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  
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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

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 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

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 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
                                   54

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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.
                                    59

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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.
                                    60

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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.
                                  61

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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.
                                  62

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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.
                                  63

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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.
                                   64

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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., (
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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.
                                  67

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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.
                                   69

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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.
                                 70

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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),
                                    71

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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
                                    73

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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:
                                    74

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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.
                                   75

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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.
                                   76

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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
                                    77

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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
                                    78

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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
                                     79

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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
                              80

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