Page 62 - IJPS-8-2

International Journal of
Population Studies Projecting sex ratio at birth in Pakistan

which modeled the sex ratio transition of multiple For ∈{1,…n}, the likelihood can be written as:
countries, including Pakistan. The standard deviations 1
ωδ
of prior distribution are set such that the CV is 0.1. The p ( i | v Φ pi   i ,, pi  α , pi   i ) ∝
,t
,t
2
informative priors assist the provincial-level modeling      σ +ω 2
i
of the sex ratio transition in Pakistan by exploiting the  2 
v
corresponding information at the national level. For p ∈ exp  − ( i –b Φ pi   i − δ pi  α pi   i )  .
,t
,t
 
 

{1,…,k}, we have:  2 2 ) 
 2 ( σ +ω 
i
 
 0, 0060 006., . 2 ,
p
Posterior Density
11 01 1 ,
 , 0 ., . 2 The posterior density for V , the true SRB on the log scale

1p
p,t
for province p at time t, up to proportion is:
76 08 ,
2p  , 0 ., . 2  , ωΦ, δ , ,t ξ , λ , λ , λ , V

1: ,0:k
  ,, h , 1: ,0:k |v h 1:k 0,1:k 1:k 1,1:k 2,1:k 3,1:k  p  ∝
ρ σ µ σ ,
16 11 6 .
3p  , 0 ., . 2  π  π 1:n k k 

( −1 /2
( − 1 ρ 2 ) kh ) ∏ π ∑ p =1 δ p ( − 1 π ) k − ∑ p =1 δ p
2.2.2. Data quality model δ ∈{0,1} p p ×
p
( −1
r is the i observed SRB in province p[i] in year t[i], σ ò kh ) σ π ∏ k   h ( − ρ1 2t ) ∏ t k /2   n (σ i 2 ω + 2 ) 1/2
th
i
where i indexes all SRB observations across the provinces  =1  i =1
over time. r is assumed to follow a normal distribution

i
on the log scale with mean of log pi ,ti and variance   k 2 µπ − π 2 p n ( i –b Φ v pi   i δ − pi  α pi   i )  2 
,t
,t
 
p
π

 
2
of σ : exp   =1 2 σ π 2 − ∑ i =1 (σ 2 2 ω + 2 ) ∑  p ×
i
 i 
( ) Θ log r i | pi   i ∼  ( ( Θ log pi   i ) σ i 2 ω + , 2 ) , k h 1 2 , 2 k 2 2
,t
,t
t
2
 
 

.

fori ∈ , , },n exp 2 , pt t 2 p 0  , p 0
{1
2
p t1 1 2  1 p 1 21

2
Where, n = 531 is the total number of observations. σ
i
is the sampling error variance of log (r ), which reflects the 2.2.4. Statistical computing and Bayesian Inference
i
uncertainty in log-scaled SRB observations because of the We obtained posterior samples of all the model parameters
2
survey sampling design. σ is calculated using a jackknife and hyperparameters using a Markov chain Monte Carlo
i
method (Appendix A.1). ω is the non-sampling error (MCMC) algorithm, implemented in the open source software
2
variance representing the uncertainty contributed by non- R 4.2.1 (R Core Team, 2022) and JAGS 4.3.0 (Plummer,
responses, recall errors, and data input errors. We assume 2003), using R-packages R2jags (Su & Yajima, 2015) and
that ω is immeasurable and is estimated using the model rjags (Plummer, 2018). Results were obtained from 10 chains
2
by assigning a vague prior: with a total of 5000 iterations in each chain, while the first
1000 iterations were discarded as burn-in. After discarding

00 5,. . burn-in iterations and proper thinning, the final posterior

sample size for each parameter by combining all chains is
2.2.3. Posterior distribution 25,000. The convergence of the MCMC algorithm and the
sufficiency of the number of samples obtained were checked
Likelihood through visual inspection of trace plots and convergence
For the i observation r, let v = log (r) and V pt, log pt, . diagnostics of Gelman & Rubin (1992), implemented in the
th
i
i
coda R-package (Plummer et al., 2006).
i
The likelihood on log-scale up to proportion is: 2.2.5. Post-modeling process
i
v V 2 2.2.5.1. Identifying provinces of Pakistan with SRB imbalance
i 1 pi ti SRB imbalance in a Pakistan province is detected if δ =
,
pv V| , , exp . p
2
pi ti
2

2 2 2 1 for more than 95% of the posterior samples (indicating
i
i
SRB inflation).
Volume 8 Issue 2 (2022) 56 https://doi.org/10.36922/ijps.v8i2.332