Estimating sexual behavior parameters from routine sexual behavior data.The advent of the HIV/AIDS HIV/AIDS Human Immunodeficiency Virus/Acquired Immune Deficiency Syndrome epidemic epidemic, outbreak of disease that affects a much greater number of people than is usual for the locality or that spreads to regions where it is ordinarily not present. has greatly stimulated the study of sexual behavior sexual behavior A person's sexual practices–ie, whether he/she engages in heterosexual or homosexual activity. See Sex life, Sexual life. . Empirical behavior studies such as surveys (e.g., ACSF ACSF Afghan Civil Society Forum ACSF Artificial Cerebrospinal Fluid 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
ability of an agent to infect. , to the epidemiology epidemiology, field of medicine concerned with the study of epidemics, outbreaks of disease that affect large numbers of people. Epidemiologists, using sophisticated statistical analyses, field investigations, and complex laboratory techniques, investigate the cause of HIV HIV (Human Immunodeficiency Virus), either of two closely related retroviruses that invade T-helper lymphocytes and are responsible for AIDS. There are two types of HIV: HIV-1 and HIV-2. HIV-1 is responsible for the vast majority of AIDS in the United States. and other sexually transmitted diseases Sexually transmitted diseases Infections that are acquired and transmitted by sexual contact. Although virtually any infection may be transmitted during intimate contact, the term sexually transmitted disease is restricted to conditions that are largely (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 Mixing patterns refer to systematic tendencies of one type of nodes in a network to connect to another type. For instance, nodes might tend to link to others that are very similar or very different. (Gupta, Anderson Anderson, river, Canada Anderson, river, c.465 mi (750 km) long, rising in several lakes in N central Northwest Territories, Canada. It meanders north and west before receiving the Carnwath River and flowing north to Liverpool Bay, an arm of the Arctic , & 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 Watts, residential section of south central Los Angeles. Named after C. H. Watts, a Pasadena realtor, the section became part of Los Angeles in 1926. Artist Simon Rodia's celebrated Watts Towers are there. & May, 1992) parameters of interest with respect to sexual behavior include the rate of acquisition of new partners', and concurrency Operations that are performed simultaneously within the computer. For example, dual-core CPUs provide complete overlapping of two independent processes. See dual core, hyperthreading, multiprocessing, multitasking, multithreading, SMP and MPP. concurrency - multitasking 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 serial monogamy Noun the practice of having a number of long-term romantic or sexual partners in succession Noun 1. 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 estimation In mathematics, use of a function or formula to derive a solution or make a prediction. Unlike approximation, it has precise connotations. In statistics, for example, it connotes the careful selection and testing of a function called an estimator. of all relevant parameters: Detailed information about the timing of the last three sex partners is solicited from respondents In the context of marketing research, a representative sample drawn from a larger population of people from whom information is collected and used to develop or confirm marketing strategy. (Johnson, Wadsworth, Wellings, & Field, 1993). Surveys using a similar or more extensive local sexual-network design have been performed in the United States United States, officially United States of America, republic (2005 est. pop. 295,734,000), 3,539,227 sq mi (9,166,598 sq km), North America. The United States is the world's third largest country in population and the fourth largest country in area. (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 as·cer·tain tr.v. as·cer·tained, as·cer·tain·ing, as·cer·tains 1. To discover with certainty, as through examination or experimentation. See Synonyms at discover. 2. 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. METHODS Assumptions 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 Null; void; without force or effect; lacking in authority. For example, a will that has not been properly witnessed is invalid and unenforceable. INVALID. In a physical sense, it is that which is wanting force; in a figurative sense, it signifies that which has no effect. . For example, if the reason for interviewing is the detection of an STD (Subscriber Trunk Dialing) Long distance dialing outside of the U.S. that does not require operator intervention. STD prefix codes are required and billing is based on call units, which are a fixed amount of money in the currency of that country. , then it may be reasonable to assume a "flurry Flurry A drastic volume increase in a specific security. " 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 sexual intercourse or coitus or copulation Act in which the male reproductive organ enters the female reproductive tract (see reproductive system). has occurred in ongoing partnerships during that period. For some African countries where male migrant workers A migrant worker is someone who regularly works away from home, if they even have a home.[] Although the United Nations' use of this term overlaps with 'foreign worker', the use of the term within the United States is more specific. often reside in the cities for prolonged pro·long tr.v. pro·longed, pro·long·ing, pro·longs 1. To lengthen in duration; protract. 2. To lengthen in extent. 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 spu·ri·ous adj. Similar in appearance or symptoms but unrelated in morphology or pathology; false. spurious simulated; not genuine; false. estimates of the rate of partner acquisition. We could find estimates of concurrency in excess of one, which are clearly incompatible incompatible adj. 1) inconsistent. 2) unmatching. 3) unable to live together as husband and wife due to irreconcilable differences. In no-fault divorce states, if one of the spouses desires to end the marriage, that fact proves incompatibility, and a divorce 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 A Poisson process, named after the French mathematician Siméon-Denis Poisson (1781 - 1840), is a stochastic process which is used for modeling random events in time that occur to a large extent independently of one another (the word event . 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. Notation notation: see arithmetic and musical notation. How a system of numbers, phrases, words or quantities is written or expressed. Positional notation is the location and value of digits in a numbering system, such as the decimal or binary system. 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 Expected value The weighted average of a probability distribution. Also known as the mean 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 The same. Contrast with heterogeneous. homogeneous - (Or "homogenous") Of uniform nature, similar in kind. 1. In the context of distributed systems, middleware makes heterogeneous systems appear as a homogeneous entity. For example see: interoperable network. 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 mul·ti·ply 1 v. mul·ti·plied, mul·ti·ply·ing, mul·ti·plies v.tr. 1. To increase the amount, number, or degree of. 2. Mathematics To perform multiplication on. 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 (1) Any value passed to a program by the user or by another program in order to customize the program for a particular purpose. A parameter may be anything; for example, a file name, a coordinate, a range of values, a money amount or a code of some kind. [Alpha], we have used a Taylor expansion, in which the variance The discrepancy between what a party to a lawsuit alleges will be proved in pleadings and what the party actually proves at trial. In Zoning law, an official permit to use property in a manner that departs from the way in which other property in the same locality of [Alpha] is related to the variance of [Alpha], the variance of C, and the covariance Covariance A measure of the degree to which returns on two risky assets move in tandem. A positive covariance means that asset returns move together. A negative covariance means returns vary inversely. 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 infinity, in mathematics, that which is not finite. A sequence of numbers, a1, a2, a3, … , is said to "approach infinity" if the numbers eventually become arbitrarily large, i.e. . 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 spouse A legal marriage partner as defined by state law 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 Not the same. Contrast with homogeneous. heterogeneous - Composed of unrelated parts, different in kind. Often used in the context of distributed systems that may be running different operating systems or network protocols (a heterogeneous network). Behavior More attractive estimators can be obtained by permitting population heterogeneity het·er·o·ge·ne·i·ty n. The quality or state of being heterogeneous. heterogeneity the state of being heterogeneous. using Empirical Bayes estimators In decision theory and estimation theory, a Bayes estimator is an estimator or decision rule that maximizes the posterior expected value of a utility function or minimizes the posterior expected value of a loss function. (See also prior probability. (Cox & Hinkley, 1974). In Method 2, we assume that the rate of partner acquisition, [Alpha], varies between individuals according to according to prep. 1. As stated or indicated by; on the authority of: according to historians. 2. In keeping with: according to instructions. 3. a gamma distribution with parameters r and p/(1-p). The skewness Skewness A statistical term used to describe a situation's asymmetry in relation to a normal distribution. Notes: A positive skew describes a distribution favoring the right tail, whereas a negative skew describes a distribution favoring the left tail. 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 A statistical method developed by the 18th century French mathematician S. D. Poisson, which is used for predicting the probable distribution of a series of events. For example, when the average transaction volume in a communications system can be estimated, Poisson distribution is used with parameter [[Alpha].sub.0]. Under these assumptions the population ([n.sub.1] - [n.sub.2]) follows a negative binomial distribution In probability and statistics the negative binomial distribution is a discrete probability distribution. The Pascal distribution and the Polya distribution are special cases of the negative binomial. with parameters r and p (Bain & Engelhardt, 1987). For the negative binomial binomial (bī'nō`mēəl), polynomial expression (see polynomial) containing two terms, for example, x+y. The binomial theorem, or binomial formula, gives the expansion of the nth power of a binomial (x+ 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 In statistics, the average amount a number varies from the average number in a series of numbers. (statistics) standard deviation - (SD) A measure of the range of values in a set of numbers. 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. APPLICATIONS 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 sentinel /sen·ti·nel/ (sen´ti-n'l) one who gives a warning or indicates danger. sentinel a recording mechanism, such as an animal, a farm or a veterinarian, posted explicitly to record a possible occurrence or series of surveillance in Nairobi (National AIDS and STDs Control Programme & National Council for Population and Development, 1996). Application 1 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 VICE VERSA. On the contrary; on opposite sides. , 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 unstable, adj 1. not firm or fixed in one place; likely to move. 2. capable of undergoing spontaneous change. A nuclide in an unstable state is called radioactive. An atom in an unstable state is called excited. 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 See parenthesis. parentheses - See left parenthesis, right parenthesis. . N = 1,383. Application 2 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 marital status, n the legal standing of a person in regard to his or her marriage state. , 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 heterosexual /het·ero·sex·u·al/ (-sek´shoo-al) 1. pertaining to, characteristic of, or directed toward the opposite sex. 2. one who is sexually attracted to persons of the opposite sex. partnerships as single when they are de facto [Latin, In fact.] In fact, in deed, actually. This phrase is used to characterize an officer, a government, a past action, or a state of affairs that must be accepted for all practical purposes, but is illegal or illegitimate. Jiving in a stable, consensual CONSENSUAL, civil law. This word is applied to designate one species of contract known in the civil laws; these contracts derive their name from the consent of the parties which is required in their formation, as they cannot exist without such consent. 2. 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 o·ver·es·ti·mate tr.v. o·ver·es·ti·mat·ed, o·ver·es·ti·mat·ing, o·ver·es·ti·mates 1. To estimate too highly. 2. To esteem too greatly. 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 stratum /stra·tum/ (strat´um) (stra´tum) pl. stra´ta [L.] a layer or lamina. stratum basa´le 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. DISCUSSION We described two methods to estimate the rate of partnership acquisition, the average duration of partnerships and concurrency of partnerships from retrospective LAW, RETROSPECTIVE. A retrospective law is one that is to take effect, in point of time, before it was passed. 2. Whenever a law of this kind impairs the obligation of contracts, it is void. 3 Dall. 391. information about the total number of sex partners in two overlapping intervals. The first method assumes homogeneity Homogeneity The degree to which items are similar. 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 INTERCOURSE. Communication; commerce; connexion by reciprocal dealings between persons or nations, as by interchange of commodities, treaties, contracts, or letters. ) 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 Pe´ter`man n. 1. A fisherman; - so called after the apostle Peter. , 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 veracity (v  ras´itē),n 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 con·ceal tr.v. con·cealed, con·ceal·ing, con·ceals To keep from being seen, found, observed, or discovered; hide. See Synonyms at hide1. 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 proportional values expressed as a proportion of the total number of values in a series. proportional dwarf the patient is a miniature without disproportionate reductions or enlargements of body parts. " 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. REFERENCES ACSF Investigators. (1992). AIDS and sexual behavior in France. Nature, 360. 407-409. Bain, L. J., & Engelhardt, M. (1987). Introduction to probability and mathematical statistics Mathematical statistics uses probability theory and other branches of mathematics to study statistics from a purely mathematical standpoint. Mathematical statistics is the subject of mathematics that deals with gaining information from data. . Boston: Duxbury Press. Blower, S. M., Anderson, R. M., & Wallace, P. (1990). Log-linear models log-linear model a statistical model which models frequency counts in contingency tables by using an analysis of variance approach. , sexual behaviors and HIV. Epidemiological epidemiological emanating from or pertaining to epidemiology. epidemiological associations the associative relationships between the frequency of occurrence of a disease and its determinants, its predisposing and precipitating implications of heterosexual transmission. Journal of AIDS, 3, 763-772. Catania, J. A., Coates, T. J., & Turner, H. (1992). Prevalence of AIDS related risk factors and condom 1. condom - The protective plastic bag that accompanies 3.5-inch microfloppy diskettes. Rarely, also used of (paper) disk envelopes. Unlike the write protect tab, the condom (when left on) not only impedes the practice of SEX but has also been shown to have a high failure use in the United States. Science, 258, 1101-1105. Catania, J. A.. Gibson, D. R., Chitwood, D. D., & Coates, T. J. (1990). Methodological problems in AIDS behavioral behavioral pertaining to behavior. behavioral disorders see vice. behavioral seizure see psychomotor seizure. research: Influences on measurement error and participation bias in studies of sexual behavior. Psychological Bulletin, 108, 339-362. Cleland, J., & Ferry, B. (1995). Sexual behavior and AIDS in the developing world. Geneva Geneva, canton and city, Switzerland Geneva (jənē`və), Fr. Genève, canton (1990 pop. 373,019), 109 sq mi (282 sq km), SW Switzerland, surrounding the southwest tip of the Lake of Geneva. : WHO. Cox, D. R., & Hinkley, D. V. (1974). Theoretical statistics. London: Chapman and Hall Chapman and Hall was a British publishing house, founded in the first half of the 19th century by Edward Chapman and William Hall. Upon Hall's death in 1847, Chapman's cousin Frederic Chapman became partner in the company, of which he became sole manager upon the retirement of . Garnett, G. P., & Johnson, A. M. (1997). Coining a new term in epidemiology: Concurrency and HIV. AIDS, 11, 681-683. Ghani, A. C., Swinton, J., & Garnett, G. P. (1996). The role of sexual partnership networks in the epidemiology of gonorrhea gonorrhea (gŏnərē`ə), common infectious disease caused by a bacterium (Neisseria gonorrhoeae), involving chiefly the mucous membranes of the genitourinary tract. . Sexually Transmitted Diseases, 24, 45-56. Gupta, S., Anderson, R. M., & May, R. M. (1989). Networks of sexual contacts: Implications for the spread of HIV. AIDS, 3, 807-817. Johnson, A. M. (1996). Sources and uses of empirical observations to characterise Verb 1. characterise - be characteristic of; "What characterizes a Venetian painting?" characterize differentiate, distinguish, mark - be a distinctive feature, attribute, or trait; sometimes in a very positive sense; "His modesty distinguishes him from his networks of sexual behavior. In V. Isham & G. Medley med·ley n. pl. med·leys 1. An often jumbled assortment; a mixture: "That night he dreamed he was traveling in a foreign country, only it seemed to be a medley of all the countries he'd ever been to and (Eds.), Models of infectious human diseases: Their structure and relation to data (pp. 253-262). Cambridge, England: Cambridge University press Cambridge University Press (known colloquially as CUP) is a publisher given a Royal Charter by Henry VIII in 1534, and one of the two privileged presses (the other being Oxford University Press). . Johnson, A. M., Wadsworth, J., Wellings, K., & Field, J. (1993). Sexual attitudes and lifestyles. Oxford, England: Blackwell Scientific. Kault, D. A. (1993). Assessment of sexual mixing patterns. Mathematical Biosciences. 115, 33-64. Laumann, E., Gagnon, J., Michael, R., & Michaels, S. (1994). The social organization of sexuality. Chicago: University of Chicago Press The University of Chicago Press is the largest university press in the United States. It is operated by the University of Chicago and publishes a wide variety of academic titles, including The Chicago Manual of Style, dozens of academic journals, including . Morris, M. (1993). Telling tales explain the discrepancy DISCREPANCY. A difference between one thing and another, between one writing and another; a variance. (q.v.) 2. Discrepancies are material and immaterial. in sexual partner reports. Nature, 365, 437-440. Morris, M. (1995). Data driven network models for the spread of infectious disease Infectious disease A pathological condition spread among biological species. Infectious diseases, although varied in their effects, are always associated with viruses, bacteria, fungi, protozoa, multicellular parasites and aberrant proteins known as prions. . In D. Mollison (Ed.), Epidemic models The introduction to this January 2007 provides insufficient context for those unfamiliar with the subject matter. Please help [ improve the introduction] to meet Wikipedia's layout standards. You can discuss the issue on the talk page. : Their structure and relation to data (pp. 302-322). Cambridge: Cambridge University Press. Morris, M., & Kretzschmar, M. (1995). Concurrent partnerships and transmission dynamics in networks. Social Networks, 17. 299-318. Morris, M., & Kretzschmar, M. (1997). Concurrent partnerships and the spread of HIV. AIDS. 11, 641-48. Morris, M., Podhisita, C., Wawer, M. J., & Handcock, M. S. (1996). Bridge populations in the spread of HIV/AIDS in Thailand. AIDS, 10, 1265-1271, Morris, M., Sewankambo, N., Wawer, M., Serwadda, D., & Lainjo, B. (1995). Concurrent partnerships and HIV transmission in Rakai district Rakai is a district in south-central Uganda. Like other Ugandan districts, it is named after its 'chief town'. The district borders the northwestern Tanzanian district of Kagera to the south. As a result, it experiences much cross border commercial traffic to and from Bukoba. , Uganda. Paper presented at the IXth International Conference on AIDS and STD in Africa, Kampala. Moses, S., Muia, E., & Bradley, J. E. (1994). Sexual behavior in Kenya: Implications for sexually transmitted disease sexually transmitted disease (STD) or venereal disease, term for infections acquired mainly through sexual contact. Five diseases were traditionally known as venereal diseases: gonorrhea, syphilis, and the less common granuloma inguinale, transmission and control. Social Science and Medicine, 39, 1649-1656. National AIDS and STDs Control Programme & National Council for Population and Development. (1996). AIDS in Kenya; Background. Projections, Impact, Interventions. Nairobi: National AIDS and STDs Control Programme. Peterman, T. A. (1995). Can we get people to participate in a study of sexual behavior? Sexually Transmitted Diseases, 23, 191-196. Rothenberg, R. (1997). Model trains of thought. Sexually Transmitted Diseases, 24, 201-203. Van Zessen, G., & Sandfort, T G. M. (1991). Seksualiteit in nederland: seksueel gedrag, risico en preventie van AIDS. [Sexuality in the Netherlands: Sexual behavior, risk and prevention of AIDS]. Amsterdam: Swets and Zeitlinger. Watts, C. H., & May, R. M. (1992). The influence of concurrent partnerships on the dynamics of HIV/AIDS. Mathematical Biosciences, 108, 89-104. Wawer, M. J. (1990). Behavioral research for AIDS prevention in Thailand. Unpublished manuscript manuscript, a handwritten work as distinguished from printing. The oldest manuscripts, those found in Egyptian tombs, were written on papyrus; the earliest dates from c.3500 B.C. . Wawer, M. J. (1993). Ugandan sexual network/behaviors study for HIV prevention. Unpublished manuscript. World Health Organization. (1989). Guidelines guidelines, n.pl a set of standards, criteria, or specifications to be used or followed in the performance of certain tasks. of qualitative approaches to study KABP and partner relations in the context of HIV/AIDS. GPA/WHO. Geneva: WHO. World Health Organization Global Programme on AIDS. (1994). Evaluation of a national AIDS programme: A methods package. 1. Prevention of HIV injection. Geneva: WHO. Yorke, J. A., & Hethcote, H. W. (1978). Dynamics and control of the transmission of gonorrhea. Sexually Transmitted Diseases, 5, 51-56. Manuscript accepted December 30, 1997. This study was supported by grants from the Health, Family Planning family planning Use of measures designed to regulate the number and spacing of children within a family, largely to curb population growth and ensure each family’s access to limited resources. , and AIDS program (DG VIII) and the Science and Technology for Developing Countries (STD) program (DG XII) of the Commission of the European Communities European Community: see European Union. European Community (EC) Organization formed in 1967 with the merger of the European Economic Community, European Coal and Steel Community, and European Atomic Energy Community. , Brussels, Belgium. Direct correspondence to Mrs. Carina Carina (kərē`nə) [Lat.,=the keel], southern constellation, representing the keel of the ancient constellation Argo Navis, or Ship of the Argonauts. Carina contains Canopus, the second brightest star in the sky. van Vliet, Centre for Decision Sciences in Tropical Disease Tropical diseases are infectious diseases that either occur uniquely in tropical and subtropical regions (which is rare) or, more commonly, are either more widespread in the tropics or more difficult to prevent or control. Control, Department of Public Health, Erasmus University Erasmus University Rotterdam is a university in the Netherlands, located in Rotterdam. The university is named after Desiderius Erasmus Roterodamus, a 15th century humanist and theologian. Rotterdam, P.O. Box 1738, 3000 DR Rotterdam, The Netherlands; e-mail:vanliet@mgz.fgg.eur.nl |
|
||||||||||||||||
Printer friendly
Cite/link
Email
Feedback
Reader Opinion