biased downO
A*
ward to a considerably greater degree than 8 ,
1
•
Although the reason for
the bias in the estimate of X using the square root model is more
o
difficult to identify, it is thought to be analogous to that described
-26-
for. the quadratic modeL
The effect of the omission of
o from
the square
root model upon the bias in X is not known, as we did not fit"models
o
without 0 in this study.
It is interesting to apply the procedures in Section 3 to the
problem of determining X •
O
Applying (3.5) to (1.5), we obtain
XO(exponential)
where r
= dl /d 2 •
=
R.n(l +r) + R.n R.n(l + r) - R.n(p/d )
2
R.n(l + r)
(5.1)
Similarly applying (3.7) and (3.8) to (1.9), we obtain
-
Xo(quadratic)
=
1 + r -
Z+
II
r -
+
r
p/d
-
Z
(5.Z)
2/1 + r
Finally if we apply the results for the yield plateau example in (3.13)
to (1.9) we obtain
XO(quadratic; plateau)
-* =
=X
o
41 + 54r - 70p/d
Z
10(1 + 2r)
Results are presented below for selected values ofr and p/d :
Z
•
X
o
X
o
-*
X
o
LO
0.5
O.Z
0.1
0.5
L5
2.8
3.8
0.5
2.0
2.8
3.1
0.8
2.0
2.7
2.9
2
LO
0.5
0.2
0.1
L1
1.7
2.6
3.2
1.4
1.9
2.2
2.3
1.6
2.3
2.7
2.8
3
1.0
0.5
0.2
0.1
1.2
1.7
2.4
2.9
1.5
1.8
1.9
2.0
1.9
2.4
2.7
2.8
4
1.0
0.5
0.2
0.1
1.3
1.7
2.3
2.7
1.5
1.7
1.8
1.8
2.1
2.5
2.7
2.8
r
p/d
1
Z
(5.3)
-27-
••
It is noteworthy that the estimate of X based on yield data which
o
reach a plateau for X S 2 may be even smaller than the estimate of X
o
based on the exponential model.
As indicated in Section 3(b), the use of
a quadratic model for data exhibiting this yield plateau usually will
cause the maximum point to be moved to the right.
Hence, we are
suspicious that the exponential function will produce a too large X '
O
especially for small price ratios.
The Iowa data, much of the ilorth
Carolina data, and a recent set of data by Engelstad and 2arks (1971)
display the yield plateau problem.
It is clear from the results of our study that the exponential
model, which was considered the best choice from the point of view of
unbiased estimates of soil nutrient, may not be the best choice when
o.
considering the estimate of X
However, based upon our limited research
on this problem, we cannot recommend an alternative model which can be
o•
expected to produce better estimates of X
6.
Summary.
Various models have been considered for the estimation of 0, the
existing level of a single soil nutrient, based upon the response to
several added levels of the nutrient.
Comparisons of these models have
involved both theoretical considerations and results from statistical
analyses of a number of soil fertilizer response experiments.
The exponential model, n
= a[l - e -y (X+o) ] has been used as a
standard for comparison, because estimates of 0, based on this model,
seemed to be both realistic and theoretically justifiable.
•
Comparisons
of estimates using this exponential model and several polynomial models
are made for a number of fertilizer experiments.
Two polynomial models
-28-
investigated in detail are:
quadratic:
square. root:
For a variety of experimental situations, 0* is usually less .thEm half of
o and
0 is considerably larger than O.
If there is a yield plateau at
small levels of applied nutrient, the estimate of 0 often is as much as
twice
o.
Results of fitting a number of polynomial models of the form
in which p ranged from 0.25 to 2.00, demonstrated that the bias in the
estimation of 0 could not be removed by choice of the polynomial model.
Hence it
wo~ld
appear that, if possible, one should use the exponential
model itself when estimating
o.
An ad hoc procedure was introduced for removing some of the bias
in estimating 0 when the quadratic model is fitted for situations
involving a yield plateau for small X.
Some preliminary research was conducted to ascertain the effect of
choice of model.upon X ' the estimate of the optimum amount of nutrient
O
to be applied.
Comparisons of estimates based on the polynomial and
exponential models are highly dependent upon the value of the price ratio,
p.
For small p, X based on the exponential model is usually larger than
o
X based on the quadratic ,model; these results are usually ,reversed. for
o
large p.
If there isa yield plateau for small X, there is some evidence
that the use of either the exponential or the quadratic model, based on
all.the data, will produce an overestimate of X '
O
-29The exponential
~ode1
which was considered the best choice from the
point of view of unbiased estimates of soil nutrient may not be the best
when estimating X •
O
However, based upon our limited research on this
phase, we cannot recommend an alternative model which can be expected to
produce better estimates of X •
O
It might be argued that if we are
interested in obtaining the best value of X ' we should develop an
O
estimating procedure
bas~d
residual sum of squares •
•
on obtaining a good X rather than a minimal
o
-30References Cited
u_
_
Anderson, R. L. (1956). A Comparison of Discrete and Continuous Models
in Agricultural Production Analysis. Methodological Procedures in
the Economic Analysis of Fertilizer Use Data. Edited by E. L. Baum,
et a1. Iowa State College Press, Chap. 3:39-61.
Anderson, R. L. (1957). Some Statistical Problems in the Analysis of
Fertilizer Response Data. Econ. and Tech. Analysis of Fertilizer
Innovation and Resource Use. Edited by E. L. Baum, et ale Iowa
State College Press, Chap. 17:187-206.
Baird, B. L. and Fitts, J. W. (1957). An Agronomic Procedure Involving
the Use of a Central Compositei:esign for Determining Fertilizer
Response Surfaces. Econ. and Tech. Analysis of Fertilizer Innovation and Resource Use. Edited by E. L. Baum,et ale Iowa State
College Press, Chap. 13:135-143.
Dean, L. A.
(1970).
Personal Communication.
Engelstad, O. P. and Parks, W. L. (1971). Variability in Optimum N
Rates for Corn. Agronomy Journal 63:21-23.
Gregory,W. C. and Anderson, R. L. (1970). Design Procedures and Use
of Prior Information in the Estimation of a Non-Linear 11odel.
Tech. Report No. 14, Department of Statistics, University of
Kentucky.
Heady, E. 0., Pesek, J. T., Brown, W. G. (1955). Crop Response
Surfaces and Economic Optima in Fertilizer Use. Res. Bull. 424,
Agricultural Exp't. Sta., Iowa State College, Ames.
Hurst, D. C. (1962). Modifications of Response Surface Techniques
for Biological Use. Unpublished Ph.D. thesis, North Carolina State
University, Raleigh.
Hurst,D. C. and Mason, D. D. (1957). Some Statistical Aspects of the
TVA-North Carolina Cooperative Project on the Determination of
Yield Response Surfaces for Corn. Econ. and Tech. Analysis of
Fertilizer Innovation and Resource Use. Edited by E. L. Baum,
et ale Iowa State College Press, Chap. 18:207-216.
Paez, Gilberto A. (1971). Some Contributions to Soil-Fertilizer
Response Modeling. Unpublished Ph.D. thesis, North Carolina State
University, Raleigh •
•
·'
Table 1.
LOCREP
B
O
B
1
B
2
551-1
2
3
4
68.9
88.5
65.7
80.8
76.0
8.2
14.3
5.6
12.7
10.2
4.8
9.6
552-1
2
3
4
91.0
80.3
84.9
Avg.
553-1
2
3
4
Avg.
Avg.
554-1
2
3
4
Avg.
555-1
2
3
4
Avg.
556-1
2
3
4
Avg.
557-1
2
3
4
Avg.
•
Estimated Regression Coefficients and Usable Soil Hutrients for 25
Fertilizer Experiments on Corn in Eas tern North Carolina t1955-5 7. 1
559-1
2
3
4
Avg.
D
s
2
LOCREP
B
O
B
1
B
2
D
s
2
6.4
6.3
65.2
23.5
77.3
43.3
52.3
45.8
186.7
78.3
102.0
103.2
650-1 112.1
2 105.3
3 106.5
4 111.9
Avg.
109.0
13.7
11.3
15.6
17.0
14.4
8.6
6.1
8.1
7.3
7.5
56.4
82.8
49.0
57.7
61.5
69.9
169.6
53.0
74.8
91.8
10.6
5.7
4.7
9.9
3.3
3.9
33.6
135.7
132.9
142.7
73.4
130.0
651-1
2
3
4
85.4
7.0
5.7
100.8
115.4
Avg.
86.4
81.5
83.3
83.7
83.7
8.7
13.4
9.1
7.6
9.7
7.0
5.3
4.9
3.2
5.1
59.4
53.4
Su.1
128.3
80.3
102.0
51.4
123.8
94.9
93.0
97.8
94.0
94.9
83.0
92.4
16.9
15.2
12.4
7.6
13.0
9.0
6.5
6.0
5.7
6.8
30.3
50.4
67.3
74.8
55.7
105.0
81.6
144.6
46.1
94.3
652-1
2
3
4
103.7
107.4
100.5
94.2
101.5
13.3
18.0
15.7
15.6
15.6
4.4
9.9
7.7
6.1
7.0
98.7
32.1
45.8
83.2
65.0
82.7
97.7
50.0
87.2
79.4
32.8
35.5
40.7
53.3
40.6
2.4
5.0
2.0
7.9
4.3
1.4
3.4
3.6
5.3
3.4
126.7
37.2
69.4
32.4
66.4
94.1
125.7
134.9
57.6
103.1
80.1
82.8
78.5
69.6
77.7
10.1
7.2
8.2
10.5
9.0
8.4
5.4
6.6
4.1
6.1
33.6
82.0
55.0
65.1
59.0
66.8
65.7
86.0
76.6
73.8
78.2
83.0
91. 7
102.9
88.9
10.9
11.0
20.0
18.7
15.1
6.8
8.6
8.9
8.6
8.2
42.9
33.2
17.6
34.0
79.9
73.5
78.6
83.7
78.9
14.4
15.7
13.4
11.1
13.6
6.3
6.3
6.9
6.7
6.6
37.3
24.3
33.6
49.9
36.3
101.1
25.3
26.7
40.3
48.4
93.5
96.1
83.2
73.1
86.5
10.4
10.7
7.5
4.9
8:4
5.1
6.1
4.2
4.2
4.9
86.8
74.1
102.8
100.8
91.1
118.8
42.7
65.8
36.7
66.0
110.0
102.8
105.7
102.6
105.3
14.9
16.1
11.6
10.1
13.2
7.9
3.9
3.8
5.0
5.2
57.0
92 .0
122.5
101.1
93.1
73.3
39.2
73.0
71. 7
64.3
53.7
65.2
65.4
62.6
61.7
1.5
7.0
6.6
5.0
5.0
4.8
6.2
3.7
8.2
5.7
75.7
45.6
87.0
29.3
59.4
64.4
84.9
118.0
212.7
120.0
115.8
118.2
116.4
115.8
116.5
10.6
13.7
14.4
11.2
12.5
6.7
7.3
8.1
7.1
7.3
89.6
74.8
62.3
82.4
77 .4
53.2
93.0
40.3
46.0
58.1
76.8
80.5
80.0
65.8
75.8
8.3
10.1
11.9
6.9
9.3
4.6
6.6
7.4
8.1
6.7
79.9
50.8
36.2
28.6
48.9
53.7
86.2
76.8
206.0
105.7
96 .0
93.2
100.5
91.5
95.3
15.3
17.8
18.8
16.5
17.1
7.4
7.5
8.4
6.8
7.5
44.8
33.6
32.5
41.0
37.9
31.8
48.7
34.1
90.0
51.2
4.~
39.2
100.1
86.0
90.0
31.9 78.8
Avg.
653-1
2
3
4
Avg.
654-1
2
3
4
Avg.
655-1
2
3
4
Avg.
656-1
2
3
4
Avg.
657-1
2
3
4
Avg.
•
Ta~'le
1 (concluded)
LOC-
B
O
B
1
B
2
89.7
95.1
96.4
93.8
93.8
3.3
8.2
7.0
7.2
2
D
s
6.4
1.5
3.3
6.7
3.1
3.6
298.9
140.6
82.0
152.8
168.6
122.7
62.9
105.1
67.4
89.5
112.1
108.1
106.0
103.6
107.5
17.7
1.8.5
20.4
21.2
19.4
7.9
8.4
8.9
7.5
8.2
50.1
40.5
30.8
35.2
39.4
109.7
104.0
52.1
132.6
99.6
660-1
99.7
2 103.0
93.1
3
4
99.0
Avg.
100.0
16.4
17.7
14.8
10.5
14.9
8.0
9.6
6.6
3.6
7.0
40.4
34.6
49.6
124.5
62.2
37.8
42.3
52.8
661-1
2
3
4
103.0
100.2
97.7
89.2
97.5
16.0
17.9
17.7
17.7
17.3
7.6
8.8
8.1
6.3
7.7
48.7
31.5
3?f.2
3;3.3
J:J • .L
62.9
77 .9
78.9
81.8
75.4
662-1 103.0
2 101.4
3
96.3
4
87.8
Avg.
97.1
11.3
14.9
17.1
14.4
7.7
5.6
6.8
5.6
6.4
62.5
69.7
44.5
54.8
57.9
67.2
95.9
75.6
112.1
87.7
REP
658-1
2
3
4
Avg.
659-1
2
3
4
Avg.
Avg.
1
Model:
1'4:4
')
-
B + B x - B x2 ; x
1
2
O
and Y in bushels per acre.
y
~
D
~
"
~
62.5
LOC-
4l f.6
e·
751-1 126.2
2 126.4
3 131.4
4 123.8
Avg.
126.9
19.0
19.1
24.0
22.3
21.1
8.8
7.5
8.7
7.9
8.2
53.5
63.8
46.3
49.5
53.3
88.8
90.7
76.1
59.0
78.7
752-1
2
3
4
105.5
84.8
70.0
75.0
83.8
16.4
15.3
13.2
3.0
12.0
7.5
5.9
9.0
5.1
6.9
50.6
44.6
9.1
97.2
50.4
103.4
269.6
198.8
348.0
230.0
753-1 104.7
2 102.1
98.0
3
4
75.4
Avg.
95.0
19.7
23.7
16.4
15.1
18.7
8.0
6.7
13.9
8.9
9.4
37.0
32.3
8.2
11.5
22.2
156.2
77.0
340.3
84.9
164.6
754-1
2
3
4
7.0
13.0
5.1
4.7
7.4
6.7
5.5
3.6
5.5
5.3
80.7
87.8
138.3
98.8
101.3
70.2
216.1
54.0
78.3
104.7
Avg.
96.1
107.8
84.9
87.8
94.1
s
D
X - 2, each X represents 62.5 1bs. of N per acre
j
~01 +M
+40 0;_ 2
2B
0
2
2
B2
Avg.
45.l}
B
O
B1
REP
,
•
Table 2.
d
1
d
0
0
0.58
0.85
0.26
0.32
0.40
0.16
0.22
0.25
0.11
0.17
0.09
0.12
0.14
n =1
0
0.77
0.92
0.37
0.40
0.45
0.24
0.27
0.29
0.17
0.21
0.14
0.16
0.17
2
1
5
1
2
5
1
3
5
1
4
1
3
5
1
1
2
2
2
3
3
3
4
4
5
5
5
Comparisons of 0, 0 and 0* for Selected Experiments.
0*
0.23
0.30
0.08
0.09
0.11
0.04
0.05
0.05
0.02
0.03
0.01
0.02
0.02
-
I
31
9
40
25
12
47
23
16
52
22
57
30
21
1*
60
65
70
72
74
76
78
79
80
82
83
84
85
d
1
1
1
2
2
3
3
3
4
4
4
5
5
5
d
2
1
5
1
5
1
3
5
1
3
5
1
3
5
1
0
1.58
2.80
0.77
1.34
0.50
0.74
0.86
0.37
0.53
0.63
0.29
0.41
0.49
n =4
0
2.50
3.40
1.25
1.64
0.83
0.99
1.07
0.62
0.73
0.79
0.49
0.58
0.63
1
1
3
1
3
1
3
5
1
5
1
2
5
1
5
1
3
1
3
1
1
2
2
2
3
3
4
4
5
5
•
-
0.69
0.45
0.53
0.33
0.39
0.26
0.30
0.33
1.32
2.22
0.63
0.81
1.06
0.40
0.68
0.29
0.42
0.23
0.32
n =3
0
1.98
2.58
0.98
1.10
1.26
0.65
0.82
0.48
0.57
0.38
0.45
0.20
O.ll
0.14
0.07
0.08
0.04
0.06
0.06
0.79
1.08
0.32
0.38
0.45
0.18
0.26
0.02
0.15
0.08
0.10
41
48
54
27
59
31
63
34
24
50
16
55
37
19
60
22
64
34
68
37
48
57
64
67
68
71
72
75
76
40
51
49
53
57
55
62
60
64
64
68
1
1
2
2
2
3
3
3
4
4
5
5
1 0 given by (3.9), 0 by (3.10) and 0* by (3.18);
-
-
I = 100(0-0)/0;
-
I
1*
1.04
1.45
0.45
0.63
0.26
0.33
0.37
0.17
0.22
0.24
0.12
0.16
0.17
58
19
62
22
66
35
24
69
38
27
34
48
42
53
48
55
58
53
59
61
57
62
64
1.27
1.81
0.56
0.66
0.80
0.33
0.43
0.47
0.22
0.31
0.16
0.22
65
22
68
46
25
o)~
72
40
29
n0=5
1.~ 0.52
1.00
0.46
0.29
0.42
0.21
0.30
0.16
0.23
0.26
-
0
n0 =2
1
2
3
3
4
4
5
5
5
1
1* = 100(0-0*)/0 .
1
5
1
2
5
1
3
5
1
4
1
4
1.81
3.32
0.89
1.17
1.60
0.58
0.87
1.03
0.43
0.70
0.34
0.54
2.g-r4.06
1.50
1.71
2.00
1.00
1.21
1.31
0.75
0.94
0.60
0.74
71
38
27
74
34
77
36
30
46
37
44
50
43
51
54
48
56
52
59
•
Table 3.
LOCREP
551-1
4
552-3
553-1
2
3
4
554-2
4
555-1
2
3
4
556-1
2
3
557-2
3
559-1
3
650-1
2
3
4
651-2
3
4
652-1
2
3
4
653-1
2
3
4
654-1
2
3
4
•
D(p) and
8 for
Selected Sets of the North Carolina Data.
o
A
.50
.75
1.25
1.50
2.00
0
13
23
10
57
7
15
21
27
7
6
41
23
91
16
30
41
48
18
16
25
21
10
20
56
47
50
30
106
21
37
50
57
25
21
32
25
12
25
67
57
82
31
66
62
25
42
63
36
45
43
61
106
79
25
34
LI2
23
64
41
48
27
17
25
37
65
43
133
30
50
67
75
37
32
43
33
18
34
87
74
103
46
87
80
36
56
83
49
58
53
80
128
99
32
46
83
34
82
55
65
37
24
34
50
28
15
41
12
22
22
29
10
9
18
20
10
5
36
3
8
11
16
3
2
6
6
2
5
17
15
28
4
16
17
3
10
17
8
12
15
16
L;3
26
6
8
13
3
19
9
9
6
2
5
8
13
11
4
10
31
26
44
10
30
29
8
19
30
16
21
23
28
63
41
12
15
22
7
31
18
19
11
6
10
16
71
23
55
52
19
35
53
30
38
38
51
94
68
21
28
36
17
53
33
39
22
13
20
30
1('l
.L ...
28
28
48
9
24
30
11
26
32
23
32
39
31
78
52
22
23
35
10
33
21
21
20
12
17
21
LOCREP
o
655-1
2
3
4
656-1
2
3
4
657-1
2
3
4
658-2
3
4
659-1
2
3
4
660-1
2
3
4
661-1
2
3
4
662-1
2
3
4
751-1
2
3
4
752-1
2
3
753-1
2
4
754-1
2
4
.75
1.25
1.50
2.00
0
11
26
35
27
20
14
11
17
8
5
5
8
50
19
43
8
20
41
54
43
34
26
21
29
15
10
10
14
36
65
86
70
59
47
39
52
28
20
19
26
104
54
107
31
22
18
25
24
21
31
94
30
18
20
21
41
48
29
37
36
43
30
33
32
27
5
22
21
4
51
56
63
38
43
75
99
81
70
57
47
63
34
25
24
31
117
64
123
38
23
22
29
30
26
37
106
37
23
25
27
49
56
35
44
43
51
36
39
39
33
6
27
25
6
62
57
92
123
101
90
75
62
82
45
34
32
41
141
82
153
51
40
31
35
40
35
50
124
49
31
34
38
63
70
45
55
54
64
46
49
51
45
9
37
32
11
81
88
99
27
60
62
47
33
26
26
28
25
18
18
24
87
33
53
23
4
10
6
6
9
49
9
4
5
5
13
19
10
14
13
16
10
12
9
6
1
5
6
1
16
17
21
13
A
1~(p) = B [X+D(p)] + S2[X+D(P)]P; ~ = ~[1 _ e-Y(X + 0)].
1
A
.50
4
Ave.
A
1
71
32
67
16
9
3
16
12
12
17
67
17
9
10
10
23
30
17
22
22
26
17
20
17
13
2
11
12
2
29
31
36
22
c:VI'7
75
1+6
60
13
16
29
19
21
25
92
25
16
19
18
30
45
28
36
34
38.
29
33
26'
22
6
19
24
4
21
31
27
28