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		L.3

Model	Fitting	Information	for

Formaldehyde (Bag	1)

This appendix summarizes model fitting for Formaldehyde (Bag 1). Model-fitting techniques and 

approaches are described in Section 8.7. Features of the data and modeling for this compound are listed

below.

Fuel-parameter matrix: FULL

No. measurements: 956

No. vehicles: 15

No. censored values: 0

Modeling approach: MIXED MODEL
2

Models fit for Formaldehyde (Bag 1).

Model Term Notation Model

Full Model FM4 FM5 FM6 FM7 Linear Effects

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 ● ● ×

T90 × T90 ZZ99 ● ● ● ● ×

T50 × T90 ZZ59 ● ● ● ● ● ×

RVP × T90 ZZr9 ● ×

Fitting history for Bag-1 Formaldehyde – with “FM7” selected as the reduced model.

Fit Parameters Test with respect to

Full

Test with respect to

Previous Model

Model p -2lnL BIC

1 Dev.1

d Pr>χ

2 Dev. d Pr>χ2

Full 17 -102.866 -48.705

FM4 13 -101.651 -58.322 1.215 4 0.876

FM5 12 -100.336 -59.715 2.530 5 0.772 1.315 1 0.252

FM6 11 -99.040 -61.127 3.826 6 0.700 1.296 1 0.255

FM7210 -95.848 -60.643 7.018 7 0.427 3.192 1 0.074

Linear Effects 5 -13.166 8.498 89.700 12 0.000000 82.682 5 0.000000

1 A lower value indicates a better fit.

2

Selected as “best fit” with respect to the 17-term full model.

Note that based on the history, FM6 is actually the “best fit,” 

however, we have retained FM7 due to the decision to exclude the 

T90×T90 term from final model fitting. See Section 7.2.1.

1 The deviation is the difference in

the -2loglik statistics for the 

nested and reference models,

respectively, per Error! 

Reference source not 

found..
3

Coefficients and Type-III Tests of Effect for the Full and Best-Fit Models – Formaldehyde (Bag 1).

Effect Full Model Best-Fit Model (FM7)

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

Intercept -5.9771 0.1498 15 -39.9 0.000000 -5.9771 0.1498 15 -39.9 0.000000

Ze 0.2279 0.01234 898 18.5 0.000000 0.2299 0.009640 898 23.8 0.000000

Za 0.03528 0.008841 898 3.99 0.000071 0.02822 0.007979 898 3.54 0.00043

Zr -0.05202 0.01234 898 -4.21 0.000028 -0.04718 0.008457 898 -5.58 0.000000

Z5 0.1577 0.01738 898 9.07 0.000000 0.1672 0.01001 898 16.7 0.000000

Z9 0.1357 0.01064 898 12.7 0.000000 0.1302 0.007360 898 17.7 0.000000

ZZee -0.01498 0.01475 898 -1.02 0.31

ZZ55 0.05026 0.01251 898 4.02 0.000064 0.05262 0.008341 898 6.31 0.000000

ZZea 0.02017 0.01241 898 1.63 0.10 0.01651 0.007340 898 2.25 0.025

ZZer 0.004100 0.008366 898 0.490 0.62

ZZe5 -0.03686 0.01594 898 -2.31 0.021 -0.01627 0.008177 898 -1.99 0.047

ZZe9 0.02181 0.01023 898 2.13 0.033 0.02004 0.008838 898 2.27 0.024

ZZar 0.007384 0.01535 898 0.481 0.63

ZZa5 -0.006739 0.01645 898 -0.41 0.68

ZZa9 -0.01036 0.008437 898 -1.23 0.22

ZZ99 0.02104 0.01435 898 1.47 0.14

ZZ59 0.03974 0.01080 898 3.68 0.00025 0.03489 0.009322 898 3.74 0.00019

ZZr9 -0.003140 0.009498 898 -0.331 0.74

2  veh 0.3360 0.3358

2 

0.1395 0.1406