Table Of ContentLecture Notes in Statistics
Vol. 1: R. A. Fisher: An Appreciation. Edited by S. E. Fienberg and D. V. Hinkley.
xi, 208 pages, 1980.
Vol. 2: Mathematical Statistics and Probability Theory. Proceedings 1978. Edited
by W. Klonecki, A. Kozek, and J. Rosinski. xxiv, 373 pages, 1980.
Vol. 3: B. D. Spencer, Benefit-Cost Analysis of Data Used to Allocate Funds.
viii, 296 pages, 1980.
Springer Series in Statistics
L. A. Goodman and W. H. Kruskal, Measures of Association for Cross Classi
fications. x, 146 pages, 1979.
J. O. Berger, Statistical Decision Theory: Foundations, Concepts, and Methods.
vi x, 420 pages, 1980.
Lecture Notes in
Statistics
Edited by S. Fienberg, J. Gani, J. Kiefer,
and K. Krickeberg
3
Bruce D. Spencer
Benefit-Cost Analysis of
Data Used to Allocate Funds
Springer-Verlag
New York Heidelberg Berlin
Professor Bruce D. Spencer
Northwestern University
The School of Education
Evanston, Illinois 60201/USA
AMS Subject Classification: 62P25
Library of Congress Cataloging in Publication Data
Spencer, Bruce D
Benefit-cost analysis of data used to allocate
funds.
(Lecture notes in statistics; 3)
Based on the author's thesis, Yale.
Bibliography: p.
1. Revenue sharing-United States-Cost
effectiveness. I. Title. II. Series.
HJ275.S59 336.1 '85 80-19589
ISBN-13: 978-0-387-90511-2 e-ISBN-13: 978-1-4612-6099-6
001: 10.10071978-1-4612-6099-6
All rights reserved.
No part of this book may be translated or reproduced in any form
without written permission from Springer·Verlag.
© 1980 by Springer-Verlag New York Inc.
Softcover reprint of the hardcover 1s t edition 1980
9 8 7 654 321
Preface
This monograph treats the question of determining how much to
spend for the collection and analysis of public data. This difficult
problem for government statisticians and policy-makers is likely to
become even more pressing in the near future. The approach taken here
is to estimate and compare the benefits and costs of alternative data
programs. Since data are used in many ways, the benefits are hard
to measure. The strategy I have adopted focuses on use of data to
determine fund allocations, particularly in the General Revenue
Sharing program.
General Revenue Sharing is one of the largest allocation programs
in the United States. That errors in population counts and other data
cause sizable errors in allocation has been much publicized. Here
we analyze whether the accuracy of the 1970 census of population and
other data used by General Revenue Sharing should be improved. Of course
it is too late to change the 1970 census program, but the method and
techniques of analysis will apply to future data programs. In partic
ular, benefit-cost analyses such as this are necessary for informed
decisions about whether the expense of statistical programs is justi
fied or not. For example, although a law authorizing a mid-decade
census was enacted in 1976, there exists great doubt whether funds will
be provided so a census can take place in 1985. (The President's
Budget for 1981 allows no money for the mid-decade census, despite
the Census Bureau's request for $1.9 million for planning purposes.)
Is this decision in the national interest? Explicit benefit-costs
analyses are severely needed.
Much of the present work is technical. The reader interested solely
in policy implications is encouraged to read chapter 0 lightly and
then proceed directly to chapter 7. For a thorough but non-technical
course, I suggest chapter 0; chapter 1, first four sections at least;
chapter 3 sections 1 and 2, then other sections as interest dictates;
the first section of chapter 4; the first section or two of chapter 5;
then chapters 6 and 7, skipping proofs all the while. The manuscript
iii
-iv-
is heavily cross-referenced, so the reader should feel free to skip
around, since work that is used later will be referenced at that time.
In addition, guideposts placed throughout the work indicate parts that
can be skipped on a first reading. An overview of the work is given
in chapter O.
Acknowledgments
This monograph has arisen from my doctoral dissertation for the
Department of Statistics at Yale University. I wish to express my
indebtedness to I. Richard Savage, who led me to the problem and
provided continuing guidance and stimulation. I am grateful also to
William Kruska1, Stephen Dresch, Francis Anscombe, and Thomas Jabine
for reading the manuscript and for helpful comments.
This research was largely carried out while I was at Yale, where
I was partly supported by university fellowships and by a National
Science Foundation grant (SOC 75-15614). I wrote the final draft while
at the Committee on National Statistics of the National Research
Council - National Academy of Sciences and while at Northwestern Uni
versity. I am grateful to colleagues of all three places for their
encouragement and support.
Evanston, Illinois B. D. S.
March 1980
TABLE OF CONTENTS
Preface iii
List of Tables vii
Chapter 0 Introduction 1
Chapter 1 Loss Function and Benefit Measurement 9
§1.0 Introduction 9
1.1 Utility and Social Welfare 12
1.2 Equity 18
1.3 A Simple Loss Function for
Errors in Allocation 25
1.4 Estimating the Parameters of the
Loss Function L(f.e} for GRS 31
1.5 Fisher-Consistency and Other
Properties of the Loss Function
L(t.e} 33
1.6 Hore General Loss Functions:
~(~,~) , Lw(~'~} 41
1.7 Exponential Loss Functions 47
Chapter 2 The Delta Method 53
§2.0 Outline 53
2.1 Notation 53
2.2 Description of the Del ta Meth cxl 55
2.3 ApplicabilIty of the Delta Method
to GRS 60
2.4 Application of the Delta Method
to Calculating Expected Loss 63
Chapter 3 Data Used in General Revenue Sharing 67
§3.l Introduction 67
3.2 Population (state level) 76
-v-
-vi-
3.3 Urbanized Population 83
3.4 Population (subs tate level) 88
3.5 Total Money Income. Per Capita
Income (state level) 97
3.6 Per Capita Income (substate level) 103
3.7 Personal Income 117
3.8 Net State and Local Taxes 120
3.9 State Individual Income Taxes 123
3.10 Federal Individual Income Tax
Liabilities 126
3.11 Income Tax Amount (state level) 128
3.12 Adjusted Taxes (subs tate level) 132
3.13 Intergovernmental Transfers
(substate level) 136
3.14 Interrelationships Among State-
level Data Elements 138
3.15 Interrelationships Among Substate
Data Elements 143
3.16 Outline-Symbol Glossary 147
Chapter 4 Interstate Allocation in GRS 151
§4.l Introduction 151
4.2 Notation 154
4.3 Conclusions 158
4.4 Assumptions and Degrees of
Approximation 166
4.5 Derivations and Proofs 169
Chapter 5 Intrastate Allocations in GRS 177
§5.l Overview 177
5.2 Determination of Substate
Allocations 177
5.3 Notation 190
5.4 Assumptions and Degrees of
Approximations 197
5.5 Conclusims 199
-vii-
Chapter 6 Computations and Analyses 207
§6.l Introduction 207
6.2 Errors in Allocation to States 210
6.3 Substate Errors in Allocation 224
6.4 Benefit Analysis 239
6.5 Summary of Findings 247
Chapter 7 Policy Perspectives and Recommendations 250
§7.0 Introduction and Summary 250
7; 1 Redundancy of Allocation Formulas 252
7.2 Data Burdens Imposed by Legislation 254
7.3 Minimizing Large Errors in
Allocation 255
7.4 Uniform Biases, Non-uniform
Variances 256
7.5 Adjusting for Undercoverage or
Other Suspected Biases 257
7.6 Improving the Coverage of the
Decennial Census 260
7.7 State Errors vs. Substate Errors 262
7.8 Allocation Formulas as
Approximations 264
7.9 Other Uses of Data 265
Appendix A Tables of Biases in Data 266
Appendix B Determination of General Revenue
Sharing Allocations 271
Technical
Appendix 275
Bibliography 284
LIST OF TABLES
3.1 Target Variables - State Population 77
3.2 Target Variables - Subs tate Population 90
3.3 Target Variables - State Money Income, State Per Capita Income 98
3.4 Target Variables - Personal Income 118
3.5 Target Variables - Net State and Local Taxes 121
3.6 Target Variables - State Individual Income Taxes 124
3.7 Target Variables - Federal Individual Income Tax Liabilities 126
3.8 Target Variables - Subs tate Adjusted Taxes 133
4.1 Locator for Description of Data 154
5.0 Substate Units Constrained in New Jersey, EP 1 189
5.1 Data Element References 190
5.2 Subs tate Allocation Formulas 193
6.0 Total Allocations for GRS, EP's 1-11 209
6.1 Error Models Used for Sensitivity Analysis of State Allocations 212
6.2 Moments of Errors in GRS Allocations to States, EP 1 215
6.3 Moments of Errors in GRS Allocations to States, EP 6 216
6.4 Sums of Mean Absolute Errors in GRS Allocations to States, EP 1 217
6.5 Sums of Mean Absolute Errors in GRS Allocations to States, EP's 3,4,6 218
6.6 GRS Allocations to States Based on 1970 Census Reported Data and
1970 Census Data Corrected by the Basic Synthetic Method 222
6.7 GRS Allocations to States Based on 1970 Census Reported Data and
1970 Census Data Corrected by a Modified Synthetic Method 223
6.8 Moments of Errors in New Jersey County-Area Shares, EP 1 225
6.9 Moments of Errors in New Jetsey County-Area Shares, EP 6 228
6.10 Changes in GRS Allocations to County Areas in California After
Adjusting for Underenumeration 229
6.11 Moments of Errors in Allocation to Places in Essex County, EP 1 235
6.12 Moments of Errors in Allocation to Places in Essex County, EP 6 236
6.13 Sums of Mean Absolute Errors in Allocation to County Areas in
New Jersey 237
6.14 Sums of Mean Absolute Errors in Allocation to Places in Essex County 238
A1 Estimates of State-level Relative Biases in Population, Urbanized
Population, and Money Income 267
A2 Estimates of Undercoverage Rates in New Jersey Counties 269
A3 Estimates of Undercover age Rates in Places and Municipalities in
Essex County, New Jersey 270
Chapter O. Introduction
In 1970 the U.S. census cost more than $220 million and
failed to count an estimated 5.3 million people. The 1980 census
will cost a billion dollars and the size of the undercount, or the
number of persons missed, is also certain to be in the millions.
If the undercount were evenly distributed among geographic regions
and population subgroups, matters would not be as serious as they
are. In fact, the undercount is spread unevenly, with certain
minority groups and geographic regions undercounted more than
others. Census data are used in more than 100 formulas that
allocate over $50 billion annually in federal aid. A greater
than average undercount for a group or region can deny it its
fair share of these funds. Since census data also determine
apportionment of seats in the House of Representatives, under
count can cause a state to gain or lose a seat. National ethnic
organizations worry that under counts diminish their political
influence by failing to indicate their true numbers.
Errors in census data clearly have significant impact, but
how much should be spent to get good data? The 1970 census missed
an estimated 2.5 percent of the population. Would it have been
worthwhile to spend an extra $10 million to reduce the undercount?
An extra $100 million?
Similar questions, not just about censuses but about other
data collection activities as well, are relevant to statistical
agencies in general, already hard pressed by increasing demands
for a multitude of data. And the present conflict between demands
and scarce resources can be expected to worsen as the demands
for d~ta outstrip the resources for obtaining and providing the
data. The preferred way to cope with this problem is rational
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Description:This monograph treats the question of determining how much to spend for the collection and analysis of public data. This difficult problem for government statisticians and policy-makers is likely to become even more pressing in the near future. The approach taken here is to estimate and compare the
