Separating two sources of dependence in repeated influenza outbreaks

SUMMARY Two or more responses measured on a randomly selected individual can be dependent for two quite different reasons. In the first place they can be dependent within, that is conditional upon, each individual. Secondly, dependence is created simply from the heterogeneity between individuals. This paper addresses a rare instance of being able to measure separately the dependence from the two sources. The data comprise frequencies of the infection profiles for a sample of individuals exposed to four influenza outbreaks. Two of the outbreaks were caused by the same virus and gave rise to negative withinindividual dependence because infection at the first occasion leads to a reduced probability of infection at the second. This negative dependence cannot be estimated from the 2 x 2 table of frequencies for these two outbreaks because it is masked by positive dependence due to between individual heterogeneity. However the negative dependence can be estimated using the joint frequencies for three or four outbreaks, the two caused by the same virus and one or both of the other two.