Estimating the size of hard-to-reach populations: a novel method using HIV testing data compared to other methods

Abstract

Objective: 

To estimate population size of hard-to-reach groups such as injecting drug users and men who have sex with men.

Design: 

Several different methods were used to estimate the size of these populations in Canada's three largest cities (Toronto, Montreal and Vancouver).

Methods: 

A novel method (referred to as the indirect method) was developed for use in Toronto and Vancouver that combines HIV serodiagnostic information with data on HIV testing behavior. Population size estimates were obtained by dividing the number of injecting drug users or men who have sex with men recorded in HIV serodiagnostic databases in a given year by the proportion of the corresponding group that reported being tested in a 1-year period. Results of this method were compared with four other methods: (1) population surveys; (2) capture-recapture (for injecting drug users only); (3) a modified Delphi technique; and (4) a method based on the proportion of never-married men aged 45 and over (for men who have sex with men only). Only these other methods were used in Montreal.

Results: 

The survey method gave the lowest estimates which are best viewed as minimum estimates given the relative inability of surveys to access these populations and the reluctance of participants to admit to sensitive behaviors. The indirect method produced results more closely comparable with those obtained by other methods, but they are probably slight overestimates, at least for injecting drug users, due to possible underestimation of the proportion tested for HIV. Point estimates using the indirect method were 17 700 and 17 500 for injecting drug users in Toronto and Vancouver, respectively, and 39 100 and 15 900 for men who have sex with men. In Toronto, results for the other methods ranged from 12 300-13 360 for injecting drug users and 18 800-35 000 for men who have sex with men. For Vancouver, these ranges were 6400-11 670 and 7000-26 500, respectively. In Montreal, ranges were 4300-12 500 for injecting drug users and 18 500-40 000 for men who have sex with men.

Conclusions: 

This novel method provides estimates of population size of hard-to-reach groups such as injecting drug users and men who have sex with men that are comparable with results derived by other methods. These estimates may be useful for the purposes of planning, implementing and evaluating prevention and care services, especially when they are combined with the results of other estimation methods to improve the degree of confidence in the resulting estimates.

Introduction

In most societies around the world, there are a number of populations that are difficult to access and are at increased risk for health problems. The identity of these hard-to-reach groups varies from country to country, but they usually include injecting drug users (IDU), men who have sex with men (MSM), sex workers, and migrant populations. Generally, the reason that these groups are hard-to-reach is that their activities are illegal and/or viewed as socially unacceptable. The health problems they experience vary from group to group and from country to country, but frequently include higher-than-average burdens of tuberculosis, hepatitis B and C, and sexually transmitted infections, including HIV. The social stigma associated with many of these health problems, particularly HIV, serves to further alienate these hard-to-reach groups and makes it more difficult to access them and to provide them with the health services necessary for appropriate prevention and care.

To effectively plan, implement and evaluate health programs for these groups, estimates of the size of specific hard-to-reach populations are needed. These estimates are also needed to better understand the absolute burden of disease, to better inform policy and program development, and to assess population trends in the behaviors themselves. A variety of methods have been used to estimate population size ^[1-3], but each method has its limitations and the task remains a difficult one. The use of conventional surveys is one method, but it often underestimates population size as individuals in the population of interest are less likely to be reached by such surveys (particularly telephone surveys) ^[2,4]. Furthermore, if the population of interest is reached, illegal or socially unacceptable activities may be underreported ^[5,6]. Ethnographic surveys using nominative techniques ^[2,7], snowball sampling ^[8], or privileged access interviews ^[9] can overcome some of the limitations of conventional surveys, but are themselves limited in terms of generalization ^[8-10]. Multiple capture models and other variations of capture-recapture methods have also been used for these hard-to-reach populations. These methods are themselves limited by the availability of independent databases in which individuals in the target population have an equal chance of being represented, the ability to identify individuals, difficulties defining the target population between databases in a consistent and comparable manner, and other problems ^[11-13].

In this article, a novel method is described that was developed to estimate the population size of hard-to-reach groups at risk for HIV infection, namely IDU and MSM, in Canada's three largest cities (Toronto, Montreal, and Vancouver). This method combines information on HIV testing behavior with data from HIV serodiagnostic databases. To assess the method's relative validity, population estimates obtained with this method were compared with estimates obtained using the population survey method, ancillary methods such as a modified Delphi technique, and capture-recapture methods ^[14] (for IDU only). This novel method provides estimates of population size of hard-to-reach groups that could prove useful for the purposes of planning, implementing and evaluating prevention and care services.

Materials and methods

The methods used were intended to estimate population sizes in 1996, or as close to 1996 as the available data would allow. These methods are described in the following sections.

Indirect method

The novel method that was developed (hereafter referred to as the indirect method) combines data from provincial HIV serodiagnostic databases with information obtained from surveys and epidemiological studies on the HIV testing behavior of IDU and MSM. Provincial serodiagnostic databases contain information on all individuals who come forward for HIV testing, including information on geographic residence and HIV-related risk behaviors. In a given city, the population size for IDU (or MSM) in the year 1996 was estimated by the formula N/p where N is the number of IDU (or MSM) who had been tested for HIV in 1996 and p is the proportion of the corresponding group (IDU or MSM) that reported being tested for HIV during a 1-year period. Note that data for N were obtained from the provincial HIV serodiagnostic databases and data for p were obtained from surveys and studies on HIV testing behavior. In essence, this method uses the number of IDU (or MSM) in 1 year of the respective HIV serodiagnostic databases as a sample of the IDU (MSM) community; the proportion that this sample represents of the entire community is given by the proportion of IDU (MSM) that report having been tested in a 1-year period. As this method depends only on the number of HIV tests done and not on whether the test results are positive or negative, it is not affected by any potential biases associated with high-risk individuals being more or less likely to be tested for HIV.

Individuals who were diagnosed with HIV infection prior to 1996 would probably not have been tested for HIV in 1996, and so would not be present in the 1996 serodiagnostic database. However, this indirect method does include such individuals in its estimate of population size. This is because the data on the proportion of the group that has been tested for HIV in a 1-year period are obtained from sources such as surveys that include both HIV-positive and HIV-negative individuals (data sources explained more fully below). These data sources ask about the history of HIV testing in the year before the interview. Diagnosed HIV-positive individuals will be less likely to have been tested as most will have been diagnosed more than 1 year prior to the interview and so would have had no reason to be tested in the prior year. Therefore, the data sources on the proportion tested for HIV take into account the under-representation of diagnosed HIV cases in the HIV serodiagnostic database for a given year, such as 1996.

Sufficient data were not available to use this method for Montreal, and only results from the other methods are presented for Montreal. For Toronto and Vancouver, data on the number of individuals that were tested for HIV in 1996 were examined for duplicate records. Duplicate tests for the same individual were ruled out for all data in Toronto and for all HIV-positive test results in Vancouver. For HIV-negative test results in Vancouver, no estimate was available of the extent of duplicate records. Therefore, to adjust the Vancouver data, information on duplication rates within the serodiagnostic database for Toronto was used to reduce the overall raw numbers of IDU by 12% and of MSM by 8% ^[15]. The geographic boundary of Toronto was defined as the metropolitan area encompassing the six central health units (1996 population approximately 2.3 million) ^[16], an area that constitutes approximately two-thirds of Toronto's census metropolitan area (CMA). Vancouver was defined by six health regions (1996 population approximately 1.8 million) ^[16], an area that is nearly identical to Vancouver's CMA. Risk categories (IDU, MSM) were assigned based on information provided on the HIV test requisition forms. However, approximately half of the forms did not contain risk information and these cases were assigned to risk categories in the same proportion as among those with known risk information. A study in Ontario (including Toronto) designed to investigate this problem of inadequate risk information found that among HIV-positive tests, the distribution of risk categories among the known group (information obtained from test requisitions) was very similar to the unknown group (information obtained by physician call-back) (P = 0.48) ^[15]. There were no comparable data available for HIV-negative tests, and it was assumed that the risk distribution of these individuals was also the same between the known and the unknown groups.

These serodiagnostic data were then combined with data on HIV testing behavior of IDU and MSM. Two sources of data were used for the proportion of these groups tested during a 1-year period. The first source was two national telephone surveys of persons aged 15 years and over conducted by the Canada Health Monitor (CHM) in 1995-1996 and 1997 [17; C. Archibald, unpublished data, 1997]. Random sampling was used in these surveys with stratification by province and community size according to census population estimates; each survey had a sample size of approximately 3000. The definition of IDU in the 1995-1996 CHM survey was injecting non-prescription drugs since 1978; MSM were defined as having had oral or anal intercourse with other men since 1978. For the 1997 CHM survey, IDU were defined as injecting in the past year and MSM as having had anal intercourse with other men in the past year. In these surveys, the boundaries of Toronto and Vancouver were defined as noted above.

The second source of data on testing behavior was epidemiological studies on IDU and MSM in Toronto and Vancouver. In addition, because of limited data, this source was supplemented by testing behavior data from studies conducted in other cities, including Montreal (see references in Table 1). In these studies, the definition of IDU varied from injecting in the past 1 month to injecting in the past 6 months, and the definition of MSM varied from self-identification as gay to self-identification as having affective and/or sexual relations with other men.

Table 1: The proportion of injecting drug users (IDU) and men who have sex with men (MSM) that were tested for HIV in lifetime and in the past year. Results for Canada are from Canada Health Monitor surveys in 1995-1996 and 1997; results for cities are from the sources indicated.

To establish a range of uncertainty associated with these indirect estimates, we calculated upper and lower limits for the input parameters N and p (defined above). Given the methods used to obtain N and p, it was not possible to calculate rigorous confidence intervals. For N, plausible upper and lower 95% certainty limits were estimated by using statistical information on the extent of duplicates and risk category misclassification ^[15] together with our subjective assessment of the likely magnitude of uncertainty. For p, a similar process was carried out based on sample size information provided in the source surveys and studies (references in Table 1). A Monte-Carlo simulation was then carried out ^[18] to obtain ranges for the final population estimates (simulations done using commercial software, Crystal Ball, Decisioneering, Aurora, Colorado, USA). Normal distributions were assumed for input parameters and 10 000 iterations were performed. Results for the indirect method are presented as point estimates and associated ranges of uncertainty. The simulation process was also used to assess the sensitivity of the final estimate to variation in the two input parameters, N and p.

Survey method

This method used data from population-based surveys to estimate population sizes of IDU and MSM in Toronto, Montreal and Vancouver. The proportion of the surveyed population reporting IDU behavior in the past year was multiplied by total population size aged 15 years and over to estimate the total number of IDU; a similar procedure was used for MSM except that the calculations applied only to the population of males aged 15 years and over. Sources of data were the 1997 CHM survey and a similar survey conducted in 1994 (the 1995-1996 CHM survey did not ask about these behaviors in the past year). The 1994 survey defined IDU as injecting in the past year and MSM as having had oral or anal intercourse with another man in the past year. In addition, the 1994 and 1997 surveys also asked about IDU behavior in lifetime and the 1994 survey asked about MSM behavior in lifetime. In these surveys, the boundaries of Toronto and Vancouver were defined as noted above and Montreal was defined as Montreal island (1996 population approximately 1.8 million) ^[16]. Results for the survey method are presented as point estimates and 95% confidence intervals calculated according to exact binomial methods ^[19]

Capture-recapture method

A study using capture-recapture methods to estimate the number of IDU in these three cities was recently conducted by Remis et al.^[14]. This method used the number of individuals found in two or more IDU databases to mathematically estimate total population size ^[11]. The definition of IDU used in this method was injecting in the past year; city definitions were the same as noted above. Population estimates are presented as point estimates and associated plausible ranges ^[14].

Ancillary methods

Two ancillary methods were used in these three cities to estimate IDU and MSM population size. One method used was a modification of the Delphi technique described by Dalkey ^[20]. For this method, experts in each city were asked their opinion of the number of IDU or MSM in their city. Experts were individuals in these three cities with extensive experience and knowledge in the fields of either drug use or MSM sexual behavior; three experts were used for each risk category in each city. City boundary definitions were as described above. Unlike the classic Delphi process, an iterative process was not used to refine these expert opinions; results are presented as a midpoint and range of the individual experts' point estimates.

The other ancillary method was used only for MSM and was based on the proportion of men aged 45 years and over that have never been married ^[21]. For each city, this proportion was obtained from census data and multiplied by the census population of adult men (aged 15 years and over) to obtain point estimates of the number of MSM. Toronto, Montreal and Vancouver were defined as noted above. Although this method obviously would include some never-married heterosexual men and exclude some married MSM, these two biases will partially cancel each other out ^[21,22]. This method has been used by Holmberg ^[23] to estimate the MSM population size in American cities.

Results

To illustrate the indirect and survey methods, an example is provided of estimating the number of IDU in Toronto. For the indirect method, the number of IDU in Toronto who were tested for HIV in 1996 was 4047. Data on the proportion of IDU that reported testing for HIV in lifetime and in the past year are shown in Table 1. As there are no data from Toronto on testing in the past year, this proportion was estimated as follows: the proportion of IDU tested in their lifetime in Toronto (70%) was midway between the proportions for Vancouver (83.4%) and Canada as a whole (51.5-61.6%), so it was assumed that the proportion tested in the past year in Toronto was similarly midway between the proportions for Vancouver (25.6%) and Canada as a whole (19.2-21.2%), or 22.9%. Therefore, the point estimate of the number of IDU in Toronto in 1996 is 4047/0.229 = 17 700 (rounded to the nearest 100).

For the survey method, the 1996 population of Toronto aged 15 years and older was 1 815 000. The proportion of this population that reported IDU in the past year was 0.4-0.96% with a midpoint of 0.68% (Table 2). Therefore, the survey estimate of the number of IDU (injecting in past year) in Toronto is 1 815 000 × 0.68% = 12 300 (rounded to the nearest 100).

Table 2: The proportion of the general population aged 15 years and older that reported injecting drug use (IDU) and of men aged 15 years and older that reported having sex with other men (MSM) in lifetime and in the past year. Results from Canada Health Monitor surveys in 1994, 1995-1996 and 1997.

Table 2 also shows the data on IDU behavior for Montreal, Vancouver and for Canada as a whole. Overall, the ranges are 0.6-3.4% for lifetime IDU and 0-0.96% for IDU in the past year. Tables 1 and 2 show the available data from Toronto, Montreal, Vancouver and Canada on the proportion of MSM that tested for HIV and the proportion of the general population aged 15 years and over that report MSM behavior, respectively. For the estimates of MSM testing in the past year in Toronto and Vancouver, data were interpolated as in the example described above for the indirect estimate of IDU in Toronto.

The results from the indirect method were compared with those from several other methods: population surveys, a modified Delphi technique, a method based on the proportion of never-married men aged 45 years and over (for MSM only), and a capture-recapture method ^[14] (for IDU only). The final estimates for IDU and MSM population sizes are presented in Tables 3 and 4, respectively.

Table 3: Estimates of population size of injecting drug users (point estimates and ranges) for Toronto, Montreal and Vancouver using four different methods.

Table 4: Estimates of population size of men who have sex with men (point estimates and ranges) for Toronto, Montreal and Vancouver using four different methods.

In Toronto, the survey method gave the lowest estimate (12 300) for IDU and the indirect method the highest estimate (17 700). The range of the Delphi method (10 000-15 000) and the capture-recapture point estimate (13 360) were both close to the survey estimate and were slightly lower than the indirect estimate. For MSM, the survey estimate (18 800) was again the lowest and the range of the Delphi method (25 000-45 000) included both the estimate using the proportion of never-married men (35 000) and the indirect estimate (39 100).

In Vancouver, the survey method gave the lowest estimate (6400) for IDU and the indirect method the highest (17 500), whereas the capture-recapture estimate (11 670) and the range for the Delphi method (8000-13 000) were between these two estimates. For MSM, the survey estimate (7000) was about half of the indirect estimate (15 900). The highest estimate was derived from never-married men (26 500) and the range of the Delphi method (10 000-20 000) was intermediate and included the indirect estimate.

For IDU in Montreal, the survey method gave an estimate of 4300 which was substantially less than the range associated with the Delphi method (10 000-15 000) and the capture-recapture point estimate (11 680). Similarly for MSM, the survey estimate (18 500) was less than the estimate derived from never-married men (37 000) and the range of the Delphi method (30 000-50 000).

Discussion

In this article, HIV testing behavior information was combined with HIV serodiagnostic data to estimate the population size of hard-to-reach groups, in particular IDU and MSM. To our knowledge, this is the first reported use of such data to produce population size estimates. The concept underlying this method is the use of one data set (HIV serodiagnostic data in our case) as a sample of the population of interest, in combination with another data source to estimate the proportion of the total population likely to be captured by the first data set (surveys and studies to provide data on the proportion tested for HIV in a 1-year period). This general concept is not new and has been used by others to estimate population size of hard-to-reach groups. Wolf et al.^[5] used a data set on substance abuse treatment together with the proportion of needle-exchange attendees who reported treatment to estimate the number of current IDU in Boston. Similarly, Hartnoll et al.^[26] estimated the number of heroin users in London by combining a data set from one drug treatment clinic with the proportion of a sample of heroin users and their friends that had attended that clinic in the past year. The flexibility of this relatively simple concept may be especially useful in countries where data sources are few and resources for other estimation methods are limited.

Our indirect estimation method produced results that were, in general, higher than those produced by the more traditional survey method, but comparable with results obtained by other methods for MSM and slightly higher than results obtained by other methods for IDU. Although it was not possible for us to absolutely validate any of the methods presented here, some aspects of the relative validity of these different methods can be assessed by considering their respective limitations and the relative ranking of the absolute size of the estimates. The population surveys used in this paper to provide data for the survey method were of relatively small sample size, especially considering that the behaviors under investigation are relatively rare. As a result, the ranges of the estimated population size of IDU and MSM using the survey method are very broad. Nonetheless, the values that were found for the proportion of IDU and MSM in the general urban population are comparable with those found in other surveys in North America. We found a range of 0-0.96% of individuals aged 15 years and over in these three cities had injected drugs in the past year. Holmberg ^[23] reported a range of 0.4-1.3% for American cities, Wolf et al.^[5] found a value of 0.3% in Boston, and Adrien et al.^[27] found 0.1% for the province of Quebec (including both urban and rural areas). For MSM behavior in the past year, our surveys found values up to 4.2% in these three cities and comparable results have been found in other surveys: 3.7% for Quebec ^[27] and 3.7% for the general population in the USA ^[28].

Additional limitations of the survey method are the lack of an adequate sampling frame, inconsistent wording of questions between surveys, the reluctance of participants to admit to sensitive behaviors, and the relative inability of such surveys to access these populations. This last bias would be expected to be particularly true for IDU as many of them either do not have a permanent place of residence and/or do not own a telephone. Therefore, a priori, the estimates derived from population survey data can be expected to be underestimates of the true size of these hard-to-reach groups. Indeed, this appears to be true in the current study as the survey method estimates were consistently the lowest of all the methods examined. These limitations will be even more pronounced in countries where the social and legal environments are less tolerant of these behaviors.

The use of the proportion of never-married men (aged 45 years and over) yielded estimates of MSM population size that were the highest or close to the highest of all methods. This may indicate that the bias of including never-married heterosexual men in the calculation is greater than that of excluding married MSM, resulting in an overestimate of population size. Moss and Wiley ^[29] found that the MSM population size estimate using the never-married method in San Francisco was about 30% higher than the estimate obtained by a method based on a household random sample, stratified by age and adjusted for response rate. In the present study, the never-married estimate of MSM in Toronto was very close to the indirect estimate, but in Vancouver it was substantially higher than the indirect estimate, as would be expected if it is an overestimate.

The Delphi method and its variations are really structured ways to arrive at a consensus or range of possible values for a parameter, in this case for population size of various risk groups. The existence of a consensus or a narrow range of possible values does not mean that the truth has been found, and there is no way to validate the result without reference to observable data. The main value of this method in the present context is the increase in confidence we have in the results since the Delphi estimates are higher than the survey estimates (which are expected to be underestimates) and are fairly close to those of the other, empirically-based methods.

Remis et al.^[14] state that one of the most significant limitations in their capture-recapture analysis was the difficulty in assessing whether the source databases were truly independent, especially for Toronto and Vancouver. They point out the probability that some individuals present in one database were probably over-represented in one of the other databases for that city, which would result in the population size estimates for IDU being underestimates. In these cities, the capture-recapture estimate was higher than both the Delphi and survey estimates, and lower than the indirect estimate. If the capture-recapture estimate is indeed an underestimate, then the true population size for IDU in Toronto and Vancouver probably lies between the capture-recapture estimate and the indirect estimate.

The indirect method relied on HIV serodiagnostic data and on testing behavior information obtained from general population surveys and epidemiological studies. The HIV serodiagnostic data used in this study have a number of limitations. Some individuals may have been tested more than once for HIV in the same calendar year. However, duplicate tests for the same individual in the same year were ruled out for all HIV tests in Toronto and for the positive HIV tests is Vancouver. For negative HIV tests in Vancouver, the duplicate rate was estimated using data from Toronto and the raw numbers were adjusted accordingly prior to their use in the estimation procedure. This limitation would become more significant the longer the time interval considered, and so the use of serodiagnostic data was restricted to a 1-year period.

Another limitation of serodiagnostic data is the extent of incomplete risk information. We adjusted for this by assigning risk categories to reports with missing information in the same proportion as among reports with complete risk information. This assumption may have slightly underestimated the number of HIV tests that were carried out among IDU and MSM because of the potential reluctance of some tested individuals to admit to these behaviors. This possible bias would be at least partially offset by the fact that risk information in the HIV serodiagnostic database does not indicate whether or not the risk behavior was recent, which by itself would tend to overestimate the number of current IDU or MSM.

The other data source required for the indirect method is testing behavior information and this source also has a number of limitations. Information on HIV testing behavior of IDU and MSM was scarce, and not all of it was from the three cities being studied. We found that the proportion tested for HIV was generally higher in epidemiological studies than in the surveys, both for testing in lifetime and in the past year (Table 1). This difference is probably due to several factors. Respondents in the epidemiological studies were from urban areas whereas the surveys were conducted among the general population, both urban and rural, and HIV testing in Canada is more common among urban populations ^[17]. In addition, the epidemiological studies tended to recruit participants from inner city service agencies where HIV testing was generally readily available to the studied population, and probably more available than for IDU and MSM in the general population. Our testing rates were comparable to rates found in American studies, both for testing in lifetime (55-82% of IDU ^[30,31] and 61-94% of MSM ^[30,32,33]) and in the past year (50% of MSM ^[34]).

The Monte Carlo simulation analyses indicated that the indirect estimate was most sensitive to variation in the proportion tested for HIV and so any bias in the final estimate would be most influenced by bias in this parameter. Based on anecdotal information, we think that the data on proportion tested may tend to underestimate the true proportion due to the fact that individuals are more likely to forget that they were tested during a specific time period than they are to incorrectly state that they were tested. Therefore, if there is a bias associated with the indirect method, it probably is in the direction of producing slight overestimates since the proportion tested is in the denominator of the formula N/p. Indeed, the relative comparison of the different estimates described earlier supports the idea that the indirect method produces a slight overestimate, at least for IDU population size. It is possible that the IDU group is more subject to this type of recall error than the MSM group due to their itinerant, less-structured lifestyle. We are currently doing further studies to assess potential biases for the indirect method and to measure the magnitude of their effect on population size estimates. Although each of the estimation methods has its own limitations, they are based on different assumptions and yet yield results that are comparable to the indirect method (or if different, the differences can be explained). This fact provides some reassurance regarding the relative validity of the estimates obtained by the indirect method.

Although we have illustrated the use of the indirect method to estimate population sizes within cities, it could also be used to estimate population sizes of hard-to-reach groups at the provincial or national level. However, the larger the geographic area, the more one should consider building the final estimate by combining estimates from smaller areas to account for any geographic variation in HIV testing patterns or test availability.

In principle, the indirect method could also be used to estimate population size of various subgroups within MSM and IDU, for example drug users who inject heroin. In practice, however, such subgroup analysis is limited by the information available on the HIV test requisition form, and this does not usually include type of drug injected. Information from separate sources would be needed to subdivide by drug type the population size estimate of IDU obtained via the indirect method.

The concept of using one data set in conjunction with another to indirectly estimate population size is advantageous even in resource-limited countries since much of the required data may already be available. The usefulness of the indirect method will be limited in countries where HIV testing data is not collected and merged at least at the local level. However, as the new HIV antiretroviral treatments become more widely available, HIV case reporting will need to replace AIDS case reporting as the main surveillance tool to monitor the HIV epidemic ^[35]. Thus HIV serodiagnostic databases will likely soon become available in many countries and could be used to help estimate the population size of hard-to-reach risk groups. It is important to be able to estimate the proportion of duplicate test reports within a specified time period (such as 1 year) and to obtain risk information that is as complete as possible on individuals being tested for HIV. If such data are not readily available within the existing HIV database, these data could be obtained by studying a sample of those testing for HIV. Data on HIV testing behavior of hard-to-reach groups are also needed to use this method, and could be collected in many situations by adding questions to existing studies rather than by starting studies de novo. Once we gain a better understanding of the extent to which the components of uncertainty influence the indirect method in different settings, such an approach could be very useful in resource-limited situations.

In conclusion, this indirect method of combining HIV serodiagnostic information with data on HIV testing behavior can provide estimates of population size for hard-to-reach groups such as IDU and MSM, and these estimates are comparable to those obtained using other methods such as capture-recapture techniques. This indirect method may produce slight overestimates of population size in situations where extensive information on the proportion tested for HIV is not available, and the validity of the method will improve with more accurate testing behavior information. The estimates produced by all the methods presented here are necessarily imprecise, given the difficulties associated with studying hidden, hard-to-reach populations. Any single method on its own can only provide a relatively uncertain estimate of population size, and to improve the degree of certainty it is important to use different estimation methods wherever possible. The indirect method we have developed is a useful one to consider in this process of trying to obtain population size estimates of these hard-to-reach groups for the purposes of planning, implementing and evaluating prevention and care services.

Acknowledgements

The authors thank the following individuals: Tanya Degazio, Rob MacDougall and Norm Cameron for their assistance in obtaining HIV serodiagnostic data; and Rob Palmer for help with the Canada Health Monitor surveys.