Page 132 - IJPS-2-2

Utilization of institutional delivery services across successive births in India

natively, multilevel logistic regression analyses account for variations due to the hierarchical struc-
ture of the data — allowing the simultaneous examination of the effects of village-level and individ-
ual-level variables while accounting for the non-independence of observations within groups.
The units of analysis are births occurring within five years of the survey date, which are nested
within mothers, and are further nested within communities (PSU). First, we applied a three-level
model considering children at the lowest level. The estimation was based on the first-order predictive
quasi-likelihood procedure (the second-order approximation methods did not converge). Second,
because births of the same woman are correlated, the binomial assumption (over-dispersion of data)
was assessed. For the purpose of model diagnostics, we included additional binomial multilevel
2
models and tested the assumption of independent Bernoulli trails by keeping σ as unconstrained.
e
Finally, we used multivariate (two or more dependent variables) multilevel models to allow for the
strong dependence between successive outcomes in the same unit — but do not require long series of
repeated measures — as suggested by Cox (1972) for handling data with dependent binary responses.
In this case, the assumption is that for a mother the response y for each individual pregnancy is
ijk
one component of a multivariate binary response at the mother level j (Griffiths, Brown, and Smith,
2004). This model does not assume conditional independence of the individual responses and speci-
fies the correlation structure between pregnancies to the same mother. We modified the general
model by Yang et al. (2000) and considered a maximum of three consecutive births per woman. The
number of women having four and five births during the five-year study period was 102 and 3, re-
spectively, and therefore excluded from the analysis due to limited sample size.
Response y has two categories: 1 indicates institutional delivery for birth i nested within
ijk
women j of PSU k, and 0 indicates home delivery.
y ijk ~binomial(π ijk ,1) where i = 1,2,3 (1)
var(y ijk π ijk ) = π ijk (1−π ijk ) (2)

 π 
ln ijk =  b + bx + b x + + b x + ϑ (3)
 1−π  0 11ijk 2 2ijk m mijk 0k
 ijk 
where there is one constant item and m explanatory variables.
ϑ is a PSU random effect.
0k
The extension of equation (2) in the case of three births per women gives
y ijk = π ijk + ez (4)
ijk ijk
e  1 jk   σ 2 1 e 
   
var e  2 jk  = σ  21 σ 2 2 e  (5)
     2 
 e 3 k j   σ  21 σ e 32 σ 3 e  
Unlike independent univariate distributions, the multivariate distribution gives a complex form of
2
the variance of e . In this model, to assess the binomial assumption, the variance parameter σ
ijk
ei
(i=1, 2, and 3) is unconstrained (not fixed at 1) and estimated to compare the value from 1. In the
2
unconstrained model, if the value of σ is close to 1, it indicates that the model follows the bino-
ei
2
mial assumption. If we consider the variance parameter σ equal to 1, it gives the required bi-
ei
nomial variation for each pregnancy outcome to mothers. From the data we can also estimate the cova-
riance, which identifies whether the women’s behavior is independent or not, across different births
in relation to the institutional delivery. Finally, we calibrated a three-level multivariate multilevel
regression model that had a structure of pregnancies (three for each woman) at Level 1 nested within
women at Level 2 and finally nested within PSU at Level 3.

126 International Journal of Population Studies | 2016, Volume 2, Issue 2