Using Markov Chain Monte Carlo for Modeling Correct

Enumeration and Match Rate Variability 

Andrew Keller 

U.S. Census Bureau 

Washington, DC 20233/andrew.d.keller@census.gov 

This report is released to inform interested parties of ongoing research and to encourage discussion of work in

progress. The views expressed on statistical, methodological, technical, or operational issues are those of the

author and not necessarily those of the U.S. Census Bureau. 

Abstract

The Census Bureau conducted the Accuracy and Coverage Evaluation (A.C.E.) with the goal of producing estimates 

of the net coverage error of Census 2000. The A.C.E. used dual system methodology to estimate the net coverage 

error. Dual system estimates were created for population subgroups called post-strata. A problem in model-based 

evaluation of coverage with respect to smaller geographies is that the variance between blocks and within those 

geographies needs to be specified in order to estimate a coverage correction factor. To improve coverage estimates, 

effort has been made towards advancing models involving smaller geographies. This paper offers Markov chain 

Monte Carlo (MCMC) methods as a computer intensive method to generate these estimates. Specifically, random 

effects models of the correct enumeration and match rates at the block level are developed to specify this variance 

with an accompanying statement of precision. 

Background 

The A.C.E. used two samples to evaluate coverage for Census 2000, the population sample (P sample) and the 

enumeration sample (E sample). The P sample estimated persons that should have been enumerated in the census at 

that location according to census residence rules but were not. The P sample consisted of people rostered from a 

sample of housing units in a specific location (independent of the census) from a sample of census block clusters 

(from now on referred to as blocks). It was populated based on the results from a person interview, independent 

from the census enumerations in the sample blocks. 

The E sample estimated census erroneous enumerations that should not have been included anywhere in the census 

or at the specific location. The E sample consisted of census enumerations. It was identified in the same set of 

census blocks selected for the P sample. E-sample enumerations who matched to P-sample people were counted as 

correct enumerations. Nonmatched E-sample enumerations underwent a follow up interview to determine whether 

they were correct enumerations for the specific location. 

The A.C.E. divided the population into 416 post-strata where smaller groupings were combined or collapsed to 

produce more stable estimates. A post-stratum was a group of people sharing demographic and geographic 

characteristics that were assumed to have the same probabilities of inclusion in the census (U.S. Census Bureau 

2004). A post-stratum was composed of an E-sample post-stratum and P-sample post-stratum pair. Within a single 

post-stratum, the dual system estimate (DSE) formula was defined as: 

CEk/ Ek DSE = census × DDRATE × (1) k k k

Mk/ Pk

where: 

k : Post-stratum 

censusk: The census count within post-stratum k
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

DDRATEk: The ratio of data defined census records to all census records within post-stratum k

 1

. 

CEk: Weighted estimate of correct enumerations in post-stratum k

Ek: Weighted estimate of enumerations in post-stratum k

Mk: Weighted estimate of matches in post-stratum k

Pk: Weighted estimate of P-sample records in post-stratum k

The model development described in this paper focuses on the third term for Formula 1, the ratio of the correction 

enumeration (CE) rate and match rate. The CE rate quantifies the ratio between total E-sample enumerations and a 

smaller subset of correct E-sample enumerations. The match rate quantifies the ratio between total P-sample people

and a smaller subset of P-sample people who matched to a census enumeration. This ratio drives the calculation of 

DSE

(1), and the resulting coverage correction factors, CCFk=

k

. 

censusk

The goal of this research is to eventually model CE and match rate variances between blocks within a specified 

geography. From the model-based approach used here, coverage estimates can be made from the modeled CE rate 

and match rate. The model-based rates, pCE and pM

, respectively can be used for computation of new DSEs in (2) 

and coverage corrections factors. That is, for each block b and post-stratum k : 

pCE ,bk DSE = census × DDRATE × (2) bk bk bk

pM ,bk

The model was applied to selected states by including each A.C.E. sample block with between 3 and 79 housing 

units. Traditionally, to approximate meaningful variances of the CE and match rates, it has been necessary to move 

up the geographical hierarchy. That is, variance estimates were computed on larger geographies where sufficient 

data was present to calculate a design-based variance of the respective rates. By developing the model, the goal was 

to generate variances with corresponding statements of precision for smaller levels of geography. 

Methodology 

The model development was geared towards CE rates. In the future, an analogous process will be followed to 

develop a model for match rates. 

Data Development 

For the E sample, each census record was assigned a CE probability based on A.C.E. processing (U.S. Census 

Bureau 2004). To create the data, the CE probability was compared to a uniform number between 0 and 1, U (0,1) . 

Based on that comparison, the record received a binary CE value. For each block in the A.C.E. sample with 

between 3 and 79 housing units, the number of correctly enumerated person records and total person records were 

aggregated and a CE rate was generated. 

Model Development 

This initial model development incorporated between block variability using random effects. Initially, the model 

used a single fixed effect. Let p b(, w) represent the owner CE rate in a given block b . Similarly, let p b(, r) 

represent the renter CE rate in a given block b . Since the p b(, •) are sample proportions between 0 and 1, logistic 

1

 In 2000, the census required two characteristics for a record to be data defined. Relationship, sex, race, Hispanic 

origin, and either age or year of birth counted as characteristics. A valid name also counted as one characteristic. 

To be considered valid by the census, a name had to have at least three characters in the first and last name together. 

These data defined census records were eligible for A.C.E. processing. 
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

regression was used for the model. Because home ownership has traditionally been a good predictor of correct 

enumeration, that was included in the logistic model as a fixed effect. Also, since each block has unique 

characteristics, a random block effect was included in the model. As a result, the owner and renter CE rates were 

μ α+ s +ε () b

written as p b( , ) • =

e

i

μ α i

+ s +ε ( ) b

1 + e

μ : Intercept term 

α : Ownership effect 

s

i

: Binary variable indicating whether the record i was designated as an owner ( s

i

= 1) or a renter ( s

i

= 0 ) 

ε ( ) : b Block effect 

Recall that each person had a binary CE value. Let δ ,, 

= 1 refer to each correctly enumerated owner record in an ibw

arbitrary block b with nbw total owner enumerations. Consequently, δ ,, 

= 0 refers to each erroneously ibw

enumerated owner record. Analogously, let δ ,, 

= 1 refer to each correctly enumerated renter record in an arbitrary ibr

block b with nbr total renter enumerations and δ ,, 

= 0 refer to each erroneously enumerated renter record. ibr

Assuming owner correct enumerations follow a binomial model with a CE rate pbw ( , ) , the likelihood function was 

nbw

δ 1−δ ibw ,, ibw , , written as L( ( , )) pbw = ∏ pbw ( , ) (1 − pbw ( , )) . Similarly, assuming renter correct enumerations follow a 

i=1 

binomial model with a CE rate pbr ( , ) , the likelihood function was written as 

n +n

δ 1−δ

bw br

i b, ,r i b, ,r L pbr ( ( , )) = ∏ pbr ( , ) (1 − pbr ( , )) . Combining these likelihoods together, 

i nbw = +1

n n +n bw bw br

δ 1−δ δ 1−δ i b, ,w i b, ,w i b, ,r i b, ,r L pbw ( ( , )) × L pbr ( ( , )) = ∏ pbw ( , ) (1 − pbw (, )) × ∏ pbr ( , ) (1 ( , )) − pbr . (3) 

i=1 i=nbw +1 

The CE rates were modeled to involve the parameters of interest: μ,, α ε ( ) b . Also, substituting the logistic 

expression for the respective CE rates into the likelihood function resulted in the following modification to (3): 

n μ α+ s +ε () b μ α+ s +ε () b n +n μ α+ s +ε () b μ α+ s +ε ( ) b

bw i i bw br i i

e e δ 1−δ e e δ 1−δ

ibw ,, ibw , , ibr ,, ibr , , L( , , ( )) μαε b = ∏( ) (1 − ) × ∏ ( ) (1 − )

μ α+ s +ε () b μ α+ s +ε () b μ α+ s +ε () b μ α+ s +ε ( ) b i i i i

=1 1 e i nbw i + e 1 + = +1 1 + e 1 + e

By applying what is known about si

 for owners and renters and applying exponential properties, the model was 

further simplified. Therefore, 

n μαε + + ( ) b n +n μ ε + ( ) b

bw bw br e δ 1 1−δ e δ 1 1−δ i b, ,w i b, ,w i b, ,r ibr , , L(μαε , , (b)) = ∏( ) ( ) × ∏ ( ) ( )

μαε + + ( ) b μαε + + () b μ ε + () b μ ε + ( ) b

i=1 1 + e 1 + e i=n +1 1 + e 1 + e bw

n +n n +n

bw bw

μαε + + ( ) b (1−δ ) μ +ε () ∑ ibr δ

, ,

) 

n n bw br bw br

δ

b δ

,, ∑ (1− ∑ ∑ ibr e i b, ,w i b,, w 1 e 1 

i n= +1 i=n +1

i=1 i=1 bw bw = ( ) ( ) ( ) ( )

μαε + + ( ) b μαε + + () b μ ε + ( ) b μ ε + ( ) b

1 + e 1 + e 1 + e 1 + e

By definition of δ , ,•

,

i b

n n n +n n +n bw bw bw br bw br

∑δ = nce , (1 ∑ − δ ) = nee , ∑ δ = nce , ∑ (1 − δ ) = nee ibw ,, bw ibw ,, bw ibr ,, br ibr , , br

i=1 i=1 i=n +1 i=n +1 bw bw

where: 
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

ncebw : Number of correctly enumerated owner records in an arbitrary block b

neebw : Number of erroneously enumerated owner records in an arbitrary block b

ncebr : Number of correctly enumerated renter records in an arbitrary block b

neebr : Number of erroneously enumerated renter records in an arbitrary block b

As a result, the likelihood function for block b was finally written as: 

μαε + + ( ) b μ ε + ( ) b

e nce 1 nee e nce 1 nee bw bw br br L( , , ( )) μαε b = ( ) ( ) ( ) ( )

μαε + + ( ) b μαε + + () b μ ε + () b μ ε + ( ) b

1 + e 1 + e 1 + e 1 + e

The final likelihood was: ∏L( , , ( )) μαε b . In addition, it was initially assumed that the block effects were 

b S ∈

1 ε

2

( ) 

normally distributed. This led to an augmented likelihood model, ∏L(μαε , ( )) , b × exp(− ) . (4) 

b

∈ 2πσ 2σ

2 

b S

Markov Chain Monte Carlo 

In general, let θ be a vector of unobservable population parameters and y denote observed data. Markov chain 

Monte Carlo (MCMC) methods iteratively generate dependent samples in the parameter space that converge to a 

target distribution, p( | ) θ y .

Bayesian techniques allow inference about θ conditional on the observed data y . Using Bayes’ rule, the posterior 

p( , ) θ y p y( | θ ) ( ) p θ

distribution is p(| θ y) = = ∝ p y( | θ ) ( ) p θ . The likelihood function from above is another 

p y() () p y

way of writing p y(| θ ) since it is proportional to the probability given unknown parameters. The p( ) θ term is 

called the prior distribution. 

Metropolis-Hastings Algorithm 

The Metropolis-Hastings Algorithm generates samples from a posterior distribution. The Metropolis-Hastings 

Algorithm uses an acceptance-rejection scheme to draw samples from a candidate distribution. The algorithm works 

as follows: 

1. Start with an initial value θ (t = 0) where the posterior density is greater than 0. 

2. For subsequent iterations t = 1, 2,..., T −1,T ; sample a candidate value θ t

*

() from a candidate distribution, 

C( ( ) | θ t

*

θ (t −1)) . This candidate distribution may not be symmetric. 

θ t

* ( ( θ ( )) t* 

p( ( )| y) ×C θ t −1) | 

3. Calculate the Metropolis-Hastings ratio, Ratio = . If the candidate 

p( (t −1) | y) × ( ( ) θ t

*

θ C | θ (t −1)) 

distribution is symmetric, then the Metropolis-Hastings ratio above is simplified to a ratio of the posterior 

densities. 

4. If Ratio > U (0,1) , then θ () t is accepted and θ t = θ ( ) . If not, θ t = θ (t −1) .

*

() t

* () After the candidates 

are determined to converge to the target distribution, inference is completed. 
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Gibbs Sampler 

For multi-parameter Markov chain applications, the Gibbs sampler is used to cycle through all the parameters. If θ

is composed of multiple parameters, then for each iteration, a single parameter θ is drawn from a conditional 

g

distribution given all other parameters of θ . The Gibbs sampler is a special case of the more general MetropolisHastings Algorithm where all candidates are accepted and the target distribution is known. 

Model Specifics 

This analysis used the Metropolis-Hastings algorithm within the Gibbs sampler. It used the Gibbs sampler to draw a 

single parameter from a conditional distribution given all other parameters. However, since the target distribution 

was unknown, the Metropolis-Hastings Algorithm was used to accept and reject candidates. 

For each iteration t , B + 3 parameters were processed, where B was the number of random block effects in the 

model. The remaining three parameters corresponded to the mean ( μ ), ownership effect (α ), and variance 

between the block effects (σ

2 ). The process cycled through each parameter conditional on the values of the other 

B + 2 parameters and the data by evaluating their Metropolis-Hastings ratios. The σ

2 parameter was the only 

parameter where a non-constant prior distribution was assumed. Its prior distribution was assumed to be halfCauchy. To properly check for convergence multiple sequences (chains) were run. With respect to following 

notation, z refers to a chain. 

Block Effects. For the block effect, the candidate value was sampled by drawing from a normal distribution, 

* 2 *

ε ( , , ) ~ bzt N( ( , , ε bzt −1), σ (, z t −1)) . The normal distribution is symmetric in ε ( , , bzt ) and ε ( , , bzt −1) . As a 

result, the Metropolis-Hastings ratio with respect to the block effects was simplified to the ratio of posterior 

densities and the decision to accept was based solely on the ratio of the posterior densities. 

Intercept. For the intercept term, the candidate value was sampled by drawing from a normal distribution, 

μ(, *) = μ z t −1) + e where e N . The normal distribution is symmetric in μ z t * z t (, ~ (0,1) (, ) and μ( ,z t −1) . As a 

result, the Metropolis-Hastings ratio with respect to the intercept was simplified to the ratio of posterior densities 

and the decision to accept was based solely on the ratio of the posterior densities. 

Ownership Effect. For the ownership effect, the candidate value was sampled by drawing from a normal 

distribution, α(, *

) = α z t −1) + e where e N . The normal distribution is symmetric in (, * z t ( , ~ (0,1) α z t ) and

α( ,z t −1) . As a result, the Metropolis-Hastings ratio with respect to the ownership effect was simplified to the 

ratio of posterior densities and the decision to accept was based solely on the ratio of the posterior densities. 

Variance Between Block Effects. For the variance between the block effects, the candidate value was sampled by 

drawing from a gamma distribution, 

γ ~ Gamma[ ω ψ, (z t, )] 

ω = ⎢⎡(B −1) / 2 ⎤⎥

1 B

2 ψ (,) z t = ∑ε ( , , ) bzt

2 b=1 

2 * 1

σ ( ,z t ) =

γ

1

The posterior density for the variance between block effects included a half-Cauchy prior term . The 

1 +σ

2

( z t,

*

) 

conditional posterior density was expressed as: 
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

B

2 

B−1 ∑ε ( , bzt,) 

− 1

σ z t * 1, , ),..., ( Bzt z t ( , ), ] z t * 2 −

b=1

p[ ( , ) | ε (b = z t ε b = , , ), μ( , ), α z t y ∝ σ ( , ) exp[ ]× 2 * 2 * 2 ( , ) 1 σ z t +σ (, ) z t

Since the gamma distribution is not symmetric, then the Metropolis-Hastings ratio with respect to the variance 

between block effects included the candidate distribution. The candidate distribution was: 

B

B−1 ε

2 ∑ ( , , ) bzt

* * 2 b=1 C( ( , ) | σ (, z t −1)) ∝ σ ( , ) 

−

σ z t z t exp[ − ]

2 (, σ

2 z t *

) 

B

2 

B−1 ∑ε ( , bzt,) 

The logistic terms had no effect because they had no 2 b=1 σ dependence. The σ

−

exp[ − ] terms cancelled 

σ

2 * 2 (z t, )

because they were on opposite sides of the quotient. The simplified Metropolis-Hastings ratio was expressed as: 

1 

* * 2 * 

p( ( , ) | σ z t others) ×C( (, σ z t −1) | σ (, z t )) 1 +σ (, z t )

Ratio = =

( ( z t, −1) | o he s) C σ z t *

p σ t r × ( ( , ( , )| σ z t −1)) 1

1 +σ

2

( z t, −1) 

Convergence Analysis

This analysis employed the Gelman and Rubin Method (Gelman et al. 2000) as its convergence diagnostic. To 

monitor convergence, a potential scale reduction factor was calculated for every parameter. For this analysis, the 

parameter vector subject to convergence monitoring was comprised of the random block effects of each block, the 

intercept term, the ownership term, and the variance between the block effects. That is, 

( ( b = 1), ε (b = 2),..., ε (b = B −1), ( b = B), , , 2

θ = ε ε μ α σ ) .

To begin, Z = 10 starting values for each parameter were chosen as initial values. For those initial values, dispersed 

starting points were used. This was done to determine if problems existed with the model’s convergence and to 

ensure that the parameter space was thoroughly searched to uncover possible modes. To complete inference, the 

potential scale reduction factor was calculated at intervals of one hundred iterations. When all parameters had a 

potential scale reduction factor close to 1, the MCMC method was thought to have converged at that iteration, 

t = τ . 

Inference 

After τ was determined, the parameter values were used to calculate modeled owner and renter correct enumeration 

totals for every iteration between τ + 1 and 2τ for each block within each chain. That is, 

μ ( ,) z t +α ( , ) z t +ε ( bzt , , )

e

CEmdl ( , , ) b z t = ( ) × n owners μ ( , ) z t +α ( , ) z t +ε (bzt , , ) bw 1 + e

(5) 

μ ( ,) z t +ε ( , , ) bzt

e

CEmdl ( , b z t,) = ( ) × n renters μ ( ,) z t +ε ( , , ) bzt br 1 + e

were used as draws from the joint posterior distribution. 

Model Checking 

Binomial trials were run to produce new samples. Simple means and standard error estimates from the new samples 

were compared to corresponding statistics from the observed sample to assess model fit. To do this, for each 

chain/block/iteration grouping between τ + 1 and 2τ , nbw trials were run to get a sample of the number of correctly 
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

enumerated owners in that block. Similarly, nbr trials were run to get a sample of the number of correctly 

enumerated renters in that block. The inputted Binomial probabilities were based off the modeled correct 

enumeration totals for owners and renters in (5). The correct enumeration totals for the binomial trials were 

computed as follows: 

CEmdl ( , , ) b z t

owners CEbin ( , , ) ~ b z t Bin( , n p = )

owners bw

nbw

CEmdl ( , , ) b z t

renters CEbin ( , , ) ~ b z t Bin( , n p = ) .

renters br

nbr

CEbin b z t ( , , ) = CEbin ( , , ) b z t + CEbin ( , , ) b z t

owners renters

Next, a mean (meanCERATEbin) and standard error (seCERATEbin) over the blocks were calculated for each 

iteration between τ +1 and 2τ for each chain. They were calculated as follows: 

B

[∑CEbin b( , , ) z t ] 

b=1

(,) = BmeanCERATEbin z t

[∑n + n ]

bw br

b=1 

B

[∑CEbin b( , z t, )] − CEbin b( , z t, ) 

b=1

( , , ) = B

repCERATEbin b z t

[∑n + n ] −[n + n ]

bw br bw br

b=1 

B −1 B

2

seCERATEbin(,) z t = ∑(repCERATEbin( , , ) b z t − meanCERATEbin( , )) z t

B b=1 

The means and standard errors for all iterations between τ +1 and 2τ for each chain were combined and then 

sorted from smallest to largest. That resulted in two vectors of size 10τ . That is, 

meanCERATEvect = [meanCERATEbin(1, τ +1),..., meanCERATEbin(10, 2 ) τ ]

seCERATEvect = [seCERATEbin(1, τ +1),..., seCERATEbin(10, 2 ) τ ]

sortedmeanCERATEvect = sort(meanCERATEvect) 

sortedseCERATEvect = sort(seCERATEvect) 

Then, the 5% coverage values were created by taking the mean and standard error for the 0.05 ×10τ sorted iteration. 

Similarly, the 95% coverage values were created by taking the mean and standard error for the 0.95 ×10τ sorted 

iteration. That resulted in the following coverage intervals: 

mean _ cvg _ itrvl = [sortedmeanCERATEvect(0.05 ×10 ), τ sortedmeanCERATEvect(0.95 ×10 )] τ

(6) 

se __ cvg _ itrvl = [sortedseCERATEvect(0.05 ×10 ), τ τ sortedseCERATEvect(0.95 ×10 )] 

As an example, suppose a MCMC model is run for 5000 iterations over 50 blocks with 10 chains. Suppose that, 

from the Gelman and Rubin Method, the model converges at τ = 2000 . For all chains, the means and standard 

errors for iterations between τ + = 1 2001 and 2τ = 4000 are combined and then sorted from smallest to largest. 

That results in vectors of size 20000 for the mean and standard error. The coverage intervals (6) are then formed 

by: 
 

 

 

 

 

 

 

 

 

mean _ cvg _ itrvl = [sortedmeanCERATEvect(0.05 × 20000), sortedmeanCERATEvect(0.95 × 20000)] 

se __ cvg _ itrvl = [sortedseCERATEvect(0.05 × 20000), sortedseCERATEvect(0.95 × 20000)] 

The coverage intervals were compared to the values observed from the 2000 A.C.E. sample. They were computed 

as follows: 

B

[∑ncebw + ncebr ] 

b=1 meanCERATEobserved = B

[∑n + n ]

bw br

b=1 

B

[∑nce + nce ] − [nce + nce ]

bw br bw br

b=1

repCERATEobserved() b = (7) 

B

[∑n + n ] − [n + n ] bw br bw br

b=1 

2 

1 

1 

( ( ) ) 

B

b

B

seCERATEobserved repCERATEobserved b meanCERATEobserved

B =

−

= ∑ −

Results 

The goal of this research was to determine the feasibility of applying MCMC techniques to model correct 

enumeration and (eventually) match rates. To assess this, model-based mean and standard error coverage intervals 

of the CE rate from (6) were compared to the mean and standard error of the observed sample from (7). One of the 

key aspects in developing the model was determining if the assumption that the random block effects had a normal 

distribution from (4) was accurate. The following table sets compare the model-based mean and standard error 

coverage intervals to the mean and standard error from the observed sample. This section provides results for eight 

states in each of the four census regions. Recall that, within each state, only a subset of blocks was taken to model 

CE rate variability. As a result, model results may not be illustrative of the whole state. 
 

 

 

 

 

 

 

 

 

 

 

 

 

Model 1: No Random Block Effect 

Initially, the challenge was to justify the inclusion of a random block effect in the model to account for 

heterogeneity between blocks. To do this, it was necessary to empirically show that omitting a random block effect 

would not be sufficient to model the observed sample. The results were as follows: 

Table 1.A – CE Rate Coverage Intervals 

Domain Observed Value Coverage Interval 

Lower Bound 

Coverage Interval 

Upper Bound 

Interval Covers 

Observed Value? 

1 0.9476 0.9357 0.9589 Yes 

2 0.9413 0.9341 0.9481 Yes 

3 0.9722 0.9677 0.9764 Yes 

4 0.9447 0.9389 0.9505 Yes 

5 0.9243 0.9103 0.9383 Yes 

6 0.9449 0.9365 0.9533 Yes 

7 0.9369 0.9316 0.9423 Yes 

8 0.9519 0.9446 0.9588 Yes 

Table 1.B – SE(CE Rate) Coverage Intervals 

Domain Observed Value Coverage Interval 

Lower Bound 

Coverage Interval 

Upper Bound 

Interval Covers 

Observed Value? 

1 0.0128 0.0037 0.0072 No 

2 0.0066 0.0028 0.0043 No 

3 0.0033 0.0017 0.0024 No 

4 0.0061 0.0023 0.0030 No 

5 0.0116 0.0058 0.0108 No 

6 0.0072 0.0031 0.0048 No 

7 0.0067 0.0025 0.0033 No 

8 0.0085 0.0027 0.0039 No 

Table 1.A indicates that the model-based coverage intervals for the CE rate cover the observed CE rate. However, 

Table 1.B shows that the model-based coverage intervals for the standard error consistently underestimate the 

observed standard error of the CE rate. As a result, it was determined that modeling the random block effect was 

needed to account for heterogeneity between blocks. 
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Model 2: Normal Random Block Effect

When it was clear that a random block effect would need to be included, the initial thought was to model the random 

block effect with a normal distribution. The results were as follows: 

Table 2.A – CE Rate Coverage Intervals 

Domain Observed Value Coverage Interval 

Lower Bound 

Coverage Interval 

Upper Bound 

Interval Covers 

Observed Value? 

1 0.9476 0.9362 0.9579 Yes 

2 0.9413 0.9344 0.9479 Yes 

3 0.9722 0.9679 0.9763 Yes 

4 0.9447 0.9390 0.9500 Yes 

5 0.9243 0.9103 0.9378 Yes 

6 0.9449 0.9365 0.9530 Yes 

7 0.9369 0.9317 0.9421 Yes 

8 0.9519 0.9450 0.9584 Yes 

Table 2.B – SE(CE Rate) Coverage Intervals 

``Domain Observed Value Coverage Interval 

Lower Bound 

Coverage Interval 

Upper Bound 

Interval Covers 

Observed Value? 

1 0.0128 0.0096 0.0176 Yes 

2 0.0066 0.0058 0.0081 Yes 

3 0.0033 0.0026 0.0041 Yes 

4 0.0061 0.0054 0.0071 Yes 

5 0.0116 0.0084 0.0152 Yes 

6 0.0072 0.0060 0.0092 Yes 

7 0.0067 0.0059 0.0075 Yes 

8 0.0085 0.0067 0.0102 Yes 

Table 2.A indicates that the model-based coverage intervals for the CE rate cover the observed CE rate. Table 2.B 

shows that the model-based coverage intervals for the standard error now cover the observed standard error of the 

CE rate. Because of the results, it can be inferred that the assumption that the random block effects are normally 

distributed was correct. 

Conclusions and Future Work 

This work represents a beginning in studying the feasibility of applying MCMC methods to estimate variance of 

coverage estimates over smaller geographies. Although the initial results above are promising, this model has only 

been applied to a subset of the 2000 A.C.E blocks sampled within each state. It will need to be determined if the 

inclusion of small blocks with less than three households or large blocks with greater than 79 households will 

necessitate a change to the model. 

As mentioned earlier, future work will include modeling match rates using the same paradigm. Furthermore, since 

CE and match rates are thought to be correlated within a block, the future model will have to incorporate that 

relationship. Additionally, a model for data defined rates will be constructed using similar techniques. 

References 

Gelman, A.B., Carlin, J.S., Stern, H.S., and Rubin, D.B. (2000) 

Bayesian Data Analysis. Washington D.C.: Chapman and Hall/CRC. 

U.S. Census Bureau Technical Paper 2004: Accuracy and Coverage Evaluation of Census 2000: Design and 

Methodology.