Spatial Change of Support Models for

Differentially Private Decennial Census Counts of

Persons by Detailed Race and Ethnicity

Andrew Raim1, James Livsey1, Kyle Irimata1, Ryan Janicki1, and

Scott H. Holan23

1Center for Statistical Research and Methodology, U.S. Census Bureau

2University of Missouri

3Associate Directorate for Research and Methodology, U.S. Census Bureau

2022 FCSM Research and Policy Conference

October 27, 2022
Disclaimer

This report is released to inform interested parties of ongoing research

and to encourage discussion of work in progress. The views expressed are

those of the authors, and not those of the U. S. Census Bureau.
Decennial Census

According to Abowd (2018), JASON (2020),

I In 2010, the U. S. Census counted a total population of over 308

million people

I At least 7.7 billion statistics were published from the collected data

I At least 25 published statistics per person

I In 2018, Census succeeded in reconstructing, from published 2010

census data, geographic location, sex, age and ethnicity for 46% of

the U. S. population

I Census was able to link 38% of the reconstructed micro data to

information in commercial databases
Differential Privacy

The U. S. Census Bureau will implement differential privacy for the 2020

Census.

A statistic, T, is (, δ)-differentially private if for any two data sets X and

X

0 differing by a single element and any A in the range of T,

P (T(X) ∈ A) ≤ e

P (T(X0

) ∈ A) + δ

I Laplace noise: (, 0)−differentially private

I Gaussian noise: (, δ)−differentially private
Consequences of differential privacy

Accuracy/privacy tradeoff

I Published estimates will be noisy

I Fewer estimates may be published

Our research goal: use model-based methods to

I Produce estimates which are more precise than those based on

differentially private measurements

I Produce estimates when no differentially private measurement is

available
Notation

Source support

I Let A1, . . . , Am be a set of non-overlapping geographies representing

a “source”: Counties in this example

Target support

I Let B1, . . . , Bn be a second set of geographies representing a

“target”: American Indian and Alaska Native (AIAN) areas in this

example
Notation, continued

I For each area Ai

in the source support we assume access to noisy

measurements Z(Ai) of an unobservable true value Y (Ai), as well as

a set of vector of predictors, x

T (Ai).

I For each area Bj

in the target support, we have only knowledge of a

set of predictors, x

T (Bj). We do not have noisy measurements,

Z(Bj), on the target support.

I Our goal is prediction of the true values, Y (Ai) and Y (Bj), on both

the source support and the target support, using the observations

{Z(Ai)}, and the predictors x

T (Ai)

	

and x

T (Bj)

	

.
Counties and AIAN areas in Oklahoma

33.5°N

34.0°N

34.5°N

35.0°N

35.5°N

36.0°N

36.5°N

37.0°N

102°W 100°W 98°W 96°W

Counties in Oklahoma

33.5°N

34.0°N

34.5°N

35.0°N

35.5°N

36.0°N

36.5°N

37.0°N

100°W 99°W 98°W 97°W 96°W 95°W

AIAN areas in Oklahoma

33.5°N

34.0°N

34.5°N

35.0°N

35.5°N

36.0°N

36.5°N

37.0°N

102°W 100°W 98°W 96°W

Combined couties and AIAN areas in Oklahoma
Statistical modeling framework

For a general region, A, the observed noisy measurements Z(A) satisfy

Z(A) = Y (A) + ε(A),

where ε(A) is a draw from a known differentially private distribution and

Y (A) is the unobservable true count.

Assume the true counts, Y (A), can be aggregated from a point-level

process

Y (A) = Z

A

Y (s)ds,

Further assume that the point-level process can be decomposed as

Y (s) = μ(s) + γ(s)

where μ(s) represents the fixed effects, and γ(s) represents the random

effects, which account for spatial dependencies and residual variation.
Fixed effects

The model for the fixed effects, μ(s), is

μ(s) = x

T

(s)β,

so that

μ(A) = Z

A

μ(s)ds =

Z

A

x

T

(s)βds = x

T

(A)β.

In our examples, x includes

I An intercept

I The count from the previous census

I The American Community Survey (ACS) 5-year estimate
Random effects

We use a basis expansion for the random process, γ(s) (Cressie and

Johannesson, 2008; Bradley et al., 2017):

γ(s) = X∞

k=1

ψk (s)ηk ≈

Xr

k=1

ψk (s)ηk + ξ(s),

where {ψk (s)} is a collection of spatial basis functions, and ηk are

independent, mean-zero Gaussian random variables and ξ(s) is a residual

random effect.

γ(A) = Z

A

 Xr

k=1

ψk (s)ηk + ξ(s)

!

ds =

Xr

k=1

ψk (A)ηk + ξ(A).
Random coefficients

We assume a multivariate normal distribution for the random effects:

η = (η1, . . . , ηr)

T ∼ Nr



0, σ2

ηK



I K is a known covariance matrix, constructed to induce spatial

dependencies using a conditional autoregressive structure (Hughes

and Haran, 2013)

I σ

2

η

is an unknown parameter
Construction of K

K is constructed to

I induce spatial dependencies

I reduce rank compared to a conditional auto regressive process

Let

I PX = I − X



XTX

−1

XT

I A the adjacency matrix for counties

I S the first r eigenvectors of PX APX
Construction of K, continued

Let u

T = (u1, . . . , um) be an intrinsic conditional autoregressive process,

with precision matrix 1

σ2 Q, so that

ui| uj, j 6= i, σ2 ∼ N

 X

j∼i

uj

ni

,

σ

2

ni

!

,

where niis the number of neighbors of area i. Then

K = arg minC





Q − SC−1S

T







F

,

where the minimization is over the space of r × r positive definite

matrices.
Basis functions

Bisquare basis functions

ψk (s) = 1 −

ks − cjk

2

w2

!2

I (ks − cjk < w).

I (c1, . . . , cr) is a collection of equally-spaced knots

I w is 1.5 times the minimum distance between any two knots

The integral

ψk (A) = Z

A

ψk (s)ds

is approximated numerically for each region A. (Bradley et al., 2017;

Pebesma, 2018; Raim et al., 2021)
Source support model

Data model:

Z(Ai) = Y (Ai) + ε(Ai)

Process model:

Y (Ai) = x

T

(Ai)β +

Xr

k=1

ψk (Ai)ηk + ξ(Ai), i = 1, . . . , m,

η = (η1, . . . , ηr)

T ∼ Nr



0, σ2

ηK



, ξ (Ai)

i.i.d. ∼ N



0, σ2

ξ



Parameter model:

β ∼ Np (0, 10Ip×p), σ2

η ∼ IG(1, 1), σ2ξ ∼ IG(1, 1)

This model can be fit using a Gibbs sampler (Choi and Hobert, 2013).
Change of support

The true count, Y (Bj), in area Bj can be estimated using

Yˆ (Bj) = E



Y (Bj) | {Z(Ai)}

n

i=1

, x(Bj)



.

The covariates x

T (Bj) are assumed known, and the basis functions

ψk (Bj) are approximated using numerical integration. The distribution of

[Y (Bj) | Z] = Z

θ

[Y (Bj) | θ] [θ | Z] dθ

can be approximated using the output of the Gibbs sampler for fitting the

source support model.
Example: estimation of the number of Choctaw persons in

counties and AIAN areas in Oklahoma

I Let Y (Ai) be the Census 2010 count of the number of Choctaw

persons in county i in Oklahoma

I Let Y (Bj) be the Census 2010 count of the number of Choctaw

persons in AIAN area j in Oklahoma

I x(s) includes an intercept, the Census 2000 count, and the 2009

ACS 5-year estimate.

I We generate Z(Ai) = Y (Ai) + ε(Ai), where ε(Ai)

i.i.d. ∼ Lap(48)

I 1000 data sets were created

I Our goal is estimation of Y (Ai) and Y (Bi) from the observed Z(Ai)
25°N

30°N

35°N

40°N

105°W 100°W 95°W 90°W

V1

V2
Estimation of the number of Choctaw persons in counties

in Oklahoma

MOD DP

RMSE 38 68

RMAE 4.5 10.1

MAX 241 346

Coverage 95% 95%

The metrics used are

RMSE =

vuut

1

m

Xn

j=1



Yˆ

j − Yj

2

, RMAE =

1

m

Xm

j=1

 

| Yˆ

i − Yi

|

Yi

!

.

MAX = max

i=1,...,m







Yˆ

j − Yj






Estimation of the number of Choctaw persons in counties

in Oklahoma

0

2

4

6

8

Log Count

Log of 2010 Census count of Choctaw persons in counties in Oklahoma

0

2

4

6

8

Log Count

Log of predicted number of persons in counties in Oklahoma

0

2

4

6

8

Log Count

Log of DP measurements of number of persons in counties in Oklahoma
Estimation of the number of Choctaw persons in AIAN

areas in Oklahoma

MOD AREAL

RMSE 145 1064

RMAE 0.35 1.97

MAX 592 5631

Coverage 91% NA

Comparison of model-based predictions with simple proportional

allocation (Prener et al., 2019)

Yˆ

j =

Xn

i=1

Z(Ai)

|Bj ∩ Ai|

|Bj|
Estimation of the number of Choctaw persons in AIAN

areas in Oklahoma

0.0

2.5

5.0

7.5

10.0

Log Count

Log of 2010 Census count of Choctaw persons in AIAN areas in Oklahoma

0.0

2.5

5.0

7.5

10.0

Log Count

Log of predicted number of Choctaw persons in AIAN areas in Oklahoma

using the change of support model

0.0

2.5

5.0

7.5

10.0

Log Count

Log of predicted number of Choctaw persons in AIAN areas in Oklahoma

using areal interpolation
Questions which need to be addressed

I Sensitivity analyses

I Should we include additional covariates?

I Class of basis functions

I How many basis functions to use?

I Choice of tuning parameters

I How many geographic regions to include in the model?

I Log transformation gives a better fit to some data sets, but doesn’t

allow for change of support

I Test on other data sets
References I

John M. Abowd. The U. S. Census Bureau adopts differential privacy. In

Proceedings of the 24th ACM SIGKDD International Conference on

Knowledge Discovery & Data Mining (KDD ’18), New York, NY, USA,

2018. Association for Computing Machinery. URL

https://doi.org/10.1145/3219819.3226070.

Jonathan R. Bradley, Christopher K. Wikle, and Scott H. Holan.

Regionalization of multiscale spatial processes by using a criterion for spatial

aggregation error. Journal of the Royal Statistical Society, Series B, 79:815 –

832, 2017.

Hee Min Choi and James P. Hobert. Analysis of mcmc algorithms for bayesian

linear regression with laplace errors. Journal of Multivariate Analysis, 117:

32–40, 2013. ISSN 0047-259X. doi:

https://doi.org/10.1016/j.jmva.2013.02.004. URL https:

//www.sciencedirect.com/science/article/pii/S0047259X13000183.

Noel Cressie and Gardar Johannesson. Fixed rank Kriging for very large spatial

data sets. Journal of the Royal Statistical Society, Series B, 70:209 – 226,

2008.

J. Hughes and M. Haran. Dimension reduction and alleviation of confounding

for spatial generalized linear mixed models. Journal of the Royal Statistical

Society, Series B, 75:139 – 159, 2013.
References II

JASON. Formal privacy methods for the 2020 Census. Technical report, The

MITRE Corportation, 2020.

Edzer Pebesma. Simple Features for R: Standardized Support for Spatial

Vector Data. The R Journal, 10(1):439–446, 2018. doi:

10.32614/RJ-2018-009. URL https://doi.org/10.32614/RJ-2018-009.

Prener, Christopher, Revord, and Charles. areal: An R package for areal

weighted interpolation. Journal of Open Source Software, 4(37), 2019. doi:

10.21105/joss.01221. URL https://doi.org/10.21105/joss.01221.

Andrew M. Raim, Scott H. Holan, Jonathan R. Bradley, and Christopher K.

Wikle. Spatio-temporal change of support modeling with R. Computational

Statistics, 36:749 – 780, 2021.
Thank You!

I ryan.janicki@census.gov