DOT HS 810 847 October 2007

The Role of Vertical Roof Intrusion and 

Post-Crash Headroom in Predicting 

Roof Contact Injuries to the Head, 

Neck, or Face During FMVSS No.216 

Rollovers; An Updated Analysis

This document is available to the public from the National Technical Information Service, Springfield, Virginia 22161
This publication is distributed by the U.S. Department of Transportation, National Highway Traffic 

Safety Administration, in the interest of information exchange. The opinions, findings, and 

conclusions expressed in this publication are those of the authors and not necessarily those of the 

Department of Transportation or the National Highway Traffic Safety Administration. The United 

States Government assumes no liability for its contents or use thereof. If trade or manufacturers’ 

names are mentioned, it is only because they are considered essential to the object of the publication 

and should not be construed as an endorsement. The United States Government does not endorse 

products or manufacturers. 
Technical Report Documentation Page 

1. Report No. 2. Government Accession No. 3. Recipients's Catalog No. 

DOT HS 810 847 

4. Title and Subtitle 5. Report Date

The Role of Vertical Roof Intrusion and Post-Crash October 2007 

Headroom in Predicting Roof Contact Injuries to the 

Head, Neck, or Face During FMVSS No. 216 6. Performing Organization Code 

NVS-421 Rollovers: An Updated Analysis 

7. Author(s) 8. Performing Organization Report No.

Alexander Strashny, Ph.D. 

9. Performing Organization Name and Address 10. Work Unit No. (TRAIS)n code

Mathematical Analysis Division, Office of Traffic Records and Analysis 

National Center for Statistics and Analysis 

National Highway Traffic Safety Administration 11. Contract of Grant No. U.S. Department of Transportation 

NVS-421,1200 New Jersey Avenue SE.

Washington, DC 20590 

12. Sponsoring Agency Name and Address 13. Type of Report and Period Covered

Mathematical Analysis Division, Office of Traffic Records and Analysis NHTSA Technical Report National Center for Statistics and Analysis 

National Highway Traffic Safety Administration 14. Sponsoring Agency Code U.S. Department of Transportation 

NVS-421, 1200 New Jersey Avenue SE.

Washington, DC 20590

15.Supplementary Notes 

I thank Chou-Lin Chen, Rajesh Subramanian, Rory Austin, Larry Blincoe, Stephen Summers, Chuck Kahane, 

and the peer reviewers for their help. 

16. Abstract

The purpose of this report is to use the National Automotive Sampling System Crashworthiness Data System (NASS CDS) data for 

years 1997 through 2005 to determine whether there was a statistically significant relationship between the maximum severity of

head, neck, and face injuries due to occupant roof contact that occurred in rollovers that were likely to be covered by Federal Motor

Vehicle Safety Standard No. 216 and either post-crash headroom or vertical roof intrusion. The report uses the ordered probit and

binary probit models, both unadjusted and adjusted for potentially confounding factors, to establish the existence of statistically 

significant relationships. The report presents estimates of 24 different models, 12 for intrusion and 12 for headroom. In all 24 

models, the relationship between injury severity and the explanatory variable (intrusion or headroom) was statistically significant. 

17. Key Words 18. Distribution Statement

Post-crash headroom, vertical roof intrusion, Document is available to the public through the National 

rollover, quarter turns, roof crush, end-over-end, Technical Information Service, 

Springfield, VA 22161 rollover severity, roof impact, roof-to-ground 

contact, roof-to-ground exposure, ordered probit 

19. Security Classif. (of this report) 20. Security Classif. (of this page) 21. No of Pages 22. Price

 Unclassified Unclassified 

Form DOT F1700.7 (8-72) Reproduction of completed page authorized
Table of Contents 

Executive Summary...................................................................................1

Introduction ................................................................................................2

Ordered Probit Model ................................................................................8

Binary Probit Model .................................................................................14

Conclusion................................................................................................16

References................................................................................................17

List of Tables 

Table 1. Distribution of the maximum injury severity of all head, neck, and face injuries from 

contact with a roof component................................................................................................ 5 

Table 2. Ordered probit models of maximum injury severity for head, neck, or face injuries from 

roof contact. .......................................................................................................................... 10 

Table 3. Probit models for the probability of maximum injury severity for head, neck, or face 

injuries from roof contact being either AIS 0 or 1................................................................ 15 
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Executive Summary 

Objective. The purpose of this report is to use the National Automotive Sampling System 

Crashworthiness Data System (NASS CDS) data for years 1997 through 2005 to determine 

whether there was a statistically significant relationship between the maximum severity of head, 

neck, and face injuries due to occupant roof contact that occurred in rollovers that were likely to 

be covered by Federal Motor Vehicle Safety Standard No. 216 (FMVSS No. 216) (“relevant 

injuries”) and either post-crash headroom or vertical roof intrusion. The report is not intended to 

make specific recommendations for changes to FMVSS No. 216, nor is it intended to calculate 

benefits from any such change. Instead, the conclusions in this report are meant to provide a 

basis and rationale for further analysis. 

Method. The report uses NASS CDS data for years 1997 through 2005. It also uses data 

furnished by Consumers Union to estimate pre-crash headroom. It uses the ordered probit and 

binary probit models, both unadjusted and adjusted for potentially confounding factors, to 

establish the existence of statistically significant relationships. 

Result. The report presents estimates of 24 different statistical models, 12 for intrusion and 12 

for headroom. In all 24 models, the relationship between injury severity and the explanatory 

variable (intrusion or headroom) was statistically significant. In 22 of the 24 models, the p-value 

was less than 0.0001; in two of the models, the p-value was 0.0009 or less. 

Conclusion. This report shows that a statistically significant relationship existed between both 

vertical roof intrusion and post-crash headroom on the one hand and maximum injury severity of 

head, neck, or face injury from roof contact on the other hand. The relationship remained 

regardless of the statistical model used. In particular, the report used both unadjusted and 

adjusted models, both with continuous and dichotomous versions of the explanatory variables, 

and with multilevel or bilevel versions of the dependent variable, and still the relationship 

remained statistically significant. 
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Introduction 

Background and purpose. Austin et al. (2005) use data from the NASS CDS for the years 1997 

through 2001 to illustrate the existence of a statistically significant relationship between 

maximum severity of head, neck, and face injuries due to occupant roof contact that occurred in 

rollovers that were likely to be covered by FMVSS No. 216 and the headroom remaining over 

the occupant after the crash (“post-crash headroom” or simply “headroom”). Relevant occupants 

were age 13 and older, sat in the two front outboard seats, were belted, and were neither 

completely nor partially ejected. Relevant rollovers were single-vehicle passenger-vehicle 

rollovers that involved roof-to-ground exposure (that is, two or more quarter turns in sideways 

rollovers), as well as end-over-end rollovers, excluding convertibles and vehicles before model 

year 1987. A draft of that report was peer-reviewed by Flannagan et al. (2005). The final report 

addressed and incorporated reviewers’ comments. The comments were also specifically 

addressed by NHTSA in a separate document, NHTSA (2005). 

Austin et al. (2005) find a statistically significant relationship between injury severity and 

whether post-crash headroom was positive or negative. They did not, however, find a statistically 

significant relationship between injury severity and headroom as a continuous variable, nor did 

they find a statistically significant relationship between injury severity and the amount of vertical 

roof intrusion over the occupant seating position (“intrusion”). 

The purpose of this report is to extend Austin et al. (2005) using more recently available data, 

namely, the purpose is to use NASS CDS data for years 1997 through 2005 to determine whether 

there was a statistically significant relationship between the maximum severity of relevant 

injuries and either headroom or intrusion, both as dichotomous and as continuous variables. The 

report is not intended to make specific recommendations for changes to FMVSS No. 216, nor is 

it intended to calculate benefits from any such change. Instead, the conclusions in this report are 

meant to provide a basis and rationale for further analysis. 

Data. Much of the following discussion of the data is taken verbatim from Austin et al. (2005). 

Some adjustments have been made, such as for more recent years available in the database. Also, 

more detailed discussions of certain issues are presented. 

The primary data set used for this analysis is NASS CDS for years 1997 through 2005. NASS 

CDS is a complex, random sample of crashes involving at least one passenger car or “light truck 

or van” (LTV), defined by a gross vehicle weight rating of 4,536 kilograms (10,000 pounds) or 

less, that was towed due to damage. The beginning year of 1997 was selected because it was the 

first year that NASS CDS coded continuous intrusion measures. Prior to that year, NASS CDS 

coded intrusion in categories with ranges that were too wide to be of use in this study. The 

ending year of 2005 was the most current year available at the time of this analysis. Note that the 

database codes intrusion in centimeters. To make results in this report comparable to those in 

Austin et al. (2005), measurements have been converted from centimeters to inches. This study 

focuses on automobiles, utility vehicles, light pickup trucks, and light vans that were involved in 

single-vehicle rollover crashes. The decision to restrict the analysis to single-vehicle crashes is 

to minimize confounding factors and to increase the purity of the analysis. In addition, 
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convertibles and vehicles with a model year earlier than 1987 have been excluded because of the 

lack of (pre-crash) headroom measures. 

Additional restrictions were applied to the data to identify crashes that are most closely related to 

FMVSS No. 216, “Roof crush resistance.” Vehicles that experienced any fixed-object collisions, 

such as a collision with a tree, causing damage to their tops or that rolled only one-quarter turn to 

the side, and thus did not have roof-to-ground contact, were excluded. Both of these situations 

are outside of the scope of the rollovers currently addressed by FMVSS No. 216. Following 

Austin et al. (2005), “fixed object” means CDS classifications of “fixed object,” other than 

shrubbery, bush, or embankment, or CDS classification of “nonbreakaway pole or post,” other 

than ditch, culvert, or ground. Two other restrictions were that the vehicle must not have been 

towing a trailing unit or that it must not be a multistage or certified altered vehicle because 

changes may have been made to the roof structure. 

The occupants of interest for this analysis were seated in one of the two outboard front seats 

(seating positions 11 and 13). The analysis was further restricted to belted occupants who were 

not ejected, either totally or partially, and who were 13 or older. It is believed that the injury 

patterns for ejected (both totally and partially) and unbelted occupants differ substantially from 

those of belted occupants, and their inclusion would complicate the analysis. The primary reason 

for excluding children 12 and under is due to NHTSA recommendation that children 12 and 

under should be placed in the back seat whenever possible. Secondarily, there is uncertainty in 

reasonably estimating the pre-crash headroom for children 12 and under due to variables such as 

child restraint system characteristics. 

Only occupants who are known not to have been ejected were included in the analysis. 

Occupants who were totally ejected, partially ejected, ejected to an unknown degree, or whose 

ejection status is unknown, were not included in the analysis. The two main reasons why only 

non-ejected occupants were included in the analysis are: 

1. This report was done to support prospective rulemaking for FMVSS No. 216. NHTSA is 

currently working under a SAFETEA-LU mandate for ejection mitigation. The ejected 

occupants in rollover will be considered separately in this rulemaking. The Office of 

Management and Budget (OMB) does not permit “double-counting” of safety benefits. 

Ejected occupants are more appropriately considered in the context of the ejection 

mitigation rulemaking. 

2. The report is meant to perform analysis that is comparable to the analysis performed in 

Austin et al. (2005), which does not include partially ejected occupants in its analysis. 

One rationale for only considering restrained occupants was that a reduction in roof intrusion is 

less likely to prevent occupant-to-roof contacts for unrestrained occupants. The consideration of 

restrained occupants only makes the safety estimates more reliable, but also more conservative. 

This is consistent with NHTSA’s best practices. 

The effects of a rollover are different on restrained and unrestrained occupants. Bidez et al. 

(2005), citing other reports, state the following regarding this difference (emphases added): 
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Although some industry representatives still dispute any relationship between vehicle 

design and injury causation in rollover crashes, according to Digges et al. (1994), 

“...significant portions of the HARM (in rollover crashes) are associated with head/brain 

and spinal injuries from contact with the roof and upper interior structure.” Moreover, 

restrained occupants are at higher risk of HARM from roof and rail/header 

contacts than unrestrained occupants. (Digges et al., 1994) “Reductions in the 

severity of contacts which produce injury can be achieved by reducing the relative 

displacement between the occupant and the surface contacted and by providing a 

yielding ‘friendly’ surface.” ... 

Published biomechanical studies suggest that cervical spine preflexion yields a greater 

incidence of lower cervical compression and burst fractures than neutrally positioned 

spines (Pintar et al., 1990, 1995). Restrained occupants in rollover collisions are 

typically pre-flexed, due to the initial locking of their belt restraint system, and thus 

are at higher risk of cervical spine injury in the presence of an intruding roof 

structure when compared to unrestrained occupants. Ford, for instance, recognized 

the increased risk to belted occupants in rollover crashes as early as the 1970s when 

it first began serious investigations into vehicle rollover countermeasures. This higher 

risk of serious head and/or spine injury with roof impact for restrained versus 

unrestrained occupants in rollovers was confirmed with U.S. field accident data 

(Digges et al., 1994).

Finally, another simple reason for only considering restrained occupants is that Austin et al. 

(2005), which this report is building upon, does likewise. 

Post-crash headroom was estimated as the estimated pre-crash headroom minus intrusion. Precrash headroom was estimated from the data obtained from the Consumers Union,1

 adjusted by 

seated occupant height. Following Austin et al. (2005), seated occupant height was estimated as 

0.48 times total occupant height. 

The dependent variable considered in this report is the maximum severity of head, neck, and face 

injuries due to contact with a roof component. Using the NASS CDS occupant injury file, 

identified injuries to the head, neck, or face are identified using the body region variable. The 

intention of using head, neck, and face injuries related to roof contact was to identify injuries that 

were most likely to be affected by a reduction in headroom for belted occupants. While not as 

common as head or neck injuries, facial contact with a roof component, particularly the A and B 

pillars, can occur during a rollover. The injuries include direct contact injuries as well as 

induced or indirect contact injuries.2

 For each injury, NASS CDS codes an injury source from 

the findings of NHTSA’s trained crash investigators. The decision is based upon inspection of 

the vehicle, a review of the medical records, and interviews with surviving occupants. The 

injury was classified as contact with a roof component if the injury source was the A pillar, B 

 

1

 Consumers Union is an independent nonprofit organization whose stated mission is “to test products, inform the 

public and protect consumers.” For more information, see http://www.consumerreports.org . 2

 A direct contact injury results from a force impact directly on the injured body region by the injury source 

component. An indirect contact injury results from a force transmitted from the injury source component through 

another body region to the injured region. For example, an indirect contact neck injury may occur when the force is 

transmitted through the head. 
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pillar, front or rear header, roof side rail, or the roof itself. The A- and B-pillars are included 

because dynamic rollover testing indicates frequent occupant-to-roof impacts at the top of the Aand B-pillars where they connect with the roof. Also in “matchbox” type failures where the roof 

shears sideways, the occupant contact related to the roof crush is often with the vehicle pillars or 

side rail. Injury severity is measured on the Abbreviated Injury Scale (AIS). The AIS is a sevenlevel scale for classifying injuries, based on the threat to life, with higher levels associated with 

more serious injuries. In order of increasing severity, the injury scale is: not injured (0), minor 

(1), moderate (2), serious (3), severe (4), critical (5), and maximum (untreatable) (6). 

When the values of certain variables that are required in the analysis are unknown, the entire 

observation must be removed from the analysis. Following Austin et al. (2005), the sample 

weights of the remaining observations are inflated such that the estimated number of cases does 

not change after the removal of observations. 

When considering the dependent variable of maximum injury severity of all head, neck, and face 

injuries from contact with a roof component, occupants who did not have a roof contact injury to 

the head, neck, or face were assigned an AIS of 0 (uninjured). The distribution of the dependent 

variable is shown in Table 1. For comparison, the table also shows the distribution as reported in 

Austin et al. (2005). 

Table 1. Distribution of the maximum injury severity of all head, neck, and face injuries from contact with a 

roof component. 

 Maximum Severity of Head, Neck, or Face Injury From Roof Contact 

 

Current Report (using CY 1997-2005 

data) 

Austin et al. (2005) (using CY 1997-

2001 data) 

Weighted Weighted 

Weighted Annual Sample Weighted Annual Sample 

 Percent Average Size Percent Average Size

Not Injured 

(AIS 0) 80.96% 159,181 1,907 84.42% 127,633 522

Minor Injury 

(AIS 1) 15.55% 30,574 588 12.20% 18,444 158

Moderate Injury 

(AIS 2) 2.62% 5,145 191 2.78% 4,206 51

Serious Injury

(AIS 3) 0.43% 848 60 0.38% 573 20

Severe Injury

(AIS 4) 0.12% 237 24 0.16% 246 12

Critical Injury 

(AIS 5) 0.05% 96 13 0.02% 36 6

Maximum 

(untreatable) 

Injury (AIS 6) 0.27% 537 56 0.03% 50 4

Total 100% 196,618 2,839 100% 151,187 773

Source: NHTSA, NCSA, NASS CDS 1997-2005; Austin et al. (2005). 

As the table shows, based on the annual average, there were fewer and fewer cases at each higher 

injury severity. The only exception is that at AIS 6, the highest possible injury severity, there 

were more cases than at AIS 5. The proportion of cases at each AIS level as used in this report 
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and as reported in Austin et al. (2005) are very similar. For example, at AIS 3, there was an 

estimated 0.43 percent of the cases according to this report and an estimated 0.38 percent of the 

cases according to Austin et al. (2005). However, the sample size in this report is markedly 

higher than it was in Austin et al. (2005). For example, the sample size for AIS 6 was only 4 

observations in Austin et al. (2005) compared to 56 observations in the current report. The 

sample size for all AIS levels was 773 observations in Austin et al. (2005) compared with 2,839 

observations in this report. 

Methodology. To achieve the report’s purpose, it estimates several statistical models. Since the 

dependent variable is measured on an ordinal scale and has more than two levels, an appropriate 

statistical model for analyzing it is the ordered probit model (also known as the “cumulative 

probit model”). In addition to modeling multiple levels of injury severity, Flannagan et al. (2005) 

suggested also modeling whether the injury was AIS 0 or 1 on the one hand or AIS 2 or greater 

on the other hand. This is done using the binary probit model. The SAS procedure called 

SURVEYLOGISTIC was used to estimate these models. 

While Austin et al. (2005) use cumulative logistic regression in its analysis, there is little 

difference between logistic and probit models, both in theory and in practice. Analyzing data 

with either model produces practically the same results. The logistic model, however, could be 

seen as an approximation of the probit model. Many papers that model injury severity that have 

been published in the scientific literature in recent years use the ordered probit model. These 

papers analyze data from a variety of databases on a variety of crash types that have occurred in 

a variety of localities. Here are a few examples. Khattak et al. (2003) apply the ordered probit 

model to data from the Highway Safety Information System (HSIS) to model injury severity of 

occupants involved in single-vehicle large truck rollovers in North Carolina. Austin and Faigin 

(2003) apply the ordered probit model to data from NASS CDS to investigate occupant injury 

severity across different age groups. Kockelman and Kweon (2002) apply the ordered probit 

model to data from the National Automotive Sampling System General Estimates System (NASS 

GES) to model injury severity in all types of crashes, two-vehicle crashes, and single-vehicle 

crashes. Pai and Saleh (2007) apply the ordered probit model to data from United Kingdom’s 

STATS19 database to model injury severity of motorcycle riders in junction-type crashes. 

Quddus et al. (2002) use the ordered probit model with crash data from Singapore to model 

injury severity in motorcycle crashes. Duncan et al. (1998) use the ordered probit to model injury 

severity of passenger car occupants in rear-end collisions between heavy trucks and passenger 

cars. Abdel-Aty (2003) uses the ordered probit to model driver injury severity in crashes that 

occurred in central Florida. Finally, Zajac and Ivan (2003) use the ordered probit to model 

pedestrian injury severity in crashes that occurred in rural Connecticut. 

As mentioned, the purpose of this report is to see whether a statistically significant relationship 

exists between the severity of relevant injuries and either post-crash headroom or intrusion. 

Therefore, the report uses each one of these as an explanatory variable. They are coded as 

continuous variables, simply stating the amount, measured in inches, as well as dichotomous 

variables. A dichotomous variable is set to equal to 1 if the corresponding continuous variable is 

positive and 0 otherwise. The dichotomous variables are used for two reasons. First, since Austin 

et al. (2005) were able to establish a relationship between injury severity and dichotomous postcrash headroom, and since this report builds on that report, dichotomous variables are of interest. 

Second, the values of the dichotomous variables correspond to important physical events – 
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dichotomous intrusion indicates whether the roof intruded; dichotomous headroom indicates 

whether the roof hit the occupant. 

Some models used only one explanatory variable (intrusion or headroom, whether continuous or 

dichotomous). These are called “unadjusted” as they have not been adjusted for potentially 

confounding factors. “Adjusted” models, on the other hand, control for a number of potentially 

confounding factors. One set of adjusted models control for all the variables that Austin et al. 

(2005) controls for, namely, number of quarter turns to the side (0 in case of an end-over-end 

rollover), whether the rollover was end-over-end, occupant age, occupant sex, and whether the 

vehicle was a passenger car or a light truck or van. 

One reason for using adjusted models is to control for rollover severity. According to Eigen 

(2003), while there is no universally accepted measure of rollover severity, some studies do use 

the number of quarter turns for this purpose. This is what Austin et al. (2005) does, and that is 

why this report uses the number of quarter turns in one set of adjusted models. Eigen (2003) also 

states that rollover severity as experienced by occupants varies with the row in which the 

occupants are seated. Since this report only considers front seated occupants, there is no need to 

control for the row. 

Following Eigen (2005), define the number of roof-to-ground exposures in a sideways rollover 

as the number of times that the vehicle roof faced downward, toward the ground, regardless of 

the number of times that the roof physically contacted the ground. Specifically, in case of an endover-end rollover, the number of roof-to-ground exposures is set to 0. In case of a sideways 

rollover, the number of roof-to-ground exposures is defined as 0.25 times the number of quarter 

turns, rounded to the nearest unit. Thus, 2, 3, 4, and 5 quarter turns correspond to 1 roof-toground exposure; 6, 7, 8, and 9 quarter turns correspond to 2 roof-to-ground exposures, and so 

on. To describe this quantity, defined in terms of quarter turns, other studies, such as Digges and 

Eigen (2003) and Eigen (2005), use words such as “impact” or “contact”. To avoid confusion, 

and to emphasize that a roof exposure is not necessarily the same as an actual physical roof-toground contact, this report uses the word “exposure”. 

Digges and Eigen (2003) and Eigen (2005) state that the number of roof-to-ground exposures is a 

good measure of rollover severity. Specifically, Digges and Eigen (2003) find that “for belted 

occupants and unbelted ejected occupants in single vehicle crashes, the number of roof impacts 

is an appropriate severity indicator.” Since this report considers belted occupants in singlevehicle crashes, according to Digges and Eigen (2003), the number of roof exposures is a good 

measure of rollover severity. Therefore, another set of adjusted models controls for all the 

variables used in Austin et al. (2005), except that instead of using the number of quarter turns to 

the side, they use the number of roof-to-ground exposures. The adjusted models that use quarter 

turns are called “quarter turn adjusted”; the models that use roof-to-ground exposures are called 

“roof exposure adjusted”. 

It might appear that the number of quarter turns variable and the end-over-end indicator might be 

highly correlated and that using both of them as explanatory variables in the same model might 

cause multicollinearity. This is because there is a relationship between the two variables: 

whenever the end-over-end indicator is equal to 0, the number of quarter turns is greater than or 

equal to 2; whenever the end-over-end indicator is equal to 1, the number of quarter turns is 
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equal to 0. In fact, the correlation coefficient between the two variables is just -0.12, which 

means that the variables are not strongly correlated at all and using both of them as explanatory 

variables at the same time would not cause multicollinearity. Likewise, the correlation 

coefficient between the number of roof exposures and the end-over-end indicator is -0.17, which 

means that these two variables are not strongly correlated either and can both be used in the same 

model as explanatory variables without causing multicollinearity. 

Ordered Probit Model 

In this section, the ordered probit is used to model the maximum severity of head, neck, or face 

injury from roof contact as a function of a number of explanatory variables. Suppose a response 

variable has k possible ordered responses. In our case, injury severity has k = 7 possible ordered 

responses, namely, AIS 0 through AIS 6. Let α i be the intercept for category i, β′x be a linear 

combination of coefficients and covariates for a particular case, Φ( ) be the standard normal 

cumulative distribution function, and pi the probability of category i. Then, in the ordered probit 

model, the probabilities of each possible outcome are determined as follows: 

p ( ) +Φ= βα ′x 1 1 , (1) 

p ( )( ) x x i i +Φ= ′ i +Φ− βαβα ′ −1 for 1 << ki , and (2) 

p ( ) x k k +Φ−= βα ′ 1 −1 . (3) 

Note that Austin et al. (2005) use expressions of the form α i − β ′x rather than α i + β′x . Because 

of this, when comparing the estimates of the β coefficients (the coefficients on the explanatory 

variables, not the intercepts) between the two reports, the estimates from one of the reports 

should be negated. As used in this report, a positive coefficient means that as the explanatory 

variable increased, injury severity tended to decrease. An easy way to remember this is that a 

positive coefficient is “good,” as an increase in the variable tended to improve injury outcomes. 

The explanatory variables that are of primary interest are the amount of roof intrusion and postcrash headroom, all measured in inches. Both the dichotomous variables (1 if the value of the 

variable was positive, 0 if it was not) and continuous variables (which simply stated the value of 

the variable in inches) are considered. 

Two types of models are used, “unadjusted” and “adjusted.” The unadjusted models are bivariate 

models in which injury severity is modeled as a function of one of the above explanatory 

variables. The adjusted models are multivariate models which control for a number of potentially 

confounding factors. One set of these adjusted models control for all the variables that Austin et 

al. (2005) controlled for, namely, number of quarter turns to the side (0 in case of an end-overend rollover), whether the rollover was end-over-end, occupant age, occupant sex, and whether 

the vehicle was a passenger car or a light truck or van. Another set of adjusted models control for 

the same variables, except that instead of controlling for the number of quarter turns, they control 

for the number of roof exposures. As mentioned in the previous section, the number of quarter 

turns and the end-over-end indicator were not highly correlated, which means that both could be 
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9

used as explanatory variables in the same model without danger of multicollinearity. The same 

applies to the number of roof exposures and the end-over-end indicator. 

Table 2 shows coefficient estimates for all the models. Each column represents a different 

model; each row represents a different variable. Estimated coefficients that were statistically 

significant at the 0.05 level are shown in bold. Note that the p-values for each of the four 

explanatory variables of interest (continuous and dichotomous intrusion and headroom) were less 

than 0.0001, indicating that both intrusion and headroom were statistically significant. 

Percent concordant and c are two popular measures of association between estimated 

probabilities and observed outcomes. Percent concordant can be anywhere between 0 and 100; c 

can be anywhere between 0 and 1. Higher values of each measure indicate better association. 
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Table 2. Ordered probit models of maximum injury severity for head, neck or face injuries from roof contact. 

 Unadjusted 

 Intrusion 

Post-crash

headroom

 cont dich cont dich 

Intrusion cont (inches) -0.075 

 dich -0.736 

Post-crash

headroom cont (inches) 0.060 

 dich 0.822

Intercepts AIS 0 1.20 1.37 0.85 0.35

 AIS 1 2.21 2.35 1.83 1.36

 AIS 2 2.81 2.93 2.42 1.95

 AIS 3 3.06 3.18 2.68 2.21

 AIS 4 3.18 3.30 2.79 2.33

 AIS 5 3.24 3.35 2.85 2.38

% concordant 58.9 29.7 61.7 36.3

c 0.637 0.583 0.633 0.609

 Quarter turn adjusted Roof exposure adjusted 

 Intrusion 

Post-crash

headroom Intrusion 

Post-crash

headroom

cont dich cont dich cont dich cont dich 

Intrusion cont (inches) -0.078 -0.075 

 dich -0.745 -0.709 

Post-crash

headroom cont (inches) 0.068 0.066 

 dich 0.903 0.873

# quarter turns 

(end-over-end 

= 0) -0.083 -0.085 -0.079 -0.082 

# roof 

exposures

(end-over-end 

= 0) -0.42 -0.42 -0.42 -0.41

End-over-end

(1= yes / 0 = 

no) -0.87 -1.01 -0.86 -0.98 -1.09 -1.21 -1.08 -1.18

Occupant age (years) -0.008 -0.005 -0.010 -0.010 -0.009 -0.006 -0.010 -0.010

Occupant sex

(1 = male / 0 =

female) 0.14 0.10 0.30 0.26 0.11 0.08 0.27 0.23

Vehicle 

(1 = LTV / 0 = 

car) 0.21 0.19 0.17 0.24 0.17 0.15 0.13 0.20

Intercepts AIS 0 1.60 1.70 1.19 0.64 1.83 1.92 1.45 0.89

 AIS 1 2.65 2.72 2.22 1.70 2.89 2.95 2.48 1.95

 AIS 2 3.28 3.34 2.84 2.33 3.53 3.57 3.11 2.59

 AIS 3 3.54 3.59 3.10 2.59 3.79 3.82 3.38 2.86

 AIS 4 3.66 3.71 3.22 2.71 3.91 3.94 3.50 2.97

 AIS 5 3.72 3.77 3.28 2.77 3.97 3.99 3.56 3.03

% concordant 64.3 61.1 64.1 63.8 63.9 60.4 63.7 63.5

c 0.655 0.623 0.652 0.648 0.65 0.618 0.648 0.646

Note: Bolded estimates are statistically significant at the 0.05 level. 

Source: NHTSA, NCSA, NASS CDS 1997-2005. 
 NHTSA’s National Center for Statistics and Analysis, 1200 New Jersey Avenue SE., Washington, DC 20590

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As expected, coefficient estimates for intrusion were negative, indicating that an increase in 

intrusion tended to be associated with an increase in the level of injury severity; the coefficient 

estimates for post-crash headroom were positive, indicating that an increase in headroom tended 

to be associated with a decrease in the level of injury severity. In most cases, for both intrusion 

and headroom, coefficient estimates for the adjusted models were greater in absolute value than 

the estimates for the corresponding unadjusted models. This means that adjusting for potentially 

confounding factors usually increased the estimated effect of intrusion and post-crash headroom 

on injury severity. 

The number of quarter turns, the number of roof exposures, and the end-over-end indicator were 

statistically significant in all of the adjusted models. The estimated coefficients on the number of 

quarter turns and on the number of roof exposures were negative, indicating that the more quarter 

turns or roof exposures, the worse the injury severity, as expected. Digges and Eigen (2003) used 

the CDS database for years 1995 to 2001 to study factors that were associated with injury 

severity in single-vehicle rollovers for belted non-ejected occupants and found the same effect. 

Accounting for the different sign conventions between the two reports, the sign of the coefficient 

on the end-over-end indicator is the same as it is in Austin et al. (2005). An interpretation of the 

relationship between the coefficients on the number of quarter turns and the end-over-end 

indicator is discussed below. 

Occupant age was statistically significant in the quarter turn and roof exposure adjusted 

continuous intrusion, continuous headroom, and dichotomous headroom models. In all cases, the 

coefficient was negative, indicating a tendency toward higher injury severity with increasing age, 

as expected. This is consistent with Austin et al. (2005). Strashny (2007) studies the probability 

of fatality in single-vehicle rollovers using the Fatality Analysis Reporting System and GES 

databases. It finds that higher occupant age was associated with a greater probability of fatality. 

Occupant sex was statistically significant in the continuous and dichotomous headroom models. 

The positive coefficient indicated that, other things being equal (including post-crash headroom), 

males tended to have lower injury severity than females. However, according to Austin et al. 

(2005), males tended to have higher injury severity, though the result was not statistically 

significant. According to the Strashny (2007), males tended to have higher probability of fatality. 

One possible explanation for this result is as follows. Males tend to be taller than females. Thus, 

an equal amount of post-crash headroom implies a generally lower amount of intrusion for a 

male as compared to a female occupant. If injury is primarily associated with the amount of 

intrusion as opposed to the amount of post-crash headroom, then, given the same amount of 

headroom, the generally lower amount of intrusion over the male occupants is associated with a 

lower risk of injury. This explanation is consistent with the fact that, in the intrusion models, the 

occupant sex variable was not statistically significant. 

Finally, vehicle type was only statistically significant in the dichotomous headroom models. The 

positive coefficient indicated that occupants in light trucks and vans (LTVs) tended to have 

lower injury severity than occupants in passenger cars. According to Austin et al. (2005), 

occupants in LTVs tended to have higher injury severity, though the result was not statistically 

significant. According to Strashny (2007), occupants in LTVs tended to have lower probability 

of fatality, agreeing with the current report. 
 NHTSA’s National Center for Statistics and Analysis, 1200 New Jersey Avenue SE., Washington, DC 20590

12

Ten of the 12 models shown in Table 2 had about the same association between estimated 

probabilities and observed outcomes, with percent concordant ranging between about 60 and 65 

and c near 0.65. The two exceptions with lower associations were the unadjusted dichotomous 

intrusion model and the unadjusted dichotomous headroom model. The model with the best 

association, according to both percent concordant and c measures, was the quarter turn adjusted 

continuous intrusion model. 

Recalling Equations 1-3, and given the estimates in Table 2, here are some sample calculations 

of probabilities of the various severities of relevant injury. The first example uses the unadjusted 

continuous intrusion model, the estimates for which are given in the first column with estimates 

in Table 2. To estimate the probabilities using this model, all one needs to know is the amount of 

intrusion, measured in inches. For this example, suppose intrusion was 3 inches. Recall that 

Φ( ) y can be calculated in Excel using the expression =NORMDIST(y,0,1,1). Then, the 

estimated probabilities are as follows: 

y −=−= 226.03*075.0

( )( ) AISp y =+Φ= %6.8320.10

( ) ( )( ) AISp 21.21 y y =+Φ−+Φ= %0.1420.1

( ) ( )( ) AISp 81.22 y y =+Φ−+Φ= %86.121.2

( ) ( )( ) AISp 06.33 y y =+Φ−+Φ= %27.081.2

( ) ( )( ) AISp 18.34 y y =+Φ−+Φ= %07.006.3

( ) ( )( ) AISp 24.35 y y =+Φ−+Φ= %03.018.3

() ( ) AISp y =+Φ−= %13.024.316 . 

The second example uses the quarter turn adjusted continuous intrusion model. To estimate the 

probabilities using this model, one needs to know not only the intrusion, measured in inches, but 

also the values of the other covariates, namely, the number of quarter turns to the side, whether 

the rollover was end-over-end, occupant age, occupant sex, and vehicle type. For this example, 

suppose intrusion was 3 inches, there were 4 quarter turns, the rollover was not end-over-end (0), 

occupant age was 30 years old, occupant sex was male (1), and vehicle type was a car (0). Then, 

the estimated probabilities are as follows: 

y −−−−= −=++ 677.00*21.01*14.030*008.00*87.04*083.03*078.0

( )( ) AISp y =+Φ= %2.8260.10

( ) ( )( ) AISp 65.21 y y =+Φ−+Φ= %4.1560.1

( ) ( )( ) AISp 28.32 y y =+Φ−+Φ= %97.165.2

( ) ( )( ) AISp 54.33 y y =+Φ−+Φ= %26.028.3

( ) ( )( ) AISp 66.34 y y =+Φ−+Φ= %07.054.3

( ) ( )( ) AISp 72.35 y y =+Φ−+Φ= %03.066.3

() ( ) AISp y =+Φ−= %12.072.316 . 

α i + β′x , the expression on which the standard normal cumulative distribution function Φ( ) in 

Equations 1-3 operates, is called the latent response. For instance, as is seen from the sample 
 NHTSA’s National Center for Statistics and Analysis, 1200 New Jersey Avenue SE., Washington, DC 20590

13

calculations, the greater the latent response for i = AIS0 , the greater the probability of no injury 

(AIS 0). Let β Q be the coefficient on the number of quarter turns variable and β E be the 

coefficient on the end-over-end indicator. Hypothetically, suppose two occupants were in two 

relevant rollovers that were identical in every way except that one of them was an end-over-end 

rollover while the other was a sideways rollover with q ≥ 2 quarter turns. Let z be the 

difference in the latent response of the occupant who was in a sideways rollover minus the latent 

response of the occupant who was in an end-over-end rollover. This difference is 

Q qz −= ββ E . 

The difference z illustrates the estimated hypothetical effect on the latent response of having 

changed an end-over-end rollover to a sideways rollover with q quarter turns. For example, 

using the quarter turn adjusted continuous intrusion model, the difference z was estimated to be 

qz +−= 87.0083.0 . 

Other things being equal, a sideways rollover with q quarter turns would have been 

hypothetically equivalent to an end-over-end rollover in terms of injury severity if the difference 

z was 0. This occurs when the number of quarter turns is 

Q

E q β

β =∗ . 

For example, using the quarter turn adjusted continuous intrusion model, q∗ = 10.6 , indicating 

that, other things being equal, a sideways rollover with about 11 quarter turns was hypothetically 

equivalent to an end-over-end rollover in terms of injury severity. Sideways rollovers with fewer 

than q∗

 quarter turns were better than end-over-end rollovers because the difference z was 

positive, indicating a higher probability of no injury in the sideways rollover as compared with 

an end-over-end rollover. Sideways rollovers with greater than q∗

 quarter turns were worse than 

end-over-end rollovers because the difference z was negative. 

For the four quarter turn adjusted models in Table 2, q∗

 ranged from 10.6 to 12.0. Since over 99 

percent of all occupants who were involved in relevant sideways rollovers were in rollovers with 

between 2 and 10 quarter turns, this means that, according to the models, other things being 

equal, an end-over-end rollover was almost always worse than a sideways rollover in terms of 

injury severity. 

The same logic can be applied to the roof exposure adjusted models to determine the number of 

roof exposures that were hypothetically equivalent to an end-over-end rollover. Let β R be the 

coefficient on the number of roof exposures and let r ∗

 be number of roof exposures that were 

hypothetically equivalent to an end-over-end rollover. Then, 
 NHTSA’s National Center for Statistics and Analysis, 1200 New Jersey Avenue SE., Washington, DC 20590

14

R

E r

β

β =∗ . 

For the four roof exposure adjusted models, r ∗

 ranged between 2.6 and 2.9, indicating that, other 

things being equal, a sideways rollover with 2 roof exposures was hypothetically less severe than 

an end-over-end rollover, while a sideways rollover with 3 roof exposures was more severe. 

Binary Probit Model 

This section performs the same analysis as the previous section, except that it uses dichotomized 

injury severity at the AIS 2+ cutpoint. For consistency of coefficient signs between these models 

and those estimated above, probability of an AIS 0 or AIS 1 injury is modeled. Thus, as before, a 

positive coefficient means that increasing the value of the explanatory variable tended to 

decrease injury severity. Results are shown in Table 3. Estimated coefficients that were 

statistically significant at the 0.05 level are shown in bold. 
 NHTSA’s National Center for Statistics and Analysis, 1200 New Jersey Avenue SE., Washington, DC 20590

15

Table 3. Probit models for the probability of maximum injury severity for head, neck, or face injuries from 

roof contact being either AIS 0 or 1. 

 Unadjusted 

 Intrusion 

Post-crash

headroom

 Cont dich cont dich 

Intrusion cont (inches) -0.055 

 dich -0.654 

Post-crash

headroom cont (inches) 0.045 

 dich 0.670

Intercept 2.08 2.28 1.81 1.42

% concordant 59.7 29 62 38

c 0.641 0.584 0.637 0.619

 Quarter turn adjusted Roof exposure adjusted 

 Intrusion 

Post-crash

headroom Intrusion 

Post-crash

headroom

cont dich cont dich cont dich cont dich 

Intrusion cont (inches) -0.056 -0.052 

 dich -0.683 -0.592 

Post-crash

headroom cont (inches) 0.050 

0.046

 

 dich 0.718 0.650

# quarter 

turns

(end-over-end 

= 0) -0.17 -0.17 -0.17 -0.17 

# roof 

exposures

(end-over-end 

= 0) -0.71 -0.71 -0.71

-0.69

End-over-end

(1= yes / 0 = 

no) -1.63 -1.73 -1.62 -1.69 -1.86 -1.94 -1.86

-1.90

Occupant age (years) -0.010 -0.007 -0.011 -0.011 -0.011 -0.008 -0.012 -0.011

Occupant sex

(1 = male / 0 =

female) 0.08 0.03 0.21 0.16 0.01 -0.02 0.13

0.09

Vehicle 

(1 = LTV / 0 = 

car) 0.25 0.24 0.22 0.28 0.19 0.17 0.16

0.21

Intercept 2.98 3.15 2.69 2.23 3.27 3.39 3.00 2.56

% concordant 64 62.1 63.5 64.6 63.5 61.6 62.9 63.7

c 0.651 0.631 0.646 0.656 0.647 0.63 0.642 0.65

Note: Bolded estimates are statistically significant at the 0.05 level. 

Source: NHTSA, NCSA, NASS CDS 1997-2005. 

The p-values for all the intrusion and headroom variables were less than 0.0001. The two 

exceptions were the quarter turn adjusted dichotomous intrusion model, in which the p-value was 

0.0005, and the roof exposure adjusted dichotomous intrusion model, in which the p-value was 

0.0009. This indicates that, in all cases, the intrusion and headroom variables were statistically 

significant. 

The coefficients on the intrusion variables were negative, indicating that higher intrusion was 

associated with a higher probability of an AIS 2+ injury. The coefficients on headroom were 
 NHTSA’s National Center for Statistics and Analysis, 1200 New Jersey Avenue SE., Washington, DC 20590

16

positive, indicating that higher post-crash headroom was associated with a lower probability of 

AIS 2+ injury. 

In the eight adjusted models, the number of quarter turns, the number of roof exposures, whether 

the rollover was end-over-end, and occupant age were statistically significant, whereas occupant 

sex and vehicle type were not statistically significant. The signs on the coefficients are the same 

as in the ordered probit analysis. The only exception is that in one of the models, the coefficient 

on occupant sex is negative. 

Ten of the 12 models shown in Table 3 had about the same association between estimated 

probabilities and observed outcomes. As before, the two exceptions with lower associations were 

the unadjusted dichotomous intrusion model and the unadjusted dichotomous headroom model. 

The coefficients on intrusion and headroom were lower in absolute value in the bilevel models 

shown in Table 3 than in the multilevel models shown in Table 2. Flannagan et al. (2005) found 

the same effect when using the AIS 2+ cutpoint in a quarter turn adjusted dichotomous headroom 

model. 

Below is a sample calculation using the unadjusted continuous intrusion model. For the purposes 

of the example, suppose intrusion was 3 inches. Then, the estimated probabilities are: 

y −=−= 164.03*055.0

( )( ) AISp y =+Φ= %3.9708.21,0

() ( ) AISp y =+Φ−=+ %77.208.212 . 

Conclusion 

This report shows that a statistically significant relationship existed between both vertical roof 

intrusion and post-crash headroom on the one hand and maximum injury severity of head, neck, 

or face injury from roof contact on the other hand. The relationship remained regardless of the 

statistical model used. In particular, the report used both unadjusted and adjusted models, both 

with continuous and dichotomous versions of the explanatory variables, and with multilevel or 

bilevel versions of the dependent variable, and still the relationship remained statistically 

significant. 
 NHTSA’s National Center for Statistics and Analysis, 1200 New Jersey Avenue SE., Washington, DC 20590

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DOT HS 810 847

October 2007