Application of multivariate extended Poisson distribution

Patel, H. I.

Trivedi, S. J.

English

NASA Technical Reports Server

. 
NASA CONTRACTOR 
REPORT 
Report No. 61297 
AN APPLICATION OF THE MULTIVARIATE EXTENDED POISSON 
DISTRIBUTION IN 2 X 2 CONTINGENCY TABLES 
By H. I. Patel and S. J. Trivedi 
The UniversiQ of Georgia 
Athens, Georgia 
September 10, 1969 
Prepared for 
N A S A - G E O R G E  C .  MARSHALL S P A C E  F L I G H T  C E N T E R  
Marshall Space Flight Center, Alabama 35812 
https://ntrs.nasa.gov/search.jsp?R=19690026637 2020-03-12T05:14:38+00:00Z
1 .  REPORT NO. 2. GOVERNMENT ACCESSION NO. 3, RECIPIENT'S CATALOG NQ. 
NASA CR-61297 
1. T I T L E  AND SUBTITLE 5. REPORT DATE 
AN APPLICATION OF THE MULTIVARIATE EXTENDED September 10, 1969 
NASA-George C. Marshall Space F l ight  Center 
Marshall Space F l ight  Center,  Alabama 
POISSON DISTRIBUTION I N  2 X 2 CONTINGENCY TABLES 
7 A J T d 3 R f S j  
H. I. P a t e 1  and S. J. Tr ivedi  
9. FERF3RMING ORGANIZATION NAME AND ADDRESS 
The University of Georgia 
Athens, Georgia 
12. SPONSORING AGENCY NAME AN0 ADDRESS 
(Final)  
(4. SPONSORING AGENCY CODE 
6. PERFORMING ORGANIZATION CCOE I 
I 
8 .  PERFORMING ORGANIZATION REPOR r # 
Technical Report No. 40 
(0 .  WORK UNIT NO. 
129-04-04-01-62 
NAS8-11175 
1 1 .  CONTRACT OR GRANT NO. 
13. TYPE OF REPORII' & PERIOD COVERED 
Contractor  Report 
1 
Is. SUPPLEMENTARY NOTES 
This research  w a s  performed under contract  wi th  the  Aerospace Environment Divi- 
s ion ,  Aero-Astrodynamics Laboratory, MSFC. Mr. L. W. F a l l s  i s  the  con t r ac t  monitor. 
21. NO. OF PAGES 
12 
In a series of 2x2 contingency tables, suppose that n ( > l ) ,  the total 
number of individuals, varies according to a Poisson distribution. 
these situations, to test the independence of the two characteristics in- 
volved, the likelihood ratio test is developed which also makes use of the 
additional information that the distribution of n is known to be Poisson. 
The unconditional distribution of the cell frequencies is obtained, and 
the first approximates for the iteration procedure to obtain the maximum 
likelihood estimates of the parameters of the distribution under the 
hypothesis of independence of the two Characteristics are given. 
In 
22. PRICE 
17. KEY WORDS 
s t a t i s t i c a l  ana lys i s  
frequency d i s t r i b u t i o n s  
random v a r i a b l e s  
I 
/ 
120. SECURITY CLA I U 19. SECURITY CLASSIF.  (of thi8 r e p a t )  U 
18. DISTRIBUTION STATEMENT 
, PUBLIC RELEASE 
/ :- A/. / & K - d L -  
E. D. GEISSLER, D i r ,  Aero-Astrodynamics 
Laboratory 
1 I I 
m F C  - F o r m  3292 1969) 
AN APPLICATION OF THE MULTIVARIATE 
EXTENDED POISSON DISTRIBUTION IN 2 x 2 
CONTINGENCY TABLES 
by 
H. I. Pate1 
and 
S .  J. Trivedi 
1. Introduction 
The various methods for making a combined test of independence of 
two characteristics in the data consisting of a series of 2 x 2 tables 
obtained under different situations have been described by Cochran [l]. 
Let n, the total number of individuals in a 2 x 2 table, be divided as 
follows : 
xoo xol 
xlo xll 
1 x.  x * O  r 
TI and TI (such that roo + TI + + = 1) 
If Too’ nO1’ 10 11 01 10 11 
are the corresponding probabilities, the probability distribution of 
(Xoo, Xol, X ) will be multinomial with parameters n, TI 10 f and f 00’ 01 10 
Now let us consider a situation where n (>1) varies according to a Poisson 
distribution. 
a certain high-way may be divided into a 2 x 2 contingency table according 
to major and minor accidents and day-time and night-time accidents. 
Applications of perhaps greater interest to a ,ospace scientists might 
involve thunderstorm accompanied by rain versus thunderstorm without 
rain; thunderstorm within, say, a ten mile radius of the observer versus 
thunderstorm at a 
Here we may assume the Poisson distribution for the number of occurrences. 
In such situations, we can make use of the additional information that the 
For example, the number of accidents observed per week on 
specific point, and other similar atmospheric phenomena. 
distribution of n is known. 
in this paper ,  the likelihood ratio test is developed. 
To test the independence in these situations, 
2 .  Unconditional Distribution of  the Cell Frequencies (X oo’xol’ x 10 1.  
For fixed n, the probability distribution of  (Xoo,XolyX1o) will be 
u criven hy a m z l t i r . ~ ~ ~ i a ?  distribution with parameters n, floo,~ol and fl10’ 
X ) in terms of Z1,Z2 oo’xol’ 10 T’ne probability generating function of (X 
and Z3  will be given by 
+ T (  z y  P(Z 1’ z 2’ z 3 In) = (1 - roo - Tr o1 + Trl0 + 7100z1 + Tr01Z2 10 3 
I f  we assume that n is a random variable with truncated Poisson dis- 
t r ibut ion 
then the unconditional probability generating function of (X 00’ xol’ X I  10 
will be given by (Khatri [3]) 
Hence the unconditional distribution of (Xoo,Xol,X1o) will become (Cohen 
[ 21 1 f (XOO JO1 ,X10;ml,m*’m3, e )  
1-8 f o r  Xoo = Xol = Xl0 = 0 i 
-(m +m +m ) X Xol Xl0 1 2 3 m 0 0 m  
, otherwise .. .. 1 2 3 e 
-(m +m +m ) 
1 1 2 3  !X ! [l-e xoo!xol 10 
2 
00 
01 
10 
m = Xn 
m = XIT 
m = X R  
1 
2 
3 
Where 
00 1 =01 IT 
1-exp[-m -m -m ] 1 2 3  and e =  1-exp(-A) 
71 0. 
This may be considered as an extended multivariate 
3 .  Formulation of Hypotheses: 
Taking IT and  IT^ as the marginal probabilit .o 
Poisson distribution. 
es of given two charac:er- 
- istics, under the assumption of independence, roo =  IT.^ x roe ,  re1 -  IT.^ x no. 
and nlo = R x IT .o 1. 
10 IT 
IT .o I .1 R 
R 1. 
1 .o 
Thus we have, 
: Cell-probabilities can be written as the 
product of the marginal probabilites as shown 
above 
HO 
and H1: Cell probabilities cannot be written as the 
product of the marginal probabilities 
Under Ho, the probability distribution (2.1) becomes (after having 
the relation X = (m3+ml)(m2+ml) /ml) 
3 
m -  OOm, xol- 111, xlo 
, otherwise 1 L 5 Xoo!Xol! Xl0! [exp (Bj-exp (-a) ]
.... (3.1) 
Where a = m m /m and B = m +m +m 
convenience; our  interest is still lying only in m lym2 and m3. 
Here a and B are introduced only f o r  2 3  1 1 2 3’ 
4. Maximum Likelihood Estimates of the Parameter of distribution (2.1). 
We have a series of groups of observations and each group is divided 
into a 2 x 2 contingency table as described before. 
Xoo = i, Xol = j and Xl0 
Let fijk = frequency for 
= k. Then the likelihood function based on (2.1) will 
be given a5 
i j k  i j k  e exp (- B) mlm2m3 
L = (1 -e )  I 
i j k  i!j!k! [1-exp(-B)] 
i+ j +k>O 
Writing c = logi! + logj! + logk!, we get 1ogL = fooo10g(l-8) + c ,E fijk $+I + L O  
+ k 1 ogm3 - c - 1 o g [ 1 - exp ( - 6) 3 1 
Differentiating logL partially with respect to 0 ,  m 
A = l-exp(-B), we obtain 
m and m3 and writing 1’ 2 
I 4 
i 1  
i j k  
1 - a i 0  L = C C C f  j 1  
am2 i j k ijk 2 
k 1  
i j k  i j k  
Equating t h e  system (4.1) t o  zero,  we g e t  
A A 
t he  number of  2 x 2 tables,A = 1 - exp(-B) and n 
f i j k ,  
Where N = 
i j k  
A A A 
From last t h r e e  equations of (4 .21 ,  w e  g e t  
= ml + m2 + m3' 
A A 
m = m T  
1 2 00 / To1 
(4.3) 
S u b s t i t u t i n g  (4.3) i n  t h e  last  equation of  (4.2) w e  g e t  
5 
e 
Expanding the exponential term, step by step, first upto linear term in m2, 
then UPto quadratic term in m 
estimates of m2. 
of m become nearly equal. Finally ml and m will be obtained. 
* 
and so on, we shall get successive improved 2 
This method is t o  be repeated ytil two successive values 
A * 
2 3 
5 .  Maximum Likelihood Estimates of the parameters of distribution (2.1) 
Under Ho, the log likelihood will be given by 
+ C C C 
i j k  
fijk [ilogm +jlogm +klogm3-c], 1 2 
i+ j +k>O 
where c = logi! + logj! + logk! (from (3.1)) 
Differentialing partially with respect to m m and m we get 1' 2 3 
exp(B)-exp(-a)m m /m2 
I-exp(-a)m2m /m - { exp(bj - expt-a - 2  }N + N~oo/ml 
alogL= 000 
am 1 1- exp (-a) 3 1  
exp ( c 1 + exp ( - 3 )  m /n 
IN + Nyol/m2 alogL= f O O O  exp(-a)m3/mll - { 3,  1 am 2 1-exp (-a)' exp (8 )  -exp (-a) 
. . . . . .  (5.1) 
Let us  define  ai0 L = gi(ml,m2,m3). If ml, -0  m -0  and m3 *O are the first 2 i 
approximate values of the estimates of ml, m2 and m3 respectively, then 
according t o  the Taylor Series expansion, we Ret t h e  following equations. 
(Expansion is considered only upto the first power of (mi - mi)). 
A 
-0 
6 
J 
A n  .. 
By iteration, we can solve these equations for ml,m2, and m3. 
6 .  First Approximates under H of m m and m by using moments of first 0' 1' 2 3 
order and zero cell frequency. 
X ) is given by oo'xol' 10 Under Ho, the probability distribution of (X 
[l-exp(-a)] /[exp(B) - exp(-a)] 
for Xoo = Xol = Xl0 = 0 
xlo m 'Om Olm 
x ~ ~ ! x ~ ~ ! x ~ ~ !  [e p (B) - exp ( - a > j  , otherwise 
3 1 2  i f (Xoo BX01 'XlO;ml,m2 9m3> = 
Where CI = m m /m and f3 = m + m + m 2 3  1 1 2 3  
It can be shown that 
...... ( 6 . 1 )  
Now equating the probability of Xoo = Xol = Xl0 = 0 to the ratio of zero 
frequency to the total frequency in the sample, we get 
n .. . 
fOOO [exp(B) - exp(-a>] = N[l-exp(-a)] 
Equating (6.1) to the corresponding sample means, we get 
- - 
Hence, m = X m / Xoo 2 01 1 
...... (6.3) 
... (6.4) 
A A A 
Substituting exp(B) - exp(-a) = N[exp(B) -l]/(N-f 
first equation of (6.3), we obtain 
) (from (6.2))in the 000 
1 A - 
m + m + m = log xo0 - log [Too-ml+T 1 2 3 
Substituting (6.4) in (6.5), we get 
. . .  
It can be seen that equations (6.4) and (6.6) are equivalent to the equations 
(4.3) and (4.4). Thus we can use the maximum likelihood estimates obtained in 
the general case as the first approximates for the iteration procedure for 
obtaining the maximum likelihood estimates under H 0 '  
7. Likelihood Ratio Test. 
Let (KI,z2, ..., X ) be a sample from the probability density function 
(2.11, where X .  is 3-dimensional vector having the components (Xooi,Xoli,~loi)- 
Under HOy the p.d.f. (2.1) reduces to p.d.f. (3.1) containing only 3 parameters. 
Let Q4 be the 4-dimensional parameter space for which ml>O, m ' 0 ,  m3>0, 8>0. 
Let w3 be the subset of Rq for which H (Ho: the hypothesis of in- 0 
dependence of two given characteristics). Thus for this test the likelihood 
-P 
-1 
2 
is true. 
8 
r a t i o  h is given by 
l ikel ihood funct ion 
SUP 
u7 
J A =  
sup l ikel ihood function 
R4 
Where t h e  est imates  of the  parameters given i n  (5) are used t o  obta in  t h e  
numerator and those given i n  (4) are used t o  obta in  t h e  denominator. 
For l a rge  number of groups (number of 2x2 t a b l e s ) ,  t h e  d i s t r i b u t i o n  
of -210gA is approximately X2 w i t h  1 d.f .  i f  Ho is  true [Wilks (4 ) ] .  
8. A Numerical I l l u s t r a t i o n : -  
A random sample of s ize  31 was drawn from a Poisson population with 
mean h = 3.3333. The following a r e  t h e  observat ions.  
2,5,8,5,4,3,8,2,6,5,6,1, 2 , 5 , 3 , 6 , 4 , 3 , 3 , 2 , 5 , 2 , 4 , 6 , 2 , 3 , 1 , 2 , 4 , 4 , 4 .  
Each of t h e  above observations was considered as t h e  t o t a l  s ize  of a 2x2 
contingency t a b l e .  Thus, 31 t a b l e s  were constructed.  The marginal 
p r o b a b i l i t i e s  ao. and n. 
This gave t h e  following t a b l e  under the  hypothesis of independence. 
were selected t o  be 1/3 and 2/5 respec t ive ly .  0 
n = 1/3 0' IT = 3/15 01 aO0 = 2/15 
T = 2/3 1' n = 6/15 11 nl0 = 4/15 
1 
 IT.^ = 2/5 = 3/5 I /  1.0 
The t o t a l  s i ze  of a t a b l e  was divided randomly i n t o  4 cells according 
t o  t h e  multinomial d i s t r i b u t i o n  w i t h  t h e  parameters given by t h e  above 
t a b l e .  The following 2x2 t a b l e s  were observed: 
9 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
In the space R ,  the maximum likelihood estimates of ml, m2, m3and 0 were 
obtained using (4.21, (4.3) and (4.4). 
ml, m2 and m 
Using these as initial estimates of 
the maximum likelihood estimates of the parameters under the 3’ 
10 
1 0 3 1 5 
0 1 0 1 2 
1 0 1 2 4 
1 2 2 1 6 
1 0 0 1 2 
0 1 1 1 3 
0 0 0 1 1 
0 1 1 0 2 
0 1 2 1 4 
0 1 2 1 4 
1 1 1 1 4 
null hypothesis were obtained by carrying out the iteration procedure (5.2) 
on IBM 7090/7094 using Fortran IV.  
decimals in 3 cycles. 
The solution converged to five places of 
In the space R ,  we obtained 
m = 0.53610, m 1 2 = 0.85146, m3 = 1.07221 and 8 = 0.93550 
In the space w, we obtained 
n ,. 
m = 0.54197, m2 = 0.85561, m3 = 1.07796 1 
These results gave 
-210g X = 0.02422 e 
Considering -210gA as approximat ly x2 with 1 d.f., the valu of 
-210g~ is not at all significant resulting i n  the acceptance of H 1.e .  the 
hypothesis of independence. 
0 
2 Combining all the tables into one, x (usual test) was obtained as 
0 . 4 2 4 3 .  This value, too, is not significant. It should, however, be noted 
that the value of x 2  obtained according our test is much lower as compared 
to the one obtained by combining all the tables. 
test developed here may be more efficient. 
the efficiency aspect are necessary before reaching a final conclusion. 
This suggests that the 
However, further studies into 
Acknowledeement 
We wish to express our sincere thanks to Dr. A .  Clifford Cohen, Jr. 
of the University of Georgia for his suggestions in preparing this paper. 
We are also thankful to Dr. B. J .  Williams and Dr. James E. Norman of the 
University of Georgia for  the useful discussion'we had while preparing this 
paper. 
REFERENCES 
[l] Cochran, W. G. (1954). Some Methods for Strengthening the Common 
Chi-square Tests. Biometrics, 10, pp 417-51. 
[ 2 [  Cohen, A .  C.,Jr. (1960). An extension of a truncated Poisson Distribution. 
Biometrics, 16, pp 446-450 
[3] Khatri, C. G. (1961). On the distributions obtained by varying the 
number of trials in a binomial distribution. Annals of the Institute of 
Statistical Mathematics, Vol. XIII, No. 1 
[4] Wilks, S. S. (1962). Mathematical Statistics. John Wiley & Sons, Inc. 
12 


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