*2.2. Statistical Model*

Let *yijm* denote the contraceptive use status of woman, *i* from stratum *j* and cluster *m*, with *i* = 1, 2, 3, ... , 6847, *j* = 1, 2, 3, ... , 60 and *m* = 1, 2, 3, ... , 492. The outcome variable is defined as a dichotomous variable such that *yijm* = 1 if the women *i* is currently using any type of contraceptive method and *yijm* = 0 if the women *i* is not currently using any type of contraceptive method. The contraceptive use status among women of reproductive age is a binary outcome in the current study and hence it is assumed to follow a Bernoulli distribution:

$$y\_{ijm} \sim Bernoulli(p\_{ij}) \tag{1}$$

where *pij* is the probability that a woman *i* from district *j* is currently using a contraceptive method and 1 − *pij* is the probability that a woman *i* from district *j* is not currently using a contraceptive method. Therefore, the use of contraceptive methods among women of reproductive age can be associated with the explanatory variables using an appropriate link function from a generalized linear models approach as follows:

$$\text{logit}\left(p\_{ij}\right) = \log\left(\frac{p\_{ij}}{1 - p\_{ij}}\right) = \mathcal{W}\_{ij}^{\prime}\mathcal{S}\tag{2}$$

Model (2) is known as binary logistic regression, where *Wij* is the vector of explanatory variables and *β* is the vector of coefficient parameters. However, classical Generalized Linear model (GLM) has a rigid assumption that all observations are independent; but this assumption is sometimes not satisfied, as some observations may have, for instance, spatial dependence, or may have nonlinear effects. Hence, there is a need to include nonlinear and spatial variability in model (2) and it is given by:

$$\text{logit}\left(p\_{\vec{i}\vec{j}}\right) = \mathcal{W}\_{\vec{i}\vec{j}}^{\prime}\boldsymbol{\beta} + \sum\_{k=1}^{q} f\_k\left(\mathbf{x}\_{\vec{i}\vec{k}}\right) + f\_{\text{spat}}\left(\mathbf{s}\_{\vec{j}}\right) \tag{3}$$

where *βi* is the vector of fixed effect corresponding to categorical variables, *fk* is the appropriate smooth function of continuous variables such as mother's current age and mother's age at first cohabitation and *fspat sj* are the parameters of random effects, which capture unobserved spatial heterogeneity at district *sj*.
