Descriptive analysis was conducted to get information about the distribution of the variables. The data consisted of 9147 women of which about 7842 (85.5%) were closed interval and the rest 1323(14.5%) were open interval. Summary of results of socio-economic and demographic variables of this study are presented in Table 1.

According to Table 1, 5856 (64%) of the women and 4067 (44.5%) of the husbands have not had any formal education, 2288 (25%) of the women and 3220 (35.2%) of the husbands have at least primary level education, 1003 (11%) of the women and 1785 (19.5%) of the husbands have a formal educational level of at least secondary school and above. About 5775 (63.2%), 1466 (16%) and 1906(20.8%) of the women belonged to poor, middle and rich households, respectively. The output shows that 1106 (12%) of the women had first birth before the age of 16, while 5463(59.7%) and 2055(22.5%) of the women had their first birth in the ages 16–20, 21–25 years respectively. 523 (5.7%) of the women had their first birth above 26 years. 7465 (81.6%) were rural dwellers, while 1682(18.4%) were urban.

The median survival time of time-to-birth interval for rural women, is 33 months with 95% CI [32.62, 33.68] is less than that of urban women, 34 months with its 95% CI [33.23, 34.77] and its *P* value is 0.0213. The median survival time of time-to-birth interval for contraceptive users i.e., 38 months with [37.13, 38.87], was greater than that of non-users (32 months) with 95% CI [31.585, 32.415] and its *P* value is < 0.001. Women who have a job had a median survival time of 34 months with 95% CI [33.48,34.52] which was greater than that of jobless women (33 months) with 95% CI [32.54,33.46] based on *P* value is < 0.001. The median time of birth intervals for illiterate (who can’t read and write) women was 30 months with 95% CI [29.58, 30.42] which is less than that of women having primary education (36 months) with 95% CI [35.06, 36.94] and secondary school and above education (79 months) with 95% CI [76.96, 81.04] and its *P* value is < 0.001. The overall mean and median survival time of birth intervals for Ethiopian women is 37 months with 95% CI [40.20, 41.41] and 32 months with 95% CI [32.66, 33.34], respectively.

The survival plot of time-to-birth interval by contraceptive use is given in Fig. 1. This plot showed that as compared to non-user contraceptive women, women who use contraceptive better timing of their birth interval. The log rank test also revealed that contraceptive use had significant association with time-to-birth interval of women (*P* < 0.001).

The survival plot of time-to-birth interval of women by access to mass media is shown in Fig. 2. The plot indicates that the probability of giving birth after birth is similar both for women who had access to mass media and who hadn’t. However, the difference becomes visible towards the middle of the curve and gets closer towards the end. At the middle of the curve, the survival plot of the birth interval for women having access to mass media had better birth interval timing than women who didn’t have mass media access.

### Determinants of successive birth interval accelerated failure time (AFT) model

For time-to-birth interval data, multivariable AFT models of Weibull, log-logistic, and log-normal distribution were fitted including all the covariates significant in the univariate analysis at 10% level of significance. To compare the efficiency of different models, Akaki information criteria (AIC) which is the most common applicable criterion to select model was used. Based on the AIC, the model having a minimum AIC value was selected. Accordingly, Log-logistic AFT model (AIC = 58,784.19) was the best one of the time-to-birth interval data set of the given alternative among the covariates significant in the univariate analysis.

Covariates insignificant in the multivariate analysis were removed from the model using backward elimination technique. Accordingly, Knowledge of ovulatory cycle, family’s access to mass media, contraceptive and employment status of women were excluded. The final model kept the covariate age of women at first birth, family wealth index, women’s marital status, religion and educational level of both spouses.

As shown in Table 2, using age at first birth of less than or equal to 15 as reference, in the log-logistic AFT model, when the effect of other factors is kept constant, the estimated acceleration factor for the age ranges at first birth of 16–20, 21–25 and greater than or equals to 26 was 1.04, 1. 2 and 1.4 with 95% CI [1.03, 1.07], [1.17, 1.23] and [1.32, 1.43], respectively. This indicates that compared with the women whose age was less than or equals to 15, women whose age at first birth was 16–20, 21–25 and ≥ 26 had longer birth interval. Using women who have orthodox religion as reference, the acceleration factor for Muslim women and other follower was 1.04 and 1.11, respectively. This result indicates that Muslim women and other follower had longer survival of time-to-birth interval than orthodox women based on the significance value which is less than the level of significance value (α = 0.05).

Using uneducated women, who cannot read and write, as reference, the acceleration factors for women attending primary education and secondary school and above level of education are estimated to be 1.002 and 1.11, respectively. This implies that women attending secondary school and above level of education had longer survival of time-to-birth interval, while for women attending primary level education, it was insignificant. Likewise, using uneducated husbands as reference, the acceleration factor for husbands attending primary education and secondary school and above level of education are estimated 1.5 and 2.8 with 95% CI (1.45, 1.5) and (2.71, 2.8) respectively and *P* value was less than the level of significance which implies that husbands attending secondary school and above level of education and primary level of education had longer survival of time-to-birth interval.

Using single marital status as reference, the acceleration factor for women whose marital status are married, divorced, widowed and separated are estimated to be 1.04, 1.1, 1.07 and 1.01 respectively. Moreover, using singles as reference, women whose marital status is married, divorced, widowed, and separated have longer survival of time to birth interval. The acceleration factors for families whose wealth index are middle and rich are 1.029 and 1.029 respectively. It also indicated that compared to families of poor wealth index, middle and rich wealth index families had longer survival of time to birth interval.

To identify baseline distribution and associated risk factors and to analyze the survival of time to birth interval, three AFT models were fitted and compared. The log-Logistic AFT model was selected based on AIC value. The main focus of this study was to investigate risk factors associated with the survival of time to birth interval using parametric shared frailty models. For the data on time-to-birth interval, the three parametric baseline distributions with Gamma frailty and inverse Gaussian distribution were fitted by using regional states of the women as frailty term. The effect of random component (frailty) was significant for both log-normal-gamma shared frailties, log-logistic gamma shared frailty models, and Weibull-gamma shared frailty model. The AIC value for all parametric frailty models is summarized and the given log-logistic gamma shared frailty model had smaller AIC value (58,793). This indicates that log-logistic gamma shared frailty model is a more suited and efficient model to describe time to birth interval dataset.

### Log-logistic inverse gaussian frailty model results

This model is the same as the log-Logistic AFT model discussed previously, except that a frailty component has been included. The frailty in this model is assumed to follow an inverse Gaussian distribution with mean 1 and a variance equal to theta (θ). The estimated value of theta ((theta)) is 0.0013. A variance of zero ((theta) = 0) would indicate that the frailty component does not contribute to the model. A likelihood ratio test for the hypothesis (theta) = 0 shown in at the bottom of Table 3 indicating a chi-square value of 9616.02 with thirty two degrees of freedom resulted in a highly significant *P* value of < 0.001. This indicates that the frailty component had significant contribution to the model but the associated Kendall’s tau ((tau)), which measures dependence with in clusters (region) is estimated to be 0.00065. The estimated value of the shape parameter in the log-logistic inverse Gaussian frailty model is 3.00 ((rho = 3.00)). This value showed the shape of uni-modal hazard function because the value is greater than a unit implies that it increases up to its maximum point and then begins to decrease.

From Table 3, the confidence interval of the acceleration factor for all significant categorical covariates does not include one at 5% level of significance. This shows that they are significant factors for determining the survival of time-to-birth interval among women in Ethiopia. However, taking the uneducated women as reference and based on the covariate, women who have primary level of education is not significant at (*P* value = 0.81, ϕ = 1.002, 95% CI = 0.99, 1.02).

The age of women at first birth was statistically significant in determining the time-to-birth interval of Ethiopian women. The acceleration factor for all categories of age at first birth was 1.05, 1.20 and 1.37, with 95% confidence interval [1.03, 1.07], [1.17, 1.23] and [1.01, 1.43]) respectively, yielding a significant *P* value for all age categories was less than the level of significant (α = 0.05). As compared to women whose age group was less than or equal to 15, all the categories of age of women at first birth 16–20, 21–25, and ≥ 26 respectively have shorter birth interval. Additionally, the confidence interval did not include one indicating that the age of women at first birth was statistically significant for the survival of time-to-birth interval. Using orthodox religion of women as reference, the acceleration factors for Muslim women were 1.02, 1.012 and 1.07. This indicates that Muslim women and other follower had longer survival of time-to birth interval than women who follow orthodox religion, respectively.

Using uneducated women as reference, the acceleration factors for women attending primary level of education and secondary school and above level of education are estimated to be 1.002 and 1.114 respectively. This implies that women have secondary school and above level of education have longer survival of time-to-birth interval. Among women in their primary level of education however, the interval was not significant. Taking uneducated husbands as reference, the acceleration factors for husbands attending primary level of education and secondary school and above level of education are estimated to be 1.48 and 2.77 respectively. This implies that husbands attending secondary school and above level of education and primary level of education have longer survival of time-to-birth interval. Taking single marital status as reference, the acceleration factor for the marital status of women such as married, divorced, widowed and separated are estimated to be 1.04, 1.10, 1.074 and 1.12, respectively. This indicates that compared to the single women, women who were married, divorced, widowed and separated have longer survival of time-to-birth interval. The acceleration factor of families belonging to middle and rich wealth index were 1.03 and 1.04, respectively which imply that families who have middle and rich wealth index have longer survival of time to birth interval.

### Comparison of log-logistic AFT and log-logistic inverse Gaussian frailty model

From Table 4, it can be seen that the results from the Log-Logistic AFT and Log-Logistic Inverse Gaussian frailty model are quite similar though not identical. AIC was used to compare the efficiency of the models. Table 4 shows that compared to log-logistic shared inverse Gaussian frailty model (AIC = 58,792.16), the log-logistic AFT model has a lower AIC (58,784.19) indicating that log- logistic AFT model fitted the survival of time-to-birth interval data better than the log-logistic shared inverse Gaussian frailty model, which took the clustering effect in to account. The estimated value of coefficients of the covariate are altered with the inclusion of the frailty component, and the confidence interval for the acceleration factor is slightly narrower for log-logistic inverse Gaussian frailty model. In general, for modeling time-to-birth interval dataset, log-logistic AFT is preferred over Log-logistic inverse Gaussian shared frailty model. The graphical method of model checking was also used to strengthens the decision made by AIC value that log-logistic baseline distribution is appropriate for the given data.