DATA EVALUATION AND METHODS RESEARCH

SyntheticEstimation of

State Health Characteristics

Basedon the

Health Interview Survey

This report discusses thevarious methods that have beenproposed

or used for obtaining estimates of health characteristics for local

areas. Particular emphasis is given to discussion and evaluation of

synthetic estimation procedures developed originally at the

National Center for Health Statistics for purposes of estimating

levels of health characteristics obtained from the Health Interview

Survey for each State and the District of CoIumbia.

DHEW Publication No. (PHS) 78-1349

Series 2

Number 75

U.S. DEPARTMENT OF HEALTH, EDUCATION, AND WELFARE

Public Health Service

National Center for Health Statistics

Hyattsville, Md. October 1977
.. .. .

Library of Congress Cataloging in Publication Data

United States. National Center for Health Statistics.

Synthetic estimation of State health characteristics based on the health interview survey.

(Vital and health statistics: Series 2, Data evaluation and methods research; no. 75)

(DHEW publication; (PHS) 78-1 349)

Bibliography

1. Health surveys-Statistical methods. 2. Estimation theory. 3. Health surveys–United

States–States. I. Title. II. Series: United States. National Center for Health Statistics. Vital

and health statistics: Series 2, Data evaluation and methods research; no. 75. 111. Series:

United States. Dept. of Health, Education, and Welfare. DHEW publication; (PHS)

78-1349. R.4409.U45 no. 75 312’.07’23s [614.4’2] 77-22219

ISBN 0-8406 -0107-7

For sale by the Superintendent of Documents, U.S. Government Printing Office

Washington, D.C. 20402

Stock t?o. 017-022 -00596-4
NATIONAL CENTER FOR HEALTH STAT STICS

DOROTHY P. RICE, Director

ROBERT A. ISRAEL, Deputy Director

JACOB J. FELDMAN, Ph.D., Associate Director for Analysis

GAIL F. FISHER, Associate Director for the Cooperative Health Statistics System

ELIJAH L. WHITE, Associate Director for Data Systems

JAMES T. BAIRD, JR., Ph.D., Associate Director for International Statistics

ROBERT C. HUBE~ Associate Director for Management

MONROE G. SIRKEN, Ph.D., Associate Director for Mathematical Statistics

PETER L. HURLEY, Associate Director for Operations

JAMES M. ROBEY, Ph.D., Associate Director for Program Development

PAUL E, LEAVERTON, Ph.D., Associate Director for Research

ALICE HAYWOOD, Information Officer

OFFICE OF THE ASSOCIATE DIRE=TOR FOR MATHEMATICAL STATISTICS

MONROE G. SIRKEN, Ph.D., Associate Dwector

Vital and Health Statistics-Series 2-No. 7!5

DHEW Publication No. (PHS) 78-1349

LiibrarY of congresscatalogcard Number 77-22219
PREFACE

The investigations on which this report is based have been supported by con.

tracts between the Measurement Research Laboratory, Office of Research,

National Center for HeaIth Statistics (NCHS), and the School of Public He~fi,

University of Illinois at the Medical Center, Chicago. Dr. Monroe G. Sirken,

Associate Director for Mathematical Statistics, NCHS, served as project officer

for the Center on these investigations. Dr. Sirken and Ms. Patricia Royston,

mathematical statistician, Office of Mathematical Statistics, NCHS, provided

considerable input throughout the course of fiis project. Dr. Wesley L. Schaible

and Dr. Dwight Brock, Office of Research, NCHS, reviewed an earlier draft of

this report and made many helpful suggestions.

...

Ill
CONTENTS ‘

prefa~~ .om............................. .............................. ............ .... ........ ...... ........ .... ...... .. ...... ...............................

~~rod”~ti~n ......................................... ....... .................. ....... ......................... ..........................................

R~view of the Literature ..........................................................................................................................

Alternative Methods of Obtshing State Estimates ....................................................................................

Background .........................................................................................................................................

Nearly Unbiased Estimator ..................................................................................................................

Synthetic Estimator .............................................................................................................................

Regression-Adjusted Estimator ...........................................................................................................

Evaluation of Synthetic Estimates ............................................................................................................

Background .........................................................................................................................................

Estimation of the Me~ %we Emor (MSE) and Avemge Mean Square Error (AMSE) of Synthetic

Estimates ..........................................................................................................................................

Evaluation of HIS SYnthetic Estimates for Alternative a-Cell Grids ....................................................

References

Appendix:

.............................................................................................. ..... ...........................................

Proofs of Lemmas and Theorems .......... ..................................................................... ..........

LIST OF TEXT TABLES

A. Interpretation of he components of tie squ=e of the bii of the “nearly” Unbias;d estimate ..........

B.

c.

D.

E.

Dktnbution of percentage absolute differences between the nearly unbiased estimate and the true

vahre among 42 States in the North, Centml, South, and West Regions for total deaths, deaths

from major cardiovascular-renal diseases, and deaths from motor vehicle accidents, 1960 ................

Maximum contribution to the relative varkmce of .$?~of sampling variation in the Psa ...... ..................

Mean proportional absolute differences between the synthetic estimate produced by the total 50aceU grid and that produced by other a-cell grids for selected Health Interview Survey (HIS) variables, 1969-71 ....................................................................................................................................

Maximum proportional absolute differences between the synthetic estimate produced by the total

50 a-cell grid and that produced by other a-cell grids for selected Health Interview Survev fHIS)

variables, 1969-71

. . .

.............................................................................................................................

...

lu

1

2

3

3

3

6

11

12

12

12

13

16

17

5

6

10

14

F. Correlation coefficients between the synthetic estimate produced by the total 50 a-cell grid and that

produced by other a-cell grids for selected Health Interview Survey (HIS) variables, 1969-71 .......... 15
SYNTHETIC

OF STATE HEALTH

BASED ON THE HEALTH INTERVIEW SURVEY

ESTIMATION

CHARACTERISTICS

Paul S. Levy, SC.D., School of Public Health, University of Illinois at the Medical Center, Chicago, and

Dwight K. French, Statistical Methods Staff, National Center for Health Statistics

INTRODUCTION

Statisticians, demographers, economists, and

others have long been aware of the critical need

for accurate smalI area statistics. While the U.S.

decennial census provides accurate local statistics of many characteristics once every 10 years,

the accuracy of these statistics becomes questionable as time elapses from the Iast census and,

in addition, characteristics other than those

found on the census questionnaire are often

desired.

Although a rather extensive system of

ongoing general purpose surveys is conducted by

Federal agencies, they are a.Imost always

designed to produce estimates for the United

States as a whole or, at most, for rather large

geographic regions” or divisions. For reasons of

sampIe size and design, direct estimates for such

subdivisions as cities, counties, States, or other

minor civil divisions, which are so criticality

needed, can rarely be obtained from these surveys.

The National Center for Health Statistics,

one of the Federal agencies responsible for maintaining a system of sample surveys and other

data collection systems, has Iong recognized the

need for good small area statistics, and for the

past decade has investigated alternate strategies

for obtaining such estimates. In particular,

NCHS has developed a procedure known as

“synthetic estimation” for obtaining small area

statistics. This procedure obtains small area estimates of characteristics by combining national

estimates of the characteristics specific to demographic subgroups with estimates of the proportional distribution of the local population into

the subgroups. The subgroups would be chosen

for their reIevance to the characteristic being

estimated. For example, if it were desired to

estimate the prevalence of the sickle cell trait in

a particular county having a racial distribution

of 30 percent white and 70 percent bIack, and

if a hypothetical national survey estimated that

the trait was prevalent among 10 percent of U.S.

blacks and virtually nonexistent among U.S.

whites, one would estimate that 7 percent of the

population in the county had the trait (30% X

O% + 70% X 10%). This is a synthetic estimate.

The advantages of the synthetic-estimation

approach to local estimation are its intuitive

appeal, its simplicity, and its low cost relative to

a direct survey of the local population. A major

disadvantage is its lack of sensitivity to certain

local characteristics. For example, in the above

illustration, it may happen that the white population in that area are all of Meditemanean descent and have more than a negligible amount of

persons with the sickle celI trait.

Much research on the synthetic estimation

procedure has emerged since an NCHS report on

synthetic estimates of disability for States was

published in 1968.1 The purpose of this report

is to examine critically the various methods for

1
obtaining local estimates that are in the literature and, in particular, to examine synthetic

estimation from a methodological point of view.

REVIEW OF THE LITERATURE

The need for methods of obtaining valid and

reliable estimates of characteristics of local

populations has been recognized for a long time

by statisticians and demographers. In particular,

much effort has been expended by statisticians

associated with the U.S. Bureau of the Census

and their contractors, especially in the use of

symptomatic variables such as births, deaths,

and school enrollment, which are available on a

local level, to measure changes in population size

since the most recent decennial census. Methods

such as the vital rates technique, censal ratio

method, Census Bureau Component Methods I

and II, ratio correlation method, and others have

been described extensively in the Literature.z

Basically, these methods use the relationship

between the population size of the local area at

the most recent census and the measure of the

symptomatic variable or variables for that year,

in conjunction with the value of the symptomatic variable(s) at the date for which the estimate is desired, to produce the desired local

estimate of population size or change. An

elaboration of the use of techniques based on

symptomatic variables has been developed

recently by Ericksen .3-5 His elaboration involves

use of sample data from the Current Population

Survey in conjunction with symptomatic variables to obtain estimates of population size for

l~cal areas.

Although health statisticians have long felt

the need for valid and reliable estimates of

health characteristics for local areas, only in the

past decade has serious attention been given to

the development of methodology for obtaining

local area estimates of such health characteristics as morbidity, mortality, disability, and

utilization of health care services. The methods

developed by demographers for estimating local

population sizes could not, however, be directly

applied to the estimation of health characteristics for local areas; hence, methodology for

estimating health conditions for local areas has

developed along different lines from those

discussed above for local estimation of population size.

A major advance in estimation of health

characteristics for 10CZI areas came with an

experiment conducted by Walt R. Simmons

and his staff at the National Center for Health

Statistics (NCHS) during the mid-1 960’s and

published in 1968.1 In this experiment, three

different estimation techniques were used to

produce estimates of long- and short-term disability for each State in the United States for

the 2-year period beginning July 1, 1962, and

ending June 30, 1964. The NCHS data used for

estimating disability were from the Health Interview Survey (HIS), and the population data were

from the Current Population Survey update of

the 1960 Decennial Census.

One of the methods used was proposed

originally by Woodruff to produce local estimates of retail trade;6 the other two, namely,

the synthetic estimator and the nearly unbiased

estimator, were developed at NCHS. These

methods will be discussed in greater detail. Of

the three methods investigated, the synthetic

estimator was judged to be the most promising .,

for estimating disability on the State Ievel, and

the estimates finally published were obtained

by this method.

The NCHS publication on synthetic

estimates of disability seemed to stimulate

further efforts to apply and evaluate synthetic

estimation. Within NCHS, an evaluation of the

synthetic estimation procedure was conducted

in which synthetic estimates of death rates in

1960 from four causes (motor vehicle accidents,

major cardiovascular-renal diseases, suicide, and

tuberculosis) were calculated for each State and

for the District of Columbia.7 These synthetic

estimates of death rates were then compared to

the known true death rates for each State and,

in general, agreement between synthetic estimates and true death rates was good for one of

the causes examined (major cardiovasculw-renal

diseases), fair for another (suicides), and poor

for the other two (motor vehicle accidents and

tuberculosis). The general conclusion from the

study was that the validity and reliability of

synthetic estimates might differ from characteristic to characteristic. -

As part of the NCHS study of death rates, an

2
alternative estimation procedure was developed.

The resulting estimator, called the regressionadjusted estimator, uses the synthetic estimate

in combination with ancillary data available on

the State level and thought to be correlated with

the health characteristic to be estimated.7 This

estimate, for at Ieast one of the causes of death

examined, seemed to be an improvement over

the synthetic estimator.

After the NCHS publication on synthetic

estimates of disability, the Bureau of the Census

produced synthetic estimates of unemployment

rates and number of dilapidated housing units

that had all plumbing facilities for States,

SMSA’S, and counties.s-l 0 In addition, extensive

studies were undertaken to evaluate the synthetic estimates. An important result of these

studies was the emergence of a criterion, caUed

the average mean square error (AMSE), as a

proposed measure of the accuracy of a set of

synthetic estimates, and the development of a

method for estimating the AMSE.9~11 These

methods will be discussed in greater detail later

in this report.

Most recently, Namekata, Levy, and

O’Rourkel 2 investigated the use of synthetic

estimation in obtaining estimates of complete

and partial work disability for States based on

data from the 1970 census. The synthetic estimates were obtained and compared with the

direct estimates that were available from the

1970 Decennial Census for each State. Their

general conclusions were that the synthetic

estimation technique was fairly good for partial

work disability but fairly poor for complete

work disability.

ALTERNATIVE METHODS OF

OBTAINING ESTIMATES

Background

In the original NCHS investigation of

alternative procedures for small area estimation

of health characteristics from the Health Interview Survey (HIS), several procedures were considered.1 In this section, we will discuss in detail

two of the methods, namely, the nearly

unbiased estimator and the synthetic estimator.

In addition, we will discuss an estimator, called

the regression-adjusted estimator, not considered

in the original investigation but developed in a

later study.7

One of the problems in obtaining estimates

for States of health characteristics based on HIS

data is the fact that the basic design of the HIS

does not lend itself to unbiased estimates for

States. In the basic HIS design, a primary sampling unit (PSU), which is generally a county or

SMSA, is chosen to represent a stratum consisting of one or more demographiczdly similar

PSU’S. Those strata consisting of more than one

PSU are called non-self-representing strata, and

their component PSU’S may not be from the

same State, although they would be from the

same census region (Northeast, North Central,

South, or West). Thus, the estimate from a sample PSU when inflated to represent the entire

stratum might cut across State boundaries, and

hence, it would be impossible to combine the

unbiased estimates for strata into unbiased estimates for States.

Nearly Unbiased Estimator

One of the methods considered in the original NCHS investigation is calIed the nearly unbiased estimator and, yields an estimate for a

State that is technically nearly unbiased. Basically, this procedure takes the usmd HIS stratum

estimate for an aggregate and allocates it to a

State in relation to the proportion of the totzd

stratum population coming from the State. In

other words, the nearly unbiased estimate ~~ of

the mean level of characteristic X for State s is

given by

where

(1)

~~ = unbiased HIS estimate for the mean level

of X for stratum j,

‘s.. = i ‘sj. j= 1

= the number of persons in States,

3
= the number of persons in that portion of

stratum j which is in States,

n~ji = the number of persons in the ~th stratum,

sth State, ith PSU; s=l,..., S; j=l,..., J;

i= 1,..., l~j> ~d

I,j = the number of PSU’S in stratum j that are

in State s.

Some properties of the nearly unbiased estimate

~~ are given below. The proofs are presented in

appendix I.

Lemma 1: The expectation E(~J) of the nearly

unbiased estimator ~j, is given by

where

= average level of X in stratum j,

(2)

= number of persons in stratum j,

= average level of X in that portion of

stratum j which is in States, and

~~ji = average level of characteristic X in that

portion of State s which is in PSU i of

stratum j.

Lemma 2: The bias 11(~~) in the nearly unbiased estimate ~j can be expressed

by

or equivalently by

(3)

Lemma 3: Let us assume that the ratio nsji/n.j.

is the same for all PSU’S in the same

State, which implies that

‘* =+ fori=l, .....lsj.

n.S3. sj

(’5)

Then the bias J3(~j) in the nearly unbiased, estimate ~~ has the form given by

n ..

Theorem 4: If # = # for i=l,..., Isj, then

ti.

~z (X;), th~ square of the bias in

~J, is given by the expression

4
Table A. Interpretation of the components of the square of tha bias of the “naarly” unbiased estimate

Component lntarpretation

J 2

-fl~j.-x.)z ‘“z 2 J.

...........................................................

j= I ‘s..lsj

J 2 2..

2. ,gn: :

............. .......... ............ .................... .....*.... .............

322 ~ ~~~.j. ‘z.$ji~ ~ (~~-~’~,)...........

j= I k <j ‘S..’ssksk &J i=l

1. J 2. SJ

--- F.je ) @sjit - X.j-} . . . . . . . . . . . . . . . . . . . . . . . . . .

4. z~~ +~z%l

- -Y1=7i’+ j=lns l..

where

Theorem 4 implies that under the condition

that nsjilnsj. is the same for all PSU’S within the

same stratum and belonging to the same State,

the square of the bias of the nearly unbiased

estimate Xl consists of four components specified in table A.

It can be shown that the third component

of B* (~j) can be transformed to the equivalent

algebraic form given by

* & ~ ‘sj.:k.

(Xsj.-x.j.) (Xsk.-x.k.)”

j=l k<j s..

Thus, an equivalent expression for the square of

the bias in the neady unbiased estimate ~j is

given by

+

Represents difference batwaen tha average level of X for a stratum

and the average level of X for the portion of the straum that is in

Statas.

Represents variance in X among PSU’S belonging to tha same State

and stratum.

Represents a “batween-strata” coveriance.

Represents a “Qetwean-PSU” covariance.

Varzance and mean square error of the nearly unbiased estimate.–The variance a:: can be .

obtained directly from its definition~ “formula.

This is given by

(9)

where a~~ is the variance of ~~, the unbiased

1

estimate of the mean level of X in stratum j.

It follows that the mean squme error MSEX~

of the nearly unbiased’ estimate ~j is given by

where CJ$: is given by relation (9), and B2 (~j )

is given by reIation (8) (under the condition

that n~ji/nsj. = l/lsj)”

Evaluation of the nearly unbiased estimator. —In the original NCHS investigation of “

methods for obtaining local estimates, the nearly
unbiased estimate did not emerge as the method

of choice for producing these local area HIS estimates of health characteristics primarily because

examination of the estimates produced by this

method showed evidence that they were unstable.l

A later study was performed at NCHS to

determine the extent to which the nearly unbiased estimator might be biased. The data base

chosen for this study was mortality data in 1960

for the 42 States in the North Central, South,

and West Regions of the United States. In particular, nearly unbiased estimates of total

deaths, deaths from motor vehicle accidents, and

deaths from major cardiovascular-renal diseases

were obtained from each State using the same

stratification that is used in the Health Interview

Survey. These nearly unbiased estimates were

then compared with the true number of deaths

in each State in the three regions examined, and

the biases of the estimators were evaluated bv

the percentage absolute difference 1OOIX;-XJ

x,.. “

Table B. Distribution of percentage absolute differences between

the nearly unbiased estimate and the true value among 42

States in the North Central, South, and West Regions for total

deaths, deaths from major cardiovascular-renal diseases, and

deaths from motor vehicle accidents, 1960

Percentage absolute difference

between nearly unbiased

estimate and true value

Total ..........................

0.0-0.9

1.0-1.9

2.0-2.9

3.0-3.9

4.0-4.9

5.0-5.9

6.0-6.9

7.0-7.9

8.0-8.9

9.0-9.9

Median

.......................................

.......................................

.......................................

.......................................

.......................................

.......................................

.......................................

.......................................

.......................................

.......................................

percentage absolute difference ....................................

Cause of death

T

Major

cardioMotor

Al I

vascular- vehicle

causes renal accidis- dents

II eases I

42

16

6

6

5

1

1

3

0

2

2

Frequency

42 I 42

15

8

5

3

2

2

2

1

0

4

12

6

4

1

3

3

1

2

5

5

Percen+

1.78 1.70 2.70

The distribution of percentage absolute difference is given in table B. The median percentage

absolute difference was 1.78 percent for total

deaths, 1.70 percent for major cardiovascularrenal deaths and 2.70 percent for motor vehicle

accident deaths. The small biases obtained from

this empirical study would yield the interpretation that for the stratification used in HIS, the

nearly unbiased estimator is in fact an estimator

having small bias.

However, in a given year, the number of

households interviewed in a particular stratum

might be quite small for the Health Interview

Survey. It is, therefore, anticipated that a:: and

hence u+, the sampling variance of the n~arly

unbiased ;stimator might be quite large. In addition, the ok, are difficult to estimate from the

i

data. Hence; in terms of sampling variance, the

nearly unbiased estimate ~~ might not be the

method of choice.

Synthetic Estimator

Background.-The other method for obtaining local estimates of health characteristics

investigated in the original NCHS studyl is

known as synthetic estimation and was the

method finally chosen for producing local estimates of HIS health characteristics for States in

1963-64 and again in 1969-71.13 The underlying rationale for synthetic estimation is that the

distribution of a health characteristic does not

vary among populations of States except to the

extent that States vary in demographic composition. In other words, the method assumes that

the incidence or prevalence of a health characteristic would be the same for two States if their

composition were the same with respect to such

demographic variables as age, sex, race, family

income, family size, place of residence, ar~d industry of the head of the family.

Conceptually, synthetic estimation uses the

model given by

where

-?. = mean level of characteristic X for the sth

State,

6
P = proportion of the population who are

‘a members of population cell a (alpha),

which is the socioeconomic demographically bounded class of specified age, sex,

race, income, etc. The sum over all o! cells

of P~a = unity.

~a = mean level of characteristic X for persons

in cell a in the United States as a whole,

and

k = number of u cells utilized.

In the original NCHS investigation, the ~a

were national estimates of HIS variables for the

period July 1962-June 1964. The population

estimates P~a were obtained from tabtiations of

a 5-percent sample questionnaire of the 1960

Decennial Census of Population for the 50

States and the District of Columbia. The population a cells were defined by cross-classifications

of the following variables:

1.

2.

~3.

‘ 4.

5.

6.

7.

Color: white; all other

Sex: male; female

Age group: under 17 years; 17-44 years;

45-64 years; 65 years and over

Residence: standard metropolitan statistical area (SMSA)-centra.l city; SMSA~

not central city; not SMSA

Family income: under $4,000; $4,000

and over

Family size: fewer than seven members;

seven members or more

Industry of head of family: Standard Industrial Classification codes 1 through

17 (Forestry and Fisheries, Agriculture,

and Construction) and codes 19 and over

(AII Other Industries)

The 384 possible cross-classification cells were

collapsed to 78 so that reliable estimates could

be obtained from the Health Interview Survey

for each CYcell.

For the synthetic estimates for the years

1969-71, HIS data from the three surveys of

1969, 1970, and 1971 were used to obtain the

rates or percentages of the health characteristics

measured. The populations of the 50 States and

the District of Columbia were obtained from a

sample described in a publication of the U.S.

Bureau of the Census entitled Public Use Sanzples of Basic Records From the 1970 Decennial

Census, Description and Technical Documentation, published in 1972. Of six such samples,

the one used was the State PubIic Use Sample

from the 5-percent questionnaires. Persons in

the military or confiied to institutions were not

included in the population estimates produced

for each State. Thus the restriction of the HIS

samples to the civikm noninstitutionalized population was carried over to these synthetic estimates. Of the seven variables used to produce

the 78 o! cells for the original report,l only six

were available in the Public Use Sample used to

produce these synthetic estimates. The variable

that was not avaiIable was residence in standard

metropolitan statistical areas. The six variables

can produce’ a possible 128 cells of data. These

were coIIapsed to 50 ~ cells for which reliable

national estimates from the Health Interview

Survey could be provided. A regional adjustment

(as specified below) was employed for the State

estimates within each of the four geographic regions of the United States to make these estimates consistent with the regional estimates

produced by the probability design of the

HeaIth Interview Survey.

In summary, the synthetic estimates produced for HIS health characteristics for the

years 1969-7113 use the same basic method as

was used in the originzd NCHS investigation.

However, in addition to estimates of long- and

short-term disability, estimates of utilization

of medical services were provided as well as estimates for subdomains of the population of each

State (age, sex, color, and family income). AIso,

a methodology has been developed for providing

sampling variances of synthetic estimates.

Detailed synthet~ estimate. -The detailed

synthetic estimate X5 of the mean level of

characteristic X for States is given by

(11)

~ EtPrt

t=l

where

~

X. = final synthetic estimate of the mean level

of characteristic X for States,

7
~~ = the usual HIS final estimate of the mean

level of characteristic X for region r

(where region r contains State s),

Prt = proportion of the population (from the

1970 Decennial Census) of region r which

is in State t (t=l,..., T), ‘

T = number of States in region r,

= first-stage synthetic estimate of characteristic X for States,

~~ = final HIS estimate of the mean level of

characteristic X for demographic cell a for

the United States,

P’

sa

= estimated proportion of the 1970 population in State s belonging to cell a (as estimated from the 1970 U.S. Census 1-

Percent Public Use Tapes), and

k = number of cxcells.

Synthetic estimates %Sw for subdomains

(age, sex, color, and income) are given by the

estimator

where

?,u=the final synthetic estimate for the mean

level of characteristic X for subdomain u

within States,

= preliminary synthetic estimate for the

mean level of characteristic X in subdomain u of States,

Pja = the estimated proportion (as estimated for

the U.S. Census 1970 l-Percent Public

Use Sample Tapes) of the population of

States belonging to cell a, and —

= the estimated proportion of the population of State s belonging in subdomain u,

except for synthetic estimates of workIoss days per person per year in age

groups. By definition, HIS excludes all

persons under 17 years of age fro:m the

employed population. Therefore, the factor f~uinthe denominator of the ratio adjustment for these statistics is redefined as

the estimated proportion of the population age 17 and over in State s that belongs to subdomain (age group) u.

The synthetic estimates for subdoma.ins as

given by equation (12) are ratio adjusted so that

the aggregates are consistent with the final synthetic estimates for the State as a whole.

The a variables were limited to those listed

below:

Color: white; other than white.

Sex: male; female.

Age group: under 17 years; 17-44 years; 45-

64 years; 65 years and over.

Family income: under $5,000; $5,000 and

over.

Family size: fewer than seven members;

seven members or more.

Industry of head of family: Standard Industrial Classification Codes 1 through 17 (ag-riculture, forestry, fisheries, mining, and construction); 18 and above (all others).

The 128 cross-classification cells produced by

these variables were collapsed into 50 a cells

for which reliable national estimates from the

Health Interview Survey could be made.

The ratio adjustment ~~/~ Prt~t was int=l

eluded in order to reflect a regional component

in final estimates. It is the ratio of the published

regional figure to the preliminary, derived regional rate calculated from the State estimates.

Estimation of sampling errors of synthetic

estimates for HIS characterz”stics 1969- 71.—The
synthetic estimates presented for the 1969-71

HIS data are subject to sampling variability from

two sources because they are based on HIS estimates and estimated population proportions.

(When synthetic estimates are computed from

known proportions and population means they

are not subject to sampling error, since there

wotdd be no sampling involved in the synthetic

estimation procedure.) The sampling variance of

-.

a synthetic estimate ~~ (ignoring the regional adjustment) is given by

Theorem 5: If the P~a are independent of the

~~, then the variance of %~ given

by equation (13) reduces to

@ +~ P2C#’ +@R(l-P.a)P.a

x~ SCi -t

a=l Xa %a=l a

for large values of n..

The first and third te=ms of equation (14)

represent the variance of X~ if the Pja were not

subject to sampling variation. Thus, the effect

of sampling variation of the Pja on cr~ is measS

ured by the expression

k

If Psm= minimum (PSI, .... P~k) and I% = RelXs

variance of ~~, we have

~ p2 *2, + 2 ~ PsaPsalCov (x; , x:)

VT =“=1 ‘a ‘a a’<a

s [WS)12

But

and

therefore,

i ~:psa(1-%)<1;7[E(?j]z

~=1 s

and the rel-variance V% of the synthetic estix~

mate ?S satisfies the inequality given by

(16)

9
Since the first term in the right-hand side of

the relation (16) is the relative variance of ?$

under conditions that the P~a are not subject to

sampling error, the effect of sampling error in

-.

the P~a on the relative variance of ~, would

therefore be less than (1 - P~ )/(P~n, ), the

second term in the right-hand side of relation

(16). This is summarized in table C.

As is seen in table C, for all but one or two

of the smallest States, the effect of sampling

variance in the P~u on the relative variance of

the synthetic estimator ~~, would be quite

small. ~

The approximate variance of X,U, the synthetic estimator for subdomains, can be expressed in a form parallel to expression ( 14) as

Sampling variances of synthetic estimates of

HIS health characteristics for the 3-year period

1969-71 were obtained based on equations (14)

and (17 ) with the following two modifications

made to simplify the calculations:

1. Psa A pa for all a, where Pa represents

the proportion of the U.S. population

in cell a. That is, the proportion of the

populatiori in any 0! cell is approximately

the same for all States.

Table C. Maximum contribution to the relative variance of Es of

sampling variation in tha Psd

1

P?

“s .0001 .001 .01 .05 .10

I

Maximum contribution

P

the

ralative variance of s

1,000 ................. 10.00 1.00 .10 .019 .009

10,000 ............... 1.00 .10 .01 .0019 .0009

100,000 ............. 0.10 .01 .001 .00019 .00009

2. Cov (x:, X:J) = O for all a < a’, so that

the third term of equations (14) and

(17) drops out. - “

Under these assumptions equation (14) reduces

to

and equation (17 ) becomes

where

a w

Equations (18) and (19) were the expressions used to compute sampling errors for the

estimates in the report.1 3 AImost all the estimates had sampling errors that were very small

relative to the size of the estimates themselves.

The relative standard error (RSE), defined by

and

was 5 percent or less for virtually all statistics in

the report, even for the smallest States. The only

important exceptions occurred for estimates of

the proportion of persons in certain population

subgroups who were unable to carry on major

activity. The most variable subgroup was the

10
under-45 age group, where the RSE ranged from

7.4 percent for the entire United States to 10.4

percent for Alaska. The highest single RSE was

11.6 percent for white persons in Alaska. Although it may seem strange for State estimates

to have such small sampling errors, these estimates were essentially weighted averages of

national HIS estimates based on 3 years of data

collection.

Bias of synthetic estimator.–The synthetic

est.mater is a biased estimator with the bias

B( k ~) given by

where ~~a is the true mean level of characteristic

X for demographic cell a in States.

Regression-Adjusted Estimator

Background.-One of the &sic limitations

on the synthetic estimator X~ is that it is

adjusted only for the specific set of demographic

cells (or o! cells) taken into consideration. If the

parameter being estimated is influenced by variables other than those taken into consideration

by the a cells, then the synthetic estimator will

not reflect this influence. Often it is not possible

to include in the ct-cell array all the variables

thought to be important because data on these

variables are not available in sufficient demographic detail. However, although a particular

variable might not be able to be used in the

synthetic estimator, it can often be taken into

consideration in other ways. In an earlier article,7 a method was proposed to take into consideration such variables.

Method of estimation.–The method ~resented below uses the synthetic estimator X~ in

conjunction with a set of ancillary variables

‘%2..., ZS-m to produce an adjusted synthetic

estimator. In particular, we assume the Iinear

model given by

~=a+fil z$l+. ..+pmzm ++. (21)

where ~, the percentage difference between the

synthetic estimate ~~ and the true value ~~ of

characteristic X for States is given by

+~ = term representing random error,

‘s1,..., ‘sm = values of variables Z1,..., am for

States, and

% &,_, ~m = refpession coefficients to be estimated

if estimates & of a, & of /31, .... and /?m of &

were available and substituted into the righthand side of equation (21), aJgeb~tic manipulation would result in an estimator X~ of ~~ given

(22)

Equation (21) states that the percentage

difference ~ between the synthetic estimate ~~

and the true value ~~ is a linear function of a set

of variables z~l , .... Z~m.For example, z~l might

be the proportion of persons in State s living in

SMSA’S; z~z, the proportion of persons in State

s having family income below the poverty level,

and so forth. Equation (21) expresses the concept that except for random variation the percentage difference between a synthetic estimate

and a true value is a linear function of a set of

variables Z$l, . . . , Zm. The estimator given by

equation (22) is called the regression-adjusted

estimator, and it was used and evaluated by

LevyT in computing State estimates of deaths

from motor vehicles for the year 1960. In that

study, it was found to be an improvement over

the synthetic estimator. However, it can be used

ordy when relevant ancilkiry data are avaiIable.

11
EVALUATION OF SYNTHETIC

ESTIMATES

Background

A fundamental problem of the synthetic

estimation procedure has been the difficulty in

evaluating the estimates produced by this

methodology. Although expressions have been

derived for estimating the sampling variance of

synthetic estimates, it is much more difficult to

estimate the bias of a synthetic estimate, and

since sampling errors are often small for synthetic estimates, the bias may often make the

largest contribution to the total mean square

error. A method has been developed, however,

by investigators at the U.S. Bureau of the

CensusgY14 for estimating the mean square error

of synthetic estimates by somewhat indirect

means.

Another consideration of importance in obtaining synthetic estimates is their sensitivity to

the particular set of a cells used in producing

them. Although a more detailed et-cell grid

should produce synthetic estimates having lower

bias, the potential reduction in bias may in fact

be smaI1 and may not justify the cost of obtaining the detailed a-cell grid. This issue has been

addressed in an empirical study using the synthetic estimates of disability, utilization of

health services, and limitation of activity based

on 1969-71 data from the Health Interview Survey and is discussed in a later section.

Estimation ot Mean Square Error (MSE)

and Average Mean Square Error (AMSE)

of Synthetic Estimates

A procedure has been developed by Waksberg and Gonzalezg which enables the mean

square error of a synthetic estimate to be estimated provided that an unbiased estimate of

the same characteristic exists for the same

population which is uncorrelated with the synthetic estimate. This procedure is developed by

means of theorems presented below:

Theorem 6: Let 2$ estimate a ~m-ameter x$

with bias given by l?(~~) and let ~~

be an unbiased estimate of ~s

which is uncorrelated with X~.

Then the following relation is true:

As

where

MSE~ = the mean square error of ?,

s

Theorem 7: If ~S is an e#imate of ~~ with bias

given by B(X, ), if ~~ is an unbiasexd

estimate of x$ uncorrelated with X$

and if &, is an unbiased estimate

Xs

of o~,” then the estimate M#E~

x~ s

given by

M$E; = (~; - ?,)2 - a;, (24)

s s

is an unbiased estimate of MSEfi .

.$

Investigators at the Census Bureau have used

the relationship given by equation (24) to evaluate synthetic estimates for certain variables such

as unemployment where independent estimates

are available. However, a serious limitation on

the use of the estimated mean square error A

M~E; as given in equation (24) is its likely instab~.;y since the unbiased estimate ~~ and the

estimate 62 of its variance tie both likely to

,=

have large %riances themselves, since, in all likelihood, they would be based on relatively small

sample size. Aware of this, Gonzalez and Waksberg have introduced the concept of evaluating

synthetic estimates not by their individual mean

square error, but by the average mean square

error (AMSE) of a set of synthetic estimates.

Specifically, the AMSE of a set of S synthetic

estimates is given by

12
m~ estimated without bias by the expression,

4MSE, given by

This statistic has been used with certain elaborations by Census Bureau investigators as the

major criterion for evaluating synthetic estimates. A shortcoming of this criterion, however,

is that it does not yield an estimate of the mean

square error for a specific synthetic estimate

(e.g., estimated unemployment in Ohio, 1976).

Rather, it gives the average mean square error a

set of synthetic estimates.

Evaluation of HIS Synthetic Estimates for

Alternative a-Celi Grids

Investigators at NCHS originally hoped to

evaluate the 1969-71 HIS synthetic estimates by

means of the AMSE criterion. However, there

was no unbiased estimate uncorrelated with the

synthetic estimate that could be used in equation (26). Although it is thought that the bias of

the neaxly unbiased estimate discussed above is

likely to be small for HIS variables, and that the

correlation between the synthetic estimate and

nearly unbiased estimate is also likely to be

small, the task of obtaining a reasonable estimate of its variance is difficult since it is often

based on data from one or two primary sampling

units.

The main thrust in the evaluation of the

1969-71 HIS synthetic estimates was an empirical investigation comparing synthetic estimates

based on the 50 a-cell grid used to obtain the

published 1969-71 synthetic estimates with

those obtained by collapsing the 50 cells into a

smaUer grid. In particular, synthetic estimates

were obtained for the 50 States and the District

of Columbia based on (1) the total 50 o!-cell

grid, (2) a 2 &-cell grid based only on sex, (3) a 4

a-cell grid based on age alone, (4) an 8 al-cell grid

based on sex and age, (5) a 16 cwcell grid based

on color, sex, and age, and (6) a 16 ~-cell grid

based on family income, sex, and age. The

synthetic estimates produced by each of the

collapsed grids (sex, age, sex by age, sex by age

and color, and sex by age and income) were

compared with IIWsynthetic estimates produced

by the total 50 a-cell grid by use of the following summary statistics:

1.

2.

3.

The mean over all 50 States and the

District of Columbia of the proportional

L%,ga- 2s1 between

absolute difference

x,

the synthetic estimate ~~, based on a

particular grid and the syn& etic estimate

%$ based on the totzd 50 cr-cell grid

(table D).

The maximum over all 50 States and the

District of Columbia of the proportional

absolute difference defined above (table

E).

The correlation coefficient over all 50

States and the District of Columbia between the synthetic estimate produced

by a collapsed grid with that produced

by the total grid (table F).

The mean proportional absolute difference

(table D) is a measure of the average relative

difference between synthetic estimates produced

by a collapsed grid and those produced by the

total grid. For the HIS variables considered in

this study, synthetic estimates produced by each

of the collapsed grids a~eed closely by this criterion with synthetic estimates produced by the

detailed 50 a-cell grid. For most of the 14 varibles in this study, the mean proportional absolute difference was less than 5 percent, and the

worst agreement by this criterion was shown by

the synthetic estimates produced by the e+cell

#d based on sex. In most cases, the synthetic

estimates based on age by sex, age by sex and

color, and age by sex and income did not show

substantially better agreement by this criterion

with those based on the total detailed grid than

the synthetic estimates based on age alone.

The maximum proportional absolute difference (table E) gives a feeling of the extent that

a single synthetic estimate based on a collapsed

grid might differ ”from the comparable synthetic

estimate based on the detailed 50-cell grid: The

magnitudes of some of the statistics shown in

table E imply that in individual States, the grid

used to compute the synthetic estimates might

affect the size of the estimate, even though the

13
Table D. Mean proportional absolute differences between the synthetic estimate produced by the total .50 a-cell grid and that produced

by other a-cell grids for selected Health Interview Survev (HIS) variables, 1969-71

HIS variable

Restricted ectivity days ...................................................................... .......................................

Bad disability days .................. ........ .............. ........ .......................................................... ........ ..

Work loss days .............................................. ................ ............................................................

Hospital discharges per 100 person years ...................... ...... ......................................................

Average length of hospitalization ............. .................................................................................

Percent of persons having one or more hospital episodes in a year .... ............................ ............

Percent of persons having one or more physician visits in a year ...............................................

Number of physician visits per person year ................... ...... ......................................................

Percent of persons having one or more dental visits in a year ....................................................

Number of dental visits per person year ...................................... ..............................................

Percant not limited in activity . ............................................................................................... ...

Percent limited in activity .. ...... ............................................ .................................... .................

Percent limited in amount or kind of major activity ...... ........ ...................................................

Percent unable to carry on major activity ........ .................................................. .......................

a-cell grid

I

.028

,029

.039

.021

.035

.018

.008

.021

.031

.056

.008

.061

.064

.094

!an proportional absolute

difference

.017

.026

.026

.006

.034

.008

.008

.016

.031

.055

.003

.021

.024

.058

.017

.025

.025

.008

.034

.008

.006

.016

.031

.055

.003

.021

.024

.061

.024

.032

.032

.008

.028

.007

.005

.015

.016

.032

.003

.029

.025

.081

.020

.021

.021

.008

.023

a .006

.007

.017

.027

.062

.004

.026

.027

.057

Table E. Maximum proportional absolute differences between the synthetic estimate producad by the total 50 a-call grid and that

moduced bvother o+cellmidsfors electedHeaith interview Survev (HIS)variables, 1969-71

HIS variable

Restricted activity days ..................................... ........................................................................

Bed disability days ....................................................................................................................

Work loss days ....................................................................... ...................................... .............

Hospital discharges per 100 person years ...................................................................................

Average length of hospitalization ..............................................................................................

Percent ofparsons having oneormore hospital episodes in a year ............................................

Percent of persons having one or more physician visits in a year ....................... ........................

Number of physician visits per person year .............. ...... .................................................. ..........

Percent of persons having one or more dental visits in a year ............. ........ ........ .......................

Number of dental visits per person year ................... ............................ .....................................

Percent not limited in activity ....................................... .................................................. ..........

Percent limitad inactivity ....................... ................ .................................. ...................... ..........

Percent limited in amount or kind of major activity ........... .................................................... ..

Percent unable tocarry on major activity ............................................................... ..................

Sex

a-cell grid

Sex

by

Age

Sex

by

age

Sqx

by

color

by

age

income

by

age

Maximum proportional absolute

.103

.134

.216

.127

.189

.103

.042

.104

.221

.362

.037

.454

.493

.695

.073

.088

.047

.055

.201

.057

.041

.064

.235

.393

.013

.094

.088

.233

difference

.073

.089

.140

.070

.203

.073

.042

.068

.237

.398

.013

.092

.091

.192

.099

.114

.071

.034

.083

.026

.019

.050

.058

.101

.014

.130

.122

.264

.136

.170

.211

.024

.266

.020

.027

.048

.241

.402

.021

.184

.154

.494

14
Table F. Correlation coefficients between thesynthetic estimate produced bythetotal 50a<ell grid andthat produced byother a-cell

grids for selected Health interview Survey (HIS) variebles, 1969-71

a-cell grid

Sex Sex by

HIS variable Sex by

Sex Age by

incolor

come

age by by

age

age

Restricted activity days ...................................................................... .................... ..... .............. .89

Bad disability days .. ........ .................. ....................................... .............................. ................... .94

Work loss days ......................................................................... ................................................. .81

Hospital discharges per 100 person years ................. ...... .................... .............................. ........ .. .88

Average length of hospitalization ....... ........... .................................... ...... .................................. .89

Percent of persons having one or more hospital episodes in a year .................................... ........ .90

Percant of persons having one or more physician visits in a yaar ......... ...................................... .81

Number of physician visits per person year ........... ........ ............ .................... ............................. .93

Parcent of persons having one or more dental visits in a year ... .............................................. ... .91

Number of dental visits per person year ............. ...... ...... ............................................................ .96

Percent not limited in activity ............... ........ ........................ .. ...................... ............................ .63

Percent limited in activity .. ..................... ......................... .................................................. ....... .61

Percent limited in amount or kind of major activity ........... .... ................i ............................. .... .50

Percent unable to carry on major activity ............................ ................................... .................. .85

Correlation coefficient

.96

.95

.89

.98

.89

.98

.82

.96

.90

.95

.95

.95

.94

.94

.96

.96

.90

.97

.89

.98

.81

.96

.90

.95

.96

.95

.94

.94 ——

.94

.95

.96

.99

.96

.99

.95

.98

.98

.99

.95

.93

.94

.93

.93

.94

.85

.99

.87

.99

.91

.98

.92

.96

.92

.93

.93

.91

average proportioned absolute differences are that is particularly appropriate when it is desired

small: - - to rank a set of estimates and when the absolute

The correlation coefficient between syn- values of the estimates are of secondary interest.

thetic estimates based on the detailed ,50-ceII In general, the correlations were quite high, with

grid and those based on a less detailed grid (table estimates based on the sex grid showing the

F) measures the strength of the relationship be- Iowest correlations with those based on the detween the two sets of estimates. It is a measure tailed grid.

000

15
REFERENCES

1National Center for Health stati~tic~: Synthetic

State Estimates of Disability. PHS Publication No. 1759.

Public Health Service. Washington. U.S. Government

Printing Office, 1968.

2U.S. Bureau of the Census: The Methods and

Materials of Demography, by H. S. Shryock, J. S. Siegel,

and associates. Washington. U.S. Government Printing

Office, 1973.

3U.S. Bureau of the Census: Developments in statistical estimation for local areas, by E. P. Ericksen.

Statistical Methodology of Revenue Sharing and Related

Estimated Studies. Census Tract Papers, Series GE40,

No. 10. Washington. U.S. Government Printing Office,

1974. pp. 51-56.

4Ericksen, E. P.: A method for combining sample

survey data and symptomatic indicators to obtain population estimates for local areas. Demopaphy 10:

137-160, 1975.

5Ericksen, E. P.: A regression method for estimating

population changes of local ‘areas. J. Am. Stat. Assoc.

69:867-875, 1974.

6Woodruff, R. S.: Use of a regression technique to

produce area breakdowns of the monthly national

estimates of retail trade. j. Am. Stat. Assoc. 61:

496-504, 1966.

7Lew, p. &: The use of mortality data in evaluating

synthetic estimates, in Proceedings of the American

Statistical Association 1971, Social Statistics Section.

Washington. American Statistical Association, 1974.

pp. 328-331.

8Gonzalez, M. E., and Hoza, C.: small Area Estimation of Unemployment. Presented at meeting of International Statistical Institute, Warsaw, Poland, 1975.

9Gonzalez, M. E., and Waksberg, J. E.: Estimation of

the Error of Synthetic Estimates. Presented at the first

meeting of the International Association of Survey

Statisticians, Vienna, Austria, Aug. 1973.

10Simmons, W. R.: Adjustment of Data-Synthetic

Estimates. Presented at the first meeting of the International Association of Survey Statisticians, Vienna,

Austria, Aug. 1973.

11U.S. BWeau of the Census: Use ad ev,~uation of

synthetic estimates, by M. E. Gonzalez. Statistical

Methodology of Revenue Sharing and Related Estimated

Studies. Census Tract Papers, Series GE-40, No. 10.

Washington. U.S. Government Printing Office, 1974.

pp. 46-50.

12Namekata, T., Levy, P. S., and O’Rourke, T. W.:

Synthetic estimates of work loss disability for each State

and the District of Columbia. Public Health Rep. 90:

532-538,1975.

13 National Center for He~th Statistics: State Esti.

mates of Disability and Utilization of Medical Services,

United States, 1969-71. DHEW pub. NO. (HRA)

77-1241. Health Resources Adrninktration. Washington,

U.S. Government Printing Office, Jan. 1977.

14 Gonz~ez, M. E.: Small mea estimation of unemployment, in Proceedings of the American Statistical

Association, Social Statistics Section. Washington.

American Statistical Association, 1975.

15Hansen, M., Hurwitz, W., and Madow, W.: Sample

Survey Methods and Theory? Vol. 1. Methods and

applications. New York. John Wiley & Sons, Inc., 1953.

16
APPENDIX

Proofs of Lemmas and Theorems ......................................................................................... .................. ..

L3nma I .................................................. ...................................... .....................................................

I%mna 2 ............. ........ .................................................................. ............................ ..........................

Lemma 3 ...................... .................................................................. .. .... .. .............................................

Theorem 4 ............ .............................. ...... ............ ...... .. .................. .. .................... ...... ...... ...................

Theorem 5 ....... ................................................ ............ ........ ................................................................

Theorem 6 ...... ........ .......................................... .................. .................................................................

Theorem 7 ..... .. ............ .............................. .............................................. .... ........ ...............................

18

18

18

19

19

20

21

22

“

17
APPENDIX

PROOF OF LEMMAS AND

Lemma 1: The expectation l?(~j ) of the nearly

unbiased estimator ~j is given by

We

E(~j:

where

= ~ n.j. ‘.j.in... j= 1

= average level of X in stratum j,

= the number of persons in stratum j,

I~j

l?~j. = ~ n,jix~jilnsj.

~= I

by

(1A)

THEOREMS

%

Proof

n~te that the expectation of ~j is give

= ~ ~5j. E(~)/n...

j= 1

= f ‘s-j.‘.j.lns..

j= 1

since Xj is

Lemma 2:

= average level of X in that portion of

stratum j which is in State s, ‘(X1 ) = Z ‘sj. (X.j. - ‘sj. )/nS..

j= 1

an unbiased estimator of ~.j..

QED

The bias B(~~ ) in the nearly unbiased estimate ~~ can be expressed

by

J’

and

or equivalently by

~5ji = the average level of characteristic X in J~. ‘Si ~ .

that portion of State s which is in PSU i B(X:)’=J2 5 Z( +5iix

)

sji “

of stratum j. s.. i=. 1 sj ‘sj-

(24)

(3A)

18
Proof

Since, by definition, the bias B(~j) of ~J is

equal to

where

J

l?~-. = ~ n~j. ‘T.fj. Ins..

j= 1

= average level of X in State S, relation (2A)

follows directly from lemma 1 and the

definition of ~~,..

We note that

(4A)

Therefore, relation (3A) follows from relation

(2A) by substitution of relation (4A) into relation (2A).

QED

Lemma 3: Let us assume that the ratio nSji/nSj.

is the same for all PSU’S in the same

stratum in the same State, which implies that

‘sji _ 1

‘–1— fori=l, . . . ,Isj.

%j. sj

(5A)

Then the bias ll(~j) in the “nearly” unbiased

estimate ~~ has the form given by

B(.X~) = ~ +s.. +sj i-1f (X.j. - Xsji). (6A)

j= 1

Proof

The results folIow direcdy from relation

(5A) and lemma 2.

Theorem 4:

QED

If?=+ for +1, . . . . Z’j, then

Sj. Sj

B* (~~), the square of the bias in

~S, is given by the expression

where

‘$ji = + $ (Xsji - ‘Sj .)2.

sj i- 1

+2X J ~ y..y.

~ (-yj ‘-r,j,) S (-y~.-xs~i)

j=l k<j s.. sj sk i=l .

(7A)

i= 1

Proof

By adding and subtracting ~Sj to the right-

..

hand side of relation (6A) we obtain

B(x;) =}g & ++

-. Sj

19
Squaring the right-hand side of relation (8A), we obtain

{

Jn. l%

/

B2(X:)= ~~ :1— X_ [(x.j.-xsj.) + (Xsj.-xsji)l 2

s.. Sji–1

\

2

(X.j.-Xsj_) + (Xsj.-J@] + 2 i z

[--1[: :

~f~~~ ~, (X.j ‘Xsji) Z (y.k.-~$ki) j=l k<j s.. sjs

‘,$ ‘$+ $ (X,j.‘Xsj. +X Sj.- ‘sji)’ + 2,$ #~ f X (X.j.-xsji) (X.j.-xsji’)

.. ~j ~= “-

. . sj i-1 i’<i

Ig- isk

“ ‘xsji)~l (X.k.-xski)” + 2 k x yin: “ z (z]. (9A)

j= 1 k<j ‘,s.. sj sk i=l

But the first term in equation 9A

by

[ 1

=,$ $ + $ (X,j. - Xoj.)’ + : (Xsji-xsje)’

.. sj i=l ● i= 1

= i * + [(xsj. - ‘j.)’ + ‘;jil -

(1OA)

j=l s.. sj

Therefore, by appropriate substitution of relations (1OA) into (9A), we obtain the form specified by equation (7A).

QED

Theorem 5: If the Pja are independent of the

X:, then the variance of ?S given

‘i=var(ilp’ffx’)

reduces to

+ 2 ~ P5aPsafCov (x:,x:’). (12A

U<a’

for large values of n

20
Proof

The second term in the right-hand side of

equation (11A) is given by

Cov (P;a x:, P;a, x:,)

= E(P:a ~: P;aI ~:1) - E(P:a X:)E(p;aI ~:’)

= E(P;a P:.) E(~: %)

- E(P;a )E(P;a,)E(~: )E(~:f) (13A)

since Pja is independent of ~~.

But

E (P;. ),= p~a (the true proportion of States faUing into cell a), and

where

n~ = the sample size in State s used for estimating the PS&(e.g., for a State having 1 million

persons and for the I-Percent Public Use

SampIe Tapes, n, = 10,000).

Therefore,

E(P:a P:a,) = cov (P;a , P:aI) +E(p;a) E(p&I)

n~-1

=—P$a P,a, (15A)

ns

and for large n$

E(P;a P;aI) A Psa Psa’. (16A)

Therefore, from equations (13A) and (16A)

Cov (P:a x;, P:a, ~:t) =P~a P~a, [E(~a ~:t)

- E(~: )E(~:t)J = PSaP$a#COV(-$?:,~;I). (17A)

Hansen, Hurwitz, and Madowl 5 show that

Var (P;a x:) + P:a X:

which reduces to

Since

and this reduces further to a form given by

Substituting equations (1 7A) and (19A) into

equation (11A),

+ 2 ~ P,a Psa,Cov (x:, X:,).

Cl <C?’

QED

Theorem 6: Let ~S estimate a p~ameter ~S

with bias given by 13(XS) and let

~j be an unbiased estimate of ~,

which is uncorrelated with ?~.

Then the foIIowing relation is true

NOTE: A list of references follows the text.

21
where

MSE~, = the mean square error of ?,

and

Proof

E(x; -Yy =E [(x: -x.) + (x. -2,)]2

=L7(jf; - 5?,)2 +E(x, - 5J2

~

- 2E(X; - x,) (x$ - x.)

but

and

AlSO, it can be shown that

~

E(i; -~, ) (~, ‘~~)=cov (~;,~,)

+B(x:) I@, ) = o

since ~J and *, are uncorrelated and ll(~j ) = O.

Theorem 7:

QED

‘If ?$ is an estimate of ~, with bias

-.

given by 11(~~), if ~~ is an unbiased~estimate of x$ uncorrelated

with X*, and if & is an unbiased

z;

estimate of 02 , then the estimate

z;

M$E; given by

s

M$E~$ = (~; - X$)2 - 6;, (21A)

s

is an unbiased estimate of MS13~.

s

Proof

Proof follows directly from theorem 6.

E(X, - ~s )2 = MSE~; QED

000

22

IW3.GOVERNMENT PRINTINGOFFICE 1977-260-937:35

m
series1.

series 2.

Sers”es3.

Sers”es4.

Sers”es10.

Sers”es11.

VITAL AND HEALTH STATISTICSPUBLICATIONS SERIES

Formerly l%blic Haslth Setvsce Publication No. 1000

programs and Collection hocedures. –Reports which daaibe the general programs of the National

Center for Health Statistia and its offices and divisions, data collection methods used, defiitiona, and

other material necessary for understanding the data.

Data Evaluation and Methods Research. –Studies of new statistical methodology including experimental

teats of new survey methods, studies of vital statistics collection methods, new analytical techniques,

objective evaluations of reliability of collected data, contributions to statistical theory.

Andy tical Studies. –Reports presenting analytical or interpretive studies baaed on vital and health

statistics, carrying the analysis further than the expository types of reports in the other series.

Documents and Committee Reports. -Final repofis of major committees concerned with vital and

health statistia, and documcnts such as recommended model vital registration laws and revised birth

and death certificates.

Data from the Health Interos”ew Surwy.-Statistics on illness; accidental injuries; disability; use of

hospital, m&kal, dental, and other services; and other”health-related topics, based on data collected in

a continuing national household isiterview survey.

Data from the Health Examination Survey.-Data from direct examination, testing, and measurement

of national samples of the civilian, nonixkutionzlized population provide” the b&& for two types of

reports: (1) estimates of the medically defined prevalence of 6pecKlc diseases m the United States and

the distributions of the popukkon with respect to physical, physiological, and psychological characteristic; and (2) analysis of relationships among the various measurements without rekrence to an

explicit ftite universe of persons.

Sers”es12. Data from the Institutionalized Population Surveys. –Discontinued effective 1975. Future repmts from

these surveys will be in Series 13.

Sers”es13. Data on Health Resources Utilization. –Statistics on the utilization of health manpower and facilities

providing long-term care, ambulatory care, hospital care, and family pkuming services.

Series 14. Data on Health Resources: Manpower and Facilities. –Statistics on the numbers, geographic distribution, and characteristics of health resources including physicians, dentists, nurses, other health occupations, hospitals, nursing homes, and outpatient facilities.

Seri”es20. Data on Mortality. –Various statistia on mortality other than ss included in regular annual or monthly

reports. Special an~yses by cause of death, age, and other demographic variables; geographic and time

series analyses; and statistics on characteristics of deaths not available from the vital records, based on

sample surveys of those records.

Serses 21. Data on Natality, Marrhge, and Divorce. –Various statistia on natality, marriage, and divorce other

than as included in regular annual or monthly reports. Special analyses by demographic variables;

geographic and time series analys=; studies of fertility; and statistics on characteristics of births not

available from the vital records, based on sample surveys of those records.

Series 22. Data from the National Mortality and Natality Surveys. –Discontinued effective 1975. Future reports

from these sample surveys based on vital records will be included in Series 20 and 21, respectively.

Series 23. Data from the National Survey of Family Growth. –Statistics on fertility, family formation and dissolution, family planning, and related maternal and infant health topics derived from a biennial survey of

a nationwide probability sample of ever-married women 1544 years of age.

* For a list of titles of reports published in these series, write to: Scientific and Technical Information Branch

National Center for Health Statistics

Fublic Health Service

Hyattsville, Md. 20782

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