Estimating the Measurement Error in the Current Population Survey Labor Force–A Latent Class Analysis Approach 

With Sample Design1

Bac Tran, Justin Nguyen 

U.S. Census Bureau, 4700 Silver Hill Road, Washington, D.C. 20233-1912

Bac.Tran@census.gov, Justin.D.Nguyen@census.gov

Key Words: Latent Class Models, Panel Data, Unemployment,

Rotation Group, Month-in-Sample, Classification Probability, 

Interview, Census Bureau. 

I. Introduction

The Current Population Survey (CPS) is a U.S. national household

survey conducted by the U.S. Census Bureau for the Bureau of

Labor Statistics. It is designed to generate national and state-level 

estimates of labor force characteristics such as: employed (E), 

unemployed (UE) and not in the labor force (NILF); demographic

characteristics; and other characteristics of the noninstitutionalized civilian population. Previous papers (Biemer 

2000, Tran 2003, Tran 2004) applied traditional first-order Latent

Markov models to estimate measurement error in CPS labor force.

However, those models could not deal with the unobserved 

heterogeneity that meant that there were groups of sample persons 

having different transition and error probabilities. Furthermore, 

the CPS sample design was not taken into account in the analysis. 

This resulted in overestimating the measurement error by a

substantial amount. This conclusion is supported by both our 

empirical analysis of the complexity of CPS data as well as by our 

simulation results. Also, the analysis showed that month-insample 1 has more measurement error in estimating the 

unemployment rate than the other months in sample (2-8). Past 

research (Causey, 1976) indicated that month-in-sample 1 likely 

produces less bias in estimating the labor force. 

This paper will present a validation of applying Latent Markov

models to estimate measurement error by a thorough simulation. 

This paper introduces the Mover-Stayer Latent Markov model (a

mixture latent Markov model) and its application to the CPS data

in order to estimate measurement error for the labor force status. 

The analysis used LatentGold4.5, software developed by

Statistical Innovations, to implement the model estimation.

The CPS uses addresses from the most current U.S. Census, 

adding new construction, as the frame. The total sample size is

about 72,000 assigned households per month. The CPS uses a 4-

8-4 rotating panel design, i.e. 4 months in, 8 months out, and 4

months in. In the CPS, the same respondents are interviewed at

several points following the pattern 4-8-4. For any given month, 

the CPS sample is grouped into eight sub-samples corresponding

to the eight rotation groups. 

This report is released to inform interested parties of research 

and to encourage discussion. Any views expressed 

on statistical, methodological, technical, or operational issues are those 

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

II. Mover-Stayer Latent Markov Model- A Mixture

Latent Markov Model 

Latent Class Analysis (LCA) treats the true classification of the

labor force status as an unobserved variable. The observed

variables (A, B, C, D: labor force for four consecutive months in

our study, see Figure 1) obtained from the CPS survey in a panel

survey are fallible indicators of the latent variable X. LCA 

suggests a relationship between observed variables and latent

variables through a mathematical equation. Under the equation 

the table of observed data is viewed as a partial table from a full 

table of observed and unobserved data. The Markov assumption is

employed (see below), hence the so-called Markov Latent Class 

Analysis (MLCA). With the homogeneity in the distribution of

the labor force in the population we have one Markov chain, as in 

a traditional first-order Markov chain. With heterogeneity we 

have more than one chain. A simple case is two chains as in our 

study, the Mover-Stayer model. 

As mentioned above, a small portion of the population that 

becomes unemployed and stays unemployed for a long time could

violate the Markov assumption. First-order Markov models

cannot deal with the heterogeneity of the underlying latent class. 

Before presenting the Mover-Stayer Markov model (M-S model)

we would like to show the traditional first-order Markov model 

(MLC). The MLC has the following form (Van De Pol and 

Langeheine, 1990): 

where T +1 is the number of time points (0 ≤ t ≤ T), yi

is response

vector of i

th observation of length T+1 (in our application T=3), yit

is the tth component of yi

, xt denotes a possible value of a latent

variable at time t where xt = 1, 2, .., K (in our study K=3, 

representing the three categories of labor force Employed, 

Unemployed, and Not In Labor Force which are abbreviated by E, 

U, and NILF). The Markov assumption states that 

P (xt

| xt-1)= P (xt| xt-1, xt-1, ..., x0). 

The assumptions of the first-order Markov Latent Class in the 

equation (1) are:

1. xt

is independent of xt-2, xt-3, ..., x0.

2. There is no unobserved heterogeneity. 

3. Classification errors are independent across time points. 

The unknown model probabilities to be estimated are: 

P (x0): initial latent state probabilities, 

P(xt

| xt-1 ): transition probability, and

P(yit|xt): classification error probabilities. 

They are parameterized as follows: 

1
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

where αs

, γ

t

rs, and βrs are parameters from the logit models for 

P(x0=s), P(xt=r| xt-1 =s) and P(yit=m| xt=s) respectively.

The Mover-Stayer model, a two-class mixed Markov Latent Class 

model, assumes that there are two unobserved subgroups with 

different transition probabilities. The M-S model has the form

2 K K T

P( yi) = ... P(w)P(x | w) P(x | x ,w) P( y | xt ∑∑∑ ∑ ,w) (2) 0 ∏ t t−1 ∏ it

w=1 x =1 x =1 x =1 0 1 T

where w=1 or 2 denotes two classes of latent variables and w=2

represents the stayer class, i.e.

for r ≠ s, otherwise 

In our study we assume classification error is time homogeneous. 

Therefore, in the parameterization equations there is no t term for 

the βs. The path diagram for model (2), applied to the CPS labor 

force with four consecutive time periods, looks like below.

Figure 1 

X01 X11 X21 X31

W Ai

 Bj Ck Dl 

X02 X12 X22 X32

The latent variable W represents two groups of sample persons 

(Unemployed and the other). The Xs are latent variables represent

the true labor force status at four time points. A, B, C, D are the

observed labor force variables used as the indicators of the Xs. 

III. Validation Process 

The main purpose of the use of MLCA is to estimate classification 

errors, and then estimate the true labor force status distribution

under the correct model. We validated the method through a

simulation. Simulate data was created from CPS labor force data 

collected over time. Basically, we considered three different 

model-type settings: 

1. Traditional first-order homogeneous Markov

2. Mixture model with violation of homogeneity

3. Second-order Markov model violating the Markov

assumption 

The three settings will answer the question on how sensitive 

estimates from Markov Latent Class models are when certain 

model assumptions are violated. The first and second models

were presented above. The third Markov model has the form: 

(3) 

The simulate data sets were created based on seven sets of

transition probabilities and three sets of misclassification 

probabilities: 

1. One set is in agreement with a homogeneous first-order

Markov process. 

2. Three sets are in agreement with a second-order Markov

process, where the three sets represent small, moderate, 

and large violations of the first order Markov assumption. 

3. Three sets are in agreement with a Mover-Stayer process, 

where the three sets represent small, moderate, and large 

violations of the homogeneous Markov assumption.

Three sets of misclassification errors correspond to the conditions

of small, moderate, and high proportions of misclassification.

These three conditions are based on the summary presented by

Tran and Winters (2003). In total, we have 21 design cells and

generated 1,000 replicates for each cell. Each replicate contains 

eight rotations, each rotation with about 6,000 sample units. 

Therefore, each replicate has about 48,000 sample units. For each

of the data sets, the parameters of setting # 1 were estimated.

When this is not the correct model, the correct model is also 

estimated. Depending on the design cell, this is either the MoverStayer model or the second-order Markov LC model (settings #2

or 3). 

The four questions we want to answer with the simulation study

are: 

1. Is it is possible to detect whether model assumptions are 

violated?

2. Are the estimated misclassification probabilities unbiased

when the correct model is specified?

3. Are the estimated misclassification probabilities biased 

when model assumptions are violated when an incorrect 

model is specified?

4. Are the estimated class sizes biased when model 

assumptions are violated when an incorrect model is

specified?

The results from the simulation give the answers as follows: 

1. Yes, it is possible to detect that model assumptions are

violated, but only for large violations. 

2. Yes, estimates of the misclassification probabilities are 

unbiased when the right model is specified. 

3. Yes, there is an upward bias in the estimates of the 

misclassification probabilities, but it is surprisingly small.

Only with a very extreme (and unrealistic) second-order 

process do we see substantial bias in the estimated
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

misclassification probabilities obtained with an incorrect

first-order model. 

4. Yes, estimates of the class sizes are biased downwards. 

With weak violations, this bias is negligible. 

IV. Application to the CPS Data 

We used CPS data from June 2006 to September 2006. The labor

force status for four months was used as indicators of the latent

variable X. The survey variables used were rotation, proxy, and

interview mode. The grouping variable used was SEX 

(Male/Female). As in Tran (2004) the rotation variable was 

collapsed into two categories, rotation 1 versus the other rotations

(2-8). The proxy variable had two categories, self and proxy. 

Variable MODE contained CATI, CAPI personal visit, and CAPI

telephone.

We ran eight different models that are summarized as follows: 

Table 1

Model LL/BIC(LL)/#parameters/ 

 dof 

pvalue 

1 X0, X1|X0, X2|X1, X3|X2, A|X0, 

B|X1, C|X2, D|X3 

-215234.214 

430785.640 

26 

216 

0 

2 X0|S,

 X1|X0S, X2|X1S, X3|X2S,

A|X0, B|X1, C|X2, D|X3

-212863.559 

426141.936 

34 

450 

0 

3 X0, X1|X0, X2|X1, X3|X2, 

A|X0PRM{AX0,AP,AR,AM} 

B|X1PRM{BX1,BP,BR,BM} 

C|X2PRM{CX2,CP,CR,CM} 

D|X3PRM{DX0,DP,DR,DM} 

-214718.750 

430047.5254 

50 

106584 

1 

4 X0|S,

 X1|X0S, X2|X1S, X3|X2S 

A|X0PRM{AX0,AP,AR,AM} 

B|X1PRM{BX1,BP,BR,BM} 

C|X2PRM{CX2,CP,CR,CM} 

D|X3PRM{DX0,DP,DR,DM} 

-212323.593 

425354.8146 

58 

179026 

1 

5 W X0|W,

 X1|X0W, X2|X1W,

X3|X2W, A|X0, B|X1, C|X2, D|X3

-215099.933 

430553.6795 

29 

213 

0 

6 W|S X0|WS,

 X1|X0WS, X2|X1WS, 

X3|X2WS, A|X0, B|X1, C|X2, D|X3 -212695.790 

425879.6005 

40 

444 

0 

7 W X0|W,

 X1|X0W, X2|X1W,

X3|X2W,

A|X0PRM{AX0,AP,AR,AM} 

B|X1PRM{BX1,BP,BR,BM} 

C|X2PRM{CX2,CP,CR,CM} 

D|X3PRM{DX0,DP,DR,DM} 

-214593.312 

429833.2497 

53 

106581 

1 

8 W|S X0|WS,

 X1|X0WS, X2|X1WS, 

X3|X2WS 

A|X0PRM{AX0,AP,AR,AM} 

B|WX1PRM{BX1,BP,BR,BM} 

C|WX2PRM{CX2,CP,CR,CM} 

D|X3PRM{DX0,DP,DR,DM} 

-212149.942 

425080.7164 

64 

179020 

1 

Notes: 

CATI: Computer Assisted Telephone Interviewing

CAPI: Computer Assisted Personal Interviewing

X0, X1, X2, and X3: latent variables for labor force 

W: latent mover-stayer variable 

A, B, C, D: labor force indicators for June – September 2006

P: Proxy (Proxy/Self) 

R: Rotation (1 versus 2-8), rotation 1 is not conducted in CATI 

M: Mode (CATI, CAPI personal visit, CAPI telephone) 

We used LatentGold4.5 (LG4.5), software developed by Statistical 

Innovations (2007), to estimate the model parameters. All the 

models were identifiable. We use the following criteria to identify

a good model. 

• The model is identifiable. 

• The p-value of the likelihood ratio p-value should be

greater than 0.01. 

• The Bayesian information criterion (BIC), defined as L

2

– 

log (N) degrees of freedom, should be the smallest among

all competing models. 

Based on those criteria model, the best model for these data is

model 8. 

V. Classification Probabilities 

We compared our estimates of the CPS classification probabilities

with similar estimates obtained from previous papers (Biemer & 

Bushery 2000, Tran & Winters 2003, and Tran & Mansur 2004). 

The results are summarized in Table 2. 

The correct classification probability Pr(observed= Unemployed| 

True = Unemployed)) for the Unemployed category is estimated 

as 79.16 percent. This figure is close to 81.81 percent, as found in

Biemer and Bushery (1993 data). However, in this study we use 

the sample design with weight (see VI). The estimates and their 

standard errors are included in Table 2. Below we give the

probabilities Pr(response| Proxy, Rotation, Mode, True status) 

Output 1 

Probability 

P R M X E U NILF 

2 1 2 1 0.9822 0.0094 0.0084 

2 1 2 2 0.0385 0.7666 0.1950 

2 1 2 3 0.0388 0.0240 0.9373 

1 1 3 1 0.9961 0.0026 0.0013 

1 1 3 2 0.1464 0.7657 0.0879 

1 1 3 3 0.0063 0.0064 0.9873 

2 1 3 1 0.9901 0.0039 0.0059 

2 1 3 2 0.0895 0.7127 0.1978 

2 1 3 3 0.0283 0.0135 0.9582 

1 2 2 1 0.9900 0.0064 0.0035 

1 2 2 2 0.0237 0.8557 0.1206 

1 2 2 3 0.0041 0.0068 0.9891 

2 2 2 1 0.9739 0.0096 0.0165 

2 2 2 2 0.0134 0.7359 0.2507 

2 2 2 3 0.0187 0.0145 0.9668 

1 1 2 1 0.9920 0.0062 0.0018 

1 1 2 2 0.0647 0.8462 0.0891 

1 1 2 3 0.0087 0.0115 0.9798 

2 2 3 1 0.9842 0.0041 0.0117 

2 2 3 2 0.0320 0.7056 0.2623 

2 2 3 3 0.0135 0.0081 0.9784 

1 2 3 1 0.9948 0.0027 0.0025 

1 2 3 2 0.0565 0.8177 0.1258 

1 2 3 3 0.0030 0.0038 0.9933 

2 2 1 1 0.9827 0.0062 0.0111 

2 2 1 2 0.0442 0.7880 0.1678 
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 2 2 1 3 0.0207 0.0211 0.9582 

1 2 1 1 0.9935 0.0041 0.0023 

1 2 1 2 0.0728 0.8521 0.0751 

1 2 1 3 0.0046 0.0100 0.9854 

Looking at the correct classification for the Unemployed category

alone extracted from Output 1, we have:

P R M Pr(response=UE|X=UE,P,R,M) 

1 1 2 0.8462 

1 1 3 0.7657 

1 2 1 0.8521 

1 2 2 0.8557 

1 2 3 0.8177 

2 1 2 0.7666 

2 1 3 0.7127 

2 2 1 0.7880 

2 2 2 0.7359 

2 2 3 0.7056

We see that, in terms of measurement error, self-reporting is better 

than proxy reporting. If considering self-reporting as more 

accurate, then month-in-sample 2-8 has less measurement error 

than month-in-sample 1. This also means that month-in-sample 

one overestimates the unemployment rate more than the other 

months in sample. 

Table 2: Classification Probabilities 

Classification Previous Estimates Current 

Estimate/

(s.e) 

True 

(estimated)

Reported Biemer&

Bushery 

MLCA

Tran& 

Winters 

MLCA 

(1996-

1999)

Tran& 

Mansur 

LCA 

(Jan2002-

Dec 2003)

Tran 

&Nguyen 

MoverStayer 

(June06-

Sep06) 

EMP EMP 98.77 (1993)

98.73 (1995)

98.73 (1996)

98.74 97.35 98.89 

(0.1) 

UE 0.34 (1993)

0.49 (1995)

0.37 (1996)

0.37 0.35 0.51 

(0.06)

NILF 0.89 (1993)

0.78 (1995)

0.79 (1996)

0.89 2.29 0.60 

(0.08)

UE EMP 7.06 (1993)

7.86 (1995)

8.57 (1996)

9.87 11.39 

5.01 

(1.44)

UE 81.81 (1993)

76.09 (1995)

74.42 (1996)

71.38 71.51 79.16 

(1.82)

NILF 11.13 (1993)

16.04 (1995)

17.00 (1996)

18.75 17.10 15.83 

(1.74)

NILF EMP 1.41 (1993)

1.11 (1995)

1.13 (1996)

1.26 8.93 1.15 

(0.12)

UE 0.75 (1993)

0.69 (1995)

0.87 (1996)

0.72 1.85 1.06 

(0.09)

NILF 97.84 (1993)

98.20 (1995)

98.00 (1996)

98.03 89.22 97.79 

(0.15)

VI. Sample Design With Weight 

A sample unit, taken from a finite population, in this research is 

identified by an identification code and the following four

characteristics: 

1. Stratum

2. Primary Sampling Unit (PSU) 

3. Sampling weight

The model parameters of a mixture latent Markov model are 

estimated by means of pseudo-Maximum Likelihood (PM)

estimation (Skinner, Holt, and Smith, 1989) through the 

linearization variance estimator. Readers will find the technical 

details in the Technical Guide from Vermunt & Magidson (2007). 

VII. Limitation

There are limitations when using maximum likelihood procedure 

with missing values: The procedure can deal with missing values

on response variables, but not with missing values on covariates, 

and it assumes that the missing data are missing at random

(MAR). The data prepared for this study were four-consecutive 

month data (June 2006 to September 2006) from the CPS. We

need to use more data to fit the model. This study applied a 

simple Mover-Stayer model in which there were two classes, 

mover and stayer. There could be more than two classes. The 

weight we used for the analysis was the average of the second

stage weights from four-month CPS data. We need to figure a 

weighting scheme that is better than the averaging.

VIII. Conclusion 

This paper utilized the suggestion from Tran (2003) that 

heterogeneity of the underlying class of the labor force is

necessary for the model. That idea leads to the investigation of 

Mover-Stayer Markov Latent Class models. This study validated 

the method by an extensive simulation, while Tran 2003

simulation was just a partial case. Furthermore, the sample design

with weight is taken into consideration for this paper. With these 

features and investigations we are able to estimate the 

measurement errors by using Markov Latent Class Analysis, 

specifically Mixed Markov Latent Class Analysis. 

References 

Agresti, A. (2002) Categorical Data Analysis, New York: Wiley 

Bailar, B.A.(1975). “The Effect of Rotation Group Bias on

Estimates from Panel Surveys,” Journal of the American 

Statistical Association, Vol.70, pp. 23-30. 

Biemer, P.P (2004). “The Twelfth Morris Hansen Lecture Simple 

Response Variance: Then and Now”, Journal of Official

Statistics, 20 (3): 417-439

Biemer, P. and Wiesen, C. (2002). “Measurement Error 

Evaluation of Self-Reported Drug Use: A Latent Class Analysis of 
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

the US National Household Survey on Drug Abuse,” Journal of

Royal Statistical Society, Part 1, 165, pp. 97-119. 

Biemer, P. and Bushery, J. (2000). “On the Validity of Markov

Latent Class Analysis for Estimating Classification Error in Labor 

Force Data,” Survey Methodology, Vol. 26, No. 2, pp. 139-152.

Causey, B. (1976). Draft for the Record, Internal Memorandum,

dated June 9, 1976, “Findings on CPS Rotation Group Bias,”

Statistical Research Division, Bureau of the Census, Washington, 

D.C. 

Goodman, L.A. (1961). “Statistical Methods for the Mover-Stayer

Model,” Journal of the American Statistical Association, 81, 354-

365 

Hagenaars, J.A, and McCutcheon A.L. (2002). Applied Latent

Class Analysis. Cambridge University Press. 

Little, R.J., and Rubin, D.B. (1987). Statistical Analysis With

Missing Data. New York: Wiley. 

Mansur, K. and Shoemaker, Jr. H. (1999). “The Impact of

Changes in the Current Population Survey on Time-in-Sample 

Bias and Correlations between Rotation Groups,” Proceedings of

the Section on Survey Methods, American Statistical Association, 

pp. 180-183. 

McCutcheon, A.L. (1987). Latent Class Analysis. Newbury Park, 

CA: Sage. 

Skinner, C.J., Holt, D., and Smith, T.M.F. (eds.) (1989), Analysis 

of Complex Surveys, New York: Wiley. 

Shockey, J. (1988). “Adjusting of Response Error in Panel

Surveys, A Latent Class Approach,” Sociological Methods and 

research, Vol.17, No.1, August, pp. 478-488. 

Tran, Bac and Winters, Franklin (2003). “Markov Latent Class 

Analysis and Its Application to the Current Population Survey in

Estimating the Response Error,” 2003 Proceedings of the 

American Statistical Association, Statistical Computing Section 

[CD-ROM], Alexandria, VA: American Statistical Association. 

Tran, Bac and Mansur, Khandaker (2004). “Analysis of the

Unemployment Rate in the Current Population Survey- A Latent

Class Approach,” 2004 Proceedings of the American Statistical

Association, Statistical Computing Section [CD-ROM], 

Alexandria, VA: American Statistical Association.

Van De Pol, F., and De Leeuw, J. (1986). “A Latent Markov 

Model to Correct for Measurement Error,” Sociology Method and

Research, 15, 118-141

Van De Pol, F., and Langeheine, R. (1990). “Mixed Markov

Latent Class Models,” Sociology Methodology, 213-247

Vermunt, J. (1997). Log Linear Models for Event Histories. 

Thousand Oaks: Sage 

Vermunt, J.K., Langeheine, R., and Bockenholt, U. (1999). 

“Latent Markov Models With Time-Constant and Time-Varying

Covariates,” Journal Education and Behavioral Statistics, 24,

178-205

Vermunt, J.K. (2003). “Multilevel Latent Class Models,” 

Sociology Methodology, 33, 213-239

Vermunt, J.K. and Magidson, J. (2007) Technical Guide to Latent

Gold 4.5. Belmont Massachusetts: Statistical Innovations Inc.