K - College of Natural Resources - University of California, Berkeley

Descriptions of all outbreaks and issues specific to each dataset are outlined later in the
SOM.
For the Poisson, geometric and negative binomial models, the maximum
likelihood estimate of the basic reproductive number (R0 or R) was the sample mean
(Rice 1995). For the negative binomial distribution, the shape parameter k was also
estimated by maximum likelihood estimation (mle). A numerical algorithm was used
for maximization, which required that an upper bound be set for the range of k sampled.
This upper bound was set to k=1000, and when this bound was returned as the mle
estimate then ˆ was set to infinity, because a NegB(R0,k=1000) distribution is
k mle
indistinguishable in practice from Poisson(R0).
Having computed the maximum likelihood scores for each dataset, we compared
the Poisson, geometric and negative binomial models using Akaike’s information
criterion (AIC) (Anderson et al. 2000):
where ( L(
ˆ | data )
( L(
ˆ | data ) K
AIC = -2 ln θ + 2
ln θ is the log-likelihood maximized over the unknown parameters (θ),
given the model and the data, and K is the number of parameters estimated in the model.
Because some of our datasets are small, we used the modified criterion AICc, which
reduces to the conventional expression as sample size N becomes larger (Anderson et al.
2000):
AIC c
= -2 ln θ
( ( ˆ
2K
) ( K + 1)
L | data + 2K
+
129
N − K −1