1

Assessing	the	Effect	of	Five	Gasoline	Properties	on	Exhaust	Emissions	from	LightDuty	Vehicles	certified	to	Tier-2	Standards

Analysis	of	Data	from	EPA	Phase	3

(EPAct/V2/E-89)

Appendix	I.4n

Final	Model	Fitting

Methane (CH4)	(Bag	2)

No. Observations: 956

No. Vehicles: 15

No. censored measurements: 0

No. missing measurements: 0

No. measurements removed: 0

Model Type: Mixed model
2

I.4n.1			Model	fitting	with	respect	to	the	11-term	design	model.

Design Model (11-terms): Bayesian Information Criterion (BIC) vs. number of terms in model (p) 

for all possible models respecting hierarchy. 

Design Model (11-terms): Bayesian Information Criterion (BIC) vs. number of terms in model (p) 

for all possible models respecting hierarchy (CLOSEUP of previous figure). 
3

Design Model (11-terms): Bayesian Information Criterion (BIC) vs. number of terms in model (p) 

for all possible models respecting hierarchy (CLOSEUP of previous figure). 
4

CH4 (Bag 2): Number of terms (p), Goodness-of-fit (BIC) and terms included in the 35 best-fitting 

candidate models (out of a total of 294 possible models with hierarchy). (Terms included in models 

ranked 1-6 comprise the “superset” for final model-fitting).
5

Models fit for CH4 (Bag 2): (all models include an intercept term).

Model Term Notation Model

Superset SM2

1

etOH Ze ● ●

Arom Za ● ●

RVP Zr ● ●

T50 Z5 ● ●

T90 Z9 ● ●

etOH × etOH ZZee --- ---

T50 × T50 ZZ55 ● ●

etOH × Arom ZZea ● ●

etOH × RVP ZZer ● ×

etOH × T50 ZZe5 --- ---

etOH × T90 ZZe9 ● ×

1 Represents “Superset minus 2,” etc.

CH4 (Bag 2): Model fitting history, starting with the 9-term superset model.

Fit Parameters Test with respect to

Previous Model

Model p -2lnL BIC

1 Dev. d Pr>χ2

Superset 10 -691.760 -659.263

SM128 -687.712 -660.632 4.048 2 0.132

1 A lower value indicates a better fit.

2 Best fit with respect to the 16-term extended model.
6

CH4 (Bag 2): Coefficients and Tests of Effect for the Superset and Reduced Models, with respect to 

the 11-term design model.

Effect Full Model (superset) Reduced Model (SM2)

Estimate Std. Err. d.f. t -value Pr>t Estimate Std. Err. d.f. t -value Pr>t

Intercept -5.7080 0.2722 15 -21.0 0.000000 -5.7075 0.2722 15 -21.0 0.000000

Z e 0.05859 0.006829 941 8.58 0.000000 0.05860 0.006839 941 8.57 0.000000

Z a-0.09846 0.005308 941 -18.5 0.000000 -0.09836 0.005317 941 -18.5 0.000000

Z r-0.02053 0.006080 941 -3.38 0.00076 -0.02049 0.006036 941 -3.39 0.00072

Z 5 0.04321 0.007399 941 5.84 0.000000 0.04394 0.007393 941 5.94 0.000000

Z 9 0.02539 0.005359 941 4.74 0.000002 0.02575 0.005290 941 4.87 0.000001

ZZee --- --- --- --- --- --- --- --- --- ---

ZZ55 0.01248 0.005381 941 2.32 0.021 0.01227 0.005379 941 2.28 0.023

ZZea 0.008520 0.005183 941 1.64 0.10 0.008769 0.005185 941 1.69 0.091

ZZer -0.007004 0.005258 941 -1.33 0.18

ZZe5 --- --- --- --- --- --- --- --- --- ---

ZZe9 -0.006794 0.005260 941 -1.29 0.20

1.1109 1.1108

0.02507 0.02518

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I.4n.2 Model	Fitting	with	respect	to	the	16-term	extended	Model.

Extended Model (16-terms): Bayesian Information Criterion (BIC) vs. number of terms in model 

(p) for all possible models respecting hierarchy. 

Extended Model (16-terms): Bayesian Information Criterion (BIC) vs. number of terms in model 

(p) for subset of models respecting hierarchy (CLOSEUP of previous figure).
8

Extended Model (16-terms): Bayesian Information Criterion (BIC) vs. number of terms in model 

(p) for subset of models respecting hierarchy (CLOSEUP of previous figure).
9

CH4 (Bag 2): Number of terms (p), Goodness-of-fit (BIC) and terms included in the 35 best-fitting 

candidate models (out of a total of 2,964 possible models with hierarchy). (Terms included in 

models ranked 1-5 comprise the “superset” for final model-fitting).

Rank p BIC

etO

H

AromR

VP

T50T90etO

H × etO

H

T50 × T50etO

H × Arom

etO

H × R

VP

etO

H × T50

etO

H × T90

Arom × R

VP

Arom × T50Arom × T90T50 × T90R

VP × T90

1 10 -667.041 • • • • • • • • •

2 9 -666.822 • • • • • • • •

3 8 -666.263 • • • • • • •

4 9 -665.701 • • • • • • • •

5 11 -665.585 • • • • • • • • • •

6 11 -665.274 • • • • • • • • • •

7 11 -665.265 • • • • • • • • • •

8 10 -665.124 • • • • • • • • •

9 11 -665.117 • • • • • • • • • •

10 10 -665.026 • • • • • • • • •

11 10 -665.024 • • • • • • • • •

12 9 -665.023 • • • • • • • •

13 10 -664.96 • • • • • • • • •

14 11 -664.827 • • • • • • • • • •

15 9 -664.797 • • • • • • • •

16 11 -664.614 • • • • • • • • • •

17 8 -664.601 • • • • • • •

18 11 -664.568 • • • • • • • • • •

19 10 -664.563 • • • • • • • • •

20 10 -664.543 • • • • • • • • •

21 10 -664.491 • • • • • • • • •

22 11 -664.448 • • • • • • • • • •

23 12 -664.436 • • • • • • • • • • •

24 9 -664.377 • • • • • • • •

25 9 -664.302 • • • • • • • •

26 10 -664.246 • • • • • • • • •

27 12 -664.207 • • • • • • • • • • •

28 8 -664.16 • • • • • • •

29 10 -664.151 • • • • • • • • •

30 10 -664.114 • • • • • • • • •

31 10 -664.016 • • • • • • • • •

32 11 -663.984 • • • • • • • • • •

33 9 -663.976 • • • • • • • •

34 10 -663.884 • • • • • • • • •

35 12 -663.861 • • • • • • • • • • •

Design Terms Extended Terms
10

Models fit for CH4 (Bag 2): (all models include an intercept term).

Model Term Notation Model

Superset SM1

etOH Ze ● ●

Arom Za ● ●

RVP Zr ● ●

T50 Z5 ● ●

T90 Z9 ● ●

etOH × etOH ZZee --- ---

T50 × T50 ZZ55 ● ●

etOH × Arom ZZea ● ●

etOH × RVP ZZer ● ×

etOH × T50 ZZe5 --- ---

etOH × T90 ZZe9 --- ---

Arom × RVP ZZar --- ---

Arom × T50 ZZa5 --- ---

Arom × T90 ZZa9 --- ---

T50 × T90 ZZ59 ● ●

RVP × T90 ZZr9 ● ●

1 Denotes “Superset minus 1,” etc.

CH4 (Bag 2): Model fitting history, starting with the 15-term superset model.

Fit Parameters Test with respect to

Previous Model

Model p -2lnL BIC

1 Dev. d Pr>χ2

Superset 11 -700.789 -665.585

SM1

2

10 -699.537 -667.041 1.252 1 0.263

1 A lower value indicates a better fit.

2 Best fit with respect to the 16-term extended model.
11

CH4 (Bag 2): Coefficients and Tests of Effect for the Superset and Reduced Models, with respect to 

the 16-term extended model.

Effect Full Model (superset) Reduced Model (SM1)

Estimate Std. Err. d.f. t -value Pr>t Estimate Std. Err. d.f. t -value Pr>t

Intercept -5.7078 0.2722 15 -21.0 0.000000 -5.7076 0.2722 15 -21.0 0.000000

Z e 0.06063 0.006873 941 8.82 0.000000 0.06076 0.006877 941 8.84 0.000000

Z a-0.09260 0.005608 941 -16.5 0.000000 -0.09211 0.005595 941 -16.5 0.000000

Z r-0.02135 0.006033 941 -3.54 0.00042 -0.02082 0.006018 941 -3.46 0.00056

Z 5 0.04424 0.007413 941 5.97 0.000000 0.04477 0.007403 941 6.05 0.000000

Z 9 0.02510 0.005324 941 4.71 0.000003 0.02445 0.005295 941 4.62 0.000004

ZZee --- --- --- --- --- --- --- --- --- ---

ZZ55 0.01445 0.005496 941 2.63 0.0087 0.01398 0.005483 941 2.55 0.011

ZZea 0.01015 0.005202 941 1.95 0.051 0.01047 0.005198 941 2.01 0.044

ZZer -0.005907 0.005277 941 -1.12 0.26

ZZe5 --- --- --- --- --- --- --- --- --- ---

ZZe9 --- --- --- --- --- --- --- --- --- ---

ZZar --- --- --- --- --- --- --- --- --- ---

ZZa5 --- --- --- --- --- --- --- --- --- ---

ZZa9 --- --- --- --- --- --- --- --- --- ---

ZZ59 0.01241 0.005994 941 2.07 0.039 0.01374 0.005879 941 2.34 0.020

ZZr9 -0.01034 0.005802 941 -1.78 0.075 -0.009921 0.005794 941 -1.71 0.087

1.111 1.1109

0.02483 0.02486

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