The advent of the HIV/AIDS epidemic has greatly stimulated the study of sexual behavior. Empirical behavior studies such as surveys (e.g., ACSF investigators, 1992; Catania, Coates & Turner, 1992; Cleland & Ferry, 1995; Moses, Muia & Bradley, 1994) have gained popularity and have been recognised by the World Health Organization as an important health related activity (World Health Organization; 1989). Mathematical models relating sexual behavior and disease parameters, such as the duration of infectivity, to the epidemiology of HIV and other sexually transmitted diseases (STDs) have also gained interest beyond academia (Rothenberg, 1997). Such models are useful tools in understanding more precisely the epidemiology of these infections, in predicting their impact and the success of interventions, and thereby assisting the design of more effective public health policy. Mathematical models have also indicated the need for an improved understanding of sexual behavior and the role of sexual networks and mixing patterns (Gupta, Anderson, & May, 1989; Kault, 1993; Morris, 1995; Morris, Podhista, Wawer, & Handcock, 1996; Yorke & Hetchcote, 1978).
In many mathematical models for the spread of STDs (Ghani, Swinton, & Garnett, 1996; Morris & Kretzschmar, 1995; Watts & May, 1992) parameters of interest with respect to sexual behavior include the rate of acquisition of new partners', and concurrency that is, the number of simultaneous sexual partnerships an individual has at a given time. In routine surveys of sexual behavior, however, questions on these variables are not always asked. Instead, individuals are often requested to report the total number of sex partners they have had over a specific period T (e.g. the previous year, or the previous 3 months). Under serial monogamy, the number of new sex partners an individual has during T equals either the total number of sex partners during T minus 1 if the individual was in a partnership at the beginning of the period T, or the total number of sex partners during T if the individual was not in a partnership at the beginning of this period.
However, in many societies serial monogamy is neither norm nor practice. Evidence for high levels of concurrency in some African populations has emerged from recent sexual behavior studies in Uganda, where about 40% of the total population reports at least one concurrent partnership when the last three sexual partners were examined (Morris, Sewankambo, Wawer, Serwadda & Lainjbo, 1995). In Western societies, where serial monogamy might be the norm for large sections of the population, partnerships may overlap: For instance, a survey in The Netherlands showed that 6.4% of men had sex outside of a steady partnership in the previous year (Van Zessen & Sandfort, 1991). Concurrency in partnerships has been shown to have a considerable impact on the spread of HIV and other STDs (Garnett & Johnson, 1997; Moms & Kretzschmar, 1995, 1997; Watts & May, 1992).
For concurrent partnerships, the sexual behavior parameters relevant to mathematical modelling usually cannot be derived directly from traditional surveys of sexual behavior or from data now routinely collected in clinical settings. Awareness of the importance of the rate of partner acquisition and concurrency for understanding sexual behavior has led to improved surveys of sexual behavior. For example, the British National Survey of Sexual Attitudes and Lifestyle uses innovative instruments for data collection that allow the direct estimation of all relevant parameters: Detailed information about the timing of the last three sex partners is solicited from respondents (Johnson, Wadsworth, Wellings, & Field, 1993). Surveys using a similar or more extensive local sexual-network design have been performed in the United States (Laumann, Gagnon, Michael, & Michaels, 1994), Uganda (Wawer, 1993), and Thailand (Wawer, 1990). Furthermore, the World Health Organization has developed questionnaires asking about types of partnerships and concurrency, which are to be used in the evaluation of national AIDS control programs (World Health Organization Global Programme on AIDS, 1994).
However, this is not the type of data that is likely to be collected on a routine basis in clinic settings and other public-health surveillance studies. In these settings, questions on total numbers of partners in specified intervals are simple and quick to ask and relatively easy to understand. But questions related to the timing of events may be difficult to ask routinely in some societies, particularly in non-Western societies. Unfortunately, error rates are likely to be the greatest among those with the most sexually varied lifestyles who are of the greatest interest in sexual-network studies (Johnson, et al., 1993). Data obtained from such clinical settings and other routine sources have limited value for mathematical modeling. However, if in a survey information about the total number of partners is ascertained not just over a single time interval, but over two intervals (e.g., both over the previous 3 months and over the previous year), then information about the loss of partnerships is available (see, for example, Blower, Anderson, & Wallace, 1990; Johnson, 1996). This information can be exploited to derive estimates of rate of partner acquisition and concurrency. Using two different methods we derived estimates of both concurrency and the rate of acquisition of new partners from this type of information.
We assume that, for the individuals in the sample, sexual behavior has not changed over the periods involved. We will call this assumption the steady-state assumption. This assumption may not be valid if there were a substantial age effect--for example, if the longer of the two periods is very long, such as would be the case if the number of lifetime sex partners is assessed. Also, if the reason for interviewing individuals relates to recent sexual activity then this assumption may be similarly invalid. For example, if the reason for interviewing is the detection of an STD, then it may be reasonable to assume a "flurry" of sexual activity in the recent past.
We also assume that the shorter of the two periods is long enough to guarantee that sexual intercourse has occurred in ongoing partnerships during that period. For some African countries where male migrant workers often reside in the cities for prolonged periods, sometimes up to a year, without returning to the rural homes of their wives, this may require long periods.
In addition, we assume that concurrency can occur in the population studied. As our methods cannot test this assumption, application of these methods in a population for which serial monogamy is the rule may result in spurious estimates of the rate of partner acquisition. We could find estimates of concurrency in excess of one, which are clearly incompatible with serial monogamy.
For both methods, we suppose that the rates of acquisition and loss of partnerships for each individual can be adequately modeled by an immigration-death process, in which new partnerships are started randomly, independent of the number of present or past partnerships, In this case, the times at which new partnerships are initiated follow a Poisson process. We also assume that the duration of partnerships is independent of the number of partnerships present. These assumptions are probably a gross simplification, as the rate of acquisition of new partners and the loss of existing partners is likely to depend on the current number of partnerships. In fact, some form of "crowding out" is likely to occur: Individuals may be more inclined to enter into new sexual partnerships when they have no or few sex partnerships than when they already have several. However, the higher dimensionality of more realistic models is likely to lead to problems in identifying parameters.
We will call the two periods over which the total number of sex partners is ascertained [T.sub.1] and [T.sub.2], with [T.sub.1] [is greater than] [T.sub.2] and the numbers of sex partners reported, respectively, [n.sub.1] and [n.sub.2], with [n.sub.1] [is greater than] [n.sub.2]. Note that in this case, ([n.sub.1] - [n.sub.2]) partnerships must have ended in the period [T.sub.1, 2] = [T.sub.1] - [T.sub.2]. Figure 1 graphically shows an example of the process of acquisition and loss of sex partners for a situation where concurrency exists.
[Figure 1 ILLUSTRATION OMITTED]
The rate of partnership acquisition--the number of new sex partnerships per year--is equal to [Alpha], and the average duration of a partnerships is [Delta] years. The average concurrency, defined as the average number of partnerships of an individual at a given moment, will be called C. This measure C is not a very precise indication of the amount of overlap in partnerships: For example, both an individual with no partners half of the time, and two partners during the other half of the time, and an individual in a stable monogamous partnership have a value of 1 for this concurrency measure. To indicate the expected value of a variable x we will use the notation E(x); the expectation of the variable x conditional on the variable y will be denoted by E(x | y). The average value of a variable x will be denoted by [bar] x.
Method 1: Estimating Dynamics for a Population With Homogeneous Behavior
In Method 1 we assume that the sexual behavior of all individuals in a given population is driven by the same rate of acquisition of new partners, [Alpha]; average duration of partnerships, [Delta]; and average concurrency measure, C. For an individual with [n.sub.1] partners in [T.sub.1] and [n.sub.2] partners in [T.sub.2, ([n.sub.1] - [n.sub.2]), partnerships must have ended in the period [T.sub.1,2] = [T.sub.1] - [T.sub.2]. Therefore, the expected number of partners lost is equal to E([n.sub.1] - [n.sub.2]), and the expected rate of partner loss is thus E([n.sub.1] - [n.sub.2]) / [T.sub.1,2]. The rate of partner loss is also equal to C/[Delta] because persons are on average in C partnerships at a given time that end at a rate of 1/[Delta] per partnership. Therefore:
(1) C/[Delta] = E([n.sub.1] - [n.sub.2]) / [T.sub.1,2].
Because of the steady-state assumption, the rate of partner acquisition, [Alpha], is equal to the rate of partner loss:
(2) C/[Delta] = [Alpha].
Furthermore, the total number of partnerships over the period [T.sub.1], [n.sub.1], equals the total number of partnerships existing at the beginning at interval [T.sub.1] plus the total number acquired during that interval, resulting in:
(3) E[n.sub.1]) = C+ [Alpha][T.sub.1]
These three equations can be used to derive estimates for [Alpha], [Delta] and C. Using Equations 1 and 2, we obtain:
(4a) [Alpha] = E([n.sub.1] - [n.sub.2])/[T.sub.1,2],
leading to the estimator:
(4b) [Alpha] = ([n.sub.1] - [n.sub.2])/[T.sub.1,2]
Using Equations 3 and 4a it follows that:
(5a) C = E([n.sub.1] - [T.sub.1] [multiplied by] E([n.sub.1] - [n.sub.2])/[T.sub.1,2],
leading to the estimator:
(5b) C = [n.sub.1] - [T.sub.1] [multiplied by] ([n.sub.1] - [n.sub.2]/[T.sub.1,2].
Finally, Equations 2, 4, and 5a result in:
(6a) [Delta] = E([n.sub.1]) [multiplied by] [T.sub.1,2]/E([n.sub.1] - [n.sub.2]) - [T.sub.1],
leading to the estimator:
(6b) [Delta] = [n.sub.1] [multiplied by] [T.sub.1,2] / ([n.sub.1] - [n.sub.2] - [T.sub.1].
Our estimate of the rate of partner acquisition, [Alpha], does not depend on the assumption of concurrency. It is equally valid under conditions of serial monogamy. However, our estimates of [Alpha] and C are not, because serial monogamy imposes additional restrictions on the values of these parameters.
For the estimates of parameters [Alpha] and C, standard errors can be calculated directly on the basis of Equations 4b and 5b, using standard statistical methods. To obtain standard errors for the estimate of parameter [Alpha], we have used a Taylor expansion, in which the variance of [Alpha] is related to the variance of [Alpha], the variance of C, and the covariance of [Alpha] and C.
In principle, Method 1 can also be applied to estimate parameters per individual by filling in the individual's value of [n.sub.1] for [bar] [n.sub.1] and [n.sub.1] - [n.sub.2] for [bar] [n.sub.1] - [n.sub.2]. In this way the assumption of homogeneous behavior is not stringent. However, it is unattractive that the method yields a zero rate of acquisition of new partners for individuals reporting the same number of sex partners during the two periods [T.sub.1] and [T.sub.2], i.e. [n.sub.1] - [n.sub.2] = 0. Further, for these individuals, the average duration of their partnerships will be estimated as infinity. These estimates are unrealistic, especially when [n.sub.1] = [n.sub.2] happen to be high.
This method can be made somewhat more realistic by assuming that partnerships are of two types: marriages and affairs. If we assume that married people are married forever, and could be married only to one person, then applying this method for married people--subtracting an individual's spouse from his/her reported number of sex partners--will provide a model for the dynamics of extra-marital affairs.
Method 2: Estimating Dynamics for a Population With Heterogeneous Behavior
More attractive estimators can be obtained by permitting population heterogeneity using Empirical Bayes estimators (Cox & Hinkley, 1974). In Method 2, we assume that the rate of partner acquisition, [Alpha], varies between individuals according to a gamma distribution with parameters r and p/(1-p). The skewness of the gamma distribution makes it a realistic model for sexual behavior: Most people have, acquire, and lose few sex partners, but a minority on the tail of the distribution are highly sexually active (Morris, 1993).
Furthermore, Method 2 assumes that for an individual with a specific rate of partner acquisition, [[Alpha].sub.0], the reported number of new sex partners and the reported number of lost partnerships have a Poisson distribution with parameter [[Alpha].sub.0]. Under these assumptions the population ([n.sub.1] - [n.sub.2]) follows a negative binomial distribution with parameters r and p (Bain & Engelhardt, 1987). For the negative binomial parameters, we have the expectation of [n.sub.1] - [n.sub.2]:
(7) E([n.sub.1] - [n.sub.2]) = r(1 - p)/p,
while the variance of [n.sub.1] - [n.sub.2] is equal to:
(8) var([n.sub.1] - [n.sub.2]) = r(1 - p)/[p.sup.2].
Thus the parameters r and p can easily be estimated from the observed mean, M, and variance of [n.sub.1] - [n.sub.2], using information from the whole population. The idea underlying Empirical Bayes estimators is that both the average behavior in a population and the reported individual behavior over a limited period are useful for predicting the long-term behavior of an individual.
Applying the statistical Bayes rule yields
(9a) E([Alpha] | [n.sub.1] - [n.sub.2] = ([n.sub.1] - [n.sub.2] + r)M/([T.sub.1,2](M + r)).
Thus, for persons who have lost [n.sub.1] - [n.sub.2] partners, we can apply the conditional estimator [Alpha] | [n.sub.1] - [n.sub.2]:
(9b) [Alpha] | [n.sub.1] - [n.sub.2] = ([n.sub.1] - [n.sub.2] + r)M/([T.sub.1,2] (M + r))
Using Equation 3, the average concurrency, C, conditional on the value of [n.sub.1] - [n.sub.2] can then be estimated as:
(10) C | [n.sub.1] - [n.sub.2] = ([bar] [n.sub.1] | [n.sub.1] - [n.sub.2]) - ([Alpha] | [n.sub.1] - [n.sub.2]) [T.sub.1]
For the estimate of the average partnership duration, [Delta] conditional on [n.sub.1] - [n.sub.2] ,we apply on the basis of Equation 2:
(11) | [n.sub.1] - [n.sub.2] = ([Alpha] | [n.sub.1] - [n.sub.2])/(C | [n.sub.1] - [n.sub.2]
Standard errors of the estimates for parameter [Alpha] | [n.sub.1] - [n.sub.2] can be obtained from the following formula for the variance of [Alpha] | [n.sub.1] - [n.sub.2] (Bain & Engelhardt, 1987):
var([Alpha] | [n.sub.1] - [n.sub.2]) = (([n.sub.1] - [n.sub.2] + r)[M.sup.2]/([T.sub.1,2] [(M + r)).sup.2].
However, it is much more difficult to calculate standard errors for the estimates of the parameters C | [n.sub.+] - [n.sub.2] and [Delta] | [n.sub.1] - [n.sub.2] because these involve the covariance of [n.sub.1] | [n.sub.1] - [n.sub.2] and [Alpha] | [n.sub.1] - [n.sub.2], which can not be estimated directly. Therefore, we have approximated this covariance by its upper limit: the standard deviation of [Alpha] | [n.sub.1] - [n.sub.2] times the standard deviation of [n.sub.1] | [n.sub.1] - [n.sub.2]. This implies that the real standard errors of C | [n.sub.1] - [n.sub.2] and [Delta] | [n.sub.1] - [n.sub.2] are probably lower than the values we report.
Although in Method 2 heterogeneity in rate of partner acquisition can be taken into account, no relation between rate of partner acquisition and partnership status is assumed. In practice, in many societies it is likely that those married or in a long-term sexual partnership differ from those without such a partnership in their rates of partner acquisition. Therefore, in using Method 2, we should distinguish between those two subpopulations and, if possible, apply the method separately in those two subpopulations.
We applied our models using data from Nairobi, Kenya. First, we used data from males attending an STD clinic. Second, we used data from females attending a family-planning clinic. In Kenya the HIV/AIDS problem is substantial: In 1995, 24.6% of pregnant women were found to be HIV-infected in the HIV sentinel surveillance in Nairobi (National AIDS and STDs Control Programme & National Council for Population and Development, 1996).
Participants were 1,420 men attending an STD clinic in Nairobi, Kenya. The participants were asked about the number of sex partners they had in the previous two weeks,the previous one month, and the previous three months. As sexual behavior in recent weeks may have been influenced by their STD or vice versa, we used the latter two variables to infer their rate of partner acquisition. The assumption that sexual behavior has not changed over the periods addressed in the survey remains somewhat questionable, even while using the two longest periods.
Of these 1,420 men, 37 were excluded from the analyses because they did not respond to either or both of the two variables. The average number of sex partners in the previous 3 months and previous month were 2.18 (SD = 2.08) and 1.21 (SD = 1.10) respectively; the average of [n.sub.1] - [n.sub.2] was 0.97 (SD = 1.61). The data are shown in Tables 1a and 1b.
Table 1a. Distribution of Reported Number of Sex Partners in the Previous 3 Months and in the Previous Month, and the Distribution of the Number of Partners Lost During the First 2 of the Previous 3 Months.
Number of Partners During: Number of Partners Previous 3 Months Previous Month 0 85 289 1 495 707 2 443 269 3 185 80 4 68 18 5 40 10 6 17 4 7 6 2 8 16 2 9 4 0 10 13 2 More than 10 11 0 Number of Partners During: Partners Lost in Number of Partners First 2 Months 0 645 1 466 2 154 3 52 4 21 5 20 6 5 7 8 8 2 9 4 10 2 More than 10 4
Note. Data from 1,383 men attending an STD clinic in Kenya.
Table 1b. Reported Number of Sex Partners in the Previous 3 Months and in the Previous Month
Number of Partners in Previous Month Number of Partners in 0 1 2 3 4 Previous 3 Months 0 85 0 0 0 0 1 144 351 0 0 0 2 43 234 166 0 0 3 11 76 67 31 0 4 3 26 16 16 7 More than 4 3 20 20 33 11 Number of Partners in More Previous 3 Months than 4 0 0 1 0 2 0 3 0 4 0 More than 4 20
Note. Data from 1,383 men attending an STD clinic in Kenya.
When we applied Method 1, thereby assuming homogeneous behavior, we estimated that the males attending the STD clinic had an average of 5.80 new sex partners per year (SE = 0.25). Furthermore, the average duration of partnerships, [Delta], was 0.13 (SE = 0.01) year and average concurrency, C, was 0.74 (SE = 0.03).
When we applied Method 2, thereby allowing heterogeneity in behavior, we obtained different estimates of [Alpha], [Delta], and C for individuals with different values of [n.sub.1] - [n.sub.2], as shown in Table 2. The group of males attending the STD clinic engaged in many brief sexual partnerships. Although the average concurrency, C, seems to have increased with [n.sub.1] - [n.sub.2], the estimated average duration of partnerships was short for all levels of [n.sub.1] - [n.sub.2], except 0. As the high standard errors indicate, estimates of [Delta] and C are quite unstable for high values of [n.sub.1] - [n.sub.2], because they were based on very small numbers.
Table 2. Empirical Bayes Estimates of Sexual Behavior Parameters for Men Attending an STD Clinic inn Kenya.
Number of Average Duration of Partners ([n.sub.1] - New Partners Partnerships in [n.sub.2]) Acquired ([Alpha]) Years ([Delta]) 0 2.17 0.35 (0.11) (0.05) 1 5.93 0.08 (0.21) (0.02) 2 9.68 0.07 (0.49) (0.02) 3 13.44 0.07 (0.98) (0.03) 4 17.20 0.10 (1.75) (0.06) 5 20.95 0.12 (1.98) (0.06) 6 24.71 0.08 (4.30) (0.07) 7 28.47 0.11 (3.66) (0.08) 8 32.22 0.03 (7.78) (0.10) 9 35.98 0.03 (5.81) (0.05) 10 39.73 0.11 (8.64) (0.09) 11 43.49 0.08 (9.04) (0.08) 17 66.03 0.05 (15.75) (0.07) 27 103.59 0.04 (19.73) (0.06) Number of Partners ([n.sub.1] - Average [n.sub.2]) Concurrency (C) 0 0.75 (0.06) 1 0.47 (0.10) 2 0.69 (0.21) 3 0.91 (0.40) 4 1.75 (0.84) 5 2.61 (0.97) 6 2.02 (1.28) 7 3.26 (1.96) 8 0.94 (2.94) 9 1.01 (1.45) 10 4.57 (2.66) 11 3.63 (2.76) 17 3.50 (3.94) 27 4.11 (4.93)
Note. Standard errors are shown in parentheses. N = 1,383.
In a family-planning clinic in Nairobi, Kenya, women attending the clinic for the first time were interviewed about their sexual behavior. Questions asked were about marital status, lifetime number of sex partners, and number of sex partners in the previous year and in the previous three months. In addition, women were asked about the number of new sex partners in the previous year. In this application we used data only from single women in the sample (n = 121). There was no nonresponse. The average number of sex partners in the previous year and previous three months were 1.36 (SD = 0.80) and 1.02 (SD = 0.62), respectively; the average of [n.sub.1] - [n.sub.2] was 0.34 (SD = 0.62). The data are shown in Tables 3a and 3b.
Table 3a. Distribution of Reported Number of Sex Partners in the Previous Year and in the Previous 3 Months, and the Distribution of the Numbers of Partners Lost During the First 9 Months of the Previous Year.
Number of Partners During: Number of Partners Previous Year Previous 3 Months 0 5 17 1 81 89 2 27 12 3 5 2 4 1 1 5 2 0 Number of Partners During: Partners Lost in Number of Partners First 9 Months 0 89 1 24 2 7 3 1 4 0 5 0
Note. Data from 121 single women attending a family-planning clinic in Kenya.
Table 3b. Reported Number of Sex Partners in the Previous Year and in the Previous 3 Months.
Number of Partners in Previous 3 Months Number of Partners 0 1 2 3 4 in Previous Year 0 5 0 0 0 0 1 9 72 0 0 0 2 3 13 11 0 0 3 0 4 0 1 0 4 0 0 0 1 0 5 0 0 1 0 1
Note. Data from 121 single women attending a family-planning clinic in Kenya.
When we applied Method 1, assuming homogeneous behavior, we estimated that the females attending the family-planning clinic had an average of 0.45 new sex partners per year (SE = 0.08). For the average duration of partnerships, [Delta], we obtained an estimate of 2.00 years (SE = 0.43), and for the average concurrency, C, we found an estimate of 0.90 (SE = 0.06).
To obtain estimates of the rate of sex partner acquisition per year [Alpha], average duration of partnerships [Delta], and average concurrency C under the assumption of heterogeneity in behavior, we applied the empirical Bayes estimators of Method 2. Estimates are presented in Table 4. For comparison, the average reported number of new sex partners in the past year is also included in Table 4.
Table 4. Empirical Bayes Estimates of Sexual Behavior Parameters Compared to Reported Numbers of New Sex Partners for Single Women Attending a Family-Planning Clinic in Kenya.
Number of New Partners Average Duration Partners Acquired of Partnerships ([n.sub.1] - ([Alpha]) in Years [n.sub.2]) ([Delta]) 0 0.39 1.79 (0.03) (0.33) 1 0.57 2.20 (0.07) (0.71) 2 0.74 2.40 (0.14) (0.90) 3 0.91 4.33 (0.41) (2.35) Number of Average Mean number of Partners Concurrency (C) new partners ([n.sub.1] - reported [n.sub.2]) 0 0.70 0.11 (0.08) 1 1.26 0.63 (0.26) 2 1.82 0.43 (0.34) 3 4.06 1.00 (0.41)
Note. Standard errors are shown in parentheses. N = 121.
The proposed empirical Bayes procedure grossly overestimates [Alpha] for [n.sub.1] - [n.sub.2] = 0. This is probably due to the classification of many women in stable heterosexual partnerships as single when they are de facto Jiving in a stable, consensual union. These women are likely to have a low rate of acquisition of new partners. Estimating their rate of acquisition of new partners using a mean, M, from a group that includes many truly single women leads to an overestimation of their [Alpha]. We tested this explanation by separating women reporting only one partner and those reporting more than one partner over the previous year ([n.sub.1]). Those reporting [n.sub.1] = 1 and [n.sub.1] - [n.sub.2] = 0 (72 women) reported an average of 0.03 new sex partners over the previous year, whereas the 12 women reporting [n..sub.1] [is greater than] 1 and [n.sub.1] - [n.sub.2] = 0 reported an average of 0.67 new sex partners over the previous year. This result indicates the need to carefully select the population stratum within which the methods can be applied. Unfortunately, we do not know which women in our sample lived in a stable, consensual union and which did not. Thus, we cannot explore this idea of separating the two groups of women with this data set.
We described two methods to estimate the rate of partnership acquisition, the average duration of partnerships and concurrency of partnerships from retrospective information about the total number of sex partners in two overlapping intervals. The first method assumes homogeneity in sexual behavior, whereas the second method considers the possibility of heterogeneity in sexual behavior. In Method 2, the estimate for an individual is provided by a compromise between his reported behavior and the population mean. In many societies, however, the difference between persons with and without a steady partnership are so large that the heterogeneity taken into account in Method 2 cannot fully capture these differences. For instance, the British National Survey of Sexual Attitudes and Lifestyle found an adjusted (for age, social class, and age at first intercourse) odds ratio of reporting two or more partners in the previous year of the order of 10 for unmarried men and women when compared with married men and women (Johnson, et al., 1993). A similar pattern was found in many developing countries (Cleland & Ferry, 1995). Therefore, researchers should ideally distinguish individuals in stable partnerships, such as marriages, from those outside such partnerships, while applying our methods. Distinguishing these subgroups may also be helpful in interpreting and understanding the rather abstract measure of average concurrency. As indicated before, individuals with no partners half of the time and two partners during the rest of the time have the same value for this concurrency measure as individuals in stable, monogamous partnerships. In terms of risk-taking behavior, however, these groups are very different.
Sexual behavior surveys are likely to be faced with problems of non-response, recall bias and social-desirability bias (Catania, Gibson, Chitwood, & Coates, 1990; Peterman, 1995). In a sense, poor validity (e.g., systematic underreporting as a result of a tendency to report desirable behavior) is worse than nonresponse: Poor validity may go unnoticed whereas nonresponse cannot. Although there was little nonresponse in both study samples used to illustrate our methods, the veracity of responses is unknown. In fact, an apparent good response may correlate with poor veracity, because underreporting the number of sex partners and nonresponse are competing strategies for concealing one's true behavior.
Of course, the quality of estimates derived from data cannot be better than the data themselves. For example, it individuals report exactly half the number of sex partners they have had during both intervals, then from Equation 4b and 5b, [Alpha] and C in Method 1 will also be estimated at half their true values. By contrast, [Delta] will not be affected by this "proportional" underreporting. Effects of other forms of underreporting can similarly be explored, but unfortunately not easily remedied, using the equations presented.
As our methods are based on several stringent assumptions, such as the possibility of concurrency in the population, which cannot be tested from the data, they should not be preferred to the direct measurement of relevant model parameters. Where possible, surveys of sexual behavior should aim at measuring relevant parameters directly. However, much data on sexual behavior has been, and will continue to be, collected without this aim. Our methods were developed to help mathematical modelers to utilize data that would otherwise be of limited value.
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Manuscript accepted December 30, 1997.
This study was supported by grants from the Health, Family Planning, and AIDS program (DG VIII) and the Science and Technology for Developing Countries (STD) program (DG XII) of the Commission of the European Communities, Brussels, Belgium.
Direct correspondence to Mrs. Carina van Vliet, Centre for Decision Sciences in Tropical Disease Control, Department of Public Health, Erasmus University Rotterdam, P.O. Box 1738, 3000 DR Rotterdam, The Netherlands; e-mail:firstname.lastname@example.org…