Statistical data integration using multilevel 

models to predict employee compensation

Andreea L. Erciulescu, Jean D. Opsomer, Benjamin J. Schneider, Westat

2021 FCSM Research and Policy Conference 

November 3, 2021
Contributions

Wage, benefits, and total employee compensation estimates

◮ Bureau of Labor Statistics (BLS)

◮ 242,686 domains defined as geography x occupation

◮ metropolitan statistical areas (MSAs) and balance of state areas

(BOSs); example:

◮ 6-digit standard occupational classification codes (SOC6); example:

◮ SOC2: 15-0000, Computer and Mathematical Occupations

◮ SOC4: 15-2000, Mathematical Science Occupations

◮ SOC6: 15-2041, Statisticians

Statistical data integration methodology

◮ Erciulescu A.L., Opsomer J.D., Schneider, B.J. (2021), “Statistical data integration

using multilevel models.” Under review.

2
Data

National Compensation Survey (NCS)

◮ wage and benefits survey estimates; in $/hr

◮ point estimates: y

NCS

i = (y

NCS

1,i

, y

NCS

2,i

)

◮ variance-covariance estimates, adjusted: Σ

NCS

i

◮ levels: MSA/BOS/census division/nation x SOC6/SOC2/no SOC

◮ variations: original scale, log scale, sum

◮ small sample

Occupational Employment Statistics (OES) program*

◮ wage survey estimates; in $/hr

◮ point estimates: y

OES

1,i

◮ variance estimates, adjusted: (σ

OES

1,i

)

2

◮ levels: MSA/BOS/census division/nation x SOC6/SOC2/no SOC

◮ variations: original scale, log scale

◮ large sample

Prediction space: the set of domains for which there are sample data

available in at least one of the two surveys; May 2019 as reference time

Occupational Employment and Wage Statistics Program - as of spring 2021

3
Need for data integration: distinct wage estimates

Domain-level wage survey estimates, MSA/BOS x SOC6

Two (large) domain-level NCS wage estimates were removed to improve visualization

4
Need for small area estimation: small sample data

Summary of sample sizes of domains in the prediction space, by level of

aggregation; pseudo-effective sample sizes for NCS

Level NCS OES

Minimum Median Maximum Minimum Median Maximum

MSA/BOS x SOC6 0 0 61 0 6 14,826

Census division x SOC6 0 1 191 1 236 68,810

Census division x SOC2 1 49 423 449 11,254 127,475

Nation x SOC6 0 8 796 21 2,272 366,362

Nation x SOC2 7 488 2,208 10,446 112,978 661,453

◮ median NCS sample size is 1 in NCS-only domains and 1 in all NCS

domains

◮ median OES sample size is 5 in OES-only domains and 6 in all OES

domains

Currently, BLS publishes employee compensation statistics at levels of

aggregation defined using either geography or occupation

(https://www.bls.gov/web/ecec/ececrse.htm).

5
Need for data integration and small area estimation:

incomplete sample data

Number of domains in the prediction space, by level of aggregation

Level Prediction Space Subset

NCS-only NCS-and-OES OES-only

MSA/BOS x SOC6 186 19,509 222,991

Census division x SOC6 0 4,358 2,565

Census division x SOC2 0 198 0

Nation x SOC6 0 721 50

Nation x SOC2 0 22 0

◮ small number of domains with benefits estimates

◮ large number of domains with two wage estimates

◮ very large number of domains with wage estimates from only one of

the two sources

6
Hierarchical modeling estimation

Domain-level: MSA/BOS x SOC6-level survey estimates and associated

variance estimates

◮ NCS-only domains (sNCS ), NCS-and-OES domains (sNCS−OES ),

OES-only domains (sOES )

Bivariate: wage and benefits

◮ borrow strength from the strong relationship

Hierarchical Bayes: sampling levels, smoothing (latent) level, prior

distributions

◮ borrow strength across surveys, across domains, and from covariates

◮ covariates xi defined in terms of area type (MSA or BOS), census

division, and their two-way interactions

◮ link the NCS and OES wage estimates

◮ maintain the relationship between wage and benefits

Multi-fold: MSA/BOS x SOC6, SOC6

◮ borrow strength from the nested structure

7
Domain-level bivariate hierarchical Bayes multi-fold model

Sampling Level

y

NCS

i,log |(θi,log , Σ

NCS

i,log ) ∼ N(θi,log , Σ

NCS

i,log ), i ∈ sNCS ∪ sNCS−OES

y

OES

1,i,log |(θ1,i,log , σOES1,i,log ) ∼ N



θ1,i,log ,(σ

OES

1,i,log )

2



, i ∈ sOES ∪ sNCS−OES

Smoothing Level

θi,log |(β, uI, Σb) ∼ N(x

′

i β + uI

, Σb), i ∈ sNCS ∪ sNCS−OES ∪ sOES , i ∈ I

uI|Σu ∼ N(0, Σu), i ∈ sNCS ∪ sNCS−OES ∪ sOES , i ∈ I

Prior Distributions

β ∼ N(0, 104), component-wise

(Σb, Σu) ∼ inverse-Wishart(I2, 3), component-wise

◮ i indexes MSA/BOS x SOC6 domains

◮ I indexes SOC6 domains

8
Model fit, assumptions checks, prediction

Fit

◮ R JAGS

◮ Markov chain Monte Carlo (MCMC): 3 chains, 10,000 samples,

3,000 burn-in, thinning every 10th sample: 2,100 samples for

inference

◮ SOC2-specific: 22 models

Assumptions checks

◮ MCMC diagnostics: Rˆ, MC effective sample size, MC standard error,

autocorrelation

◮ model specification: posterior predictive checks

Prediction

◮ posterior distribution

[θi,log |y

NCS

log , y

OES

1,log , Σ

NCS

log , σOES1,log , x, β, Σb, Σu], i ∈ sNCS∪sNCS−OES∪sOES

◮ transformations: exponential, sum

9
Comparison of NCS and model: point estimates

Domain-level wage and benefits estimates, MSA/BOS x SOC6

10
Comparison of OES and model: point estimates

Domain-level wage and benefits estimates, MSA/BOS x SOC6

11
Comparison of NCS and model: standard errors

Domain-level wage and benefits estimates, MSA/BOS x SOC6

12
Comparison of OES and model: standard errors

Domain-level wage and benefits estimates, MSA/BOS x SOC6

13
Comparison of NCS, OES, and model: coefficients of

variation

Summary of coefficients of variation (%) of compensation estimates for the

MSA/BOS x SOC6 domains in the prediction space

Estimation Approach Wages Benefits Total Compensation

Median % ≥ 30 Median % ≥ 30 Median % ≥ 30

Survey, NCS; adj. s.e. 49 77 90 92 58 83

Survey, OES; adj. s.e. 17 27 N/A N/A N/A N/A

Model, HB 9 0 28 44 11 1

Recall there are 242,686 domains in the prediction space

14
Summary

◮ Methodological developments in statistical data integration, as

extensions to small area estimation

◮ Incomplete survey data on two strongly-related variables

◮ one variable collected on two surveys, the other collected only on the

smaller survey

◮ domains of interest represented by the union of the domains with

sample data available for either variable and from either survey

◮ Complete set of wage, benefits, and total compensation estimates

for all domains of interest, with associated uncertainty measures

◮ granular levels lower than the levels at which current official statistics

are available

◮ Hierarchical model estimates of improved precision, compared to the

survey direct estimates

15
Selected references

Erciulescu, A.L., and J.D. Opsomer. 2019. “Task Order 5: Developing a Small

Domain Estimation Methodology for the Office of Compensation and

Working Conditions: Subtask 9: Final Report.” Report Prepared for Bureau

of Labor Statistics’ Office of Compensation and Working Conditions.

Goodman, L. A. 1960. “On the Exact Variance of Products.” Journal of the

American Statistical Association 55 (292): 708-13.

Guciardo, C. J. 2001. “Estimating Variance in the National Compensation

Survey, Using Balanced Repeated Replication.” Accessed November 9, 2020.

https://www.bls.gov/osmr/research-papers/2001/pdf/st010110.pdf

Hájek, J. 1971. “Comment on a paper by D. Basu.” Foundations of statistical

inference, 236.

Lettau, M. K., and D. A. Zamora. 2013. “Wage estimates by job characteristic:

NCS and OES program data.” Monthly Labor Review, U.S. Bureau of Labor

Statistics, August. https://doi.org/10.21916/mlr.2013.27.

Myers, M., and D. A. Zamora. 2015. “Revisiting the Dilemma of Review for

Modeled Wage Estimates by Job Characteristic.” Monthly Labor Review,

U.S. Bureau of Labor Statistics, September.

https://doi.org/10.21916/mlr.2015.36.

16
Thank you!

AndreeaErciulescu@westat.com

17
JAGS two-fold model specification

17