August 16, 1994

LOGISTIC REGRESSION ANALYSIS OF CPS OVERLAP SURVEY SPLIT PANEL

 DATA

Robin Fisher

Robin Fisher, Demographic Statistical Methods Division, Bureau of the Census, Washington,

DC 20233

KEY WORDS: Complex Survey Data, CPS

The complex nature of survey data changes the distributions of statistics associated with logistic

regression models. The variances in the traditional logistic regression models are generally too

small, leading to overly liberal tests. Roberts, Rao, Kumar (1987) provide a method for

constructing approximate tests when a separate estimator for the covariance matrix is available. 

I apply this method with some modifications and the variance estimator described in Fisher et al

(1993) to the CPS Survey split panel data.

I. Introduction

The Census Bureau undertook an investigation (Thompson (1994)) into the effects of

some changes in interviewing methods on Labor Force (LF) estimates. A part of the

study was focused on whether there is an effect on LF estimates from the use of personal

computers during interviewing by telephone (CATI) vs. traditional interviewing, in

person on paper (PAPI). We will refer to this as a combined centralization and

automation effect. In this paper I apply a logistic regression analysis to get more

information about the combined centralization and automation effect. I was also

interested in gaining understanding of the performance of this technique with CPS and

related data. In that context, this paper should be viewed as a first step in an attempt to

implement logistic regression techniques in these analyses. This approach is in contrast

to that of Thompson (1994), where the effect was tested separately in each of several

subgroups.

The problem with the usual logistic regression is that the complex nature of the survey

means the distributions of the estimators of the parameters is different than we would

expect. The variances and p-values in the traditional logistic regression are generally too

small. One possibility is to expand the model to explicitly incorporate the features of the

complex sample design. This approach can be daunting in the face of a design as

complicated as CPS. An alternative is to separately estimate the variation of pertinent

statistics and incorporate that knowledge into the testing procedure. Two notable

approaches are the bootstrap and related methods, where the test statistic may be

replicated with a jackknife or bootstrap (see, e.g. Fay (1985)), and methods which use
2

completely separate estimates of variance to adjust the test statistics or get

approximations for the sampling distributions. Rao and Scott (1981, 1984) and Bedrick

(1983) provide examples of the latter method for getting approximate tests in

contingency tables. Different statistics are given depending on whether the whole

covariance matrix for cell probabilities can be consistently estimated or just the

variances. When only variance estimators are available, they give simple "first-order"

adjustments to the Pearson chi-squared test statistic (X2) and the likelihood ratio statistic

(G2). When an estimator for the whole covariance matrix is available, they gave

adjustments to these statistics based on the Satterthwaite approximation to a weighted

sum of X2 random variables as well as a Wald statistic. They apply these tests to the

Canadian Health Survey. Thomas and Rao (1987) and Thomas, Singh, and Roberts

(1991?) showed the Wald statistic can be unstable when the number of cells is large and

the number of sample clusters is small. Fay's jackknifed tests and the Rao-Scott

connections performed well, however.

Roberts, Rao, and Kumar (1987) (RRK) assume a logistic regression model for cell

proportions and provide corrections to the X2

 and G2 goodness of fit statistics analogous

to the Rao-Scott adjustments described above, a Wald statistic, and an F statistic. They

further give the statistics for testing nested hypotheses. Rao, Kumar, and Roberts (1989)

extend the results, giving weighted least squares estimators for generalized linear models

with singular covariance matrices along with a Wald statistic. Nguyen and Alexander

(1989) consider tests of independence of hierarchial log-linear models when cell and

marginal design effects are fixed. Upton (1991) discusses exploratory data analysis for

survey data using log-linear models. Graubard and Korn (1991) present another

approach using BRR. Choi and McHugh (1989) use modifications to a chi-squared

statistic based on modeling and estimation of correlations in each stage of multi-stage

surveys. Shimizu and Choi (1992) and Choi (1992) apply the technique to particular

survey applications. Thomas, Singh, and Roberts (1991) consider the power of tests for

RxC tables for two-stage cluster sampling.

There have been some efforts to compare different approaches. Thomas, Singh, and

Roberts (1991?) present a Monte Carlo study to compare the approaches of Rao-Scott

(1981 and 1984) and Fay (1985) for small samples. Parsons (1992) compares some of

the common approaches on surveys of the National Center for Health Statistics, including

various simplifications of the design structure used in the derivations of test statistics. 

The most extreme example is to assume simple random sample variances. Carlson,

Cohen, and Monheit (1992) explore alternative pieces of software on the basis of

computational considerations. Their study includes SUDAAN, RTILOGIT, SAS PROC

LOGIT, and SURREGR. SUDAAN (eg. Shah (1989)) in particular is

computationally expensive.

Section II gives some background about the split panel study as it relates to this paper. 

In Section III I describe the use of GVFs and a simple modelling of the covariance
3

matrix for the logistic regression model of RRK. Section IV describes the results. 

Finally I conclude in Section V.

II. Background

Sample Design

The CPS is a monthly survey of 60,000 eligible households. These households are

selected to represent the population of the Nation and of each State. The probability

sample of housing units is drawn using a multistage stratification procedure. The

sampled households are located in 729 selected geographic areas. The largest

metropolitan areas within each State are always included; the remaining areas of a State

are sampled with probability of selection proportionate to the population of the area1. 

The sample is designed to provide a 1.8 percent monthly coefficient of variation2 on the

estimated national unemployment rate, assuming a 6 percent rate. It was also designed to

meet specific reliability criteria for the monthly estimates of unemployment for 11

States; the remaining 40 have fixed levels of reliability for an annual average. At the

national level, this means that a month-to-month change of 0.2 percentage point in the

estimated unemployment rate is significant at a 90-percent confidence level.

Data Collection Design

In an effort to balance respondent burden with improved estimates of change, households

are interviewed for 4 consecutive months, not interviewed for the next 8 consecutive

months, and then interviewed for another 4 consecutive months.

Each month, a new household panel of approximately one-eighth the total monthly

sample size (60,000/8 = 7,500 households) is initiated, and the panel which received its

eighth interview the previous month is dropped. Thus, each month, eight different panels

are being interviewed for the 1st, 2nd,..., and 8th time. This rotating panel structure

means that three-quarters of the sample in a given month is retained in the sample the

next month, improving the estimates of month-to-month change. In the CPS, first and

fifth month-in-sample households are interviewed through personal visits. For

subsequent months, the majority of interviews are conducted by telephone.

Prior to January 1994, most of the CPS data were collected with a paper survey

instrument and translated into computer readable form using FOSDIC3 technology. 

 

 1Following each decennial census, a new sample of areas is selected. The current sample is

based on the 1980 census.

 2

The coefficient of variation of an estimate is defined as the standard error of the estimate

divided by the estimate.

 3

Film Optical Sensing Device for Input to Computers.
4

Approximately 9 percent of the data was collected by interviewers working in two

centralized facilities using computer-assisted telephone interviewing. Only households in

a subset of the CPS sample areas were eligible for centralized computer-assisted

telephone interviewing (CATI). These areas were purposely chosen based on operational

considerations and were generally large metropolitan areas.

Centralized computer-assisted telephone interviewing had been used in the CPS since

January 1989, when a centralized facility in Hagerstown, Maryland, was opened. In

order to minimize any potential effects on published CPS estimates, the percent of

sample cases interviewed from CATI was originally kept small. Over the 5-year time

period, the percent of the CPS sample interviewed from CATI gradually increased to the

9 percent level used in the 1993 CPS4.

From January 1991 through December 1992, the Bureau of the Census and the Bureau of

Labor Statistics jointly conducted a special study in the CPS CATI-eligible areas to

measure the effects of centralized telephone interviewing combined with computerassisted interviewing on CPS data. Findings from this study showed that inclusion of

CATI produced a 0.8 percentage point higher unemployment rate (Shoemaker, 1993). 

However, this difference could not be attributed to CATI alone. The paper-and-pencil

questionnaire itself was not administered from a centralized location; rather, it was a

computerized version, with modified wording of the lead-in question to the labor force

section5. Thus, it was impossible to distinguish whether this difference was due to

centralization, computer-assisted interviewing, or a slightly modified questionnaire.

Hypotheses and Experimental Design

This analysis is a contrast study. To study the effect of a possible "treatment," the CPS

sample was randomly split into two "independent" groups (split panels). Each panel is

statistically representative of the parent sample. The treatment is administered to

respondents in one of the two split panels. The treatment is excluded from the other

panel. The difference between the estimates from the two panels gives an estimated

difference of the "treatment effect."

Panel definitions:

CATI Panel Households in this type of panel are eligible for interview at one of the

centralized telephone facilities. Not all households in the panel will be

interviewed by CATI. To be interviewed by CATI, a respondent must

 

 4To accommodate the increased CATI sample, a second telephone facility was opened in

Tucson, Arizona, in 1992.

 5

See Rothgeb (1994) for a more detailed discussion of the lead-in question and possible

influences of computer-assisted interviewing and centralization.
5

have a telephone and speak English or Spanish. More important, during

the personal visit interviews (usually MIS 1 and MIS 5) the household

must agree to be interviewed in subsequent months by telephone. If not,

the household's subsequent interviews will be completed by a field

representative, either by personal visit or by telephone. Generally, if the

household has not been interviewed from a centralized telephone facility

by mid-week, then the interview is transferred to a field representative for

interviewing.

NonCATI

Panel All households in this type of panel are ineligible for CATI interviewing. 

Thus, even if a household meets all of the basic requirements for CATI,

the interview will be completed by a field representative (decentralized

interviewing only).

The Month in Sample (MIS) refers to the number of months that a housing unit has been

in sample. This is usually the same as the number of interviews that a household has

undergone. For example. MIS 1 refers to the first interview. MIS 1 and MIS 5

interviews are always conducted by personal visit.

The analysis here centers on the CATI effect. 

1. Description

Tests of these hypotheses are based on data from the CPS. The PAPI

questionnaire was used by the CPS field representatives (decentralized

interviewing). A computerized version of the questionnaire with a slightly

modified wording of the lead-in labor force question was used by the CPS CATI

interviewers (centralized and computer-assisted interviewing). The centralized

telephone interviewing effect was combined with computer-assisted interviewing,

because the old questionnaire did not have a computerized version outside of the

CATI environment. This study was a continuation of the study presented in

Shoemaker (1989 and 1993) with respect to data collection. In the Shoemaker

(1989) study, data was collected from June 1985 until December 1988, but the

portion which measured the effect of CATI on labor force estimates started in

August 1986. In the Shoemaker (1993) study, data was included from 1991 and

1992. In the present study, data was collected from October 1992 until December

1993, as part of the CPS Overlap Study (Bureau of the Census (1994)).

2. Experimental Design

The sample within the CPS CATI-eligible areas was randomly split into two

representative panels: CATI-eligible (Panel A) or nonCATI (Panel B). The
6

number of households in Panel A each month increased over time. Thus, the

composition of both panels changed on a monthly basis. The only areas included

in this study were those that had sample in both Panel A and Panel B.

Note that these panel estimates were not nationally representative, because they

used data from a non-random group of sample areas. The population covered by

the CPS CATI-eligible sample areas was approximately 12 percent black and 11

percent Hispanic.

Data obtained from the first and fifth interviews were excluded from the panel

estimates for testing this hypothesis. Estimated monthly sample sizes of persons

16+ and unemployed persons for each panel are presented below.

CPS Panel Persons 16+ Unemployed persons

CPS CATI Panel A 14,580 730

CPS NonCATI Panel B 16,250 740

The data were collected every month from October 1992 to December 1993. The data

from March 1993 were omitted from the study: a CATI facility was closed during the

March interview week because of a blizzard. The estimates of cell proportions were

formed from estimates of totals averaged over these 14 months.

3. Limitations

The confounding caused by the mix of CATI and non-CATI interviews in the

CATI Panel A estimates was also present in these tests. Additionally, the Panel A

interviews which were not completed at a CATI facility were conducted with a

slightly different wording of the lead-in labor force question.

III. Methods

We have a five-dimensional table of probabilities defined by the four binary predictors,

race, sex, ethnicity, and test status, and the binary response variable, unemployment

status. These four variables are defined below. The cell probability of an unemployment

status in a race-sex-ethnicity classification is the probability of that status given the racesex-ethnicity classification. The estimated number of population individuals in the

studied areas in a cell is the sum of the weights of the sample individuals in the cell,

where the weights are a combination of the CPS base-weights and the probabilities of

membership in the test or control panel. See Bureau of the Census (1994) for a

discussion of the weighting for this study.
7

The model I used was a natural one where a cell is defined by the binary variables test,

race, sex, ethnicity, which are defined below.

Variable Values Meaning of Values

 Test 1 Test: CATI eligible

 -1 Control: Non-CATI eligible

 Sex 1 Male

 -1 Female

 Ethnicity 1 Hispanic

 -1 Not Hispanic

 Race 1 Black

 -1 Not Black

We are mostly interested in the variable 'test'; that is, whether CATI eligibility has an

effect on measures of Labor Force, in this case proportion unemployed. The analysis can

focus on whether terms with 'test' in them need to be present.

 

The log-odds ratio for cell i is

logit(pi) = Xi

T

ß (1)

where Xi is the vector of the indicator variables for the cell i, i=(1,...,I). Here Xi has

dimension s.

The methods described by RRK are derived by finding the asymptotic distribution of X2

and G2 and using that to obtain an adjustment which makes them approximately chisquared. The simpler adjustment depends on the covariance matrix of the estimated cell

probabilities only through its diagonals. A more precise adjustment requires an estimator

for the whole covariance matrix. Another possibility, which also requires the whole

covariance matrix, is to simulate the asymptotic distribution of X2

 and G2. 

In this analysis, I estimate the covariance matrix of the cell proportions with the 

Generalized Variance Function (GVF) methods described by Fisher, et al (1993). The

method had to be extended to estimate correlations between cells. To get an estimator

for the covariance matrix, I decomposed it by conditioning on the total. Let X be the

estimator for the vector of levels of a characteristic in a population.

Var(X) = E(var(X3ΣXi)) + var (E(X3ΣXi)) (2)
8

 

Now say

E(var(X3ΣXi)) = M

where M is the covariance matrix for a weighted multinomial random variable. If W is

the average weight,

[M]ii = var(Xi3ΣXi) ≈ DEW(ΣXi)pi(1-pi) and

[M]ij = -DEW(ΣXi)pipj,

where pi is the fraction of the population represented by cell i and DE is an average

design effect. Say the other term is

var(E(X3ΣXi

)) = pp' var(ΣXi).

That is, E(XΣXi)=pΣXi, the multinomial expectation. Now the estimated correlation

becomes

cor(Xi,Xj) = -DEW(ΣXi)pipj+pipjvar(ΣXi) 

(DEW(ΣXi)pi(1-pi)+pi

2

var(ΣXi))1/2(DEW(ΣXj)pj(1-pj)+pj

2

var(ΣXj))1/2 (3)

The covariance matrix of X was estimated with

where 

Install Equation Editor and doubleclick here to view equation. and [cor(X)]ij = cor(Xi

,Xj). Notice this

covariance matrix is not typically nonsingular.

The variance for cell i is 

Install Equation Editor and doubleclick here to view equation. was calculated with a

Generalized Variance Function (GVF) method. We can decompose

total variance for cell i 

cov( $

Install Equation Editor and doubleclick here to view equation.
9

Here n is the sample size, 

Install Equation Editor and doubleclick here to view equation. and DEF is a design

effect. Note our model here has sample size n and

Install Equation Editor and doubleclick here to view equation. expressed as random variables. Also

note the design effect is only applied to the variance of

Install Equation Editor and doubleclick here to view equation. conditioned on Install Equation Editor and doubleclick here to view equation. . This portion corresponds to the variance expression we usually

see for CPS, that is, a variance like a simple random sample.

 In this study, however, we use estimates where there has been

no raking procedure to force various population estimates to

match census estimates of population. The other term has been

added to account for the extra variation. Now assume W = SI,

the sampling interval in the survey where SI = N/E(n). Then

Install Equation Editor and doubleclick here to view equation. can be written

BRR estimates from 1987 were used to estimate DEF and 

Install Equation Editor and doubleclick here to view equation. .

 Further adjustments are needed to make the estimates

applicable to the weighting and sample size in the split panel

study. The final expression for the variance estimator for

panel s in the split panel study is

Here, 

Install Equation Editor and doubleclick here to view equation. is the 1987 estimate of population

size, 

Install Equation Editor and doubleclick here to view equation. is the split-panel estimate of

population size for 

Install Equation Editor and doubleclick here to view equation. , panel s and

Install Equation Editor and doubleclick here to view equation.

Install Equation Editor and doubleclick here to view equation.
10

Install Equation Editor and doubleclick here to view equation. is the probability of including

Install Equation Editor and doubleclick here to view equation. in panel s.

A variance for an estimator for a cell proportion, say 

Install Equation Editor and doubleclick here to view equation. is

calculated with the well-known Taylor expansion expression,

The last step depends on the relationship

which is met, for example under simple random sampling without replacement.

I estimated the parameter vector ß with a pseudo-likelihood method, where

Install Equation Editor and doubleclick here to view equation. is the solution to the equation

where XT=(X1,...,XI), D=diag

Install Equation Editor and doubleclick here to view equation. , Install Equation Editor and doubleclick here to view equation. is the weighted sum of the whole sample,

Install Equation Editor and doubleclick here to view equation. is the weighted sum from the ith cell,

Install Equation Editor and doubleclick here to view equation. are the estimated cell probabilities of

Install Equation Editor and doubleclick here to view equation.

Install Equation Editor and doubleclick here to view equation.

Install Equation Editor and doubleclick here to view equation.
11

being unemployed in the model, and q is the vector of observed

cell probabilities of unemployment.

RRK preserve the familiar forms of the test statistics by providing adjustments to the

traditional X2 and G2 statistics. The simpler adjustment to these statistics are where

Install Equation Editor and doubleclick here to view equation. is the average eigen value of

Install Equation Editor and doubleclick here to view equation. Where

H is an Ix(I-s) matrix of rank I-s such that 

Install Equation Editor and doubleclick here to view equation.

and

The matrix 

Install Equation Editor and doubleclick here to view equation. has been dubbed the "generalized

design effect matrix." They point out that the adjustments

should work well when the coefficient of variation of the

Install Equation Editor and doubleclick here to view equation. is small. RRK give an improved

adjustment, based on the well-known Satterthwaite

approximation, where

or

Install Equation Editor and doubleclick here to view equation.

Install Equation Editor and doubleclick here to view equation.

Install Equation Editor and doubleclick here to view equation.

Install Equation Editor and doubleclick here to view equation.
12

These statistics are approximately chi-squared with (I-s)/

Install Equation Editor and doubleclick here to view equation.

degrees of freedom. 

Here,

This adjustment takes the variation of the 

Install Equation Editor and doubleclick here to view equation. into

account.

Another alternative is to simulate chi-squared random variables from the asymptotic

distribution of 

Install Equation Editor and doubleclick here to view equation. and Install Equation Editor and doubleclick here to view equation. . 

These statistics have the same as the distribution as

where the 

Install Equation Editor and doubleclick here to view equation. 's are independent chi-squared

random variables if the asymptotic theory of RRK holds. We

can get many realizations of random variables from these

distributions and estimate p-values. The result is the exact

asymptotic p-value except for the variation in the simulation,

which we can make small by using a lot of replications. I

used 10,000, so my estimated p-values have standard deviation

Install Equation Editor and doubleclick here to view equation.

Install Equation Editor and doubleclick here to view equation.

Install Equation Editor and doubleclick here to view equation.

Install Equation Editor and doubleclick here to view equation.
13

I calculated the test statistics Xc

2

, Gc

2

, Xs

2

, and Gs

2

, to test the goodness of fit of each

model along with the p-values based on the approximate distributions, also given by

RRK. I also provide the simulated p-values mentioned above. I also test some nested

hypotheses, where the model is partitioned as

and the

hypothesis

under test is

H:ß2=0 conditioned on this model. The test statistics are RRK's X2(231), G2(231), with

p-values associated with their approximations and simulated p-values.

 

IV. Results

The first test I performed was for the model with every effect except for the four-way

interaction and the test-ethnic-sex interaction. This model had the goodness-of-fit pvalues in table 1.

Table 1

Test Statistic P-value

Xc

2 .83

Xs

2 .83

Gc

2

 .82

Gs

2

 .82

X2

(sim) .83

G2(sim) .82

The model fit. In my search to find models with fewer parameters to still fit the data, I

tried two major reductions in the data. First, I tried the reduction where all terms

containing the test term were eliminated and performed the test of the reduced model

given the larger model was true. The goodness-of-fit p-values are in table 2.

Table 2

Test Statistic P-value

Xc

2

 .030

Xs

2

 .030

Gc

2

 .081

Gs

2

 .082

X2(sim) .033

G2(sim) .085

Install Equation Editor and doubleclick here to view equation.
14

which led me to reject the null hypothesis of the reduced model; apparently there is

evidence that some test effect is non-zero.

The test of the model with no test terms given the model in test 1 also led me to reject the

null hypothesis that the model in test 2 is sufficient.

The other model I tried was that with only main effects. The p-values are given in table

3.

Table 3

Test Statistic P-value

Xc

2 .97

Xs

2 .98

Gc

2 .97

Gs

2

 .97

X2(sim) .97

G2

(sim) .96

The model cannot be rejected at significance .10. Indeed, it is the model I chose. 

Examinations of the normalized residuals revealed nothing pathological; there did not

appear to be any outlying cells.

The tests of the main effects models vs. any of those available with these variables do not

lead me to reject the hypothesis that the main effects model is sufficient. Further, the test

of each model with a main effect absent rejects the hypothesis of an adequate model.

Estimates of the coefficients for the main effects model are in table 4.

Table 4

Parameter Estimate Standard Error

Intercept -2.52 .035

Test -0.07 .020

Race -0.40 .026

Ethnicity -3.17 .029

Sex 0.19 .020

These results are mostly in accord with the previous studies mentioned. The Bureau of

the Census (1993) tested differences of unemployment rates separately in several

categories. The results of those tests are reprinted in table 5.
15

Table 5

EFFECT OF CENTRALIZED TELEPHONE AND COMPUTER-ASSISTED

INTERVIEWING

 Unemployment Rate

(14 Month Average, 10/92 - 12/93 excluding 3/93)

CPS CATI

Panel C

CPS NonCATI

Panel D

Difference

C-D P-value

Total

 Men

 Women

 7.55

 7.78

 7.28

 6.54

 6.81

 6.24

1.00

0.96

1.04

0.00*

0.00*

0.00*

White

 Men

 Women

 6.64

 6.91

 6.32

 5.57

 5.80

 5.30

1.07

1.11

1.03

0.00*

0.00*

0.00*

Black

 Men

 Women

13.42

14.10

12.84

12.30

13.90

10.98

1.12

0.20

1.86

0.18

0.76

0.04*

CPS CATI Panel C = CPS sample that can be sent to CATI, but includes nonCATI

sample. NonCATI sample interviewed with paper and

pencil (PAPI).

CPS NonCATI Panel D = CPS sample that cannot be sent to CATI. All sample

interviewed with paper and pencil (PAPI).

MIS 1 and MIS 5 not included in Panel C or Panel D.

*Differences significant at the 10% level.
16

The only group in which they failed to detect a CATI effect was in Blacks and Black

Males. The analyses were organized differently but are not in disagreement.

V. Conclusion

We have detected effects on unemployment due to Race, Sex, Ethnic Group, and CATI

status. These results are similar to those in the Bureau's original investigation. That

analysis was not designed to detect interactions.

The test statistics seem well behaved in this study; we might be encouraged to use

similar methods on other related projects like other parts of the mode effects study

(Bureau of the Census (1993)).

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