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.4m

Final	Model	Fitting

Methane (CH4)	(Bag	1)

No. Observations: 956

No. Vehicles: 15

No. censored measurements: 0

No. missing measurements: 0

No. measurements removed: 0

Model Type: Mixed model
2

I.4m.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 1): 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

V

P

etO

H × T50

etO

H × T90

1 10 -506.33 • • • • • • • • •

2 7 -505.646 • • • • • •

3 9 -505.54 • • • • • • • •

4 11 -505.305 • • • • • • • • • •

5 9 -505.278 • • • • • • • •

6 8 -504.925 • • • • • • •

7 12 -504.911 • • • • • • • • • • •

8 11 -503.724 • • • • • • • • • •

9 10 -503.703 • • • • • • • • •

10 8 -503.404 • • • • • • •

11 8 -503.314 • • • • • • •

12 8 -503.053 • • • • • • •

13 10 -503.043 • • • • • • • • •

14 10 -503.037 • • • • • • • • •

15 10 -502.965 • • • • • • • • •

16 9 -502.745 • • • • • • • •

17 9 -502.734 • • • • • • • •

18 9 -502.266 • • • • • • • •

19 11 -501.58 • • • • • • • • • •

20 11 -501.43 • • • • • • • • • •

21 9 -500.816 • • • • • • • •

22 9 -500.73 • • • • • • • •

23 10 -500.091 • • • • • • • • •

24 10 -500.077 • • • • • • • • •

25 6 -498.114 • • • • •

26 8 -498.028 • • • • • • •

27 7 -496.925 • • • • • •

28 10 -496.179 • • • • • • • • •

29 8 -496.142 • • • • • • •

30 9 -496.084 • • • • • • • •

31 9 -495.926 • • • • • • • •

32 7 -495.891 • • • • • •

33 9 -495.731 • • • • • • • •

34 7 -495.578 • • • • • •

35 7 -495.529 • • • • • •

Design Terms
5

Models fit for CH4 (Bag 1): (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 1,” etc.

CH4 (Bag 1): 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 12 -542.823 -504.911

SM1210 -538.826 -506.330 3.997 2 0.136

1 A lower value indicates a better fit.

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

CH4 (Bag 1): 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 -3.0072 0.1380 15 -21.8 0.00000 -3.0074 0.1381 15 -21.8 0.00000

Z e 0.06938 0.00841 941 8.25 0.00000 0.06994 0.008428 941 8.30 0.00000

Z a-0.1055 0.005865 941 -18.0 0.00000 -0.1053 0.005876 941 -17.9 0.00000

Z r-0.03439 0.006698 941 -5.13 0.00000 -0.03275 0.006627 941 -4.94 0.00000

Z 5 0.07516 0.008449 941 8.90 0.00000 0.07554 0.008465 941 8.92 0.00000

Z 9 0.003600 0.005867 941 0.61 0.54

ZZee 0.02814 0.01129 941 2.49 0.013 0.02844 0.01131 941 2.51 0.012

ZZ55 0.05185 0.008457 941 6.13 0.00000 0.05170 0.008474 941 6.10 0.00000

ZZea 0.02049 0.005737 941 3.57 0.00037 0.02088 0.005745 941 3.63 0.00029

ZZer 0.008907 0.005852 941 1.52 0.13 0.01082 0.005779 941 1.87 0.062

ZZe5 0.02983 0.01212 941 2.46 0.014 0.03048 0.01213 941 2.51 0.012

ZZe9 0.01138 0.005759 941 1.98 0.048

0.2853 0.2855

0.03001 0.03014

2 veh

2 

2 veh

2 
7

I.4m.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 1): 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 1-8 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 15 -534.546 • • • • • • • • • • • • • •

2 16 -532.551 • • • • • • • • • • • • • • •

3 16 -532.341 • • • • • • • • • • • • • • •

4 13 -532.065 • • • • • • • • • • • •

5 14 -531.548 • • • • • • • • • • • • •

6 14 -531.484 • • • • • • • • • • • • •

7 13 -531.384 • • • • • • • • • • • •

8 15 -531.224 • • • • • • • • • • • • • •

9 12 -530.726 • • • • • • • • • • •

10 17 -530.397 • • • • • • • • • • • • • • • •

11 11 -530.391 • • • • • • • • • •

12 14 -529.83 • • • • • • • • • • • • •

13 14 -529.57 • • • • • • • • • • • • •

14 13 -529.491 • • • • • • • • • • • •

15 14 -529.458 • • • • • • • • • • • • •

16 14 -529.368 • • • • • • • • • • • • •

17 14 -529.357 • • • • • • • • • • • • •

18 15 -529.338 • • • • • • • • • • • • • •

19 15 -529.277 • • • • • • • • • • • • • •

20 13 -529.262 • • • • • • • • • • • •

21 15 -529.258 • • • • • • • • • • • • • •

22 16 -529.107 • • • • • • • • • • • • • • •

23 15 -528.932 • • • • • • • • • • • • • •

24 14 -528.751 • • • • • • • • • • • • •

25 14 -528.529 • • • • • • • • • • • • •

26 12 -528.297 • • • • • • • • • • •

27 14 -528.194 • • • • • • • • • • • • •

28 12 -528.121 • • • • • • • • • • •

29 12 -528.07 • • • • • • • • • • •

30 13 -528.062 • • • • • • • • • • • •

31 13 -528.024 • • • • • • • • • • • •

32 12 -527.945 • • • • • • • • • • •

33 12 -527.773 • • • • • • • • • • •

34 12 -527.751 • • • • • • • • • • •

35 15 -527.366 • • • • • • • • • • • • • •

Design Terms Extended Terms
10

Models fit for CH4 (Bag 1): (all models include an intercept term).

Model Term Notation Model

Superset SM2

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 ● ×

CH4 (Bag 1): 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 17 -581.850 -530.397

SM2215 -580.583 -534.546 1.267 2 0.531

1 A lower value indicates a better fit.

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

CH4 (Bag 1): 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 (SM2)

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

Intercept -3.0065 0.1380 15 -21.8 0.000000 -3.0068 0.1380 15 -21.8 0.000000

Z e 0.09137 0.009954 941 9.18 0.000000 0.08877 0.009439 941 9.40 0.000000

Z a-0.09725 0.006246 941 -15.6 0.000000 -0.09816 0.006199 941 -15.8 0.000000

Z r-0.02406 0.008404 941 -2.86 0.0043 -0.02455 0.008296 941 -2.96 0.0032

Z 5 0.10813 0.01109 941 9.75 0.000000 0.1059 0.01067 941 9.92 0.000000

Z 9 0.006944 0.006434 941 1.08 0.28 0.008573 0.006196 941 1.38 0.17

ZZee 0.03479 0.01160 941 3.00 0.0028 0.03133 0.01113 941 2.81 0.0050

ZZ55 0.06151 0.009467 941 6.50 0.000000 0.05882 0.008899 941 6.61 0.000000

ZZea 0.03944 0.009387 941 4.20 0.000029 0.03977 0.009181 941 4.33 0.000016

ZZer 0.005566 0.006365 941 0.87 0.38

ZZe5 0.03118 0.01226 941 2.54 0.011 0.02883 0.01205 941 2.39 0.017

ZZe9 0.02377 0.007526 941 3.16 0.0016 0.02655 0.006892 941 3.85 0.00013

ZZar 0.02957 0.01017 941 2.91 0.0037 0.02791 0.009975 941 2.80 0.0052

ZZa5 0.02268 0.01133 941 2.00 0.046 0.02585 0.01075 941 2.41 0.016

ZZa9 0.03032 0.006365 941 4.76 0.000002 0.03072 0.006248 941 4.92 0.000001

ZZ59 0.02094 0.008467 941 2.47 0.014 0.02280 0.007557 941 3.02 0.0026

ZZr9 -0.005353 0.007195 941 -0.744 0.46

0.2853 0.2853

0.02879 0.02883

2

veh

2



2

veh

2

