May 2003 DOT HS 809 639

Traffic Safety Facts

Research Note

Belt Use Regression Model - 2003 Update

by Jing-Shiarn Wang and Lawrence Blincoe of The Office of Planning, Evaluation, and Budget

This research note describes 

modifications to NHTSA’s BELT USE 

regression model. BELT USE is a 

second-degree polynomial model 

that was created in 1994 (Blincoe, 

1994). In this note, that model is 

referred to as the 1994 version. 

Basically, this model uses the 

national observed belt use rate to 

predict lives-saved by safety belts. 

The 19-City Surveys and the Fatality 

Analysis Reporting Systems (FARS) 

from 1983 to 1991 were fitted to 

generate the model. NHTSA has 

used the 1994 version to estimate 

the lives that would be saved due 

to increased safety belt use in the 

general public. 

In 2001, NHTSA updated the BELT 

USE model (Wang & Blincoe, 2001) 

from its 1994 version. In the 2001 

update (2001 version), 8 more 

years of historical data from 1992 

to 1999 were added to refine the 

modeling process. For continuity of 

belt use trends and to include more 

data points, the 2001 version used 

the State Observational Surveys, 

instead of the National Occupant 

Protection Use Survey (NOPUS), as 

the basis for national belt use rates. 

The 2001 version still has the format 

of a second-degree polynomial, but 

there were three major changes in 

the process:

(1) The 2001 version modeled 

the relationship between the 

observed belt use rates and the 

belt use rates among fatalities in 

potential fatal crashes (UPFCs) 

as opposed to modeling the 

relationship between the 

observed belt use rates and 

lives-saved rate.

(2) The 2001 version used the 

weighted average belt effectiveness among passenger cars 

(45 percent) and light trucks (60 

percent) to calculate UPFCs as 

opposed to a single safety belt 

effectiveness rate of 45 percent.

(3) The 2001 version abandoned 

the constraint that, in the 1994 

version, forced UPFC to be 

100 percent when observed 

use was 100 percent.

Recently, NHTSA’s National 

Center for Statistics and Analysis 

(NCSA) has re-examined all the 

methodologies and mathematical 

formula that are used to estimate 

lives saved by safety belts and 

other restraint systems. NCSA 

has decided to improve the 

methodologies by incorporating 

the impact of air bags and using 

the updated effectiveness rates. 

For consistency, this update 

(2003 version) adapts these 

improvements. The 2003 version 

also changes the baseline 

population that is used to calculate 

belt usage and related statistics. 

The “Modifications” section details 

these changes and improvements. 

Following it is the “Model Results” 

section, which describes the initially 

derived model and factors such as 

time of day and seating position 

that might influence the outcome 

of the prediction. One question 

remaining is whether the model 

properly predicts UPFC at much 

higher observed belt rates given 

that the historical belt use data 

were all lower than 75 percent. The 

“Alternatives” section examines 

5 alternative approaches to this 

issue. Finally, the “Discussion” 

section recommends the model 

that NHTSA will use to estimate the 

impact of safety belts in the future. 

That section also discusses factors 

such as time of day and seating 

position and their impacts on the 

accuracy of the prediction. 

Modifications

This section describes the 

modifications that are made in the 

2003 version. The changes include: 

(1) directly modeling the relationship 

between UPFCs of all passenger 

vehicle occupants age 5 and older 
and observed belt use, (2) adding 

2 more years (2000 and 2001) of 

FARS data and state observed 

belt use to fit the model, (3) using 

updated safety belt effectiveness 

rates, and (4) taking into account 

the air bag impact.

All Passenger Vehicle 

Occupants 5 and Older

The 2001 model describes the 

relationship between the UPFCs in 

front-outboard passenger vehicle 

occupants 5 and older and the 

observed use rates. The rationale 

was that the observed belt use 

survey was conducted for frontoutboard occupants only. By limiting 

to frontal-outboard occupants, the 

model would eliminate the seating 

bias. However, the regression model 

is used to estimate overall lives 

saved by safety belts regardless 

of the seating positions. Thus, the 

2002 version includes all passenger 

vehicle occupants 5 and older. 

This way, the model reflects the 

true relationship between UPFCs 

for all occupants and the observed 

belt use rates. Hence, the model 

would better predict the overall lives 

that would be saved at a specified 

observed safety belt use rate.

Two Additional Years of Data

The process includes two more 

years of data: 2000 and 2001 

FARS and state observed belt use 

surveys. The observed belt use 

rates were 73 and 75 percent for 

2000 and 2001, respectively. In 

total, 19-pairs (from 1983 to 2001) 

of data points were used to fit 

the regression model. These two 

new data points gave the model a 

slightly better fit at the higher belt 

use rates. Table 1 lists all the data 

used to fit the model. Note that data 

under the column titles “% UPFC” 

and “% Observed Belt Use” in 

Table 1 were used in the modeling 

process. The data under the column 

“% Use in FARS” were used to 

derive UPFCs. 

Updated Belt Effectiveness 

Rates

The 2001 model used the weighted 

effectiveness of 45 percent for 

passenger car occupants and 60 

percent for light trucks/vans. These 

effectiveness rates represented 

the seating positions and vehicles 

that were included in the state 

observational surveys. The weights 

were the number of potential fatal 

injuries in passenger cars and light 

trucks/vans. Several NHTSA studies 

(Kahane, 1996 & 2000, Morgan 

1999) examine the effectiveness of 

the restraint systems. These studies 

show that the effectiveness of safety 

belts varies depending on many 

factors such as type of restraint 

systems (2-point, 3-point, lap/

shoulder, and etc), seating position 

(front, mid, and rear), and vehicle 

types (passenger cars and light 

trucks/vans). The 2003 version uses 

these updated effectiveness rates.

Table 2 lists all the updated safety 

belt effectiveness rates used to 

generate the 2003 version. For each 

seating position, the effectiveness 

of the unknown restraint type is set 

Table 1 

Belt Use Trends 

From 1983 to 2001

% %

 Use in % Observed

Years FARS UPFC Belt Use

1983 3 6 14

1984 5 8 14

1985 9 16 21 

1986 15 24 37

1987 18 29 42

1988 20 32 45

1989 22 34 47

1990 23 36 49

1991 24 37 59

1992 26 40 62

1993 29 43 66

1994 34 49 67

1995 34 50 68

1996 36 52 66

1997 37 53 67

1998 38 54 69

1999 38 54 70

2000 39 56 73

2001 40 57 75

Table 2 

Effectiveness of Safety Belts Against Occupant Fatalities by Vehicle 

Type, Restraint Type, and Seating Position

 Front - Front - Rear- Rear -

Passenger Cars Driver Right Middle Outboard Middle Other

 2-point 32 32 na na na na

 3-point 48 37 na na na na

 Lap/Shoulder 48 37 na 44 na na

 Lapbelt 32 32 19 32 32 32

 Unknown Type 32 32 19 32 32 32

Light Trucks/ Vans

 2-point na na na na na na

 3-point 61 58 na na na na

 Lap/Shoulder na na na 73 na na

 Lapbelt na na 32 63 63 63

 Unknown Type 61 58 32 63 63 63

na – not applicable

2
and has the general format , 

where f is the fatalities and 

e is the effectiveness rate. A higher 

effectiveness results in a larger PF 

and higher benefits. In practice, 

fatalities were first partitioned 

into small subsets based on their 

restraint types, seating positions, 

and vehicle types, i.e., fatalities 

were partitioned into “driver fatalities 

with lap/shoulder belt in passenger 

cars”, “front-outboard passengers 

with lap/shoulder belt in passenger 

cars,” etc. These partitions were 

generated to correspond to the 

proper effectiveness rates of the 

restraint system.

Total PF was the sum of all these 

subsets. The formula is: 

PF = where f

i

 is the

fatalities in the i

th partition, ei

 is the 

effectiveness rate.

The 1994 and 2001 model versions 

used the belt only effectiveness for 

to be equal to the smallest number 

among those of the known types. 

As shown in Table 2, safety belts 

have a greater effectiveness against 

light truck/van occupant fatalities. 

With increasing numbers of light 

trucks/vans on the roadway, the 

use of new and refined effectiveness 

rates should improve the model’s 

predictability of belt benefits.

Air Bag Impact

The methodology and theory for the 

1994 version of BELTUSE model 

(Blincoe, 1994) was developed when 

air bags were not prevalent. Thus, the 

model did not take into account the 

safety impact of air bags, nor was this 

addressed in the 2001 version. Wang 

& Blincoe addressed the air bag effect 

on lives saved estimates by belt use 

in their 1999 research note (Wang & 

Blincoe, 1999). When air bag vehicles 

comprised a very small portion of the 

operational fleet, its impact on safety 

belt benefits would have been minimal. 

However, the number of vehicles 

equipped with air bags has steadily 

increased since 1990. The portion 

of air bag equipped vehicles in fatal 

crashes has increased to 43 percent 

in 2001 from only 1 percent in 1990. 

Since all new light vehicles now have 

air bags, the on-road fleet is gradually 

moving towards 100 percent air bag 

penetration. The impact of air bags can 

no longer be ignored.

The impact of air bags on safety 

belt benefit estimates is due to the 

interaction of air bags and safety 

belts in saving lives. This interaction 

modifies our estimates of the belted 

potential fatalities (PF) and thus 

impacts UPFCs. Its impact only 

applies to belted frontal-outboard 

occupants in air bag equipped 

vehicles. Air bags are 11 percent 

effective against belted driver 

fatalities and 14 percent against 

unbelted driver fatalities regardless 

of vehicle types (The Fifth/Sixth 

Report to Congress, 2001). These 

f

1 – e air bag effectiveness rates also 

apply to the front-right passenger 

fatalities. Table 3 lists the combined 

effectiveness of safety belts and air 

bags. The combined effectiveness 

(e) was derived based on the 

formula:

e = eb + ea – eb * ea

where eb = belt effectiveness and 

ea = air bag effectiveness.

Only drivers and front-right occupants have air bags, therefore, 

the combined effectiveness 

rates impact the front-outboard 

occupants only. The effectiveness 

rates for other occupant positions 

are the same as the safety belt 

effectiveness that are shown 

in Table 2.

To consider the air bag effect, the 

combined effectiveness rates of 

safety belts and air bags were used 

to derive (PF). PF is a function of 

fatalities and effectiveness rates 

Table 3 

The Combined Effectiveness of Safety Belts and Air Bags* Against 

Occupant Fatalities

 Front - Front - Rear- Rear -

Passenger Cars Driver Right Middle** Outboard** Middle** Other**

 2-point 39 39 na na na na

 3-point 54 44 na na na na

 Lap/Shoulder 54 44 na 44 na na

 Lapbelt 39 39 19 32 32 32

 Unknown Type 39 39 19 32 32 32

Light Trucks Van

 2-point na na na na na na

 3-point 65 63 na na na na

 Lap/Shoulder na na na 73 na na

 Lapbelt na na 32 63 63 63

 Unknown Type 65 63 32 63 63 63

** Air bag effectiveness is 11 percent against belted occupant fatalities regardless of vehicle type.

** Same as belt only effectiveness

 fi

1 – ei

∑

i

3
ei

 (with the 1994 version based on 

passenger car effectiveness and the 

2001 version based on weighted 

effectiveness for both passenger cars

and light trucks/vans). For example, 

for the subset “drivers with lap/

shoulder belts in air bag equipped 

passenger cars”, safety belts are 48 

percent effective and air bags are 

11 percent effective against belted 

fatalities. The 2001 version used 

48 percent for ei to calculate the 

historical PF (belt only consideration) 

and subsequent UPFCs for data 

fitting. The 2002 model, by contrast, 

uses the combined effectiveness 

of 54 (=0.48 + 0.11 - 0.48 * 0.11) 

percent. The older versions would 

underestimate PF about 13 percent 

for the belted driver fatalities in air 

bag equipped passenger cars.

Model Results

Based on the graphic presentation 

of the relationship of UPFC and 

observed use rate, the regression 

process first uses the second 

polynomial format u = a1 * uo + a2 * 

uo

2

 to fit the data. In this format, u 

represents UPFC, a1 and a2 are the 

parameters, and uo is the observed 

use rate. Note that, the format 

does not have a constant term 

because the regression process 

was extended to include an extra 

data point (0,0). This inclusion was 

to force the model to predict 0 

percent UPFC at the 0 observed 

belt use rate (0 restriction rule). The 

regression model is:

u = 0.56782 * uo + 0.25642 * uo

2 

(Model 1)

The model has an adjusted R2 = 

0.9950 and sum of error square 

(SSQ) = 0.0145. The R2 represents 

the squared correlation of predicted 

UPFCs and the actual UPFCs. The 

SAS procedure PROC REG (SAS, 

1994) was used as the modeling 

tool to derive the model parameters 

a1 and a2. This model predicts 82 

percent UPFC at the 100 percent 

observed belt use rate. Ideally, 

UPFC would be 100 percent when 

the observed belt use rate reached 

100 percent. However, as previously 

noted, observational belt use 

surveys are conducted for frontoutboard occupants during daytime 

hours. These use patterns do not 

reflect belt use at nighttime and 

among rear-seated occupants. 

Thus, a model that forces the 100 

percent observed rate to correspond 

to the 100 percent UPFC might 

overestimate the UPFCs and 

benefits of safety belts.

Daytime vs Nighttime

Table 4 shows the UPFC trend 

from 1983 to 2001 for all passenger 

vehicle occupant fatalities age 

5 and older. UPFC in daytime is 

consistently higher than in nighttime – about 17-19 percentage 

points higher in recent years. UPFC 

is a function of both belt use rates 

among fatalities and effectiveness 

rates. Both belt use and effectiveness affect the magnitude of 

UPFCs and thus could contribute to 

the discrepancy between day and 

night UPFC rates. Specific studies 

are not available to assess nighttime effectiveness rates, but 

a sense of the relative importance of 

Table 4 

UPFC and Belt Use

Day vs Night

 UPFC Belt Use Among Fatalities

 Night Night

 % Points /Day % Points /Day

Year Daytime Nighttime Difference Ratio Daytime Nighttime Difference Ratio

1983 7 4 3 .57 4 2 2 .50

1984 10 6 4 .60 6 3 3 .50

1985 21 12 9 .57 12 7 5 .58

1986 30 18 12 .60 19 11 8 .58

1987 36 22 14 .61 23 13 10 .57

1988 41 23 18 .56 27 14 13 .52

1989 42 25 17 .60 28 15 13 .54

1990 44 27 17 .61 30 17 13 .57

1991 45 28 17 .62 30 17 13 .57

1992 48 31 17 .65 32 19 13 .59

1993 51 34 17 .67 36 21 15 .58

1994 57 38 19 .67 41 25 16 .61

1995 58 39 19 .67 42 25 17 .60

1996 59 42 17 71 43 27 16 63

1997 61 42 19 .69 44 28 16 .64

1998 62 43 19 .69 46 28 18 .61

1999 62 43 19 .69 45 28 17 .62

2000 64 46 18 .72 47 30 17 .64

2001 65 46 19 .71 48 30 18 .63

Source: 1983-2001 FARS

4
the difference in usage compared to effectiveness can be 

gathered by examining use rates 

among fatalities.

As shown in Table 4, the difference 

between daytime and nighttime 

belt use among fatalities follows 

the same pattern as UPFCs. In 

general, belt use in the daytime was 

higher than in the nighttime – about 

17 percent points higher in recent 

years. The 17 percentage points 

difference is close to that of UPFCs. 

This indicates that effectiveness of 

the restraint systems has a smaller 

variation between daytime and 

nighttime, and thus affects UPFCs 

less. Most of the impact on day 

vs night is due to lower nighttime 

usage rates.

To further verify that the UPFC difference between daytime and nighttime

was not greatly inflated by different 

average effectiveness rates, an alternative methodology was used to 

assess the effectiveness rates. This

analysis uses a double-pair comparison analysis and fatality reduction process as stated in Kahane’s 

report (Kahane, 2000) to estimate 

the difference of effectiveness 

between daytime and nighttime. 

Only the 1996 to 2001 FARS were 

used for this purpose. Data are 

limited to cars with both a driver and 

a right-front (RF) occupant. At least 

one of them was killed and both 

were at least 14 years old. There 

were a total of 33,951 fatalities 

that occurred in the daytime and 

28,194 occurred in the nighttime. 

Table 5 tabulates the results and 

lists the driver-RF and RF-driver risk 

ratios. These risk ratios were used 

to derive effectiveness rates (or 

reduction rates) of safety belts.

Two double-pair comparisons – one 

using belted RF as the control 

group and the other using the 

unbelted RF – are used to derive 

the belt effectiveness against driver 

fatalities. Similarly, two double-pair 

comparisons using belted and 

unbelted drivers as control groups 

determine the belt effectiveness 

against RF fatalities. Readers can 

consult Kahane’s (2000) report for 

details of that process. Based on 

the two double-pair comparisons, 

the fatality reduction rates for driver 

in daytime are 66 and 70 percent. 

The corresponding rates at nighttime are 66 and 66 percent 

– identical. The fatality reduction 

rates for RF are 60 and 65 

percent for daytime and 63 and 

63 for nighttime. After applying 

weights (driver 77%, RF 23%) 

and the Universal Exaggeration 

Factor (UEF)1 (Kahane, 2000) of 

1.369, the fatality reduction rate 

for daytime is 55 percent and 52 

percent for nighttime. The belt 

effectiveness at daytime is similar 

to that at nighttime. Thus, the effect 

of belt effectiveness rates on the 

discrepancy between daytime and 

nighttime UPFCs is minimal.

Given that belt use is the dominate 

factor influencing the difference in 

day vs night UPFCs, and that belt 

use in daytime was consistently 17 

percentage points higher than that 

in the nighttime, state observational 

belt use surveys conducted during 

1 UEF is an adjustment factor to measure the overly reported belt use by survivors in FARS in response to belt use laws. UEF = (1 – e1)/(1 – e2), where e1 is the 

unbiased 3-point belt effectiveness (i.e., before the laws) and e2 is the biased 3-point belt effectiveness. 

Table 5 

Front-Outboard Occupant Risk Ratios

 Driver/RF RF/Driver Risk 

 Driver RF Total Risk Ratio Ratio 

 Daytime Fatalities Died Died Died (Driver/RF) (RF/Driver)

1. Driver Not Belted, 5,972 6,207 12,179 0.962 1.039

 RF Not Belted

2. Driver Belted, 863 2,633 3,496 0.328 3.051

 RF Not Belted

3. Driver Not Belted, 1,748 642 2,390 2.723 0.367

 RF Belted

4. Driver Belted, 7,195 8,691 15,886 0.828 1.208

 RF Belted

 Total 15,778 18,173 33,951 0.868 1.152

Nighttime Fatalities

1. Driver Not Belted, 7,573 7,409 14,982 1.022 0.978

 RF Not Belted

2. Driver Belted, 739 2,098 2,837 0.352 2.839

 RF Not Belted

3. Driver Not Belted, 1,945 714 2,659 2.724 0.367

 RF Belted

4. Driver Belted, 3,736 3,980 7,716 0.939 1.065

 RF Belted

 Total 13,993 14,201 28,194 0.985 1.015

Source: 1996 to 2001 FARS

5
the daytime overestimate the overall 

belt use rates.

Front-Seat vs Rear-Seat

Belt use rates also are different by 

seating position. Table 6 lists the 

belt use trend from 1983 to 2001 for 

front-outboard and rear-outboard 

passenger vehicle occupant 

fatalities age 5 and older. The first 

half of the table is for daytime and 

the other is for nighttime trends. As 

shown in the table, belt use rates 

are higher among front-outboard 

occupant fatalities than among rearoutboard occupants. From 1997 to 

2001, on average, belt use among 

front-outboard occupant fatalities 

during the daytime was about 19 

percentage points higher than rearoutboard occupants. The difference 

during the nighttime was about 

12 percentage points. This also 

indicates that state observational 

surveys of front-outboard 

occupants might not be appropriate 

for rear-seating occupants.

Alternatives

As described in the “Model Result” 

section, the naturally fitted model 

0.56782 * uo + 0.25642 * uo

2 only 

predicts 82 percent UPFC at 100 

percent observed belt use rate. This 

study examines two approaches to 

modify the original model to better 

address the 100 percent issue. The 

first approach is to segment the 

potential fatalities into four portions 

by their seating position (front, rear) 

and time of crash (daytime, nighttime). Their corresponding UPFCs 

are: UPFCfront, day, UPFCfront, night, 

UPFCrear, day, and UPFCrear, night. Then, 

the approach hypothesizes that at 

the 100 percent observed belt use 

rate, the UPFC among front seating 

occupants at daytime would reach 

100 percent, i.e., UPFCfront, day = 

Table 6 

Day vs Night Belt Use and UPFC

Front-Outboard vs Rear-Outboard Occupant Fatalities

Daytime

Fatalities Belt Use Potential UPFC

Among Fatalities Fatalities

Year Front Rear Front Rear Points Front Rear Front Rear Points

 Diff- Difference erence

1983 11235 593 4% 4% 0% 11661 605 8% 6% 2%

1984 12202 577 6% 7% -1% 12853 597 11% 10% 1%

1985 12550 656 13% 13% 0% 13953 701 21% 18% 3%

1986 13325 759 20% 14% 6% 15689 821 32% 21% 11%

1987 14013 781 24% 19% 5% 17056 869 38% 27% 11%

1988 14560 848 28% 22% 6% 18206 963 42% 32% 10%

1989 14618 901 30% 20% 10% 18541 1013 45% 29% 16%

1990 14141 839 31% 20% 11% 18145 950 46% 30% 16%

1991 13421 852 32% 24% 8% 17262 1002 47% 35% 12%

1992 13136 862 35% 20% 15% 17247 1015 50% 32% 18%

1993 13771 924 38% 23% 15% 18501 1131 53% 37% 16%

1994 14788 1032 44% 27% 17% 20820 1312 60% 43% 17%

1995 15177 920 44% 29% 15% 21692 1188 60% 45% 15%

1996 15542 999 45% 27% 18% 22635 1265 62% 42% 20%

1997 15729 1116 47% 29% 18% 23328 1490 63% 47% 16%

1998 15664 1069 48% 30% 18% 23797 1443 65% 48% 17%

1999 15875 1105 48% 28% 20% 24229 1509 65% 47% 18%

2000 15573 1057 50% 32% 18% 24493 1504 66% 52% 14%

2001 15508 1051 51% 31% 20% 24909 1488 67% 51% 16%

Nighttime

Fatalities Belt Use Potential UPFC

Among Fatalities Fatalities

Year Front Rear Front Rear Points Front Rear Front Rear Points

 Diff- Diff-

 erence erence

1983 13926 706 3% 2% 1% 14227 711 5% 2% 3%

1984 14037 745 4% 4% 0% 14489 766 7% 7% 0%

1985 13626 750 7% 5% 2% 14499 770 13% 7% 6%

1986 14900 911 11% 10% 1% 16415 958 20% 14% 6%

1987 15079 959 14% 11% 3% 16949 1021 23% 17% 6%

1988 15486 892 15% 11% 4% 17548 945 25% 16% 9%

1989 14905 882 16% 12% 4% 17053 946 26% 18% 8%

1990 14585 936 18% 11% 7% 16930 1012 29% 18% 11%

1991 13510 879 19% 12% 7% 15781 961 30% 19% 11%

1992 12666 830 20% 11% 9% 15020 902 33% 18% 15%

1993 12568 842 23% 14% 9% 15199 944 36% 23% 13%

1994 12168 851 26% 15% 11% 15216 961 41% 24% 17%

1995 12841 887 27% 18% 9% 16297 1030 42% 29% 13%

1996 12814 984 29% 15% 14% 16771 1137 45% 27% 18%

1997 12559 949 30% 18% 12% 16548 1129 45% 31% 14%

1998 12296 911 30% 21% 9% 16447 1114 45% 35% 10%

1999 12387 886 30% 16% 14% 16804 1057 46% 30% 16%

2000 12697 960 32% 20% 12% 17657 1162 48% 34% 14%

2001 12510 989 32% 21% 11% 17561 1253 48% 38% 10%

6
100 percent. Other UPFCs would be 

adjusted proportionally to UPFCfront, 

day. The second approach is to 

investigate other model formats 

such as higher degree polynomials. 

First Approach

This approach assumes that the 

observed belt use rate is truly a 

reflection of the belt use among 

potential front-outboard occupant 

fatalities that occurred during daytime. In order words, UPFCfront, day

= 100 percent when the observed 

belt use rate reached 100 percent. 

Two methods are used to estimate 

other UPFCs at the 100 percent 

observed belt use level. The first 

method assumes that the absolute 

difference between UPFCfront, day and 

other UPFCs is constant, i.e., for 

example, UPFCfront, day – UPFCfront, night

= constant. To better predict the 

future trend, the latest five years 

(1997-2001) averages were used. 

These averages are:

(UPFCfront, day – UPFCfront, night) = 

19 percent points

(UPFCfront, day – UPFCrear, day) = 

16 percent points, and 

(UPFCfront, day – UPFCrear, night) = 

31 percent points.

At 100 percent observed belt use, 

UPFCfront, day is assumed to be 100 

percent. Thus,

UPFCfront, night = 81 (=100-19) 

percent,

UPFCrear, day = 84 (=100-16) percent, 

and

UPFCrear, night = 69 (=100-31) percent.

Using their potential fatalities as 

weights, the overall weighted UPFC 

would be 91 (= 0.552*100 + 0.388*81 

+ 0.034*84 + 0.026* 69) percent.

Table 7 illustrates this process.

Table 7 

UPFC Adjustment Process Using 1997-2001 Average Data

Potential Fatalities Method 1 Method 2

 At 100 At 100

 Percent Percent Percent 

 (Used as Point Observed Percent Observed

Category Number Weights) Difference Belt Use Ratio Belt Use

Front, Daytime 24151 55.2% 0 100% 100.0% 100%

Front, Nighttime 17003 38.8% 19% 81% 71% 71%

Rear, Daytime 1487 3.4% 16% 84% 75% 75%

Rear, Nighttime 1143 2.6% 31% 69% 52% 52%

Total 43784 100.0%

Weighted UPFC 91% 87%

Note: Figures in Table 6 were used to derive the 1997-2001 averages.

Table 8

Summary of Regression Models and Related Statistics

Associated Statistics

 Predicted UPFC 

Models With a Square Term at 100%

(Model Number) R2 Value SSQ Observed Belt Use

0.56782 * uo + 0.25642 * uo

2 (1) 0.9950 0.01450 82%

0.43751 * uo + 0.47249 * uo

2 (2) 0.9941 0.01819 91%

0.49829 * uo + 0.37171 * uo

2 (3) 0.9950 0.01555 87%

Models With Degree Greater Than 2

0.63018 * uo + 0.24192 * uo

3 (4) 0.9951 0.01422 87%

0.64326 * uo + 0.25829 * uo

3.5 (5) 0.9952 0.01410 90%

0.65223 * uo + 0.28341 * uo

4 (6) 0.9952 0.01399 94%

Table 9

Predicted UPFCs for Observed Belt Use Rates 

75* Percent and Higher

Observed

Belt Use Model 1 Model 2 Model 3 Model 4 Model 5 Model 6 

Rate (%) (82%) (91%) (87%) (87%) (90%) (94%)

75 57 59 58 57 58 58

80 62 65 64 63 63 64

85 67 71 69 68 69 70

90 72 78 75 74 76 77

95 77 84 81 81 83 84

100 82 91 87 87 90 94

*2001 national level

7
Table 10 

Summary of Regression Models and Corresponding Predicted UPFC Results

Models with Square Term

Model 1 (100% ~ 82%) Model 2 (100% ~ 91%) Model 3 (100% ~ 87%)

0.56782*uo + 0.25642*uo

2 0.43751*uo + 0.47249*uo2 0.49829*uo + 0.37171*uo2

Year UPFC Predicted Error* Predicted Error Predicted Error

 UPFC UPFC UPFC

1983 6 8 2 7 1 8 2

1984 8 8 0 7 -1 8 0

1985 16 13 -3 11 -5 12 -4

1986 24 25 1 23 -1 24 0

1987 29 28 -1 27 -2 27 -2

1988 32 31 -1 29 -3 30 -2

1989 34 32 -2 31 -3 32 -2

1990 36 34 -2 33 -3 33 -3

1991 37 42 5 42 5 42 5

1992 40 45 5 45 5 45 5

1993 43 49 6 49 6 49 6

1994 49 50 1 51 2 50 1

1995 0 50 0 52 2 51 1

1996 52 49 -3 49 -3 49 -3

1997 53 50 -3 51 -2 50 -3

1998 54 51 -3 53 -1 52 -2

1999 54 52 -2 54 0 53 -1

2000 56 55 -1 57 1 56 0

2001 57 57 0 59 2 58 1

Models with Degree Greater Than 2

Model 4 (100% ~ 87%) Model 5 (100% ~ 90%) Model 6 (100% ~ 94%)

0.63018*uo + 0.24192*uo

3 0.64326*uo + 0.25829*uo3.5 0.65223*uo + 0.28341*uo4

Year UPFC Predicted Error* Predicted Error Predicted Error

 UPFC UPFC UPFC

1983 6 9 3 9 3 9 3

1984 8 9 1 9 1 9 1

1985 16 13 -3 14 -2 14 -2

1986 24 25 1 25 1 25 1

1987 29 28 -1 28 -1 28 -1

1988 32 31 -1 31 -1 31 -1

1989 34 32 -2 32 -2 32 -2

1990 36 34 -2 34 -2 34 -2

1991 37 42 5 42 5 42 5

1992 40 45 5 45 5 45 5

1993 43 49 6 48 5 48 5

1994 49 49 0 49 0 49 0

1995 50 50 0 50 0 50 0

1996 52 49 -3 48 -4 48 -4

1997 53 49 -4 49 -4 49 -4

1998 54 51 -3 51 -3 51 -3

1999 54 52 -2 52 -2 52 -2

2000 56 55 -1 56 0 56 0

2001 57 57 0 58 1 58 1

* Difference between UPFC and predicted UPFC

8
We re-ran the regression process 

and forced the 100 percent 

observed belt use rate to predict a 

91 percent UPFC, the model (Model 

2 - 91% model) became

0.43751 * uo + 0.47249 * uo

2

 with R2

= 0.9941 and SSQ = 0.01819.

The second method assumes that 

the ratios of other UPFCs and 

UPFCfront, day are constants, i.e., for 

example,UPFCfront, night / UPFCfront, day

= constant. As shown in Table 7, 

UPFCfront, night / UPFCfront, day = 71 

percent, UPFCrear, day / UPFCfront, day

= 75 percent, UPFCrear, night / 

UPFCfront, day = 52 percent. At the 

100 percent observed belt use rate, 

the overall weighted UPFC is 87 

(= 0.552*100 + 0.388*71 + 

0.034*75 + 0.026* 52) percent.

We re-ran the regression process 

and forced the 100 percent observed belt use rate to predict a 87 

percent UPFC, the model (Model 3 

- 87% model) became 0.49829 * uo 

+ 0.37171 * uo

2

 with R2 = 0.9950 

and SSQ = 0.01555.

Second Approach

The quadratic format might be too 

conservative to predict the future 

UPFCs. The research note examines 

three different models with a higher 

degree of polynomials: (1) a1 * uo + a2

* uo

3

, (2) a1 * uo + a2 * uo

3.5, and (3) a1

* uo + a2 * uo

4

. With the 0 restriction 

rule, the three models are:

(1) 0.63018 * uo + 0.24192 * uo

3

, with 

R2 = 0.9951 and SSQ = 0.01422 

(Model 4),

(2) 0.64326 * uo + 0.25829 * uo

3.5

with R2 = 0.9952 and SSQ = 

0.01410 (Model 5), and

(3) 0.65223 * uo + 0.28341 * uo

4

 with 

R2 = 0.9952 and SSQ = 0.01399 

(Model 6).

Note that these higher power 

polynomials were also fitted with a 

square term, i.e, polynomials with 

format as a1 * uo + a2 * uo

2 + a3 * uoi

, 

where i = 3, 3.5, and 4. These 

models have a large SSQ and a much 

smaller R2, thus the results were not 

as good as these models without 

the square term.

Table 8 summarizes the six models 

introduced here – the original hyperbolic model and the alternatives. 

Figure 1 depicts these models. As 

shown in the Figure 1, the variations 

of predicted values from these 

models are very small until the 

observed belt use rate reaches 85 

percent. Table 9 lists the predicted 

UPFCs for observed belt use rates 

75 percent and higher. Table 10 

compares the actual UPFCs and 

predicted UPFCs from these models.

Figure 1

Predicted UPFCs

Discussion

All 6 models introduced in this 

analysis provide a good fit to the 

historical belt use data. In each case 

R2 values exceed .99 and SSQs 

are less than .02, with differences 

between models that are trivial. 

However, selection of a preferred 

model must be based on judgments 

regarding their reliability to predict 

UPFC values at belt use levels that 

have not yet been achieved. From 

Figure 1, it is apparent that there 

is very little difference in predicted 

UPFCs among these models until 

observed use reaches about 85 

percent – roughly 10 points beyond 

current use levels. From that point 

they begin to disperse until there is 

as much as a 12 percentage point 

difference between their predictions 

at 100 percent observed usage.

100%

Observed Belt Use Rate

100%

90%

80%

70%

60%

50%

40%

60% 70% 80% 90%

Predicted UPFC

Model 1 (82%)

Model 3 (87%)

Model 5 (90%)*

Model 2 (91%)

Model 4 (87%)*

Model 6 (94%)*

*With degree greater than 2

9
There are really 2 basic issues that 

are of interest in selecting the best 

model. The first is, how accurate 

will it be in predicting UPFC over 

the span of predicted observed 

usage rates that are likely to 

occur in the next few years. This 

distinction is of interest because, 

over time, additional years data will 

become available and there will be 

opportunities to revise the algorithm 

in light of these added data. 

Therefore, its most immediate use is 

of principle interest. In this regard, it 

can be noted that while the national 

observed use rate in 2001 is 75 

percent, some states were already 

achieving rates of 85-90 percent. 

Indeed, there is a large discrepancy 

between states that have passed 

primary belt laws and those that 

haven’t, with individual state usage 

rates ranging from 52 percent to 

91 percent. The majority of states 

fall within the 70-85 percent range, 

but the states that still haven’t 

passed primary laws, and for which 

legislative action would be most 

meaningful, average roughly 10 

points below this. Large increases 

of 10 points or more are possible 

with the passage of primary belt 

use laws, but raising belt use 

becomes more difficult at higher 

levels because the remaining pool of 

belt non-users tends to include the 

least risk-averse parts of the driving 

population. As such, they are the 

least likely to respond to campaigns 

that appeal to their own safety, or to 

campaigns that threaten them with 

legal sanctions. This means that the 

expected increases for states with 

current high rates will be relatively 

small, while for states that have low 

rates, the potential is fairly large. The 

range of observed use from about 

65 to 90 percent is therefore likely to 

be the focus of most programs over 

the next few years. 

The last 6 years of data appear to 

be the most relevant to this range 

of usage rates. During this time, 

the SSQs for Models 2 and 3 are 

about half that of the 3 higher order 

models, and roughly 25-30 percent 

less than Model 1, with Model 2 

being about 20 percent lower than 

Model 3. Although the differences 

for all models are small, for the most 

recent time period, there appears to 

be an advantage to the predictions 

of Model 2.

The second issue of concern is 

the model’s ultimate prediction 

of UPFC when observed usage 

is 100 percent. This is of interest 

because it is a measure that is 

often cited in discussions of the 

lifesaving potential of safety belts, 

but it is also relevant because it 

can be an indicator of the accuracy 

of predictions at lower levels. 

Because use is currently only about 

75 percent, the behavior of the 

curve at higher levels is necessarily 

somewhat speculative. The relevant 

issue in addressing expected UPFC 

at 100% is the behavior of those 

who are most resistant to belt use. 

As mentioned above, at higher 

observed usage levels the remaining 

pool of belt non-users tends to 

include the least risk-averse parts 

of the driving population. As such, 

they are the least likely to respond 

to campaigns that appeal to their 

own safety, or to campaigns that 

threaten them with legal sanctions. 

The question becomes, at what 

point do these most reticent users 

begin to consistently buckle up, 

and how many of them will still be 

unbuckled even after observed 

use reaches 100 percent. The 

answer determines the degree 

of curvilinearity that would be 

expected, especially at the upper 

range of the curve. 

Model 1 is the least curvilinear 

of these models – it projects a 

continuation of current trends 

without significant change as 

observed usage approaches 100 

percent. It essentially seems to 

predict that there will be no relative 

change in the behavior of those who 

are the least risk averse. Although 

this is theoretically a possible 

outcome, it seems unlikely in light of 

surveys that indicate that most nonuse reflects lapses by occasional 

users rather than hard-core refusal 

to use belts. In NHTSA’s 2000 

Motor Vehicle Occupant Safety 

Survey, only 4 percent of drivers 

indicated that they either never or 

rarely wore belts. About 77 percent 

claimed to wear belts all the time 

while the remaining 19 percent 

represent drivers who wore belts 

either most of the time or some 

of the time. The most common 

reasons given for occasionally 

not wearing belts were that they 

felt it was unnecessary for short 

distance trips, or that they forgot. 

Both of these factors should be 

well represented in any statistically 

reliable daytime observation survey, 

and if we obtain a statistically 

reliable observed use rate of 100 

percent, then the behavior of 

these occasional users will have to 

shift to more closely match those 

who always wear their belts. The 

behavior of the 4 percent who rarely 

or never use belts would also have 

to change, although since they 

represent a relatively small part 

of the driving population, there is 

some chance that they might be 

missed by the survey. In any case, 

the implication is that as observed 

daytime usage approaches 100 

percent, UPFC for daytime crashes 

should approach the observed 

usage rate. Model 1, which reflects 

the sampling error at current and 

10
based on the ratio of increased 

belt use at nighttime vs. that 

experienced in daytime. While 3 

different scenarios could produce an 

increasing Day/Night ratio, only one 

– which assumes equal percentage 

point growth for both day and night, 

was consistent with the patterns 

found in the historical data in Table 

4. This would seem to imply that 

most new belt wearers during this 

period either became full time users, 

or that whatever level of occasional 

use they brought into the mix was 

offset by improved belt use habits of 

others. In either case, the result is 

a consistent percentage point gap 

between daytime and nighttime use 

rates as UPFC increases. For this 

reason, as well as for the accuracy 

advantage it gives at current use 

levels, Model 2 (u = 0.43751*uo

+ 0.47249*uo

2 ) is selected as the 

preferred model. 

Note that, based on this model, the 

observed belt use u0 can be derived 

from known UPFC (u) by rearranging 

the quadratic equation as follows:

0.47249 * uo

2

 + 0.43751 * uo – u = 0.

The observed belt use rate (as a 

function of UPFC) would be the 

solution of the above quadratic 

equation. The solution is:

uo =

i.e.,

uo =

previous use rates in all seating 

positions at all hours, does not 

predict this condition. 

Models 4, 5, and 6 involve the 

same basic interactions as model 

1. However, they were designed 

to force higher estimates of UPFC 

as observed use approaches 100 

percent and this is reflected in their 

results. Unfortunately, as noted 

above their added curvilinearity 

also causes more variation in UPFC 

predictions at current use levels.

By contrast, Models 2 and 3 are 

specifically designed to reflect a 

daytime UPFC that is equal to 

daytime observed usage, while 

maintaining the relative gap in 

nighttime usage that has existed 

for over a decade. Model 2 does 

this by maintaining the percentage 

point gap between daytime front 

outboard and other positions, while 

Model 3 does it by maintaining the 

ratios between them. From Table 

4, the gap between day and night 

UPFC has been a fairly stable 

17-19 points since 1988. During 

this time (through 2001), daytime 

UPFC has increased by 24 points 

and nighttime UPFC has increased 

by 23 points. However, the 

Nighttime/Daytime ratio has steadily 

increased. A number of day and 

night belt use increase scenarios 

were analyzed to determine which 

would be consistent with this 

experience. These scenarios were 

– 0.43751 + √(0.43751)2 – 4 * 0.47249 *(–u)

2 * 0.47249

– 0.43751 + √0.191415 + 1.88996 *u

0.94498

It is unknown whether this daytime/

nighttime pattern will continue. An 

alternative scenario that could occur 

would be one in which daytime 

usage increases at a faster rate 

than nighttime usage. This might 

occur if the most risk adverse 

drivers wear belts during daylight 

hours to avoid penalties for non-use 

under primary law enforcement, 

but frequently ride unbelted at night 

when such enforcement is difficult. 

Another scenario that might create 

this result would be if those most 

reluctant to buckle up also have a 

higher tendency to both drink and 

drive at night and don’t buckle up 

when they are inebriated. Both of 

these scenarios may already be 

occurring to some extent, but they 

do not appear to be a dominant 

factor in the relative increase in day 

and night belt use. If they were, 

the absolute difference would be 

increasing while the night/day ratio 

would be decreasing. The patterns 

of observed belt use and UPFC will 

be monitored as usage increases 

to determine whether such a 

shift is occurring, and whether a 

further revision in the algorithm is 

appropriate.

11
These reports and 

additional information are 

available from your State 

Highway Safety Office, 

the NHTSA Regional Office 

serving your State, or from 

NHTSA Headquarters, Office 

of Planning, Evaluation and 

Budget, ATTN: NPO-320, 

400 Seventh Street, S.W., 

Washington, DC 20590; 

or NHTSA’s web site at

www.nhtsa.dot.gov

References

Blincoe, L., Estimating the Benefits 

from Increased Safety Belt Use, 

NHTSA Technical Report, DOT HS 

808 133, June 1994.

Block, A., 2000 Motor Vehicle 

Occupant Safety Survey, Volume 

Two, Seat Belt Report, November 

2001, DOT HS 809 389

Kahane, C., Fatality Reduction 

by Safety Belts for Front-Seat 

Occupants of Cars and Light 

Trucks, Updated and Expanded 

Estimates Based on 1986-99 FARS, 

NHTSA Technical Report, DOT HS 

809 199, December 2000.

Kahane, C., Fatality Reduction 

by Air Bags, Analysis of Accident 

Data Through Early 1996, NHTSA 

Technical Report, DOT HS 809 470, 

August 1996.

Morgan, C., Effectiveness of 

Lap/Shoulder Belts in the Back 

Outboard Seating Positions, NHTSA 

Technical Report, DOT HS 808 845, 

June 1999.

Fifth/Sixth Report to Congress, 

Effectiveness of Occupant 

Protection Systems and Their Use, 

DOT HS 809 442, November 2001.

SAS/Stat User’s Guide, Volume 2, 

GLM-VARCOMP, Version 6, Fourth 

Edition, SAS Institute, June 1994.

Wang, J. and Blincoe, L., The 

Estimated Air Bag Effect on 

Lives Saved by Belt Use, NHTSA 

Research Note, July 1999.

Wang, J., and Blincoe, L., BELTUSE 

Regression Model Update, NHTSA 

Research Note, June 2001.