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	G.4

Final	Model	Fitting

Oxides	of	Nitrogen (NOx)	(Bag	2)

No. Observations: 893

No. Vehicles: 14 (drop Cobalt) 

No. censored measurements: 0

No. missing measurements: 0

No. measurements removed: 0

Model Type: Mixed model
2

G.4.1			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

NOx (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 1-10 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 3 1105.17 • •

2 4 1106.57 • • •

3 4 1107.66 • • •

4 4 1107.67 • • •

5 4 1107.68 • • •

6 4 1107.74 • • •

7 5 1108.17 • • • •

8 5 1109.03 • • • •

9 5 1109.06 • • • •

10 5 1109.11 • • • •

11 5 1109.20 • • • •

12 5 1110.20 • • • •

13 5 1110.20 • • • •

14 5 1110.21 • • • •

15 5 1110.21 • • • •

16 5 1110.23 • • • •

17 5 1110.25 • • • •

18 5 1110.26 • • • •

19 5 1110.29 • • • •

20 5 1110.33 • • • •

21 6 1110.68 • • • • •

22 6 1110.69 • • • • •

23 6 1110.71 • • • • •

24 6 1110.81 • • • • •

25 2 1111.53 •

26 6 1111.53 • • • • •

27 6 1111.58 • • • • •

28 6 1111.58 • • • • •

29 6 1111.62 • • • • •

30 6 1111.66 • • • • •

31 6 1111.67 • • • • •

32 6 1111.71 • • • • •

33 6 1111.73 • • • • •

34 6 1111.80 • • • • •

35 6 1112.75 • • • • •

Design Terms
4

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

Model Term Notation Model

Superset SM4

1

SM6

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

denotes “Superset minus 4,” etc.

NOx (Bag 2): 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 9 1089.365 1118.395

SM4 5 1089.700 1108.173 0.335 4 0.987

SM6

2

3 1091.971 1105.166 2.271 0.321

1 A lower value indicates a better fit.

2 Best fit with respect to the 11-term design model.
5

NOx (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 (SM6)

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

Intercept -4.5680 0.1844 14 -24.8 0.00000 -4.5692 0.1842 14 -24.8 0.000000

Z e 0.05813 0.01952 879 2.98 0.0030 0.06299 0.01444 879 4.36 0.000014

Z a 0.04469 0.01492 879 3.00 0.0028 0.04407 0.01465 879 3.01 0.0027

Z r-0.01729 0.01653 879 -1.05 0.30

Z 5-0.00245 0.02024 879 -0.12 0.90

Z 9 0.00491 0.01481 879 0.33 0.74

ZZee -0.00447 0.01525 879 -0.29 0.77

ZZ55 --- --- --- --- --- --- --- --- --- ---

ZZea 0.00478 0.01455 879 0.33 0.74

ZZer 0.01418 0.01455 879 0.97 0.33

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

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

0.4730 0.4720

0.1830 0.1836

2

veh

2



2

veh

2


6

G.4.2 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).
7

NOx (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 1-8 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 3 1105.17 • •

2 4 1106.57 • • •

3 4 1107.66 • • •

4 4 1107.67 • • •

5 4 1107.68 • • •

6 4 1107.74 • • •

7 5 1108.17 • • • •

8 5 1109.03 • • • •

9 5 1109.06 • • • •

10 5 1109.09 • • • •

11 5 1109.11 • • • •

12 5 1109.2 • • • •

13 5 1109.34 • • • •

14 5 1109.75 • • • •

15 5 1110.2 • • • •

16 5 1110.2 • • • •

17 5 1110.21 • • • •

18 5 1110.21 • • • •

19 5 1110.23 • • • •

20 5 1110.25 • • • •

21 5 1110.26 • • • •

22 5 1110.29 • • • •

23 5 1110.33 • • • •

24 6 1110.61 • • • • •

25 6 1110.66 • • • • •

26 6 1110.68 • • • • •

27 6 1110.69 • • • • •

28 6 1110.71 • • • • •

29 6 1110.75 • • • • •

30 6 1110.81 • • • • •

31 6 1111.3 • • • • •

32 6 1111.42 • • • • •

33 2 1111.53 •

34 6 1111.53 • • • • •

35 6 1111.58 • • • • •

Design Terms Extended Terms
8

NOTE: the model-fitting process for the extended model gives identical results to those from the 

design model, as presented above.