In recent years, there has been increasing interest in the spread of HIV within stable sexual partnerships [1], since the HIV epidemic has matured in many countries and it is believed that the proportion of new infections occurring within couples has risen [2]. Evidence has shown that across countries, a sizeable proportion of couples with any HIV infection are discordant [2]. A serodiscordant couple is a couple in which one partner has tested positive for HIV and the other has not.

Evidence suggests that women have the greatest risk of contracting HIV within a marital relationship. Fewer attempts have been made to understand a man’s risk. But regions with generalised epidemics, high rates of serodiscordance among heterosexual couples in which the woman is HIV positive suggest that a man’s risk of marital HIV acquisition could also be substantial [3].

In Mozambique, in among 1.6 million people living with HIV/AIDS in 2009, around one in every ten couples were discordant. In 5% of all couples, both members were HIV positive (concordant positive), while in 10% of all couples, one member was HIV positive whilst the other was HIV negative (female or male discordant). The estimates of 2011 showed that the proportion of male discordant couples was similar to the proportion of female discordant couples [1]. By female discordant we refer to a couple where the woman was HIV positive while the man was HIV negative. A couple where the man was HIV positive whilst the woman was HIV negative is referred to as a male discordant couple.

Despite the empirical evidence pointing to their programmatic importance, serodiscordant couples are often overlooked or, at best, only vaguely addressed in many national prevention plans. This omission may stem not only from sensitivity surrounding HIV within couples but also from misperceptions about the extent of serodiscordance and failure to understand that it is possible to prevent transmission within a stable union once one partner has become infected [4].

Statistical methods and models to investigate risk factors associated with HIV serodiscordance among couples can be very helpful. Simple methods are based on the use of descriptive statistics as well as bivariate associations. Extensions of such descriptive analyses have been covered by (standard) logistic regression models. Kaiser et al. [5] developed two different models to assess factors associated with couple status, one for HIV discordance as the outcome and a second one with HIV concordance as the binary outcome. In the first model, concordant HIV-infected couples were excluded from the denominator and in the second model concordant uninfected couples were excluded from the denominator. Based on the 2007 Kenya AIDS Indicator Survey, they found that the following factors were independently associated with HIV-discordance: young age in women, increasing number of lifetime sexual partners in women, HSV-2 (herpes simplex virus type 2) infection in either or both partners, and lack of male circumcision. Independent factors for HIV-concordance included HSV-2 infection in both partners and lack of male circumcision. Fishel et al. [1] based their findings on an analysis including two logistic regression models. The first model compared concordant positive couples with all discordant couples, regardless of whether the discordant couple is male discordant or female discordant. The second multinomial logistic regression compares concordant positive couples separately to male discordant and female discordant couples. They concluded that HIV discordance of couples in Mozambique does not appear to have a strong association with whether or not the couple is polygynous, the amount of time passed since the couple last had sexual intercourse with each other, or risk factors for non-sexual transmission of HIV. Although there are some differences in the HIV status of couples by characteristics of interest, the differences do not outline a profile that makes discordant couples easy to distinguish from the general population.

Our approach is different in two ways. Instead of using two separate logistic models, we propose to use genuine bivariate models for correlated binary data, allowing one to fit separate logistic regression models in one single analysis while taking into account the correlation between both binary responses (HIV status of man and woman within a couple). Moreover we propose to reparametrize the model such that serodiscordance itself is a model parameter, allowing to investigate the factors affecting the serodiscordance in a direct way. The particular measure of serodiscordance proposed in this paper is inspired by the conditional synchrony measure (CSM) as introduced by [6] to measure synchrony in neuronal firing.

There exist different families of models for correlated binary data: so-called marginal, population averaged models and conditional models. Full likelihood models such as the bivariate Dale model [7] and alternative models estimated by the method of moments (Generalized Estimating Equations, GEE, [8]) are of the first type. Generalized linear mixed models [9], such as the logistic-normal model, are based on the use of random effects and constitute a second type of models. Which model family or combination of types is to be preferred depends on the design of the study and on the research question(s) of interest.

In this paper, we will propose the combination of a reparametrized version of the bivariate Dale model together with multivariate random effects with different covariance structures to take into account the design of the study. Focus is on the specification and estimation of appropriate covariate models for all model components including the serodiscordance parameter. The models were designed to investigate in particular the relationship/association between the HIV status of woman and man within a couple and the risk factors associated with HIV serodiscordance among couples.

The paper is organized as follows. A first section on materials and methods provides information on the INSIDA survey with some details about the survey design and the variables used in this study. The statistical models are introduced together with a new measure of serodiscordance. The use of survey weights and the model building strategy is briefly discussed. The second section summarizes the results of the application of the proposed methodology on the INSIDA data and the paper ends with conclusions and a final discussion.

As the selected set of covariates for the two univariate logistic regression models for π_F and π_M only differs in one covariate for each model component, it was decided to keep the set equal for both, allowing a more complete comparison and interpretation of the estimated effects on the probability to be HIV positive for the female and the male partner within the same couple. The covariates involved are: the HIV prevalence of the province, the union number of the woman and that of the man, the condom use by the man, and the wealth index. The stepwise procedure to identify the factors influencing the serodiscordance measure SDM led to significant effects of the HIV prevalence of the province, and the union number of the woman.

The model components Eq (8) with these selected sets of covariates were then fitted simultaneously with different (multivariate) random effects structures, capturing the heterogeneity across the EAs. Table 2 shows the values of -2×log-likelihood, the number of parameters, the AIC and BIC goodness of fit measures of all fitted models: the joint marginal model without random AE effects and a subset of all random effects models, with different and with partly common slopes for the models for π_F and π_M. The last two columns shows the models’ ranks according to AIC and BIC. The independent random effects and the full or partial random effects models did not convergence. So the results and the discussion is limited to the full-, partial-shared and partial equal random effects models.

According to AIC and BIC the best model is the partial-equal random effects model with common effects (CE-PE) for the HIV prevalence of the province and the wealth index, rather closely followed by the partial-shared random effects model with the same common effects (CE-PS). Based on the likelihood ratio test (LRT), the null hypothesis of equal random effects σ_M = σ_F cannot be rejected at level 0.05 (p-val = 0.19). So, the CE-PE model is taken as final model.

The estimates of all 16 parameters of the CE-PE model and their standard error estimates are shown in Table 3. In the following discussion 95% confidence intervals for the odds ratio (OR-CI) are also shown. First of all, the probability to be HIV positive increases with the HIV prevalence in the province, for both women and men in the same way, as to be expected. As compared to a province with less than 5% prevalence, the odds to be HIV positive increases with a factor of about 6.9 when living in a province with more than 15% prevalence (OR-CI [4.10,11.62]). The effect of wealth index is also identical for women and men: as compared to the “richer” reference category, the odds to be HIV positive is estimated to be about 32% lower for the “middle” category, and 46% lower for “poorer” category (OR-CI [0.48,0.96] and [0.38.0.75] respectively). In Mozambique it is quite usual that one (or both but especially the male) partner within a richer couple has multiple extramarital partners or even visits sex workers, increasing the risk for HIV infection [16], [17]. The effects of union number and condom use are different for women and men. Whereas condom use has no significant effect on the odds for men to be HIV positive, not using a condom is estimated to increase the odds for women to be HIV positive with about 90% (OR-CI [1.16,3.12]). That the male partner has been married or lived with a female partner more than once has no significant effect on the odds for the woman to be HIV positive, but it increases the odds for the man to be HIV positive with about 45% (OR-CI [1.09,1.93]). A female partner being married or having lived with a male partner more than once has a much higher odds to be positive (increase of about 121%, OR-CI [1.61,3.02]) and the same holds for her male partner, though with a more limited increase of about 56% (OR-CI [1.13,2.16]). For most combinations of the covariate values the estimated probability to be HIV positive is higher for the female partner, in line with what is known. For the female partner the fitted probability to be HIV positive varies between 1.55% and 48.50% depending on the covariate pattern; for the male partner between 1.55% and 31.86% which is less but still substantial. The above discussed OR estimates and confidence intervals are for a given EA, and the estimated probabilities for a “central” EA with specific EA-effect equal to 0.

The null hypothesis H₀: σ = 0 is rejected at 5% level, using a χ 0 , 1 2 mixture, indicating a significant EA effect on the probability to be HIV positive (common to both members of the couple in that EA) and implying highly varying probabilities to be HIV positive across EAs. For an EA with an extreme negative EA-effect - 2 σ ^ = - 1 . 356 the estimated probability to be HIV positive for the female partner varies between 0.4% and 19.50% (depending on the covariate pattern); for the male partner between 0.4% and 10.75%. On the other hand, for an EA with an extreme positive EA-effect 2 σ ^ = 1 . 356 the estimated probability to be HIV positive for the female partner varies between 5.77% and 78.52% (depending on the covariate pattern); for the male partner between 5.77% and 64.48%.

The odds for a couple to be serodiscordant decreases with the HIV prevalence of the province. As compared to a province with less than 5% prevalence, the odds for a couple to be serodiscordant decreases with 66% when living in a province with more than 15% prevalence (OR-CI [0.14,0.84]). In other words, given that at least one of the two partners is positive, the probability that both are positive increases with the HIV prevalence of the province where the couple is living. The odds of serodiscordance also decreases when the woman has been married or living with a man more than once, with an estimated decrease of about 43% (OR-CI [0.36,0.89]). Given that one of the two partners is positive, the probability that the other is not varies between 50.50% and 84.16%. Note that, according to the CE-PE model, the probability to be serodiscordant does not vary across EAs, in contrast to the probabilities for men and women to be HIV positive.

In this paper, a new serodiscordance measure SDM is defined as the conditional probability that both partners differ in their HIV status, given that one of them is positive. Together with the probabilities to be HIV positive for man and woman, the SDM parameter is embedded in a bivariate statistical model with fixed and random effects for covariates and design variables on each of the three parameters. A (final) model with significant fixed effects for the HIV prevalence of the province, the union number of woman and man, condom use and wealth index, and a significant random effect for the EAs was fitted using data from the 2009 INSIDA survey. An extension of the current analysis with more recent survey data and extending the model to investigate any time trend is a very interesting and relevant topic for further research.

In Zambia, a retrospectively study of 65 couples to estimate the likely origin of HIV infection, found that at least one quarter of cases of HIV infection in recently married men were acquired from extramarital partnerships, and for both men and women, less than one half of cases of HIV infection were acquired from their spouse/husband [4]. In addition, they report that many infections in married men, even in those with HIV-infected wives, could be acquired from outside the marriage [4]. Furthermore, a study conduct in South Africa to investigate who was infecting whom among migrant and non-migrant within concordance as well as discordance couples, found that non-migrant men were 10 times more likely to be infected from outside their regular relationships than inside [18].

Hence, it seems also relevant for Mozambican public health authorities to get more insights in this matter and to launch national prevention plans promoting regular HIV testing for couples (even prior to getting married or living together) and to support couples to prevent transmission once one partner has become infected. A similar recommendation was given by Kaiser at al. in Kenya, where they recommended that prevention interventions should begin early in relationships and include mutual knowledge of HIV status [5].

An interesting methodological topic for further research is the modification and application of the model to same-sex couples. Two approaches seem to be readily applicable, although further research is needed. A first possibility is the use of another unique identifier for both dimensions of the bivariate distribution. For instance, instead of y₁ and y₂ referring to the HIV status of woman and man respectively, one could use y₁ to be the HIV status of the youngest of the two same-sex partners and y₂ the status of the oldest of the two partners. Exactly the same model could be applied but now the two marginal success probabilities would represent the probabilities for the youngest partner and the oldest partner to be HIV positive. Any other unique identifier could be used, if available for all paired observations and if allowing interpretations of interest. A second approach would fully reflect the exchangeability. The HIV status of both partners could be put in any order to be y₁ and y₂. This would only make sense if both models for both probabilities would be taken exactly the same with all effects in common (not only partial as in our model), such that all data contribute to the estimation of that same probability π = P(y₁ = 1|x) = P(y₂ = 1|x). This would result in a model with only two components; the first two components of Eq (2) would be identical (h(π) = β^T x). As the SDM is symmetric, its definition Eq (6) can remain unchanged, but it reduces to (as both marginal probabilities are identical): SDM = 2 ( π - π 11 ) 2 π - π 11 . Similar modifications are to be applied to the joint probabilities Eq (3). What the potential as well as limitations of such an approach are, needs still to be investigated.

proc nlmixed data = couples qpoints = 100;

/* Eq (8): probability model for P(y1 = 1) on logit scale (with random effect a) */

eta1 = b0_1 + b1_1*prev_d2 + b2_1*prev_d3 + b3_1*N_union_dW2 + b7_1*N_union_dM2 + b4_1*po + b5_1*mid + b8_1*Condom_dM1 + a;

/* probability model for p1_1 = P(y1 = 1) on probability scale */

p1_1 = 1 / (1 + exp(-eta1));

/* Eq (8): probability model for P(y2 = 1) on logit scale (with random effect a) */

eta2 = b0_1 + b1_1*prev_d2 + b2_1*prev_d3 + b3_2*N_union_dW2 + b7_2*N_Union_dM2 + b4_1*po + b5_1*mid + b8_2*Condom_dM1 + a;

/* probability model for p1_2 = P(y2 = 1) on probability scale */

p1_2 = 1 / (1 + exp(-eta2));

/* Eq (8): probability model for SDM on logit scale (without random effect) */

eta3 = b0_3 + b1_3*prev_d2 + b2_3*prev_d3 + b3_3*N_union_dW2;

SDM = 1/(1 + exp(-eta3));

/* probability model for SDM on probability scale */

/* Eqs (3)&(7) joint probabilities as function of p1_1 = P(y1 = 1), p1_2 = P(y2 = 1), SDM */

p11 = ((1-SDM)*(p1_1 + p1_2))/(2-SDM);

p10 = p1_1 − p11;

p01 = p1_2 − p11;

p00 = 1 − p1_1 − p1_2 + p11;

/* weighted log-likelihood for multinomial distribution (weights of survey design) */

ll = HIV_W*HIV_M*log(p11) + HIV_W*(1-HIV_M)*log(p10) + (1-HIV_W)*HIV_M*log(p01) + (1-HIV_W)*(1-HIV_M)*log(p00);

ll = ll*weight;

model ll ~ general(ll);

random a ~ normal(0,sigma2_a) subject = EA;

run;