01.12.2013 | Research article | Ausgabe 1/2013 Open Access

# A multi-state model to estimate incidence of heroin use

- Zeitschrift:
- BMC Medical Research Methodology > Ausgabe 1/2013

## Electronic supplementary material

## Competing interests

## Authors’ contribution

## Background

## Methods

### Incidence estimation: the multi-state model with immigration

_{ t }denote the expected number of people entering state 1 (heroin use) at t. The number p

_{ t }denotes the probability that a given heroin user initiates their first treatment ever at time t, given that they were in state 1 at the previous time. We assume this probability independent of the era when their heroin use began. The number q

_{ t }denotes the probability of an individual leaving heroin use at time t, given that they were in state 1 at the previous time. The cause could be death or other permanent cessation of heroin use. We want to estimate the parameters h

_{ t }, the expected number of new heroin users at time t. The probability q

_{ t }is assumed known.

_{ ij }be the observed number of individuals that start heroin use in year i and enter first treatment in year j. Let μ

_{ ij }be the expected value of N

_{ ij . }A short computation shows that:

_{ ij }is the product of the expected number of new heroin users in year i (h

_{ i }), the probability that a given heroin user remains in state 1 from time i to time j-1 and the probability of a transition from state 1 to state 2 at time j. It also follows that each N

_{ ij }is Poisson distributed. This gives us the following simple expression for the likelihood:

_{ t }and p

_{ t. }

### Treatment data

### Assumptions about parameters

#### First treatment data for the period 1971–1990

_{ t }, first treatment data was available for t between 1991 and 2006, thus restricting estimates to this period. For the preceding part of the study period (t in 1971–1990), we made an educated guess of p

_{ t }based on general heroin use information in Spain. Based on the first appearances of admissions for heroin use in the emergency units in Spain in 1982 [17–19]; we assumed probabilities of entering treatment (p

_{ t }) as low as 0.01 between 1971 and 1981, as there were still no specific treatments available. Thereafter we assumed a linear increase to the value estimated for the parameter p

_{ t }in 1991.

#### Mortality for heroin users

#### Cessation rates

### Analysis

_{ t }, t ranging from 1971 to 2006). As explained, the model also estimates the probability of entering first treatment (p

_{ t }, t in range 1991 to 2006). We considered the yearly cessation rate of 0.04 and the non-modified mortality rate derived from the local cohort studies. Note that the probability of leaving drug use without having ever been registered for first treatment (q

_{ t }) is the sum of the cessation and mortality rates for each year from 1971 to 2006.

_{ ij }by 0.5.

_{ ij }with their observed values N

_{ij}, we have drawn their curves stratified by year of heroin use onset (i).

_{ t }, that are reflected in four curves of estimated incidence rates. These combinations were: firstly and as a matter of choice, the available mortality rates and a yearly cessation rate of 0.04; secondly, the same mortality rates and a yearly cessation rate of 0.02; thirdly, the same mortality rates modified by adding 0.01 to the rate for each year and a yearly cessation rate of 0.04; and finally, the modified mortality rates and a yearly cessation rate of 0.02.

_{ t }in the period from 1971 to 1990, and the cessation rate for all years were sampled from gamma distributions. The shape and scale parameters were derived from the mean and standard deviation, taking the mean as the “best-guess” value, and the standard deviation was established as 0.01.

## Results

### Estimates of incidence and probability of entering treatment

_{ t }) which exhibited an overall increasing trend from 0.08 (95% CI 0.07-0.09) in 1991 to 0.29 (95% CI 0.23-0.49) in 2005. Incidence estimates of general heroin use and by route of administration with 95% confidence intervals are plotted in Figure 2. For general heroin use, the highest incidences were between 1985 and 1990 with rates around 1.5 new heroin users per 1,000 inhabitants aged 10–44, followed by a steep decline from 1991 to 1997, then a more gradual decrease from 1998 (0.24 per 1,000) to 2005 (0.05 per 1,000).

_{ ij }with their observed values N

_{ij}, by year of heroin use onset (i), we assessed that the fit was good (Figure 4).

_{ i,j-i }and the expected values as μ

_{ i,j-i }, where j-i represents the lag time between drug use onset i from 1991 to 2004 and first treatment ever j, conditional on treatment starting before 2006 (Figure 5).

### Sensitivity analysis

## Discussion

_{ t }, the probability of entering first treatment, was independent of the era when a person's first heroin use began. Actually, this would be not entirely true if lag time between the drug use onset and first treatment followed a determined pattern, as previous studies assumed [1, 6]. However, if we observe Figure 5, the lag time distribution for the observed values N

_{ i,j-i }and for the expected values μ

_{ i,j-i }(j-i represents the lag time), for each year of heroin use onset from 1991 to 2004 all fitted well. So, to modify the equation 1 including the probability of entering first treatment conditional on the initiation of heroin use would be too complex and may not have great practical importance. Therefore, bias can be inherent in both the independence assumption and assuming a determined pattern of lag time, although the direction of such bias cannot be determined.