40 CFR Part 86, Appendix XVIII to Part 86 - Statistical Outlier Identification Procedure for Light-Duty Vehicles and Light Light-Duty Trucks Certifying to the Provisions of Part 86, Subpart R

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Appendix XVIII to Part 86—Statistical Outlier Identification Procedure for Light-Duty Vehicles and Light Light-Duty Trucks Certifying to the Provisions of Part 86, Subpart R
Residual normal deviates to indicate outliers are used routinely and usefully in analyzing regression data, but suffer theoretical deficiencies if statistical significance tests are required. Consequently, the procedure for testing for outliers outlined by Snedecor and Cochran, 6th ed., Statistical Methods, PP. 157-158, will be used. The method will be described generally, then by appropriate formulae, and finally a numerical example will be given.
(a) Linearity is assumed (as in the rest of the deterioration factor calculation procedure), and each contaminant is treated separately. The procedure is as follows:
(1) Calculate the deterioration factor regression as usual, and determine the largest residual in absolute value. Then recalculate the regression with the suspected outlier omitted. From the new regression line calculate the residual at the deleted point, denoted as (yi − yi′). Obtain a statistic by dividing (yi − yi′) by the square root of the estimated variance of (yi − yi′). Find the tailed probability, p, from the t-distribution corresponding to the quotient (double-tailed), with n-3 degrees of freedom, with n the original sample size.
(2) This probability, p, assumes the suspected outlier is randomly selected, which is not true. Therefore, the outlier will be rejected only if 1 − (1-p)n < 0.05.
(3) The procedure will be repeated for each contaminant individually until the above procedure indicates no outliers are present.
(4) When an outlier is found, the vehicle test-log will be examined. If an unusual vehicle malfunction is indicated, data for all contaminants at that test-point will be rejected; otherwise, only the identified outlier will be omitted in calculating the deterioration factor.
(b) Procedure for the calculation of the t-Statistic for Deterioration Data Outlier Test.
(1) Given a set of n points, (x1, y1), (x2. y2) * * * (xn, yn).
Where:
xi is the mileage of the ith data point.
yi is the emission of the ith data point.
Assume model:
y = a β(x − x) ∈
(2)(i) Calculate the regression line.
ŷ = a b(x − x)
(ii) Suppose the absolute value of the ith residual
(yi − ŷi) is the largest.
(3)(i) Calculate the regression line with the ith point deleted.
ŷ′ = a′ b′(x − x)
(ii)
Where:
y1 is the observed suspected outlier.
ŷ′i is the predicted value with the suspected outlier deleted.
(x is calculated without the suspected outlier)
(iii) Find p from the t-statistic table
Where:
p = prob (|t(n-3) | ≥ t)
t(n-3) is a t-distributed variable with n-3 degrees of freedom.
(iv) yi is an outlier if 1-(1-p)n < .05
x y ŷ y − ŷ
1 Suspected outlier.
8 59 56.14 2.86
6 58 58.17 −0.17
11 56 53.10 2.90
22 1 53 41.96 11.04
14 50 50.06 −0.06
17 45 47.03 −2.03
18 43 46.01 −3.01
24 42 39.94 2.06
19 39 45.00 −6.00
23 38 40.95 −2.95
26 30 37.91 −7.91
40 27 23.73 3.27
(4)(i) Assume model:
y = a β(x − x) ∈
y = 45 − 1.013(x − x)
(ii) Suspected point out of regression:
y = 44.273 − 1.053(x − x)
y = 44.273 − 1.053(22 − 18.727) = 40.827
yi − ŷ′i = 12.173
[62 FR 31270, June 6, 1997; 62 FR 45289, Aug. 26, 1997, as amended at 63 FR 987, Jan. 7, 1998]

Title 40 published on 2015-07-01.

The following are only the Rules published in the Federal Register after the published date of Title 40.

For a complete list of all Rules, Proposed Rules, and Notices view the Rulemaking tab.

  • 2015-07-28; vol. 80 # 144 - Tuesday, July 28, 2015
    1. 80 FR 44863 - Greenhouse Gas Emissions and Fuel Efficiency Standards for Medium- and Heavy-Duty Engines and Vehicles—Phase 2; Notice of Public Hearings and Comment Period
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      DEPARTMENT OF TRANSPORTATION, ENVIRONMENTAL PROTECTION AGENCY, National Highway Traffic Safety Administration
      Notice of public hearings; extension of comment period.
      NHTSA and EPA will jointly hold a public hearing on Thursday, August 6, 2015, beginning at 9:00 a.m. local time, and a second hearing on Tuesday, August 18, 2015, beginning at 9:00 a.m. local time. EPA and NHTSA will make every effort to accommodate all speakers that arrive and register. Each hearing will continue until everyone has had a chance to speak. If you would like to present oral testimony at one of these this public hearings, please contact the person identified under FOR FURTHER INFORMATION CONTACT by August 3, 2015, for the first hearing, or by August 11, 2015, for the second hearing. In order to provide commenters 30 days after the last public hearing, the comment period for the proposal is being extended through September 17, 2015.
      40 CFR Parts 9, 22, 85, 86, 600, 1033, 1036, 1037, 1039, 1042, 1065, 1066, and 1068

Title 40 published on 2015-07-01

The following are ALL rules, proposed rules, and notices (chronologically) published in the Federal Register relating to 40 CFR Part 86 after this date.

  • 2015-09-08; vol. 80 # 173 - Tuesday, September 8, 2015
    1. 80 FR 53756 - Greenhouse Gas Emissions and Fuel Efficiency Standards for Medium- and Heavy-Duty Engines and Vehicles—Phase 2; Extension of Comment Period
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      DEPARTMENT OF TRANSPORTATION, ENVIRONMENTAL PROTECTION AGENCY, National Highway Traffic Safety Administration
      Proposed rule; extension of comment period.
      The comment period for the proposed rule published July 13, 2015 (80 FR 40139) is extended. Written comments for both documents must be received on or before October 1, 2015.
      40 CFR Parts 9, 22, 85, 86, 600, 1033, 1036, 1037, 1039, 1042, 1065, 1066, and 1068