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.4o

Final	Model	Fitting

Methane (CH4)	(Bag	3)

No. Observations: 893

No. Vehicles: 14 (drop Corolla)

No. censored measurements: 0

No. missing measurements: 0

No. measurements removed: 0

Model Type: Mixed model
2

I.4o.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 3): 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 1-7 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

1 7 -353.062 • • • • • •

2 8 -352.229 • • • • • • •

3 8 -351.155 • • • • • • •

4 8 -351.109 • • • • • • •

5 9 -350.906 • • • • • • • •

6 8 -350.865 • • • • • • •

7 8 -350.778 • • • • • • •

8 9 -350.017 • • • • • • • •

9 6 -349.815 • • • • •

10 9 -349.594 • • • • • • • •

11 9 -349.567 • • • • • • • •

12 7 -349.505 • • • • • •

13 7 -349.226 • • • • • •

14 9 -349.218 • • • • • • • •

15 9 -349.116 • • • • • • • •

16 9 -349.068 • • • • • • • •

17 10 -349.003 • • • • • • • • •

18 9 -348.894 • • • • • • • •

19 10 -348.842 • • • • • • • • •

20 6 -348.693 • • • • •

21 9 -348.572 • • • • • • • •

22 7 -348.469 • • • • • •

23 10 -348.27 • • • • • • • • •

24 8 -348.16 • • • • • • •

25 7 -348.083 • • • • • •

26 8 -348.027 • • • • • • •

27 7 -347.969 • • • • • •

28 10 -347.863 • • • • • • • • •

29 7 -347.691 • • • • • •

30 10 -347.481 • • • • • • • • •

31 10 -347.384 • • • • • • • • •

32 8 -347.236 • • • • • • •

33 10 -347.21 • • • • • • • • •

34 11 -347.144 • • • • • • • • • •

35 10 -347.142 • • • • • • • • •

Design Terms
5

Models fit for CH4 (Bag 3): (all models include an intercept term).

Model Term Notation Model

Superset SM21SM3 SM4

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 1,” etc.

CH4 (Bag 3): Model fitting history, starting with the 10-term superset model.

Fit Parameters Test with respect to

Previous Model

Model p -2lnL BIC

1 Dev. d Pr>χ2

Superset 11 -380.510 -346.200

SM2

2

9 -379.936 -350.907 0.576 2 0.750

SM3 8 -378.619 -352.229 1.317 1 0.251

SM427 -376.814 -353.062 1.805 1 0.179

1 A lower value indicates a better fit.

2 Best fit with respect to the 11-term design model.
6

CH4 (Bag 3): 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 (SM4)

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

Intercept -4.4735 0.1789 14 -25.0 0.000000 -4.4742 0.1789 14 -25.0 0.000000

Z e 0.02354 0.009267 871 2.54 0.011 0.02805 0.007619 871 3.68 0.00025

Z a-0.09565 0.006507 871 -14.7 0.000000 -0.09578 0.006502 871 -14.7 0.00000

Z r-0.008909 0.007373 871 -1.21 0.23

Z 5 0.02400 0.009333 871 2.57 0.010 0.03025 0.007967 871 3.80 0.00016

Z 9 0.01811 0.006477 871 2.80 0.0053 0.01691 0.006375 871 2.65 0.0081

ZZee -0.000506 0.01230 871 -0.0411 0.97

ZZ55 0.01093 0.009364 871 1.17 0.24

ZZea 0.01463 0.006349 871 2.30 0.021 0.01528 0.006289 871 2.43 0.015

ZZer --- --- --- --- --- --- --- --- --- ---

ZZe5 -0.01724 0.01321 871 -1.30 0.19 -0.02079 0.006491 871 -3.20 0.0014

ZZe9 0.004783 0.006314 871 0.758 0.45

0.4478 0.4477

0.03425 0.03439

2

veh

2



2

veh

2


7

I.4o.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 3): 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).

NOTE: Model fitting with respect to the 16-term extended model gives identical results to those 

obtained for the 11-term design model, as presented above.

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 7 -353.062 • • • • • •

2 8 -352.229 • • • • • • •

3 8 -351.155 • • • • • • •

4 8 -351.109 • • • • • • •

5 9 -350.906 • • • • • • • •

6 8 -350.865 • • • • • • •

7 8 -350.778 • • • • • • •

8 10 -350.633 • • • • • • • • •

9 8 -350.531 • • • • • • •

10 8 -350.516 • • • • • • •

11 8 -350.429 • • • • • • •

12 9 -350.017 • • • • • • • •

13 6 -349.815 • • • • •

14 9 -349.727 • • • • • • • •

15 9 -349.715 • • • • • • • •

16 9 -349.66 • • • • • • • •

17 9 -349.639 • • • • • • • •

18 9 -349.594 • • • • • • • •

19 9 -349.567 • • • • • • • •

20 7 -349.505 • • • • • •

21 7 -349.226 • • • • • •

22 9 -349.218 • • • • • • • •

23 7 -349.155 • • • • • •

24 9 -349.116 • • • • • • • •

25 9 -349.068 • • • • • • • •

26 10 -349.003 • • • • • • • • •

27 9 -348.894 • • • • • • • •

28 10 -348.842 • • • • • • • • •

29 8 -348.707 • • • • • • •

30 6 -348.693 • • • • •

31 9 -348.636 • • • • • • • •

32 11 -348.629 • • • • • • • • • •

33 9 -348.62 • • • • • • • •

34 9 -348.594 • • • • • • • •

35 9 -348.574 • • • • • • • •

Design Terms Extended Terms