Duration and pattern of postpartum amenorrhea of Nepalese mothers: a statistical analysis

Arjun Kumar Gaire; Arjun Kumar Gaire

doi:10.48130/stati-0025-0002

Postpartum amenorrhea (PPA) is the absence of menstrual periods in the mother after the birth of a child. The length of PPA can vary widely among women, ranging from a few weeks to several months or even years. Duration of PPA is significant in the fertility life of women since it is one of the natural methods of contraception. In this research, the status of the PPA of Nepalese mothers and the differential mean duration of PPA among different sub-populations of Nepal are evaluated. The variation in the duration of PPA and the probabilistic distributional pattern is fitted by using Log-Logistic and some generalization of this distribution. The AIC and BIC at the maximum log-likelihood value are used to validate the model fitting. A dataset of NDHS-2022 is taken for these findings. The mean duration of PPA for Nepalese mothers is found to be 5.89 months. These findings can have significant implications for the health of mothers and children. It would also be helpful for the research personnel, medical personnel, and policymakers who are interested in women's general and reproductive health to develop policies and programs related to family planning, reproductive health, and women's empowerment.

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Introduction

Postpartum amenorrhea (PPA) is the temporary absence of menstruation following childbirth. It is a natural physiological response to pregnancy and childbirth, influenced by hormonal changes that regulate reproductive functions. During pregnancy, elevated levels of estrogen and progesterone suppress ovulation and the menstrual cycle. After childbirth, these hormone levels gradually decline, allowing the body to return to its pre-pregnancy menstrual pattern. However, the duration of PPA varies significantly among women, ranging from a few weeks to several months or even years. Several factors influence the length of PPA. Breastfeeding is one of the most critical determinants, as the production of prolactin, a hormone necessary for lactation, suppresses ovulation, and delays the return of menstruation. Other factors such as maternal age, parity, and health status also contribute to variations in PPA duration. The resumption of menstruation marks the end of PPA and signals the restoration of fertility.

Understanding the duration and determinants of PPA is essential for reproductive health planning and family planning strategies. In this context, this study aims to assess the status of PPA among Nepalese mothers, examine the differences in its mean duration based on various socio-demographic factors, and identify a suitable probability distribution to describe its pattern. Researchers in Asian countries pay little attention to modeling demographic variables; however, modeling these variables has received significant attention among demographers around the world. The distribution used to fit the pattern of PPA research depends on the specific study and the characteristics of the population being used. However, some skewed models and mixture distributions for bi-model-shaped curves are commonly used to model the duration of PPA. Several statistical distributions have been applied to describe the pattern of PPA, Talwar^[1] utilized the Triangular distribution. Lesthaeghe & Page^[2] applied the Logit model, whereas Potter & Kobrin^[3] used a mixture of binomial and geometric distribution. Ford & Kim^[4] modeled the bimodal nature of amenorrhea duration in Bangladesh and India using a mixture of type I Extreme value distribution. Additionally, Rahman^[5] adopted the modified exponential distribution to analyze the PPA data of Bangladesh. More recently, Holman et al.^[6] used the four mixture models to fit the PPA distribution in Bangladesh, including a mixture of log normal, a mixture of Weibull, a mixture of gamma, and a mixture of Gumbel distribution. Aryal^[7] estimated the PPA duration indirectly through breastfeeding duration, while Aryal^[8] explored the Type I extreme value distribution for Nepalese PPA data. Mishra et al.^[9] applied a mixture of Gumbel distribution to smooth PPA data across four states of India, utilizing the expectation-maximization algorithm for parameter estimation.

From the observations it was found that the PPA duration of Nepalese mothers is found to be uni-modal and right-skewed in nature. To model such right-skewed nature data of the PPA of Nepalese mothers, the Log-logistic distribution and its generalized versions have been employed to fit the PPA data of Nepal. The density function of LLog distribution is uni-modal with heavy right tails, the hazard function also increases fast along with the uni-modality. It decreases slowly, creating a right skew curve. The Rayleigh Generated Log Logistic (RG-LLog) distribution has been identified as a suitable model for Nepalese PPA data, effectively capturing its distributional characteristics.

Three important proximate determinants of fertility change are nuptiality or marriage, contraception, and postpartum fecundability^[10]. Additionally, three key variables－intercourse, conception, and gestation－affected the entire fertility dynamics of women. Demographical and ecological factors also influence the duration of PPA. Among these, breastfeeding has been widely recognized as the most significant determinant. Other factors include maternal age, income level, maternal anthropometric measures, and parity, with studies indicating that women who have had more children tend to experience a shorter duration of PPA malnourished women, on the other hand, may experience longer durations of PPA. Jain et al.^[11] identified age, education, place of residence, and cultural variations as significant factors influencing breastfeeding behaviors, which in turn affect the duration of PPA. Huffman et al.^[12] found that maternal nutritional status and age were major determinants of PPA among Bangladeshi breastfeeding women. Similarly, Hajian-Tilaki^[13] noted that the pattern of breastfeeding frequency of suckling, and duration of breastfeeding significantly delayed the resumption of menstruation and ovulation.

In Ethiopia, Ahmed^[14] highlighted that PPA plays a crucial role in natural fertility control. The literature consistently supports the notion that the intensity of breastfeeding is positively associated with the duration of PPA^[12,13,15]. Aryal^[8] examined the duration of PPA in relation to maternal and child characteristics. Additionally, postpartum hemorrhage, smoking, and hormonal contraceptive use have been associated with variations in PPA durations, with smoking and postpartum hemorrhage linked to shorter durations, while hormonal contraceptives have been found to extend the period of amenorrhea. Cultural beliefs and postpartum recovery practices also influence PPA duration. In this study, a chi-square test of independence was conducted to assess the statistical association between PPA duration and key socio-demographic variables, including maternal education, family income, and place of residence in ecological regions of Nepal. The findings provide further evidence of the complex interplay between biological, behavioral, and socio-economic factors in determining PPA duration among Nepalese women.

Materials and methods

In this article, we use quantitative data on the duration of PPA (moths) of Nepalese women after the delivery to analyze the distributional pattern, so the ontological position of this research is objectivism. We try to establish the probabilistic relationship between the duration of PPA as the independent variable and the probability of occurrence of PPA at that duration as the dependent variable. So, the epistemological position of this research is positivism. To answer the research question of this study, we propose the objective reality, the duration of PPA for Nepalese mothers, which is the realistic ontology. The reality of this research is the status, differential in the duration of PPA across sub-populations, determinants, and the distributional pattern of duration of PPA of Nepalese mothers. The PPA period and the distributional pattern are real and universal phenomena that can be observed using scientific methods. This research proposed a new mathematical model and applied it to real data sets. So, it is a theory-testing, and the ontological position of this research is objectivism.

The duration of the PPA of women is independent of our knowledge about them. The data on the duration of PPA are a reality, and a survey of women can obtain that; this can be observed as the reality of the cause and effect between the duration of PPA and the probability or relative frequency of occurrence of PPA. The duration of PPA is the effect of their previous demographic, socio-economic, and cultural background. These are the social realities of mothers, which are real and the main cause of the present duration of PPA. Their background may have a different effect on the present situation. This research aims to find the probabilistic causal relationship between the duration of PPA and the probability of occurrence of PPA at particular months among mothers. By this, we claim to attain objectivity. We will choose the quantitative method as a research tool and the deductive method to test the hypotheses that the distributional pattern follows a specified probability distribution and that there is a significant difference in the mean duration of PPA among sub-populations of Nepal. We believe there is a single reality, and valid and reliable statistical tools can measure the reality of PPA, so our research philosophy is positivism.

Data
To analyze Nepalese mothers' PPA duration, we consider the secondary data set from the national representative Nepal Demographic and Health Survey^[16]. It includes 721 women who provide data from Nepalese mothers' PPA duration. Among them 37.38 % of women provide the information at the first birth of a child, 34.47% are of the second birth of a child, 16.34 % are of the third birth of a child, and the remaining more than the third birth. It also includes data regarding the birth order, age at menarche, body mass index of mother, sex of children at birth, total children ever born, income level of family, duration of breastfeeding, smoking habit of mothers, size of the child at birth, educational attainment of females, place of resident in ecological regions, provinces, etc.

Test statistics
Descriptive statistics and a box and whisker plot are used to present the status of the duration of PPA of Nepalese mothers. To test the significant difference between the mean duration of PPA for Nepalese sub-populations, the t-test statistic is used to compare the mean duration of PPA among subgroups. Since the duration of PPA data showed a right-skewed in nature, the Independent-Samples Kruskal-Wallis Test (a non-parameter test) is used to compare the difference in mean among sub-populations. The Bonferroni correction test is applied to control the family-wise error rate. The Karl Pearson correlation coefficient and chi-square test of independence are used to find the factors that are significantly associated with the duration of PPA.

Probability distribution models
The mathematical expressions of the proposed models to fit the distributional pattern of duration of Nepalese mothers have been expressed in the following section. The Log-Logistic (LLog) and its generalized distributions are chosen to fit the duration of the PPA data of Nepalese mothers. Since the density function of the LLog is a uni-modal with heavy right tails. Similarly, the hazard function increases fast along with the uni-modality and decreases slowly, creating a right-skewed nature curve. Also, the data on the duration of PPA shows outliers and a right-skewed nature; to capture such data, LLog distribution and its generalized versions were used.

Log-Logistic distribution

If X denotes the duration of PPA of the mothers that follow the Log-Logistic (LLog) distribution, then the density function with three parameters α, β, and γ is given as:

$ f_1\left(x\right)=\dfrac{\dfrac{\alpha}{\beta}\left(\dfrac{x-\gamma}{\beta}\right)^{\alpha-1}}{\left(1+\left(\dfrac{x-\gamma}{\beta}\right)^{\alpha}\right)^2},\mathrm{\ \ \ f}\mathrm{o}\mathrm{r}\; x \gt \gamma $

(1)

where, α > 0 is the shape parameter, and β > 0 is the scale parameter. And the third parameter γ is a threshold parameter.

Rayleigh Generated Log-Logistic distribution

The PDF of Rayleigh Generated Log-Logistic distribution proposed and studied by Gaire & Gurung^[17] as:

$ f_2\left(x\right)=\dfrac{2\alpha\theta}{\beta\left\{1-exp\left(-\theta\right)\right\}}\times\dfrac{\left(\dfrac{x-\gamma}{\beta}\right)^{2\alpha-1}exp\left(-\mathrm{\theta}\dfrac{\left(\dfrac{x-\gamma}{\beta}\right)^{2\alpha}}{\left(1+\left(\dfrac{x-\gamma}{\beta}\right)^{\alpha}\right)^2}\right)}{\left(1+\left(\dfrac{x-\gamma}{\beta}\right)^{\alpha}\right)^3} $

(2)

Skew Log-Logistic distribution

The density and the distribution function of the Skew Log-Logistic (SLLog) distribution introduced by Gaire & Thapa^[18], and studied by Gaire & Gurung^[19] are expressed as:

$ f_3\left(x\right)=\frac{2\alpha}{\beta}\dfrac{\left(\dfrac{x-\gamma}{\beta}\right)^{2\alpha-1}}{\left(1+\left(\dfrac{x-\gamma}{\beta}\right)^{\alpha}\right)^3},\mathrm{\ \ \ f}\mathrm{o}\mathrm{r}\; x \gt \gamma $

(3)

Kumaraswamy Log-Logistic distribution

The density and the distribution function of Kumaraswamy LLog (KuLLog) distribution introduced by De-Santana et al.^[20] as:

$ {f}_{4}\left(x\right)=\dfrac{\alpha \theta \lambda }{\beta }{\left(\dfrac{x}{\beta }\right)}^{\alpha \theta -1}{\left(1+{\left(\dfrac{x}{\beta }\right)}^{\alpha }\right)}^{-(\theta +1)}{\left[1-{\left\{\dfrac{{\left(\dfrac{x}{\beta }\right)}^{\alpha }}{1+{\left(\dfrac{x}{\beta }\right)}^{\alpha }}\right\}}^{\theta }\right]}^{\lambda -1} $

(4)

Transmuted LLog distribution

Finally, the Transmuted Log-Logistic (TrLLog) distribution proposed by Aryal^[21] has the density and distribution functions as:

$ {f}_{5}\left(x\right)=\dfrac{\alpha }{\beta }{\left(\dfrac{x-\gamma }{\beta }\right)}^{\alpha -1}\left\{\dfrac{\left(1+\lambda \right)+(1-\lambda ){\left(\dfrac{x-\gamma }{\beta }\right)}^{\alpha }}{{\left(1+{\left(\dfrac{x-\gamma }{\beta }\right)}^{\alpha }\right)}^{3}}\right\} $

(5)

Model validation tools

The mean duration of PPA for Nepalese mothers has been compared by using t-test statistics for different groups such as ecological regions, provinces, ethnicities, and educational level attainment, and data for these groups, and total data are presented by using box-and-whisker plots. Using the above-listed probability distributions, the duration of PPA data is fitted. The negative log-likelihood (NLL) value of the probability distribution, Akaike's Information Criteria (AIC), the Bayesian Information Criteria (BIC), the chi-square test statistics, the SSE, and the coefficient of determination (R²) between the observed and theoretical duration of PPA of women, which have been used to test the validity and suitability of proposed models. To fit the model, numerical simulation was used to minimize the SSE. The Chi-square test statistics are calculated as follows:

$ \chi^2=\sum_{i=0}^n\left(\dfrac{\left(O_i-E_i\right)^2}{E_i}\right) $

(6)

where, Q_i is the observed number of women with PPA at different ages and E_i is the expected or theoretical number of women having PPA at different ages obtained from the fitting distribution for Nepalese women.

Similarly, the formulae of AIC and BIC for the fitted models are given as:

$ AIC= 2v- 2LL $

(7)

$ BIC=vLn(n) - 2LL $

(8)

where, v is the number of parameters associated with the probability distribution, n is the number of observations, and LL is the log-likelihood function at the maximum likelihood estimate of that distribution.

Supplementary Table S1 Descriptive measure of duration of PPA for Nepalese mothers (provinces).
Supplementary Table S2 The mean duration of PPA for different ecological regions of Nepal.
Supplementary Table S3 Mean duration of PPA for different ethnic groups of Nepal.
Supplementary Table S4 The mean duration of PPA (Educational attainment groups).
Supplementary Table S5 Multiple Comparison of mean duration of PPA by t-test (Provinces).
Supplementary Table S6 Multiple comparisons of mean duration of PPA for different ecological regions.
Supplementary Table S7 Multiple comparisons of mean duration of PPA for different educational levels.

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Duration of PPA	Observed	Expected number of women
Duration of PPA	Observed	LLog	SLLog	Tr-LLog	KuLLog	RG-LLog
0−1	102	84.095	74.572106	99.611	99.391	93.792
1−3	175	184.719	190.463529	173.489	180.143	182.120
3−6	183	167.887	155.884902	179.800	176.676	174.085
6−9	101	88.191	79.859153	99.240	100.989	96.196
9−12	53	51.009	47.416113	53.422	58.688	55.688
12−15	31	32.226	31.142130	30.082	35.198	34.150
15−18	28	21.790	21.923420	17.997	21.841	22.055
18−21	11	15.518	16.226545	11.399	14.006	14.885
21−24	13	11.507	12.471816	7.585	9.261	10.424
24−27	6	8.812	9.871832	5.260	6.298	7.531
27−30	7	6.926	7.999844	3.776	4.394	5.587
30−33	11	5.562	6.608797	2.791	3.138	4.240
α		1.370	1.072	2.049	0.568	1.210
β		4.386	2.006	7.723	42.028	8.777
γ		0.000	0.000	−	0.000	0.000
θ		−	−	0.478	10.632	1.000
λ		−	−	1.071	−	−
χ²		17.161	30.266	12.474	21.698	12.252
R²		0.980	0.950	0.993	0.995	0.995
NLL		−49.889	−49.649	−48.060	−52.046	−47.040
AIC		105.778	105.298	104.121	112.093	102.081
BIC		107.233	106.753	106.060	114.032	104.021
The values of the Chi-square test statistic, AIC, and BIC for the RGLLog distribution are found to be minimum, while the values of R² and NLL are found to be maximum. These results are presented in boldface.

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Duration and pattern of postpartum amenorrhea of Nepalese mothers: a statistical analysis