FORTRAN 2 computer program calculates maximum estimates of a monotonic non-decreasing response function. The program uses the method of reversals algorithm which applies to the analysis of univariate or multivariate sensitivity experiments

Alexander, M. J.

English

NASA Technical Reports Server

May 1967	 Brief 67-10136 
NASA TECH BRIEF 
NASA Tech Briefs are issued to summarize specific innovations derived from the U.S. space program, 
to encourage their commercial application. Copies are available to the public at 15 cents each 
from the Clearinghouse for Federal Scientific and Technical Information, Springfield, Virginia 22151. 
Computer Program Calculates Monotonic Maximum
Likelihood Estimates Using Method of Reversals 
The problem: 
To devise a method for rapidly calculating the 
maximum likelihood estimates of a monotonic non-
decreasing response function. The methodmust apply 
to the analysis of univariate or multivariäte sensitivity 
experiments. In univariate sensitivity experiments, the 
threshold at which an increase in stimulus produces a 
measured response has an assumed monotonicity. 
That is, with increasing stimulus, the -probability of a 
response is at least nondecreasing. Commonly used 
methods for the design and analysis of such experi-
ments use the binomial distribution as a model for 
analyzing the data generated at a particular stimulus 
level. However, if these estimates exhibit a "reversal" 
(response decreases with increasing stimulus) then a 
procedure is necessary for "correcting" the estimates 
so that they-conform to the monotonicity assumption. 
To avoid tedious and time consuming study of the 
response and individually replacing estimates which 
are nondecreasing, it is desired to find initially the 
estimates with maximum likelihood properties within 
the framework of monotonicity. 
The solution: 
A Fortran II computer program for the complete 
ordering of sensitivity response probabilities by using 
an algorithm called the method of reversals. 
How it's done: 
Univariate response probabilities are completely or 
linearly ordered, and the estimates can be found by 
the method of reversals. Multivariate response prob- 
abilities are only partially ordered, and a second, more 
complicated algorithm, based on the known partial
ordering, is given to find a complete ordering. The 
method of reversals may then be applied as in the 
univariate case. The multivariate response estimates 
are at a maximum likelihood under the initial assump-
tion that the sensitivity parameters are monotone non-
decreasing in each stress variable. However, a com-
plete ordering resulting from the application of the 
second algorithm need not be unique. Any ordering 
arrived at through the second algorithm will give iden-
tical estimates, which are at a maximum of the likeli-
hood function, when the method of reversals is sub-' 
sequently applied. 
Notes: 
1. Available information concerning this program 
includes the mathematical analysis of the basic 
theory, a:. program flowchart, and a complete listing 
of the computer program. 
2. Inquiries concerning this innovation may be di-
rected to:
COSMIC 
Computer Center 
The University of Georgia 
Athens, Georgia 30601 
Reference: B67-10136 
Patent status: 
No patent action is contemplated by NASA. 
Source: M. J. Alexander 
of North American Aviation, Inc. 
under contract to 
Marshall Space Flight Center 
(M-FS-1 516) 
Category 01 
This document was prepared under the sponsorship of the National 
Aeronautics and Space Administration. Neither the United States 
Government nor any person acting on behalf of the United States
Government assumes any liability resulting from the use of the 
information contained in this document, or warrants that such use 
will be free from privately owned rights.
https://ntrs.nasa.gov/search.jsp?R=19670000136 2020-03-11T20:44:58+00:00Z


Computer program calculates monotonic maximum likelihood estimates using method of reversals

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