Measuring the Performance of Vaccination Programs Using Cross-Sectional Surveys: A Likelihood Framework and Retrospective Analysis

Justin Lessler and colleagues describe a method that estimates the fraction of a population accessible to vaccination activities, and they apply it to measles vaccination in three African countries: Ghana, Madagascar, and Sierra Leone.


Individual Vaccination Probability and Likelihood Formulation
The probability that an individual is vaccinated at age x is one minus the probability that they avoid vaccination during every vaccination activity to which they are exposed. Assume that there is some portion of the population, ρ, that is accessible to vaccination activities: g(x; ρ) = 1 − (1 − ρ) + ρ m j=1 Pr(not vaccinated in V j | accessible) where V 1 , ..., V m are all vaccination activities to which the child might have exposed. Let f (V j ) be the probability of not being vaccinated in activity j given that you are in the target population for that activity and in the accessible population. Let z ij = 1 if person i is in the target population for campaign j, and z ij = 0 otherwise. Hence: The probability of not being vaccinated given that you are in the accessible population, f (V j ) should be some function of the number of doese nominally distributed in campaign j, v j , and the size of the accessible target population for that activity, ρN j .
If all nominally distributed doses go into a unique vacinee in the target population, then f (v j , ρN j ) = 1 − v j /(ρN j ). However, it seems we can assume that all nominally distributed doses do not result in a unique vaccinee within the target population. If we consider our doses to be a sequence, k = 0 . . . (v j − 1), it further seems reasonable to assume that the chances of the first dose in this sequence is more likely to result in a unique vaccinee than later doses. This effect can be captured by the equation: where ψ is a discount factor on how much the effective denominator changes on additional doses. That is, the term −k(1 − ψ) denotes how much the effective denominator (i.e., the number of people competing for doses) decreases because k doses have been given. If a campaign is perfect, then ψ = 0, and each dose in the sequence decreases the denominator by exactly 1 (and f (v j , ρN j ) = 1 − v j /(ρN j )). If a campaign is effectively at random (i.e., the fact that doses have been previously distributed does not increase a new person's chance of receiving the next dose) then ψ = 1, and the probability of receiving (or avoiding) a dose remains constant. We would expect most vaccination activities to fall somewhere in this range. However, while it may be unlikely, we can even imagine a situation where there are "vaccine hungry" individuals who try to get vaccinated as many times as possible. In this case ψ > 1, and subsequent doses are even less likely to result in a unique vaccinee. Because values of ψ less than 0 are nonsensical (no dose can result in more than 1 vaccinee), we restate ψ in terms of α: We will use e α instead of ψ throughout the supplement. Equation 1 now becomes: This equation can be further simplified by finding a closed form solution for the inner product as detailed in section 2 below.
In a cross sectional survey we observe a set of individuals with ages x = {x 1 , · · · , x n } and corresponding vaccination statuses y = {y 1 , · · · , y n }, where y i = 1 denotes having ever been vaccinated, and y i = 0 denotes having never been vaccinated. If we assume all y i are independent events, then the likelihood of observing the cross sectional data given ρ and α is:

Derivation of Simplified Form for Vaccination Probability
The probability that person i is vaccinated is: Let the portion of this equation that depends on v j and ρN j be designated f (v j , ρN j ): Dropping the subscripts and taking ρN j = N for convienence, note that: Hence, by the rectangular quadrature formula: Therefore (see limit calculations below): Note that the above expression is undefined when α = 0. However: Therefore, for large N : And: Note that this convergence appears to occur very quickly. Empirically, it appears that this value is accurate to three decimal places for N > 100 in sample scenarios.

Individual Campaign Coverage
Denote the actual coverage of a campaign j to be c j . Note that c j is the probability of a person covered only by campaign (or pseudo-campaign) j being vaccinated. Hence:

Routine Vaccination
Routine vaccination differs from campaigns in that children are vaccinated over a much larger time scale than is true of campaigns. However, routine vaccination can be modeled within our framework as a special type or vaccination activity. Consider R years of routine vaccination activitiy, 1...R. Denote the event of a member of the accessible population having the "opportunity" for vaccination during year j of routine vaccination as O j and assume that each individual only has one routine vaccination opportunity. Further, assume that if the routine vaccination opportunity occurs during a given year then the probability of avoiding vaccination during that opportunity follows the same general form for activities: If we let Pr(Ō) be the probability of having not yet had the opportunity for routine vaccination, then: If we assume that each child has a probability F R (x) be the probability of having had your routine vaccination probability by age x. The the probability that person i is not vaccinated in a routine campaign is: where x ij is person i's age at the beginning of routine vaccination year j and l j is the length of vaccination year j (12 months for all years except for the year the data was collected). In other words, routine vaccination becomes a pseudo-campaign representing the weighted sum of the coverage in all of the years of routine vaccination, where the weights represent the probability that routine vaccination happened in that year: And the probability for vaccination for a given individual becomes: where m now represents the number of proper campaigns v R and N R are the number of doses distributed during routine vaccination activities.