Multilevel Logistic Regression Model
Multilevel logistic regression model is appropriate for research designs where data forrespondents are organized more than one level (i.e, nested data). The units of analysis are usually individuals at a lower level (women) who are nested within aggregate units at a higherlevel (regions). A multilevel logistic regression model is also referred to as a hierarchical logistic regression model, or as a random effects (mixed effects) logistic regression model. Themultilevel logistic regression extends from single level logistic regression model byincluding random effects from the model (Snijders and Bosker, 1999).
Multilevel logistic regression analysis can be employed in the simplest case without explanatory variables, (usually called the empty model) and also with explanatory variables by allowing only the intercept term or both the intercept and slopes (regression coefficients) to vary randomly. In this study, multilevel logistic regression model taking into account the data to be analyzed on the case of two-levels. We note that extensions to the case of three or higher levels is straight forward. In this study, women are considered as level-1 and regions is considered as level-2 (Snijders and Bosker, 1999).
3.6.1 A Two Level Logistic Regression Model
Multilevel analysis is a methodology for the analysis of data with complex patterns of variability, with a focus on nested sources of variability. The best way to analysis multilevel data is an approach that represents within-group as well as between group relations within a single analysis, where ‘group’ refers to the units at the higher levels of the nesting hierarchy. Very often it makes sense to use probability models to represent the variability within and between groups, in other words, to conceive of the unexplained variation within groups and the unexplained variation between groups as random variability. For example, a study of women’ within regions means that not only unexplained variation between women, but also unexplained variation between regions’ is regarded as random variable. This can be expressed by statistical models called random coefficient model. Multilevel analysis is an approach to the analysis of such data including the statistical techniques as well as the methodology of how to use these (Snijders and Bosker, 1999).
Testing heterogeneous proportions
The most commonly used test statistic to check for heterogeneity of proportion between groups (regions) which is proper application of multilevel analysis is chi-square test statistic. To test whether there are indeed systematic differences between the groups (regions), the chi-square test can be used and written as:
X^2=?_(j=1)^g??nj ?((y.) ?j-(p.) ?)?^2/((p.) ?(1-(p.) ?))? ……………………………………………….…….….(3.6)
Where, Y ?_(.j)is group average, obtained asY ?_(.j)=1/n_j ?_(i=1)^(n_j)?Y_(ij ) is the proportion of successes in group j which is an estimate for the group-dependent probability ?? and (P.) ? is the overall average, i.e.P ?_.=Y ?_(..)=1/n ?_(j=1)^g??_(i=1)^(n_j)?Y_ij is the overall proportion of successes. The decision is based on chi-square distribution with g-1 degrees of freedom (Agresti, 1996).
Estimation of between and within-group variance
Consider a population having two-levels, the basic data structure of two-level logistic regression analysis is a collection of N groups (units at level-two (regions)) and within group j (j= 1, 2, …,N) a random sample of ?? level-one units (women). The outcome variable is dichotomous and denoted by ???,(? = 1,2, … , ??,? = 1,2, … ,?) for level-one unit? in group . The total sample size is. M=?_(j=1)^N?n_j .
Then, the theoretical variance between the groups (regions) dependent probabilities, i.e., the population value ofVar (??), can be estimated by:
? ?=S^2 between-(S^2 within)/n ? ,
Where n ?=1/(N-1) (M-(?_(j=1)^N??n_j?^2 )/M)=n ?-(S_((n_j))^2)/(Nn ? )
For dichotomous dependent variable, the observed between- groups variance is closely related to the chi-squared test statistic (Snijders and Bosker, 1999). They are given by the formula:
S_between^2=(p ?(1-p ?))/(n ?(N-1)) ?^2
Where,?2is as given by equation (3.1), and the within- group variance in the dichotomous case is a function of the group: