Contamination probability distribution models for spacecraft - microbiological assays and sterilization of spacecraft component

Cornell, R. G.

English

NASA Technical Reports Server

PROGRESS REPORT: NASA GRANT NGR-10-004-029 to  
FWRIDA STATE UNIVEBsmt 
PROJECT, TITLE: Biostatistics of Space  E x p l o r a t i o n :  
Microbio logy  and S t e r i l i z a t i o n  
PRINCIPAL INVESTIGATOR: Richard G. Cornel1 
August 24, 1966 _- 
Department of S t a t i s t i c s  
F l o r i d a  S t a t e  U n i v e r s i t y  
T a l  l a h a s  see, F l o r i d a  
https://ntrs.nasa.gov/search.jsp?R=19660026768 2020-03-16T20:32:47+00:00Z
* pRoQ3SS REPORT: NASA GRANT NGR-10-004-029 t o  FIIIRIDA STATE UNIVERSITY 
PROJSCT TITLE: B i o s t a t i s t i c s  of Space Exploration: Microbiology and 
S t e r i l i z a t i o n  
PRINCIPAL INVESTIGATOR: Richard G. Cornel1 
August 24, 1966 
Introduct ion 
The p.Jrpose of t h i s  gran t  i s  t o  provide continuing research support 
on s ta t i s t ica l  Froblems a r i s i n g  i n  invest igat ions on the  decontamination of 
spacecraf t  and r e l a t ed  s tudies  of microbial l i f e .  The work under t h i s  
g r a n t  i s  of two d i f f e r e n t  types, namely, s t a t i s t i c a l  consul ta t ion t o  Mr. 
Lawrence B. Hall, Planetary Quaranthe Officer,  t o  the Spacecraft  S t e r i l i e a -  
t i o n  Advisory Conanttee and t o  NASA contractors ;  and basic  research on 
contaminatioa probabi l i ty  models and on microbial assay. 
With regard to consul ta t ion on research problems, the pr inc ipa l  
inves t iga tor  has attended two meetings of the Spacecraft  S t e r i l i z a t i o n  
C o z r m i t t e e  during the l a s t  six months. 
1966 coosul t ing with the perscnnel of t he  S t e r i l i z a t i o n  Group of the  Jet 
Propulsion Laboratory and he has evaluated repor t s  on decontamination 
H e  a l s o  spent a week during March, 
probabi l i ty  models fo r  Mr. Lawrence B.Hal1.  
Ear Mr. B a l l  a standard nomenclature of mathematical terms t o  be used i n  the 
computation of p robab i l i t i e s  of planetary contamination. 
is given i n  the enclosed Technical Report N u m b e r  5.  
H e  recent ly  agreed to  prepare 
This nomenclature 
The remainder of t h i s  report  w i l l  be concerned with a descr ip t ion  
of progress made on research of i n t e r e s t  t o  NASA. 
descr ip t ion  of the research p ro jec t  and its relevance t o  the  space quarantine 
program w i l l  be given. The cu r ren t  status of each pro jec t  w i l l  be indicated. 
I n  each instance a b r i e f  
- 2 -  
I n  t h i s  and subsequent progress reports, a de t a i l ed  descr ip t ion  of research 
w i l l  be given only f o r  pro jec ts  which a r e  complete enough f o r  t he  issuance 
of t echnica l  r epor t s  and usual ly  subsequent submission f o r  publ icat ion i n  
profess iona l  journals .  For such projects, technical  repor t s  o r  r e p r i n t s  
w i l l  be enclosed i f  not  s en t  earlier, o r  w i l l  be c i t e d  i f  copies  have already 
been sent ta NASA, For orher projects target dates  for &a cmple r io=  of 
t echnica l  repor t s  w i l l  be given. 
Spacecraf t  S t e r i l i z a t i o n  Probabi l i ty  Models 
One way t o  represent  the information ava i lab le  on the probabi l i ty  
of contaminating Mars i s  i n  the  form of a probabi l i ty  d i s t r i b u t i o n  function 
which would r e l a t e  a probabi l i ty  based on curren t  knowledge with any 
spec i f ied  i n t e r v a l  on a contamination probabi l i ty  scale. For instance, if 
one were to ask: "Based on our curren t  knowledge, what is the l ikel ihood 
t h a t  the  probabi l i ty  of contaminating Mars (on a given probe o r  during a 
given period of time) is less than 10'4?", then the  answer would require 
t h a t  the  area under the  probabi l i ty  d i s t r i b u t i o n  funct ion t o  the  l e f t  of 
be determined. It should be emphasized t h a t  under spec i f ied  conditions, 
the  t r u e  probabi l i ty  of contaminating Mars is constant.  The d i s t r i b u t i o n  
funct ion proposed would not involve the  t r u e  probabi l i ty  of contaminating 
Mars as a random variable,  but wauld ins tead  be a d i s t r i b u t i o n  OIJ our 
knowledge of the  probabi l i ty  of contaminating Mars. 
The d i s t r i b u t i o n  function being devised would have t o  be defined 
over the domain from ze to  to one, the possible  range of a contamination 
probabi l i ty ,  and would have t o  be f l ex ib l e  enough t o  g ive  d i f f e r e n t  pa t t e rns  
as our information i s  increased and refined. The parameters of the  
Y 
- 3 -  
d i s t r i b u t i o n  should have meaning i n  themselves. The d i s t r i b u t i o n  should be 
of simple funct ional  form, o r  a t  l ea s t  be extensively tabled. 
be sens i t i ve  and adaptable to new information, t h a t  is, it should be such 
chat  methods can be developed t o  modify i t  i n  accord with new research 
f ind ings  and new s t r a t e g i e s  fo r  Martian exploration. 
It should 
Work has  been begun here t o  develop a probabi l i ty  d i s t r ibu t ion  
funct ion on our knowledge on the probabi l i ty  of contaminating Mars. 
approach has been t o  develop appropriate d i s t r ibu t ions  f o r  spacecraf t  p a r t s  
o r  coaponents which a re  of a type which can be subjected t o  microbial  assay. 
Our 
Later methods w i l l  be developed t o  combine such d i s t r ibu t ions  determimd f o r  
diZferent  components i n  order t o  obtain a d i s t r ibu t ion  on our information 
on the  probabi l i ty  t h a t  the  spacecraft  as a whole is contaminated. Then 
ways w i l l  be sought t o  modify t h i s  spacecraft  p robabi l i ty  d i s t r i b u t i o n  i n  
accord with knowledge of missions to  be undertaken. 
It is  common prac t ice  and reasonable, based on the  assumption t h a t  
the nGmber of contaminating pa r t i c l e s  on a componnet of a specif ied kind 
w i l l  follow a Poisson d is t r ibu t ion ,  t o  assume t h a t  the probabi l i ty  that the  
component i s  contaminated is given by p = (1-e”) where X is the mean number 
-Ad of contaninants per component, or, more generally,  by p = (1-e 
is a dosage i n  area, t i m e  o r  d i lu t ion  factor ,  and X is a mean per  dosage 
uni t .  Put t ing  a Probabi l i ty  d i s t r ibu t ion  on our knowledge of p over the  
domain from zero t o  one i s  equivalent t o  determining a probabi l i ty  d i s t r ibu -  
t i o n  on our knowledge of X over the domain from zero t o  i n f in i ty .  
f lex ib le ,  unimodal d i s t r ibu t ion  which can be used t o  descr ibe information 
on X i s  the gamma-distribution with densi ty  function 
), where d 
A 
- 4 -  
where here h i s  regarded a s  a random variable  even though it is  our 
information about A which is  ac tua l ly  variable.  I n i t i a l l y ,  the parameters 
a and p would be chosen t o  represent current  evidence concerning the  conta- 
iiifnation load rm a cmpcnent, or 3 unit eree of a co=lpooent. Io case no 
such information i s  available,  the i n i t i a l  values taken f o r  CY and p would 
be such t h a t  g(1) would be very diffuse.  
the type under consideration are examined and x of t h e m  a r e  found t o  be f r ee  
of contamination. The probabi l i ty  that  t h i s  occurs given X is 
Next suppose t h a t  n components of 
using the binomial d i s t r ibu t ion .  
with g(X) as  given by (1) using Bayes' formula t o  give a new probabi l i ty  
d i s t r i b u t i o n  on our knowledge of h which would replace (1) and would be 
This probabi l i ty  function can be combined 
responsive t o  the experimental evidence. I f  the new densi ty  function f o r  
A should be of the  same form as (l), t h a t  is, be a gamma densi ty  function, 
but possibly with d i f f e r e n t  parameter values, then t h i s  approach would 
allow us t o  e a s i l y  incorporate new evidence in to ' au r  probabi l i ty  d i s t r ibu -  
t i o n  on our information on A. 
Mr. Andres Petrasovi ts  has worked on t h i s  problem as a general 
method of analysis  fo r  s e r i a l  d i lu t ion  experiments where i n  addi t ion t o  
modifying the densi ty  function (1) a t  each s tage of experimentation, he 
a l s o  estimates X by A. 
h 
He a l so  considers the poss ib i l i t y  of an experiment 
containing several  d i lu t ion  (dosage) levels ,  di, i = 1,. . . , k. 
formulated i n  Technical Report Number 6, which i s  a preliminary report .  
€lis work is 
- 5  - 
6 
H i s  es t imate  of A. is given by formula (2.10) or (4.4) i n  t h a t  report, and 
i s  too  complicated f o r  p rac t i ca l  use, although he shows i n  the  l a s t  pa r t  of 
t he  r epor t  t h a t  it is  a r a t i o  of r a t iona l  combinations of polygamma funct ions 
which are  tabled. Also, the modified densi ty  function he obtains  f o r  our 
knowledge of A is  not  of the same form as  (l), but  is  much more complicated. 
H e  c a l l s  t h i s  densi ty  function the  poster ior  densi ty  of X and it is given io 
formula (2.9). 
Since he wrote the  essay which cons t i t u t e s  Technical Report Number 6, 
h 
M r .  Pe t rasovi t s  has succeeded i n  approximating A by a s imple function of n and 
x and the parameters of (1) and he a l so  has been able  t o  c lose ly  approximate 
the  pos te r ior  densi ty  of A by a gamma densi ty  of the  form of (1) but  with 
parameters modified by the  experimental evidence. H i s  der iva t ion  and 
eva lua t ion  of h i s  approximate method is mathematically involved, bu t  the 
r e s u l t s  are simple and can be incorporated i n t o  the component l eve l  of the 
genera l  contamination probabi l i ty  level we  a r e  seeking. H i s  evaluat ion of 
h i s  r e s u l t s  is  not complete, but numerical comparisons with the exact  but  
cumbersome formulas given i n  Technical Report Number 6 i nd ica t e  t h a t  h i s  
approximate approach i s  highly e f f i c i e n t  i n  terms of a comparison of r e l a t i v e  
r i sks ,  i n  fac t ,  usua l ly  more than 984 e f f i c i e n t  f o r  most parameter values 
invest igated.  It is ant ic ipated that  these r e s u l t s  w i l l  be presented to MASA 
i n  t he  form of a technical  repor t  by January 1, 1967. Mr. Pet rasovi t s  is 
being employed under the NASA grant  t h i s  summer but w i l l  w r i t e  a repor t  on 
the  recent  r e s u l t s  t h i s  f a l l  a f t e r  he r e tu rns  t o  h i s  job with the  
Department of Agriculture of Canada. 
I n  a manner s imi la r  t o  t h a t  employed by Mr .  Petrasovi ts ,  it is 
expected t h a t  count da ta  on contamination w i l l  be incorporated i n t o  our 
probabi l i ty  model. 
than t h a t  encountered w i t h  the binomial type da ta  already considered. 
t h i s  is  done, then consideration w i l l  be given t o  combining probabi l i ty  
d i s t r i b u t i o n s  of the  information avai lable  on d i f f e r e n t  components and to  
incorporating non-experimental evidence i n t o  our model. 
Actually, i t  is thought t h a t  t h i s  i s  an e a s i e r  problem 
After  
i n  addi t ion to the work done t h i s  summer by Pir. Petrasovits and tke 
pr inc ipa l  invest igator ,  D r ,  Duane Meeter has been carrying out research and 
taking p a r t  in seminars on astronaut ics  and exobiology a t  the Ames  Research 
Center during the l a s t  two months w h i l e  being supported by t h i s  grant.  
thought t h a t  h i s  experience there  w i l l  add f resh  in s igh t  i n t o  our work under 
t h i s  g ran t  on the development of contamination probabi l i ty  models. 
research he has s t a r t e d  t h i s  sunmter w i l l  be reported on l a t e r .  
It is 
The 
However, 
enclosed is Technical Report Number 7 which was prepared by Dr. Meeter before 
he l e f t  f o r  Ames, Although t h i s  report  as i t  stands is a highly theo re t i ca l  
inves t iga t ion  of nonlinear estimation, it is  thought t h a t  fu r the r  considera- 
t i o n  of i ts  implicat ions w i l l  be helpful  i n  more applied nonlinear estima- 
t i o n  problems, including bioassay. 
/- I n  the  February, 1966 Progress Report, reference was made to the  
d i s s e r t a t i o n  research of Mr. William E. Lever on a decis ion theory approach 
to  testing hypotheses with l ikelihood r a t i o  tests which w i l l  be appl icable  
t o  making decis ions with regard t o  the achievement of spacecraf t  s t e r i l i z a t i o a  
probabi l i ty  requirements. 
and h i s  research i s  v i r t u a l l y  completed. 
w i l l  be prepared t h i s  f a l l .  
cons i s t s  of paying f o r  p a r t  of the  computer expenses encountered i n  h i s  
research. 
Mr. Lever i s  present ly  wr i t ing  h i s  d i s s e r t a t i o n  
A technical  repor t  on t h i s  p ro jec t  
The only support being given Mr .  Lever by NASA 
.I 
I .  
- 7 -  
S t a t i s t i  
- 
1 Procedures f o r  Microbial Assays and Re1 
E s t i m a t i o a  Techniques 
ted Nonlinear S ta  i c a  1 
The work of the  pr inc ipa l  inves t iga tor  with Mr. Petrasovi ts ,  which 
has already been described, is r ea l ly  a bioassay est imat ion technique, bu t  
has  been described i n  the  previous sec t ion  because of its close re l a t ionsh ip  
t o  our  contamination probabi l i ty  model. Similar ly ,  although Dr. Meeter is 
concentrating on model building considerations t h i s  sumer ,  h i s  work i n  
Technical Report Number 7 i s  on nonlinear estimation. I n  t h i s  sec t ion  
considerat ion w i l l  be given t o  research on nonlinear s t a t i s t i c a l  es t imat ion 
which is d i r e c t l y  appl icable  t o  microbial assays and which i s  being ca r r i ed  
out without reference t o  more general contamination probabi l i ty  models. 
The February 1966 Progress Report mentioned work of the  pr inc ipa l  
i nves t iga to r  with John J. Beauchamp on simultaneous est imat ion f o r  severa l  
nonlinear regression equations. 
w i t h  t h a t  Progress Report and later r e p r i n t s  of the published manuscript 
(Beauchamp, J. J. and Cornell, R. G . ,  "Simultaneous Nonlinear Estimation, 
Technometrics (1966), 319-326) were sen t  t o  NASA. That paper d e a l t  with 
general ized least squares estimation. Mr. Beauchamp and I are now working 
with two other  es t imat ion schemes for s iml t aneous  nonlinear es t imat ion 
which are p a r t i c u l a r l y  appl icable  t o  the  descr ip t ion  of the movement of a 
tracer, say microbial  contamination, through a cunpezknental system. Research 
on this projec t  i s  almost completed and Mr. Beauchamp is  wr i t i ng  up the 
r e s u l t s  i n  d i s s e r t a t i o n  form. It i s  an t ic ipa ted  t h a t  two addi t iona l  
technica l  r epor t s  w i l l  be prepared from t h i s  work during the next six months. 
The only NASA support  being given Mt. Beauchamp is  the  payment of some of the 
cost of using a computer i n  h i s  research. 
A technical  repor t  on t h i s  work was included 
/ f I  
- 8 -  
Another pro jec t  which was mentioned i n  the previous Progress Report 
and which w i l l  only require  support from NASA for computing costs,  is t h a t  
being car r ied  out  j o i n t l y  with Mr. Donald C. Martin. This pro jec t  involves 
evaluat ing the s m a l l  sample variance and mean square errors of various 
est imators  of the  instantaneous die-off rate X i n  microbial  assays for the 
binomial model given i n  equation (2), but  without specifying a d i s t r i b u t i o n  
on A as w e  previously d id  i n  (1). 
but  has  been delayed becsrse Mr. Martin, who is  not being paid from the  
This work has been s t a r t e d  successful ly  
grant ,  has been completing a &.D. d i s se r t a t ion  on another topic. Workwil l  
be resumed on t h i s  pro jec t  i n  the  f a l l  and it is an t ic ipa ted  t h a t  a Technical 
Report w i l l  be wr i t t en  ea r ly  i n  1967 summarizing the  r e su l t s .  A graduate 
s tudent  a s s i s t an t ,  Mr. Richard Gero, has  been employed, s t a r t i n g  on September 
I, t o  work --quarter t i m e  on NASA research projects .  H e  is  an experienced 
computer programmer and he w i l l  i n i t i a l l y  spend most of h i s  t i m e  on t h i s  
p ro jec t  . 
Dr. Myles Hollander is  current ly  developing nonparametric, 
d i s t r ibu t ion- f ree ,  procedures for t es t ing  paral le l ism of regression l ines ,  
a problem which of ten  occurs i n  bioassay. D r .  Hollander has  developed e a s i l y  
applied procedures under very general assumptions fo r  comparing two regression 
l i n e s  when the  two samples, from which the two l i n e s  are calculated,  are 
of equal size. 
although work thus f a r  ind ica tes  tha t  he has developed procedures w h i c h a r e  
highly e f f i c i e n t  r e l a t i v e  to the corresponding normal-theory t-test. 
an t i c ipa t e s  completing t h i s  work and perhaps extending it t o  unequal sample 
s i z e s  during the  coming year. 
He has  not completed the  evaluation of these procedures 
Be 
A Technical Report w i l l  be prepared when this 
- 9 -  
work i s  completed. D r .  Hollander has received a major port ion of h i s  support 
during the pas t  year from funds from the NASA grant, but the pr inc ip le  
invest igator ,  who has been more ac t ive ly  engaged i n  the work under t h i s  
grant ,  has  only been supported f o r  four months by t h i s  grant.  
Work by the pr inc ipa l  investigator,  D r .  Richard G. Cornell, is 
underway on the  est imat ion of parameters i n  a bioassay model with iogarithmic 
die-off  and with microbial  counts observed fo r  each p l a t e  studied. Specif ical ly ,  
a t t e n t i o n  is focused on tk estimation of t h e  parameters a and B i n  the model 
-vi 
E(Yi) = are , i = 0,1, ..., k-1; a,@ > 0, 
where E denotes "expected values of" and where it is assumed t h a t  the Y 
independently d i s t r ibu ted  Poisson variables. 
W i l l i a m a s  (Williams, E. J. F i t t i n g  a geunetric progression t o  frequencies, 
are i 
i' For equal ly  spaced t 
I1 )I 
B i o m e t r i c s  - 17 (1961), 584-606), considers the problem of estimation and the  
s ign i f icance  of departures  from the  m o d e l .  The purpose of the  cur ren t  work is 
t o  extend t h i s  work t o  the s i t ua t ion  where the t a r e  equal ly  spaced on a i 
logarithmic scale .  For t h i s  model, the maximum likelihood estimate is a 
function of s u f f i c i e n t  s t a t i s t i c s ,  so only maximum l ikel ihood estimation has 
been studied. 
obtained, but  they can only be solved i t e r a t ive ly .  
l ikel ihood estimates have been computed fo r  severa l  spacings of dosages on 
the logarithmic scale.  The l a rge  sample propert ies  of the est imators  have 
been studied. 
s tudied next through the use of e m p i r i c a l  polynomial regression equations f o r  
the estimates. 
Equations for  the maximum likelihood est imates  have been 
Tables of maximum 
The small sample d i s t r ibu t ions  of the est imators  w i l l  be 
Goodness-of-fit tests of the  underlying model w i l l  a l s o  be 
- 10 - 
invest igated.  It is expected that t h i s  work w i l l  be completed t h i s  f a l l  
with the help of Mr. Gero i n  obtaining the empirical equations for the 
maximum likelihood estimates from the tables  already prepared. 
report on this work should be completed before the end of t h i s  year. 
A technical 
Y 


Biostatistics of Space Exploration - Microbiology and Sterilization

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