Estimating number of users based on number of deaths
One method for estimating the number of injecting drug users in a country, is to multiply the number of deaths from overdose with a constant:
number of users = konstant * number of overdose deaths
or:
nu = k * nod
The intuition is simple. In any given year a fixed share of the drug users will dive from overdose, and this means that we can multiply the number of deaths by a constant to get the total number of users. For instance, assume the probability of dying from overdose is 1% in a given year. If 100 people dies, this means that there are 10 000 users.
Note that the constant in the calculation (k) is one divided by the probability of experiencing death from an overdose for a random user.
k=1/pu
It is important to note this because typically p is very small. Compared to the total number of drug users, relatively few people die from an overdose every year. This means that k is very sensitive to small changes in the estimated probability of death from overdose. This, in turn, means that the estimated number of drug users is very sensitive to the estimated probability of dying from an overdose in a given year. To correctly estimate the number of drug users, it is important that k is correct.
One factor that could affect the probability of death from an overdose, is the introduction of different types of treatments. For instance, drug users who get substitution therapy are believed to have a significantly lower probability of dying. If fully successful, a person on substitution therapy would quit illegal drugs and never die from an overdose. In practice, however, some people who start substitution therapy end up also using drugs and overdose deaths, but on average for all those entering treatment, the probability of overdosing and dying, should go down.
To model this, split the population into three groups:
- number of ordinary drug users (outside treatment): no
- number of of individuals in substitution treatment who are clean and do not use other illegal drugs: nc
- number of individuals in treatment who still sometimes use illegal drugs i.w. a mix: nm
The probability of dying from an overdose among users is then the average of the probability of overdose in the two groups, weighted by the size of the groups. There are no overdose deaths from those in treatment who do not use drugs, so the probability of dying is:
pu = po * no/(no + nm) + pm * nm/(no + nm)
Recall that the equation for estimating the number of drug users was
nu = k * nod = (1/pu) * nod
and that that this estimation was sensitive to the choice of k, which in terms depends on the probability of dying from an overdose for a drug user.
After introducing treatment with partially compliant participants, the average probability of death from overdose became a more complicated expression (see pu above). It depends, both, on the number of individuals in treatment who continue to use drugs (nm) and the extent to which being in treatment affect their probability of dying from an overdose (pm).
In short, since pu is an important factor for the multiplier used when estimating the number of drug users based on deaths from overdoses, and since pu will change when treatment is introduced, this means that k should also be adjusted to get the correct number of drug users. If k is assumed constant over time, the estimated number of drug users will be too low in a situation where the drug users on average have a lower than previous probability of dying from an overdose. This effect could be large because k is very sensitive to errors in pu, but some numerical examples are necessary to explore this further.
number of users = konstant * number of overdose deaths
or:
nu = k * nod
The intuition is simple. In any given year a fixed share of the drug users will dive from overdose, and this means that we can multiply the number of deaths by a constant to get the total number of users. For instance, assume the probability of dying from overdose is 1% in a given year. If 100 people dies, this means that there are 10 000 users.
Note that the constant in the calculation (k) is one divided by the probability of experiencing death from an overdose for a random user.
k=1/pu
It is important to note this because typically p is very small. Compared to the total number of drug users, relatively few people die from an overdose every year. This means that k is very sensitive to small changes in the estimated probability of death from overdose. This, in turn, means that the estimated number of drug users is very sensitive to the estimated probability of dying from an overdose in a given year. To correctly estimate the number of drug users, it is important that k is correct.
One factor that could affect the probability of death from an overdose, is the introduction of different types of treatments. For instance, drug users who get substitution therapy are believed to have a significantly lower probability of dying. If fully successful, a person on substitution therapy would quit illegal drugs and never die from an overdose. In practice, however, some people who start substitution therapy end up also using drugs and overdose deaths, but on average for all those entering treatment, the probability of overdosing and dying, should go down.
To model this, split the population into three groups:
- number of ordinary drug users (outside treatment): no
- number of of individuals in substitution treatment who are clean and do not use other illegal drugs: nc
- number of individuals in treatment who still sometimes use illegal drugs i.w. a mix: nm
The probability of dying from an overdose among users is then the average of the probability of overdose in the two groups, weighted by the size of the groups. There are no overdose deaths from those in treatment who do not use drugs, so the probability of dying is:
pu = po * no/(no + nm) + pm * nm/(no + nm)
Recall that the equation for estimating the number of drug users was
nu = k * nod = (1/pu) * nod
and that that this estimation was sensitive to the choice of k, which in terms depends on the probability of dying from an overdose for a drug user.
After introducing treatment with partially compliant participants, the average probability of death from overdose became a more complicated expression (see pu above). It depends, both, on the number of individuals in treatment who continue to use drugs (nm) and the extent to which being in treatment affect their probability of dying from an overdose (pm).
In short, since pu is an important factor for the multiplier used when estimating the number of drug users based on deaths from overdoses, and since pu will change when treatment is introduced, this means that k should also be adjusted to get the correct number of drug users. If k is assumed constant over time, the estimated number of drug users will be too low in a situation where the drug users on average have a lower than previous probability of dying from an overdose. This effect could be large because k is very sensitive to errors in pu, but some numerical examples are necessary to explore this further.
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