Modeling the Determinants of Time-to-Premarital Cohabitation among Women of Ethiopia: A Comparison of Various Parametric Shared Frailty Models

Background : Premarital cohabitation is defined as the state of living together and having a sexual relationship without being married. It has become more prevalent globally in recent decades. The main objective of this study was modeling the potential risk factors of time-to-premarital cohabitation among women of Ethiopia by using parametric shared frailty models where regional states of the women were used as a clustering effect in the models. Methods : The data source for the analysis was the 2016 EDHS data. The Gamma and Inverse-Gaussian shared frailty distributions with Exponential, Weibull, Log-logistic and Lognormal baseline models were employed to analyze risk factors associated with age at premarital cohabitation. All the fitted models were compared by using AIC values. Results : The median age of women at premarital cohabitation was 18 years. Based on AIC values, Log-logistic-Gamma shared frailty model has smallest AIC value among the models compared. The clustering effect was significant for modeling the determinants of time-to-premarital cohabitation dataset. The results showed that women’s education status, occupation, pregnancy and place of residence were found to be the most significant determinants of age at premarital cohabitation whereas wealth status and religion were not significant at 5% level. Conclusions : The Log-logistic-Gamma shared frailty model described the premarital cohabitation dataset better than other distributions used in this study. There is heterogeneity between the regions of women. Further studies should be conducted to identify other factors of age at premarital cohabitation of women in Ethiopia that were not included in this study.


Conclusions:
The Log-logistic-Gamma shared frailty model described the premarital cohabitation dataset better than other distributions used in this study. There is heterogeneity between the regions of women. Further studies should be conducted to identify other factors of age at premarital cohabitation of women in Ethiopia that were not included in this study.
Keywords: Survival Analysis; Premarital cohabitation; Women; Frailty; Acceleration factor Background In this study, cohabitation is defined as the state of living together and having a sexual relationship without being married [24]. It has become more prevalent globally in recent decades [25]. Different studies has been conducted in the world like in North America, Latin America, Western Europe and Eastern Europe and observed that couples exercise premarital cohabitation either as an alternative or stepping stone to marriage [4,9,11,12,14,18,19,23]. Premarital cohabitation is associated with higher risks of divorce compared to married couples and cohabiting couples have poor relationship skills and less predictable attitudes about marriage [6,15]. Thus, the experience of cohabitation may change people's attitude about marriage and make them less strongly committed and devoted to the institution. Studies conducted previously were identified covariates of premarital cohabitation by using logistic regression. But Logistic regression does not account the censoring observations, that is, it does not hold for time-to-event data. Thus, this research aims to address the shortage of literatures by exploring covariates of time to premarital cohabitation in Ethiopia by using parametric shared frailty models. In this study, clustering (frailty) has an effect on modeling the determinants of time to premarital cohabitation, which might be due to the heterogeneity in regions. Frailty is common to all individuals in the cluster and responsible for creating dependence between event times [21]. To best of my knowledge, there is no any study which has been conducted in Ethiopia regarding time to premarital cohabitation. Finally, this study plays its crucial role on the formulation of effective policy to give awareness for women about the risk of premarital cohabitation on their health and marriage.

Data source
The data for this study was extracted from the published reports of Ethiopian Demographic and Health Survey (EDHS-2016) which is obtained from Central Statistical Agency [5]. It is the fourth survey conducted in Ethiopia as a part of the worldwide DHS project. The 2016 EDHS was designed to provide estimates for the health and demographic variables of interest for the following domains. Ethiopia as a whole; urban and rural areas (each as a separate domain); and 11 geographic administrative regions (9 regions and 2 city administrations), namely: Tigray, Affar, Amhara, Oromia, Somali, Benishangul-Gumuz, South Nations Nationalities and Peoples (SNNP), Gambela and Harari regional states and two city administrations, Addis Ababa and Dire Dawa. The principal objective of the 2016 EDHS was to provide current and reliable data on fertility and family planning behavior, child mortality, adult and maternal mortality, children's nutritional status, use of maternal and child health services, knowledge of HIV/AIDS, and prevalence of HIV/AIDS and anemia.

Study population and variables
The total number of sample 15,683 women between the ages of 10 years and 43 years were identified in the households of selected clusters (regions). Thus, the analysis presented in this study, on the risk factors of women's premarital cohabitation, was based on the 15,683 women.
The response variable in this study is time-to-premarital cohabitation among women of Ethiopia which is measured in years. It is measured as the length of time from birth to age at first cohabitation. Cohabited women were event of interest for this research. The women who did not experience the event (cohabit) were censored. The following variables were considered for their influence on the time-to-premarital cohabitation of women: place of residence, education status, occupation, wealth index, pregnancy and religion.

Shared Frailty Model
Conditional on the random term, called the frailty denoted by , the survival time in cluster (1 < < ) are assumed to be independent, the proportional hazard model assumes Where indicates the ℎ cluster, indicates the ℎ individual in the ℎ cluster, ℎ 0 ( ) is the baseline hazard, is the random term for all subjects in cluster , is the vector of covariates for subject in cluster , and is the vector of regression coefficients. We assumed that , ℎ = ( ) has the gamma or inverse Gaussian distribution so that the hazard function depends upon this frailty that acts multiplicatively on it. The main assumption of a shared frailty model is that all individuals in cluster share the same value of frailty ( = 1, 2, … , ), that is why the model is called the shared frailty model. The survival time is assumed to be conditionally independent with respect to the shared (common survival times) frailty. This shared frailty is the cause of dependence between survival time within the clusters [8,13,21]. In order to investigate the effect of the candidate covariates on time-to-premarital cohabitation of women, first univariable analysis has been done by fitting a separate model for each candidate covariates. Covariates that were found to be significant in the univariable analysis were included in the multivariable analysis. The multivariable survival analysis in the study was done by assuming the Exponential, Weibull, Log-normal and Log-logistic distributions for the baseline hazard function; and the Gamma and Inverse-Gaussian frailty distributions. It was performed using all the significant covariates in the univariable analysis namely place of residence, education status, occupation, wealth index, pregnancy and religion.

Descriptive Statistics
A total of 15,683 women from nine regional states and two city administrations were included in the study. The time interval between the women's date of birth and the time cohabitation took place was an interest of this study. Out of 15,683 women considered, 11,405 (72.7%) women were experienced the event (cohabited) and the rest 4278 (27.3%) did not experience the event.
The median age at premarital cohabitation was 18 years while the minimum and maximum observed event time was 10 years and 43 years, respectively. The highest prevalence of premarital cohabitation occurred at age of 15 years, 1653 (10.5%) women. From

AIC=Akaike's Information Criteria
Analysis based on Log-logistic-Gamma shared frailty model showed that women's education, occupation, pregnancy and place of residence were significant whereas wealth index and religion were not significant covariates of premarital cohabitation. The survival time of premarital cohabitation of women shortened by a factor of (ɸ=0.714, 0.455 and 0.385) for primary, secondary and higher education respectively at 5% level of significance. In other words, women who have attended at least primary school education have smaller survival time to premarital cohabitation than the reference category. The survival time of premarital cohabitation shortened by a factor of (ɸ=0.826) for pregnant women respectively at 5% level of significance. An acceleration factor of greater than 1 indicates prolonging the time of premarital cohabitation.

Diagnostic Plots of the Parametric Baselines
To check the adequacy of baseline hazard, the exponential is plotted by the − ( ( )) with the time of the study; the Weibull is plotted by (− ( ( )) with the logarithm of time of the study; the log-logistic is plotted by (̂( the given baseline distribution is appropriate for this dataset (Fig. 1). The plot of log-logistic was more linear than other plots. Fig.1: Graphical evolution of Exponential, Weibull, Log-normal and Log-logistic assumption

The Cox Snell Residual Plots
The Cox-Snell residuals are one way to investigate how well the model fits the data. The plot of Cox-Snell residuals is obtained for log-logistic baseline distribution to premarital cohabitation dataset (Fig.2). The plot makes straight lines through the origin for log-logistic baseline

Discussion
The main aim of this study was modeling the determinants of time-to-premarital cohabitation among women of Ethiopia by using shared frailty models. The comparison of the models was done using the AIC criteria, where the model which has smallest AIC is chosen as best model [16]. Thus, the Log-logistic-Gamma frailty model is chosen as the best model to describe the dataset of time-to-premarital cohabitation since it has smallest AIC value of 65144.36. The dataset is obtained from EDHS 2016 after registering and requesting. In this study, region was used as a clustering (frailty) effect. The clustering effect was significant (p value < 0.001) for Log-logistic-Gamma shared frailty model. This showed that there is heterogeneity between regions by assuming women within the same region share similar risk factors towards age at .5 1 1.5 Cox-Snell residual premarital cohabitation. In this study, the adequacy of baseline distributions and shared frailty models were checked by using graphs (Fig.1 and Fig.2). From the plots of baseline distributions, the plot of baseline Log-logistic was more linear than the other plots. The Cox-Snell residuals plot of Log-logistic distribution showed straight line through the origin suggesting that the model is appropriate for time-to-premarital cohabitation dataset. These findings were consistent with [3,7,17] for log-logistic model. The results of this study suggested that place of residence was significant predictive factor for time-to-premarital cohabitation among women of Ethiopia. This shows that women who lived in rural areas had longer survival with respect to premarital cohabitation than women who lived in urban areas. A similar study has been conducted in sub-Saharan Africa by [18] and revealed that urban women were more likely to cohabit compared to their rural counterparts. A significant association was found between women education and timeto-premarital cohabitation. Women who attended primary school, secondary school and higher education had significant shorter time-to-premarital cohabitation as compared to women who were not educated. This is consistent with the study conducted in Spain, the United States, sub-Saharan Africa and Italy by [2,4,18,22] respectively and revealed that women who attended at least primary school had higher risk to be cohabited than uneducated women. The results of this study were also revealed that occupation was significant predictive factor for time-to-premarital cohabitation among women of Ethiopia. This shows that women who had occupation had longer survival with respect to premarital cohabitation than women who had not occupation. A similar study that has been conducted by [2,22] revealed that women who had occupation were less vulnerable towards premarital cohabitation. A significant association was also found between pregnancy and time-to-premarital cohabitation. Women who were pregnant had significant shortened time-to-premarital cohabitation as compared to women who were not pregnant. This is consistent with the study conducted in Spain by [2] and revealed that pregnant women were at higher risk to be cohabited than women who were not pregnant.

Conclusions
This study was based on a dataset of age at premarital cohabitation which was obtained from Central Statistics Agency with an aim of modeling the determinants of time-to-premarital cohabitation of women by using parametric shared frailty models. The estimated median age of women at premarital cohabitation was 18 years. Based on AIC values, the Log-logistic-Gamma shared frailty model was chosen as the best model to describe premarital cohabitation dataset.
There is a frailty effect on age at premarital cohabitation dataset that arises due to heterogeneity between the regions of women. Based on result of chosen model, women's educational level, place of residence, occupation and pregnancy were significant factors of age at premarital cohabitation. Further studies should be conducted to identify other factors of age at premarital cohabitation of women in Ethiopia that were not included in this study.

Consent for publication: Not applicable
Availability of data and material: The dataset is available via http://www.dhsprogram.com/ by registering and requesting the datasets.