Approximations to Bayes estimate for quantal assay with simple exponential tolerance distributio

Cornell, R. G.

Petrasovits, A.

English

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APPF@XIMATIOIVS TO THE BAYES ESTIWUE FOR A QUANTAL ASSAY 
WITH SIMPLE EXPONENTIAL TOLERANCE DISTRIBIITION 
Andrgs Petrdsovits 
Canada Department of Agriculture 
and 
Richard G. Cornell 
Florida S ta te  University 
Techntcal Report Number 21 
r/ 
NASA Grant Number NGR-10-004-029 
artment of S t a t i s t i c s  
ri& Sta te  University 
Tallahassee, Florida 
32306 
https://ntrs.nasa.gov/search.jsp?R=19700031925 2020-03-11T22:39:55+00:00Z
Approximations t o  the Bayes E s t i m a t e  f o r  a Quantal Assay 
w i t h  Simple Exponential Tolerance Distribution 
BY 
Andr6s PetrBsovits 
Canada Department of Agriculture 
and 
Richard G. Cornell 
Florida S ta t e  University 
SUMMARY 
Approximations t o  t he  computationally cunibersome Bayes estimate o f  t he  
parameter X i n  t h e  exponential tolerance d is t r ibu t ion  F(d) = l-exp'(-hd), d 2 0; are 
developed f o r  dichotomus response quantal assays. A gamma p r io r  d i s t r ibu t ion  
and a quadratic loss function for  X and a binomial d i s t r ibu t ion  of posi t ives  
a t  each dose d f o r  Sxed  h, are assumed. The first approximation is  for a 
single-dose assay, i s  eas i ly  computed and has average risk only s l igh t ly  
l a rge r  than t h e  Bayes r i s k  for small samples. The select ion of d and n 
are considered when t h i s  approximation is  t o  be used as an estimate of  A. The 
second approximation is  a modification of the first,  i s  somewhat more 
d i f f i c u l t  t o  compute than the  first and retains  the  re la t ive ly  s m a l l  average 
r i s k  of  t h e  first f o r  s m a l l  samples but, unlike the first, is a consistent 
estimate of  A. 
assay with several doses. 
The t h i r d  approximation is an extension of  the first t o  an 
b 
-2- 
1. INTRODUCTIOM 
Approximations t o  the Bayes estimate are presented i n  t h i s  paper for a 
dichotomous response quantal assay w i t h  exponential tolerance d is t r ibu t ion  
F(d) = 1 - ea( -Ad) ,  0 < A ,  d C 0 0 ,  (1.1) 
where d is  the dose and A the parameter t o  be estimated. Attention is  focused 
on single-dose experiments for the  most pa r t ,  although an extension t o  several  
doses i s  suggested i n  Section 6. A single-dose experiment i s  reasonable when 
p r i o r  information i s  available (see Mantel ( 1 9 6 7 ) ) .  A table containing 
optimum doses i s  provided. A t  the  dose u t i l i zed ,  n subjects are exposed with 
r of them responding posi t ively.  It i s  assumed tha t  these n subjects  react 
independently so t h a t ,  for given A ,  r i s  binomially dis t r ibuted w i t h  expectation 
nF( a) ,  where F( d)  i s  given by (1.1). Hence the  l ikel ihood of r i s  
-3- 
The loss Function for  A i s  assumed t o  be quadratic. 
X i s  taken to  be gamma with probabili ty density function 
The pr ior  dis t r ibut ion of 
and with mean and variance 
respectively. This dis t r ibut ion i s  i n  the  family of extended natural  conjugate 
p r i o r s ,  it has the desirable property t h a t  it does not depend on the  design 
parameter d, and it is the natural  conjugate pr ior  for  the corresponding counting, 
as opposed to quantal, response experiment based upon the same tolerance distribution. 
The maximum l ikelihood estimate of X for  a single-dose assay is  developed by 
Cocbran (1950) and cal led the "most probable number". 
the selection of dose level. 
Cochran also considers.. 
In  Section 2 formulas for  a Bayes analysis are presented f o r  a single-dose 
The Bayes estimate fi  is d i f f i c u l t  t o  compute, par t icular ly  i f  a digital  assay. 
computer i s  not available. 
Section 3 which lead t o  an eas i ly  calculated approximation t o  the Bayes estimate 
given i n  Section 4. 
and it is shown empirically that it nearly achieves the minimum (Bayes) r i sk  fo r  
s m a l l  sample sizes. 
table can be used fo r  designing an assay, that is ,  for select ing n and d. 
In Section 5 a s l igh t ly  more cumbersome a,pproximati.on t o  the  Bayes estimate for 
a single-dose assay is presentedwhich retains  the desirable s m a l l  sample 
Bounds for  the Bayes estimate are developed i n  
A table of average r i sks  fo r  t h i s  estimate are displaJred 
Average risks along with optimum dosages given i n  the same 
-4- 
Properties of the f irst  approximation but which, unlike t h e  first, is a consistent 
estimator of A .  
with multiple-dose assays. 
Then i n  Section 6 the f i r s t  approximation is  extended fo r  use 
2. BAYES RESULTS 
Bayes results are presented here i n  order t o  display fomulas used i n  
calculations tabled i n  Section 4 and t o  show the  need f o r  a simple approximation 
t o  the Bayes estimate, Derivations fo r  any number of doses are given by P e t d s o v i t s  
(1970) 
The Bayes estimate of A f o r  a single-dose assay given the l ikelihood (1.2) 
and the pr ior  density (1.3) is  
where 
The posterior probabili ty density function f o r  X i s  
the  w-th poster ior  moment of A about the or igin i s  
the marginal dis t r ibut ion of r i s  
-5- 
and the  Bayes r i s k  is 
Optimum dosages d which minimize the  Bayes r i s k  are given i n  Table 1 i n  
Section 4 fo r  pr ior  mean ~1 =1 and fo r  pr ior  coeff ic ients  of variation ranging 
from 0.1 t o  2.0 and for  sample sizes n = 1, 5 and 10. 
0 
Corresponding Bayes 
r i sks  evaluated at  do, B(do), are ah:o g-iven 4n tha t  t ab l e  and can be used t o  
se lec t  n i f  an exact Bayes analysis i s  to  be computed. 
for pr ior  mean  CI # 1 can be obtained by multiplying corresponding en t r i e s  i n  
Table 1 by p-' and p29 respectively. 
Values of do and B(do) 
3. BOU"1DS FOR THE BAYES ESTIMATE 
for a single-dose assay has been found t o  l i e  i n  the  The Bayes estimate 
in te rva l  
r+*l < A^  < *a+l -
dn+ B d( n-r)+B 
A derivation of these bounds for integral  values of  a is given i n  t h i s  section. 
In extensive calculations (3.1) has also been found t o  hold fo r  nonintegral 
values of  a. 
namely, t o  propose approximations t o  
approximations are presented i n  Section 4 and -5. 
This analyt ical  and empirical evidence is adequate for  our purpose, 
i n  the  in te rva l  given'by (3-1). These 
In order t o  see the  rat ionale  leading t o  (3.11, first define 
By induction, f o r  integral  values of y, it can be shown t h a t  
-6- 
Moreover, T(h) can be expressed as 
where 
since 
Or-1 
h=O k M  
1 S(a-h)T(h+l) = f S(a+l-k)T(k) - S(a+l)T(O) 
= ( ~ l + l ) S (  a+2) - S( a+l )T(  0 ) .  
Using (2,1), ( 3 . 6 )  can be rearranged t o  give 
Hext the  numerator of i i n  (2.1) may be rewrit ten using ( 3 . 3 )  t o  obtain 
-7- 
where 
m 
R 7 [. 1 S(ob+l-h)T(h)]/[aS(Cl)]* 
h=l  
From ( 3 . 3 )  and ( 3 . 4 1 ,  
ob-1 
k=O 
R = 1 S ( ~ - k ) T ( k + l ) / [ d S ( c l ) ]  
ob-1 ob-1 
S(a-h)U(h+l)]/[ 
h=o h=O 
= (c%/d)((n-r*B/d) f [ 1 1 S(cr-h)T(h+l)])-l 
Substi tution of (3 .9)  in to  (3.8) y ie lds  
ob/[d(n-r) $. B ]  + T(O)/d. (3.10) 
Now by s e t t i n g  the  index of summation i n  each t e r m  of (3.2) equal t o  i t s  
upper and lower limits, it can eas i ly  be seen t h a t  
(r+l)(n+B/d)-' T(0) (r+l)(n-r+B/d)-'. 
'R-iis inequality,  together with (3.7) and (3.10), yields  (3.1). 
4. S?,lALL SA?"PLE APPROXIMAYIOX TO !l!XE BAYES ESTIMATE 
Because o f  the form of  t h e  bounds (3.1) on t h e  Bayes estimate iy consideration 
is given i n  t h i s  section t o  the  family of approximations t o  X given by 
A 
Now 4 ( D )  is a continuous and mnotonically decreasing function of D i n  
-8- 
[O,d] and equals the lower and upper bounds given by (3.1) for  D = d and 0, 
respectively. 
f n f 
Therefore, for  some unique D , where 0 c D c d, ^X(D ) = i. 
One reasonable procedure would be t o  se lec t  D t o  minimize the average r i s k  
of  i, (D) given by 
where E denotes the expectation with respect t o  t h e  Joint  dis t r ibut ion of  
A,r 
A and r. It can be  shown t ha t  
n 
R ( D )  = 1 
F O  
i1(D)S(a+2) + (a+2)(~~+1)/6~ 
where S(y)  is defined i n  (3.3). 
in te rva l  [O,d] and a t ta ins  a minimum on [O,d] at ,  say,D . 
indicates tha t  D 
Values of D 
The function R(D)  is  continuous i n  D on the  
n 
Numerical evidence 
8 
i s  unique and i s  an i n t e r i o r  point of t ha t  interval.  
and corresponding R(D ) values have been computed for  sample 
8 Q 
sizes 1, 5 and 10 and fo r  pr ior  coefficients of 
variation 
each of these calculations d has been taken equal t o  d,, the dose which minimizes 
ranging from 0.1 t o  1.0 i n  s teps  of 0.1 as well as 1 .5  and 2.0. In 
t@e 
the  
u 
Bayes r isk.  
Two important empirical relationships were found. In every instance 
f 
r a t io  of D t o  d was between 0.3 and 0.4 and the  r a t i o  of the Bayes 0 
n 
@inimum)risk at  d t o  R ( D  ) a t  do w a s  greater than or equal t o  0.969. 
l a t t e r  ra t ios  show tha t  i l ( D  
for  s m d l  samples. 
tables  of D are  available. 
requiring tables  of D may be overcome by se t t i ng  D equal. t o  a constant proportion 
The 
is  a good al ternat ive t o  the Bayes estimate 
0 
f 
It has the advantage of being easi ly  calculated i f  appropriate 
% 
The former ra t ios  indicate t h a t  the disadvantage of 
f 
-10- 
V 
0.1 
0.2 
0.3 
0.4 
0-  5 
0.6 
0.7 
0.8 
0.9 
1.0 
1.5 
2 .o 
1.5850 
1.5593 
1 * 5179 
1.4626 
I.. 3961 
1.3215 
1 2418 
1.1602 
1.0789 
1.0000 
0.6717 
0.4586 
Table 1. Values of the  optimum dose (d ), the average 
r i sk  ( R )  of x,at d = do and tge r a t i o  (E)  at 
d of the Bayes r i s k  t o  R by prior coefficient 
09 variation ( v )  and sample s i ze  (n) .  
n = l  
R 
0. oogg 
0.0390 
0.0854 
0.1468 
0.2210 
0.3066 
0.4026 
0.5087 
0.6244 
0.7502 
1 5352 
2.5961 
E 
1.0000 
1.0000 
1.0000 
1.0000 
1.0000 
1.0000 
1.0000 
0 9999 
0 9999 
0 - 9998 
0 9991 
0.9985 
1.5844 
1.5524 
1.4910 
1.4019 
1.2944 
1.1792 
1.0651 
0 9572 
0.8583 
o - 7693 
0.4510 
o .2921 
n = 5  
R 
0.0097 
0.0356 
0.0712 
0.1111 
0 * 1533 
0.1974 
0.2439 
0 2937 
0.3470 
0.4044 
0.7615 
1.2481 
E 
0 9998 
0.9995 
0 9992 
0.9989 
0.9982 
0.9971 
0.9957 
0 -9943 
0.9928 
0.9915 
0.9871 
0 9833 
1.5838 
1.5451 
1.4674 
1 * 3577 
1.2320 
1.1021 
0.9824 
0,8722 
0,7737 
o . 6871 
o 3969 
0.2494 
n = 10 
R 
0.0094 
0.0321 
0.0590 
0.0860 
0.1125 
0.1396 
0.1682 
0.1987 
0.2318 
0.2677 
0 4937 
0.8028 
E 
0.999 3 
0.9981 
0.9968 
0.9946 
0.9912 
0.9873 
0.9830 
0,9790 
0 -9753 
0.9721 
0.9610 
0.9557 
-11- 
5. LAp.GE SAi’PLE APPROXIXATIO!T To TIE BAYES ESTIMATE 
The estimate h^ proposed i n  Section 4 has been found t o  be an excellent 1 
apporximation t o  ‘the Bayes ‘estimate f o r  sample sizes of 10 and ‘ less ,  which are the  
sample sizes  of ten  used i n  assays. However, A is  not a consistent estimate of  
A .  
which approximate ^x w e l l  f o r  small samples and would be consistent i s  given by 
A 
1 
.i An a l te rna t ive  family of estimates which could be used t o  construct estimates 
r 
j=o 
i 2 ( D )  = d-’ 1 (j+n-r+B/d)-’ + a[d(n-r)+Dr+B]-’, 0 < D d. (5.1) 
For s m a l l  samples D could be set equal, for  instance,  t o  0.35d as i n  Section 4. 
The consistency of  i ( D ) ,  which holds fo r  a l l  D i n  [O,d], i s  gained a t  the expense 
of m&&g 
2 
h 
the approximation of  A computationally more d i f f i c u l t .  O f  some help 
though is the f ac t  that  the  sum i n  the  f irst  term on the  r igh t  o f  (4.1), defined 
as T ( 0 )  by equation (3.2) , can be expressed as the difference of two digamma 
functions. 
h 
Thus a l te rna t ive  expressions for X2(D) ’ are 
where the digamma funceion 
and Stegm (1964) and r( x )  is the gamma function r (  x )  = 1; e t dt.  
Y ( x )  = d[lnr(x)] /dx i s  tabled,  f o r  instance, by Abramowitz 
-t x 
h A 
The rat ionale  fo r  suggesting A2(D) i s  the same as that  fo r  X (D) i n  Section 4 1 
given by (3 .7 )  and (3.10) instead of  
This relat ionship between 
except tha t  i , (D)  i s  based on t h e  bounds for 
the  wider in te rva l  derived from them and given by (3.1). 
i l ( D )  and i , (D)  shows tha t  i , (D)  w o u l d  perform well fo r  s m a l l  samples f o r  suitably 
selected D and is the  reason f o r  suggesting that D be taken equal t o  0.35 as i n  
-12- 
the construction of A i n  Section 4. In the  remainder of t h i s  section it w i l l  
be shown t h a t ,  conditionally on 
h (D) coneverge; i n  probabili ty t b  A ,  that i s ,  that  A2(D)  is  a consistent 
est$nator o f  A .  
1 
f o r  any D i n  [O,d] as n + ..,the estimate 
6. e 
2 
The sum T ( 0 )  i n  equation (5.2) can be expressed using the  Maclaurh-Euler 
summation formula given by Jordan (1947, pp. 261-2) t o  obtain 
(5.4) -1 T ( 0 )  = ln(l+n+B/d) -Y(n-r+B/d) + (n-r+S/d) + Rn 
where 
But from C r a m &  (1946, p. 130) it follows t h a t  
Y( x )  = In( x+1)-2-’( x+l)-l - P1( z)( x + l + ~ ) - ~ d z  (5.6) 
where P1( z) = [ 2]-zt1/2 and [ z] denotes the  grea tes t  integer  $2. 
1 P1( z) 1 S 1/2,  (5.6) can be rewrit ten as 
Noting t h a t  
Thus from (5.21, (5.4) and (5.7), 
where in I ,< 1 and Rn is  given by ( 5.5). 
i n  probabili ty t o  exp(-Xd) , it follows readily from (5.8) t ha t  &(D) converges 
i n  probabili ty t o  X fo r  fixed X and any D i n  [O,d]. 
Noting t h a t ,  as n+-$ l-rh converges 
6. EXTENSION TO SEVERAL DOSES 
I n  t h i s  section a family of approximations t o  the  Bayes estimate i s  proposed 
for  a k-dose assay with dose leve ls  di when ri subjects respond posi t ively and 
independently i n  samples of s i z e  ni, i = 1,2,. . . 
Section 1 i s  s t i l l  assumed t o  hold at each dose level .  
appro ximated by 
k. The model described i n  
The Bayes estimate may be 
k k k 
i=l i=l i=l 
X3(D) = ( 1 ri + a + 1)[ 1 di(ni-ri) + D 1 ri+Bl-ly 
This is the estimate, with D = 0.35do, tha-t would be obtained i f  an experimenter were 
t o  perform a sequence of k simple-dose assays a t  doses close t o  a s t ab le  optimum 
d determined from p r i o r  information using the approximate Bayes estimate $(O. 35d0), 
given by (4.1), a f t e r  each assay. 
which does not change appreciably when accumulated experimental information is  used 
a f t e r  each successive single-dose assay t o  modiw the p r i o r  dis t r ibut ion upon which 
0 
By a s tab le  optimum d is meant an optimum dose 0 
the  optimum f o r  the  next assay is  based. Since i w a s  found i n  Section 4 t o  be 1 
c. robust relative t o  the  Bayes estimate for departures of d from d another reasonable 0’ 
L. 
approximation t o  A even when d i s  not en t i r e ly  stable could be constructed by 
subst i tut ing 0.358 fo r  D i n  ( 6 ~ ) ~  where a is the  mean of  the k doses used in  t h e  
assay. 
0 
-14- 
The authors acknowledge f inancial  support received for  t h i s  research from grant 
nunher NGR-10-004-029 from The Rational Aeronautics and Space Administration. 
Pa r t i a l  support f.or the  Florida State  University Computing Center, whose f a c i l i t i e s  
were used i n  t h i s  research, w a s  received from the  National Science Foundation. ' 
Additional computing f a c i l i t i e s  were provided by the  Canada Department of  Agriculture. 
RETZRENCES 
Abramowitz, M. and Stegun, I. A. (Editors) (1964). Handbook of 
Mathematical Functions. Washington: National Bureau of  
Standards. 
Cochran, W. G. (1950). Estimation of bac te r ia l  densit ies by means 
of the  smst probable number. Biometrics 16, 105-16. 
Cram&, H. (1946). Mathematical Methods o f  S t a t i s t i c s .  Princeton: 
Princeton University Press. 
Jordan, C. (1947). Calculus of Fini te  Differences. New York: Chelsea. 
Mantel, N. (1967). Adaption of  Karber's method for  estimating the  
exponential parameter from quantal data, and i t s  relatiGnship 
t o  b i r th ,  death and branching processes. Biometrics 23, 739-46. 
PetrGsovits, A. (1970). Approximations t o  Bayes Procedures 
for Quantal Assays with Simple Exponential Tolerance Distributions. 
Tallahassee : Florida S ta t e  University disser ta t ion.  


Approximations to the Bayes estimate for a quantal assay with simple exponential tolerance distribution

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