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I AGRICULTURE HANDBOOK NO. 25 6

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CERTAIN NONLINEAR FUNCTIONS

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with emphasis on agricultural

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Washington, D. C.
CONTENTS

Pagei

I. INTRODUCTION 1

II. THE PEARL-REED (LOGISTIC) FUNCTION 1

Method 1: A Transformation of the Dependent Variable 2

Method 2: A Difference Equation for the Dependent Variable 2

III. THE SPILLMAN FUNCTION 3

Method 1: A Transformation of the Dependent Variable 3

Method 2: A Difference Equation for the Dependent Variable 3

IV. THE GOMPERTZ FUNCTION 4

Method 1: A Transformation of the Dependent Variable 4

Method 2: A Difference Equation for the Dependent Variable 4

V. THE FUNCTIONS AS EXPLICIT SOLUTIONS OF DIFFERENCE

EQUATIONS 5

The Spillman 5

The Gompertz 5

The Logistic 5

VI. EXAMPLES FOR THE SPILLMAN AND GOMPERTZ FUNCTIONS 5

VII. THE RELATIVE MERITS OF LINEAR TRANSFORMATION VS.

ITERATIVE METHODS 12

LITERATURE CITED 14

APPENDIX 15

August 1963

For sale by the Superintendent of Documents, U. S. Government Printing Office, Washington, D.C. 20402

ii
Errata Sheet

Agriculture Handbook No. 256: Simple Methods of Estimating Certaf^^,

Nonlinear Functions with Emphasis on Agricultural Data

Page 2, Equation (2)

Page 2, Equation (4)

Substitute a for a

Substitute y... for y.+l •^ 1+1 ^ 1

Page 4, first paragraph of Part IV, The Gompertz Function; third

sentence should read:

"As yield response to nitrogen at low levels of nutrient

application frequently manifest this phenomenon, the

Gompertz curve is particularly useful. It also may be

applicable for estimating trends in technological diffusion,

a process that typically exhibits an increasing rate in its

early stages and a declining rate in its latter stages (4)."
V

SIMPLE METHODS OF ESTIMATING CERTAIN NONLINEAR

FUNCTIONS WITH EMPHASIS ON AGRICULTURAL DATA

By

Richard H. Day, Agricultural Economist

Farm Production Economics Division,

Economic Research Service^'

I. INTRODUCTION

This paper presents two elementary methods for fitting three different nonlinear

functions to empirical data by means of simple linear regression* Iterative least

squares methods which have been developed for estimating parameters of nonlinear

functions (-^y^^i^ sonaetimes lead to certain difficulties in application. Because this

is the case the much simpler methods developed in this handbook are useful tools for

application. ^The relative merits of this approach versus the nonlinear iterative

approach ar« briefly described^in-the concluding-section.

The Spillman, Gompertz, and Pearl-Reed (logistic) functions a^^ considered.

The two methods presented for the Pearl-Reed function are already well known and are

given first. Tttèn analogous methods -a^f^ derived for the Spillman and the closely

related Gompertz curves; these apparently have not been presented in the literature.] ^.^. /

U. THE PEARL-REED (LOGISTIC) FUNCTION

The logistic curve has been applied by Griliches (4) to the estimation of the

rate of adoption of hybrid seed corn, and the author of this handbook has found suitable

use for it as a production function.1^ The logistic, like the Gompertz, is an

S-shaped or sigmoid curve, and is therefore particularly applicable to the estimation

of single input production functions manifesting increasing returns to scale, and to

trends in technological diffusion (4, 2^). The equation for this function is

(1) y= ^ ,.. ß, y>o, o<a<l

1 + ße ax

in which y is the "dependent" and x the "independent" variable. The parameter 7

is the maximum value for y, and it is approached assymptotically as x increases;

a determines the rate at which y increases with x, and ß determines the value of y

when X is zero.

1/ A part of the research on which this report is based was financed by the National

Cotton Council. ^.^ -, .^

2/ Underscored figures in parentheses refer to Literature Cited, p 14

3/ Day R. H. Recursive Programing and Production Response. North-Holland

Pub. Co ' Amsterdam, Contributions to Economic Analysis, 1963.
Method 1: A Transformation of the Dependent Variable

Equation (1) may be placed in the form

(2) Z - looO'/y - 1) = logß- ix

If the data are suitable a graphical estimate of T may be obtained, 7. Letting

Z^ = log {y I y^-l), i = l,...,h, where h is the number of observations, the

regression equation (3) may be fitted by classical least squares.

(3) Z = A + Bx

Estimates of oe and ß can now be obtained from the expressions, a = -B, p = e^ =

antilog A. A similar transformation due to Berkson il) was used by Griliches.

Note that at and ß are not least squares estimates of a and /?, Instead, they

are derived from the least squares parameters A and B.

Method 2: A Difference Equation for the Dependent Variable

Suppose data are selected in such a way that x^ - mi, i = j, . . ., j+h-1. That is,

there are h observations available at even intervals of width m for the independent

variable. Then equation (1) may be written as

y

(4) y.-

and y

^; . ' '' ^^ " ^ , n -oem(i+l) i' r I 1 + pe

By combining these two expressions the following is obtained,

(5) 1 _± d-e-""') + e-«"' _!_

Let Z- = l/y^ i=j, . . ., j+h-1. Then the difference equation

(6) Z^^^ = A + BZmaybe fitted by least squares. From equation (5) it is seen that A = (l-e"""^) /?' and

B = e~^^. Therefore the estimates of a and y are Û = -(log B) /m and 9= (l-B) /A

This procedure is due to Rhodes (_1_3) who also suggests a method for estimatinj

ß. He utilizes the transformation (2) as follows:

Í7) log ß. = (^ /y.-l) +âmi, i=j, ...,j+h-l.

He then takes the mean of the right hand side of this expression as the estimate of

log ß. Thus,

(8) ß = antilog [I(^/y.-l) /h + am(h-l)/2]..
Freauenttv T It .^ ^''^"^^'^^ ^^^^^ '^ extremely poor fits for some data.

Frequently, a much bet er estimator of ß is simply the value which will force the

curve through the first (or last or any other) data point (y., x-). Thus

A 3 3

A ^-

<8> è=e«^n^/y,-l]

A ,A

1

IfXj=mj=o, then p={9/y-l).

A discussion of various methods for estimating tne parameters of the logistic

may be found m Nair (11), s ^

III. THE SPILLMAN FUNCTION

The SpiUman curve has been used by Spillman (M) and Ibach (7) for estimating

the yield response of field crops to nitrogen. The latter has developed an ingenious

method for graphical estimation of its parameters (6) . Paschal and French (12) have

presented an iterative least squares method due to Stevens (^5) for fitting the ftTnction.

The Spillman curve is expressed algebraically as

^9) y = y -ßa^ 0< a < í; y, ß>o.

The parameters a, ß, and >'have the same general interpretation as the correspondmg parameters for the Pearl-Reed curve. Two methods, analogous to those presented for the logistic, are now derived for estimating these parameters.

Method 1: A Transformation of tne Dependent Variable

From equation (9) we obtain

(10) Z = log ( 7-y) = log/8+(loga) x.

If data are distributed fortunately a good graphical estimate of y may be obtained,

say y . Now let (y-, x-), i = 1, .... h be a set of h observations of the variables y

and x. Let Z¿ = log (^-y-), i = 1, . . ., h. The linear equation

(11) Z = A+ Bx

may now be fitted to the points (Z¿, X^) i = 1, , . ., h, giving least squares estimates

of A and B. From equation (10) it is seen that Ù = e^ = antilog B and ^ e^ =

antilog A,

Method 2: A Difference Equation for the Dependent Variable

Let us suppose that tne observations (y^, x^) are available in such a way that

Xi = mi, i = j, . , ., j + h-1. That is, the variable x is spaced at even intervals of

width m. Then equation (9) may be written

(12) y. = y-ßa^\ i =j, ...,j+h-l,

and also y.^^ = y -/JcT^^^^^' i - j, . . ., j+h-1.

By a suitable combination of these two equations the following expression is obtained.

(13) y.^. = (1 - a"") r+ a"' V.,. i ^ .1 Hh,

3
The linear equation

(14) Yi+l = A + By.

may now be derived from tne resulting least squares estimates A and B. Thus

0= B-^and^ = A/(l-B).

In a manner analogous to Rhodes' method for estimating ß for the logistic the

following estimator of ß for the Spillman can be obtained.

(15) ^= antilog Cr(^- yj) /h - m log ¿^(h-l)/2].

Or by forcing the curve through (y., Xj) the first data point, one obtains

(15») ^ = (^-y.) ce-^j

or ^ = ^-y^if x^ =o.

IV. THE GOMPERTZ FUNCTION

The Gompertz function was found by French (_3) to be superior to a great variety

of alternative production function forms for the analysis of yield response of corn to

nitrogen for a series of Nebraska experiments. The Gompertz is an S-shaped or

sigmoid curve, that is, when used as a production function it may show increasing

returns to scale. As yield response to nitrogen at low levels of^nutrient^ap^

frequently manifest this phenomenon, the Gompertz curve also ouggcatc thatlt^may be

applicable fe^estimating trends in technological diffusion, a process that typically exhibits a^ increasing rate in its early stages anda declining rate in its latter stages (4).

The Gompertz function is closely related to the Spillman function as can be

observed in its algebraic expression.

Thus y' = log y will have the Spillman form if y has the Gompertz form. For this

reason the results derived for the forrtier function are directly applicable here.

Method 1: A Transformation of the Dependent Variable

By letting y^ = log y. equation (10) can be obtained from (9). Then Z^ = log

{^ -yt.) = log (^ - log y.) = log B + (logoe)x. The least squares estimates of A and B

are obtained from (11). Then the estimates a = e^ and^ = e"^ are derived just as

before. As in the Spillman case an exogenous estimate of Y must be made, probably

by a graphical analysis. The only difference is therefore the conversion of the

observations of the dependent variable y to logarithnas in the case of the Gompertz

curve.

Method 2: A Difference Equation for the Dependent Variable

Converting y to logarithms also enables the direct application of method 2 for

the Spillman to be used for estimating the parameters of the Gompertz. Thus, if

y*- = log y., i = j, .. ., j+h-1 equation (13) may be derived from (16). By fitting (14)

the estimates of a and y are obtained from the least squares parameters A and B.

Thus¿^= B"^ and ^ = A/(l-B). To estimate ß , (15) or (150 maybe used.
V. THE FUNCTIONS AS EXPLICIT SOLUTIONS OF DIFFERENCE EQUATIONS

We may note that in the search for a linearizing transformation of the nonlinear

functions explicit solutions to several difference equations have been found. These

follow from the "even interval data" methods.

The Spillman

The linear, nonhomogeneous difference equation y._^. = a + b y, has as its

explicit solution the Spillman function in the form

(5.1) y =^^ß(bt)

in which ß'is a constant determined by initial conditions on y.

The Gompertz

The nonlinear difference equation exp(y^^^) = exp(a+by.) has as its explicit

solution the Gompertz function in the form

(5.2) y = exp[y§^ ^ß b^]

in which ß is determined by initial conditions on y, and in which exp(y) = e^.

The Logistic

The nonlinear difference equation y^_^^^ = a + by^^ has as its explicit solution the

logistic function in the form

(5.3) y "-"'

in which^ is determined by initial conditions.

VL EXAMPLES FOR THE SPILLMAN AND GOMPERTZ FUNCTIONS

The data presented in tables 1 and 2 are used here to illustrate the methods

derived above for fitting the Spillman and Gompertz functions. The literature already cited contains examples of the methods described in section II for the PearlReed function. First, three examples are given using the simple production

function or relation of one output (oat yield) to one input (nitrogen). Second, two

examples are given in which the trend of an aggregate index of mechanical power and

machinery is estimated.

The examples demonstrate the utility of these simple methods, but suggest

caution in their application. For some types of data, extremely close fits will be

obtained; for others, poor results will follow. A graphical presentation of the results

will indicate the success or failure of the method. For this reason graphs of both

the derived nonlinear curve and the regression on the transformed variables are

presented for each example. (Fig. 1 to fig. 10, inclusive. )
Table 1.-Average yields for oats-nitrogen response experiments, Mississippi

Delta, 1951-1957

Low rate experiments

Nitrogen Average yield

Pounds

0. 0

7. 5

15. 0

22. 5

30. 0

37. 5

45. 0

Bushels

13.6

23.4

34.7

36.8

43.5

49.5

54.8

: High rate exp eriments

: Nitrogen : Average yield

: Pounds Bushels

: 0 15.4

: 45 54.4

: 60 60.3

: 75 66.5

: 90 68.0

: 105 66.6

: 120 67.5

Source: Perrin H. Grissom (_5).

Table 2. -Index of mechanical power and machinery inputs, 1933-1958

(1947-49 = 100)

Year Index : : Year Index : : Year Index

.. — — • : 1941 61 : : 1951 127

-- : : 1942 66 : : 1952 133

1933 44 : : 1943 69 : : 1953 134

1934 44 : : 1944 70 : : 1954 135

1935 45 : : 1945 74 : : 1955 136

1936 48 : : 1946 80 : : 1956 137

1937 52 : : 1947 89 : 1957 138

1938 55 : : 1948 100 : 1958 137

1939 55 : : 1949 111 : : --

1940 58 : : 1950 118 : : --

Source: Ralph A. Loomis and Glen T. Barton (9, table 13).

In the discussion of sections II, III, and IV, natural logarithms or logs to the

base "e" were used. However, Naperian logarithms, logs to the base 10, will serve

just as well. The latter were in fact used for all computations contained in this

section.

Example 1: The high rate data from table 1 are used to illustrate method 1 for

the Spillman function. The estimated curve and the original data are presented in

figure 1. These data indicate a strong nonlinear relationship, and a good " eyeball "

estimate of the upper asymptote, y , apparently can be made, y was estimated by

eye to be 68. 1. Figure 2 shows the simple regression of the transformed yield

variable Zj = log (68. 1 - y.) on nitrogen level. The transformation is strongly linear

except for the z value at 90 pounds of nitrogen. This "flyer" is due to the fact that

the logs of values between zero and 1 are negative, and decrease very rapidly to

-coas z gets close to zero. As, z^ = 68.1 - 68 = 0.1, this strong negativity results.
Spillman Funcfîon, Estimation Method 1

YIELD RESPONSE OF OATS TO NITROGEN

High Rate Experiments, See Appendix Sheet 4

BU. OATS

^y^^^<^

• 1

An

•

1

50

40

30

20

in

X

•

/

/

/

\Ê

0 20 40 60 80 100 120 140 160

LB. NITROGEN

U. S. DEPARTMENT OF AGRICULTURE NEC. ERS 1687-63(1) ECONOMIC RESEARCH SERVICE

Figure 1

SpiWrnon Function, Estimation Method 1

SIMPLE REGRESSION

High Rate Experiments, See Appendix Sheet 3

Zj = lc>g(y-Yj)

1 _

^ _ I J

41 —- ií

0 -

-2

A

i»

o 15 30 45 60 75 90 105 120

Xi=LB. NITROGEN

U. S. DEPARTMENT OF AGRICULTURE NEG. ERS 1688-63(1) ECONOMIC RESEARCH SERVICE

Figure 2
Such a flyer demonstrates the extreme care one should take in applying this particular

transformation. In this example and those to follow all such flyers were omitted in

the computation of the regression coefficients A and B.

Example 2: Figure 3 shows the result of applying method 2 for the Spillman

function for the low rate data of table 1. Unlike the high rate data, these 7 observations are distributed at even nitrogen intervals of 7. 5 pounds and method 2 is

applicable. Figure 4 shows the simple regression of yield on "lagged" yield

Yi+l ^ ^^^ y*- ^^^^ relation is also strongly linear. The flyer problem is not

present for these data and all seven observations were used. The strong linear relationship shown in figure 4 is equivalent to serial correlation in a time series

variable. The serial correlation of yields spaced at equal nitrogen levels is due to a

causal relation between yield levels and nitrogen applications. It is exploited here to

estimate the causal or structural relation represented by the nonlinear Spillman

function.

Method 2 yields estimates of y and a , The ß coefficient was estimated by using

equation 15' with j = 0, that is, ^ = ^ - y^

Example 3: To illustrate the use of the Gompertz function using method 1, the

low and high rate experiments of table 1 were combined. The data and the estimated Gompertz production function are shown as figure 5. Substituting 10 for e in the

various Gompertz equation logarithms to the base 10 were used. Thus the upper

asymptote was estimated as 68 = 10^ so that Twas chosen as 1.83251. Figure 6

illustrates the simple regression of the transformed variables,

z^ = log (y - log y^) = A + Bxn where x^ was the nitrogen level, A close linear relation is evident in this transformation but the use of logs, as in example 1, introduces

a flyer in the transformed data. At 90 pounds of nitrogen the experimental yield is

68.0, the same as the eyeball estimate of lOT. Consequently, the antilogarithm

is -CO. This data point was omitted. The occurrence of this flyer suggests, as in

example 1, that the transformation for method 1 is linearizing for a considerable

range of the data, but not for points close to the maximum attainable value. A

superior estimate might be obtained by choosing only data points in the 0-60 pound

nitrogen level range. Examination of the graph of the transformed variable is again

seen to be an important precautionary supplement to a straightforward application of

method 1.

Example 4: The data from table 2 were used to illustrate method 2 for the

Gompertz function. Figure 8 indicates a strong linear serial correlation of the

Zj = log y^ variables, that is, the logs of the aggregate power and machinery index.

However, the derived nonlinear relationship shown in figure 7 is extremely poor.

While this might not have been guessed fronci a graphical analysis of the transformed

data (fig. 8), the reason manifests itself in the original data. The latter do form a

sigmoid or S-shaped curve, and the Gonnpertz is itself a function of this kind.

However, the Gompertz best represents data which increase rapidly at first and then

taper off into a long and more gradual approach to the upper asymptote. Yield response data typically follow such a pattern. The index considered, however, has an

opposite pattern, that is, a gradual increase climaxed by an abrupt approach to an

asymptote.
Spillman Function, Esfîmafîon Method 7

D 1 1

YIELD RESPONSE OF OATS TO NITROGEN

tow Rafe Exper'imenH, See Appendix Sheet 6

BU.

70

60

50

40

30

20

10

(

U. S. DE

WIMID

^^^^

/•'^

X

/

4

/

•

/

1

) 20 40 60 80 100 120 140 160

LB. NITROGEN

PARTMENT OF AGRICULTURE NEC. ERS 1689-63(1) ECONOMIC RESEARCH SERVICE

Figure 3

Spi//mon funtlion, Estimation Mefhod 2

SIMPLE REGRESSION

tow Rate Experiments, See Appendix Sheet 5

50

40

30

20

10

10 20 30 40 50 60

. S. DEPARTMENT OF AGRICULTURE

NEC. ERS 1690-03 (I) EC'ONOMIC RESEARCH SERVICE

Figure 4

9
Gompertz Function, Estimation Method 1

YIELD RESPONSE OF OATS TO NITROGEN

High and Low Rate Experiments Combined, See Appendix Sheet 8

6U. OATS 1 . 1 I

60

50

40

30

20

lO'

(

U. S, DE

-—'•

/•^

/

/

/.

/

/

/

D 20 40 60 80 100 120 140 160

LB. NITROGEN

PARTMENT OF AGRICULTURE NEC. ERS 1691-63(1) ECONOMIC RESEARCH SERVICE

Figure 5

Gompertz Function, Esfîmatîon Method I

SIMPLE REGRESSION

High and Low Rate Experiments Combined, See Appendix Sheet 7

0

-2

LOG rr-log

1

^j

"».«

1

»

« ^^•^*x^„^^ ift

0 15 30 45 60 75 90 105 120

LB. NITROGEN

U. S. DEPARTMENT OF AGRICULTURE NEG. ERS 1692-63(1) ECONOMIC RESEARCH SERVICE

Figure 6

10
Gomperfz Function, Estimation Method 2

TREND OF MECHANICAL POWER AND

MACHINERY INPUTS

See Appendix Sheet 10

1935 1940 1945 1950 1955 1960 1965 1970

J9Í7-49 =100

U. S. DEPARTMENT OF AGRICULTURE NEC. ERS 1693-63(1) ECONOMIC RESEARCH SERVICE

Figure 7

Gompertz Function, Estimation Method 2

SIMPLE REGRESSIO N

See Appendix Sheet 9

LOG INDEX YE AR t+l

/

.^

r

2.1 ,/ ./

./

•

2.0 • y^

/

1.9

1.8 /

1.7 /

/

•

1.6 1.7 1.8 1.9 2.0 2.1 2.2

LOG INDEX YEAR t

U. S, DEPARTMENT OF AGRICULTURE NEC. ERS 1694-63 (1 ECONOMIC RESEARCH SERVICE

Figure 8

11
The logistic curve, which is asymmetric sigmoid function, would better

represent the data, but even it would not be adequate to describe the asymmetry of

this particular example. Again we are reminded to supplement the application of

the techniques of this paper by graphical analysis of the data.

Example 5: The Spillman function is not sigmoid but could be used to represent

the upper portion of an S-shaped curve. As the Gompertz and Logistic curves are

not suited for representing the index of power and machinery over the period 1933-

1958, the Spillman could be used to represent the upper portion of the trends.

Figure 9 shows the result using method 2. As in example 1, the strong serial correlation of the variable enables the parameters of the Spillman to be estimated with

some accuracy. Then a close fit to the index for the sub-period is obtained.

VIL THE RELATIVE MERITS OF LINEAR TRANSFORMATION

VS. ITERATIVE METHODS

The fitting of nonlinear functions to empirical data has been greatly facilitated

in recent years by the advent of computer programs ^1 for iterative least squares

estimation techniques {2, ^). These techniques generally begin with an initial guess

for the parameters of the particular nonlinear function to be fitted. They then

choose increments by which to modify the initial guess in such a way as to diminish

the sums of squares of deviations of the data from the function fitted. One such

modification is called an iteration. For well behaved functions and well behaved

data a series of such iterations will converge to the least squares estimates.

Such techniques, however, have been found through extensive practical use to

have four disadvantages.

(1) Certain types of data lead to convergence difficulties. Briefly, it has been

found that if data are not sufficient to approximate an extensive portion of the function, numerous local optima in a least squares sense may be found by beginning with

various initial guesses for the parameters.^' One is thus faced with the problem of

finding all such least squares estimates and picking the best one.

(2) The computer programs involved usually require a specially programed

subroutine for the particular function being used. Not all research workers who

desire to use nonlinear functions will be interested in, or have the time to program

such subroutines.

(3) The iterative procedures may consume a substantial amount of computer

time relative to the linear transformation method.

4/ The following references describe several such programs: Booth, G. W.,

and Peterson, T. I. Non-linear Estimation (Princeton - IBM), in^ Mathematics

and Applications Department, Internatl. Bus. Mach. Corp., Forecasting by Generalized Regression Methods (WL NLI), White Plains, N. Y., Feb. 1959; Gardner,

R. S. A Least Squares Iteration. U. S. Naval Ord. Test Sta., IBM SHARE Distrib.

No. 934, Prog. No. 145-20, China Lake, Calif., June 1960; and Wood, P. B. Nonlinear Least Squares - (OR NLLS). IBM Appl. Sei. Spec. Prog. Group, SHARE

Distrib. No. 837, Union Carbide Nuclear Co., Oak Ridge, Tenn.

_5/ See Gardner, footnote 4, A Least Squares Iteration.

12
Spillman function. Estimation Method 7

INDEX*

TREND OF MECHANICAL POWER AND

MACHINERY INPUTS

See Appendix, Sheet 12

60 'I'll J—\—\ I I \ \ I I I I J \ \ L

1945 1950 1955 1960 1965 1970

* 1947-49 =100

U. S. DEPARTMENT OF AGRICULTURE NEC. ERS 1695-63(1) ECONOMIC RESEARCH SERVICE

Figure 9

Spillman Function, Estimation Method 2

SIMPLE REGRESSION

See Appendix., Sheet H

130

110

90

70

50

EX IN YEAR t +1

• y

•

•

jr%

^

/

50 70

. s. DEPARTMENT OF AGRICULTURE

90 110 130 150

INDEX IN YEAR t

NEC. ERS 1696-43(1) ECONOMIC RESEARCH SERVICE

Figure 10

13
(4) Finally, iterative techniques are not well adapted for use on desk

calculators, a disadvantage that may be serious to research workers who do not

have an adequate scientific computer at their disposal.

The chief advantage of iterative techniques is their convergence if the data are

suitable to least squares parameter estimates when the error or shock term is

additive.

The disadvantages of transformation methods are (1) that they do not give

least squares estimates for additive error terms, and (2) that some of the parameters may have to be exogenously estimated either "by eye" or some other means,

perhaps one statistically arbitrary. But their ease of application, particularly to

data which fall short of the range of values needed to approximate a sufficient range

of the function, may frequently justify their use. Finally, one may observe that the

methods presented in this handbook may be used to find initial parameter values

which may then be improved by the iterative methods to which we have referred.

LITERATURE CITED

(1) Berkson, Joseph.

1953. A Statistically Precise and Relatively Simple Method of Estimating

the Bio-Assay with Quantal Response, Based on the Logistic

Function. Jour. Amer. Statis. Assoc. 48:565-599, illus.

Washington, D. C.

(2) Curry, H. B.

1944. The Method of Steepest Descent for Non-linear Minimization

Problems. Quart. Appl. Math. 2(3):258-261.

(3) French, B. L.

1956. Functional Relationships for Irrigated Corn Response to Nitrogen.

Jour. Farm Econ. 38(3):736-747, illus.

(4) Griliches, Zvi.

1957. Hybrid Corn: An Exploration in the Economics of Technological

Change. Econometrica 25(4):501-522, illus.

(5) Grissom, P. H.

1958. Sources and Rates of Nitrogen. Miss. Agr. Expt. Sta. Inform.

Sheet 575. February.

(6) Ibach, D. B.

1956. A Graphic Method of Interpreting Response to Fertilizer. U. S.

Dept. Agr. Agr. Handb. 93, 27 pp., illus. January.

(7)

1958. Use of Standard Exponential Yield Curves. U. S. Agr. Res. Serv.

ARS 43-69, 24 pp., illus. February.

14
(8) Levenberg, Kenneth.

1944. A Method for the Solution of Certain Non-Linear Problems in Least

Squares. Quart. Appl. Math. 2(2): 164-168.

(9) Loomis, R. A., and Barton, G. T.

1961. Productivity of Agriculture, United States, 1870-1958 U S Dept

Agr. Tech. Bui. 1238, 63 pp., illus. April.

(10) Mansfield, Edwin.

1961. Technical Change and the Rate of Imitation. Econometrica

29(4):741-766, iUus. October.

(11) Nair, K. R.

1954. The Fitting of Growth Curves. In Kempthorne, O., and others (eds )

Statistics and Mathematics in Biology. Iowa State Col. Press

Ames, pp. 119-132.

(12) Paschal, J. L., and French, B. L.

1956. A Method of Economic Analysis Applied to Nitrogen Fertilizer Rate

Experiments on Irrigated Corn, U. S. Dept. Agr. Tech

Bui. 1141, 73 pp., illus. May.

(13) Rhodes, E. C.

1940. Population Mathematics. Jour. Royal Statis. Soc. 103(3):362-387,

illus. London.

(14) Spillman, W. J.

1933. Use of the Exponential Yield Curve in Fertilizer Experiments.

U. S. Dept. Agr. Tech. Bui. 348, 67 pp., illus.

(15) Stevens, W. L.

1951. Asymptotic Regression. Biometrics 7:247-268.

APPENDIX

The computations involved in the methods described in this paper require only

a desk calculator and a table of natural or common logarithms. Computation forms

were prepared to facilitate the various steps in estimating the coefficients. These

forms are shown in this appendix. Forms for fitting the logistic function are included, though empirical examples were not prepared. The data and all computations

are then provided on forms appropriate for each example presented in section VI.

Flyers described in the text, which were omitted in the computation of the

parameters are indicated by asterisks.

15
COMPUTATION SHEET

for

Estimation of Logistic Parameters

(Method 1)

Sheet la

Description of Data: Sample computation sheet for ^

fitting the Pearl-Reed or logistic y _

curve by means of Method 1.

(1) (2) (3) (^) (5) (6)

variable

Y.

J

variable

A

y

(3) - 1.0 Log (Col. 4) (5) X (1)

ÏJ (2)

XZj = XXj. XXjZj

rxj2 (XXj)' XXj £2j

B =

XX,Z. - n XX. XZ.

J J J J

XX.^ - i (XX.)2

J n J

A = i [X Z. - BXX.1

n J J J

A

oe -B =

A

ß - Antilog A

16
COMPUTATION SHEET

for

Estimation of Logistic Parameters

(Method 2)

Sheet lb

Description of Data- ^^P^^ computation sheet for fitting Pearl-Reed or lo-

^ * gistic curve by means of Method 2.

(1) (2) (3) (4) C = (scale factor)^

^J Zj = Q^ 2i+ 1 V ^

Col. 2 «Col. 3

2j • 2j>i

n-1

V 7 -

j=l J

n-1

r (z ) - - --

j=i J

/^f^>, r7 \ ¿1D

S-^l^J^ "

n-1

^2iZ._^l _

j=l ^ ^

n-1 n-1

r z - r ^ -u-n J J+1 -

B =

n-1 ]

n-1 n-1

ü Z. Xz.

n-1 ,

rz' n-1 2

(A) Sf,V^

A =

n-1 n-1

A_ -log B -

a - M

'^ A

^Scale factor introduced for computational convenience. Note appearance of

"c" in equation for the constant, A.

17
Sheet 2

TABLE FOR COMPUTING VALUES OF LOGISTIC FUNCTION

Description of Data: sample computation

sheet for computing values of logistic

curve. Parameters from sheets la and lb.

ß

y

(1) (2) (3) (^) (5) (6)

X

variable

a . X Antilog

Col. (2)

ß °Col. 3 1.0 - Col. 4 r-^-Col. 5

18
Sheet 3

COMPUTATION SHEET

for

Estimating Spillman Parameters

(Method 1)

Description of Data: Computations for Example 1, Figure

taken from Table 1, high rates. ^ -

Columns 2 and 3

68.1

(1) (2) (3) (4) (5) (6)

observation

number j variable variable y - Col. 3

Log Col. 4 Col. 5 X

Col. 2

1 0 15.4 52.7 1.72181 0

2 45 54.4 13.7 1.13672 51.2

3 60 60.3 7.8 .89209 53.5

4 75 66.5 1.6 .20412 15.3

*5 90 68.0 .1 -2.303 -207.3

6 105 66.6 1.5 .17609 18.5

k-l 120 67.5 .6 -.511 -61.3

E Zj = 4.13083

ZXj2 = 22,275.0

E Xj = 285.0

(SXj)2 = 81,225.00 E Xj E Z .

138.5

1177.28655

B =

-.0161

EX.2 . i (EXi)2

J n -^

A = - [E Zi - B E X. ] = 1.7438

n J J

0^ = Antilog B = .9840 'ß= Antilog A 55.44

*Flyers as described in text omitted from computations

19
Sheet 4

TABLE FOR COMPUTING VALUES OF SPILLMAN FUNCTION

Description of Data: a = .9840

Computation for Exajnple 1 Figure 1.

Parameters from Sheet 3.

ß

y

- 55.44

68.1

Log a = -.0161

(1) (2) (3) (4) (5)

X

variable Log a ' Col. 1 Antilog Col. 2 ß ' Col. 3 r - Col. 4

0 0 1.0000 55.4400 12.7

7.5 -.1208 .7572 41.9792 26.1

15.0 -.2415 .5634 31.2349 36.9

22.5 -.3622 .4274 23.6951 44.4

30.0 -.4830 .3241 17.9631 50.1

37.5 -.6038 .2458 13.6272 54.5

45.0 -.7245 .1864 10.3340 57.8

60.0 -.9660 .1073 5.9487 62.2

75.0 -1.2075 .0617 3.4206 64.7

90.0 -1.4490 .0355 1.9681 66.1

105.0 -1.6905 .0200 1.1088 67.0

120.0 -1.9320 .0117 .6486 67.4

150.0 -2.4150 .0038 .2107 67.9

20
Sheet 5

COMPUTATION SHEET

for

Estimating Spillman Function

(Method 2)

Description of Data- Computations for Example 2, Figure 4. Columns 1 and 2

^ " ' from Table 1, low rates. (M=7.5)

(1)

varlab le

B =

A =

A

13.6

23.4

34.7

36.8

43.5

49.5

(2)

Yj + 1

var iable

23.4

34.7

36.8

43.5

49.5

54.8

(3)

Col. 1 X Col. 2

318.24

811.98

1276.96

1600.80

2153.25

2712.60

n-1 n-1 n-1

2 Yj Yj+i . _i_ E Yj E Yj+i

j=l (n-1) j=l J=l

n-1

(EY.)2

n-1

j=l J (n-1) j=i J

n-1 n-1

^-i-— [ Z Yj+i - B E Yj ] = 12.4358

(n-1) j=i J j=i

E Yj = 201.5

E Yj^ = 7633.35

n-1

E

j=l

n-l

E

j=l

n-1

(E Y.)2 = 40.602.25

j=l

n-1

E Y. Y.+l = 8873.83

j = l -

n-1 n-1

E Yj E Yj+i = 48.904.05

j=l j=l

n-1

E Yj+i = 242.7

j=l

.8342

¿i= Antilog ( log B ) = .9761

M

^ = A/(l-B) = 75.0048

21
Sheet 6

TABLE FOR COMPUTING VALUES OF SPILLMAN FUNCTION

Description of Data:

2 Figure 3.

a

ß

y :

Log Ci :

= .9761

Computations for Example

Parameters from sheet 5.

61.4

75.0

= -.0242

(.1) (2) (3) (4) (5)

X

variable LogOt • Col. 1 Antilog Col. 2 ß •Col. 3 r-coi. 4

0 0 1.0000 61.4000 13.6

7.5 -.1815 .8340 51.2076 23.8

15.0 -.3630 .6956 42.7098 32.3

22.5 -.5445 .5801 35.6181 39.4

30.0 -.7260 .4838 29.7053 45.3

37.5 -.8954 .4084 25.0758 49.9

^5.0 -1.0890 .3366 20.6672 54.3

60.0 -1.4520 .2341 14.3737 60.6

75.0 -1.8150 .1628 9.9959 65.0

90.0 -2.1780 .1133 6.9566 68.0

105.0 -2.5410 .0788 4.8383 70.2

120.0 -2.9040 .0548 3.3647 71.6

150.0 -3.6300 .0265 1.6271 73.4

22
Sheet 7

COMPUTATION SHEET

for

Estimating Gompertz Function

(Method 1)

_ . ^. Computations for Example

Description of Data:

taken from Table 1.

3, Figure 6

7 =1.83251

. Column 2

= Log 68.0

(1) (2) (3) (4) (5) (6) (7)

J = variable variable Log Col. 3 Y- Col. 4

Log Col. 5

Col. 6 X

Col. 2

1 0 13.6 1.13354 .69897 -.15552 0

2 0 15.4 1.18752 .64499 -.19044 0

3 7.5 23.4 1.36922 .46329 -.33442 -2.5082

¿e 15.0 34.7 1.54033 .29218 -.53462 -;3.0193

5 22.5 36.8 1.56585 .26666 -.57349 -12.9035

6 30.0 43.5 1.63849 .19402 -.71220 -21.3660

7 37.5 49.5 1.69461 .13790 -.86012 -32.2545

8 45.0 54.8 1.73878 .09373 -1.02812 -46.2654

9 45.0 54.4 1.73560 .09691 -1.01362 -45.6129

10 60.0 60.3 1.78032 .05219 -1.28233 -76.9398

11 75.0 66.5 1.82282 .00969 -2.01368 -151.0260

*12 90.0 68.0 1.83251 0 _oo - oo

13 105.0 66.6 1.82347 .00904 -2.04383 -214.6022

14 120.0 67.5 1.82930 .00321 -2.49349 -299.2188

B =

1

J ^ = -13.2359

(2:xj)2

562.5

= 316,406.

-.0194

XZjXj:

25 XXjXZ

= -910.7165

2 = 41,793.75 .= 7445.1938

J

ÍZA -ï^h^^'i

xx/ - i (rxj)^

-.1787

A = ± [IZj -B^A. J =

A

oe = Antilog B = .95 64 Q= Antilog A = _ .6627

^See note on Sheet 3.

23
Sheet 8

TABLE FOR COMPUTING VALUES OF GOMPERTZ FUNCTION

Description of Data:

ß =

.9564

.6627

Computations for Example 3, Figure 5.

Parameters taken from sheet 7. 7=1

Log Cl = -

.8325

.0194

(1) (2) (3) (4) (5) (6)

Variable

Log a • Col. 1

Antilog

Col. 2

ß • Col. 3 r-coi. 4

Antilog

Col. 5

0 0 1.0000 .6627 1.1698 14.78

7.5 -.1455 .7087 .4696 1.3629 23.06

15.0 -.2910 .5022 .3328 1.4997 31.60

22.5 -.4365 .3634 .2408 1.5917 39.06

30.0 -.5820 .2575 .1706 1.6619 45.91

37.5 -.7275 .1863 .1235 1.7090 50.22

45.0 -.8730 .1320 .0875 1.7450 55.59

60.0 -1.1640 .0677 .0449 1.7876 61.32

75.0 -1.4550 .0347 .0230 1.8095 64.49

90.0 -1.7460 .0178 .0118 1.8207 66.18

105.0 -2.0370 .0091 .0060 1.8265 67.07

120.0 -2.3280 .0047 .0031 1.8294 67.52

150.0 -2.9100 .0012 .0008 1.8317 67.88

24
COMPUTATION SHEET

for

Estimating Gompertz Function

(Method 2)

Sheet 9

Description of Data:

Computations for Example 4,

from Table 2.

Figure 8. Column 1 talcen

(1) (2) (3) (4)

Ygvariable

Zj variable

= Log Yj

Zj + l

variable

Col. 2 X Col. 3

n-1

IZi = 47.60445

44 1.64345 1.64345 2.70093 n-1 2

44 1.64345 1.65321 2.71697 DZ,. - 91.44561

45 1.65321 1.68124 2.77944

48 1.68124 1.71600 2.88501

52 1.71600 1.74036 2.98646 n-1

55 1.74036 1.74036 3.02885 iLZ])^ = 2266.1836

55 1.74036 1.76343 3.06900 j=l

58 1.76343 1.78533 3.14830

61 1.78533 1.81954 3.24848 n-1

66 1.81954 1.83885 3.34586 LZ. Zj+n = 92.36993

69 1.83885 1.84510 3.39286

70 1.84510 1.86923 3.44892

74 1.86923 1.90309 3.55731 n-1 n-1

80 1.90309 1.94939 3.70986 rZjZZj+i = 2289.66551

89 1.94939 2.00000 3.89878

100 2.00000 2.04532 4.09064

111 2.04532 2.07188 4.23766 n-1

118 2.07188 2.10380 4.35882 XZi+i = 48.09772

127 2.10380 2.12385 4.46816 j=l

133 2.12385 2.12710 4.51764

134 2.12710 2.13033 4.53142

135 2.13033 2.13354 4.54514

136 2.13354 2.13672 4.55878

137 2.13672 2.13988 4.57232

138 2.13988 2.13672 4.57232

B =

n-1

r Z,- Z

j^x -iÀ)^j!i ^'^''

n-1 n-1

EZ^

n-1 2

rzj

n-1

Ù = Antilog ( log B ) =

M

n-1

.9813

.0553

A

y= V(l-B) = 2.9572

25
Sheet 10

TABLE FOR COMPUTING VALUES OF GOMPERTZ FUNCTION

Description of Data: a = .9813

Computations for Example 4,

Figure 7. Parameters from

Sheet 9.

ß

y

Log a

= 1.3138

= 2.9572

= -.0082

(1) (2) (3) (^) (5) (6)

variable

Log a • Col. 1 Antilog

Col. 2

ß • Col. 3 y- Col. 4

Antilog

Col. 5

0 0 1.0000 1.3138 1.6434 44.0

2 -.0164 .9630 1.2652 1.2920 49.3

5 -.0410 .9100 1.1956 1.7616 57.8

8 -.0656 .8598 1.1296 1.8276 67.2

10 -.0820 .8280 1.0878 1.8694 74.0

12 -.U984 .7946 1.0439 1.9133 81.9

15 -.1230 .7534 .9898 1.9674 92.8

18 -.1476 .7119 .9353 2.0219 105.2

20 -.1640 .6855 .9006 2.0566 113.9

22 -.1804 .6601 .8672 2.0900 123.1

25 -.2050 .6237 .8194 2.1378 137.3

30 -.2460 .5676 .7457 2.2115 162.2

35 -.2870 .5164 .6784 2.2788 190.0

38 -.3116 .4880 .6411 2.3161 207.1

26
Sheet 11

COMPUTATION SHEET

for

Estimating SpilLíiian Function

(Method 2)

r.^^^^- +• ^ T^ ^ Computations for Exajnple 5. Figure 10, Columns 1 and 2

Description of Data: ^,^ ^^^^^ ^^ y

taken from Table 2.

(1)

variable

89

100

111

118

127

133

134

135

136

137

138

(2)

variable

100

111

118

127

133

134

135

136

137

138

137

n-1

B =

A =

n-1 p

n-1

1 [ lY

Tïï^ j=i j+i

(3)

Col. 1 X Col. 2

8900

11100

13098

14986

16891

17822

18090

18360

18632

18906

18906

n-1

j =

1358

n-1 2

TYj ^ 170,474

n-1

(r Yj)2 1,844,164

J=l

n-1

L Yj Yj.^-L ^ 175,691

j=l

n-1 n-1

L Yj X Yj+-L ^ 1,909,348

n-1

1406

.n-1 n-1

XY,- XY,+1

(n-1) .1=1 ' j=l

.7504

1 n-1 2

Tir:ïT (XYj)

n-1

- B X Y i ] _

35.1783

j=l

a = Antiiog ( l'2g_B ) =

M

.7504 V = A/(l-B) =

140.9387
Sheet 12

TABLE FOR COMPUTING VALUES OF SPILLMAN FUNCTION

Description f ^^ + , Computations for Example

Figure 9. Parameters from

Sheet 11.

01 =

5, ß =

y =

Log Ct =

.7504

51.9387

140.9387

-.12471

(1) (2) (3) (^) (5)

X

variable Log üi • Col. 1 Antilog Col. 2 ß ' Col. 3 y- Col. 4

0 0 1.0000 51.9387 89.0

1 -.12471 .7504 38.3408 102.6

2 -.24942 .5631 29.2467 111.7

3 -.37413 .4225 21.9441 118.9

4 -.49884 .3171 16.4698 124.5

5 -.62355 .2379 12.3562 128.6

6 -.74826 .1785 9.2711 131.7

7 -.87297 .1340 6.9598 133.9

8 -.99768 .1005 5.2198 135.7

9 -1.12239 .0754 3.9162 137.0

10 -1.24710 .0566 2.9397 138.0

11 -1.37181 .0425 2.2074 138.7

12 -1.49652 .0319 1.6568 139.3

22 -2.74362 .0018 .0935 140.8

28

* U.S. GOVERNMENT PRINTING OFFICE : 1963 O—699-007