Biology Reference
In-Depth Information
and Chapter 9)
119,120
to fully stochastic considerations of changes in the
number of parasites within individual hosts.
63
e
Statistical Models
Here we highlight some statistical methods that are of broad appli-
cability to the analysis of parasitological data in general and of
A. lumbricoides in particular. Emphasis is placed on appropriate univariate
and multivariate (regression) analysis of overdispersed count data, often
with an excess of zeros; the use of hierarchical methods for clustered and
longitudinal data, and the closely related and burgeoning discipline of
temporal
spatial data analysis. We also discuss briefly more general
methods of fitting non-linear and population dynamics models to data.
e
Negative Binomial and Zero-inflated Negative Binomial
Distributions
In 1941, Fisher
121
successfully used the negative binomial distribution
to describe the overdispersed distribution of ticks on sheep. Since this
time, the distribution has become a ubiquitous and well-validated
description of the distribution of parasites of humans,
16,20
wildlife
122
and of adult A. lumbricoides.
57
In parasitology, the negative binomial
distribution is expressed in terms of its mean, m, and dispersion param-
eter, k, which inversely describes the degree of parasite aggregation. The
Poisson distribution is obtained as k
and the log series distribution
/N
0.
123,124
The geometric distribution is a special case of the
negative binomial distribution when k
arises as k
/
1.
Two methods are commonly used to estimate k. The moment estimator
(so called because it is based on the first two moments, the mean and the
variance s) k
¼
m
2
/(s
2
m) is derived from equating the variance of the
negative binomial distribution to the sample variance, s
2
.
125
This estimator
substantially overestimates k (underestimates the degree of over-
dispersion) at low sample sizes.
126
A refined moment estimator, which
partially corrects for this bias, is given by k
¼
e
m).
127
The
maximum likelihood estimator of k
123
is generally recommended over
moment estimators
128
owing to its reduced bias and superior asymptotic
efficiency.
129
e
131
Since k is a critical parameter in population dynamics models of
A. lumbricoides and of helminth parasites more generally, inaccurate or
biased estimation may profoundly affect the output from such models.
A straightforward method of mitigating such effects is to estimate
parameters from sufficiently large sample sizes. However, this may be
less straightforward when estimators are required for different pop-
ulation strata; obtaining sufficiently large sample sizes from different age
groups typically requires increased sampling effort for older individuals,
(m
2
s
2
/n)/(s
2
¼
e
e
