X.  PROBABILISTIC UNCERTAINTY ANALYSIS

This chapter identifies and quantifies the major uncertainties in the preliminary regulatory impact analysis. Cost-effectiveness and net benefits are two principal measurements in the economic assessment. Throughout the course of both the cost-effectiveness and net benefit analyses, many assumptions were made; diverse data sources were used; and different statistical processes were applied. The variability of these assumptions, data sources, and statistical processes potentially would impact the estimated regulatory outcomes. These assumptions, data sources, and derived statistics all can be considered as uncertainty factors for the regulatory analysis. Some of these uncertainty factors contributed less to the overall variations of the outcomes, and thus are less significant. Some uncertainty factors depend on others, and thus can be combined with others. With the vast number of uncertainties imbedded in this regulatory analysis, the uncertainty analysis identifies only the major independent uncertainty factors having appreciable variability and quantifies them by their probability distributions. These newly defined values are then randomly selected and fed back to the cost-effectiveness and net benefit analysis process using the Monte Carlo statistical simulation technique [44]. The simulation technique induces the probabilistic outcomes accompanied with degrees of probability or plausibility. This facilitates a more informed decision-making process.

The analysis starts by establishing mathematical models that imitate the actual processes in deriving cost-effectiveness and net benefits as described in the previous chapters. Each variable (e.g., cost of technology) in the mathematical model represents an uncertainty factor that would potentially alter the modeling outcomes if its value were changed. The variations of these variables are described by an appropriate probability distribution function based on available data. If data are not sufficient or not available, professional judgments are used to estimate the variability of these uncertainty factors.

After defining and quantifying the major uncertainty factors, the next step is to simulate the model to obtain probabilistic results rather than single-value estimates. The simulation process is run repeatedly. Each complete run is a trial. For each trial, the simulation first randomly selects a value for each of the uncertainty factors based on their probability distributions. The selected values are then fit into the models to forecast results. The simulation repeats the trials until certain pre-defined criteria are met and a probability distribution of results is generated.

A commercially available software package, Crystal Ball from Decisoneering, Inc., was used for this purpose - building models, running simulations, storing results, and generating statistical results. Crystal Ball is a spreadsheet-based risk analysis software which uses the Monte Carlo simulation technique to forecast results. In addition to the simulation results, the software also estimates the degree of certainty (or confidence, credibility). The degree of certainty provides the decision-maker an additional piece of important information to evaluate the forecast results.

Simulation Models and Uncertainty Factors

The spreadsheet based mathematical models were built to imitate the cost-effectiveness and net benefits process as developed in previous chapters. The cost-effectiveness measures the cost per fatality equivalent avoided. In other words, at a given discount rate, the cost-effectiveness is the ratio of the total costs of the rule and the total fatal equivalents avoided (or equivalent lives saved) at that discount level. The net benefits measures the cost difference between the total dollar value that would be saved from reducing fatalities and injuries and the total costs of the rule.

Both the cost-effectiveness and net benefit models comprise two principal components: total benefits and costs. In the cost-effectiveness model, benefits are represented by fatal equivalents. In the net benefit model, benefits are represented in dollars, which is the product of cost per life saved and fatal equivalents. Since benefits (fatalities and injuries reduced) were already expressed as fatal equivalents in the cost-effectiveness model, the net benefit model is just one step removed from the cost-effectiveness model. This analysis first describes the mathematical models for deriving fatal equivalents and quantifies their uncertainty factors. Then, in a parallel section, the analysis discusses the total cost models and quantifies their uncertainty factors.  Finally, the analysis presents and summarizes the simulated results.

Benefit Models

As described earlier, fatal equivalents (FE) are the basic benefit measurement for both cost-effectiveness and net benefit models. The estimated FE are comprised of four mutually exclusive portions: skidding/loss of control (FE1), flat tires and blowouts (FE2), stopping-distance, preventable (FE3), and stopping-distance, non-preventable (FE4). In a mathematical format, These FEs were derived using different methodologies and assumptions. As expected, these FEs have somewhat different mathematical formats and uncertainty factors. But, some of the uncertainty factors (e.g., safety target population) are shared by these FEs. Each of these common factors is consistantly represented by a mathematical symbol throughout this chapter. Also, whenever applicable the following indexes are used universally for these four FE models:

• i  represents MAIS injury severity. The severity level increases with the value of i, i.e., 1 represent MAIS 1 minor injuries and 6 represents fatalities.
• j  represents vehicle type with 1 = passenger cars (PC) and 2 = light trucks and vans (LTV).
• k  represents roadway condition with 1 = wet and 2 = dry.
• l  represents traveling speeds with 1 = 0 - 35 mph, 2 = 36 - 50 mph, and 3 = 51+ mph.

Skidding/Loss of Control

The generalized fatal equivalent model (FE1) for loss of control is:

 Where Pi = Target population with i = 1: MAIS 1; 2: MAIS 2; : and 6: fatality ei = the effectiveness of TPMS in preventing skidding/loss of control ri = injury-to-fatality equivalence ratios d = cumulative lifetime discount factor (at 3 percent or 7 percent) a1 = adjustment factor for existing TPMS system a2 = adjustment factor for response rate to TPMS warning light

In this model, Pi*ei represents the benefit from reducing severity i level injuries. The number Pi * ei * ri represents the fatal equivalents that were contributed from severity i injuries. For example, P1*e1 represents the total MAIS 1 injuries that will be reduced. Multiplying this number by its injury/fatality ratio r1 derives the contribution of MAIS 1 injuries to the total FEs. The notation represents the total initial estimated FEs. This initial estimated total FEs then were modified by multiplying the adjustment factors, d, a1, and a2 to derive the final FEs. The modification reflects the discounting level, the portion of the fleet tested that already is equipped with a TPMS, and the assumed response rate to TPMS. Each of these adjustment factors will be examined in detail in the following discussions.

Based on the FE1 model, there are six major uncertainty factors that would impact the estimated benefit outcome: target population (Pi), effectiveness of TPMS (ei), injury-to-fatality equivalence ratios (ri), cumulative lifetime discount factor (d), adjustment factor for the existing TPMS systems (a1), and adjustment factor for driver response rate to TPMS warning (a2).

The first uncertainty factor Pi, target population, is obviously important to benefit estimates because it defines the population of risk without the rule. The major uncertainties in this factor arise from, but are not limited to, the percentage of all crashes that were caused by skidding (0.77 percent was assumed in the previous chapter), demographic projections, driver/occupant behavioral changes (e.g., shifts in safety belt use), increased roadway travel, new Government safety regulations, and survey errors in NHTSA's data sampling system NASS-CDS. Based on professional judgment and the available data (Tri-Level Study), the analysis assumes that the percent of crashes caused by skidding/loss of control is uniformly distributed from 0.52 to 1.02 percent with a mean of 0.77 percent.

The impact of demographic and driver/occupant behavior changes, roadway travel, and new automobile safety regulations are reflected in the crash database. Thus, the analysis examined the historic FARS and CDS to determine whether variations resulting from these uncertainty sources would warrant further adjustment to the future target population.

Based on 1998 to 2002 FARS, there is no definite trend for this period of time. The changes among years were small with a variation within +2.0 percent. Thus, the analysis will not further adjust the FARS-derived fatalities and treats fatalities as a constant.

For injuries, the analysis considers the CDS associated survey errors and treats injuries as normally distributed. About 68 percent of the estimated target injuries are within one standard error (SE) of the mean survey injury population. Thus, the mean injury population and corresponding standard errors (as the proxy for standard deviation) were used for establishing the normal distribution for the size of the non-fatal injury target population. The standard errors were derived using the formula [45]:

, x = estimated target injuries.

Combining the variations from the percent of skidding/loss-of-control and survey errors, the final fatal target population is close to a uniform distribution from 169 to 327. The final target MAIS injuries are close to normal distributions with slightly positive skewing. Figure X-1 depicts these distributions. Note that two parameters, the maximum and the minimum values, are required to establish a uniform distribution. Mean and standard deviation (SD) are required for a normal distribution.

The second uncertainty factor in the FE1 model is ei  the effectiveness of TPMS against loss of control. The effectiveness measures to the extent to which involved vehicles would brake normally without skidding if the tire pressures had been corrected. Data were not available at this time to assess its variability. However, in order to estimate its impact on the benefit outcomes, the analysis assumes ei is uniformly distributed between 10 to 30 percent and maintains its mean at 20 percent for every injury severity i.

Figure X-1
Probability Distributions For
Skidding/Loss of Control Target Population
Injury Severity   Probability Distribution
MAIS 1 Mean: 19,679
SD: 4,497
MAIS 2 Mean: 2,193
SD: 557
MAIS 3 Mean: 933
SD: 284
MAIS 4 Mean: 90
SD: 25
MAIS 5 Mean: 58
SD: 16
Fatality
(Uniform Distribution)
Minimum: 169
Maximum: 327

The third uncertainty factor ri, injury-to-fatality equivalence ratios, affects the total fatal equivalent estimates. These ratios reflect the relative economic impact of injury compared to fatality based on their estimated comprehensive unit costs. They were derived based on the most current 2002 crash cost assessment [46]. The crash cost assessment itself is a complex analysis with an associated degree of uncertainty. At this time, these uncertainties are unknowns. Thus, the variations of these ratios are unknown and this analysis treats these ratios as constants. Table X-1 lists these ratios. Benefits in this rule are a mixture of both crash avoidance and crashworthiness benefits. Crashworthiness measures mitigate injury but do not prevent crashes. Crash avoidance measures reduce property damage and travel delay as well as injury related factors. Therefore, different ratios would be applicable to each type of benefit. To provide a conservative basis, this analysis used the more conservative crashworthiness ratios.

Table X-1
Injury-to-Fatality Equivalence Ratios (ri)*
MAIS 1
(r1)
MAIS 2
(r2)
MAIS 3
(r3)
MAIS 4
(r4)
MAIS 5
(r5)
Fatality
(r6)
0.0031 0.0458 0.0916 0.2153 0.7124 1.000
* same for each discount level

The fourth uncertainty factor d, cumulative lifetime discount factor, is treated as a constant. At the 3 percent discount rate, d = 0.8233. At the 7 percent discount rate, d = 0.6600.

The fifth uncertainty factor a1, adjustment factor for existing TPMS, is treated as a constant. Currently, about 1 percent of passenger vehicles are equipped with a TPMS meeting the proposal. Under this analysis, the initial FE was estimated assuming that no passenger vehicle was equipped with a passing TPMS. In this sense, the initial FE overestimated the actual benefits by 1 percent and had to be adjusted down to reflect this overestimation. Thus, a1 = 0.99 (= 1 - 0.01).

The last uncertainty factor a2 represents the response rate to TPMS warnings. In 2002, the agency conducted a TPMS survey. The survey included a total of 106 vehicles with a direct TPMS. Of these, 105 are applicable for analysis. Based on these 105 cases, 95 percent of drivers took the following actions: putting air into their tires, changing tires, or taking their vehicles into service stations. This factor is subject to survey errors and selection bias. The survey was terminated before its completion due to the change in the initial TPMS rule. Thus, no viable statistical survey errors can be assessed. The selection process would likely generate a survey sample that includes proportionally more high-end vehicles than make up the existing TPMS market. As a result, the reported response rate might be biased upward. However, at this time, the agency has no data to estimate the magnitude of the upward bias. Nevertheless, to consider the impact of response rate on the overall outcome and address this upward bias possibility, the analysis assumes that the response rate is normally distributed with a more conservative mean of 90 percent and a standard deviation of 1.7 percent. This indicates that the response rate is normally distributed between 85 and 95 percent with a mean of 90 percent. Figure X-2 depicts the normal distribution.

Figure X-2 Probability Distribution for
Response Rate To TPMS Warnings

Flat Tires and Blowouts

The generalized fatal equivalent model (FE2) for flat tires/blowouts is:

 Where Pi = Target population with I = 1: MAIS 1; 2: MAIS 2; : and 6: fatality ei = the effectiveness of TPMS preventing flat tires and blowouts ri = injury-to-fatality equivalence ratios d = cumulative lifetime discount factor (at 3 percent or 7 percent) a1 = adjustment factor for existing TPMS system a2 = adjustment factor for response rate to TPMS warning light

The generic form of FE2 model is identical to that of FE1 (for skidding/loss of control). So, FE2 also contains the same six major uncertainty factors as those of FE1.  Of these uncertainty factors, only the values of the target population (Pi) and effectiveness of the TPMSs (ei) against the corresponding safety population varied. The values of the remaining four major uncertainty factors, ri, d, a1, and a2 do not change. These four factors are not discussed further.

The initial target fatal population is treated as a constant and the non-fatal target populations are treated as normally distributed using the survey errors as the proxy for standard deviation. The rationales of using these types of probability distributions were described in the skidding/loss of control section, and thus are not repeated here.

In addition to the survey errors, the analysis also considers two additional sources of variations for the target population. One is the assumed percentage of the flat tires/blowouts that were caused by underinflated tires. The other source is the percentage of flat tires/blowouts that would be corrected by new tire standards in FMVSS No. 139. Both variations would impact the spread of the population distribution. However, due to insufficient data, the analysis is unable to derive its variability. Instead, the analysis assumes that the percentage of flat tires/blowouts is uniformly distributed from 10 to 30 percent with 20 percent as the mean. Similarly, the analysis assumes that percent would be corrected by the new tire standards in FMVSS No. 139 is uniformly distributed from 40 to 60 percent with 50 percent as the mean. Figure X-3 depicts the final probability distributions for the target population, which take into account the three types of variations discussed above. As shown in Figure X-3, the non-fatal target populations are close to normal distributions but slightly positive skewed. The fatal target population distribution is a combination of two triangular distributions on both tails and a uniform distribution in between.

Figure X-3
Probability Distributions for
Flat Tire/Blowout Target Population
Injury Severity   Probability Distribution
MAIS 1 Mean: 810
SD: 300
MAIS 2 Mean: 146
SD: 54
MAIS 3 Mean: 36
SD: 34
MAIS 4 Mean: 15
SD: 6
MAIS 5 Mean: 5
SD: 2
Fatality
(Uniform Distribution)
Mean: 41
Minimum: 16
Maximum: 74

The factor ei is treated as a constant. As described in Chapter IV, the target flat tires and blowouts were narrowly defined in such a way that they would be completely prevented if the involved vehicles had the correct tire pressures. Thus, ei = 1 for every i.

Stopping-Distance, Preventable Crashes

Maintaining proper tire inflation pressures can reduce stopping distance and thus prevent crashes or mitigate the severity of non-preventable crashes. Preventable crashes represent a portion of stopping distance related crashes that would be prevented if the involved vehicles had maintained the correct tire pressures to shorten its stopping distance. The benefits for this group were not only segregated by injury severity, but also by vehicle type (passenger cars, light trucks/vans) and roadway condition (wet, dry). In addition, a variety of adjustments were applied to represent factors relevant to the estimation of safety impacts. The following benefit model (FE3) reflects the process for this subset of the target population:

 Where Pi,j,k = target population with i = 1 to 6: MAIS 1 to fatality j = 1: passenger cars, 2: light trucks/vans; and k = 1: wet roadway, 2: dry roadway wi,k = adjustment factor for skidding on roadway with i = 1 to 6: MAIS 1 to fatality; and k = 1: wet, 2: dry nj = adjustment factor for average psi experience in the fleet prior to the TPMS been triggered by vehicle type with j = 1: passenger cars; 2: light trucks/vans vj = adjustment factor for no under-inflation and tire level above warning threshold by vehicle type  with j = 1: passenger cars; 2: light trucks/vans ri = injury-to-fatality equivalence ratios d = cumulative lifetime discount factor (at 3 percent or 7 percent) a1 = adjustment factor for existing TPMS systems a2 = adjustment factor for response rate to TPMS warning light a3 = adjustment factor for overlapping in target population

Based on the FE3 model, there are 9 major uncertainty factors that would impact the benefit estimate for stopping-distance, preventable crashes: target population (Pi,j,k), adjustment factor for skidding state on roadway type (wi,k), adjustment factor for average psi in a vehicle fleet before triggering a TPMS warning by vehicle type (nj), adjustment factor for no under inflation and tire pressure above warning threshold (vj), and injury-to-fatality equivalence ratios (ri), cumulative lifetime discount factor (d), adjustment factor for existing TPMS systems (a1), driver response rate to TPMS warnings (a2), and adjustment factor for overlapping target population (a3).

The target fatal population (P6,j,k) is treated as a constant for every j (passenger car or light truck/van) and k (wet or dry roadway condition). The target non-fatal population (Pi,j,k) is normally distributed for every i, j, and k. The section on skidding/loss of control already explained the rationales for determining the probability distribution for target population. Thus, these rationales are not repeated here. Table X-2 lists the fatalities and the means and standard errors for deriving the normal distribution for the non-fatal target population.

Table X-2
Means and Standard Errors for Normal Distributions
Injuries Associated with Stopping-Distance, Preventable Crashes
MAIS 1 MAIS 2 MAIS 3 MAIS 4 MAIS 5 Fatality*
PC  Wet Mean 2,361 226 101 9 6 14
SE 560 54 24 2 1 NA
PC  Dry Mean 6,680 722 310 29 19 54
SE 1,358 147 63 6 4 NA
LTV - Wet Mean 1,133 111 50 4 3 10
SE 324 32 14 1 1 NA
LTV - Dry Mean 3,546 466 192 19 14 39
SE 783 103 42 4 3 NA
* adjusted to the FARS level, thus was treated as a constant.

The uncertainty factor wi,k represents the adjustment factor for skidding state for injury severity i and roadway type k. As described in the previous chapter on benefits, only the wi,k portion of the target population Pi,j.k were skidding due to tires losing their friction capability. This portion of target population couldn't be compensated by drivers' action and thus was applicable to this TPMS rule. These adjustment factors were derived from 1995-1999 CDS. They are ratios of skidding to overall injuries. A ratio is sensitive to the frequency distribution of the survey counts. The derived ratios based on the 68 percent bounds of the CDS survey counts are similar to that from the mean survey population. Therefore, the analysis treats these adjustment factors as constants. Note that this loss of friction, can't be compensated by driver's action scenario would occur only on dry pavement (k=2). Since there is no adjustment on wet roadways, wi,1 would be 1 for every injury severity i. Table X-3 lists these values. Also note that this factor is the same for all three compliance options.

Table X-3
Stopping Distance  Preventable Crashes
(i = 6)
Injuries
(i = 1 to 5)
Wet (k=1) 1 1
Dry (k=2) 0.72 0.54
* same for all three compliance options.

The uncertainty factor nj represents an adjustment factor for average psi in the whole vehicle fleet before triggering TPMS warnings. The average estimated psi(s) before triggering TPMS warnings were different among vehicle types and compliance options (see Table V-6). Therefore, the adjustment factors differed by vehicle types and compliance options. No data were available to bound this factor. This factor was treated as a constant. Table X-4 lists the values of the adjustment factor.

Table X-4
Adjustment Factor for Average PSI Before TPMS Warnings
Stopping-Distance, Preventable Crashes
Implementation Alternative Passenger Cars
(n1)
Light Trucks/Vans
(n2)
Option 1 0.66877 0.68269
Option 2 0.62577 0.65503
Option 3 0.62577 0.65503

The uncertainty factor vj represents an adjustment factor to exclude cases with no under inflation and tire pressure above warning threshold. The analysis treats the factor as normally distributed. Its mean value and SD differ by vehicle types. For passenger cars, the mean v1 is 0.26 and SD is 0.013. For light trucks/vans, the mean v2 is 0.29 and SD is 0.014. Figure X-4 depicts these distributions.

Figure X-4
Probability Distributions for
Vehicle Type Parameters Probability Distribution
Passenger Cars
(v1)
Mean: 0.260
SD: 0.013
Light Trucks/Vans
(v2)
Mean: 0.290
SD: 0.014

The uncertainty factors ri (injury-to-fatality equivalence ratios), d (cumulative lifetime discount factor), a1 (adjustment factor for existing TPMS systems), and a2 (driver response rate to TPMS warnings) are the same as described in the previous subsection, thus are not repeated here.

The last uncertainty factor a3 represents the adjustment factor to correct two cases of overlapping between target populations: overlapping between skidding/loss of control and stopping distance, and overlapping between injuries in passenger cars and in light truck/vans within the stopping distance group. Generally, a3 is subject to survey errors inherited in the CDS systems since all the target populations were derived from 1995-1999 CDS. However, a3 is a ratio, which can be reasonably represented by the ratio derived from the mean survey population.  Thus, a3 is treated as a constant. Its value is equal to 0.9588.

Stopping Distance, Non-Preventable

The non-preventable crashes are crashes that would still occur even after the tire pressure had been corrected. The benefit of TPMS for this group would come from crash severity reduction (as oppose to prevention). Crash severity is measured by delta v. Delta v reduction is sensitive to vehicle type, roadway condition, and traveling speeds. Therefore, the benefit process for this group is further segregated by three traveling speed categories (0-35, 36-50, 51+ mph). Basically, benefits are derived by comparing the would-be injury severity distribution (corrected tire pressure condition) to the initial injury severity distribution (underinflated tire pressure condition). Injury risk curves as functions of delta v are used to induce the injury severity distributions. The following benefit model (FE4) describes the benefit process:

 Where A = d * a1 * a2 * a3 * a4, and Pi,j,k,l = target population with i = 1 to 6: MAIS 1 to fatality j = 1: passenger cars, 2: light trucks/vans; k = 1: wet roadway, 2: dry roadway, l = 1: 0-35 mph, 2: 35-50 mph, 3: 51+ mph pi(s) = probability risk of injury severity i, at delta v level s with pi(s) = 0 for s <= 0. Δvj,k,l = delta v reduction for roadway condition j, vehicle type k, and travel speed l mph wi,k = adjustment factor for skidding on roadway with i = 1 to 6: MAIS 1 to fatality; and k = 1: wet, 2: dry nj = adjustment factor for average psi experience in the fleet prior to the TPMS been triggered by vehicle type with j = 1: passenger cars; 2: light trucks/vans vj = adjustment factor for no under-inflation and tire level above warning threshold by vehicle type  with j = 1: passenger cars; 2: light trucks/vans ri = injury-to-fatality equivalence ratios d = cumulative lifetime discount factor (at 3 percent or 7 percent) a1 = adjustment factor for existing TPMS systems a2 = adjustment factor for response rate to TPMS warning light a3 = adjustment factor for overlapping among target population a4 = adjustment factor for stopping distance distribution

Based on the FE4 model, there are 12 major uncertainty factors that would impact the benefit estimate for stopping-distance, non-preventable crashes: target population (Pi,j,k), reduced dalta v (Dvj,k,l), injury risk probability for severity i at s delta v ( pi(s)), adjustment factor for skidding state on roadway type (wi,k), adjustment factor for average psi in a vehicle fleet before triggering a TPMS warning by vehicle type (nj), adjustment factor for no under inflation and tire pressure above warning threshold (vj), injury-to-fatality equivalence ratios (ri), cumulative lifetime discount factor (d), adjustment factor for existing TPMS systems (a1), driver response rate to TPMS warnings (a2), adjustment factor for overlapped target population (a3), and adjustment factor for stopping distance overestimation (a4).

The probability distributions for the first uncertainty factor Pi,j,k, target population, are the same as those of previous three target populations. The fatal target population P6,j,k is a constant, for every j and k. While the target non-fatal injury population Pi,j,k (i = 1 to 5) is treated as normally distributed. Table X-5 lists the target fatal population and the parameters (mean and standard errors) required to establish the normal distribution for target non-fatal injuries.

Table X-5
Means and Standard Errors for Normal Distributions
Injuries Associated with Stopping-Distance, Non-Preventable Crashes
MAIS 1 MAIS 2 MAIS 3 MAIS 4 MAIS 5 Fatality*
PC - Wet
0-35 mph Mean 77,340 6,930 3,172 282 167 440
SE 16,651 1,492 683 61 36 NA
36-50 mph Mean 69,807 7,044 3,076 255 165 439
SE 14,542 1,467 641 53 34 NA
51+ mph Mean 24,126 2,446 1,088 96 72 177
SE 4,690 476 212 19 14 NA
PC - Dry
0-35 mph Mean 184,795 17,408 7,790 669 448 1,175
SE 46,430 4,374 1,957 168 112 NA
36-50 mph Mean 224,077 20,930 9,332 879 490 1,549
SE 58,309 5,447 2,428 229 128 NA
51+ mph Mean 75,621 14,013 5,357 566 432 1,167
SE 16,029 2,970 1,135 120 92 NA
LTV  Wet
0-35 mph Mean 27,756 2,670 1,178 103 67 164
SE 5,410 520 230 20 13 NA
36-50 mph Mean 43,201 3,733 1,762 123 96 257
SE 8,603 743 351 24 19 NA
51+ mph Mean 9,995 1,564 661 80 67 220
SE 1,975 309 131 16 13 NA
LTV  Dry
0-35 mph Mean 100,964 11,504 4,890 475 299 847
SE 22,477 2,561 1,089 106 66 NA
36-50 mph Mean 100,620 12,242 5,078 469 348 1,104
SE 22,218 2,703 1,121 104 77 NA
51+ mph Mean 51,840 9,576 3,756 420 327 879
SE 10,572 1,953 766 86 67 NA
SE:  Standard Error; NA: Not Applicable
* constant numbers

The second uncertainty factor pi(s) represents the risk probability of injury severity i at the s delta v level. These probabilities were derived from MAIS+ injury curves, which were created through a statistical regression process. Thus, pi(s) is subject to the variations inherited in the regression process that was used to derive the MAIS+ injury curves. Table X-6 lists the mean and standard deviation for these MAIS+ injury curves.

Table X-6
MAIS+ Injury Probability Curves
Injury Severity Injury Risk Curve As a
Function of Delta V (%)
Parameters
(Mean, Standard Deviation)
MAIS 0 α: (0.0807, 0.0714)
MAIS 1+ α: (93.2210, 5.4079)
MAIS 2+ α: (0.1683, 0.0128)
β: (5.0345, 0.3362)
MAIS 3+ α: (0.1292, 0.0091)
β: (5.5337, 0.3131)
MAIS 4+ α: (0.1471, 0.0093)
β: (7.3675, 0.3344)
MAIS 5+ α: (0.1516, 0.0101)
β: (7.8345, 0.3801)
Fatality α: (0.1524, 0.0118)
β: (8.2629, 0.4481)

The third uncertainty factor Dvj,k,l represents the delta v reduction if tire pressure was corrected. Delta v is sensitive to traveling speeds and the square root of traveling distance (i.e., stopping distance). Thus, Dvj,k,l  change or reduction in delta V - is also a function of speed and the square root of change in stopping distance. Given a traveling speed category, Dvj,k,l would only be a function of the square root of the change in stopping distance. Therefore, the analysis uses the variation in the square root of stopping distance from the agency sponsored testing of 10 vehicles [47] as a proxy for Dvj,k,l. These 10 vehicles cover mid-size passenger cars, sports utility vehicles, vans, and pick-up trucks. Each vehicle was tested at 100 kph (62.1 mph) repeatedly about 10 or 20 times each on both wet and dry pavement. The mean of the square roots of these stopping distances and corresponding standard deviation is 7.4 meters and 0.3 meters on wet pavement and 7.0 meters and 0.3 meters on dry pavement. These translate to standard deviations of 4.3 percent and 3.7 percent of the mean for Dvj,k,l for wet and dry pavement, respectively. Since, the mean Dvj,k,l is small for each vehicle type and roadway condition, the small deviation does not perturb the delta v significantly. In addition, the Dvj,k,l was used to calculate pi(s). The small change in Dvj,k,l would thus alter the value of pi(s). However, these impacts are within the regression variations of pi(s). For these reasons, this analysis treats each Dvj,k,l as a constant. Table X-7 lists these values.

Table X-7
Delta V Reduction (Δvj,k,l)
0-35 mph (l=1) 3.018 1.952
36-50 mph (l=2) 4.404 2.748
51+ mph (l=3) 5.841 3.456
Light Truck Van (k=2)
0-35 mph (l=1) 3.358 2.187
36-50 mph (l=2) 4.895 3.078
51+ mph (l=3) 6.841 3.773

The remaining uncertainty factors, except for a4, are the same as those described in the FE3 model for stopping-distance, preventable crashes, and thus are not repeated here. The uncertainty factor a4 adjusts for braking distance distributions. This factor adjusts for the variable impact on delta v of braking that occurs over different stopping distances. The sources of uncertainty for this factor come from vehicle types, traveling speed, roadway condition, and the uniform probability function used for describing crash occurrence. The sensitivity study presented in the Appendix indicates that the factor is relatively stable regardless of vehicle type, traveling speed, and roadway condition (Appendix A). The only uncertainty source left is the uniform distribution used in deriving the adjustment factor. At this moment, the agency does not have data to describe the likelihood of crash occurrence at a certain point between the initial braking and the final natural stopping distance. The agency considers the uniform distribution a logical choice and there are no data to prove otherwise, therefore, the analysis does not alter the uniform distribution to calculate the adjustment factor. This factor is treated as a constant of 0.07.

The above sections discuss the FE models. FE is the basic benefit measurement for estimating cost-effectiveness. The benefit measurement in net benefits is in total dollars, which is the product of cost per fatality and FEs. Let M denote the cost per fatality. The total benefit in the net benefit calculation is equal to M*FEs. M clearly is another uncertainty factor for net benefits. Recent meta-analysis of the wage-risk value of statistical life (VSL) shows that an individual's willingness-to-pay (WTP) for reduction in premature fatalities is from \$1 million to \$10 million [48]. Thus, the agency uses this as the range for M and assumes the value of M is normally distributed with its mean equal to \$5.5 million. This value of \$5.5 million represents a central value consistent with a range of values from \$1 to \$10 million. The characteristics of the remaining factors are the same as those described in the cost-effectiveness model.

Total Cost Model

The total net cost (TC) is the product of the net cost per vehicle (NC) and the total number of vehicles (V). The net cost per vehicle consists of six cost components: technology/ countermeasure cost (C1), maintenance costs (C2), opportunity and other costs (C3), fuel savings (C4), tread wear savings (C5), and property damage/traveling delay savings (C6). The total cost model has the following generic format

Based on the TC mathematical model, the variability of these seven independent variables C1, C2, C3, C4, C5, C6, and V would perturb the estimated total costs. Thus, these seven variables are considered as major uncertainty factors. The uncertainties of these major factors come from the underlying assumptions, data sources, and statistical processing errors. Of these factors, C2, C3, C4, C5, and C6, were the costs/savings over a vehicle life and need to be discounted at the 3 and 7 percent rates. As a result, the total costs are analyzed at these two discount rates. Note again, C1 - C6 are on a per vehicle basis and all the costs are in 2001 dollars.

Technology/Countermeasure Cost (C1)

The first uncertainty factor C1 is the technology/countermeasure cost per vehicle. Basically, C1 = CTMPS * a1 where CTPMS is the TPMS cost per vehicle and a1 is the adjustment factor for existing TPMS market shares. CTPMS varies depending on the implementation of the technologies (i.e., compliance options), the maturity of the technologies/countermeasures, and potential fluctuation in labor and material costs (e.g., due to economies of scale from production volume). Based on the professional judgments of NHTSA cost analysts and contractors, the cost of a TPMS generally falls within 10 percent of the point estimate as presented in the previous chapter on costs. Each cost value within this range would have an equal chance to be the true price. Thus, this analysis treats CTMPS as uniformly distributed. Its generalized format can be represented as:

 Where Cmax = the maximum TPMS cost per vehicle, and Cmin = the minimum TPMS cost per vehicle

Two parameters are required to establish its uniform distribution: the maximum and minimum costs. Table X-8 lists these costs for CTPMS. These costs represent the investments paid now for future benefits. Therefore, there is no need to discount these costs.

Table X-8
Uniform Distribution for TPMS Cost Per Vehicle (CTPMS)
For Three Compliance Options*
(2001 Dollars)
Maximum Cost
(Cmax)
Minimum Cost
(Cmin)
The Mean Cost
(point estimate)
Option 1
\$77.65 \$63.53 \$70.59
Option 2
Direct TPMS with Warning
\$73.42 \$60.07 \$66.74
Option 3
Hybrid TPMS with Warning
\$53.81 \$44.04 \$48.92
* no discounting.

The uncertainty factor a1, adjustment factor for existing TPMS, is treated as a constant. Currently, about 1 percent of passenger vehicles are equipped with a TPMS meeting the proposal. Thus, a1 = 0.99 (= 1 - 0.01). The initial CTPMS was adjusted down by this factor to estimate the true technology cost (i.e., incremental cost) per vehicle.

Of the two factors within C1, a1 is a constant, and thus influences only the range of the costs. The other factor CTMPS dictates the type of probability distribution for C1.  As a result, C1 is uniformly distributed with a range that is slightly smaller than CTMPS. Table X-9 lists three costs (maximum, mean, and minimum) of C1 that are used to establish its uniform distribution for the three compliance options.

Table X-9
Uniform Distribution for Technology/Countermeasure Costs (C1)
For Three Compliance Options*
(2001 Dollars)
Maximum Cost
(Cmax)
Minimum Cost
(Cmin)
The Mean Cost
(point estimate)
Option 1
\$76.88 \$62.90 \$69.89
Option 2
Direct TPMS with Warning
\$72.69 \$59.48 \$66.08
Option 3
Hybrid TPMS with Warning
\$53.28 \$43.60 \$48.44
* no discounting.

Maintenance Costs (C2)

The maintenance cost per vehicle C2, is the cost for battery replacement over a vehicle's life, and thus was discounted at 3 and 7 percent. The cumulative discount factor (d) is equal to 0.8233 at 3 percent and 0.6600 at 7 percent. Obviously, C2 depends on the design of the TPMS (with or without batteries). It also varies with the labor cost, the cost of the battery, and technologies (i.e., compliance options). Since the sources of its variations (e.g., labor and material costs, etc.) are similar to those cited for C1, the analysis considers C2 to possess the same type of probability distribution as C1, i.e., C2 is uniformly distributed with its values conforming within 10 percent of the point-estimated cost presented in the chapter on costs.

Table X-10 lists the cost parameters required for establishing the uniform distribution for C2 at the 3 and 7 percent discount rates by TPMS design and three compliance options. As shown in Table X-10, under the with-battery scenario, C2 is uniformly distributed between \$36.45 and \$44.55 for Options 1 and 2, and between \$24.24 and \$29.62 for Option 3. If batteries were not required, these values would all be 0.

Table X-10
Uniform Distribution For Maintenance Costs Per Vehicle (C2)
TPMS With Batteries*
(2001 Dollars)
Maximum Cost
(Cmax)
Minimum Cost
(Cmin)
The Mean Cost
(point estimate)
At 3% Discount Rate
Option 1
\$61.58 \$50.38 \$55.98
Option 2
Direct TPMS with Warning
\$61.58 \$50.38 \$55.98
Option 3
Hybrid TPMS with Warning
\$40.95 \$33.51 \$37.23
At 7% Discount Rate
Option 1
\$44.55 \$36.45 \$40.50
Option 2
Direct TPMS with Warning
\$44.55 \$36.45 \$40.50
Option 3
Hybrid TPMS with Warning
\$29.62 \$24.24 \$29.63
* \$0 if batteries were not required.

Opportunity and Other Costs (C3)

The opportunity and other costs estimate the costs for additional tire pressure fill-ups and the air pump charges over the vehicle's life (C3). C3 varies with the estimated additional fill-up frequency (N), the time duration (in hour) for each fill-up (T), the number of occupants in a vehicle (O), the estimated cost per hour (Ch), air pump fee per vehicle over the vehicle's life (Cp), the cumulative lifetime discount rate (d), the existing TPMS market share (a1), and the TMPS response rate (a2). C3 can be represented as:

 Where N = the number of additional fill-ups per vehicle over the vehicle's life T = time needed for each fill-up O = the number of occupants per vehicle Ch = value of opportunity costs per hour Cp = lifetime air pump fee per vehicle d = cumulative lifetime discount factor (at 3 percent or 7 percent) a1 = adjustment factor for existing TPMS system a2 = adjustment factor for response rate to TPMS warning light

Of these eight factors within C3, N, T, O Ch, and Cp are newly introduced. Their variations are discussed here. The variations of these factors are based on expert judgment. The analysis assumes that

• N is uniformly distributed between 8.02 and 9.08 with a mean of 8.91.
• T is normally distributed with 0.0833 hours (=5 minutes) as the mean and 0.0167 hours (= 1 minute) as SD.
• O is normally distributed with a mean of 1.3 occupants and a standard deviation of 0.1 occupants.
• Ch is normally distributed from \$8.10 to \$14.10 per hour with a mean of \$11.10 per hour.
• Cp is uniformly distributed from \$0.55 to 0.73 per vehicle over the vehicle's life with a mean of \$0.64 per vehicle.

The values and variability of uncertainty factor d (cumulative lifetime discount factor), a1 (adjustment for existing TPMS), and a2 (response rate to TPMS warnings) were discussed in the benefit model section, and thus are not repeated here. Figure X-5 depicts the probability distributions for the uncertainty factors within C3.

After taking into account all these variations, the final outcome of C3 is close to a normal distribution. At the 3 percent discount rate, C3 has a mean of \$8.31 and a standard deviation of \$1.11. At the 7 percent discount rate, C3 has a mean of \$6.66 and a standard deviation of \$0.89. Figure X-6 depicts the probability distributions for the final outcomes of C3.

Figure X-5
Probability Distribution for Factors Within The
Opportunity and Other Costs (C3)
Factors Parameters Probability Distribution
Additional fill-ups per vehicle over the vehicle life (N) Maximum: 8.02
Minimum: 9.80
Mean: 8.91
Hours per fill-up (T) Maximum: 0.075
Minimum: 0.092
Mean: 0.0833
Occupants per vehicle (O) Mean: 1.3
SD: 0.1
Opportunity Cost per hour (Ch) Mean: \$11.10
SD=\$1.10
Lifetime Air Pump Fee per vehicle (Cp) Mean: \$0.64
SD=\$0.03
Cumulative lifetime discount factor (d) NA Constant
At 3 % = 0.8233
At 7% = 0.6600
Adjustment factor for existing TPMS system (a1) NA Constant
a1 = 0.99
Adjustment factor for response rate to TPMS warning light (a2) Mean: 0.90
SD: 1.7
Same as Figure X-2

Figure X-6
Probability Distributions for Opportunity and
Other Costs Per Vehicle Over the Vehicle Life (C3)
Discount Level Parameters Probability Distribution
At 3% Mean: \$8.31
SD: \$1.11
At 7% Mean: \$6.66
SD: \$0.89

Fuel Savings (C4)

The saving from fuel consumption over a vehicle's life, C4, also comes with certain variations. The sources of the variations come from, but are not limited to, fuel price, the 1 percent fuel efficiency equivalent psi, baseline mpg (CAFE standards), the discount factor, the existing TPMS, and the driver response rate to TPMS. Since, CAFΙ issues a different mpg standard for passenger cars and light trucks/vans and each compliance option has a different baseline mileage, C4 would depend on the vehicle types and compliance options.

The fuel price fluctuates with demand and supply cycles. The 1949-2002 retail motor gasoline prices reported by the Department of Energy [49] were used to predict the future fuel price variations. The analysis used Crystal Ball as a tool to fit these historical data into 10 different continuous probability distributions (e.g., normal, lognormal, etc.). Three goodness-of-fit tests Chi-square, Kolmogorov-Smirnov, and Anderson-Darling were used to rank each distribution [50]. None of the 10 probability distributions were found to be significant by all three of these measures. However, the logistic distribution has an overall consistent and relatively favorable ranking among these 10 probability functions. Thus, the logistic distribution was chosen to represent the variation of the fuel price. The general format of a logistic distribution is:

Mean represents the average fuel price and scale determines the spread and the shape of the probability curve. The historic motor gasoline price from 1949 to 2002 was logistically distributed with the scale equal to 9.2 percent of the mean fuel price. This mean and scale relationship was then applied to the logistic distribution used for the average fuel price, i.e., a = 0.092*m. The mean pre-tax fuel price used (Chapter V) is \$1.06, i.e., m = \$1.06. Based on the mean-scale relationship stated above, the scale, a, is equal to \$0.97. With this logistic probability distribution, the pre-taxed fuel price would range from \$0.48 to \$1.64. The estimated gasoline tax is \$0.38. Therefore, the after-tax fuel price is from \$0.86 to \$2.02. Figure X-7 depicts the logistic distribution for pre-tax fuel prices in 2001 dollars.

Figure X-7
Pre-Taxed Fuel Price Distribution
(2001 Dollars)

The 1 percent fuel efficiency equivalent psi power would also impact C4. In Chapter V, we estimated that a mean of 2.96 psi is equivalent to one percent fuel efficiency. The agency does not have sufficient data to assess its variation. However, to somewhat assess its impact on C4, the analysis assumes that the psi power is uniformly distributed with 10 percent variation from the mean 2.96 psi. In other words, 1 percent fuel efficiency would be equivalent to a range of psi from 2.66 to 3.26 psi.

Another factor, mpg , would be a constant which is based on the CAFE standards. The mpg standard is 27.5 and 22.2 mpg for passenger cars and light trucks/vans, respectively.

The remaining uncertainty factors such as d (cumulative lifetime discount factor), a1 (adjustment factor for existing TPMS), and a2 (driver response rate to TMPS warning) were discussed earlier, and thus are not repeated here.

With the consideration of the variations from the factors discussed above, the simulated final outcome of C4 is close to a normal distribution. Figure X-8 depicts these distributions by discount rates.

Figure X-8
Normal Distribution for Fuel Saving (C4)
By Three Compliance Options and Discount Rates
(2001 Dollars)
Mean & SD Probability Distribution
At 3% Discount Rate
Option 1
Mean: \$23.54
SD: \$3.26
Option 2
Direct TPMS with Warning
Mean: \$19.39
SD: \$2.69
Option 3
Hybrid TPMS with Warning
Mean: \$19.36
SD: \$2.69
Same as Option 2
At 7% Discount Rate
Option 1
Mean: \$18.67
SD: \$2.58
Option 2
Direct TPMS with Warning
Mean: \$15.37
SD: \$2.13
Option 3
Hybrid TPMS with Warning
Mean: \$15.37
SD: \$2.13
Same as Option 2

The saving from tread wear per vehicle over a vehicle's life, C5, depends on the tire materials, technologies, and roadway types, miles traveled, and response rate to TPMS. The agency does not have information at this time to discern the trends for these sources except for the miles traveled. The average number of vehicle miles traveled by motor vehicles has increased each year [51]. Consequently, the estimated tread wear saving could potentially be higher than currently estimated. On the other hand, this increased saving might be offset by future tire improvements. With no adequate data to assess the possible trends and variations, the analysis treats C5 as uniformly distributed with the overall range falling within 10 percent of the point estimate as presented in the previous chapter on costs. Figure X-9 depicts these distributions by compliance options and discount rates.

Figure X-9
Uniform Distribution for Tread Wear Savings Per Vehicle (C5)
For Three Compliance Options
(2001 Dollars)
Mean Uniform Distribution
At 3% Discount Rate
Option 1
\$4.24
Option 2
Direct TPMS with Warning
\$3.42
Option 3
Hybrid TPMS with Warning
\$3.42 Same as Option 2
At 7% Discount Rate
Option 1
\$6.03
Option 2
Direct TPMS with Warning
\$4.98
Option 3
Hybrid TPMS with Warning
\$4.98 Same as Option 2

Property Damage and Traveling Delay Savings (C6)

The property damage and travel delay savings per vehicle (C6) over the vehicle life includes savings from three areas: preventable injury crashes, vehicles in preventable PDO crashes, and non-preventable crashes. The process of C6 can be algebraically described as follow:

 Where BPi = net MAIS i benefits from preventable crashes with MAIS 6 = fatalities BNAi = net MAIS I benefits from non-preventable crashes with 6 = fatalities PDOi = PDO vehicles derived from MAIS i benefits with 6 = fatalities from preventable crashes. UCi = property damage/travel delay unit cost within MAIS i injuries with MAIS 0 = no injury and 6 = fatalities UCPDO = property damage/travel delay unit cost per PDO vehicle d = cumulative lifetime discount factor V = total number of vehicles

The variables BPi, net injury and fatality benefits, were derived from the benefit models described earlier. Values of PDOi were also derived from BPi, i > 0. Basically, the initial BPis were adjusted to its corresponding PDO vehicles by multiplying by a PDO/injury ratio. This PDO/injury ratio is a function of three variables: the number of occupants per crash involved vehicle (=1.35), property damage to injury vehicle ratio (= 5.70 for skidding/loss of control, 3.99 for others), and adjustment factor for underreporting (=1.92). These three variables were derived from NHTSA crash database such as FARS, CDS, and State Data Files. They are ratios, and thus are treated as constants. The formula for the PDO/injury ratio can be expressed as follow:

Based on the above mathematical formula, C6 varies with the derived net benefits (BPi and BNPi) and the property damage/travel delay unit costs. Since C6 was derived from BPi and BNPi, all the uncertainties for BPi and BNPi discussed in the benefit models would also apply to C6. The variability of UCi and UCPDO would also impact C6. However, as explained earlier in this chapter, the analysis does not consider its variability at this moment and treats these unit costs as constants. Table X-11 lists these unit costs. Please consult Chapter V for a detailed explanation. The factor V, the number of vehicles, also is treated as a constant of 17 million (see discussion below). Figure X-10 depicts the final outcomes of C6.

Table X-11
Property Damage and Travel Delay Unit Costs by Injury Severity*
Injury Severity Unit Costs
MAIS 0 \$1,843
MAIS 1 \$4,752
MAIS 2 \$4,937
MAIS 3 \$7,959
MAIS 4 \$11,140
MAIS 5 \$19,123
Fatality \$19,947
PDO Vehicle \$2,352

Figure X-10
Probability Distributions for Property Damage and
Travel Delay Savings Per Vehicle Over the Vehicle Life (C6)
Discount Level Parameters Probability Distribution
At 3% Discount Rate
Option 1
Mean: \$7.67
SD: \$1.67
Option 2
Direct TPMS with Warning
Mean: \$7.57
SD: \$1.76
Option 3
Hybrid TPMS with Warning
Mean: \$7.57
SD: \$1.76
Same as Option 2
At 7% Discount Rate
Option 1
Mean: \$6.15
SD: \$1.42
Option 2
Direct TPMS with Warning
Mean: \$6.07
SD: \$1.41
Option 3
Hybrid TPMS with Warning
Mean: \$6.07
SD: \$1.41
Same as Option 2

Number of Vehicles (V)

The last uncertainty factor, the number of vehicles (V) is treated as a constant of 17 million. Of these, 8 million are passenger cars and 9 million are light trucks, vans, and sport utility vehicles. Although vehicle sales have gradually increased over time, they are subject to annual variation due to changes in economic conditions, which are difficult to predict.

After the fatal equivalent (FE) and cost models were established, the cost-effectiveness model (CE) simply is the ratio of total costs (TC) to fatal equivalents. It has the format: CE = TC/FE. The net benefits (NB) has the format: NB = M*FE  TC, where M is the cost per fatality.

Modeling Results

The uncertainty analysis conducted a total of 25,000 trials before the forecasted mean results reached 99 percent precision. Even if the later criterion was reached first, the trial numbers generally are very close to 25,000. These criteria were chosen to ensure that the simulation errors () would be very close to 0.

Tables X-12 and X-13 summarize the modeling results. Table X-12 lists the results for the scenario that all the TPMSs require batteries. Table X-13 is for the no-battery scenario, which is the direction that the industry is expected to take. Therefore, the agency believes that the modeling results reported in Table X-13 are a more realistic assessment of the rule in the future than those in Table X-12.

With Batteries

As shown in Table X-12  with battery scenario, at the 3 percent discount rate, the estimated costs range from \$1,391 to \$1,973 million for Option 1, \$1,426 to \$1,975 million for Option 2, and \$851 to \$1,313 million for Option 3. These three options would save 98  296, 96  291, and 96  291 equivalent lives, respectively. As noted in Chapter VII, the most recent NHTSA study relating to the cost of crashes on valuing fatalities indicate a value of life of about \$3.5 million (in 2001 dollars).   Based on the statistics, there is almost no chance for these compliance options to produce a cost per equivalent fatality of less than \$3.5 million. If a higher \$5.5 million threshold was used (based on the midpoint of the range previously discussed), Option 3 would have 47 percent chance to meet this threshold. There is almost no chance for Options 1 and 2 to meet the \$5.5 million threshold. All three options would produce positive net benefits with different levels of certainty: 6 percent for Option 1, 5 percent for Option 2, and 44 percent for Option 3.

At the 7 percent discount rate, the estimated costs range from \$1,208 to \$1,711 million for Option 1, \$1,237 to \$1,705 million for Option 2, and \$763 to \$1,136 million for Option 3. At this discount rate, these three options would save 79  237, 77  233, and 77  233 equivalent lives, respectively. Option 3 is the only option that would produce a cost per equivalent fatality less than \$5.5 million with 32 percent certainty. The chance that each option would produce positive net benefits at this discount rate is: 3 percent for Option 1, 2 percent for Option 2, and 35 percent for Option 3.

No Batteries

For the no-battery scenario, the total compliance cost would be one-third to one-half of those with batteries. Because the compliance costs were significantly reduced, the three compliance options become much more cost-beneficial than assessed under the with-battery scenario. As shown in Table X-13, at the 3 percent discount rate, the estimated costs range from \$477 to \$977 million for Option 1, \$509 to \$986 million for Option 2, and \$247 to \$657 million for Option 3. The three options would produce a cost per equivalent fatality less than \$3.5 million with 41 percent, 34 percent, and 93 percent certainty, respectively. If the threshold were raised to \$5.5 million, all three options would have more than 90 percent chance to meet it. All three Options would also generate positive net benefits with relatively high certainty levels, 82 percent for Option 1, 79 percent for Option 2, and 97 percent for Option 3.

At the 7 percent discount rate, the estimated costs range from \$557 to \$999 million for Option 1, \$577 to \$997 million for Option 2, and \$327 to \$679 million for Option 3. The three options would produce a cost per equivalent fatality less than \$3.5 million with 8 percent, 5 percent, and 65 percent certainty, respectively. If the threshold were raised to \$5.5 million, the certainty levels for these three Options would be 68 percent, 62 percent and 99 percent, respectively. Options 1 and 2 would produce positive net benefits with about 55 percent certainty. Option 3 would have a 90 percent chance to produce positive net benefits.

Summary

The three compliance options would save a similar number of equivalent lives. However, due to its relatively low costs, Option 3 (a hybrid TPMS) is the most cost-beneficial among these three options. With technology advances, the cost of the TPMS would be reduced significantly as demonstrated in the no-battery scenario (Table X-13). This scenario probably reflects the future of TPMS design. Under this assessment, the TPMS rule is increasingly more favorable. At a 3 percent discount rate, Options 1 and 2 would have less than 50 percent chance to produce a cost per equivalent fatality less than \$3.5 million. The certainty level increased significantly to 93 percent for Option 3 to meet the \$3.5 million threshold. All three options would have a very high probability to produce a cost per equivalent fatality less than \$5.5 million and positive net benefits. Not surprisingly, with a higher discount rate of 7 percent, these options would meet the same cost-effectiveness (\$3.5 and \$5.5 million) and net benefit (>0) thresholds with less certainty. But, all three options would still produce a cost per equivalent fatality less than \$5.5 million and positive net benefits with high certainty levels.

This analysis of TPMS involved numerous data sources, methods and assumptions, most of which involve some degree of uncertainty. As a result, this uncertainty analysis includes over 100 probability distributions, comprised of a variety of uniform distributions, normal distributions, logistic distributions, and a combination of uniform and triangular distributions. Considering all of these distributions simultaneously results in very wide ranges for cost-effectiveness and net benefits as shown in Tables X-12 and X-13.

Table X-12
Simulated Cost-Effectiveness and Net Benefits
With Batteries
Compliance Option
At 3% Discount Rate 1 2 3
Range of Total Costs \$1,391 - \$1,973 M \$1,426 - \$1,975 M \$851 - \$1,313 M
Mean Net Total Cost \$1,678 M \$1,700 M \$1,081 M
90% Certainty for Total Costs \$1,498 - \$1,861 M \$1,528 - \$1,868 M \$941 - \$1,219 M

Range of Equivalent Lives Saved 98  296 96  291 96  291
Mean Equivalent Lives Saved (present value) 197 193 193
90% Certainty for Equivalent Lives Saved (present value) 140  263 142  265 142  265

Range of CE \$4.6 - \$14.9 M \$5.0 - \$15.1 M \$2.8 - \$9.9 M
Mean CE \$8.8 M \$9.1 M \$5.8 M
90% Certainty for CE \$6.5  \$14.2 M \$6.7 - \$14.5 M \$4.2 - \$9.6 M
Certainty that CE <= \$3.5 M 0.0% 0.0% 1.4%
Certainty that CE <= \$5.5 M 1.4% 0.7% 47%

Range of Net Benefits -\$1,552 to \$371 M -\$1,536 to \$366 M -\$989 to \$943 M
Mean Net Benefits -\$610 M -\$650 M -\$34 M
90% Certainty for Net Benefits -\$1,135 to \$95 M -\$1,156 to \$76 M -\$534 to \$679 M
Certainty that Net Benefits > \$0 6% 5% 44%

At 7% Discount Rate
Range of Total Costs \$1,208 - \$1,711 M \$1,237 - \$1,705 M \$763 - \$1,136 M
Mean Net Total Cost \$1,464 M \$1,475 M \$945 M
90% Certainty for Total Costs \$1,306 - \$1,617 M \$1,328 - \$1,620 M \$830 - \$1,065 M

Range of Equivalent Lives Saved 79  237 77  233 77  233
Mean Equivalent Lives Saved (present value) 158 155 155
90% Certainty for Equivalent Lives Saved (present value) 112  211 112  209 112  209

Range of CE \$5.0  - \$15.9 M \$5.3  - \$16.1 M \$3.2  - \$10.6 M
Mean CE \$9.5 M \$9.8 M \$6.3 M
90% Certainty for CE \$7.1 - \$15.4 M \$7.3 - \$15.7 M \$4.6 - \$10.5 M
Certainty that CE <= \$3.5 M 0% 0% 0.2%
Certainty that CE <= \$5.5 M 0.2% 0.1% 32%

Range of Net Benefits -\$1,381 to \$192 M -\$1,405 to \$154 M -\$880 to \$686 M
Mean Net Benefits -\$607 M -\$634 M -\$104 M
90% Certainty for Net Benefits -\$1,031 to -\$19 M -\$1,050 to -\$60 M -\$508 to \$474 M
Certainty that Net Benefits > \$0 3% 2% 35%
M: million; CE: cost per fatal equivalent

Table X-13
Simulated Cost-Effectiveness and Net Benefits
No Batteries*
Compliance Option
At 3% Discount Rate 1 2 3
Range of Total Costs \$477  \$977 M \$509  \$986 M \$247  \$657 M
Mean Net Total Cost \$726 M \$748 M \$449 M
90% Certainty for Total Costs \$570 - \$883 M \$600 - \$889 M \$323 - \$573 M

Range of Equivalent Lives Saved 98  296 96  291 96  291
Mean Equivalent Lives Saved (present value) 197 193 193
90% Certainty for Equivalent Lives Saved (present value) 140  263 142  265 142  265

Range of CE \$1.6 - \$6.9 M \$1.5 - \$7.0 M \$0.5 - \$4.5 M
Mean CE \$3.8 M \$4.0 M \$2.4 M
90% Certainty for CE \$2.6 - \$6.4 M \$2.8 - \$6.6 M \$1.5 to \$4.2 M
Certainty that CE <= \$3.5 M 41% 34%  93%
Certainty that CE <= \$5.5 M  94%  92% 100%

Range of Net Benefits -\$634 to \$1,344 M -\$653 to \$1,276 M -\$319 to \$1,556 M
Mean Net Benefits \$343 M \$300 M \$599 M
90% Certainty for Net Benefits -\$173 to \$1,069 M -\$204 to \$1,017 M \$101 to \$1,302 M
Certainty that Net Benefits > \$0 82% 79% 97%

At 7% Discount Rate
Range of Total Costs \$557  \$999 M \$577  \$997 M \$327  \$679 M
Mean Net Total Cost \$775 M \$787 M \$488 M
90% Certainty for Total Costs \$634 - \$916 M \$655 - \$916 M \$381 - \$599 M

Range of Equivalent Lives Saved 79  237 77  233 77  233
Mean Equivalent Lives Saved (present value) 158 155 155
90% Certainty for Equivalent Lives Saved (present value) 112  211 112  209 112  209

Range of CE \$2.1  - \$9.0 M \$2.3  - \$9.1 M \$0.8  - \$5.8 M
Mean CE \$5.1 M \$5.2 M \$3.2 M
90% Certainty for CE \$3.6 - \$8.5 M \$3.7 - \$8.6 M \$2.2  \$5.6 M
Certainty that CE <= \$3.5 M  8%  5% 65%
Certainty that CE <= \$5.5 M 68% 62%  99%

Range of Net Benefits -\$701 to \$889 M -\$708 to \$835 M -\$373 to \$1,113 M
Mean Net Benefits \$81 M \$54 M \$352 M
90% Certainty for Net Benefits -\$337 to \$672 M -\$358 to \$626 M -\$50 to \$911 M
Certainty that Net Benefits > \$0 59% 55% 90%
* no maintenance costs
M: million; CE: cost per fatal equivalent

[44] Any statistics books describing the Monte Carlo simulation theory are good references for understanding the technique.

[45] 1995-1997 National Automotive Sampling System, Crashworthiness Data System, DOT HS 809 203, February 2001

[46] The Economic Impact of Motor Vehicle Crashes 2000, DOT HS 809 446, May 2002

[47] Transportation Research Center Inc., "Consumer Braking Information  Finalize Test Protocol  Phase 1

[48] Mrozek, J.R. and L.O. Taylor, What determines the value of a life? A Meta Analysis, Journal of Policy Analysis and Management 21 (2), pp. 253-270.

[49] Table 5.22 Retail Motor Gasoline and On-Highway Diesel Fuel Prices, 1949-2002, Annual Energy Review, 2002, Energy Information Administration (EIA), Department of Energy (DOE)

[50] Crystal Ball 2000 User Manual

[51] Highway Statistics 2002, U.S. DOT, FHWA-PL03-010; Highway Statistics 2001, U.S. DOT, FHWA-PL02-008