United States
Environmental Protection
Agency
Motor Vehicle Emission Lab
2565 Plymouth Rd.
Ann Arbor, Michigan 48105
EPA-460/3-81-001
February 1981
Air
&EPA
Individual Manufacturer
Procedures to Establish
Fuel Economy Adjustment
Factors
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EPA-460/3-81-001
INDIVIDUAL MANUFACTURER PROCEDURES
TO ESTABLISH FUEL ECONOMY ADJUSTMENT
FACTORS
by
Falcon Research & Development Co.
One American Drive
Buffalo, New York 14225
Contract No. 68-03-2835
EPA Project Officer: Jack Schoenbaum
Prepared for:
ENVIRONMENTAL PROTECTION AGENCY
OFFICE OF AIR, NOISE AND RADIATION
OFFICE OF MOBILE SOURCE AIR POLLUTION CONTROL
EMISSION CONTROL TECHNOLOGY DIVISION
CONTROL TECHNOLOGY ASSESSMENT
AND CHARACTERIZATION BRANCH
ANN ARBOR, MICHIGAN 48105
February 1981
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This report is issued by the Environmental Protection Agency to disseminate technical
data of interest to a limited number of readers. Copies are available free of charge to
Federal employees, current contractors and grantees, and nonprofit organizations—in
limited quantities—from the Library, Motor Vehicle Emission Laboratory, Ann Arbor,
Michigan 48105, or, for a fee, from the National Technical Information Service, 5285
Port Royal Road, Springfield, Virginia 22161.
This report was furnished to the Environmental Protection Agency by Falcon Research &
Development Co., One American Drive, Buffalo, New York 14225, in fulfillment of Contract
No. 68-03-2835. The contents of this report are reproduced herein as received from
FALCON R&D i, Inc. The opinions, findings, and conclusions expressed are those
of the author and not necessarily those of the Environmental Protection Agency. Mention
of company or product names is not to be considered as an endorsement by the En-
vironmental Protection Agency.
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FALCON RESEARCH
Falcon Research & Development Co
A Subsidiary of Whittaker Corpora'on
One American Drive
Buffalo. New York 14225
716/632-4932
WhrttakeR
ERRATA
"INDIVIDUAL MANUFACTURER PROCEDURES
TO
ESTABLISH FUEL ECONOMY ADJUSTMENT FACTORS'
Falcon R&D Report 3520-4/BUF-42
Final Report
February 1981
Page 64: Last equation should read:
\ - Vu
Page 64: Second last line should read
"The v represent ..."
K
Prepared: February 24, 1981
By: S. Kaufman
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FALCON RESEARCH
Falcon Research & Development Co
A Subsidiary of Whittaker Corporation
One American Drive
Buffalo, New York 14225
716/632-4932
\\ThittakeR
INDIVIDUAL MANUFACTURER PROCEDURES
TO
ESTABLISH FUEL ECONOMY ADJUSTMENT FACTORS
Report 3520-4/BUF-42
Final Report
Prepared for
ENVIRONMENTAL PROTECTION AGENCY
ANN ARBOR, MI 48105
Prepared Under
Contract 68-03-2835
Task Order No. 4
Prepared by: S. Kaufman
Date: February 1981
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TABLE OF CONTENTS
Section Ti tle Page
1 INTRODUCTION 1
1.1 Background 1
1.2 Scope of Work 2
2 SUMMARY 4
3 FUEL ECONOMY ADJUSTMENT BASED ON VEHICLE DESIGN
PARAMETER VARIATIONS 6
3.1 Fuel Economy Mathematical Model 8
3.2 Estimation of Sensitivity Coefficient Functions
from a Fuel Economy Data Set 10
3.2.1 Derivation of Sensitivity Coefficient
Data Sets 10
3.2.2 Functional Estimation 13
3.2.3 Results for a Specific Data Set 15
3.3 Basic Considerations for Individual Manufacturer
Procedure 17
3.3.1 Accuracy 17
3.3.2 Representativeness 24
3.4 Draft Procedure for Individual Manufacturer
Coefficients 25
3.4.1 Data Requirements 26
3.4.1.1 Minimal Set 26
3.4.1.2 Additional Vehicle Tests 26
3.4.1.3 Definitions 27
3.4.2 Design Parameter Sensitivity Coefficient
Estimation 28
3.4.2.1 Test Weight 28
3.4.2.2 Axle Ratio 35
3.4.2.3 Road Load Horsepower 36
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Table of Contents (Continued)
Section Title Page
3 FUEL ECONOMY ADJUSTMENT BASED ON VEHICLE DESIGN
PARAMETER VARIATIONS (Continued)
3.5 Modified Draft Procedure if Sensitivity Coeffi-
cients are Assumed to be Constant (Parameter-
Independent) 36
3.5.1 Data Requirements 36
3.5.2 Design Parameter Sensitivity Coefficient
Estimation 36
4 FUEL ECONOMY ADJUSTMENT TO REFLECT IN-USE EXPERIENCE 39
4.1 Road Adjustment Factor Estimation from In-Use
Surveys 40
4.2 Treatment of Sample Space Heterogeneity 48
4.3 Confidence Intervals for Medians 52
4.4 Basic Considerations for Individual Manufacturer
Procedure 55
4.4.1 Accuracy 55
4.4.2 Representativeness 58
4.5 Draft Procedure for Individual Manufacturer Road
Adjustment Factors 59
4.5.1 Data Requirements 59
4.5.2 Adjustment Factor Estimation 61
5 ADDITIONAL ISSUES 63
5.1 Computation of Design Parameter Adjusted Fuel
Economy 63
5.2 Regional/Seasonal Adjustment Factors 65
5.3 Test and Parameter Adjustment Strategies 67
REFERENCES 70
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Table of Contents (Continued)
Section Title Page
Appendix A VARIABILITY OF DYNAMOMETER FUEL ECONOMY
MEASUREMENTS 71
Appendix B FUEL ECONOMY LABEL ERROR DUE TO NORMAL ERROR
BEFORE ROUND-OFF 77
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ACKNOWLEDGMENT
The author gratefully acknowledges the many hours of
detailed discussions with his colleagues at Falcon Research,
H. T. McAdams and Norman Morse, and, in particular, their
critical reading of this report. Much appreciation is also
expressed for the general guidance provided by J. D. Murrell
of EPA, for magnetic tape data supplied by John Foster of
EPA, and for technical comments received from EPA personnel
in the review process.
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1. INTRODUCTION
This report is submitted as a deliverable on Task Order No. 4,
"Fuel Economy Adjustment Factors," of Contract 68-03-2835 with the
Environmental Protection Agency (EPA).
1.1 Background
The EPA has issued an Advance Notice of Proposed Rulemaking (Federal
Register, Vol. 45, No. 190, September 29, 1980, pp. 64540-64544) with the
objective of improving "the usefulness of vehicle fuel economy labels and
the accuracy and completeness of the data used for determining corporate
average fuel economy (CAFE) levels for new passenger vehicles and light
trucks." Two of the ten regulatory options noted as being considered for
this purpose are Design Factor Labeling and Shortfall Factor Labeling.
The first would apply specific adjustment factors to normally available
laboratory measured fuel economy test results in order to more closely
estimate the fuel economy of (untested) design variations. The second
option would apply an adjustment factor to each label value to account
for the average industry difference (or "shortfall") between in-use
experience and laboratory-measured fuel economy.
In connection with these two potential regulations, the EPA is
considering developing procedures for manufacturer-specific adjustment
v
factors for EPA fuel economy labeling: both /ehicle design adjustment
factors as well as in-use road adjustment factors. These procedures
could be followed by a manufacturer if it feels that the adjustment
factors provided by EPA regulations are not appropriate for its own
vehicles.
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EPA entered into a Task Order Agreement with Falcon Research and
Development Company under which Falcon would perform the necessary
engineering and statistical analyses to develop such procedures. The
scope of work for this task follows.
1.2 Scope of Work
The proposed work will entail developing a method whereby an
automotive manufacturer may develop fuel economy adjustment factors, for
both vehicle design and road use, based upon the manufacturer's analysis
of test data representative of his vehicles. The vehicle design adjust-
ment factors will address the following technical parameters:
(a) Axle ratio;
(b) Road load horsepower;
(c) Estimated test weight.
The contractor shall develop a method for determining the quantity
and nature of fuel economy data required to constitute a representative
and statistically valid sampling of a manufacturer's vehicle fleet for
purposes of design parameter sensitivity specification. The data analysis
methods to be used by manufacturers shall also be specified by the
contractor.
The above noted parameter effects on fuel economy shall be noted for
both the EPA City and Highway cycles.
In addition, the Contractor shall review the EPA 404 Report (draft
copy) for familiarization with the effects of road use on vehicle fuel
economy. After reviewing the EPA 404 report, the contractor shall define
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the quantity and nature of in-use road fuel economy data, and analysis
techniques, required in manufacturer development of road adjustment
factors.
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2. SUMMARY
This report addresses the question of automotive vehicle fuel economy
as influenced by selected vehicle design parameters and by conditions
which differentiate the on-road environment from the test environment.
The central thrust of the report is the formulation of applicable fuel-
economy adjustment factors in the context of a specific manufacturer's
product line. The design parameter fuel economy adjustment problem is
treated in Section 3. The on-road fuel economy adjustment problem is
treated in Section 4. Section 5 considers a number of additional relevant
issues.
The background discussion in Section 3 develops a fuel economy
mathematical model in which the derivation and role of design parameter
sensitivity coefficients is clarified. A general procedure for estimation
of sensitivity coefficients from fuel economy data is presented. A
significant issue raised is whether the sensitivity coefficient for each
of the parameters (test weight, RLHP, axle ratio) should be expressed
as a linear function of the parameter value at which it is to be applied
or can be adequately represented by a constant value. The advantage of
the constant value form lies in the simplicity and relative precision of
estimation, but counterbalancing is the potential loss of accuracy of
adjustment. Analysis of 1980 General Label File data suggests that non-
zero slopes of the estimated sensitivity coefficient lines may have only
marginal statistical significance. It is recommended that EPA carefully
review its entire data set in order to decide this question.
The rationale and requirements for a procedure to estimate manu-
facturer-specific sensitivity coefficients is next presented. The section
concludes with a draft procedure which covers data requirements and data
analysis. Two alternatives are considered. The first is based on the
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assumption that each sensitivity coefficient is a linear function of its
parameter. This is the assumption on which the EPA protocol is based. The
second employs the simpler assumption that the sensitivity coefficients
are not dependent on design parameters and can thus be treated as constants.
The background discussion in Section 4 reviews the concept of a
numerical factor, derived from in-use vehicle surveys, which when multiplied
into fuel economy label values brings these more into line with actually
achieved fuel economy. The statistical objective is to achieve a match
with the median in-use fuel economy. The present EPA method for estimating
such factors for FTP (city) and HFET (highway) conditions is reviewed and
an alternative method developed which more completely utilizes survey
response data.
A critical issue in this problem is the heterogeneity of the sample
space—due to variable environmental factors which greatly influence in-
use fuel economy. A stratification procedure is recommended to ensure
representativeness of the survey data sets to be used for road adjustment
factor estimation. This leads to the employment of weighted median
estimations.
The survey requirements for use in a procedure to estimate manufacturer-
specific road adjustment factors is next presented. The section concludes
with a draft procedure which covers survey design and data analysis.
Provision is made for estimation of the factors by the present EPA approach
or by a new method developed earlier in the section.
Issues discussed in Section 5 include: (1) Procedures for applying
sensitivity coefficients to predict fuel economy of untested subconfigura-
tions; (2) A public information program to enable individuals to make their
own regional and seasonal adjustments for on-road fuel economy; and (3) The
alternative strategies available to a manufacturer of establishing a revised
mpg value for a subconfiguration by direct test vs. estimation of manu-
facturer-specific sensitivity coefficients.
5
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3. FUEL ECONOMY ADJUSTMENT BASED ON VEHICLE DESIGN PARAMETER VARIATIONS
The present method of vehicle fuel economy* labeling by EPA averages
test results over diverse configurations which, because of design/test
parameter differences, are really not expected to have the same fuel
economy. In the interest of achieving more accurate labeling, EPA is
currently developing fuel economy adjustment factors that would explicitly
account for variations in three significant vehicle design/test parameters:
vehicle test weight (inertia setting of the test dynamometer), road load
horsepower (dynamometer setting at 50 mph test speed), and axle ratio.
The EPA adjustment procedure is intended to be uniformly applicable to
all manufacturers. However, a particular manufacturer could conceivably
argue that its own vehicles are distinctly different as a class. Therefore,
manufacturers should have the option of substituting alternative adjustment
factors applicable to their own vehicles, so long as these factors satisfy
appropriate criteria.
Selection of these criteria demands a careful enunciation of the
intent of any protocol dealing with manufacturer-specific sensitivity
coefficients as opposed to those promulgated by EPA. The view taken in
this report is that whether the manufacturer-estimated coefficients are
significantly different from the EPA-promulgated values is not an issue,
nor does it need to be. Rather than considering manufacturer-specific
coefficients in a hypothesis-testing context, one simply requires that
the coefficients be estimated to some specified level of precision
consistent with the aims of the fuel-economy labeling program.
* Throughout this section the term "fuel economy" will be construed to
mean FTP and/or HFET fuel economy as measured by chassis
dynamometer testing.
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This section develops two alternative recommended procedures to be
followed by individual manufacturers who wish to challenge the EPA standard
adjustment factors. (The choice between the two alternatives hinges on the
specific form of the standard adjustment factors^) To lay the groundwork
for these procedures, a background discussion of adjustment factor
methodology is presented, along with some implications from available data.
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3.1 Fuel Economy Mathematical Model
The measured FTP or HFET fuel economy of a vehicle is viewed as
determined by the following factors:
(1) Basic Engine
(2) Engine Code
(3) Transmission Class
(4) Transmission Configuration
(5) Test Weight
(6) Road Load Horsepower
(7) Axle Ratio*
(8) Error Factor**
A unique combination of factors (1) through (7) is denoted by EPA as a
vehicle subconfiguration,and all vehicles having this combination are
essentially (though not precisely)*** identical design copies. The term
"subconfiguration" is used because "configuration," as defined in the
EPA regulations, refers to a unique combination of only factors (1) through
(4), (7), and inertia weight class (which is close to but generally not
equivalent to (5), test weight).
* The vehicle's N/V (engine rpm to vehicle speed (mph) ratio in highest
gear) is probably the more fundamental parameter. However, axle ratio
is a more accessible design parameter and, for a given transmission
class/configuration and assumed fixed tire size, axle ratio determines
N/V. The extent of variation in tire size among vehicles with same
factors (1) through (6) is believed to be small.
** Includes both measurement errors and vehicle-to-vehicle variability
within a subconfiguration.
*** For example, individual vehicle alternatives with curb weights
differing by as much as 250 Ibs. because of body differences could
have the same test weight.
8
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We shall find it convenient in this exposition to define a different
aggregation of vehicles, namely, into unique combinations of factors (1)
through (4), which we denote "design families." Note that "design family"
fixes all design characteristics of a vehicle except the three parameters
for which fuel economy adjustment factors are to be determined.
Our basic assumption is that within any design family the remaining
three factors combine multiplicatively according to the following model:
E = K • W(w) • R(r) • A(a) • (1 + e).
In this equation E is the measured fuel economy of a vehicle sampled from
the specific subconfiguration defined by: design family i, test weight w,
RLHP r, and axle ratio a. Included in the model is a random error e
whose expected value is zero. If we drop the error factor (1 + e), then
EQ = K. • W(w) • R(r) • A(a)
represents the true mean fuel economy within subconfiguration (i, w, r, a).
Taking logarithms of both sides, we obtain
In E = In Kj + In W(w) + In R(r) + In A(a).
A convenient way of representing the structural relationship between
EQ and design parameters w, r, a is to take the total differential of
In E_ with respect to w, r, a, and write
dW dR dA
0 dw . . dr da
1 = IT ' Aw + ~T" ' Ar + ~T"' Aa
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Then, defining sensitivity coefficients:
w EQ 3w W dw
«
_ _ _r 0 r dR
r EQ 9r R dr
3E0
_ _ _a_ 0 _ a dA
a E_ 8a A da
one can write:
U C* *-iW - £• i-i I .^ p. L-iQ
E w w r r a a
Each sensitivity coefficient expresses the percentage change in resulting
fuel economy per unit percentage change in design parameter value.
Knowledge of Sw, Sp, Sa therefore permits estimation of the percentage
change in EQ due to any combination of small design perturbations in
test weight, RLHP, and axle ratio. As seen from the assumed model,
Sw, Sr, Sa are functions of w, r, a, respectively. We describe next
the construction of a reasonable approach to estimation of these
sensitivity coefficient functions.
3.2 Estimation of Sensitivity Coefficient Functions from a Fuel Economy
Data Set
3.2.1 Derivation of Sensitivity Coefficient Data Sets
Each test record includes the tested vehicle's design family
designation, design parameters (w, r, a), and measured fuel economy E.
Generally, there will be both an FTP and HFET measurement, but the method
of analysis is the same for each. Partition these records into groups
10
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containing only identical subconfigurations except possibly for test weight.
Within each group, collapse all tests on vehicles with same test weight (hence,
same subconfiguration) into a single composite test by calculating a mean
fuel economy. Next, within each group containing at least two different
test weights, order according to increasing test weight, i.e., (Ep Wj, rrij),
(Eo, W2> mp), ..., (E^, w^, m^), where w, < w^ < ..., < w^ and m- is the
test multiplicity (from the above collapsing procedure) associated with
test weight Wj. Define k - 1 estimates of Sw and associated fractional
weight differences Aw as follows:
w.
Aw. =
- w.
J
w.)
- E1
*(EJ+1
w,j
Aw
J
It is important at this point to determine the relative precision of the
s**
computed S .. Our model', assumes that errors arise only from measured
w, j
fuel economy E, which has a fixed coefficient of variation a_ with
respect to the mean fuel economy of the subconfiguration to which the
vehicle belongs. (Both measurement errors and vehicle-to-vehicle differences
contribute to this variability.) Then, the variances of w-+Jh(w.+l + w.)
and w.A(w. +w.) are approximately c^/m.+1 ar>d cr^/m respectively.
J J^J- J ^^ J ^ ^ J
It can then be shown that the variance of S . is given approximately by
w, j
m.
m... + m.
J+l J.
U
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Hence, we define variance reduction factor u. given by
J
m_.. ,m.
(AW,)2
"Vl + mj
This result depends on the assumption of test-to-test error independence,
which is reasonable if all collapsed individual test results are from
different vehicles.
After completing the above operations for all comparable configuration
groups, pool the results from all of the'groups to provide a derived data set
{S ., w., u.} i = 1, ..., n
w,i i i
Repeat the above process for the other two design parameters, road
load horsepower and axle ratio, substituting r and a, respectively,
in place of w at each step in the procedure. This results in
{S ., r., u'.} i = 1, ..., n1
I j 1 1 I
{S ., a., u1.1} i = 1, ..., n"
a 5 i I 1
12
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3.2.2 Functional Estimation
At this point, it is appropriate to analyze the data qualitatively in
order to assess the likely form of the regressed functional dependence of
each sensitivity coefficient on its respective design parameter. If such
analysis supports the possibility of a linear relationship in each case,
or is at least not inconsistent with that assumption, then quantitative
linear regressions may be performed to estimate regression lines which
are of the form
S = a + 3 w
Sr = Y + 6 r
S = n + 8 a
a
Given the heteroscedasticity of the data sets, i.er, nonuniform variance
of the sensitivity coefficient estimates, an appropriate regression
procedure would be weighted least squares regression for the three
sensitivity coefficient lines, with weights {u^}, {u'^}, and {u1.1},
respectively. A detailed exposition of this procedure is provided in
Section 3. .2.1.4 as part of the recommended protocol for individual
manufacturers and will not be repeated here.
The above estimated sensitivity coefficient lines imply a general
fuel economy model of the form
EQ = K.wVa11 exp(3w + 6r + 9a)
13
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Even if the data support straight-line sensitivity coefficients,
these should be used essentially for interpolated estimation within the
range of parameter values in the data. Extrapolation appreciably out-
side this range would be highly speculative.
Since S , S , and S are, by basic physical principles, expected
to be non-positive,* should any of the lines cross the horizontal axis
into the positive value region (while still within the range of parameter
values in the data) it may be prudent to replace these positive values by
zero for general application.
If qualitative data analyses suggest no significant relationship
between each sensitivity coefficient and its respective design parameter,
or if estimates of the linear slopes, 3, 6, 6, are found to be not
significantly different from zero, then one need only estimate a mean
sensitivity coefficient in each case. Again, a weighted procedure is
most appropriate and takes the form:
E u.S
S = ——
w Z u.
Similarly for S and S . In this case the mathematical fuel economy
r a
model simplifies to:
S S S
r „ w r a
E = K.w • r -a
* It is understood that in some special cases this expectation has been
shown to be incorrect, perhaps due to some unusual engine map
characteristics.
14
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3.2.3 Results for a Specific Data Set
Fuel economy test results from the 1980 General Label data base were
provided by EPA. , The data contained records on 1673 tested subconfigura-
tions which included an unspecified number of duplications that were
eliminated in the course of the processing. Search for groups of
comparable subconfigurations and, then, estimation of sensitivity
coefficients by pairing along adjacent increasing parameter values led to
^ /\ /"S
the calculation of 53 Sw, 43 Sr and 56 Sa points. Weighted least
squares linear regressions were carried out following the procedures
described in Section 3.4.2.1.4. The results obtained are shown in Table 1.
Examination of Table 1 leads to a number of observations. The
negativity of S, the weighted mean of all sensitivity coefficient
estimates (and also the estimated coefficient at the mean design
parameter location) is generally confirmed. On the other hand, the
existence of non-zero slopes is in some cases not established with
statistical significance and, in other cases, only marginally so.
Predicted FTP axle ratio sensitivity coefficients at the lower limit
of the axle ratio range are slightly positive; this is the only instance
of interpolated positive coefficient prediction. The estimated values
of OQ, the coefficient of variation in subconfiguration fuel economy
measurement are quite consistent and in reasonable agreement with other
estimates for aQ (see Appendix A).
The marginal statistical validity of non-zero slopes in the above
numerical exercise is a situation that could conceivably also occur when
EPA estimates its standard (industry-wide) sensitivity coefficient regression
lines. It is recommended that careful attention be given to this matter. If
parameter dependence of sensitivity coefficients is not confirmed at a
suitable level of significance, then it would be prudent to choose the simpler
mode of constant (parameter-independent) sensitivity coefficients.
15
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Table 1. WEIGHTED REGRESSION OF SENSITIVITY COEFFICIENTS COMPUTED FROM 1980 GENERAL LABEL FILE
MODEL: S = ~S + b (P - P")
ERROR SOURCE: Fuel Economy Measurement with Coefficient of Variation = OQ
DESIGN PARAMETER, P
Test Weight, w
Range = [2312, 5375] Ibs
Road Load HP, r
Range = [7.2, 18.3] HP
Axle Ratio, a
Range = [2.35, 3.72]
DRIVING
SCHEDULE
FTP
HFET
FTP
HFET
FTP
HFET
n
53
53
43
43
56
56
P
4226 Ibs
4226 Ibs
9.8 HP
9.8 HP
3.06
3.06
S
- 0.266
- 0.153
- 0.127
- 0.361
- 0.234
-0.579
a
S
0.084
0.083
0.055
0.060
0.050
0.063
b
0.081/103 Ib
0.176/103 Ib
- 0.037/HP
0.004/HP
- 0.369/
Unit Ratio
- 0.562/
Unit Ratio
b
0.099/103 Ib
0.097/103 Ib
0.022/HP
0.024/HP
0.202/
Unit Ratio
0.254
Unit Ratio
°0
0.026
0.025
0.028
0.030
0.035
0.044
F*
0.68
3.28
2.73
0.02
3.34
4.90
a*
> 0.25
~ 0.08
~ 0.11
> 0.75
~ 0.08
~ 0.035
* Analysis of variance F-ratio and associated significance probability a for slope b = 0.
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3.3 Basic Considerations for Individual Manufacturer Procedure
EPA plans to promulgate standard sensitivity coefficients based on
subconfiguration test data covering all light duty automotive manufacturers,
There will be six such coefficients or coefficient lines—applicable to
each of FTP or HFET fuel economy" for each of three possible design
parameter variations: test weight, RLHP, or axle ratio. The fact that
preliminary examination of these results have revealed no consistent
patterns among individual manufacturers,1 is the basis for adopting single
(industry-wide) standards. However, no special efforts were made by
EPA to establish this conclusion with high confidence. An individual
manufacturer may have good reason to believe that one or more of the EPA
standards do not apply to its own vehicles. If that is the case, it can
propose to use alternative sensitivity coefficients derived strictly
from data on its own vehicles. Alternative coefficients (or coefficient
lines) may be proposed for any number of the six EPA standards. Each
proposed alternative should be considered independently. In order to be
accepted by EPA, such manufacturer-specific sensitivity coefficients must
meet certain accuracy and representativeness requirements. The procedures
that will be presented have been formulated to ensure that that happens.
It is appropriate at this point to consider first how the accuracy and
representativeness requirements were developed.
3.3.1 Accuracy
The process of fuel economy adjustment based on design parameter
sensitivity coefficients takes a tested vehicle subconfiguration (having
fuel economy E) as a starting point and calculates the fuel economy
E1 of a second subconfiguration which is identical to the first sub-
configuration in all respects except for a different value of single
parameter P.* The relationships used for this calculation are:
* P refers to test weight, RLHP, or axle ratio. It is also possible
for adjustments to be made for two or three simultaneous design
parameter variations, but this possibility is disregarded.
17
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E' = E + AE
AE = S • E • AP
where S is the applicable sensitivity coefficient value, AP is the
fractional change in P when going from the tested to the untested
subconfiguration, and AE is the absolute adjustment in fuel economy
(in mpg).
To begin with, it is recognized that the tested subconfiguration
fuel economy measurement E will generally be in error relative to the
true mean fuel economy for that subconfiguration, which error is carried
directly into the estimate E1 for the untested subconfiguration. A
useful statistical characterization of this error is its coefficient of
variation, i.e., the ratio of its standard deviation to the true mean fuel
economy. (The latter is adequately approximated by E for small enough
coefficients of variation.) Designate this coefficient of variation by
an- A review of available literature on fuel economy measurement errors
together with additional analysis of recent EPA test data is described in
Appendix A. It is concluded therein that a reasonable estimate for QQ
is 0.04 (4%), apparently applicable to both FTP and HFET fuel economy.
Suppose, now, that the estimated sensitivity coefficient S is
also in error relative to true value, that error being characterized by
a variance a* On the reasonable assumption of a statistical independence
of the errors, we may then express the squared coefficient of variation
of E1 by
o* = a02 + (AP)2 0-5
18
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In order to preclude any substantial increase in a relative to a_,
it is recommended that AP • a be required not to exceed 0.02. The
O
effect of that requirement would be to keep a, to within about 12% of
OQ, as indicated by the following calculation:
/ (0.04)2 + (0.02)2 = 0.0447 = 0.04(1.12).
Any more stringent requirement would lead to only slight additional
improvement in overall accuracy. On the other hand, degradation of overall
accuracy becomes increasingly more rapid with relaxation of the 0.02 criterion.
It is of interest to look further into the implications of the
requirement
AP • a. < 0.02
on the actual labeling process. Let E' be the label value assigned to
the untested subconfiguration based on the adjusted calculation E1. E'
will therefore be just E1 rounded off to the nearest whole number in
mpg. Let E" and E" be the corresponding quantities on the supposition
that true S were known precisely and the correct design adjustment
were made. Then
e = E1 - E"
EL = V - EL"
are errors (in mpg) due only to sensitivity coefficient error, e is
the unrounded error and would have standard deviation (in view of the
above requirement)
a 5 0.02 E
19
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which, for example, takes on limiting values of 0.25 mpg, 0.5 mpg, and
1 mpg at E = 12.5 mpg, 25 mpg, and 50 mpg, respectively. The second
error quantity e\_ is the difference in label values achieved under
actual and ideal (error-free) design parameter adjustment and therefore
represents the ultimate impact of sensitivity coefficient uncertainty.
E!_ is also a random value, but takes on only integer values.
On the reasonable assumptions that e is normally distributed (about
zero with variance a 2) and that the decimal component of any fuel economy
measurement is uniformly distributed between 0 and 1, probability distribu-
tions for e. have been calculated for various a . These are shown in
Table 2. The details of the calculation are given in Appendix B. Thus, the
Table 2. PROBABILITIES OF LABEL MPG ERROR FOR VARIOUS
ERROR STANDARD DEVIATIONS IN CALCULATED MPG
a£ (mpg)
0.25
0.50
1.00
LABEL MPG ERROR e, (mpg)
0
0.80
0.61
0.37
±1
0.20
0.38
0.48
±2
<0.001
0.01
0.13
±3
--
--
0.02
requirement AP • a <. 0.02 implies for E = 12.5 mpg cars at least 80%
probability that there will be no label error due to the adjustment process.
Moreover, if an error does occur, that error will rarely exceed ± 1 mpg.
For E = 25 mpg cars these probabilities shift somewhat to at least 61% no
error and no more than 38% ± I mpg error. In the case of 50 mpg cars (for
which the requirement implies a £ 1 mpg) there may be appreciable
probabilities of ± 1 mpg and even ± 2 mpg errors. It should be noted
however that for such cars ± 2 mpg is a relatively small change compared
20
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to absolute fuel economy; furthermore, in this case a_ = 0.04 implies
that the error contribution due to fuel economy measurementsin the tested
subconfiguration has a standard deviation of 2 mpg, which still dominates
the < 1 mpg a .
— e
The above analysis tends further to support the reasonableness of
the requirement
AP - QS £ 0.02
The next question to be raised then is how does this translate into
operational requirements for the manufacturer? It will be shown subsequently
(Section 3.4.2.1.4) that, after the manufacturer has obtained and processed
the appropriate data, an estimate for the maximum variance of the computed
sensitivity coefficient within the range of parameter values in the data
set* is
2
a 2 °0 f2
a
n(AP)2
v Vms
where
n is the number of individual sensitivity coefficient data points**
n is the underlying fuel economy coefficient of variation in the
manufacturer's data set
* As a consequence of the representativeness requirement, it is expected
that the parameter range will encompass most, if not all of the re-
spective parameter values over the entire set of manufacturers vehicles.
Hence, fuel economy adjustments will involve interpolated rather than
extrapolated estimates of sensitivity coefficient.
** n is not to be confused with the number of subconfiguration test results.
As previously described, each data point derives from a pair of comparable
subconfigurations from which a sensitivity coefficient has been made.
21
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is a root-mean-squared average of the fractional parameter
differences between comparable pairs of subconfigurations used
to calculate individual sensitivity coefficients
f is a parameter distribution shape factor which is defined in
Section 3.4.2.1.4. If only mean sensitivity coefficients
rather than sensitivity coefficient lines are to be estimated,
then f = 1.
One may then derive a requirement on n as follows:
AP 0.02
rms 'max - rms
An immediate simplification recommended is the identification of AP with
AP This means that the accuracy requirement on the AE adjustment
is to be imposed in the context of an "average" value for design parameter
difference between comparable subconfigurations. Admittedly, adjustments
will be made for AP > AP with correspondingly larger errors, but they
will also be made for AP < AP as well which tends, overall, to balance
out on a probabilistic basis. If this recommendation is acceptable, then
the requirement on n reduces to
The difficulty with this expression is that neither ag nor f is known
in advance but must be calculated after the data set has been assembled.
22
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The solution offered is to provide for a two-stage procedure.
A priori reasonable estimates are known for both a and f. On the
basis of these, determine a required data set size n]. Assemble a data
set of size n >^ n, and carry out the estimation procedure including
determination a of a and f. Compute a revised n? using a
and f. If n~ <_ n, then no additional data are required. If n? > n,
then n_ - n additional data points must be introduced and the estima-
tion procedure repeated with the augmented data set to yield final
estimates.
We proceed now to derive the first stage data set size requirement
n,. As previously indicated, a reasonable a priori estimate of a is
0.04. Consider, first, that EPA has selected the alternative of expressing
sensitivity coefficients as linear functions of the design parameter. As
implied by the definition of shape factor f, if the design parameter
values of the individual data points are fairly uniformly distributed over
their range, then f = 2. It is reasonable to expect that the require-
ment for representativeness of the data to all of the manufacturer's
vehicles will tend to prevent peaked or polarized distributions from
occurring.
In the preceding paragraph f was evaluated in the context of
estimated sensitivity coefficient lines. If, on the other hand, EPA
decides that estimation of mean sensitivity coefficients, independent of
parameter values, is adequate, then f = 1.
We therefore arrive at the following recommended first stage require-
ments on data set size:
23
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nl *
32 Linear Sensitivity Coefficients
8 Constant Sensitivity Coefficients
Clearly, a number of arbitrary judgments were made along the way
in arriving at this recommendation. For example, suppose there were
additional compelling arguments for keeping the probability of any label
error arising from the adjustment process to below 40% even with 50 mpg
cars. Then a
-------
the derived data set could be expected to span all test weights having
significant sales. Because of the moderate degree of correlation of RLHP
with inertia weight class, a broad span of RLHP settings could also be
expected. There is perhaps less assurance of obtaining a full span of
axle ratio values, since this parameter is not defined at the Base Level.
However, specific vehicle configurations are designated for inclusion in
emission and fuel economy test fleets based on projected sales at the
configuration level, and it is speculated that a broad span of values will
naturally be achieved even in this case. It should also be noted that the
manufacturer will have an incentive to achieve a broad spread of parameter
values (in the linear sensitivity coefficient model) in order to minimize
the value of distribution shape factor f which enters into the second
stage n requirement.
3.4 Draft Procedure for Individual Manufacturer Coefficients
This section presents a draft procedure for individual manufacturers
who wish to take exception to any or all of the six standard design
parameter sensitivity coefficients (or coefficient lines) promulgated by
EPA. It includes data requirements, first stage estimation of sensitivity
coefficients, statistical test of need for additional data, and, if
required, final estimation of sensitivity coefficients. The six cases
are categorized as FTP or HFET fuel economy sensitivities to each of:
test weight, RLHP, or axle ratio design parameters. Although each of the
six cases may be considered independently, the procedure as structured
presents parallel treatment of FTP and HFET sensitivities for each of
the three design parameters, in recognition of the fact that most sub-
configuration test results will provide both FTP and HFET fuel
economies. The procedure is presented in full detail with respect to
test weight design parameter. Application to the other two design
parameters is by reference.
25
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3.4.1 Data Requirements
3.4.1.1 Minimal Set
Fuel economy tests (FTP and HFET) conducted by the manufacturer or
by EPA on all of the manufacturer's emission data vehicles, fuel economy
label vehicles, and fuel economy data vehicles for the forthcoming model
year are to be utilized. Each fuel economy test result is associated
with a unique vehicle subconfi.guration as specified in 3.4.1.3. Similar
test results from the preceding two model years, with the exception of
discontinued basic engine-transmission combinations, are to be included.
3.4.1.2 Additional Vehicle Tests
Utilize the procedure described in Sections 3.4.2.1.1 through 3.4.2.1.3
to estimate the number of sensitivity coefficient data points that can be
generated from the minimal data set of Section 3.4.1.1. If the estimate
is less than nj, then additional subconfiguration FTP and/or HFET fuel
economy test data need to be introduced to reach this requirement.
Subconfigurations may be selected with some discretion by the manufacturer
so as to match already-tested subconfigurations and thereby generate
comparable subconfiguration groups as defined in 3.4.2.1.1. However,
their distribution among Base Levels must reasonably match projected
sales as provided for by the following rules:
In the final procedure n.. will be replaced by a specific number.
26
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(1) Base Levels with less than 1% or less than 5000 projected
sales are excluded.
(2) Add each new subconfiguration test successively from different
Base Levels starting with that with highest projected sales
and working downward.
(3) There should be no more than one new subconfiguration from
each Base Level until all eligible base Levels have contributed,
(4) Beyond this point, each additional subconfiguration added must
be from a Base Level for which the ratio of number of already-
added subconfigurations to projected (base Level) sales is
smallest.
3.4.1.3 Definitions
The variables which uniquely define a vehicle's subconfiguration
are:
Basic Engine Family (E)
Engine Code (Ec)*
Transmission Class (T)
Transmission Configuration (Tc)
Equivalent Test Weight (W)
Inertia Weight Setting (Wj)
Axle Ratio (A)
Road Load Horsepower Setting (R)
* Practical considerations may lead to deletion of Ec as a defining
variable, that is, EPA is considering the possibility of permitting
aggregation over engine code in those instances where more than one
code is compatible with a specified basic engine and transmission.
27
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The Base Level of a given subconfiguration is uniquely defined by (E, T,
Wj). Fuel economy test results (in mpg) are denoted by:
C (FTP Fuel Economy)
H (HFET Fuel Economy)
3.4.2 Design Parameter Sensitivity Coefficient Estimation
The three design parameters for which standard sensitivity coefficients
are promulgated by EPA are: test weight (W), axle ratio (A), and road
load horsepower (R). The procedure for estimating manufacturer's alternative
coefficients may be carried out independently for any or all of these
parameters.
3.4.2.1 Test Weight
3.4.2.1.1 Comparable Subconfiguration Groups
Partition the total set of tested vehicle subconfigurations (3.4.1.1
and 3.4.1.2) into groups within each of which members differ only in
test weight (W) and inertia weight (Wj), i.e., all vehicles in a given
group have identical E, Ec, T, Tc, A, and R, but different W.* Within
each such group containing more than a single member, order according to
increasing W, i.e., W^ < \\% < -•• < wk-
3.4.2.1.2 Weight Sensitivity Coefficients
For each group with k > 1 comparable subconfigurations, define
k-1 estimates of weight sensitivity coefficients and related variables
If p such tested vehicles have equal W, they must be regarded as
p samples of the same subconfiguration (even if the MI differ).
Accordingly, they are collapsed into a single test entry by taking
the mean of their C and H test results.
28
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as follows:
Average Fuel Economies:
C. . + C. H.^ + H.
r" - J+1 J u - J+l J
C0 ' 2 Hj - 2
Fractional Changes in Fuel Economy:
C. . - C. HL - H
1 AH --
C. H.
J J
Average Weight:
W.., + W.
Fractional Change in Weight:
W. - W.
AW. =
W.
J
Weight Sensitivity Coefficients:
AC. AH.
SWC . = —T71 SWH . = —r^
1 A W i A W
J J J j
j = 1, 2, ..., k-1
Further, if C. is the sample mean of p. FTP tests on different vehicles
j J
of the same subconfiguration, compute the variance reduction factor
29
-------
The variance of SWC. is then
J
V
aswc. = 77 J = l k -
J J
Note that in most instances P. = P. = 1; hence u. = (AW.)2/2. Note
also that if all FTP and HFET tests are paired, then u. applies
J
equally to SWH.; otherwise different variance reduction factors need
J
to be determined for the latter.
3.4.2.1.3 Pooling of Data
Pool the estimated sensitivity coefficients from all of the comparable
groups into aggregated data sets as follows (replacing symbol W by W):
{ SWC., W., u. }
i i i
{ SWH., W., u. }
i = 1, 2, ..., n
Thus, in the first data set there are aggregated n different determinations
of FTP fuel economy weight sensitivity coefficient, SWC-j, each estimated at
an average test weight W-j, and each with a variance reduction factor u-j
relative to basic variance in subconfiguration FTP fuel economy measurement.
Similarly for the second data set.
30
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3.4.2.1.4 Weighted Least Squares Linear Regression
Assume a linear model for dependence of SWC on W, i.e.,
SWC. = a + b (w. - W~) + e.
i v i ' i
where a and b are unknown coefficients to be estimated, W is a weighted
sample mean of the W. to be defined shortly and e. is an additive
(unbiased) error with variance:
Estimation of a and b is to be performed by weighted least squares
linear regression. Define the normalized weights
Define also:
v. = u./£u • = u./U*
SWC = zvi SWCi
(swc)2 = zvi (swc.)2
W = Zv. w.
U|2 = 7 u w
:_ i i
SWC . W = zv. SWC.W.
i i i
* In the special case of u, = ^(AW.,-)2 for all i, which applies when
none of the subconfiguration fuel economy tests are replicated, U may
be expressed as = £ . I (w )2 = fj. (AW ,2
2 n i 2 rms
where AW is a root-mean-square average of the AW,.
31
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Then unbiased estimates for a and b are:
a = SWC
C = swc-w - swc • w
w2 - (w)
Variance estimates for a and 6 require also an estimate of a 2,
the squared coefficient of variation of subconfiguration FTP fuel economy
determination. Such an unbiased estimate as given by:
S2 = |~ (SWC)2 - (SWC)2 - b2 • (WMW)2)] U/(n-2)
The variances of a and b are then estimated by:
as = °o /u
The covariance of a and b is zero by virtue of the centering of the data
around W in the linear model. This implies that for an arbitrary test
weight W, the estimated SWC at W
SWC = a + 6 (W-W)
has variance given by
/-2
= _
SWC U V
-------
The maximum variance of SIC over the range of test weights in the
data set is therefore:
2. r~"
2 0 max )i + (W - W)2
CTmax U * W. < W < W,, K rri ^
Lu 2 _
= 0
n (AW ;
rms
(see earlier footnote)
The HFET (highway) sensitivity coefficient estimates are obtained
in the same manner by repeating the above regression procedure with SWH.
in place of SWC..
3.4.2.1.5 Accuracy Check and Second Stage Estimators
Define
max r (w-w)2
f = W < W < W., < II + ~ ^~
L U U W2 - (W)
where W. and W.. are, respectively, the smallest and largest W. values
in the data set. Compute f and use it together with the previously computed
33
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estimate for basic fuel econoiny coefficient of variation, a , and
requirement o£** to check whether n satisfies the inequality:
n > 2
If it does, then the procedure is terminated. If it does not then
compute
An =
+ 1
as the additional number of data points required.(The notation [x]
denotes greatest integer less than x.) Generate the additional An
data points by introduction of a suitable number of new tested sub-
configurations in accordance with Section 3.4.1.2, continuing from the
point reached in the first stage procedure. Repeat the procedures in
Sections 3.4.2.1.1 through 3.4.2.1.4 with the augmented set of n + An
data points and discard the original (first-stage) estimates.
The above may be repeated for HFET weight sensitivity coefficients
by use of corresponding values for f and aQ.*
* If both FTP and HFET data are provided in all tested subconfigura-
tions, then f will be identical for FTP and HFET cases. However,
a« estimates will generally differ. If both FTP and HFET weight
sensitivity coefficients are being estimated, then it would be wise to
test adequancy of n for both cases together and then to generate new
data points, as required, to meet the largest deficiency (if any).
** The recommended value for ap (required on bound coefficient variation of
design parameter adjusted fuel economy) is 0.02. However, EPA may
decide to set a more or less stringent requirement.
34
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The final outputs of the estimation procedure are:
a > b , W, a, , ab , OQ
\~ U c c c
* _ 2 2 2
>s ^^ s\ s\ s\
9H' bH' W' %' %' aOH (HFET)
The manufacturer's estimated weight sensitivity coefficients are then
represented by
SWC = ac + DC (W-W)
SWH = aH + bH (W-17)
with variances:
°swc
SSWH =SIH
3.4.2.2 Axle Ratio
Repeat the procedure described in Sections 3.4.2.1.1 through 3.4.2.1.5,
substituting axle ratio for test weight. Thus, comparable subconfiguration
groups are formed on the basis of members differing only in axle ratio;
pooled axle ratio sensitivity coefficient data sets are formed; and weighted
least squares linear regression is carried out to estimate linear fit
parameters and their variances.
35
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3.4.2.3 Road Load Horsepower
Repeat the procedure described in Sections 3,4.2.1,1 through 3.4.2.1.4
substituting road load horsepower for test weight.
3.5 Modified Draft Procedure if Sensitivity Coefficients are Assumed
to be Constant (Parameter-Independent)
As previously discussed in Section 3.2.3 the assumed linear dependence
of true design parameter sensitivity coefficient on parameter value has
not received definitive statistical confirmation. If the alternative
assumption of no dependence is made,then the whole procedure of sensitivity
coefficient estimation, by EPA as well as by an individual manufacturer,
would be much simplified. This section presents the modifications that
could then be made to the draft procedure described in section 3.4.
3.5.1 Data Requirements
No procedural changes are indicated. However, the data set size
requirement, n^. will be smaller (See Section 3.3.1).
3.5.2 Design Parameter Sensitivity Coefficient Estimation
The manufacturer continues to have the option of challenging any
design parameter sensitivity coefficient for FTP or HFET fuel economy,
independently of the others. The modified estimation procedure is
presented in terms of test weight design parameter, paralleling
Sections 3.4.2.1 through 3.4.2.5. However, exactly the same modifi-
cations apply to the other two design parameters, axle ratio, and RLHR.
36
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No changes are indicated in Sections 3.4.2.1 through 3.4.2.3 except
that the data sets need no longer include parameter values, i.e., they are
of the form:
SWCi, u.
^ u. }
i = 1, —, n
In Section 3.4.2.4, the model for SWC values is
swci = ^swc + ei
where Vrur 1S the unknown constant sensitivity coefficient with respect
to weight (city) and, as before, e-j is an additive error with zero mean
and with variance
2
a
e. u.
i i
Estimation of y_wr is performed by weighted averaging. As in the
regression case, define
i = v./Zu. = U../U
SWC = Iv. SWC.
SWC2 = Iv.(SWC.)2
Then SWC provides an unbiased estimate of Vcur' An unbiased estimate
of a 2 for this case is given by
37
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aQ2 = [SWC2 - (SWC)2] U/(n-l)
The variance of the estimate SWC is then estimated by
An analogous set of values is obtained from grouping the corresponding
HFET results. Specifically, the weighted average SWH is obtained, which
provides an unbiased estimate of yc.ILI, the sensitivity coefficient with
own
respect to weight (highway). The variance of this estimate is itself
o
estimated by the quantity cr—-.
oWrl
In Section 3.4.2.5 the test for adequacy of n is revised to:
since f = 1 in this alternative estimation context. Otherwise, the
procedure for determining An and second stage estimation is unchanged,
The final outputs of the estimation procedure are:
SWC' ^' aOc
SWH, a,
38
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4. FUEL ECONOMY ADJUSTMENT TO REFLECT IN-USE EXPERIENCE
Following fuel economy label adjustment for vehicle design parameter
differences, as described in Section 3, the adjusted values still repre-
sent chassis dynamometer fuel economies (FTP and HFET). In order to
achieve label values that are more meaningful to the public, EPA is
developing a transformation of dynamometer-based fuel economy values to
correspond, on the average,* to road, i.e., actually realized in-use,
fuel economies. Two factors, ou and au are envisioned which multiply
L n
FTP and HFET fuel economies, respectively, to yield finally adjusted
label values of "city" and "highway" mpg for each vehicle configuration.
In view of the demonstrated shortfall of in-use mpg relative to EPA
(dynamometer) mpg, on the average, both factors are expected to be smaller
than 1.
In its initial implementation phase EPA plans that ou and au
L H
would be two fixed numbers uniformly applied to all light duty vehicle
configurations and a fortiori to all manufacturers. The possibility that
dynamometer-to-in-use mpg scaling is substantially different over major
vehicle design categories is also under investigation, and a possibility
for the future is that sets of distinct (ou, au) factors may be
L n
developed based on: diesel vs. spark ignition, front vs. rear wheel
drive, trucks vs. cars, manual vs. automatic transmission, and/or other
groupings shown to significantly affect the factors.
* The meaning of "average" adopted by EPA2 is the median. The rationale
is to insure that equal numbers of in-use vehicles perform above and
below their adjusted label fuel economies regardless of asymmetries in
the distribution of road mpg. If distributions are symmetric, then
the arithmetic mean and median are identical.
39
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Another possibility is that a particular manufacturer could argue
that its own vehicles are distinctly different as a class. Therefore,
manufacturers should have the option of substituting alternative
adjustment factors applicable to their-own vehicles, so long as they
are able to demonstrate that such factors satisfy appropriate criteria.
Selection of these criteria demands a careful enunciation of the intent
of any protocol dealing with manufacturer-specific adjustment factors as
opposed to those promulgated by EPA. The view taken in this report is that
whether the manufacturer-estimated factors are significantly different from
the EPA-promulgated values Is not an issue, nor does it need to be. Rather
than considering manufacturer-specific factors in a hypothesis-testing
context, one simply requires that the factors be estimated to some specified
level of precision consistent with the aims of the fuel-economy labeling
program.
First, some general methodology for determining road adjustment factors
based on in-use surveys is presented. This includes consideration of
various environmental influences on in-use fuel economy and stratification
methods to reflect these influences. The general structure of a manu-
facturer-specific data set together with criteria to be met and procedures
to be followed is then described.
4.1 Road Adjustment Factor Estimation from In-Use Surveys
By some implemented survey mechanism, responses are received relative
to the in-use experience of individual vehicles over limited driving
intervals. These responses may provide all or some of the following
information:
• Location of driving
• Time of year
40
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t Vehicle identification [which enables determination
of EPA fuel economy label values)
• Sequence of fuel purchases (gallons) 1 , .... , ,
r va ' ( (needed to calculate
t Corresponding odometer readings \ in-use fuel economy)
0 Purchase dates
• Estimated number and lengths of trips
• Estimated percent split between urban and non-urban driving.
Because of the strong dependence of in-use fuel economy on the split
between urban and non-urban driving, as well as the decision to compute
separate adjustment factors for these two modes, it is essential to have
some measure, either direct or indirect, of their relative proportions.
We therefore assume that the following data are available for each sampled
vehicle:
t EPA city fuel economy, C mpg [c = 1/C gpm]
• EPA highway fuel economy, H mpg [h = 1/H gpm]
• In-use fuel economy, R mpg [r = 1/R gpm]
• Urban (city) fraction of total driving, u; 0 < u < 1
As previously stated, u might be directly estimated by the respondent,
derived from data on trip length or miles per day or computed as a
weighted average of several such estimates.
Environmental factors, notably ambient temperature, wind speed,
road grade, road surface condition, and degree of traffic congestion, can
have an appreciable influence on in-use fuel economy.1 Since knowledge
of the location and time of year of each individual return modifies the
distribution of environmental factors impinging on the reported driving,
the statistical analysis should, strictly speaking, account for this
hetereogeneity in the data. As a first approach, we make the simplifying
assumption that the survey sample design gives equal probability to each
41
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vehicle in the U.S. fleet and each time of year. Then the sample median
of individually calculated adjustment factors provides a reasonable
estimate of the median road adjustment factor over all U.S. vehicles and
all seasons. Recall, as previously remarked, that if the distribution
of road adjustment factors is symmetric,then the median and mean
parameters are identical. Inasmuch as the actual sample is not likely
to conform to the above equal probability assumption, we shall need to
address the issue of heterogeneity in the sample space. This will be
done in Section 4.2.
We now consider two alternative methods of estimating average road
adjustment factors for both city and highway driving.
The first method, which is that described in EPA draft documents,2'3
extracts two extreme subsets of "nearly pure" highway driving and city
driving respectively from the totality of responses. This is done by
requiring u £ U and u >_ U-, respectively, where U_ is close to 0
and U, is close to 1. Specific cut-off values initially selected were
UQ = 0.2 and U = O.9.2 Within subset {u^ <_ U }, the ratio aH. =
R./H. is computed for each response and the median au is designated
11 n
the road adjustment factor for highway driving. Similarly, within subset
{u. > u-}, the ratio CXQ. = R./C. is computed for each response and the
median ar is designated the road adjustment factor for city driving.
An objection to the method just described is that it fails to use
most of the survey responses. Ideally, u-. should be very close to 0
and u, very close to 1 in order to generate subsets of reasonably pure
42
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highway driving and city driving, respectively. On the other hand, the
closer these ideal cut-off limits are approached, the fewer are the
responses actually utilized.*
We propose the following model which permits an alternative method
of estimating a,, and ou based on all of the data. Let S. and T.
H L 11
designate the (unknown) city and highway in-use fuel economies (in mpg)
for the i response (a specific car at a specific time of year) in the
survey data set. Let s. and t. designate the corresponding reciprocal
fuel consumptions (in gpm). Then
r. = u.s. + (1 - u.)t.
i 11 11
assuming r. and u. are accurately reported. Actual (r., u.) data will
of course, introduce an error term. We can also write
T. h.
yr = r" = aH +AaH.
11 i
S. c.
11 J. A
ac = c~ = T = ac Aac
L. L. Si L L.
where au and ar are the average (median) road adjustment factors which
H L
we wish to estimate. Note, therefore, that
med (Aau ) = med (Aou ) = 0
H. . L.
1111
* It is, of course, possible to use the intermediate u (mixed driving)
responses as some kind of check on the estimates derived. There is no
indication that this was done, nor is it clear how one would adjust the
estimates in the light of inconsistencies found.
43
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It follows that
h. h
t. = —-1—=
1 aH + AaH. ~a
c. c.
s. =—-J— = ^-
1 aC + aC. aC
where med (At.) = med (As.) = 0. Then, by substitution,
c.u.(—) + h.(l - u.)(—) + u.As. + (1 - u.)At.
r. i i a ' i la,, "< "> 11
1 L n
c.u. + h.(l - u.) CM. + h.(l - u.)
Define the derived data quantities
r.
..
c.u.
i i
Y
i c.u. + h.(l - u.)
11 i r
and the additive error term
u.As. + (1 - u.)At.
11 11
ei c.u. + h.(l - u.)
11 i i
44
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This leads to the simplified linear form:
_1_,
V
— + (— - r~)Y, + e.; o < i < i
The result suggests that the quantities au and <*„ may be estimated by
H L
linear regression of p.. on y.. However, one must be cautious about
applying conventional least squares since that method assumes the mean
error to be zero, and the mean behavior of e. is not known.
A reasonable assumption to make about the component deviations At.
and As. is that they are highly positively correlated in sign. For
example, an in-use highway fuel consumption above that predicted by the
median highway road adjustment factor would seem to imply (for the same
car in the same environment) an in-use city fuel consumption also above
that predicted by the median road adjustment factor. If this assumption
holds, it follows that
med (e.) = 0.
i
Some form of median linear regression of 3. on y. would then be
appropriate. The iterative method described by Mood1* is suggested.
We outline the method, but refer the reader to the reference for
additional details. (See Figure 1 for an illustrative application.)
Compute y = med (Y-)- Partition the data set into subsets S and S+
n
45
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Ab]
0
FIGURE 1. Illustrative Median Regression Procedure
46
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to the left and right of y, respectively. That is, y.. < y implies
(y.j > 8.j) is in S_, etc. Determine the median of y and 8 in each
of these subsets: (y_, 3_), (y+, g+). Compute the slope bj of the
line joining these two points. ^Compute the deviations 63, from the
line 3 = b^y. Compute the left and right medians of these deviations
6~3_, <5~3+. Compute the slope of the line joining (y_, 678) to
(y+, 678+). Add this slope to bj to yield a second approximation to
the desired slope estimate b?. Compute the deviations 6«8. from the
line 3 = b^y. Proceed as before to compute a third slope estimate b .
Continue this iteration to the desired degree of accuracy. Denote the
final slope estimate by b. (In the illustration in Figure l, the
iteration stops at bp.) The estimate for the intercept a is then the
median of the final total set of deviations. The final estimates achieved
will have the property that
med (3 • - a - by.) r med (3. - a - by.) = 0
* *+ * \tf+' 1
ieS_ ieS+
which is a necessary condition for the true median line.
Finally, we estimate our desired median road adjustment factors by:
I
a = -
H a
1
a,, = —
a + b
Several variations can be introduced as may be deemed appropriate
from a preliminary analysis of the data. For one, if the
data are partitioned into a moderate number of y interval subsets,
i.e., [0, YI] , [y^ Y2],..., [yk_r yk] , it is possible that
47
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the subset medians may show a considerably better fit to some nonlinear
form than to a straight line. This would suggest the use of nonlinear
regression. Another possibility is that relatively more scatter may be
apparent at intermediate y (away from 0 or 1). This may be due
to inherently larger errors in estimation of urban fraction u by
respondents who did substantial amounts of both urban and highway driving
in contrast to respondents who did mostly one or the other. If such a
phenomenon is evident, then one can consider weighted median linear
regression which gives more weight to the median points closer to y = 0
and Y = 1- The concept of the weighted median is defined in the next
section.
Finally, it should be observed, as previously noted, that a symmetric
error distribution implies identical mean and median regression lines.
Furthermore, if this distribution is close to normal, then ordinary least
squares regression would be preferred since it would produce efficient,
i.e., minimum variance, estimates.
4.2 Treatment of Sample Space Heterogeneity
A practical way of accounting for the effects of sample space
heterogeneity on a nonequal probability sample is through stratification
and relative weighting of responses within each stratum.
Suppose that the important environmental parameters which affect
road fuel economy are those previously listed, viz., ambient temperature
T, wind speed W, road grade G, road surface conditions S, and
degree of traffic congestion C. Now, it is presumed that a detailed
sensitivity analysis, e.g., as described in the draft 404 report,1 has
led to a stratification of each environmental parameter into a manageably
small number of intervals such that each interval can be viewed as
48
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approximately homogeneous. Denote this stratification by the parameter
intervals
[TQ, Y!. [Tr
T
[W , W ], [W W ], ..., [W W ]
w w
[60, Gj], [G1§ G2] , .... [Gj _r Gj ]
G G
[SQ, Sj], DV S2] , .... [Sj _r Sj J
o o
tco-ci]- fci'c2] ..... [V-i, ci J
L Lf
The total number of product strata, corresponding to all possible combina-
tions of the five parameters, is of course given by I = I • I.. • I. • I_ • I..
I W b b L
This suggests that the interval numbers I,., ..., If need to be as small
as possible consistent with the requirements for reasonable homogeneity
within strata.
For ease of exposition, assume that the I strata are indexed by
1 , 1 <_ i <_ I , in some specified order. Suppose, further, that given the
geographic locale x and time of year (say, month) T of a response, one
can quantify the most probable stratum (or alternatively the stratum
containing the mean value for each environmental parameter) associated
with the reported driving. Denote this stratum by i(x, T). Thus, a
response from (x, T) falls into environmental stratum i(x, T). If the
total number of survey responses is N, this is partitioned by the
function i(x, T) into N. responses associated with strata i, i = 1,
.... I, and ZN. = N. Define n = N /N.
49
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A separate analysis of the actual distribution of registered vehicles
over all U.S. locales would yield the fractional distribution over the I
environmental strata, P.., i = 1, ..., I. That is, the proportion of
cars in use throughout the U.S. over the course of a year that are in
stratum i is P... One should be careful to note that this analysis
must take account of the fact that, whereas vehicle registrations are
associated essentially with location, any given location can move
through a number of different strata with time of year.
If the survey sample design gave equal probability to each car and
time of year, then we should find a very close correspondence between P.
and n. (any differences being a consequence of the random sampling
process). In general, we would expect to find substantial discrepancies
between P. and n.. To compensate for such a biased sample, we
associate with every return from stratum i a relative weight w. = P./n..
Note that the w- will always sum to N over the total set of returns.*
Note further, that it is important for all strata to be occupied; in fact,
for good performance of this weighting procedure it is desirable that a
required minimum number or minimum fraction of returns from each stratum
be achieved, alternatively that each w. be smaller than a preselected
bound.
* The summation here is not over index i (which runs from 1 to I) but
over individual response index j. Thus,
50
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What do these relative weights mean in terms of the procedure for
calculating road adjustment factors, which are based on median estimates?
The appropriate modification is to calculate weighted medians. Suppose
we have a set of N observations and associated weights which sum to N.
After ordering according to increasing value of observation, denote
these by x.^ <_ x,, <....,<. x^ with associated weights w , w?, ...,w.
Find k such that
k-1 k
wj - ~i<
Then the weighted median equals x . If strict equality holds on the
1
left, then TT(X, + x ) is selected, in analogy with the unweighted case.
L. K— i k
To recapitulate the results of this section:
(1) The total sample space (all U.S. cars x all times of year)
is partitioned into I strata, each representing relatively
homogeneous combinations of the significant environmental
parameters. The appropriate stratum for each survey response
is determinable from its locale and time of year.
(2) The relative frequency of survey responses within each
stratum is n., i=l, ..., I.
(3) The (true) relative proportion of the population within each
stratum is P., i = 1, ..., I (determined by separate
analysis of vehicle registrations by locale).
51
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(4) Compute wi = P^ru , and use these relative weights
(associated with the I strata) to estimate median road
adjustment factors §H and a according to one or the
other of the two alternative procedures described:
(a) weighted median estimation on "pure" highway-driving
and "pure" city-driving respondents or (b) weighted median
regressions on the reexpressed survey data, (3., y.},
j = 1, .. . , N.
4.3 Confidence Intervals for Medians
Suppose that the road adjustment factors are estimated directly as
medians of separate univariate samples for city and highway driving, viz.,
{otQ.} and {a^.}. Under the assumption of large sample size N, the
probability that k observations fall below the true median is approximately
normal with mean N/2 and standard deviation /N/2. Hence, a one-sided
p-confidence interval is obtained by counting z • /N/2 indices up
(or down) from N/2 and noting the observation value at that index within
the ordered list of observations, (z is the standard normal variate with
tail probability = 1-p). Thus, for example, a 90% lower confidence bound
(z = 1.28) on the true median city road adjustment factor in a sample of
1000 would be given by the [500 - (1.28)(31.62)/2]th = 480* ordered
value, i.e., by aC/4onv Even lf weighted medians are estimated, as
previously described, to compensate for biased sample space heterogeneity,
the above confidence bound estimation procedure is generally still applicable.
However, if the heterogeneity and sample bias are so large as to cause the
median to be estimated by an order statistic a,., where j < 0.2 N or
j > 0.8 N, the confidence interval problem would have to be investigated
more carefully.
52
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Alternatively, if median linear regression on the total survey data
set is performed, as described above, then we need to develop intermediate
confidence bounds on a, the median line intercept (at y = 0), and on
a + b, the median line value at y = 1. By merely taking reciprocals, we
would then obtain corresponding confidence bounds on au and a ,
H C
respectively. An approximate procedure for a one-sided bound suggested by
Mood's discussion of the confidence interval problem1* is as follows.
The estimated median regression line, as schematically shown in the figure,
partitions the total set of N points into four approximately equal subsets.
= a + b Y
Suppose we desire to estimate a one-sided upper p-confidence bound on
intercept a (the value of B at y = 0). Rotate the line clockwise
about its Y point until N/4 - Z • /N/2 data points remain in
the upper left region. Now translate the whole line in the upward vertical
direction until Z. • /N/2 additional data points have crossed from the
two upper into the two lower regions. The new intercept is taken to be
a the upper p-confidence bound on a. An analogous procedure involving
u, p
53
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first, counter-clockwise rotation of the line followed by upward trans-
lation would establish (a + b) the upper p-confidence bound on a + b
U,p rr r
(the value of B at y = 1). If it is believed by symmetry considera-
tions (y s h and comparable dispersion on left and right sides of y)
that the magnitude of the increase from a to a should equal the
~ ~ u »P
magnitude of the increase from a + b to (a + b) , then these two
u ,p
differences could be averaged to provide symmetrically estimated upper
confidence bounds.
The procedure described above is believed to provide a conservative
estimate of confidence intervals/bounds. An exact method is not known.
As previously noted, we can now assert that
HL'P au,P
laC;L,p ~ (a
That is, the lower p-confidence bound on the median highway road adjustment
factor 6L is given by the upper p-confidence bound on the y = 0 inter-
H
cept a of the median regression line. Similarly for ac- These
regressions follow from the fact that
P =
and similarly for (a + b) . .
u ,p
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4.4 Basic Considerations for Individual Manufacturer Procedure
EPA plans to promulgate two standard road adjustment factors (for
FTP and HFET fuel economy, respectively) based on in-use experience
of a representative survey sample of vehicles covering all light-duty
automotive manufacturers. In the absence of evidence to the contrary,
it is assumed that there are no substantial statistical differences among
manufacturers with respect to the relationship of dynamometer to in-use
fuel economy. This is the basis for adopting single (industry-wide)
factors. An individual manufacturer may have reason to believe that the
EPA factors do not apply to its own vehicles. In that case, it can
propose to use alternative adjustment factors derived strictly from survey
of in-use experience of its own vehicles to replace either one or both of
the EPA factors. In order to be accepted by EPA, such manufacturer-specific
road adjustment factors must meet certain accuracy and representativeness
requirements. The procedures that will be presented have been formulated
to ensure that that happens. It is appropriate at this point to consider
first how the accuracy and representativeness requirements were developed.
4.4.1 Accuracy
One might argue that there is a fundamental limitation in the accuracy
with which road adjustment factors can be determined because of the
impossibility of defining truly objective classes of urban (city) and
highway driving conditions. An alternative point of view which has much
merit -is that what a respondent reports as his mpg and his mix of driving
conditions represents the reality to which the adjustment factors should
relate. Thus, if the respondent says he did 90% urban driving and his
average mpg (derived from his numbers) was 18.2, we should accept these
55
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numbers at face value. It may nevertheless be desirable, even in this
context, to provide each respondent with a simple qualitative definition
of "urban driving." Such an approach would reduce the chances of gross
misinterpretation while still accommodating individual perceptions.
Adopting the above position, we see that the problem of accuracy
is associated only with the sampling process. Let x be the (city or
highway) road adjustment factor for an individual vehicle, and let x
have distribution F(x) over the total vehicle population. The median
F(x), denoted by a, is the quantity we wish to determine. If a truly
random sample of N vehicles is obtained (i.e., every vehicle having
equal probability of being selected) yielding individual factors x-, x ,
..., x , and we estimate a by the sample median a, then what can be
said about the accuracy of this estimate? It is known that for large
sample sizes, a is approximately normally distributed about a with
variance5
5 4Np2(a)
where p(x) = F'(x) is the density function of the population distribution,
From published data on in-use to EPA fuel economy ratios, it appears that
x is centrally distributed mostly within the range between 0.5 and 1, and
not greatly skewed. From this we estimate that a conservative lower
bound on p(a) is 2, It follows that
* < JL
a- 16N
56
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A requirement on minimal sample size can now be established directly in
terms of desired level of precision in estimating the true population
median road adjustment factor. Observe that a~ < 0.02 is equivalent
a —
to a 2% or smaller contribution to the coefficient of variation in the
finally adjusted fuel economy value. It can be argued, in the same way
as was done in connection with design parameter adjustment factors, that
2% added (by sum of squares) to the basic 4% coefficient of variation in
fuel economy measurement results in very little increase in total error.
It is therefore recommended that the requirement
, < 0.02
a —
be adopted. This implies the following requirement on sample size (which
applies individually to the sets of "pure" urban driving responses and
"pure" highway driving responses),
N >_ 156.
If the median linear regression procedure is adopted in order to be
able to utilize all survey responses regardless of reported urban percentage,
then an f factor analogous to that developed in Section 3 needs to be
applied. On the basis of survey data already accumulated2'3 it is reasonable
to assume a fairly flat distribution of responses over "percent urban
driving." The applicable value of f is 2 and the modified recommended
requirement for N is
N > -r = 625.
— a~
a
57
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Note that the requirement of a total survey response of 625 is probably
less stringent than the requirements of 156 each at the urban and highway
driving extremes.
4.4.2 Representativeness
In the preceding section it was assumed that the sample is randomly
selected. As discussed earlier, known environmental factors
very substantially influence in-use fuel economy and it is deemed
necessary to stratify by these factors (as determined by locale and
season) to make it possible to correct for any nonuniformities in the
sample. Unfortunately, practicality dictates a moderate number of strata,
say no more than 20, and, within any single stratum, environmental
variations may still be large enough to have an appreciable differential
effect. On the other hand, it is believed that biased sampling is much
less likely to occur (intentionally or spuriously) within individual strata.
Establishment of quantitative procedural requirements that would limit the
extent of sampling bias to some prescribed level is not possible without
additional details of environmental distributions and effects. What can
be done at this point in time is to ensure a reasonably uniform probability
of representation among strata by placing an upper bound on the relative
weights w.. Ideally, w. = 1 for all i. It is recommended that none
be permitted to exceed relative weight 2 when calculated on the basis of
minimum N requirement.
That is,
P.
7- < 2
N0
58
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where NQ is the minimum requirement (not the actual number of responses.
which may be greater). Thus,
P.N
For example, if N. = 625 and Pn- = 0.05, then at least 16 returns
should come from stratum i. If that number has not been achieved, then
additional returns are required. This increases N, but the require-
ment that N. > 16 remains unchanged.
4.5 Draft Procedure for Individual Manufacturer Road Adjustment Factors
This section presents a draft procedure for individual manufacturers
who wish to take exception to the standard road adjustment factors
promulgated by EPA. It includes data requirements and estimation of
road adjustment factors.
/
4.5.1 Data Requirements
(1) All of the manufacturer's vehicle classes are to be covered in the
survey. That is, the manufacturer is not permitted to deliberately
exclude particular configurations or model types. Generally, the
survey conducted during a given calendar year will be restricted to
the most recent model year cars in order to limit the range of
odometer mileage. However, surveyed vehicles should have accumulated
at least 2000 miles.
(2) Each return should include as a minimum:
(a) Information which enables precise determination of vehicle
fuel economy label values.
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(b) Location of driving (zip code may be adequate).
(c) Time of year (month may be adequate).
(d) Three successive fuel purchases in gallons (motorist
instructed to "top off" tank each time, and to wait until
tank is at least half-empty on 2nd and 3rd fill-up).
(e) Corresponding odometer readings.
(f) Corresponding dates of purchases,
(g) Estimate of percentage urban driving.
(3) The sampling plan should be designed to make a reasonable effort at
fair representation of all regions of the U.S. and all seasons.
(4) (a) If highway road adjustment factor is to be estimated from
"pure" highway driving responses (i.e., u <_ IL), then the
total number of such valid responses received shall equal
at least N *
(b) If city road adjustment factor is to be estimated from
"pure" city driving responses (i.e., u >^ U ), then the
total number of such valid responses received shall equal
at least NQ.*
(c) (Alternative to (a) and (b)). If highway and/or city
road adjustment factors are to be estimated by the
median line regression procedure, then the total number
of valid responses received shall equal at least N.*
The recommended values of NQ and Ni based on a precision
requirement of 2% are 156 and 625, respectively.
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4.5.2 Adjustment Factor Estimation
(1) The responses are to be classified in accordance with the set of I
environmental strata defined by EPA, based on the functional relation-
ship of stratum to location and time of year of driving to be supplied
by EPA.
(2) The relative frequencies of responses from the I strata n. = N./N,
i = 1, ..., I, are calculated and, using EPA supplied data on true
proportions, P., the relative weights w. = P./n. are calculated.
(3) In order to preclude excessive deviation from equal probability
sampling which could raise questions about the validity of the
relative weighting procedure, for each stratum, N. is required
I 1
to exceed — P.N (effectively a relative weight less than 2).
If N. is insufficient, then additional returns must be obtained
from stratum i until the requirement is met.
(4) (a) Compute individual highway road adjustment factors a^. =
J
R./H. for those responses with u < u_, where R. is
J j u j
in-use average mpg* and H is HFET fuel economy of the
vehicle subconfiguration. Compute the weighted median
6L using the relative weight for each response based on
n
the stratum in which it fs classified. The manufacturer-
specific road adjustment factor for HFET (highway driving)
fuel economy is set equal to a.
* R. may need to be adjusted for variable mileage accumulation effects
b} use of a standard formula.
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(b) Compute individual city road adjustment factors ac. =
R./C. for those responses with u > u , where R is
j j — T j
in-use average mpg* and C. is FTP fuel economy of the
vehicle subconfiguration. Compute weighted median 5
road adjustment factor for FTP fuel economy in a manner
analogous to (a) above.
(c) (Alternative to (a) and (b)). Compute derived data,
quantities 6. and y. and perform a weighted median
J J
linear regression as described in Sections 4.1 and 4.2.
The outputs are estimates au and a., HFET and FTP
n L
road adjustment factors, respectively. If this method
is used, then both EPA standard factors are replaced by
manufacturer-specific factors.
(5) If a manufacturer establishes its own specific road adjustment
factor(s), then annual resurveys meeting the data requirements of
Section 4.5.1 are required to update the factor(s) in succeeding
years. Data for the three (or fewer) most recent annual surveys
are pooled and the estimation procedures as described above are
repeated. Failure to conduct a proper survey would cause denial
of the manufacturer's petition for specific alternative factor(s)
and reversion to EPA standard factors.
* RJ may need to be adjusted for variable mileage accumulation
effects by use of a standard formula.
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5. ADDITIONAL ISSUES
5.1 Computation of Design Parameter Adjusted Fuel Economy
If only a single tested subconfiguration is comparable to a given
untested subconfiguration, then the procedure for estimating its fuel
economy based on design parameter adjustment is quite clear. The total
differential formula in Section 3.1 leads to:
E1 = E (1 + S^w + S^r + SaAa)
where E is the tested subconfiguration fuel economy
E1 is th.e adjusted fuel economy for the untested subconfigurations
S , S , S are the applicable sensitivity coefficient values
Aw, Ar, Aa are the fractional design parameter increments from
the tested to the untested subconfiguration.
If the two subconfigurations differ only in test weight, for example, then
Ar = Aa = 0 and the formula is accordingly simplified.
Suppose, however, that we have K tested subconfigurations all
comparable to a given untested subconfiguration. What is the best way to
proceed? One suggestion that has been made is to adjust only from the
"closest" test result. Aside from the problem of how to measure "closeness"
in three-dimensional parameter space, it seems that such an approach
simply ignores valuable information. The approach recommended here is
that all K adjustments should be made and the final result computed as
a weighted mean of the K individual estimates.
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Let
Ek = Ek (1 + SwAwk + Vrk + SaAra} k = 1> ••" K
be the kth adjusted fuel economy estimate for the untested subconfiguration.
Note that the sensitivity coefficients (assumed to be linear functions of
their parameters) are evaluated at the untested subconfiguration point
in parameter space. This is an adequate approximation to the mean
parameter values for small increments. Assuming independence of error
contributions, we may then estimate the variance of E'/E. (equal
approximately to the squared coefficient of variation of E') as:
K
where an is the coefficient of variation in subconfiguratton fuel economy
measurement, m^ is the multiplicity of the test results that were averaged to
estimate E^, the kth subconfiguration fuel economy. The sensitivity
coefficient variances are determined from the data used to estimate the
coefficients (see Section 3.5.2.1.4). As was done in Section 3, define
and
= Vu
i
The /uT represent appropriate weighting coefficients to use for estimating
K
the untested subconfiguration fuel economy with maximum precision*,
* This property depends on the k estimates being uncorrelated. In
general, there can be substantial correlation due to the presence
of the same sensitivity coefficients in different E^ estimates. An
exact calculation of the proper weights could be made, but the
improved precision is not believed to justify the added complexity
entailed.
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VIZ. ,
E' • r \ Ek
The coefficient of variation of E1 is
V = aO//U
5.2 Regional/Seasonal Adjustment Factors
EPA is placing a great deal of emphasis on the achievement of realistic
fuel economy numbers which people can associate with automotive vehicles.
Implementation of design parameter and road adjustment factors for fuel
economy labeling will constitute a major advance toward this objective.
However, these alone cannot be sufficient because the final adjusted values
will still represent an average for the whole country (and over all times
of the year). When a person buys a car he specifies a particular sub-
configuration. That is where the label value plays a crucial role. But
he also restricts (and particularizes) the range of numerous influential
environmental factors by virtue of his specific location. The national/
yearly average of 22 mpg city label value for car X doesn't apply to that
car driven in Phoenix AZ during the summer, nor to another copy in use in
Duluth MN in January-
An obvious solution to this problem is to provide the general public
with a set of regional/seasonal adjustment factors. Given a selected
location and time of year, one looks up the indicated factor which is
then multiplied into the label value of a car to yield an adjusted fuel
economy that represents the conditional median value for the subconfiguration
at the selected location and time of year. It is, of course, recognized
that even after such adjustment, much uncertainty still remains—associated
65
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with such factors as small scale environmental detail, trip characteristics,
and driver aggressiveness. Nevertheless, major environmental factors do
have a very substantial influence and are at the same time strongly
correlated with location and time of year.
Major practical questions relative to implementation of the concept
are: how to compute such regional/seasonal factors and to what level of
spatio-temporal detail?
With respect to the first question, one could attempt detailed
analyses of environmental factor effects on fuel economy as, for example,
presented in Reference 1 and couple the results with data on the distributions
of these environmental conditions. The results could be presented in the
form of U.S. maps of constant adjustment factor contours for different
months or seasons of the year. Difficulties with this approach stems from
the complexity of the analyses required, lack of knowledge regarding
interaction of effects, and data requirements for environmental factor
distributions.
An alternative approach which circumvents much of the above analyses
yet leads to direct and meaningful results would make use of the I strata
already developed for the in-use fuel economy survey design. Essentially
all that need be done is to estimate separate median road adjustment
factors for each stratum, i.e., (
-------
Identification of the appropriate stratum for a given place and time of
year could be provided by a series of seasonal maps or by a suitable
tabulation.
The direct empirical validity of the f ..correction factors so
derived should be clear. On the other hand, it is also recognized that
one can no longer view the f's as consequences of just environmental
influences. If, for example, drivers in one particular stratum just
so happen to be very aggressive, on the average, in comparison to other
strata, then that fact will be reflected in reduced f factor values.
In the final analysis, stratification of the in-use survey sample
space is a means of discriminating systematically different locales and
times of year. Judicious choice of strata boundaries or definitions can
maximize the differences among the strata and minimize the spread of
fuel economy variations within each stratum. By giving the public access
to the differences so determined, EPA would be taking another significant
step toward the provision of realistic fuel economy numbers which are as
specifically applicable as possible.
5.3 Test and Parameter Adjustment Strategies
If a manufacturer is not satisfied with the label values derived for
some of its untested subconfigurations by application of the EPA standard
design parameter sensitivity coefficients, it has two alternatives:
(1) Estimation of manufacturer-specific sensitivity coefficients
by procedure described in Section 3.
(2) Direct FTP/HFET fuel economy tests of the subconfiguration(s)
in question.
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Alternative (2) .should always be available to the manufacturer
inasmuch as it is fully consistent with the planned new labeling
procedures. Under these procedures mandated*test subconfigurations would
have their label values determined directly by the test results. If the
number of untested subconfigurations which the manufacturer believes to
be under-rated by EPA sensitivity coefficient adjustment is small, then
the least costly strategy would likely be to test these subconfigurations
directly. It is, of course, possible that a large manufacturer may have
enough of its own comparable mandated test subconfigurations to carry
out the manufacturer-specific sensitivity coefficient estimation pro-
cedure without having to introduce additional test data. This
is even more likely if EPA adopts the constant sensitivity coefficient
model. In that case the manufacturer would most certainly check out
his own sensitivity coefficient estimates first. If he doesn't like
their implications for some of his untested subconfigurations there would
seem to be no way to prevent him from selectively testing those subconfigura-
tions for the purpose of establishing direct test fuel economy label values.
It would appear, then, that manufacturers will have considerable
flexibility in establishing fuel economy label values if they are willing
to pay the price. They could, in effect try out each of the three
available alternatives for their untested subconfigurations and choose
the largest value. Inasmuch as there are random errors in all of the
alternatives, this would be tantamount to introducing a positive bias in
fuel economy labels of the non-mandated test subconfigurations.
* As presently required for emission certification or fuel economy
label and CAFE determinations.
68
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It is recommended that EPA carefully review the process by which
manufacturers would be permitted to take exception to proposed EPA
standard design parameter sensitivity coefficients in order to preclude
biasing of fuel economy label values. This review should also consider
the question of whether to maintain the present specificity of mandated
vehicle tests at the configuration level or to raise it to the subconfiguration
level.
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REFERENCES
1 EPA, "Passenger Car Fuel Economy: EPA and Road," Draft Report
in response to Section 404 of PL95-619, January 1980.
2 EPA Informal Memo, "ECTD Deliverables--Fuel Economy Information
Rulemaking," K.- H. Hellman, dated 7/2/80 on Cover Sheet.
3 EPA Memo, "Extremes Analyses of Ford In-Use Data Base,"
K. H. Hellman, 9/3/80.
** A. M. Mood, Introduction to the Theory of Statistics, McGraw-
Hill, New York, 1950, pages 406-408.
5 E. J. Gumbel, Statistics of Extremes, Columbia University Press,
New York, 1958.
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APPENDIX A
VARIABILITY OF DYNAMOMETER FUEL ECONOMY MEASUREMENTS
The fuel economy label given to a car is generally based on un-
replicated dynamometer tests and hence subject to potentially significant
errors. These are decomposable into: (1) test repeatability errors
within a fixed test cell (dynamometer + driver + CVS apparatus),
(2) differences among test cells, (3) differences among vehicles of
same configuration, and (4) heterogeneity of the vehicle configurations
which are aggregated into a single fuel economy label value. Jn this note
we focus on (1) through (3). Components (1) and (2) are often loosely
lumped together as "measurement variability." Some published estimates for
fuel economy error standard deviations are shown in Table A-l. There seems
to be considerable disparity among investigators, with no clear historical
trend. Within-test-cell a's under the carbon balance method range from
1.2% to 4.8%. ASTM estimates that metered mpg determination can
reduce this error to 0.75%.
The correlation study by Sheth and Rice on five dynamometer test
cells suggests that the additional contribution due to between-test-cell
differences is reasonably limited, amounting to a standard deviation of
about 2%. This is consistent with some earlier (1974) data from the
Repca I correlation study,7 involving several EPA and manufacturer test
laboratories, from which a between-lab standard deviation in C02 measure-
ment of about 2.5% was deduced.
The one precise datum in Table A-l (Juneja2) that includes between-
vehicle variability is based on a single 1975 model (unique subconfiguration),
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Table A-l.
SOURCE
Simpson (1975)1
Juneja, et al (1977)2
Schumann, et al (1978)3
Sheth and Rice (1974)"
ASTM (1980)5
NHTSA (1979)6
o
(%)*
2.4
2.8**
1.2
2.8
4.8
2.7
2.9***
3.3
3.5***
1.9
0.75**
2-3.5
ERROR COMPONENTS INCLUDED
Within
Test Cell
X
X
X
X
X
X
X
X
X
X
X
Between
Test Cell
X
X
Between
Vehicle
X
X
Results shown are for FTP fuel economy tests unless otherwise stated.
* The consensus is that a is roughly proportional to true value;
hence, it is generally reported in terms of coefficient of
variation (as a percent of mean fuel economy).
** Volumetric or gravimetric procedure.
*** HFET (Highway Fuel Economy Test).
72
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with six nominally identical vehicles multiply-tested at 5000, 10,000 and
15,000 miles. The reported results imply a between-vehicle component
standard deviation of about 2.5%.
In order to be able to estimate the overall accuracy of vehicle
subconfiguration fuel economy determinations with reasonable confidence,
it would be helpful to have additional corroboration of the above reported
results, particularly with respect to the between-vehicle component of
error which is inferred from tests on only a single vehicle subconfiguration.
A recently acquired EPA report8 on replicated and multiple-vehicle
testing of a number of different 1977 models provide limited data for
such supporting analysis.
Nominally, three low-mileage (3,000 to 9,000 miles) cars drawn from
each of eleven models (representing subcompact fuel economy leaders) each
received three replicate FTP and HFET fuel economy tests. On closer
inspection of the data, however, tt was determined that: 18 different sub-
configurations were represented (due to variations in transmission, axle
ratio, etc.); the number of vehicles per subconfiguration ranged from one
to three; and the number of replications per vehicle ranged from two to
four.
A two-fold hierarchical linear model was assumed for a components of
variance analysis, viz.,9
Y. .. = y + a. + b. . + e. ..
ijk i ij ijk
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where i = model (subconfiguration) index
j = vehicle index
k = replication index
y + -a.,- = mean fuel economy of i^*1 model
b.. = random perturbation due to (i,j)th vehicle
• \J
E[b..] =0, Var[b..] - a^
,th
E[e.] =0, V.r[e
e.. = random error due to (i,j,k) test
1 JK
-ijkj - u, va- Lc.jkj - ue
Furthermore, all the {b^} and {e^-i.} were assumed to be uncorrelated.
' \J I «J K-
In this representation 05 is the (assumed common) standard deviation of
between-vehicle differences for all models and ae is the (assumed common)
measurement error standard deviation. ae includes the within-test-cell
component plus an indeterminate fraction of the between-test-cell component
due to partial test cell variation.
The analysis is somewhat complicated by the unbalanced design (unequal
numbers) but the formulae are still straightforward9 and were applied to
the data in the EPA report to estimate ab and 0e for both FTP and
HFET tests. An alternate analysis was also performed based on the
assumption of common coefficients of variation (COV) rather than common
standard deviations. This was accomplished through normalization of all
fuel economies by the appropriate estimated model mean. The resulting
estimates (and estimated standard errors) are:
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MODEL ASSUMPTION:
FTP Fuel Economy
HFET Fuel Economy
COMMON STANDARD DEVIATIONS
^.
°b
(mpg)
0.93±0.21
0.88±0.25
s\
°e
(mpg)
0.42±0.04
0.73±0.06
xs.
y
(mpg)
28.3
37.5
COMMON COEFFICIENTS
OF VARIANCE*
covb
(%)
3.3±0.73
2.2+0.65
cove
(%)
1.510.13
1.910.17
Note that the within-cell plus partial between-cell measurement error
magnitudes of 1.5-1.9% represent test cell performance which is consistent
with, though somewhat better than, most entries in the previous table.
Finally, we observe that the inferred between-vehicle variabilities of
2.2 to 3.3% tend to corroborate the previously inferred value of 2.5%.
In conclusion, the review and analysis conducted suggest that vehicle
variability, test cell variability, and test replication error all make
fairly comparable contributions to the total error in vehicle subconfigura-
tion fuel economy measurement, but with relative strengths in the order
indicated. Furthermore a reasonably conservative estimate for total error
coefficient of variation is 4%.
COV notation is used here to avoid confusion between the two model
assumptions; however it is the COV representation which corresponds
to the a's of the previous table.
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References
B. R. Simpson, "Improving the Measurement of Chassis Dynamometer
Fuel Economy," SAE Paper 750002, February 1975.
2 W. K. Juneja, D. D. Horchler, H. M. Haskew, "A Treatise on
Exhaust Emission Test Variability," SAE Paper 770136, February 1977.
3 D. Schumann, N. Krause, D. Kinne, "The Influence of Testing
Parameters on Exhaust Gas Emissions," SAE Paper 780649,
June 1978.
** N. S. Seth and T. I. Rice, "Identification, Quantification, and
Reduction of Sources of Variability in Vehicle Emissions and Fuel
Economy Measurements," SAE Paper 790232, February 1979.
5 L. J. Painter, "Review of Statistical Aspects of 'EPA Recommended
Practice for Evaluating.. .Engine Oils'," Chevron Research Company
Memorandum (No Number), March 1980.
6 NHTSA, "Review of Procedures for Determining Corporate Average
Fuel Economy," Report Nos. DOT-HS-805396 and -805397, July 1979.
7 R. E. Lowery, "Emission Laboratory Correlation Study Between EPA
and the MVMA," EPA Report (No Number), September 24, 1974.
8 F. P. Hutchins and J. Kranig, "An Evaluation of the Fuel Economy
Performance of Thirty-One 1977 Production Vehicles Relative to
Their Certification Counterparts," EPA Report 77-18 FPH (Technology
Assessment and Evaluation Branch), January 1978.
9 0. Kempthorne, The Design and Analyses of Experiments, John Wiley,
pages 103-110, 1952.
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APPENDIX B
FUEL ECONOMY LABEL ERROR DUE TO
NORMAL ERROR BEFORE ROUND-OFF
Let E and E1 be true and estimated fuel economy (in mpg) with
x = E1 - E (the error), normally distributed with mean zero and variance
a2. Let
EL = [E + 0.5]
E^ = [E' + 0.5]
be the corresponding label values, i.e., round-off to nearest whole number.
Then
y - E- - EL
is the fuel economy label error (in mpg) due to x. In contrast to x, y
can only take on integer values. It is of interest to determine the
probability distribution of y for different values of a.
A basic assumption that permits this determination to be made fairly
straightforwardly is that the decimal portion of E is uniformly dis-
tributed on [0, l] and independent of x. Then the conditional
probabilities for y given x may be expressed as:
1 - |x|; 0 < |x| < 1
= 0|X)=' 0 ; UI>1
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^x = Pr{|y| = i|x> =\2 - |x|; 1 £ |x| £ 2
0 ; |x| > 2
"|x|-(k - 1); k - l£ |x| £ k
Pklx = Pr(|y| = k|x> =^k + 1 - |x|; k £ |x| £ k + 1
0 ; elsewhere
Invoking a basic relationship in conditional probabilities:
Pk = Pr{|y| = k} = /(Pk|x) -f(x)dx
we determine:
(1 - x) • —— exp (-x2/2a2)dx
a /2rr
(x -
k-1
/2rr
(-x2/2a2)dx
+ / (k + 1 - x) exp (-x2/2a2)dx; k > 0
n /9^
k
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Rewrite PQ as:
(x - (-
-1
a /2ir
(-x2/2a2)dx
+ / (1 -
'0
exp (-x2/2a2)dx
Under transformation of variables z = x/a and u = x2/2a2 and definition
1; k = 0
2; k = 1, 2, ...
we obtain (for k = 0, 1, 2, ...):
Pk=6k
k_
a
k+1
a
- (k - 1) / ~~exp (-z2/2)dz + (k + 1)
k-1
a
L
a
2a2
a / -u .
/ e du -
e du
2a
(-z2/2)dz
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(k -f 1) (*
()) - (k - 1) (t () - 4 (^
O CT 0
/2TT
(exp( - (k + l)2/2a2) - 2 exp (-k2/2a2) + exp ( - (k - l)2/2a2))
(k
*
) + (k -
(-k2/2a2)
(-l/2a2) cosh () -
where $(•) is the standard normal cumulative distribution functi
on.
Evaluation of P , for selected a is given below:
K
a
(MPG)
0.25
0.50
1.00
PROBABILITY OF A ± k MPG ERROR IN
FUEL ECONOMY LABEL VALUE, PR
k = 0
0.80
0.61
0.37
k = 1
0.20
0.38
0.48
k = 2
< 0.001
0.01
0.13
k = 3
-
-
0.02
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