Patterns in sexual mixing across ages determine patterns of transmission of sexually transmitted infections (STIs). Consequently, sexual age-mixing has been of great interest to researchers studying the human immunodeficiency virus (HIV) since the beginning of the global epidemic. Anderson et al., 1992 used a model of partnership formation to predict that mixing between young women and older men would amplify the already-substantial effect of HIV on population growth. Garnett and Anderson, 1994 used a mathematical model to show that patterns of age-mixing could substantially influence the magnitude and timing of hypothetical epidemic trajectories, whereas Hallett et al., 2007 demonstrated that delaying sexual debut and increasing age-similar partnerships could reduce an individual’s risk of HIV infection in a highly endemic setting.

These modelling studies have been complemented by analyses of survey and population cohort data on age-mixing patterns. Gregson et al., 2002 and Schaefer et al., 2017 observed that individuals with older partners were at greater risk of HIV infection in a general-population cohort in Zimbabwe. Ritchwood et al., 2016 and Maughan-Brown et al., 2016 found that larger age differences were associated with more risky sexual behaviour in surveys of young South African people. Similarly, Akullian et al., 2017 found that partner age was an important risk factor for HIV infection in a cohort study in rural South Africa. On the other hand, Harling et al., 2014 found that age-disparate relationships were not associated greater risk of HIV acquisition in young women in South Africa.

These results underscore the importance of considering age-mixing dynamics when designing and evaluating HIV prevention strategies, and, consequently, the importance of measuring them accurately. For example, an intervention aiming to prevent new HIV infections among young women could be attenuated by high prevalence among older men. Identifying changes in sexual partner age distributions and attributing them to interventions might even be a valuable end by itself, in which case accurate measurement must be complemented by an effective modelling strategy.

Data about sexual partner age-mixing are routinely collected by long-term cohort studies (such as those that comprise the Analysing Longitudinal Population-based HIV/AIDS data on Africa, or ALPHA, Network) and large-scale household surveys (such as the Demographic and Health Surveys) (The DHS program, 2021; Reniers et al., 2016). Typically, these data consist of the respondent’s age and sex and the ages of their sexual partners in the last 12 months. These data are highly variable, skewed, and often deviate substantially from conventional parametric distributions, such as the normal distribution or the gamma distribution (Beauclair et al., 2018).

One may consider statistical modelling approaches for the distribution of partner age as a function of respondent age and sex. Some notable previous approaches to modelling partner age distributions include (Morris, 1991), who developed a log-linear modelling framework to quantify selective mixing from contingency tables, and applied the model to ordinal categorical data on husbands’ and wives’ ages in the United States. Hallett et al. used a log-logistic distribution to continuously model partner age differences for women aged 15–45 years, assuming that the partner age difference distributions did not vary over respondent age. More recently, as an input to a model of Chlamydia trachomatis, Smid et al., 2018 fit skew normal distributions to each age-/sex-specific partner age distribution and used a secondary regression model to smooth the estimated skew normal parameters across respondent age. They observed substantial changes in the estimated skew normal parameters with respect to respondent age. Although this method allows for non-linear variation across respondent age, their two-stage estimation process makes uncertainty propagation complex. Replacing this process with a single ‘distributional’ regression model, in which all distributional parameters (e.g. the location, scale, skewness, etc.) are modelled as functions of data (Kneib and Umlauf, 2017), allows for complex variation across respondent age while still robustly incorporating uncertainty. Another elegant approach has been the development of exponential-family random graph models (ERGMs) to infer full partnership networks from individuals reports of the partnerships (’ego-centric’ observations of the network) (Krivitsky and Morris, 2017). These stochastic methods, along with the broader suite of ERGMs (Hunter et al., 2008a; Hunter et al., 2008b; Krivitsky et al., 2011; Krivitsky and Handcock, 2014), can model social network data accurately with robust incorporation of covariates, and tools exist to incorporate their estimates into epidemic models (Jenness et al., 2018; Morris, 1993).

We focused on evaluating parametric models for continuously representing the distribution of sexual partner ages conditional on the respondent’s age. We were specifically interested in distributions that introduce parameters to control tail weight, which may capture intergenerational mixing that could sustain endemic HIV and STI transmission (Akullian et al., 2017; Schaefer et al., 2017; Harling et al., 2014). This led us to test the ability of the four-parameter ‘sinh-arcsinh’ distribution originally proposed by Jones and Pewsey, 2009 to fit to these data.

We hypothesised that integrating the sinh-arcsinh distribution into a distributional modelling framework would allow us to replicate observed partner age distributions more accurately than prior modelling strategies. We tested this theory by comparing a variety of candidate strategies, which varied along three dimensions: the parametrisation of the dependent variable, the choice of distribution, and the method for incorporating variability across respondent age and sex.

The AHRI data included 77,619 partnerships, Manicaland had 58,676, and the Haiti DHS had 12,447. There were 36,033 respondents reporting at least one partnership in the AHRI data, 25,024 in the Manicaland data, and 12,143 in the Haiti DHS, resulting in averages of 2.2, 2.3, and 1.0 partners per respondent, respectively. As an illustrative example of the distribution of partner ages, Figure 2 presents histograms of reported partner ages among women aged 34 years for each of our three data sets. Figure 3 shows the sex- and age bin-specific empirical moments for the three data sets. Mean partner age increased with respondent age consistently for both sexes across all three data sets: among women, mean partner age increased by 26.0, 22.7, and 23.7 years in the AHRI data, Haiti DHS data, and Manicaland data, respectively, between age bins 20–24 and 45–49. However, higher order moments were less consistent: the standard deviation of women’s partners’ ages changed by 2.3, 0.5, and 3.5 years in the AHRI data, Haiti DHS data, and Manicaland data, respectively, between age bins 20–24 and 45–49.

Figure 2

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Figure 3

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Within each data set, there is systematic variation across sex. For example, the standard deviation of partner ages in the Haiti DHS increased by 2.5 years among men and only by 0.5 years among women. These summary statistics illustrate the heterogeneity of partner age distributions across age and sex.

Probability distribution comparison

To identify the probability distribution that most accurately described the variation in sexual partner age distributions, we first determined the dependent variable with the highest ELPD for each distribution-dependent variable combination. Figure 4 illustrates each probability distribution’s best fit to AHRI data among women aged 35–39 with each of the best distribution-specific dependent variables. Results for all 36 data subsets and the 12 deheaped subsets are presented in Appendix 1—table 2–9.

Figure 4

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The best dependent variable varied across data subset and probability distribution. Table 3 provides the share of data sets for which each dependent variable has the highest ELPD given each distribution. The log-ratiodependent variable was best in 50.0% of subsets with a normal distribution, but it was best in only 27.8% of subsets with a skew normal distribution. The dependent variable that was best in a plurality of subsets in each probability distribution (i.e. the variable with the highest percentage in each column of Table 3) used a $\log$ link function. We restricted all remaining comparisons to each distribution-subset combination’s best dependent variable.

Table 3

Variable	Normal	Skew normal	Sinh-arcsinh
Age difference	22.2%	25.0%	16.7%
Linear age	8.3%	5.6%	16.7%
Log-age	19.4%	41.7%	30.6%
Log-ratio	50.0%	27.8%	36.1%

The sinh-arcsinh distribution had the highest ELPD in 35 of 36 data subsets (98%). In 29 of the 35 (83%) cases in which the sinh-arcsinh provided the highest ELPD, the absolute value of the ratio of the difference between the two best ELPDs and the estimated standard error of the difference was greater than 2, indicating that the sinh-arcsinh distribution was significantly better than the alternatives in the majority of cases. In one case, men aged 20–24 in the Haiti DHS, the skew normal distribution resulted in a slightly higher ELPD than the sinh-arcsinh distribution, but the standard error of the difference was greater than the difference. These results were not affected by deheaping the data (Appendix section ‘Age heaping’).

To summarise each distribution’s performance, we calculated the average ELPD and QQ RMSE across the three data sets (Table 4). The sinh-arcsinh distribution had the highest average ELPD and lowest average QQ RMSE in all three data sets. The sinh-arcsinh distribution was, on average, able to predict the empirical quantiles of each data set within half a year of accuracy (0.36, 0.37, and 0.44 years for the AHRI, Haiti DHS, and Manicaland data, respectively). Overlaid QQ plots are presented in Appendix 1—figure 3.

Table 4

Distribution	AHRI	Haiti 2016–17 DHS	Manicaland
ELPD
Gamma	−14847.2	−2917.9	−13152.8
Beta	−14748.0	−2896.5	−13003.5
Normal	−14593.7	−2868.4	−12856.8
Skew normal	−14505.1	−2854.0	−12778.5
Sinh-arcsinh	−14312.5	−2839.5	−12625.8
QQ RMSE
Gamma	0.83	0.82	0.95
Beta	0.99	0.82	1.11
Normal	0.82	0.68	0.97
Skew normal	0.77	0.65	0.85
Sinh-arcsinh	0.36	0.37	0.44

Distributional regression evaluation

We fit all five distributional regression specifications to all three of our data sets with sinh-arcsinh distributions and log-ratio-dependent variables and compared the ELPDs and QQ RMSEs as before (provided in Table 5). Across all three data sets, the most complex distributional model (Distributional 4) had the highest ELPD and lowest QQ RMSE. When fit to the AHRI and Manicaland data sets (but not for the Haiti DHS), the most complex distributional model was at least two standard errors better than the next best model. Notably, the largest ELPD improvements came from moving from conventional regression (Conventional) to the simplest distributional model (improvements of 1646.0 units, 361.0 units, and 2181.2 units in the AHRI, Haiti DHS, and Manicaland data, respectively). Full results are presented in Appendix 1—table 10.

Table 5

Model	AHRI	Haiti 2016–17 DHS	Manicaland
ELPD
Conventional	52689.2	4777.8	21011.3
Distributional 1	54335.2	5140.8	23192.5
Distributional 2	54794.8	5138.7	23472.1
Distributional 3	55534.2	5196.7	24313.7
Distributional 4	55841.9	5207.6	24516.1
QQ RMSE
Conventional	1.30	1.33	2.05
Distributional 1	1.15	0.98	1.89
Distributional 2	1.21	0.99	1.80
Distributional 3	0.93	0.91	1.34
Distributional 4	0.66	0.84	1.04

Figure 5 shows the posterior predictive distributions from the conventional regression model and the most complex distributional model among men aged 16 years, 24 years, and 37 years in the AHRI data to illustrate the effect of distributional regression. Not only does the distributional model capture the high peak in the youngest age more accurately, but it also allows the variance of the distributions to change appropriately (beyond the change that naturally results from the log link function).

Figure 5

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Figure 6 illustrates posterior summaries among men and women in the AHRI data for all four distributional parameters for the conventional regression model, the simplest distributional model, and the most complex distributional model. The red estimates (Conventional Regression) of the three higher order parameters were constant across age and sex, whereas the blue estimates (Distributional Model 1) included independent, linear age and sex effects. The orange estimates (Distributional Model 4) were generated sex-specific splines with respect to age, allowing for flexible variation across age and sex.

Figure 6

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The third row of plots in Figure 6, which corresponds to the skewness parameter, illustrates the impact of incorporating sex and age effects into the model. The conventional regression model estimated that neither the distrbution for men nor women exhibited much skewness; the estimated parameter value was −0.05 (95% UI: −0.06 to −0.05) regardless of age, with 0.0 corresponding to perfect symmetry. However, when we allowed independent age and sex effects in Distributional Model 1, we estimated that at age 15, women’s skewness was −0.26 (95% UI: −0.27 to −0.25) and men’s was 0.11 (95% UI: 0.10 to 0.12).

The most complex model (Distributional Model 4) inferred sex-specific, non-linear variation with respect to age in all four distributional parameters. The non-linearity was particularly dramatic in the scale parameter among men. The scale value began at 0.05 (95% UI: 0.05–0.06) among 15-year-olds, peaked among 37-year-olds at 0.11 (95% UI: 0.10–0.11), and decreased back down to 0.05 (95% UI: 0.04–0.06) at age 64.

Finally, Figure 7 presents inferred distributional parameters from Distributional Model 4 for both men and women for all three data sets. Based on those plots, the flexible model was justified for most distributional parameters in all three data sets. Were we to continue developing these models, this plot suggests that skewness might only need linear, sex-specific effects with respect to age. Interestingly, the 2016–2017 Haiti DHS and Manicaland estimates exhibit similar patterns across all four parameters, despite the different socio-cultural contexts surrounding partnerships in the two populations. We also note that the DHS does not collect data on adults aged 50 years and older, so our estimates in Haiti from age 50 to age 64 are purely extrapolated.

Figure 7

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We found that the sinh-arcsinh distribution reproduced observed sexual partner age distributions better than a number of other possible distributional assumptions across age and sex in three distinct data sets. We integrated this finding into a distributional regression framework using existing statistical modelling software. Even the simplest distributional regression in our set of candidate models far outperformed conventional regression, in which all moments except the first are estimated as constants. Our most complex distributional model fit better than all other models in all three data sets, suggesting that modelling these data benefits from the additional complexity.

These results indicate that distributional regression models with sinh-arcsinh distributions can accurately replicate age-/sex-specific sexual partner age distributions. This approach presents a number of advantages over previous methods. First, like Smid et al., it allows a unique distribution for every age-sex combination. As Figure 3 illustrates, partner age distributions can exhibit substantial, systematic variation across age and sex in any of the first four moments, so we must consider modelling strategies that allow for such variation. Second, distributional regression offers a principled method to propagate uncertainty through this estimation process.

Finally, distributional regression implemented through brms provides access to a deep set of hierarchical modelling tools that could enable estimation in a variety of low-data settings. We evaluated a small set of relatively simple distributional models in this work, but, theoretically, each distributional parameter could have its own, arbitrarily complex hierarchical regression model. Using these tools, one could estimate unique partner age distributions across levels of stratification that are substantively interesting but do not provide sufficient sample size for independent estimation (e.g. study sites or geographic areas).

We have identified several limitations in this approach. First, the amount of data required to produce usefully precise estimates is not tested. Each additional distributional parameter introduces model parameters, so this method is more complex than conventional regression. The sinh-arcsinh distribution did fail to produce the highest ELPD in our smallest data subset (N = 170), but it was not significantly worse than the best distribution. More importantly, by integrating these data into a distributional modelling framework, we gain the ability to impose structure on these parameters, which could easily offset the cost of any additional model parameters.

Interpreting the inferred model parameters in sinh-arcsinh regression can also be difficult. Although conventional regression estimates the effects of covariates on expected values, the sinh-arcsinh distribution is parametrised in terms of a location parameter. This parameter correlates closely with the central tendency of the distribution, but it is not strictly equal to the mean. We can reparametrise the distribution so that we estimate a mean (and therefore effects of covariates on the expected value), but it is not currently possible in the probabilistic programming software that underlies brms.

Third, our analysis assumed that we were operating at a level of stratification at which partnerships are basically comparable, but any number of factors could lead to fundamentally different partner age distributions. For example, we did not control for whether the partnership was same-sex or the type of the partnership (married, casual, etc.). That said, our distributional framework would allow us to incorporate data on any of those factors directly into the model.

There are two sources of possible non-independence in our data sets that were not modelled. First, in the two cohort studies, participants are eligible to participate in multiple survey rounds, in which case the same partnership could be reported multiple times in our data set. Second, many individuals report multiple partners in the same survey, and the age of one partner may be correlated ages of other partners, for example due to individual preferences for partnerships. Because we modelled multiple observations of the same individual as conditionally independent, we anticipate that these correlations may artificially increase the precision of our estimates.

Finally, our model does not address any reporting biases in self-reported partnership data. If certain relationship types are perceived as less socially acceptable, respondents might be less likely to report them, resulting in systematic missingness. Our method could still be appropriate to model the age distribution from the data reported about an under-reported partnership type, but it cannot predict whether or not a given partnership exists. However, if under-reporting correlates with partner age (or age difference), then the empirical distributions will be biased and our method will only smooth and interpolate the biased data.

Despite these limitations, we believe that the strategy we present will work well in future projects that require estimates of partner age distributions. We plan to use these methods to produce age-mixing matrices to inform epidemic models of HIV, but there are a number of additional directions that could be explored. We are specifically interested in leveraging the spatio-temporal structure of the survey data used here. Hierarchical mapping exercises with household survey data are increasingly common in epidemiology, but estimating spatially varying partner age distributions would require an evaluation of how best to model higher order moments over space. We would, for example, need to consider how the variance of partner age distributions varies by urbanicity.

Similarly, population-based studies typically collect far more detailed information on partnerships than we took advantage of here. Relationship type is a key confounder of the association between respondent age and partner age (that we ignored for the purposes of our experiments). We might expect the age distributions of casual partners to vary substantially from those of long-term cohabiting partners. Because we have built our model in an existing regression framework, incorporating new covariates into any of the distributional regression specifications is straightforward. The distribution regression framework with the sinh-arcsinh may also be a useful parametric model for continuous representation of marginal distributions within sexual mixing models or network models, such as the ERGM framework.

We believe that our framework offers a flexible, accurate, and robust method for smoothing and interpolating sexual partner age distributions, but these methods are not specific to partner age distributions. The sinh-arcsinh distribution is relatively easy to implement without incurring high computational cost, so it could be applied in many settings. Even without the distributional regression framework, we have used here, allowing the third and fourth moments of the distribution to vary from the ‘default’ normal values could be valuable across a variety of applications.

Distributional regression is also underutilised in social science applications. We often work with large surveys that would comfortably support models for higher order parameters. Data requirements will vary by application and model, but, as we have shown here, even a simple distributional model can improve fit and avoid biasing estimates.

Data from the Demographic and Health Surveys are available from the DHS Program website (https://dhsprogram.com/data/available-datasets.cfm). Data from the Africa Centre Demographic Information System are available on request from the AHRI website (https://data.ahri.org/index.php/home). Data from the Manicaland study were used with permission from the study investigators (http://www.manicalandhivproject.org/manicaland-data.html).

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Author details

1. Timothy M Wolock

   Department of Mathematics, Imperial College London, London, United Kingdom

   Contribution

   Conceptualization, Software, Formal analysis, Visualization, Methodology, Writing - original draft, Writing - review and editing

   For correspondence

   t.wolock18@imperial.ac.uk

   Competing interests

   No competing interests declared

   "This ORCID iD identifies the author of this article:" 0000-0001-5898-1014

2. Seth Flaxman

   Department of Mathematics, Imperial College London, London, United Kingdom

   Contribution

   Conceptualization, Supervision, Methodology, Writing - review and editing

   Competing interests

   No competing interests declared

   "This ORCID iD identifies the author of this article:" 0000-0002-2477-4217

3. Kathryn A Risher

   1. MRC Centre for Global Infectious Disease Analysis, School of Public Health, Imperial College London, London, United Kingdom
   2. London School of Hygiene & Tropical Medicine, London, United Kingdom

   Contribution

   Conceptualization, Data curation, Writing - review and editing

   Competing interests

   No competing interests declared

   "This ORCID iD identifies the author of this article:" 0000-0002-9588-1693

4. Tawanda Dadirai

   Manicaland Centre for Public Health Research, Biomedical Research and Training Institute, Harare, Zimbabwe

   Contribution

   Data curation, Writing - review and editing

   Competing interests

   No competing interests declared

5. Simon Gregson

   1. MRC Centre for Global Infectious Disease Analysis, School of Public Health, Imperial College London, London, United Kingdom
   2. Manicaland Centre for Public Health Research, Biomedical Research and Training Institute, Harare, Zimbabwe

   Contribution

   Data curation, Writing - review and editing

   Competing interests

   No competing interests declared

   "This ORCID iD identifies the author of this article:" 0000-0003-2707-0714

6. Jeffrey W Eaton

   MRC Centre for Global Infectious Disease Analysis, School of Public Health, Imperial College London, London, United Kingdom

   Contribution

   Conceptualization, Supervision, Methodology, Writing - review and editing

   Competing interests

   No competing interests declared

   "This ORCID iD identifies the author of this article:" 0000-0001-7728-728X

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