NISTIR 88-3803

Logistic Function

Analysis Program

LOGIT

Data

William H. Kirchhoff

U.S. DEPARTMENT OF COMMERCE

National Institute of Standards and Technology

(Formerly National Bureau of Standards)

Center for Chemical Technology

Chemical Thermodynamics Division

Gaithersburg, MD 20899

March 1989

Prepared for

U.S. DEPARTMENT OF COMMERCE

National Institute of Standards and Technology

Gaithersburg, MD 20899

NISTIR 88-3803

Logistic Function Data

Analysis Program

LOGIT

William H. Kirchhoff

U.S. DEPARTMENT OF COMMERCE

National Institute of Standards and Technology

(Formerly National Bureau of Standards)

Center for Chemical Technology

Chemical Thermodynamics Division

Gaithersburg, MD 20899

March 1989

National Bureau of Standards became the

National Institute of Standards and Technology on August 23, 1988, when the Omnibus Trade and

Competitiveness Act was signed. NIST retains

all NBS functions. Its new programs will encourage

improved use of technology by U.S. industry.

Prepared for

U.S. DEPARTMENT OF COMMERCE

National Institute of Standards and Technology

Gaithersburg, MD 20899

U.S. DEPARTMENT OF COMMERCE

Robert Mosbacher, Secretary

Ernest Ambler, Acting Under Secretary

for Technology

NATIONAL INSTITUTE OF STANDARDS

AND TECHNOLOGY

Raymond G. Kammer, Acting Director

LOGISTIC FUNCTION DATA ANALYSIS PROGRAM

LOGIT

William H. Kirchhoff

Chemical Thermodynamics Division

Center for Chemical Technology

National Institute of Standards and Technology

(formerly National Bureau of Standards)

Gaithersburg, MD 20899

Present address:

Chemical Sciences Division

Office of Basic Energy Sciences

U.S. Department of Energy

Washington, DC 20545

LOGIT

TEST INTERFACE 111 DATE: 05/30/86 STANDARD DEVIATION = 143.

ABSTRACT

A FORTRAN program has been written for the statistical analysis of

experimental data in terms of an extended logistic function that includes

non-horizontal asymptotes and asymmetry in the pre- and post- transition

portions of growth and decay curves. The program is robust in that

situations in which few or no data fall within the transition interval can be

analyzed by the program. Individual weighting of the data is allowed for

situations where the errors in experimental data are not uniform. The

primary parameters describing the transition region include a location

parameter, X0 , a width parameter, D0 , and an asymmetry parameter Q. Six more

parameters describe the two quadratic asymptotic regions. The program does

not require initial estimates for the parameters and provides statistical

estimates of all parameters. A provision is included for the identification

and exclusion of outliers. Sufficient information is given to allow the

development of companion subroutines for the graphing of the function and its

standard deviations as well as for displaying the original data with error

bars. The purpose of the program is to provide a means for systematically

parameterizing sigmoidal profiles for the comparison of measurements made

with different instruments on different systems and for the comparison of

measurements with simulation models. The program is extensively documented.

KEY WORDS: logistic function, hyperbolic function, autocatalytic function,

statistical analysis, sigmoid

i
-
LOGISTIC FUNCTION DATA ANALYSIS PROGRAM

1.0 INTRODUCTION

In its simplest form, the logistic function may be written as:

Y - L—T- (1)

1 + e'

X

in which Y progresses from 0 to 1 as X varies from -® to The

differential equation generating this function is:

dY/dX - Y(l-Y) (2)

and in this form describes a situation where a measurable quantity Y grows in

proportion to Y and in proportion to finite resources required by Y. The

logistic function was first named and applied to population growth in the

last century by Verhulst.^- It has been applied to the analysis of phenomena

in fields as diverse as demography, ^ economics,-^ bioassay, ^ and the physical

and chemical sciences where it arises deterministically and/or heuristically

in the description of autocatalysis, bimolecular reactions, oxidationreduction potentials, enzymatic catalysis,^ biomolecular phase transitions ,

and depth profile analysis. ^ Numerous descriptions have been presented for

the parametric fitting of logistic growth curves with an iterative approach

using a truncated Taylor's series expansion providing the most generally

satisfactory result. The logistic function as a distribution function and

growth curve has been extensively reviewed by Johnson and Kotz .

®

The program described herein analyzes experimental data in terms of an

extended logistic function:

Y = A + A (X-X ) + A (X-X ) s o q o

-i z

1 + e

B + B (X-X ) + B (X-X ) s o q o

1 + e -z (3a)

where z = (X-X0 )/D (3b)

and D = 2 D

q

/

[

l+eQ(X_Xo

)

] (3c)

Y is typically some form of system response or a function of system response

and X is the independent variable, such as time, temperature, or

concentration. This equation can be fit to both growth and decay curves.

Equations 3a - 3c are referred to collectively as an extended logistic

function because they include non-linear asymptotes and asymmetry in the

curvature of the pre- and post- transition regions.

The analysis program allows the user to determine which parameters are to be

varied in the least squares fit of data to Eq. 3. The program can be

directed to identify outliers and to exclude these outliers from the fit

while, at the same time, printing statistical information for the outliers.

The program is designed to calculate initial estimates of the parameters when

the parameters are not being held fixed at values provided by the user.

The program consists of a main program and 14 subroutines, written in Fortran

77 and heavily documented, that can be tailored to a variety of computational

1
environments. However, emphasis has been placed on a version that will run

on IBM XT, AT, and compatible personal computers possessing a mathematics

coprocessor (Intel 8087 and 80287). Portability has thus been sacrificed to

a certain extent in favor of user convenience by use of program - compiler - machine dependent graphics subroutines. However, since many graphics

packages contain similar commands, the inconvenience of program modification

may not be too serious and is offset by the convenience of graphical editing

of data. Memory usage depends upon the particular operating environment

(such as the Fortran compiler, the size of the Fortran function libraries,

the length of representations of floating point numbers, etc.). Typically,

250K of memory should be available for the program and its data requirements.

The purpose of this document is to guide the user in adapting the programs to

the user's computing environment and interpreting the results. A rudimentary

knowledge of error analysis and the significance of least squares fits are

necessary in order to use the program.

1.1 DESCRIPTION OF TERMS

Throughout this documentation, the regions of the sigmoidal profile will be

referred to as the pre- transition, transition, and post- transition regions.

These terms are not dependent on whether a particular transition profile is a growth or a decay curve though the analysis program must, from time to time,

distinguish between the two. The terms pre- and post- are taken in the sense

of increasing values of the independent variable X.

The nine parameters necessary to characterize the transition profile are

related to the three distinct regions of the transition profile. Three

parameters, a slope As , an intercept A, and a quadratic term A

q

are necessary

to define the pre- transition asymptote, three more, B, B s and Bq, to define

the post- transition asymptote, and two more, DQ and XQ , to define the slope

and position of the transition region. In addition, an asymmetry parameter Q

which causes the width parameter to vary logistically from 0 to 2DQ , is

introduced as a measure of the difference in curvature in the pre- and

post- transition ends of the transition region. If Q < 0, the pre- transition

region has the greatest (sharpest) curvature. If Q > 0, the post- transition

region has the greatest curvature. If Q = 0, D = Dc and the transition

profile is symmetric. The parameter Q has the dimensions of 1/X whereas D0

has the dimensions of X. The product QD0 is dimensionless and is a measure

of the asymmetry of the profile independent of its width. If the absolute

magnitude of QDC is less than 0.1, the asymmetry in the transition profile

should be barely discernible.

From time to time, it will be necessary to distinguish between the two

asymptotic limits of Y in terms of their relative values. In these

situations, they will be referred to accordingly as the high and low

asymptotes. The high asymptote may be either the pre- or post- transition

asymptote depending on whether a decay or a growth profile is being analyzed.

Mention is made of poorly structured data. Examples of poorly structured

data include, very narrow transition intervals or transition intervals

containing few or no data, data with noise comparable in magnitude to the

slope of the transition interval, and asymptotes with slopes large compared

with the change in Y through the transition interval.

2
2.0

DESCRIPTION OF THE ANALYSIS

Data in the form of X,Y pairs and saved in a text file are fit by the method

of least squares to Equation (3). Because these equations are non-linear

functions of the three transition region parameters, XQ , DQ , and Q, the least

squares fit requires an iterative solution. Consequently, Y, as expressed by

Eq. (3), is expanded in a Taylor series about the current values of the

parameters and the Taylor series is terminated after the first (that is,

linear) term for each parameter. Y(obs) - Y(calc) is fit to this linear

expression and the least squares routine returns the corrections to the

parameters. The parameters are updated and the procedure is repeated until

the corrections to the parameters are deemed to be insignificant compared to

their standard deviations.

2.1 Initial Estimates

The efficiency of the iterative procedure is sensitive to the initial

estimates of the values of the parameters. Various means of calculating

initial estimates have been tested, and the approaches selected, though they

did not always give the best set of initial parameters, were the least prone

to false starts in situations of poorly structured data. Three methods are

used. The first method calculates the initial estimate of A as the average

of the first three values of Y and the initial estimate of B as the average

of the last three values of Y. The average value of Y, <Y> *= (A+B)/2, is

calculated from the initial estimates of A and B and is taken to be the

midpoint of the transition interval. The four data with values closest to

<Y> are used to calculate a straight line. This line is assumed to be a line

tangent to the logistic curve at its midpoint. From the tangent line and

from the estimates of A and B, the transition parameters XQ and D0 are

calculated. The remaining parameters are given initial values of zero.

In the second method, the initial estimates of A and B are obtained in the

same way as the first method, but the largest change in Y is taken as spanning the midpoint of the transition interval. The tangent line is drawn

through this interval from which X0 and D0 are calculated.

In the third method, the three regions are marked in a graphical presentation

of the data. Straight lines are drawn through the data in each of the

regions giving A, As , XQ , D0 , B, and B s

.

The third method is used only when the pretransition, transition, and

posttransition regions are marked by the user. The second method is used

only when the first method fails to converge in the requisite number of

iterations

.

2.2 The Least Squares Analysis.

A cycle of up to p iterations is executed in which, at the end of each

iteration, the parameters are updated before the next iteration is performed.

The number of iterations p is chosen on the basis of experience with

particular classes of data. If p is selected to be a prime number,

oscillations between two or three local minima can be identified by

performing repeated multiples of p iterations. Generally, if convergence

3
takes longer than 11 iterations, the solution is unstable in the sense that

all of the parameters cannot be determined from the data. In most cases,

instability of the fit can be interpreted by the program and the source of

the instability removed by varying one fewer parameter in the least squares

fit. Messages keep the user informed of these situations.

All measurement errors are assumed to be normally distributed in the Y's.

The X's are assumed to be error-free. More precisely, the errors in X are

assumed to be sufficiently small that the spread in Y corresponding to the

error in X is small compared to the error in Y. If this is not the case,

uncertainties in XQ and DQ returned by the least squares program will not be

reliable

.

The least squares fit has been written to allow individual weighting of the

data to account for distributions of measurement errors that vary with the

values of X or Y. The weights should be proportional to the inverse square

of the uncertainty in the value of Y. If all data have the same expected

error, unit weights may be used and the standard deviation of the fit will be

a measure of the random error in Y. If the values of Y used in the least

squares fit are not the primary measurements, but are functions of the

primary measurements, then weights must be used and the program must be

modified to calculate the weights based on the functional dependence of Y on

the measurements. In particular, if z is a measured quantity with

uncertainty s(z), the weight of z would be w(z) = l/s(z) 2

. If Y is a function of the measurements z, the weight of Y in the least squares fit,

which should be proportional to the inverse square of the uncertainty in Y,

would be given by:

W(Y) = w(z)/[dY/dz]

2 (4)

In graphical displays of the results of this analysis, error bars should be

presented for each of the experimental points and confidence limits should be

drawn bracketing the curve calculated from the least squares parameters.

Confidence limits can be calculated using the parameters obtained from the

least squares fit as will be explained below. Confidence limits for

experimental points are simply tS/7w^ where is the weight given to the ith

datum in the fit, S is the standard deviation of the fit, and t is obtained

from the desired percentile values of the student's t distribution. ^ (For

example, for an infinite number of degrees of freedom and for a 95%

confidence interval symmetrically bracketing the data, the value of t is

1.96.)

The confidence limits for the calculated values of Y are determined from the

variance -covariance matrix, V, returned by the linear least squares

subroutine. More generally, uncertainties in all functions of the

parameters, including the calculated values of Y, are obtained from the

variance -covariance matrix. The variance -covariance matrix contains all the

information about the quality of the fit and the correlation of errors among

the parameters. In particular, if F is any function of the parameters Cj

(«A, B, ....), the variance (square of the standard deviation) of F is given

by

s

2 (F) = l (dF/dCj )Vj k

(dF/dCk ) .

' (5)

j .k

4
The square root of the variance -covariance matrix diagonals are the standard

deviations of each of the determined parameters (as can be seen by setting F = Cj in eq. 5). Equation (5) can be derived directly from the least squares

equations and from propagation of error formula:

s

2

(F) - l

(dF/dy ) 2s

2

(y ) (6)

i-1

where yi are all of the observations included in the fit. The least squares

equations in matrix form are

X

TXC *= X

tY (7)

where C is the column vector of the coefficients, Y is the column vector of

the dependent variables and X is the m x n matrix of the m independent

variables. The matrix X

TX is the least squares matrix. The

variance -covariance matrix is defined, after some manipulation of equations

(5) -(7) as

V = s

2 (XTX)1

. (8)

If we define A as the unitary transformation that transforms the set of

vectors X into an orthonormal set so that

AT(XTX)A =1, (9)

which can be rearranged to give

(XTX)

1 = AAT

, (10)

then

V = s

2AAt

(11)

This is the procedure followed in Subroutine ORTHO which performs the least

squares calculation.

The confidence limits for the logistic curve calculated from the parameters

of the least squares fit can be calculated directly from Eqs . (3) and (5).

2.3 SITUATIONS WITH FEW DATA IN THE TRANSITION REGION

The analysis becomes complicated in situations where only a few data fall

within the transition region. The parameters A, As , A~ , B, B s , and Bq are

determined primarily from the asymptotes of Equation (I), namely:

Y - A + AS (X-X0 ) + A (X-XQ ) 2

and

Y = B + B S (X-X0 ) + B

q

(X-X0 ) 2

(12a)

(12b)

while XQ , DQ , and Q are determined primarily by data in the transition

region.

5
We define the transition interval as the interval in which Equation (3)

differs from the asymptotic limits of equations (12a and 12b) by more than n

standard deviations. The number of data falling within the transition

interval will determine whether any of XQ , DQ , and Q can be determined. The

interval is given approximately by the limits Xjjj and Xpi

Xj, - x0

- Dln[ | A-B|/U - 1]

and

Xp - xc + Dln[ | A-B | /U - 1]

The subscripts m and p refer to the pre- transition (minus) and

post- transition (plus) sides of the transition interval. D is a function of

Xjh °r Xp (-= X in Eq. (3c)). U is the uncertainty in the value of Y at the

midpoint of the transition interval and is given by:

U = nS/7w (14)

where S is the standard deviation of the fit, W is the average weight (see

below) of Y in the interval between Xm and Xp and n is usually 2. When Q=0,

the interval is symmetrically placed about XQ and D is independent of X.

(These expressions are exact only when As , Aq, Bs , Bq , and Q are zero. In

the analysis program, the value of the interval, Xp - Xjjj is used only to

estimate the upper limit of DQ when Dc cannot be determined by the least

squares fit.)

The number of data falling within the transition interval, that is, the

number of data for which the measured value Y^ differs from each asymptote

(eqs. 12a and 12b) by more than nS/yw^, is counted. If no datum falls within

the interval (very sharp transitions), Q is set equal to 0 and XQ and DQ are

estimated from the separation between the two measured values of X spanning

the transition interval:

X0 = <Xi+1 + Xi)/2 (15a)

and D0 = (Xi+1 - Xi)/(2Fx ) (15b)

where Fx = the larger of ln[|A-B|/U - 1] or 2 so that DQ is always < (X^+^

- Xj^)/4. All three parameters are held fixed at these values while the other

parameters are varied.

If one datum falls within the interval, Q is held fixed at 0 and the maximum

value of DQ is estimated from the average data interval among the three

points spanning (one just below, one in, and one just above) the transition

interval

:

Do = Kxi+1 - xi-l>/2]/Fx

(16)

The value of XQ and the remaining parameters are determined from the least

squares fit, but the value of Xc is influenced very strongly by the single

datum falling within the interval.

If two data fall within the transition interval, Q is held fixed at 0, and Xc and Dc are allowed to vary.

(13a)

(13b)

6
If confidence limits are to be assigned to XQ , DQ , or Q based on the standard

deviations for these parameters returned by the least squares fit, then care

should be taken in interpreting the number of degrees of freedom used in

connection with a Student's t distribution. The appropriate number is NIN-2

or NIN-3 depending on whether Q is varied and where NIN is the number of data

falling within Xjjj and Xp. In particular, if only two points fall within the

interval, then the uncertainties in XQ and Dc must be estimated from the data

separation and not from the standard deviations returned by the fit.

2.4 POST- FITTING TESTS.

Following the cycle of p iterations, four tests are performed to judge the

quality of the fit, to test the assumptions of the determinability of XQ , DQ , and Q, and to test the determinability of the asymptotic parameters As , Aq,

Bs , and Bq. If a test is failed, the analysis is repeated holding certain

parameters constant. The user has control, however, over whether the

analysis is repeated or not.

The philosophy underlying the performance of the post- fitting tests is that

the parameters As , Aq, Bs , Bq , and Q are less critical in the analysis of the

logistic profile than the parameters D0 and XD

. In general (but not always),

the former are of a heuristic nature and have little basis in the theory

underlying the use of the logistic function in a particular application.

2.4.1 The Convergence Test.

The most recent correction to the ith parameter, 6^, is compared to the

standard deviation of the ith parameter, o If -LOG^qC I & i I /°0 ~ ^ > t*ie

corrections to the parameters are of the same order of magnitude as the

standard deviations of the parameters and the iteration cycle is considered

to have not converged. If this occurs, the analysis is repeated using the

largest jump in the value of Y as an estimate of the midpoint of the

transition interval. If the convergence test fails a second time, the

parameters As , Aq , B s , and Bq are held fixed at 0 and the data are refit to

the remaining parameters. If As , Aq , B s , and Bq had already been held fixed

at 0, the asymmetry parameter Q is held fixed at 0 . If all five had been

held fixed at 0, a lack of convergence message is printed and the calculation

continues

.

2.4.2 The Interval Test.

Under circumstances of small vertical separation between the two asymptotes

compared to the uncertainties in the values of Y, no datum may be found that

lies more than two standard deviations away from each asymptote. The program

will assume that the transition region is narrow and will estimate a small

value for DQ

. The assumption of no datum in the transition region is tested

by noting how many data overlap the value of Y at the center of the

transition interval to within one standard deviation. If one or more such

data are found, the assumption of no datum in the transition interval is

judged to be false. In general a broad transition profile can be fit

reasonably well with a profile consisting of a very narrow interval and

asymptotes with pronounced slopes and/or curvatures. The slopes and/or

curvatures allow accommodation of Eq . (3) to the broad interval.

7
Consequently, when the interval test is failed, the data are refit holding

As , Aq, B s , and constant at 0 . If As , Aq, B s , and Bg had already been

held fixed at 0, the data are refit holding Q fixed at 0, If all five had

been held fixed at 0 , a warning message is printed and the calculation

continues

.

2.4.3 The Uncertainty Test.

This test, in a sense, is a reverse of the interval test in that it tests the

assumption that the interval parameters, XQ , DQ , and Q, determined by the

least squares fit, can indeed be determined. The count of the number of data

in the transition interval is only an approximation to the number of degrees

of freedom available for the determination of the interval parameters.

Although n parameters can be obtained from n data, the uncertainties in the

resulting parameters cannot be determined. One result of fitting n

parameters to n data is large excursions in the uncertainty of the calculated

value of Y in the vicinity of the transition interval.

The 2S curves of Y (that is, the values of Y calculated from Eq . (3) ± twice

the standard deviation in Y calculated from Eq. (5)) are compared with the 2S

curves for the two asymptotes. When the calculated value of Y plus twice its

standard deviation is greater than the value of the high asymptote plus twice

its standard deviation or when the calculated value of Y minus twice its

standard deviation is less than the value of the low asymptote minus twice

its standard deviation, the uncertainties in Y in the vicinity of the

transition interval are judged to be excessive. When this situation is

encountered, the data are refit, beginning with new initial estimates, and

holding one more interval parameter constant than in the previous

calculation. This test is conducted only when the number of interval

parameters varied is equal to NIN, 'the number of data falling within the

transition interval.

2.4.4 The Asymptote Test.

The purpose of the asymptote test is to assure that the data have reached

well enough into the asymptotic region that the higher order asymptotic

parameters, As and Aq or B s and Bq , can be distinguished from the effects of

the interval parameters and can be distinguished from 0. The separation

between each value of Y and the values of the asymptotes at the corresponding

value of X is compared with the standard deviation of the measured value of

Y. The number of data falling within two standard deviations of one

asymptote or the other is counted. If more than four data fall within two

standard deviations of one asymptote or the other, the parameters defining

that asymptote are considered to be determinable. If four or fewer data fall

within two standard deviations of one of the asymptotes, that asymptote's

parameters, As and Aq or Bs and Bg , are compared with their standard

deviations. If their values are less than three times their standard

deviations, the corresponding parameter is considered to indeterminable from

0 .

8
2.5 OUTLIER IDENTIFICATION AND REJECTION

If directed to do so, LOGIT will, following completion of analysis, identify

outliers using criteria provided by the user. If any are identified and if

LOGIT is directed to do so, the data will be refit with the outliers dropped

from the data being fit. The standardized residuals are used for the

identification of the outliers. A standardized residual is the number of

standard deviations by which Yobs - Ycai c differs from its expected value of

zero, that is, the value of Yobs - Yca^ c divided by the standard deviation of

Yobs - Ycalc • ^

standard deviation of Yobs - Ycalc is given by:

s <Yobs- Ycalc) " V(SD

2 - Sc

2

) (17)

where S Q

^ - S^/W is the variance (square of the standard deviation) of an

observed value of Y, S is the standard deviation returned by the least

squares fit, W is the weight of the observation used in the fit, and S c

^ is

the variance of the calculated value of Y and is calculated from

variance -covariance matrix using equation (5)

.

The - sign appearing in

square root in equation (17) arises from the correlation between Yobs and

Ycaic (AH of the Yobs were used to calculate the parameters which in turn

were used to calculate Yca ^ c

.

)

The standardized residuals should follow a student's t distribution and therefore, for extensive data sets, a value of a standardized residual greater in absolute magnitude than 3.0 is generally an

outlier

.

9
PROGRAM LOG I 1 #-**

*** PROGRAM PLOT ***

igure 1 Subroutine structure of LOGIT and PLOT. Subroutines

enclosed in double lines are shared by LOGIT and PLOT.
3.0

PROGRAM STRUCTURE AND FLOW

The analysis program, LOGIT, consists of a main program and 14 subroutines,

OPTS, SELECT, KEY, HELP, RDDATA , LNBX, DELX, LGSTC , ESTIM8 , XYLINE , DERIV

,

FIX, ORTHO, and EPARM. The plotting program, PLOT, consists of a main

program and eight subroutines, three of which are identical to those used by

LOGIT. The structures of these programs are represented schematically in

Figure 1. Listings of each of these subroutines, which are heavily

commented, are included in Appendix A of this documentation. A brief

description of each program element follows.

3 . 1 MAIN

MAIN initiates the program and interactively directs the calculations and

output of the results. It saves the data, the parameters and the

variance -covariance matrix in an unformatted data file to be used by the

plotting program PLOT.

3 . 2 OPTS

OPTS directs a question and answer session in which the user can select which

parameters are to be determined, where the output of the results are to be

directed, which post-fitting tests are to be performed, the conditions for

outlier identification and rejection, and which data are to be included in

the least squares analysis. OPTS is called by MAIN.

3 . 3 SELECT

SELECT allows the user to select graphically which data are to be included in

the fit. It is called by OPTS in response to the user. SELECT is also used

to identify the pre- transition, transition, and post- transition regions for

those situations when the program is unable to do so. Once identified, these

three regions are used by the program to estimate the initial values of the

parameters

.

3 . 4 KEY

KEY reads the keyboard to detect if any identified keys have been pressed.

It is called by SELECT during graphical editing of the data. (KEY is also a subroutine of PLOT.)

3 . 5 HELP

HELP prints help messages on the top line of the display during graphical

editing of data.

3 . 6 RDDATA

RDDATA reads data files formatted according to the options offered by OPTS.

It performs some preliminary manipulation of the data. For example, it forms

the ratio of two columns of data when the ratios rather than the primary data

are being analyzed.

3 . 7 XYLINE

11
XYLINE calculates the slope and intercept of a line segment and is used for

forming initial estimates of the parameters.

3.8 LGSTC

LGSTC is called by the main program and directs the analysis of the data. It

is the longest and most complex of the subroutines and is consequently

divided into segments. Each segment is identified by the leading statement

number in the segment.

3.8.1 SEGMENT 400

Executes a cycle of iterative least squares fits. In each iteration

the standard deviation is calculated from the current values of the

parameters, the number of data falling within the transition interval

is counted and, based on the count, Q, DQ , or XQ may be held fixed.

Following the fit, the parameters are updated and a test for

convergence is performed. If the most recent corrections to the

parameters are less than three orders of magnitude smaller than the

corresponding standard deviations and if the interval parameters

directed to be varied have not been held fixed, the procedure is deemed

to have converged. Otherwise another iteration is performed until the

maximum number is reached.

3.8.2 SEGMENT 500

Performs the post-fitting tests:

Convergence test. If the most recent corrections to the

parameters are of the same order of magnitude as the standard

deviations of the parameters, the test fails.

Test of the number of data in the transition interval. If the

number was deduced to be 0 in SEGMENT 400, the number of data

whose uncertainties overlap the midpoint of the transition is

noted. If not also 0, the test fails.

Test for large uncertainties in Yca^ c in the vicinity of the

transition interval when the number of data falling within the

transition interval is equal to the number of interval parameters

being varied.

Test for determinability of the asymptotic parameters.

3.8.3 SEGMENT 700

Prints the data and their associated statistics. Checks for outliers

and processes them if instructed to do so by the calling program. If

outliers are found, the data are refit beginning with the determination

of their estimates by Subroutine ESTIM8 . All parameters that had been

held fixed as a result of a failure of one of the post-fitting tests

12
will again be allowed to vary.

3.9 ESTIM8

ESTIM8 determines the initial values of the parameters by various means as described in section 2.1. ESTIM8 is called by LGSTC.

3 . 10 DERIV

DERIV is called by BIX and LGSTC and calculates the value of Y from the

current values of the parameters for a given value of X. DERIV also

calculates the derivative of Y with respect to each of the parameters. These

derivatives are used as the independent vectors in the interative least

squares fit and are also used for calculating uncertainties in functions of

the parameters according to Eq.(5). (DERIV is also called by PLOT.)

3.11 FIX

FIX is called by LGSTC and forms a set of vectors to be sent to ORTHO which

performs the least squares fit. The dependent variable is Y0b s

- Yca^ c and

the independent variables are the derivatives of Yca^ c with respect to each

of the seven parameters. FIX keeps track of which parameters are not being

varied in the fit by keeping track of two sets of parameter indices, one

corresponding to the parameters in LGSTC and one corresponding to the

parameters being fit by ORTHO. FIX also excludes the data that are being

excluded from the fit. Following the fit, it readjusts the data returned by

ORTHO (the coefficients of the least squares fit, their standard deviations

and the variance -covariance matrix) to conform to the indexing of the

parameters in LGSTC.

3.12 ORTHO

ORTHO is called by FIX and performs the least squares fit using a Gram- Schmidt orthonormalization procedure on the dependent and independent

variables. It is derived from a program originally written by Walsh^® and

has been extensively tested by Wampler^. It returns to FIX the corrections

to the varied parameters, their standard deviations, and the

variance -covariance matrix.

3.13 EPARM

EPARM is called by LGSTC and prints the values of the parameters, their

standard deviations, their initial values, and the number of significant

figures in the standard deviations of the parameters that remained unchanged

in the last iteration performed. Thus, if the standard deviation of a parameter was .123 and the convergence index is given as 3, the most recent

correction to the corresponding parameter would be less than .000123. EPARM

also prints the standard deviation of the fit, the number of data fit, the

number of data falling within the transition interval, the relative deviation

(relative to the value of Y at XQ ) , and the Gram Determinant, the determinant

of the least squares matrix X^X.

3.14 DELX

13
DELX is called by the main program following the completion of the analysis.

DELX calculates the range in X in which the logistic function proceeds from

10% completion of the transition region (transition interval) to 90%

completion. DELX also calculates and prints two measures of the asymmetry of

the sigmoidal profile. The first of these is the product QDq

. The second is a parameter r) given by:

n -

- X ) + (X - X ) o o

- X ) - (X - X ) o .1 o (18)

where X ^ is the value of X where the transition is 10% complete and X 9 is

90% complete.

3.15

LNBX

LNBX returns the last non-blank character in a string and is used for display

purposes in SELECT and RDDATA. (LNBX is also called by the program PLOT).

3.16

PLOT

PLOT is the main program for plotting the results of the analysis in LOGIT.

It reads the data generated in an unformatted file by LOGIT and calculates a number of graphs from it including: 1) the original data with vertical 95%

confidence bars, the calculated curve, and 95% confidence curves for the

calculated curves; 2 ) the first derivative of the data (taken as ratios of

differences between adjacent values of X and Y) and the calculated first

derivative; 3) the calculated curve with vertical, 95% confidence bars; 4) a graph normalized to the lower or upper asymptote or to the lowest or highest

value of y; and 5) a graph of the standardized residuals. PLOT controls

which of these graphs appear on the screen through keystroke commands entered

by the user. PLOT also allows "zooming" of plots for examination of narrow

regions of data under higher resolution. A cursor routine allows reading of

calculated and observed values from the various graphs.

3 . 17 LOGPAR

LOGPAR forms strings of parameter names and values for presentation on the

various graphs.

3.18 DPLOT

DPLOT clears the graphics screen, draws the axes and experimental data with

vertical 95% confidence bars for one of the five graphs as directed by PLOT.

If the minimum value of X included in the fit is not the first value, DPLOT

marks the minimum value on the plot. If the maximum value of X included in

the fit is not the last value, DPLOT marks the maximum value on the plot.

DPLOT calculates the format for bottom line information displayed on the

graph consistent with the magnitude of the y vector being plotted. DPLOT

also determines which quadrant of the screen is the emptiest for printing of

the parameter values

.

3.19 CURSOR

14
CURSOR places a cursor on the screen following a change in the selection of

the graph being displayed. Its main function is to match as closely as possible the cursor position when the principal curve being plotted changes

from calculated values to observed values and vice-versa.

3 . 20 PPARM

PPARM writes the parameter name and value strings into the emptiest quadrant

on the screen.

3.21 TITLE

TITLE prints a help message on the top line of the graph and/or a title line

consisting of the title appearing in the first line of the original data

file, the date on which the analysis of the data file was performed, and the

standard deviation of the fit.

4.0 DATA REQUIREMENTS

LOGIT operates on ASCII text files created by the user. The data is in the

form of X,Y pairs where X is the independent variable and Y is the dependent

variable. The data must be in order of increasing X in order for the initial

estimates to be made correctly. The data should also contain at least five

points falling within each of the asymptotic limits if the asymptotic

parameters are to be determined. The first line of the data file may contain

a title up to 30 characters in length which allows for clearer identification

than the 8 character file name can convey. If a title line is not given, its

absence is recognized by the subroutine RDDATA through absence of any letters

in the first line of the file. Alternatively, if the title line contains no

letters, it will be read as data!

Currently three types of data files can be interpreted. The first consists

of X,Y pairs, one value of X and one value of Y per line. The second

consists of one value of X per line, but several values of Y. Each value of

Y is associated with the same value of X. LOGIT will fit any column of the

Y's or any ratio of two columns. For the ratios, appropriate weighting is

used assuming unit weights for each of the individual values of Y. In order

to distinguish which ratio is being analyzed, a second title line is

required . This second line contains a short (up to 8 characters) title for

each value of Y. LOGIT determines the number of Y values by counting the

number of titles in the second line of the data file. The titles are

separated by spaces (any number of spaces). The third type of data file

contains for each value of X a value for Y and a weight, W.

5.0 MISCELLANEOUS CONSIDERATIONS

The current versions of LOGIT and PLOT have been written for IBM XT, AT, and

compatible computers possessing the mathematics coprocessors. It may be

possible to compile the programs and their subroutines with a compiler that

does not require the mathematics coprocessors. However, the graphics

routines will not be realizable. The only departure from Fortran 77 is in

the printing of the iteration number during the least squares fit. The

iteration numbers are preceded by an & which causes the line being printed to

be appended to the last line printed.

15
If the program is unable to fit poorly structured data, numerical overflow or underflow may result. The error messages will not give much of a clue on how

to respond. Generally, one should try again holding more parameters constant

or allowing more parameters to vary. Occasionally, the error message "no

file handles left" may appear. In this case, the number of files that can be

simultaneously open must be increased. This is done in the CONFIG.SYS file

as described in the DOS manuals

.

The graphics programs should work with most displays but there may be

variations in performance depending on the display adapter. The graphics

displays are not in color, but will work with color displays. With some

graphics printers, the graphics displays may be printed by pressing the F10

or Shift-FlO keys. If the printer is incompatible with the graphics

subroutines, a message to this effect is printed at the top of the screen.

6.0 DESCRIPTION OF AN ANALYSIS SESSION

In the following sections, an analysis session is presented as a tutorial.

Section 6.1 deals with the configuration session in which options for the

analysis are selected by the user in response to a series of questions.

Section 6.2 describes the final results provided by the program. Section 6.3

describes the intermediate results. Section 6.4 describes the graphical

editing in LOGIT and section 6.5 the graphical display in PLOT. In sections

6. 1-6.

3

the response from the program is presented in the left hand column

and comments on the interpretation of the response are given to the right.

LOGIT and PLOT are run by entering LOGIT and PLOT from the keyboard. The

user is prompted for additional information as the program progresses.

6.1 THE SELECTION OF OPTIONS

After typing LOGIT, the user enters into an interactive configuration session

which allows the selection of:

which data files are to be analyzed

where the results of the analysis are to be stored or displayed

which parameters are to be varied

which data are to be included in the analysis

whether or not outliers are to be excluded from the fit and if so, using

what criteria

which post-fitting tests are to be applied

whether or not data for a plotting program are to be saved

During the question and answer session, the current state of each option is

presented in square brackets such as [Y] , [N] , [3.00], etc. If a carriage

return (ENTER) is entered in response to the question, the option will remain

in the state indicated in the square brackets. Thus, only changes need be

entered.

16
In general, it is not necessary to engage in an extensive question and answer

session each time LOGIT is run. Any configuration (set of options selected)

can be saved either in a default file (DFLT.OPT) or in a special

configuration file of the user's choosing (e.g. MYFILE.OPT). At the

beginning of a session, the user is given the option of changing any of the

options in the default configuration file or of loading a special

configuration file.

NATIONAL BUREAU OF STANDARDS

LOGISTIC CURVE FITTING PROGRAM, VERSION 6/10/86

RECONFIGURE - Y/N? [N] Y

or

RECONFIGURE - Y/N? [N] MYFILE DO YOU WISH TO ESTABLISH OPTIONS FILE: MYFILE

Title. Updates can be distinguished by the version

date.

Several options are available at this point. Any word of one or two letters beginning with a "y" is interpreted as yes. Any word with 3 or more letters (e.g. MYFILE) is interpreted as a special configuration file (bearing the

file name MYFILE.OPT). That file is read and options

contained in the file become the default options for the current session. If no file of that name exists, the user is prompted with a message asking if an options file

of that name is to be formed.

FINAL RESULTS OF ANALYSIS DIRECTED TO:

[1] CONSOLE

[2] PRINTER

[3] RESULTS FILE

[A] SESSION.LOG FILE

[5] USER FILE

WHICH? ENTER NUMBER 1 -> 5 [3]

If the reconfiguration is selected by entering either yes

or the name of a new options file, the following question

and answer session is entered. The first five options of

the configuration session are offered only during the

configuration session. The remainder of the options may

be changed at various times during the course of the

analysis

.

The results of the analysis can be directed to the

console, the printer, the default files "RESULTS" or "SESSION.LOG" or a file designated by the user. The

final results contain the data with associated statistics

and the determined parameters with associated statistics.

INTERMEDIATE RESULTS OF ANALYSIS DIRECTED TO:

[1] CONSOLE

[2] PRINTER

[3] RESULTS FILE

[4] SESSION.LOG FILE

[5] USER FILE

WHICH? ENTER NUMBER 1 -> 5 [1]

The intermediate results contain the parameters with

associated statistics for each step of the calculation

including the identification and dropping of outliers and

the failure of post-fitting tests.

Most users will elect to display the intermediate results on the console and print the final results or save them

in a results file.

17
IDENTIFY DATA TYPE: Three classes of data can be analyzed.

In the first, values of x and y are analyzed assuming

uniform weights.

In the second, several values of y accompany each value

of x. Each set of y values can be analyzed separately,

or ratios of the y's can be analyzed. The second line of

the data file for this case contains titles for each of

the y values. The number of titles determines the number

of values of y included in the data set.

In the third, values of the weight for each x,y pair are included in the data file.

The first line of each data file should contain a title

up to 30 characters in length. This title will appear on all output displays. If a title line is missing

(determined by the absence of letters in the first line)

the title is given the name of the file. All files are assumed to be ASCII text files. Each line contains one value of x and as many values of y or w as are appropriate for each class described above.

LOCK OPTIONS - Y/N? [N] If this is answered "y" then "LOCK" is turned on and

options will not be offered during the course of the

session. The options entered during the reconfiguration

or existing in the selected configuration file will be in

effect for the remainder of the session. This is useful

if the main program is intended to cycle through a list

of files.

AUTOSAVE DATA FOR PLOTTING - Y/N? [Y] If yes, all of the data necessary for plotting will

automatically be saved following the analysis of each

data file. If no, the user will be prompted (unless "LOCK" is on) to save or not save plotting data. Note,

that LOGIT does not contain plotting routines . Rather it saves a data file (called PLOT.DAT) for a separate

plotting program to use.

[1] X AND Y

[2] SEVERAL Y’S, TAKING RATIOS

[3] X, Y, AND WEIGHTS

WHICH? ENTER NUMBER 1 -> 3 [1]

•k -k Sr *

The above options can be selected only during the reconfiguration session. The

following options may be selected at appropriate times during the remainder of

the session if "LOCK" is off.

* * * st *

18
SELECT PARAMETERS TO BE VARIED - Y/N? [N] If yes, the user is prompted to select which of the

parameters are to be varied in the analysis.

ALLOW A TO VARY - Y/N? [Y] ALLOW B TO VARY - Y/N? [N] ENTER VALUE FOR B [0 .0] ALLOW XO TO VARY - Y/N? [Y] ALLOW DO TO VARY - Y/N? [Y] ALLOW AS TO VARY - Y/N? IN] ALLOW BS TO VARY - Y/N? IN] ALLOW AQ TO VARY - Y/N? [N] ALLOW BQ TO VARY - Y/N? [N] ALLOW Q TO VARY - Y/N? [Y]

The pretransition asymptote obeys an expression of the

form: A + AS(X-XO) + AQ(X-XO) 2

The posttransition asymptote obeys an expression of the

form:

B + BS(X-XO) + BQ(X-XO) 2

XO is the midpoint of the transition where Y“(A+B)/2

DO is the width parameter for the interval. The slope at X * XO is (AS+BS) /2 + (B-A)MDO

Q is a measure of the asymnetry of the interval.

If the answer to the first four parameter questions (A, B, XO, and DO) is no, the user will be prompted to enter

the value of the parameter (note default value in square brackets). The parameter will keep this value during the

analysis. If the answer to the last five parameter

questions (AS, BS, AQ , BQ, and Q) is no, the parameter

will be held at the value of 0.0 during the analysis.

Note that the selection of parameters to be varied and

their initial values are part of the configuration file.

SELECT POST-FITTING TESTS? - Y/N? [NJ

INCLUDE CONVERGENCE TEST - Y/N? [Y] INCLUDE INTERVAL TEST - Y/N? [Y] INCLUDE UNCERTAINTY TEST - Y/N? [Y] INCLUDE ASYMPTOTE TEST - Y/N? [Y]

If yes, the user is allowed to select which of the four

post-fitting tests are to be performed and which are not. Until the user is familiar with the purposes of the postfitting tests, it is advisable to allow all post-fitting

tests to be performed.
IDENTIFY AND REJECT OUTLIERS

OUTLIER REJECTION VALUE:

MAX. NO. OF RETRIES:

- Y/N? [N] Y

[A. 00] 3

[ 0 ] 6

ENTER NUMBER OF ITERATIONS/CYCLE: [11]

If yes is entered, the user is prompted for additional

information. If a carriage return (or any character

other than "n") is entered the current settings for the

outlier rejection criteria are neither changed nor printed to the screen.

The outlier rejection value is the number of standard

deviations by which Y(obs)-Y(calc) must differ from its

expected value of 0 before a particular value of X,Y is considered to be an outlier. In performing this

analysis, the uncertainties in both Y(OBS) and Y(CALC)

are considered. A number of retries may be necessary to identify all of the outliers if several happen to be

grouped together.

If the outlier rejection option is not selected, the

value for the identification (but not rejection) of an outlier is A. 00.

For nearly all cases encountered so far, 11 has proven to be an adequate number of iterations for the convergence

of the least squares procedure. If convergence has not been reached by 11 iterations, the data are generally not

sufficient to support the determination of all of the

parameters selected for variation. For highly precise

data, fewer iterations may be necessary. The parameters

are tested for convergence at the end of each iteration.

If the most recent correction to each of the parameters

is less than 1/1000 the value of its standard deviation,

the process is deemed to have converged and no more

iterations are performed.

*** FINISHED WITH CONFIGURATION OPTIONS *** The options selected may be saved in the default options

file (DFLT.OPT) or in the special options file (e.g.

SAVE THE OPTIONS AS DEFAULTS - Y/N? [N] MYFILE.OPT) selected by the user at the beginning of the

session. If any character other than a "y" is entered,

the options are not saved and apply to the current

analysis session alone.

ENTER DATA FILE NAME: FILENAME. EXT Enter the name of the file containing the data to be

analyzed. If the file cannot be found or the data cannot

be interpreted, the program will terminate with an error message of the following form:

UNABLE TO OPEN OR INTERPRET DATA FILE DATA FILE NAME: FILENAME. EXT DATA TITLE: LINE NUMBER: 0 CHECK DATA FILE FOR PROPER FORMAT

99

20
SELECT OPTIONS - Y/N? [N]

EDIT DATA - Y/N? [N]

SELECT PARAMETERS TO BE VARIED - Y/N? [N] SELECT POST-FITTING TESTS? - Y/N? [N] IDENTIFY AND REJECT OUTLIERS - Y/N? [Y] ENTER NUMBER OF ITERATIONS /CYCLE : [11]

BEGIN ANALYSIS AT 10:01:29.06

*** PLEASE WAIT ***

ITERATION NUMBER: 1 2 3 A 5

END ANALYSIS AT 10:01:47.96

REFIT - Y/N? [N] REINITIALIZE PARAMETERS - Y/N? [N]

SELECT OPTIONS - Y/N? [N]

BEGIN ANALYSIS AT 10:01:59.82

*** PLEASE WAIT ***

ITERATION NUMBER: 12345

END ANALYSIS AT 10:02:18.66

REFIT - Y/N? [N]

The options offered are a subset of the options offered

during reconfiguration (except for the selection of data

to be excluded from the analysis which requires a data

file to have been read) . The options are not offered if "LOCK" is on.

If yes, the data will be graphically displayed and,

through use of the cursor keys, the delete and insert

keys, and several of the function keys, selected data can be tagged to be ignored in the least squares fit. Deleted data will be displayed in the analysis of the

data but will not contribute to the determination of the

parameters. The display can also be zoomed. In section

6.4, the graphical editing of data is described in more

detail.

These options are identical to those offered in the

reconfiguration session. They are offered again in the event the user wishes to depart from the usual default

configuration or if the data warrant special treatment.

Depending of where the intermediate and final results are directed, the results of the analysis may appear on the screen. If both are directed to disk files, only the

begin and end messages appear on the screen. Depending

on the data, the wait can vary from 5 seconds to several

minutes. Printing of the intermediate results to the screen can serve as an assurance that the program is

still working productively. The count of iterations is also displayed to assure the user that the program is still functioning.

This option is presented if "LOCK” is off and allows the user to refit the data selecting different analysis

options. If yes, the user is also asked whether the

parameters are to be reinitialized. If not, the next

round of iterations begins with the current values of the

parameters

.

If the refit option is elected, the user is given the

opportunity to reset some of the options.

21
PLOT THE RESULTS - Y/N? [N] If the option of automatically saving plotting data was not selected in the configuration file, the option is offered for the current data file. Note that the data

are not plotted by LOGIT. Rather, the data necessary for the plotting program are saved. If more than one data

file is analyzed, the plotting data from the analysis of

each data file are saved in succession.

END OF ANALYSIS FOR DATA FILE: TESTDATA. 002 At this point, the user is given the option of analyzing

the data in another file. If yes, the user will be given

all of the options available for earlier data files.

Also, if yes and no entry is made when the user is

prompted to enter a file name, the previous file is assumed. Thus, for data for which ratios are being

analyzed, different ratios may be analyzed by requesting

that another file be analyzed without entering the name

of that file.

ENTER ANOTHER FILE - Y/N? [N]

FINAL RESULTS CAN BE FOUND IN THE FILE: RESULTS When all the desired data files have been analyzed, the

6.2 THE FINAL RESULTS

The final results include the original data; the calculated values of Y and

associated statistics; the values of the determined parameters and their

uncertainties; and statistics related to the overall quality of the least

squares fit. Below, the final results are presented for the analysis of the

data in the file TESTDATA. 001 . The data in this file contain five outliers

grouped near the beginning of the data and do not extend sufficiently into the

pre- and post- transition regions to allow the determination of the slopes of

the asymptotes. During the analysis, while the outliers are being identified,

three of the four post fitting tests are failed at least once. At the start of

the analysis, all parameters are directed to be varied, though, as can be seen

from the results, the slopes of the pre- and post- transition asymptotes cannot

be determined from the data. In all, 14 cycles through the least squares

program are required to arrive at a final analysis.

The final results can be printed to the screen, the printer, the "RESULTS" or "SESSION.LOG" file or to a file designated by the user.

INTERMEDIATE RESULTS CAN BE FOUND IN THE FILE: SESSION.LOG

program ends with a series of messages telling the user where the results can be found if they were written to disk files. The final "1986" indicates that the program

exited properly.

NATIONAL BUREAU OF STANDARDS

LOGISTIC CURVE FITTING PROGRAM, VERSION 6/10/86

1986

22
NATIONAL BUREAU OF STANDARDS

LOGISTIC CURVE FITTING PROGRAM, VERSION 6/10/86

ANALYSIS OF DATA IN TESTDATA. 001 DATE 6/10/86

TIME 18 : 09: 44 . 51

TEST INTERFACE #1

X(OBS) Y(OBS) Y(CALC) DIFF STANDARD

RESIDUALS

4.52 15011. 15094. -82. -1.0

5.92 15011. 15074. -62. -0.7 7.32 14959. 15035. -77. -0.9

8.72 12800. 14966. -2165. -20.1 * 10.12 14990. 14843. 147. 1.7 11.52 14761. 14637. 124. 1.5 3 12.92 14761. 14302. 458. 4.0 ir 14.32 11331. 13788. -2456. -20.3

15.73 13037. 13034. 3. 0.0 21 17.13 13037. 12016. 1021. 8.3 it 18.53 13037. 10739. 2298. 19.4 * 19.93 9157. 9267. -110. -1.4 60 21.33 7642. 7715. -73. -0.9 75 22.73 6192. 6214. -22. -0.3 63 24.13 5024. 4869. 156. 1.8 51 25.52 3878. 3742. 135. 1.5 39 26.93 2865. 2824. 40. 0.5 29 28.32 2150. 2121. 29. 0.3 22 29.72 1449. 1583. -133. -1.5 15 31.13 1123. 1176. -53. -0.6 11 32.62 770. 859. -89. -1.0 8 34.02 520. 639. -119. -1.3 5 35.42 479. 475. 4. 0.0 5 36.82 311. 353. -41. -0.4 3 38.30 267. 256. 11. 0.1 3 39.70 232. 187. 45. 0.5

41.10 167. 135. 32. 0.4 42.50 89. 95. -6. -0.1

43.90 206. 64. 142. 1.6

* NOT INCLUDED IN THE FIT

Program name and version date, title of the file, date

and time of the analysis and the title of the data

contained in the first record of the data file.

A summary of the data. The column labeled DIFF contains

the difference Y(OBS) - Y(CALC). The calculated values

of Y were calculated using the parameters in the table

that follows the data. The standardized residuals are the number of standard deviations by which the

differences, Y(OBS) - Y(CALC), differ from their expected

value of 0. The standard deviation of Y(OBS) - Y(CALC)

includes the contributions from both the observed and

calculated values of Y and include the effects of

correlation when a value of Y is included in the fit. The standardized residuals should follow a student’s t distribution and values greater than 3.00 are suspect.

The numbers appearing to the right of the standardized

residuals are the number of standard deviations by which

the calculated values of Y differ from their nearest

asymptote and, taken together, indicate how well the

interval region can be characterized. Those data not

included in the fit are flagged with an asterisk. (The

values of their standardized residuals - ranging from 4.0 to 20.3 - show clearly that these data are indeed

outliers.) Had additional outliers been found, but not

excluded from the fit, they would have been flagged with

a double asterisk and identified as possible outliers.

The number of significant figures reported for X is determined by the average separation between adjacent X

values. The number of significant figures reported for Y

is determined by the standard deviation of the fit. A

value of Y equal to the standard deviation is reported

with two significant figures.

23
CYCLE 14 ITERATION NO. 5

A - 15113. +/- 62. ( 14994. 3) B * -53. +/- 69. ( 154. 3) XO - 21.498 + /- 0.059 ( 21.454 3) DO - 3.448 +/- 0.057 ( 3.751 3) AS s 0.0 ( 0.0 *> BS - 0.0 ( 0.0 *)

Q

- -0.0315 +/- 0.0066 ( 0.0 3)

* NOT VARIED IN THE ANALYSIS

24 DATA POINTS, 16 IN THE INTERVAL

As indicated, 14 cycles of iterations were performed. On

the last cycle, 5 iterations were required to reach

convergence. This large number of cycles was required

for the identification of outliers and the response to the failing of various post fitting tests. These are detailed in the discussion of the intermediate results in

Section 6.3. Following the values for each parameter are the values of their standard deviations . Following

these, in parentheses, are the starting values for the

parameters. These starting values, if not provided by

the user, are estimated by SUBROUTINE ESTIM8 and the manner of their estimation can be found in the

documentation of that subroutine. The second nunber in

each of the parentheses is roughly the number of digits

beyond the standard deviation that remained unchanged in

the last iteration. Values of 3 or greater indicate

adequate convergence. Values of 1 or 2 are suspect.

Values of 0 cause the convergence test to fail with

subsequent refitting of the data. The parameters AQ and

BQ were not directed to be varied and they do not appear

in the parameter list. LOGIT determined that the

parameters AS and BS could not be evaluated from the data

though they were directed by the user to be varied. Hence they appear in the parameter list.

The transition interval is defined in terms of the

standard deviation of the fit (the precision in the

values of Y) and the magnitude of the separation between

the upper and lower asymptote (A-B). It is the interval

< in which an observed value of y can be expected to fall more than two standard deviations away from either

asymptote. The number of data falling within the

transition interval is a measure of the number of degrees

of freedom for the determination of TO, DO, and Q. If

less than three fall within the interval then not all

three of TO, DO, and Q can be determined.

STANDARD DEVIATION = 98.5 ( 98.5) The number in parentheses represents the standard

deviation calculated with the values of the parameters

just before the last iteration. Convergence is indicated

when these two numbers agree.

RELATIVE DEVIATION = 1.31Z 100 x the standard deviation divided by the magnitude of

a hypothetical value of Y at the midpoint of the

transition interval.

24
GRAM DETERMINANT 0.345497E-01

AT X - 14.73 TRANSITION IS 10. OZ COMPLETE AT X - 30.09 TRANSITION IS 90. OZ COMPLETE

RANGE - 15.37 +/- 0.27 ETA *= -0.1187 +/- 0 . 0245

Q*D0 - -0.1086 +/- 0.0226

The determinant of the least squares matrix. Although

the least squares matrix is never formed in SUBROUTINE

ORTHO, the value of the Gram Determinant is a measure of

the degree of independence between the various

parameters. For data typical of the logistic function,

values in the range of 0.1 to 0.0001 are to be expected.

Values less than 10 -n where n is the number of parameters

varied indicate possible problems with broad minima and

high correlation among the parameters.

These values are calculated by SUBROUTINE DELX and are additional measures of the width and asymmetry of the

transition interval. If X(f) represents the value of X

at which the transition has reached the fraction f of

completion and X(l-f) the value at which X has reached a fraction 1-f of completion, the parameter ETA is given by

ETA = -[(X(l-f)-X0)+(X(f)-X0)]/[(X(l-f)-X0)-(X(f)-X0)]

.

For f*»0.1 and for low values of Q, the product QD0 is approximately equal to ETA. Both ETA and QD0 are dimensionless measures of the asynmetry of the interval.

If Q is held fixed at 0, the values of QD0 and ETA are both zero and their values are not printed.

6.3 THE INTERMEDIATE RESULTS

During the course of the analysis, any of the four post- fitting tests may fail

and/or outliers may be identified and dropped from the fit. When these events

occur, the values of the parameters are printed and the iterative least squares

fit is redone holding certain parameters fixed or dropping outliers from the

fit. The intermediate results contain the parameters for each of the

intermediate stages as well as the final least squares fit in the same format

as they are presented in the final results. In addition, the intermediate

results contain messages about post-fitting tests failed and outliers

discovered.

The intermediate results can be printed to the screen, to a printer, to the

results file, to the session.log file, or to the file designated by the user

for the results file. The intermediate and final results can both be printed

to the same file or to two different files.

Reproduced below are the intermediate results of the analysis of TESTDATA.001

leading to the final results appearing in the preceding section.

25
NATIONAL BUREAU OF STANDARDS

LOGISTIC CURVE FITTING PROGRAM, VERSION 6/10/86

START OF SESSION LOG FOR UNIT 2 DATA FILE NAME: TESTDATA. 001 DATE 6/10/86

TIME 19:12:33.33

Program title, version number, file name, date of

analysis are printed for the session log as they are for

the results file. Timing for the analysis is indicated

by the times at which LGSTC is called and when LGSTC

returns control to the main program.

CALL LGSTC AT 19 : 12: 4*. . 10

CYCLE 1 ITERATION NO. 11

A - 12204

.

+/- 1490. ( 14994. 1) B - 1858. +/- 15944. ( 154. 0) XO - 21.7 + /- 7.6 ( 21.5 0) DO = 1.6 +/- 8.6 ( 3.8 0)

AS = -167. + /- 64. ( 0.0 1) BS = -86. +/- 665. ( 0.0 0) Q

- -0.23 + /- 0.13 ( 0.0 0)

29 DATA POINTS, 9 IN THE INTERVAL STANDARD DEVIATION =764. ( 791.) RELATIVE DEVIATION = 10.872

GRAM DETERMINANT = 0.253846E-05

When the convergence test fails, the program redoes the

calculation using a different approach for calculating

the initial estimates. In the first approach, the

average value of Y is used as the initial estimate of the

midpoint of the transition. In the second approach, the

largest jump in Y is used as the initial estimate of the

midpoint of the transition. Note the last column of

numbers in the parameter list. The 0’s and l's indicate

that the last corrections to the parameters were

comparable to the standard deviations of the parameters.

Note also the lack of agreement of the standard deviation

before and after the last iteration, again indicating

lack of convergence.

CONVERGENCE TEST FAILED

CYCLE 2 ITERATION NO. 11 Again, the analysis fails to converge. The next step is

v to conduct the analysis holding AS and BS = 0 . If this

A = 9436. +/- 3511. ( 14994. 0) too fails, then the analysis will be repeated holding

B = -770. +/- 2156. ( 154. 0) 0=0. If this fails, the analysis ends. X0 = 24.2 +/- 1.9 ( 21.4 0) DO = 2.14 + /- 0.67 ( 2.59 0) AS = -314. +/- 213. ( 0.0 0) BS = 34. +/- 108. ( 0.0 0) Q

= -0.22 + /- 0.17 ( 0.0 0)

29 DATA POINTS, 6 IN THE INTERVAL STANDARD DEVIATION =783. ( 810.) RELATIVE DEVIATION = 18.072

GRAM DETERMINANT = 0.255142E-04

CONVERGENCE TEST FAILED REFIT HOLDING AS AND BS = 0

26
CYCLE 3 ITERATION NO. 6

A 14558. +/- 392. ( 14994. 3) B

= 120. +/- 437. ( 154. 3) XO - 22.04 +/- 0.37 ( 21.45 4) DO - 3.03 +/* 0.34 ( 3.75 3) AS - 0.0 ( 0.0 *) BS - 0.0 ( 0.0 *> Q

- -0.027 +/- 0.050 ( 0.0 3)

* NOT VARIED IN THE ANALYSIS

29 DATA POINTS, 8 IN THE INTERVAL STANDARD DEVIATION = 788. ( 788.) RELATIVE DEVIATION = 10.74Z

GRAM DETERMINANT = 0.665502E-01

THE FOLLOWING HAVE BEEN IDENTIFIED AS OUTLIERS

N X(OBS) Y(OBS) Y(CALC) DIFF STANDARD

RESIDUALS

This time the analysis converges in 6 iterations. One outlier is found so the analysis is repeated with the

outlier dropped from the least squares fit and with all

parameters varied. If no outlier had been found, the analysis would have been complete. Note the agreement

between the two estimates of the standard deviation.

8 14 . 32 11331. 13762. -2431. -3.4

REPEAT, EXCLUDING ABOVE DATA FROM ANALYSIS

CYCLE 4 ITERATION NO. 11 Again the procedure does not converge. The analysis is restarted using the largest jump in Y as the measure of A = 3401. +/- 740. ( 14994. 0) the midpoint of the transition interval. Note that the

B - 1002. +/- 961. ( 154. 0) fit appears to be worse than following the first

X0 = 29.02 +/- 0.70 ( 21.45 *) iteration.

DO < 0.70 ( 3.75 *> AS = -571. + /- 51. ( 0.0 0) Note that LGSTC concluded that no data fell within the BS * -68. + /- 105. ( 0.0 0) interval and consequently held the values of X0 , DO, and

Q

- 0.0 ( 0.0 *) Q fixed at predetermined values depending in part on the structure of the data. Also note the large difference

* NOT VARIED IN THE ANALYSIS between initial and final values of the parameters.

28 DATA POINTS, 0 IN THE INTERVAL Repeat the fit using the largest jump in Y as the initial

STANDARD DEVIATION = 1528. ( 1555.

)

estimate of X0. RELATIVE DEVIATION

GRAM DETERMINANT

69.43Z

0.599094E-01

CONVERGENCE TEST FAILED

27
CYCLE 5 ITERATION NO. 11

A = 3A01

.

+/- 7A0 . ( 1A99A

.

0) B = 1002. +/- 961. ( 15A

.

0) X0 - 29.02 +/- 0.70 ( 21. A5 *) DO < 0.70 ( 3.75 *)

AS = -571. +/- 51. ( 0.0 0) BS - -68. +/- 105. ( 0.0 0)

Q

- 0.0 ( 0.0 *)

* NOT VARIED IN THE ANALYSIS

28 DATA POINTS, 0 IN THE INTERVAL STANDARD DEVIATION = 1528. ( 1555.) RELATIVE DEVIATION - 69.43Z

GRAM DETERMINANT - 0.599094E-01

CONVERGENCE TEST FAILED

REFIT HOLDING AS AND BS = 0

CYCLE 6 ITERATION NO. 7

A = 1A579 . +/- 222. ( 14994. 4) B = -176. +/- 379. ( 154. 4) Xo u 21.95 +/- 0.27 ( 21.45 4) DO = 2.88 +/- 0.27 ( 3.75 3) AS = 0.0 ( 0.0 *) BS *= 0.0 ( 0.0 *)

Q

= -0.093 +/- 0.042 ( 0.0 3)

* NOT VARIED IN THE ANALYSIS

28 DATA POINTS, 12 IN THE INTERVAL STANDARD DEVIATION =572. ( 572. ) RELATIVE DEVIATION = 7.94Z

GRAM DETERMINANT = 0.581992E-01

THE FOLLOWING HAVE BEEN IDENTIFIED AS OUTLIERS

N X(OBS) Y(OBS) Y (CALC) DIFF STANDARD

RESIDUALS

4 8.72 12800. 14578. -1778. -3.

A

8 1A .32 11331. 1A321. -2989. -A.

9

As in the second cycle, the procedure fails to converge

and so again AS and BS are held constant.

Note that both procedures for calculating the initial

estimates of the parameters gave the same starting

values

.

The procedure converges and a second outlier is discovered. Note that the standardized residual of the

first outlier which had been excluded from this fit has

increased in magnitude from 3. A to A. 9. The analysis is now repeated dropping both data from the least squares

fit

.

REPEAT, EXCLUDING ABOVE DATA FROM ANALYSIS

28
CYCLE 7 ITERATION NO. 11 The fit is repeated allowing all parameters to vary. This time the procedure converges but the analysis has

A - 5890. +/- 712. ( 14994. 6) concluded that the transition interval is extremely

B = 1845. +/- 786. ( 154. 7) narrow because no data fall within the transition

XO = 26.23 +/- 0.70 ( 21.45 *) interval. However a count of the mxnber of data whose

DO < 0.70 ( 3.75 *) standard deviations overlap the midpoint of the

AS - -526. +/- 56. ( 0.0 6) transition (NMP =3) indicate that the interval may not

BS - -118. +/- 73. ( 0.0 7) be narrow and that the sloping baselines may be

Q

- 0.0 ( 0.0 *) acconroodating a broad transition region. Hence, the fit

is repeated holding AS and BS * 0

.

* NOT VARIED IN THE ANALYSIS

(Note that when the analysis concludes that no data fall

27 DATA POINTS, 0 IN THE INTERVAL STANDARD DEVIATION = 1372. ( 1372.) RELATIVE DEVIATION * 35.46Z

GRAM DETERMINANT - 0.644768E-01

within the transition interval the maximum value for DO

is estimated and the uncertainty in XO is taken from this

value of DO .

)

1WP = 3 . INTERVAL, TEST FAILED REFIT HOLDING AS AND BS == 0

CYCLE 8 ITERATION NO. 6

A = 14928. +/- 186. ( 14994. 3) B = -98. +/- 275. ( 154. 3) X0 = 21.83 + /- 0.19 ( 21.45 4) DO = 2.97 +/- 0.19 ( 3.75 4) AS = 0.0 ( 0.0 *) BS = 0.0 ( 0.0 *)

Q

- -0.069 +/- 0.029 ( 0.0 3)

* NOT VARIED IN THE ANALYSIS

27 DATA POINTS, 13 STANDARD DEVIATION RELATIVE DEVIATION

GRAM DETERMINANT

IN THE INTERVAL =413. ( 413.

)

= 5.572

= 0.527857E-01

The procedure converges with a greatly improved value for

the standard deviation and relative deviation. Moreover,

the interval is no longer deemed to be narrow.

A third outlier is discovered as the magnitudes of the

first two outliers identified continue to increase.

THE FOLLOWING HAVE BEEN IDENTIFIED AS OUTLIERS

N X(OBS) Y(OBS) Y (CALC) DIFF STANDARD

RESIDUALS

4 8.72 12800. 14921. -2121, -4.7

8 14.32 11331. 14440. -3109, -6.8

11 18.53 13037. 11594. 1443 4.1

REPEAT, EXCLUDING ABOVE DATA FROM ANALYSIS

29
CYCLE 9 ITERATION NO. 8 The fit is repeated dropping the three outliers, and the

A C 14808. +/- 708. ( 14994. 3)

parameters AS and BS are again allowed to vary. This

time, the asymptote test fails. The asymptote test

B IB -3595. + /- 5235. ( 154. 4) requires that the value of the slope must be greater than

XO = 23.3 + /- 2.9 ( 21.5 4) three times its standard deviation if less than five

DO — 4.5 +/- 2.3 ( 3.8 4) points fall within two standard deviations of the

AS BE -11. +/- 42. ( 0.0 3) asymptote. If fewer than five data fall within two BS = 136. +/- 186. ( 0.0 4) standard deviations of an asymptote, the magnitude of the

Q

= -0.093 + /- 0.039 ( 0.0 4) slope is compared with its standard deviation. The

26 DATA POINTS, 16 IN THE INTERVAL STANDARD DEVIATION *=202. ( 202.) RELATIVE DEVIATION = 3.61Z

GRAM DETERMINANT - 0.201439E-06

magnitude of the elope must be at least three times as large as its standard deviation if it is to be judged to be significantly different from zero. When the asymptote

test fails, the fit is repeated holding the offending

slope at 0.

NU - 0 & NS6 - 0. HIGH ASYMPTOTE TEST FAILED REFIT HOLDING SLOPE ( S ) = 0

CYCLE 10 ITERATION NO. 4 A fourth outlier is discovered as the magnitudes of the

standard residuals of the previously identified outliers A = 15467. + /- 877. ( 14994. 3) continue to increase. From this point on, the asymptote

B = -96. +/- 165. ( 154. 3) tests continue to fail and one more outlier is Xo

11 21.43 +/- 0.41 ( 21.45 3) discovered. The analysis is finally complete after 14 DO = 3.28 +/- 0.13 ( 3.75 3) cycles. The large number of cycles required for AS = 29. +/- 59. ( 0.0 3) convergence resulted from a combination of exacerbating

BS = 0.0 ( 0.0 *) conditions: the clustering of the outliers, the very

Q

= -0.045 +/- 0.022 ( 0.0 3) large errors in most of these outliers and the inability

of the data to support the determination of the slopes of

* NOT VARIED IN THE ANALYSIS the asymptotes.

26 DATA POINTS, 15 STANDARD DEVIATION RELATIVE DEVIATION

GRAM DETERMINANT

IN THE INTERVAL

= 210 . ( 210 .

)

= 2.73Z = 0.507217E-03

THE FOLLOWING HAVE BEEN IDENTIFIED AS OUTLIERS

N X(OBS) Y(OBS) Y (CALC) DIFF STANDARD

RESIDUALS

4 8.72 12800. 15023. -2223 -9.4

8 14.32 11331. 14170. -2839,. -11.9

10 17.13 13037. 12406. 631 3.9

11 18.53 13037. 11049. 1988 8.1

REPEAT, EXCLUDING ABOVE DATA FROM ANALYSIS

30
CYCLE 11 ITERATION NO. 7

A = 15910. +/- 676. ( 14994. 3) B = -2390. +/- 1330. ( 154. 4) Xo H 22.18 +/- 0.88 ( 21.45 4) oo H 4.15 +/- 0.41 ( 3.75 5) AS “ 54. +/- 43. ( 0.0 3) BS - 94. + /- 50. ( 0.0 4)

Q

- -0.056 +/- 0.020 ( 0.0 4)

25 DATA POINTS, 19 IN THE INTERVAL STANDARD DEVIATION - 93.7 ( 93.7) RELATIVE DEVIATION = 1.392

GRAM DETERMINANT - 0.183880E-06

NL = 4 & NS5 = 1. LOW ASYMPTOTE TEST FAILED

NU « 0 & NS6 - 1. HIGH ASYMPTOTE TEST FAILED REFIT HOLDING SLOPE(S) = 0

CYCLE 12 ITERATION NO. 6

A = 15107. +/- 75. ( 14994. 3) B = -111. +/- 94. ( 154. 4) X0 = 21.514 +/- 0.075 ( 21.454 3) DO = 3.404 + /- 0.074 ( 3.751 3) AS = 0.0 ( 0.0 *) BS = 0.0 ( 0.0 *)

Q

- '-0.0411 + /- 0.0082 ( 0.0 3)

* NOT VARIED IN THE ANALYSIS

25 DATA POINTS, 16 IN THE INTERVAL STANDARD DEVIATION =130. ( 130.) RELATIVE DEVIATION = 1.732

GRAM DETERMINANT = 0.354953E-01

THE FOLLOWING HAVE BEEN IDENTIFIED AS OUTLIERS

N X(OBS) Y(OBS) Y (CALC) DIFF STANDARD

RESIDUALS

4 8.72 12800. 15011. -2210. -15.6

7 12.92 14761. 14425. 336. 3.0

8 14.32 11331. 13927. -2596., -16.9

10 17.13 13037. 12132. 905. 5.7

11 18.53 13037. 10813. 2224. 14.4

REPEAT, EXCLUDING ABOVE DATA FROM ANALYSIS

CYCLE 13 ITERATION NO. 6

A as 16049. +/- 620. ( 14994. 3) B s -1471. +/- 733. ( 154. 3) XO s 21.70 +/- 0.58 ( 21.45 3) DO - 3.93 +/- 0.17 ( 3.75 3) AS EE 62. +/- 38. ( 0.0 3) BS = 61. +/- 28. ( 0.0 3)

Q

- -0.039 +/- 0.016 ( 0.0 3)

24 DATA POINTS, 20 IN THE INTERVAL STANDARD DEVIATION = 64.2 ( 64.2) RELATIVE DEVIATION = 0.882

GRAM DETERMINANT = 0.200482E-06

NL = 3 & NS5 = 1. LOW ASYMPTOTE TEST FAILED

NU = 0 & NS6 = 2. HIGH ASYMPTOTE TEST FAILED

REFIT HOLDING SLOPE(S) = 0

CYCLE 14 ITERATION NO. 5

A = 15113. +/- 62. ( 14994. 3) B = -53. +/- 69. ( 154. 3) X0 = 21.498 + /- 0.059 ( 21.454 3) DO = 3.448 +/- 0.057 ( 3.751 3) AS = 0.0 ( 0.0 *) BS = 0.0 ( 0.0 *)

Q

= '-0.0315 + /- 0.0066 ( 0.0 3)

* NOT VARIED IN THE ANALYSIS

24 DATA POINTS, 16 IN THE INTERVAL

STANDARD DEVIATION = 98.5 ( 98.5) RELATIVE DEVIATION = 1.312

GRAM DETERMINANT = 0.345497E-01

THE FOLLOWING HAVE BEEN IDENTIFIED AS OUTLIERS

N X(OBS) Y(OBS) Y (CALC) DIFF STANDARD

RESIDUALS

4 8.72 12800. 14966. -2165. 1 to o

7 12.92 14761. 14302. 458. 4.0

8 14.32 11331. 13788. -2456. -20.3

10 17.13 13037. 12016. 1021. 8.3

11 18.53 13037. 10739. 2298. 19.4

RETURN FROM LGSTC AT 19:15:21.02

* •k * *

After all outliers have been identified and the determinable parameters

evaluated, the final values of the parameters and the list of outliers are

printed. This is the same table as that appearing in RESULTS.

31
H-K-MOUE

CURSOR]

[DEL-DELETE

POINT]

[INS-INCLUDE

POINT]

[F5-MARK1

[END]

32

Figure

2.

Data

Editing

Session
6.4 GRAPHICAL EDITING OF DATA

Before analysis, the data may be inspected; obvious outliers may be rejected;

and/or blocks of data may be selected for analysis. In addition, the

pretransition, transition, and posttransition regions may be marked to aid in

the initial estimation of the parameters. If, during the options session, the

user enters yes to the prompt: EDIT DATA - Y/N?, the screen will be cleared and

the data will be displayed in a manner similar to that shown in Figure 2.

The cursor position is marked with a pair of vertical bars and the direction of

cursor movement during deletion or insertion of data is indicated by an arrow

above or below the cursor. On the bottom line is given the x and y values of

the current cursor position and an indication of whether the datum at the

cursor is to be included ("IN") in or excluded ("OUT") from the fit. If the

delete key is pressed, the datum is marked as excluded and the cursor moves in

the direction indicated. If the insert key is pressed, the datum is marked as included and the cursor moves in the direction indicated. Points included in

the fit appear on the graph marked with "o" while points excluded are marked

with "+" . On the top line is an abbreviated menu to aid the user in key use:

[<->:MOVE CURSOR] [DEL: DELETE POINT] [INS: INCLUDE POINT] [F5:MARK] [END]

The function key F5 is used to mark two data values. Moving the cursor to a particular point on the graph and pressing F5 causes a double vertical bar to

be placed at that point and the help message at the top of the screen to be

changed to

:

[<-> :MOVE CURSOR] [DEL] [INS] [F5:MARK] [ SHI FT- F5 : UNMARK] [END]

with the Shift-F5 key now identified as the key to erase markers. When a second marker is placed on the screen, the help message changes to:

[ F5 : ZOOM] [ SHI FT- F5 : UNMARK] [F6: DELETE BLOCK] [SHIFT-F6: INCLUDE BLOCK] [END].

The delete and insert keys are still operative, but in addition the function

key F6 will mark all points at and between the markers as excluded from the fit

while Shift-F6 will mark all points at and between the markers as included in

the fit. If F5 is pressed again, the display will be "zoomed" to the region

including and between the markers. If the second and first marker are

coincident on the display, the display will be zoomed to half the width of the

current display with the marked datum as close to the center of the display as possible. If either the END or ENTER keys is pressed while two markers appear

on the screen, the markers are used to delimit the pretransition, the

transition and the posttransition regions. Subroutine ESTIM8 then uses these

markers to draw straight lines through each of the three regions as a basis for

the initial estimates of the parameters.

If the display is zoomed, the help message is changed to:

[<->: MOVE CURSOR] [DEL] [INS] [F5:MARK] [ SHI FT - F5 : UNZOOM] [END],

and the markers are erased. Thus, the editing session can be exited without

worry that unseen markers are marking the transition regions. The "zoomed"

display can be returned to the original display by pressing Shift-F5. As with

33
the original display, the cursor movement keys, insert and delete keys and the

function key F5 have their defined function. That is, while in a zoomed

display, one can mark regions and delete or insert the marked blocks of data,

delimit the transition regions, or zoom further.

6.5 PLOT

Once an analysis is completed, the results of the analysis can be graphically

examined with the program PLOT. PLOT works with data files generated by LOGIT.

On running PLOT, the user is prompted only to enter the limits of X values for

the display:

ENTER VALUES FOR LOWER AND UPPER LIMITS OF X

TO BE PLOTTED. PRESSING [ENTER] ALONE ACCEPTS DEFAULT VALUE

ENTER LOWER LIMIT FOR X: [ 2.0]

ENTER UPPER LIMIT FOR X: [ 45.0]

CALCULATING . . .PLEASE WAIT

As with LOGIT, default values, based on the range in X values, are displayed in

brackets. After the limits are entered, PLOT requires a few seconds to

calculate all of the calculated curves to be plotted. When complete, the

screen is cleared and a display similar to Figure 3 appears.

As in the editing of data, the options available for inspection of the data are

summarized in a help line at the top of the screen. Appearing in the display

are the original data. Data included in the fit are displayed with the symbol

"o" and data excluded with the symbol At each datum is a vertical bar

corresponding to the 95% confidence intervals for that measurement. For

uniform weighting, the vertical bar is just the standard deviation multiplied

by the one-sided 97.5 percentile values for the t distribution. (For more than

30 degrees of freedom, the 97.5 percentile values for the t distribution are

taken to be ± 2.) The solid curve is the calculated curve and it is flanked by

two dotted curves representing the 95% confidence interval centered vertically

about the calculated curve. A vertical line is drawn at the midpoint of the

transition. The values of X(min) and X(max) printed at the bottom of the

display are the minimum and maximum values of X included in the fit. If these

are other than the first and last values of X in the data set, they are marked

with arrows in the display. If the ENTER key is pressed, the help line at the

top is replaced with a title line containing the title from the title line of

the data, the date on which the numerical analysis was performed, and the

standard deviation of the fit.

Printed on the graph are the values of the varied parameters along with their

standard deviations.

Several options are available for additional inspection of the data.

If F2 is pressed, the derivative of Y with respect to X is displayed. The

experimental points in the derivative plot are the first differences of the

data and the solid curve is the derivative calculated from the values of the

parameters

.

34
tFl•

PLOT 1

tF2

-DERIU3

[F3-FLAT]

[F4-N0RM]

[F9-ERR]

[F10-PRINT]

[HOME-CURS]

[END]

1575? p1

Ill I

JULIZZt

-- -i,,, A=

14734. ±

349. ln

TT

cr» co

rt*X

it! £

i_n

TVn

rz •H£

•wX

Csj

35

Figure

3.

Data

Analysis

Plot
If F3 is pressed, the calculated curve is plotted as the primary curve with

error bars representing 95% confidence intervals. The density of calculated

values on the screen is such that the plot presents a grey band across the

screen. The observed points are also placed on the screen but without

reference to whether they were included in the fit or not. Using the cursor,

extrapolated and interpolated values are obtained from the graph and printed on

the bottom line of the display.

If F4 is pressed, the data are normalized and plotted. The user is given four

choices for normalization:

NORMALIZE PLOT ON:

[1] UPPER ASYMPTOTE

[2] LOWER ASYMPTOTE

[3] MAXIMUM VALUE OF Y

[4] MINIMUM VALUE OF Y

WHICH? ENTER NUMBER: [1]

FOR THE NORMALIZED DATA, Y(min) - 0.0178

Y(max) = 0.9985

FOR THE GRAPH:

ENTER THE MINIMUM VALUE OF Y [ 0.0178] 0

ENTER THE MAXIMUM VALUE OF Y [ 0.9985] 1

After the datum or parameter that will establish the normalization constant

(normalized to 1) has been selected and the maximum and minimum normalized

values of Y to be displayed have been entered, the data are graphed. The

normalization constant, X range and normalized Y range then remain constant for

all additional data files analyzed in the analysis session. In this way,

measurements on different quantities of materials can be compared in a systematic manner.

If F9 is pressed, the standardized residuals are plotted. The standardized

residuals for the data depicted in Figure 3 are shown in Figure 4. The

systematic variation of the standardized residuals throughout most of the range

along with the clustering of most of the standardized residuals near zero are

characteristic of the effects of a few outliers. The horizontal, dotted lines

represent the 95% confidence limits.

If F10 is pressed and the computer is connected to a compatible graphics

printer, the help line is replaced with the title line and the screen is

printed. If the printer is not compatible, the message:

PRINTER NOT READY, NOT AVAILABLE, OR NOT COMPATIBLE - ANY KEY TO CONTINUE

appears on the top line.

A few cursor-related functions are also available. In contrast to the data

editing routine, the cursor must first be turned on by pressing the HOME key.

(HOME also turns the cursor off when it is on.) The cursor appears at the

leftmost point on the display and the help line is changed to read:

36
TEST

INTERFACE

#1

DATE:

05/30/86

STANDARD

DEVIATION =

777.

ITi TT

CTi ronx

ITS x

Ln

iicwX

CnJ

37

Figure

4.

Standardized

Residuals
[F1-F4 : PLOT] [ F9 : ERR ] [HOME:CURSOR OFF) [<-->:CURSOR MOVE] [F5:MARK] [END]

As with the data editing routine, the cursor is moved with the left and right

arrow keys and the F5 key is used for marking regions of the data for zooming.

Only one marker can be placed on the screen. To zoom to a particular region,

place the marker at one end of the region and the cursor at the other end.

Pressing F5 will cause the display to be zoomed to the region between the

marker and the cursor. Shift- F5 is used to unmark a marker or to restore a zoomed plot to its original display range.

7.0 TEST DATA

Six data files, TESTDATA.001 - TESTDATA.006 accompany the program and may be

used to evaluate program performance. The results of analyses of these test

data are displayed in Figures 5-10.

TESTDATA.001 contains data from a depth profile analysis of an interface

between chromium and nickel. It has been discussed in the preceding text and

is included as an example of outlier identification and asymptotic slope

indeterminacy. In analyzing TESTDATA.001, the asymptotic slopes AS and BS and

the asymmetry parameter Q should all be allowed to vary. A maximum of 6

retries for outlier rejection should be selected and the outlier rejection

value should be set at 3.0. LOGIT should be able to identify the five outliers

and should determine that the slopes AS and BS cannot be evaluated from the

data. In Figure 5. the results of the analysis are presented along with a graph of the standardized residuals. Note the random scatter in the

standardized residuals. Also, the standardized residuals of the outliers

appear on the top and bottom line of the graph.

38
Figure 5. Results of Analysis of TESTDATA.001

TEST INTERFACE #1 DATE: 05/30/86 STANDARD DEVIATION =98.5

TEST INTERFACE #1

5

DATE: 05/30/86 STANDARD DEVIATION = 98.5

1

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39
TESTDATA.002 represents synthetic data corresponding to a broad transition

interval to which random, normal deviates amounting to a relative standard

deviation of 20% have been added. The broad profile is difficult to

distinguish from a profile with an extremely narrow transition and asymptotes

with pronounced slopes. Thus, in the top graph of Figure 5 is the result of an

analysis in which the seven parameters A, B, XQ , D0 , As , Bs , and Q are all

varied and the post-fitting, interval test is not performed. LOGIT deduced

that the transition interval was narrow because no data fell more than two

standard deviations away from its nearest asymptote. Hence, Q was set equal to

0 (its value is missing from the parameter list), TQ at the midpoint of the two

data values spanning the intervals, and D0 at a value dependent on the standard

deviation of the fit and the separation between adjacent data points. When the

post-fitting interval test is performed, the "erroneous" assumption that no

point lies in the interval is uncovered and the fit is repeated holding As and

B s fixed at zero. The results of this second fit appear in the lower graph of

Figure 6. When the fit is repeated allowing As and B s to vary but holding Q=0

(not pictured) , a fit of nearly identical quality to that in the lower graph of

Figure 6 is obtained but with substantially different parameters. In this

case, the fit cannot be distinguished from the fit appearing in the lower graph

of Figure 6, and criteria other than the quality of the fit must be used to

select between the two.

40
TEST INTERFACE #2 DATE: 06/11/86 STANDARD DEVIATION = 202.

TEST INTERFACE #2 DATE: 06/11/86 STANDARD DEVIATION = 176.

41
TESTDATA. 003 , is the error function. It is included as a test data file in

order to demonstrate the closeness of overlap between the logistic function and

the error function and as a demonstration of the effects of systematic error on

the standardized residuals. Although the fit of the data appears to be quite

good, the systematic nature of the deviations becomes quite apparent in the

standardized residuals.

42
ERROR FUNCTION DATE: 05/30/86 STANDARD DEUIATION = 0.00097

ERROR FUNCTION DATE: 05/30/86 STANDARD DEUIATION = 0.00097

43
TESTDATA. 004 , is a small set of toxicity data. The data set contains the

number of organisms tested and the number of organisms killed as a function of

the concentration of a known toxicant. In the fit of these data, the ratio of

the number killed to the number tested is analyzed (Data Type 2 in the

configuration session) . The pretransition asymptote should be held fixed at 0

and the posttransition asymptote at 1.0. This set of configurations is

contained on the configuration file LC50.0PT which is included with the data.

Therefore, typing LC50 in response to the question "RECONFIGURE Y/N? [N]" with

LC50 will invoke this configuration. Alternatively, the appropriate options

may be entered from the keyboard. Note that in this data set and the next,

TESTDATA. 005 , when ratios are being tested, the least squares fit is a weighted

fit with weights:

wi - Vdi

2 /(l+ri 2

)

' '

where is the i^ value of Y in the denominator of the ratio r^. The 95%

confidence intervals are seen to be quite large in keeping with the 3 degrees

of freedom. Figure 8 consists of 1) the standard graphical display of the data

and their error bars and the calculated curve and its 95% confidence limit, and

2) a plot of the calculated curve with 95% confidence error bars associated

with each calculated point, the observed values, and a cursor placed at X=6.52.

44
LC50 DATA KILLED/EXPOSED DATE: 06/10/86 STANDARD DEVIATION =2.45

LC50 DATA KILLED/EXPOSED DATE: 06/10/86 STANDARD DEVIATION =2.45

45
TESTDATA. 005 , is a phase transition curve for an ensemble of membrane bilayers

composed of dipalmitoylphosphatidylcholine (DPPC) . In this analysis, the

ratios of Y values labeled as 663 and 627 were analyzed. The transition is

very sharp and the width of the interval cannot be determined (though it can be

determined for the 575/627 and 575/663 ratios.) Figure 10 depicts the zoomed

region around the transition for TESTDATA. 005 . It can be seen that the 95%

confidence intervals for the posttransition data are larger than those for the

pretransition data. This is because the errors in the ratios are not normally

distributed, which the errors in the primary data are assumed to be. Hence the

fit was a weighted fit with each ratio being weighted in proportion to the

inverse square of its uncertainty as described for TESTDATA. 004 . Figure 9

includes a standard graphical display of the original data with the calculated

curves and also a "zoomed" graph of the cluster of points near the transition

region.

46
DPPC 3/31/86 663/62? DATE: 05/30/86 STANDARD DEVIATION = 0.00511

DPPC 3/31/86 663/627 DATE: 05/30/86 STANDARD DEVIATION = 0.00511

47
TESTDATA.006 contains population figures for the U.S. in decade intervals since

1790. It is an extension of the data originally analyzed by Pearl and Reed^

but to which data from 1920 through 1970 have been added. This data was

analyzed allowing the asymptotes to vary but holding the slopes of the

asymptotes fixed at zero. It serves as a demonstration of the extrapolation

capability of the analysis. Also, the plot of the standardized residuals in

Figure 10 shows a trend suggestive of the limitations of the model.

48
U.S. population 1790-1980

5

DATE: 06/10/86 STANDARD DEUIATION = 4.46

€

3

ftm

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0

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0

1 1 1 1 L_ 1 1 1 1

1 1740 X(min) = 1790. X(nax) = 1980.

Figure 10. Results of Analysis of TESTDATA.006

2170

49
REFERENCES

1. Verhulst, P.F. Acad. Brux. , 18, 1-38 (1845).

2. Pearl, R. and Reed, L. J., Proc . Nat. Acad. Sci., 6, 275-88 (1920).

3. Oliver, F. R. , Applied Statistics, 13, 57-66 (1964).

4. Berkson, J., Biometrics, 7, 327-339 (1951).

5. Reed, L. J. and Berkson, J., Journal of Physical Chemistry, 33. 760-779

(1929).

6. Kirchhoff, W.H. , Levin, I.W., and Hancock, A. J., Biophysical Journal,

45, 196a, (1984).

7. Kirchhoff, W. H. , Chambers, G. P., and Fine, J., J. Vac. Sci. Technol.

(Series A, in press, 1986).

8. Johnson, N. L. and Kotz , S. Distributions in Statistics : Continuous

Univariate Distributions, Houghton Mifflin Co., Boston, 2, Chap. 22

(1970).

9. Wilson, Jr., E. B., An Introduction to Scientific Research, McGraw-Hill

Book Co., New York, 234-46, (1952).

10. Walsh, P. J., Algorithm 127, ORTHO, Communications of the ACM, f>, 511-

516, (1962).

11. Wampler, R. H., J. Res. Nat. Bur. Stds., 73B . 59-90 (1969).

50
APPENDIX A

PROGRAM AND SUBROUTINE LISTINGS

The following pages contain the listings of the programs LOGIT and PLOT and

their subroutines. The subroutines appear in the order in which they are

called. SUBROUTINES DERIV, KEY, AND LNBX are called by both LOGIT and PLOT and

are not duplicated following the listing of PLOT. Note that although LOGIT is

the name of the executable program, the main program is stored in the file

MAIN. FOR. The program PLOT and SUBROUTINE SELECT in LOGIT call several

plotting library functions which are not generally available. If these

programs are to be adapted to a new computing environment, new plotting

routines will be required. Many plotting libraries are similar in their calls

and functions. Therefore, the listing of PLOT has beem included as a guide

since it may be possible to revise the program by little more than replacing

the calls to the various plotting routines.

The programs and subroutines have been written in FORTRAN 77 and contain

extensive comments. The main programs and subroutine OPTS contain input and

output statements of the form READ (*,...) and WRITE (*,...) with the intention

that the input is entered from the keyboard and the output appears on the

screen. For seme installations, the default input and output devices are not

the keyboard and screen, but a primary disk. If the program does not accept

input from the keyboard and/or print prompts to the screen (as outlined in

APPENDIX A) , the READ (*,...) statements should be replaced with READ (LO, . .

.

)

and the WRITE (*,...) with WRITE (LPO,...) (Note, LMzero" and IP"zero") . The

integer variable LO has been reserved as the keyboard input unit and LPO has

been reserved as the screen output unit. Both LO and LPO are defined at the

beginning of the main program. It may be necessary to redefine LO and LPO to

coincide with a particular installation's standard input and output units.

Draft Document for Review Purposes Only
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u u u u u u u u
NBS«1 14A (rev. 2»6C) U.S. DEPT. OF COMM.

BIBLIOGRAPHIC DATA

SHEET (See instructions)

1. PUBLICATION OR

REPORT NO.

NISTIR-88-3803

2. Performing Organ. Report No. 3. Publication Date

MARCH 19B9

4. TITLE AND SUBTITLE

Logistic Function Data Analysis Program: LOGIT

5. AUTHOR(S)

William H. Kirchhoff

6. PERFORMING ORGANIZATION (If joint or other than NBS. see instructions)

NATIONAL BUREAU OF STANDARDS

U.S. DEPARTMENT OF COMMERCE

GAITHERSBURG, MD 20899

7. Contract/Grant No.

I. Type of Report & Period Covered

9.

SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS (Street. City, Stole, ZIP)

10.

SUPPLEMENTARY NOTES

| 21 Document describes a computer program; SF-185, FIPS Software Summary, is attached.

11.

ABSTRACT (A 200-word or less factual summary of most significant information. If document includes a significant

bi bliography or literature survey, mention it here)

A FORTRAN program has been written for the statistical analysis of experimental

data in terms of an extended logistic function that includes non-horizontal asymptotes

and asymmetry in the pre- and post-transition portions of growth and decay curves.

The program is robust in that situations in which few or no data fall within the transition interval can be analyzed by the program. Individual weighting of the data is

allowed for situations where the errors in experimental data are not uniform. The

primary parameters describing the transition region include a location parameter, X , a width parameter, D , and an asymmetry parameter Q. Six more parameters describe

the two quadratic asymptotic regions. The prgram does not require initial estimates

for the parameters and provides statistical estimates of all parameters. A provision

is included for the identification and exclusion of outliers. Sufficient information

is given to allow the development of companion subroutines for the graphing of the

function and its standard deviations as well as for displaying the original data with

error bars. The purpose of the program is to provide a means for systematically

parameterizing sigmoidal profiles for the comparison of measurements made with differenl

instruments on different systesm and for the comparison of measurements with simulation

models. The program is extensively documented.

12.

KEY WORDS (Six to twelve entries; alphabetical order; capitalize only proper names; and separate key words by semicolon s) autocatalytic function; hyperbolic function; logistic function; sigmoid;

statistical analysis

13. AVAILABILITY

£]Unlimited

| | For Official Distribution. Do Not Release to NTIS

j | Order From Superintendent of Documents, U.S. Government Printing Office, Washington, D.C.

20402.

|y 1 Order From National Technical Information Service (NTIS), Springfield, VA. 22161

14. NO. OF

PRINTED PAGES

93

15. Price

$14.95

U SC OMM-DC 0043*P»0