.
Epidemiologists are interested in studying the sexual behavior of individuals at risk for HIV infection. Suppose 1500 gay men were surveyed and each was asked how many risky sexual encounters he had in the previous 30 days. Let
ni
denote the number of respondents reporting
i
encounters, for
i
= 1, . . . ,
16. summarizes the responses. These data are poorly fitted by a Poisson model. It is more realistic to assume that the respondents comprise three groups. First, there is a group of people who, for whatever reason, report zero risky encounters even if this is not true. Suppose a respondent has probability
α
of belonging to this group. With probability
β, a respondent belongs to a second group representing typical behavior. Such people respond truthfully, and their numbers of risky encounters are assumed to follow a Poisson(μ) distribution. Finally, with probability 1 −
α
−
β, a respondent belongs to a high-risk group. Such people respond truthfully, and their numbers of risky encounters are assumed to follow a Poisson(λ) distribution. The parameters in the model are
α,
β,
μ, and
λ. At the
tth iteration of EM, we use
θ
(t) = (α(t), β(t),μ(t), λ(t)) to denote the current parameter values. The likelihood of the observed data is given by Where
πi(
θ
) =
α1{i=0} +
βμi
exp{−μ} + (1 −
α
−
β)λi
exp{−λ} (4.82) for
i
= 1, . . . ,
16.
The observed data are
n0, . . . , n16. The complete data may be construed to be
nz,0,
nt,0, . . . , nt,16, and
np,0, . . . , np,16, where
nk,i
denotes the number of respondents in group
k
reporting
i
risky encounters and
k
=
z,
t, and
p
correspond to the zero, typical, and promiscuous groups, respectively. Thus,
n0 =
nz,0 +
nt,0 +
np,0 and
ni
=
nt,i
+
np,I
for
i
= 1, . . . ,
16. Let
N
= _16 for
i
= 0, . . . ,
16. These correspond to probabilities that respondents with
i
risky encounters belong to the various groups.
a.
Show that the EM algorithm provides the following updates:
b.
Estimate the parameters of the model, using the observed data.
c.
Estimate the standard errors and pairwise correlations of your parameter estimates, using any available method.