Web-based Supporting Materials for
Incomplete contingency tables with censored cells with application to estimating the number of
people who inject drugs in Scotland
by Overstall, King, Bird, Hutchinson and Hay
1. Equivalence of posterior inference under Poisson and multinomial models
Forster [1] shows that for complete contingency tables (i.e. when N is known), the joint posterior distribution
for  m and m is identical under the Poisson and multinomial formulations when  (0) / 1, assuming the same
prior distribution on the remaining parameters and over the model space. We extend this result here to the case
of incomplete contingency tables (i.e. when N is unknown) and in the presence of (uninformative) left censoring.
In particular, the posterior distributions are identical under the di erent model formulations under the prior
speci cations,  (0) / 1 and  (N) / N 1 (assuming identical priors on all other parameters and over model
space).
First we consider the Poisson formulation given in equation (2.1) of the main manuscript. The full set of model
parameters, under model m, is denoted by  m = fm;  
2g. We let  m =
 
 m;  2
 
(i.e. the set of parameters
excluding the intercept term). In addition, for model m (dropping the subscript notation for simplicity) we let
hi =  i   0 (i.e. the linear predictor for cell i, minus the intercept term). From equation (2.5) of the main
manuscript, and integrating out the intercept term, the (marginal) posterior distribution of  m, m, yC and yU is
given by
 ( m; m;yC ;yU jyO; zC) /  (zC jyC)( m;m)
Z
R
 (yjm;m)( 0)d 0
/  (zC jyC)( m;m)
Z
R
Qn
i=1 exp(i)
yi
iQn
i=1 yi
d 0
(recalling that  ( 0) / 1)
=  (zC jyC)( m;m)
Qn
i=1 exp(hi)
yi
Qn
i=1 yi
 
Z
R
exp(0)
N
 
 exp(0)
nX
i=1
exp(hi)
!
d 0:
1Using the substitution u = exp(0) and simplifying the expression, we obtain
 ( m; m;yC ;yU jyO; zC) /  (zC jyC)(m;m)
Qn
i=1 exp(hi)
yi (N  1)!
Qn
i=1 yi! (
Pn
i=1 exp (hi))
N : (1)
Next we consider the alternative multinomial formulation. The posterior distribution of  m, m, yC and yU is
given by,
 (m; m;yC ;yU jyO; zC) /  (zC jyC)(m;m) (yjN; m;m)(N)
/  (zC jyC) (m;m)
N !
Qn
i=1 yi!
nY
i=1
pyii  
1
N
;
substituting the probability mass function for the multinomial distribution. The result immediately follows (i.e.
the posterior distribution is identical to equation (1) in this document) by noting that
pi =
exp(hi)
Pn
j=1 exp(hj)
:
2. Weighted least squares implementation of the Metropolis-Hastings algorithm
In this section we describe the weighted least squares implementation of the Metropolis-Hastings algorithm
for GLMs [2] as applied to log-linear models. Let the current parameter values be denoted by  m and de ne
W(m) = diag fig and  i(m) = (yi   i)= i, where  i is evaluated at the current log-linear parameters,  m.
Furthermore, we let
~y(m) = Xm m +  (m);
C(m) =
 
1
 2
I + XTmW( m)Xm
  1
; and
m(m) = C( m)X
T
mW(m)~y(m):
The Metropolis-Hastings proposal parameters,  0 are simulated from,
 0  N (m( m);C( m)) ;
2and accepted with the standard acceptance probability, min(1; A), where
A =
 (0jy;  2;m)q(mj
0)
 ( mjy;  2;m)q( 
0jm)
;
in which q(0jm) denotes the multivariate normal proposal density for the proposal values, given the current
parameter values (and vice versa).
3. Reversible jump algorithm
Here we consider the reversible jump algorithm [3] to update the model within the MCMC algorithm. We let
m denote the current model with associated vector of log-linear parameters  m and design matrix Xm. Let ~m
denote the maximal model, i.e. the most complex model we are prepared to consider and ~ ~m the corresponding
posterior mode of the log-linear parameters under the maximal model  tted to the observed cell counts, yO. Note
that, for the examples we consider, the maximal model corresponds to the model with all main e ects and two-way
interactions present. We set ~ = X ~m~ ~m and de ne the (n n) matrix ~W = diag fexp(~)g.
We propose to move to a model that di ers with respect to the current model m by only a single interaction
and choose each of these models with equal probability. Suppose that we propose to move to model k, which
involves adding an interaction term (i.e. a \birth" move). We let the associated design matrix for model k be Xk.
We can write Xk = (Xm;S) where S is the column vector of the design matrix corresponding to the interaction
term that is added to the current model (note that for such moves we also re-order the  terms accordingly).
We de ne,
P k = Xk
 
XTk ~WXk
  1
XTk ~W ;
Ck =
 
ST ~W (I  P k)S
  1
; and
mk = CkS
T ~W (I  P k) ~:
We simulate u  N (mk; Ck) and set the proposed model parameters  
0
k such that,
 0k =
0
B
@
 0(1)
 0(2)
1
C
A =
0
B
@
I  
 
XTk ~WXk
  1
XTk ~WS
0 1
1
C
A
0
B
@
 m
u
1
C
A :
3Note that  0(1) denotes the proposed parameter values for the log-linear terms present in model m and  
0
(2) the
parameter value for the interaction term that is proposed to be added.
The move is accepted with probability, min(1; A), where,
A =
 ( 0k; kjy; zC ;  
2)
 (m;mjy; zC ;  2)q(u)
;
such that q denotes the proposal normal density function with mean mk and variance Ck. The Jacobian term is
simply equal to one and the probabilities of moving between models m and k cancel in the probability so that
these terms are omitted in the acceptance probability.
We now consider the case where we move from model k with parameters  0k to model m with parameters
 m which involves removing a single interaction term from the model (i.e. a \death" move). The corresponding
log-linear parameters in the proposed model are deterministically given by,
 m =  
0
(1) +
 
XTk ~WXk
  1
XTk ~WS 
0
(2);
and u =  0(2). Recall that  
0
(1) is the vector of current elements of  
0
k corresponding to the log-linear parameters
present in model m and  0(2) is the current value of the log-linear parameter that is removed from the model. This
move is accepted with probability min(1; A1), where A is given above. Finally, note that in both types of model
moves (adding or removing an interaction parameter), the hyperparameter  2 is not updated within the model
move.
4. Additional output
Web Table 1 shows the posterior means of the interaction terms for each year and for the INC-C, REM-C
and IGN-C methods. This table acts as a complement to Table 5 (showing posterior probabilities) in the main
manuscript.
Web Table 2 shows the posterior mean and 95% HPDIs for the total population size for each year under four
di erent speci cations of the prior hyperparameters, a and b, under the proposed INC-C method. The values in
this table should be compared against the corresponding values in the  rst two columns of Table 4 in the main
manuscript, where the prior hyperparameters are a = 0:001 and b = 0:001.
4Web Table 1: The marginal posterior means for each two-way log-linear interaction term for the INC-C, REM-
C and IGN-C methods. The data-sources are labelled as S1 - social enquiry reports; S2 - hospital records; S3
- Scottish Drug Misuse Database (SDMD) and S4 - HCV diagnosis data-source. An NA indicates that this
interaction cannot be identi ed with the REM-C method.
2003 2006 2009
Interaction INC-C REM-C IGN-C INC-C REM-C IGN-C INC-C REM-C IGN-C
S1  S2 0.00 0.00 0.13 -0.01 -0.00 0.00 0.00 0.01 0.19
S1  S3 -0.08 -0.09 0.08 0.12 0.14 0.19 0.07 0.09 0.26
S1  S4 -0.00 NA -0.01 0.01 NA -0.00 0.04 NA 0.01
S2  S3 0.02 0.02 0.19 -0.01 0.00 0.06 -0.04 -0.02 0.17
S2  S4 0.31 NA 0.27 0.27 NA 0.21 0.18 NA 0.07
S3  S4 0.01 NA -0.01 0.01 NA -0.00 0.01 NA -0.01
S1  Age 0.21 0.21 0.25 -0.17 -0.16 -0.21 0.05 0.05 0.19
S2  Age -0.04 -0.04 -0.00 0.13 0.13 0.08 -0.24 -0.24 -0.11
S3  Age 0.09 0.09 0.15 -0.13 -0.12 -0.18 0.01 0.01 0.16
S4  Age -0.00 NA -0.01 0.00 NA 0.01 0.03 NA -0.01
S1  Sex 0.09 0.09 0.09 -0.00 -0.00 -0.01 0.00 0.00 0.01
S2  Sex -0.00 -0.00 -0.00 0.12 0.12 0.10 -0.13 -0.13 -0.12
S3  Sex 0.00 0.00 0.00 0.01 0.01 0.00 -0.00 -0.00 0.00
S4  Sex -0.01 NA -0.00 0.00 NA 0.01 -0.00 NA -0.00
S1  Region 0.06 0.06 0.07 0.01 0.01 0.04 0.00 0.00 0.01
S2  Region -0.16 -0.17 -0.15 0.00 0.00 0.03 -0.00 -0.00 0.01
S3  Region -0.00 -0.00 0.00 0.21 0.21 0.24 0.12 0.12 0.14
S4  Region -0.14 NA -0.19 -0.00 NA -0.10 -0.01 NA -0.25
Age  Sex -0.12 -0.12 -0.12 -0.15 -0.15 -0.14 -0.16 -0.16 -0.14
Age  Region 0.19 0.19 0.18 -0.14 -0.14 -0.13 0.13 0.13 0.14
Sex  Region -0.00 -0.00 -0.00 0.00 0.00 0.00 -0.00 -0.00 -0.00
Web Table 2: Posterior mean (95% HPDI) for the total population size under the INC-C method for each year, for
di erent values of the prior hyperparameters, a and b. The analysis presented in the main manuscript corresponds
to a = b = 0:001 and the posterior mean (95% HPDI) for the total population size under this analysis (from Table
4) is also shown here for comparison.
Year a = 0:001 a = 0:001 a = 0:001 a = 0:001 Gelman
b = 0:004 b = 0:002 b = 0:001 b = 0:0005 prior
2003 16300 16500 16700 16700 16500
(14200, 20500) (14300, 20800) (14300, 20900) (14300, 20900) (14300, 20700)
2006 22800 23200 22900 23000 22900
(15700, 26600) (19800, 27000) (16300, 27000) (19300, 27600) (18700, 27800)
2009 14600 14600 15600 15200 14600
(11400, 18300) (11500, 18300) (11500, 18600) (11700, 18700) (11500, 18400)
5References
[1] Forster, J. J. (2010). Bayesian inference for Poisson and multinomial log-linear models. Statistical Methodology,
7, 210{224.
[2] Gamerman, D. (1997). Sampling from the posterior distribution in generalised linear mixed models. Statistics
and Computing, 7, 57{68.
[3] Forster, J. J., Gill, R. C. & Overstall, A. M. (2012). Reversible jump methods for generalised linear models
and generalised linear mixed models. Statistics and Computing, 22, 107{120.
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