Research Article
Volume 3 Issue 4  2016
On PoissonSujatha Distribution and its Applications to Model Count Data from Biological Sciences
Rama Shanker^{1}*, Hagos Fesshaye^{2}
Department of Statistics, Eritrea Institute of Technology, Eritrea
Received: January 29, 2016  Published: March 10, 2016
*Corresponding author:
Rama Shanker, Department of Statistics, Eritrea Institute of Technology, Asmara, Eritrea, Email:
Citation:
Shanker R, Fesshaye H (2016) On PoissonSujatha Distribution and its Applications to Model Count Data from Biological Sciences. Biom
Biostat Int J 3(4): 00069. DOI:
10.15406/bbij.2016.03.00069
Abstract
In this paper a simple method for finding moments of PoissonSujatha distribution (PSD) introduced by Shanker [1] has been suggested and hence the first four moments about origin and the variance has been given. The PSD has been fitted to the same datasets relating to ecology and genetics to which earlier Shanker & Hagos [2] has fitted PoissonLindley distribution (PLD) introduced by Sankaran [3] and Poissondistribution (PD) and the goodness of fit of PSD shows satisfactory fit in majority of datasets.
Keywords: Sujatha distribution; PoissonSujatha distribution; Lindley distribution; PoissonLindley distribution; Moments; Compounding; Estimation of parameter; Goodness of fit
Introduction
The PoissonSujatha distribution (PSD) having probability mass function
$P\left(X=x\right)=\frac{{\theta}^{3}}{{\theta}^{2}+\theta +2}\cdot \frac{{x}^{2}+\left(\theta +4\right)x+\left({\theta}^{2}+3\theta +4\right)}{{\left(\theta +1\right)}^{x+3}}\text{\hspace{0.17em}}\text{\hspace{0.17em}};x=0,1,2,\mathrm{...},\theta >0$
(1.1)
has been introduced by Shanker [1] for modeling count datasets. The PSD arises from Poisson distribution when its parameter follows Sujatha distribution introduced by Shanker [4] having probability density function
$f\left(\lambda ;\theta \right)=\frac{{\theta}^{3}}{{\theta}^{2}+\theta +2}\left(1+\lambda +{\lambda}^{2}\right){e}^{\theta \text{\hspace{0.17em}}\lambda}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}};\lambda >0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\theta >0$
(1.2)
We have
$P\left(X=x\right)={\displaystyle \underset{0}{\overset{\infty}{\int}}\frac{{e}^{\lambda}{\lambda}^{x}}{x!}}\cdot \frac{{\theta}^{3}}{{\theta}^{2}+\theta +2}\left(1+\lambda +{\lambda}^{2}\right){e}^{\theta \text{\hspace{0.17em}}\lambda}d\lambda $
(1.3)
$=\frac{{\theta}^{3}}{\left({\theta}^{2}+\theta +2\right)\text{\hspace{0.17em}}x!}{\displaystyle \underset{0}{\overset{\infty}{\int}}{\lambda}^{x}}\left(1+\lambda +{\lambda}^{2}\right){e}^{\left(\theta +1\right)\text{\hspace{0.17em}}\lambda}d\lambda $
$=\frac{{\theta}^{3}}{{\theta}^{2}+\theta +2}\cdot \frac{{x}^{2}+\left(\theta +4\right)x+\left({\theta}^{2}+3\theta +4\right)}{{\left(\theta +1\right)}^{x+3}}\text{\hspace{0.17em}}\text{\hspace{0.17em}};x=0,1,2,\mathrm{...},\theta >0$
(1.4)
Which is the PoissonSujatha distribution (PSD).
Shanker [4] has shown that the Sujatha distribution (1.2) is a three component mixture of an exponential (θ) distribution, a gamma (2,θ) distribution, and a gamma (3,θ) distributionwith their mixing proportions
$\frac{{\theta}^{2}}{{\theta}^{2}+\theta +2}$
,
$\frac{\theta}{{\theta}^{2}+\theta +2}$
and
$$\frac{2}{{\theta}^{2}+\theta +2}$$
respectively. Shanker [4] has discussed its various mathematical and statistical properties including its shape, moment generating function, moments, skewness, kurtosis, hazard rate function, mean residual life function, stochastic orderings, mean deviations, distribution of order statistics, Bonferroni and Lorenz curves, Renyi entropy measure, stressstrength reliability , amongst others along with the estimation of the parameter and applications for modeling lifetime data.
Shanker [1] has detailed study about various mathematical and statistical properties of PSD including moment generating function, coefficient of variation, skewness, kurtosis, overdispersion, hazard rate and unimodality along with the estimation of the parameter and applications. Shanker & Hagos [5,6] have obtained sizebiased PoissonSujatha distribution (SBPSD) and zerotruncated PoissonSujatha distribution(ZTPSD) and discussed their statistical properties, estimation of the parameter and applications. Further, Shanker & Hagos [7] have detailed study about zerotruncation of Poisson, PoissonLindley and PoissonSujatha distributions and their applications.
The probability mass function of PoissonLindley distribution (PLD) given by
$P\left(X=x\right)\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{{\theta}^{2}\text{\hspace{0.17em}}\left(x+\text{\hspace{0.17em}}\theta +\text{\hspace{0.17em}}2\right)}{{\left(\theta +\text{\hspace{0.17em}}1\right)}^{x+\text{\hspace{0.17em}}3}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}};$
x = 0, 1, 2,…,θ > 0. (1.5)
has been introduced by Sankaran [3] to model count data. The distribution arises from the Poisson distribution when its parameter follows Lindley [8] distribution with its probability density function
$f\text{\hspace{0.17em}}\left(\lambda ,\text{\hspace{0.17em}}\theta \right)\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{{\theta}^{2}}{\theta +\text{\hspace{0.17em}}1}\text{\hspace{0.17em}}\left(1+\text{\hspace{0.17em}}\lambda \right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}{e}^{\theta \text{\hspace{0.17em}}\lambda}$
;
$x>0\text{\hspace{0.17em}},\theta >0$
(1.6)
In this paper a simple method for finding moments of PoissonSujatha distribution (PSD) introduced by Shanker [1] has been suggested and hence the first four moments about origin and the variance has been presented. It seems that not much work has been done on the applications of PSD so far. The PSD has been fitted to the same datasets relating to ecology and genetics to which Shanker & Hagos [2] has fitted PoissonLindley distribution (PLD) introduced by Sankaran [3] and Poissondistribution (PD) and the goodness of fit of PSD shows satisfactory fit in majority of datasets.
Moments of PoissonSujatha Distribution
Using (1.3) the
$r$
th moment about origin of PSD (1.1) can be obtained as
$r$
$=\frac{{\theta}^{3}}{{\theta}^{2}+\theta +2}{\displaystyle \underset{0}{\overset{\infty}{\int}}\left[{\displaystyle \sum _{x=0}^{\infty}{x}^{r}\frac{{e}^{\lambda}{\lambda}^{x}}{x!}}\right]}\left(1+\lambda +{\lambda}^{2}\right){e}^{\theta \text{\hspace{0.17em}}\lambda}d\lambda $
(2.1)
Clearly the expression under the bracket in (2.1) is the
$r$
th moment about origin of the Poisson distribution. Taking
$r=1$
in (2.1) and using the first moment about origin of the Poisson distribution, the first moment about origin of the PSD (1.1) can be obtained as
${\mu}_{2}{}^{\prime}=\frac{{\theta}^{3}}{{\theta}^{2}+\theta +2}{\displaystyle \underset{0}{\overset{\infty}{\int}}\left({\lambda}^{2}+\lambda \right)\left(1+\lambda +{\lambda}^{2}\right)}\text{\hspace{0.17em}}{e}^{\theta \text{\hspace{0.17em}}\lambda}d\lambda =\frac{{\theta}^{3}+4{\theta}^{2}+12\theta +24}{{\theta}^{2}\left({\theta}^{2}+\theta +2\right)}$
(2.2)
Again taking
$r=2$
in (2.1) and using the second moment about origin of the Poisson distribution, the second moment about origin of the PSD (1.1) is obtained as
$${\mu}_{2}{}^{\prime}=\frac{{\theta}^{3}}{{\theta}^{2}+\theta +2}{\displaystyle \underset{0}{\overset{\infty}{\int}}\left({\lambda}^{2}+\lambda \right)\left(1+\lambda +{\lambda}^{2}\right)}\text{\hspace{0.17em}}{e}^{\theta \text{\hspace{0.17em}}\lambda}d\lambda =\frac{{\theta}^{3}+4{\theta}^{2}+12\theta +24}{{\theta}^{2}\left({\theta}^{2}+\theta +2\right)}$$
(2.3)
Similarly, taking
$r=3\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}\text{\hspace{0.17em}}4$
in (2.1) and using the third and the fourth moment about origin of the Poisson distribution, the third and the fourth moment about origin of the PSD (1.1) are obtained as
${\mu}_{3}{}^{\prime}=\frac{{\theta}^{4}+8{\theta}^{3}+30{\theta}^{2}+96\theta +120}{{\theta}^{3}\left({\theta}^{2}+\theta +2\right)}$
(2.4)
${\mu}_{4}{}^{\prime}=\frac{{\theta}^{5}+16{\theta}^{4}+84{\theta}^{3}+336{\theta}^{2}+840\theta +720}{{\theta}^{4}\left({\theta}^{2}+\theta +2\right)}$
(2.5)
Thus the variance of the PSD (1.1) can be obtained as
${\mu}_{2}=\frac{{\theta}^{5}+4{\theta}^{4}+14{\theta}^{3}+28{\theta}^{2}+24\theta +12}{{\theta}^{2}{\left({\theta}^{2}+\theta +2\right)}^{2}}$
(2.6)
Shanker [1] has shown that the PSD is always overdispersed, has increasing hazard rate and unimodal. Further, Shanker [1] has also shown that the graphs of coefficient of variation, skewness, and kurtosis of PSD are increasing for increasing values of the parameter.
Estimation of the Parameter
Maximum likelihood estimate (MLE) of the parameter: Let
$\left({x}_{1},{x}_{2},\mathrm{...},{x}_{n}\right)$
be a random sample of size
$n$
from the PSD (1.1) and let
${f}_{x}$
be the observed frequency in the sample corresponding to
$X=x\text{\hspace{0.17em}}\text{\hspace{0.17em}}(x=1,2,3,\mathrm{...},k)$
such that
$\sum _{x=1}^{k}{f}_{x}}=n$
, where
$k$
is the largest observed value having nonzero frequency. The likelihood function
$L$
of the PSD (1.1) is given by
$L={\left(\frac{{\theta}^{3}}{{\theta}^{2}+\theta +2}\right)}^{n}\frac{1}{{\left(\theta +1\right)}^{{\displaystyle \sum _{x=1}^{k}{f}_{x}\left(x+3\right)}}}{\displaystyle \prod _{x=1}^{k}{\left[{x}^{2}+\left(\theta +4\right)x+\left({\theta}^{2}+3\theta +4\right)\right]}^{{f}_{x}}}$
The log likelihood function is thus obtained as
$\mathrm{log}L=n\mathrm{log}\left(\frac{{\theta}^{3}}{{\theta}^{2}+\theta +2}\right){\displaystyle \sum _{x=1}^{k}{f}_{x}\left(x+3\right)}\mathrm{log}\left(\theta +1\right)+{\displaystyle \sum _{x=1}^{k}{f}_{x}\mathrm{log}\left[{x}^{2}+\left(\theta +4\right)x+\left({\theta}^{2}+3\theta +4\right)\right]}$
The first derivative of the log likelihood function is given by
$\frac{d\mathrm{log}L}{d\theta}=\frac{n\left({\theta}^{2}+2\theta +6\right)}{\theta \left({\theta}^{2}+\theta +2\right)}\frac{n\left(\overline{x}+3\right)}{\theta +1}+{\displaystyle \sum _{x=1}^{k}\frac{\left[x+\left(2\theta +3\right)\right]{f}_{x}}{\left[{x}^{2}+\left(\theta +4\right)x+\left({\theta}^{2}+3\theta +4\right)\right]}}$
Where
$\overline{x}$
is the sample mean.
The maximum likelihood estimate (MLE),
$\widehat{\theta}$
of
$\theta $
of PSD (1.1) is the solution of the equation
$\frac{d\mathrm{log}L}{d\theta}=0$
and is given by the solution of the following nonlinear equation
$\frac{n\left({\theta}^{2}+2\theta +6\right)}{\theta \left({\theta}^{2}+\theta +2\right)}\frac{n\left(\overline{x}+3\right)}{\theta +1}+{\displaystyle \sum _{x=1}^{k}\frac{\left[x+\left(2\theta +3\right)\right]{f}_{x}}{\left[{x}^{2}+\left(\theta +4\right)x+\left({\theta}^{2}+3\theta +4\right)\right]}}=0$
This nonlinear equation can be solved by any numerical iteration methods such as Newton Raphson, Bisection method, Regula–Falsi method etc.
Method of moment estimate (MOME) of the parameter: Let
$\left({x}_{1},{x}_{2},\mathrm{...},{x}_{n}\right)$
be a random sample of size
$n$
from the PSD (1.1). Equating the population mean to the corresponding sample mean, the MOME
$\tilde{\theta}$
of
$\theta $
of PSD (1.1) is the solution of the following cubic equation
$\overline{x}{\theta}^{3}+\left(\overline{x}1\right){\theta}^{2}+2\left(\overline{x}1\right)\theta 6=0$
Where
$\overline{x}$
is the sample mean.
Applications of PoissonSujatha Distribution
The Poisson distribution is a suitable statistical model for the situations where events seem to occur at random including the number of customers arriving at a service point, the number of telephone calls arriving at an exchange, the number of fatal traffic accidents per week in a given state, the number of radioactive particle emissions per unit of time, the number of meteorites that collide with a test satellite during a single orbit, the number of organisms per unit volume of some fluid, the number of defects per unit of some materials, the number of flaws per unit length of some wire, are some amongst others. Since the condition for the applications for Poisson distribution is the independence of events and the equality of mean and variance, this condition is rarely satisfied completely in biological and medical science due to the fact that the occurrences of successive events are dependent. Further, the negative binomial distribution is a possible alternative to the Poisson distribution when successive events are possibly dependent Johnson et al. [9], but for fitting negative binomial distribution (NBD) to the count data, mean should be less than the variance. In biological and medical sciences, these conditions are also not fully satisfied. Generally, the count data in biological science and medical science are either overdispersed or underdispersed. The main reason for selecting PLD and PSD to fit biological science data is that these two distributions are always overdispersed and PSD has some flexibility over PLD.
Applications in ecology
Ecology is the branch of biology dealing with the relations and interactions between organisms and their environment, including other organisms. The organisms and their environment in the nature are complex, dynamic, interdependent, mutually reactive and interrelated. Ecology deals with the various principles which govern such relationship between organisms and their environment. It was Fisher et al. [10] who have firstly discussed the applications of Logarithmic series distribution (LSD) to model count data in the science of ecology. Later, Kempton [11] who fitted the generalized form of Fisher’s Logarithmic series distribution (LSD) to model insect data and concluded that it gives a superior fit as compared to ordinary Logarithmic series distribution (LSD). He also concluded that it gives better explanation for the data having exceptionally long tail. Tripathi & Gupta [12] proposed another generalization of the Logarithmic series distribution (LSD) which is flexible to describe shorttailed as well as longtailed data and fitted it to insect data and found that it gives better fit as compared to ordinary Logarithmic series distribution. Mishra & Shanker [13] have discussed applications of generalized logarithmic series distributions (GLSD) to models data in ecology. Shanker & Hagos [2] have tried to fit PLD for data relating to ecology and observed that PLD gives satisfactory fit.
In this section we have tried to fit Poisson distribution (PD), PoissonLindley distribution (PLD) and PoissonSujatha distribution (PSD) to many count data from biological sciences using maximum likelihood estimates. The data were on haemocytometer yeast cell counts per square, on European red mites on apple leaves and European corn borers per plant (Table 13).
It is obvious from above tables that both PSD and PLD give much closer fit than Poisson distribution. Further, in some datasets PSD gives much closer fit than PLD while in some datasets PLD gives much closer fit than PSD and thus both PSD and PLD can be considered as important tools for modeling data in ecology.
Number of Yeast Cells per Square 
Observed Frequency 
Expected Frequency 
PD 
PLD 
PSD 
0 
213 
202.1 
234.0 
233.2 
1 
128 
138.0 
99.4 
99.6 
2 
37 
47.1 
40.5 
41.0 
3 
18 
$\begin{array}{l}10.7\\ 1.8\\ 0.2\\ 0.1\end{array}\}$

$\begin{array}{l}16.0\\ 6.2\\ 2.4\\ 1.5\end{array}\}$

$\begin{array}{l}16.3\\ 6.7\\ 2.3\\ 0.9\end{array}\}$

4 
3 
5 
1 
6 
0 
Total 

400.0 
400.0 
400.0 
Estimate of Parameter 

$\widehat{\theta}=0.6825$



$${\chi}^{2}$$


10.08 
11.04 
10.86 
d.f. 

2 
2 
2 
pvalue 

0.0065 
0.0040 
0.0044 
Table 1: Observed and expected number of Haemocytometer yeast cell counts per square observed by ‘Student’ [17].
Number Mites per Leaf 
Observed Frequency 
Expected Frequency 
PD 
PLD 
PSD 
0 
38 
25.3 
35.8 
35.3 
1 
17 
29.1 
20.7 
20.9 
2 
10 
16.7 
11.4 
11.6 
3 
9 
$\begin{array}{l}6.4\\ 1.8\\ 0.4\\ 0.2\\ 0.1\end{array}\}$

6
$\begin{array}{l}3.1\\ 1.6\\ 0.8\\ 0.6\end{array}\}$

6.1
$\begin{array}{l}3.1\\ 1.5\\ 0.7\\ 0.8\end{array}\}$

4 
3 
5 
2 
6 
1 
7+ 
0 
Total 
80 
80 
80 
80 
Estimate of Parameter 

$\widehat{\theta}$
=1.15 
$\widehat{\theta}$
=1.255891 
$\widehat{\theta}$
=1.64683 
${\chi}^{2}$


18.27 
2.47 
2.52 
d.f. 

2 
3 
3 
pvalue 

0.0001 
0.4807 
0.4719 
Table 2: Observed and expected number of red mites on Apple leaves.
Number of Bores per Plant 
Observed Frequency 
Expected Frequency 
PD 
PLD 
PSD 
0 
188 
169.4 
194.0 
193.6 
1 
83 
109.8 
79.5 
79.6 
2 
36 
35.6 
31.3 
31.6 
3 
14 
$\begin{array}{l}7.8\\ 1.2\\ 0.2\end{array}\}$

$\begin{array}{l}12.0\\ 4.5\\ 2.7\end{array}\}$

$\begin{array}{l}12.1\\ 4.5\\ 2.6\end{array}\}$

4 
2 
5 
1 
Total 
324 
324.0 
324.0 
324.0 
Estimate of parameter 

$\widehat{\theta}=0.648148$

$\widehat{\theta}=2.043252$

$\widehat{\theta}=2.471701$

${\chi}^{2}$


15.19 
1.29 
1.16 
d.f. 

2 
2 
2 
pvalue 

0.0005 
0.5247 
0.5599 
Table 3: Observed and expected number of European corn borer of Mc Guire et al. [18].
It is obvious from above tables that in table 1, PD gives better fit than PLD and PSD; in table 2 PLD gives better fit than PD and PSD while in table 3, PSD gives better fit than PD and PLD.
Application in genetics
Genetics is the branch of biological science which deals with heredity and variation. Heredity includes those traits or characteristics which are transmitted from generation to generation, and is therefore fixed for a particular individual. Variation, on the other hand, is mainly of two types, namely hereditary and environmental. Hereditary variation refers to differences in inherited traits whereas environmental variations are those which are mainly due to environment. The segregation of chromosomes has been studied using statistical tool, mainly chisquare (
${\chi}^{2}$
). In the analysis of data observed on chemically induced chromosome aberrations in cultures of human leukocytes, Loeschke & Kohler [14] suggested the negative binomial distribution while Janardan & Schaeffer [15] suggested modified Poisson distribution. Mishra and Shanker [13] have discussed applications of generalized Logarithmic series distributions (GLSD) to model data in mortality, ecology and genetics. Shanker & Hagos [2] have detailed study on the applications of PLD to model data from genetics. Much quantitative works seem to be done in genetics but so far no works has been done on fitting of PSD to data relating to genetics. In this section an attempt has been made to fit to data relating to genetics using PSD, PLD and PD using maximum likelihood estimate. Also an attempt has been made to fit PSD, PLD, and PD to the data of Catcheside et al. [16] in Table 47.
It is obvious from the fitting of PSD, PLD, and PD that both PSD and PLD gives much satisfactory fit than PD while in some datasets PSD gives much closer fit than PLD whereas PLD gives much closer fit than PSD in some datasets. Thus both PSD and PLD can be considered as important tools for modeling data in genetics
Number of Aberrations 
Observed Frequency 
Expected Frequency 
PD 
PLD 
PSD 
0 
268 
231.3 
257 
257.6 
1 
87 
126.7 
93.4 
93 
2 
26 
34.7 
32.8 
32.7 
3 
9 
$\begin{array}{l}6.3\\ 0.8\\ 0.1\\ 0.1\\ 0.1\end{array}\}$

11.2
$\begin{array}{l}3.8\\ 1.2\\ 0.4\\ 0.2\end{array}\}$

11.2
$\begin{array}{l}3.7\\ 1.2\\ 0.4\\ 0.2\end{array}\}$

4 
4 
5 
2 
6 
1 
7+ 
3 
Total 
400 
400 
400 
400 
Estimate of Parameter 

$\widehat{\theta}$
=0.5475 
$\widehat{\theta}$
=2.380442 
$\widehat{\theta}$
=2.829241 
${\chi}^{2}$


38.21 
6.21 
6.28 
d.f. 

2 
3 
3 
pvalue 

0 
0.1018 
0.0987 
Table 4: Distribution of number of Chromatid aberrations (0.2 g chinon 1, 24 hours).
Class/Exposure
$\mu gkg$

Observed Frequency 
Expected Frequency 
PD 
PLD 
PSD 
0 
413 
374 
405.7 
406.1 
1 
124 
177.4 
133.6 
132.9 
2 
42 
42.1 
42.6 
42.7 
3 
15 
$\begin{array}{l}6.6\\ 0.8\\ 0.1\\ 0.0\end{array}\}$

13.3
$\begin{array}{l}4.1\\ 1.2\\ 0.5\end{array}\}$

13.4
$\begin{array}{l}4.1\\ 1.2\\ 0.6\end{array}\}$

4 
5 
5 
0 
6 
2 
Total 
601 
601 
601 
601 
Estimate of parameter 

$\widehat{\theta}$
=0.47421 
$\widehat{\theta}$
=2.685373 
$\widehat{\theta}$
=3.125788 
${\chi}^{2}$


48.17 
1.34 
1.1 
d.f. 

2 
3 
3 
pvalue 

0 
0.7196 
0.7771 
Table 5: Mammalian cytogenetic dosimetry lesions in rabbit lymphoblast induced by streptonigrin (NSC45383), Exposure 60
$\mu gkg$
.
Class/Exposure
$\mu gkg$

Observed Frequency 
Expected Frequency 
PD 
PLD 
PSD 
0 
200 
172.5 
191.8 
192 
1 
57 
95.4 
70.3 
70.1 
2 
30 
26.4 
24.9 
24.9 
3 
7 
$\begin{array}{l}4.9\\ 0.7\\ 0.1\\ 0.0\end{array}\}$

$\begin{array}{l}8.6\\ 2.9\\ 1.0\\ 0.5\end{array}\}$

$\begin{array}{l}8.6\\ 2.9\\ 0.9\\ 0.6\end{array}\}$

4 
4 
5 
0 
6 
2 
Total 
300 
300 
300 
300 
Estimate of parameter 

$\widehat{\theta}$
=0.55333 
$\widehat{\theta}$
=2.353339 
$\widehat{\theta}$
=2.795745 
${\chi}^{2}$


29.68 
3.91 
3.81 
d.f. 

2 
2 
2 
pvalue 

0 
0.1415 
0.1488 
Table 6: Mammalian cytogenetic dosimetry lesions in rabbit lymphoblast induced by streptonigrin (NSC45383), Exposure 70
$\mu gkg$
.
Class/Exposure
$\mu gkg$

Observed Frequency 
Expected Frequency 
PD 
PLD 
PSD 
0 
155 
127.8 
158.3 
157.5 
1 
83 
109 
77.2 
77.5 
2 
33 
46.5 
35.9 
36.4 
3 
14 
$\begin{array}{l}13.2\\ 2.8\\ 0.5\\ 0.2\end{array}\}$

16.1
$\begin{array}{l}7.1\\ 3.1\\ 2.3\end{array}\}$

16.4
$\begin{array}{l}7.1\\ 3.0\\ 2.1\end{array}\}$

4 
11 
5 
3 
6 
1 
Total 
300 
300 
300 
300 
Estimate of parameter 

$\widehat{\theta}$
=0.853333 
$\widehat{\theta}$
=1.617611 
$\widehat{\theta}$
=2.034077 
${\chi}^{2}$


24.97 
1.51 
1.74 
d.f. 

2 
3 
3 
pvalue 

0 
0.6799 
0.6281 
Table 7: Mammalian cytogenetic dosimetry lesions in rabbit lymphoblast induced by streptonigrin (NSC45383), Exposure 90
$\mu gkg$
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