Assessing the Impact of Speed-Limit Increases on Fatal Interstate Crashes

Assessing the Impact of Speed-Limit Increases on Fatal Interstate Crashes

Sandy Balkin
Informed Analytics Group

J. Keith Ord*
Georgetown University

Abstract

This study investigates the relationship between speed-limit increases and increases in the number of fatal crashes on U.S. rural and urban interstates. Past studies use expected historical trends to support claims that "speed kills." Using structural modeling, we assess the change in the average of the time series after a known change in speed limit occurs. The analysis is carried out separately for urban and rural interstates for each state. The results cast doubt on the blanket claim that higher speed limits and higher fatalities are directly related. After the initial speed-limit increases in 1987, the number of fatal accidents on rural interstates increased in some states but not in all. The 1995 round of speed-limit increases generally showed smaller increases in fatalities on rural interstates and slight to no increase on urban interstates. The approach also allows identification of seasonal effects that vary across the states.

Introduction

The relationship between the speed limit and the number of traffic-related fatalities is a subject of great interest to insurance companies, to the government at all levels, and to the general public. Historically, the government has taken an active role in the determination of speed limits, starting with the establishment of the National Maximum Speed Limit (NMSL) by Congress in January of 1974. Prior to this legislation's setting the maximum speed limit at 55 miles per hour (mph), many states posted limits as high as 70 to 75 mph. In April of 1987, Congress passed legislation allowing states to increase speed limits to 65 mph on qualifying sections of interstate highways in rural areas with populations of less than 50,000. Within a few months, 38 states raised the speed limits on appropriate roads. More recently, the National Highway System (NHS) Designation Act of 1995 was signed into law on November 28, 1995. This act ended the federal government's involvement in the establishment of speed limits, putting the responsibility for speed-limit designation and compliance in the hands of the state governments. In most cases, state governments exercised their new rights and raised speed limits on rural and urban interstates.

The purpose of this study is to investigate the relationship between speed limits and traffic-related fatalities. Specifically, we aim to answer the question Does an increase in the speed limit result in a higher incidence of fatal crashes? Using a technique known as structural modeling, we are able to determine the impact speed-limit changes in the past have had on the number of fatal crashes on rural and urban interstates for each state based on its own past experiences. This method also gives information about the seasonal patterns in the number of fatal crashes.

The paper is organized as follows. The next section provides a review of the literature on the effects of speed-limit increases on the number of traffic-related crashes and fatalities. The third section presents the data and methodology used in this study. The fourth section demonstrates the analysis on a single state, and the fifth describes the results of the study for all states. The final section gives conclusions and perspectives for future work.

Literature Review

Various studies have attempted to determine the impact of speed-limit increases on the number of traffic-related crashes and fatalities. The following is a representative selection of the studies that motivated the study presented in this paper.

The report to Congress entitled "Effects of the 65 mph Speed Limit Through 1990" by the U.S. Department of Transportation (USDOT), National Highway Traffic Safety Administration (NHTSA) in May of 1992 looks at yearly interstate-fatality data split by rural and urban roadways. The analysis is based on "expected historical trends" (USDOT NHTSA 1992). These projected counts were derived from statistical models based on the historical relationship between rural-interstate fatalities and fatalities on other roadways. These results do not convey the impact of the speed-limit increase on traffic fatalities. Rather, the study relates interstate deaths to noninterstate deaths; it also assumes a stationary, or nonchanging, environment by fitting a global regression model. The authors then compare fatalities in 1986 with those in 1990 by computing percentage changes. This approach ignores historical trends and possible aberrant observations.

The paper does caution that care ought to be taken when interpreting the data. The authors note that results of individual states probably can not be generalized to the entire nation. They also point out that no statistical model is capable of controlling all factors affecting fatalities.

A 1997 paper entitled "Effect of 1996 Speed-Limit Changes on Motor Vehicle Occupant Fatalities" by Farmer, Retting, and Lund analyzes the effect speed-limit increases during and around 1996 had on interstates. This study employed linear regression models on trend and dummy variables to analyze the number of fatalities in states categorized by the time of their 1996 speed-limit increase (early, late, or none) and compares observed fatalities with projected values based on historical trends. They use percentage change between 1995 and 1996 to assess the impact of the 1996 legislation. They continually note that vehicle-miles traveled (VMT) may be able to explain the increase in fatalities but that the appropriate data are not available.

This study notes that while the national fatality toll for 1996 changed very little compared with 1995, the change in the fatality toll for individual states varied markedly between significant decreases and increases. The study also states that total interstate fatalities increased for the 11 states that had increased speed limits. The authors note that there has been an increase in the portion of the fatalities occurring on roads posted at 55 mph or greater and that some increase in fatalities on interstates is to be expected. Overall, this study presents a very thorough before and after comparison using percentage changes. A linear trend model with an intervention variable is used to compare actual 1996 fatalities with estimated 1996 fatalities based on historical trends. The restriction to annual data and the use of nonadaptive trends limit the value of the comparisons.

"The Effect of Increased Speed Limits in the Post-NMSL Era" is the title of another National Highway Traffic Safety Administration report to Congress (USDOT NHTSA 1998). This 1998 study also investigates the effect of the 1995 to 1996 speed-limit increases on rural and urban interstates.1 It groups states into "changers" (12 count) and "nonchangers" (18 count) where the latter serve as a comparison for the former. The authors modeled the logarithms of fatality counts for each year during 1990 to 1996 as functions of time and type of state. Both linear and quadratic time variables were included. The impact of the speed increase was modeled using a dummy variable equal to one in 1996 and zero in previous years. They also included an interaction term between state group and the 1996 indicator to represent the difference between pre-1996/1996 changes for the two state types while accounting for the time trend. The authors claim that if this interaction term is significant, the 1996 departure from the time trend among the states that increased limits differs from the comparison states. The foremost problem with this analysis is that linear and quadratic trend models are not appropriate for these series. Including a quadratic trend may lower the residual variance for the in-sample fit, but it will damage the predictive ability of the model. Inspecting plots of the number of fatalities or fatal crashes shows that the addition of a global quadratic trend term typically does not provide a reasonable description for the whole length of the series.

Ledolter and Chan's article "Evaluating the Impact of the 65-mph Maximum Speed Limit on Iowa Rural Interstates" (1996) examines whether a significant change in the fatal and major-injury accident rates can be detected following the implementation of a higher speed limit on rural interstates in Iowa. The authors have access to quarterly data on traffic speed, traffic volume, and traffic safety. To answer the posed question, they fit a time-series intervention model relating number of accidents to traffic volume. They also include time trend, intervention variables for the May 1987 change, and quarterly seasonality. The authors find that expected numbers of fatal accidents in Iowa rose by two incidents per quarter on rural interstates, a statistically significant increase.

Data and Methodology2

The data used in the present study are the number of fatal crashes for each month from January 1975 to December 1998 for each state separated by rural and urban interstates. We used fatal crashes rather than number of deaths since we regard the accident data as a more reliable guide to road safety conditions. The number of fatal crashes was determined from the Fatality Analysis Reporting System (FARS) and is publicly available3 and maintained by NHTSA. FARS provides monthly data on numbers of fatal crashes for each state with separate counts for rural and urban interstates.

The database was downloaded in SAS© format. It is possible to query the FARS database for yearly statistics for 1994 to 1998. Since our monthly values sum up to the yearly values reported by the online system, we are confident that we were able to successfully extract the appropriate data. Our yearly totals do not always exactly match the yearly totals given in the studies mentioned previously. These discrepancies can be attributed to the changing of the FARS database structure, to differences in opinion on which roadways were included, or to user error. Again, since our data set matches the online database query totals, we are satisfied with the quality of our data compilation.

We let yt denote the number of fatal crashes that occur in month t and use time series models to examine the impact of an increase in speed limit on the number of fatal crashes. That is, we are mainly concerned with the modeling aspect of time series analysis, looking backwards in time for structural changes in the series. Since the accident data are collected over time in regularly spaced intervals and the timing of speed-limit changes is known, we use intervention analysis to examine these effects.

Intervention analysis is a time series technique used when a change in the environment occurs at a known time and affects the phenomenon of interest.4 In this case, the known change is the speed limit. Since the change in speed limit is more or less permanent, a step intervention is most appropriate. We hypothesize that the change in speed limit results in a permanent shift in the number of accidents. To aid in the analysis and interpretation, we employ the logarithmic transformation. The use of logarithms allows us to consider percentage changes rather than absolute shifts and stabilizes the variance of the series. Since some of the months have zero fatal crashes, it is necessary to add one to each month prior to transforming the data. Thus, it is important to remember when looking at the plots of the data, as in figure 1(a), that the series is shifted up by one unit.

Motivated by some of the previous studies on this topic already discussed, we chose to employ a statistical modeling technique that could provide us with an explanation of the main features of the phenomena under investigation. Harvey and Durbin (1986) used structural time-series modeling to examine the effects of seat-belt legislation on British road casualties. In structural time-series modeling, models are set up explicitly in terms of the components of interest, such as trends, seasonals, and cycles. In addition, instead of assuming that these components remain constant over time, this approach allows them to evolve. The approach is intuitively appealing since environments that generate time series often do not remain constant and an explicit description of how these components change can provide valuable insights.

The starting point for the construction of structural models is to represent an observed value as the sum of level, seasonal, and irregular components.

lowercase y subscript {lowercase t} equals lowercase mu subscript {lowercase t} plus lowercase gamma subscript {lowercase t} plus lowercase epsilon subscript {lowercase t}; lowercase t equals 1 to uppercase t; lowercase epsilon subscript {lowercase t} is normal and identically distributed (mean equals 0, variance equals lowercase sigma superscript {2} subscript {lowercase epsilon}

where yt, as previously defined, is the observation made at time t, which in our case is after a log transformation lowercase y subscript {lowercase t} equals the natural logorithm of (lowercase x subscript {lowercase t} plus 1) and lowercase mu subscript {lowercase t}, lowercase gamma subscript {lowercase t} and lowercase epsilon subscript {lowercase t}are the level, seasonal, and irregular components. In this study, the level component is allowed to change according to a random walk process, and the seasonal component changes according to a trigonometric model. That is,

lowercase mu subscript {lowercase t} equals lowercase mu subscript {lowercase t minus 1} plus lowercase eta subscript {lowercase t}; lowercase eta subscript {lowercase t} is normal and identically distributed (mean equals 0, variance equals lowercase sigma superscript {2} subscript {lowercase eta}

lowercase gamma subscript {lowercase t} equals the summation from lowercase j equals 1 to lowercase s divided by 2 (lowercase gamma subscript {lowercase j t}

where, with s even (equal to 12 in this case) and lowercase lambda subscript {lowercase j} equals (2 times lowercase pi times lowercase j ) divided by lowercase s

1 by 2 matrix where [row 1 column 1 is lowercase gamma subscript {lowercase j t} and row 2 column 1 is lowercase gamma superscript {asterik} subscript {lowercase j t}] equals 2 by 2 matrix where [row 1 column 1 is negative cosine of lowercase lambda subscript {lowercase j}, row 2 column 1 is negative sine of lowercase lambda subscript {lowercase j}, row 1 column 2 is sine of lowercase lambda subscript {lowercase j} and row 2 column 2 is cosine of lowercase lambda subscript {lowercase j}] times 1 by 2 matrix where [row 1 column 1 is lowercase gamma subscript {lowercase j, t minus 1} and row 2 column 1 is lowercase gamma superscript {asterik} subscript {lowercase j, t minus 1}] plus 1 by 2 matrix where [row 1 column 1 is lowercase omega subscript {lowercase j t} and row 2 column 1 is lowercase omega superscript {asterik} subscript {lowercase j t}]; j equals 1 to (s divided by 2) minus 1; lowercase gamma subscript {lowercase j t} equals (cosine of lowercase lambda subscript {lowercase j}) times lowercase gamma subscript {lowercase j, t minus 1} plus lowercase omega subscript {lowercase j t}; j equals s divided by 2

and where the lowercase omega subscript {lowercase j t} times lowercase s and the lowercase omega superscript {asterisk} subscript {lowercase j t} times lowercase s are both normal and identically distributed (mean equals 0, variance equals lowercase sigma superscript {2} subscript {lowercase omega} and are independent of each other. This formulation allows the seasonal effects to vary over time.

It is possible to include a trend component within the level, but no such structure was found in any of the data used for this study, so it was omitted. Finally, in this study we are interested in testing whether a change in the speed limit results in a permanent change in the level of the number of fatal crashes for a given state on a given class of interstate. Thus, we can accommodate this type of analysis by extending the structural model to the form

lowercase y subscript {lowercase t} equals lowercase mu subscript {lowercase t} plus lowercase gamma subscript {lowercase t} plus (lowercase lambda times lowercase z subscript {lowercase t}) plus lowercase epsilon subscript {lowercase t}

where lowercase epsilon subscript {lowercase t}, lowercase eta subscript {lowercase t}, and lowercase omega subscript {lowercase t} are mutually independent of each other, and each has zero mean and constant variance and is also serially independent. Formally, we write this as

lowercase epsilon subscript {lowercase t} is normal and identically distributed (mean equals 0, variance equals lowercase sigma superscript {2} subscript {lowercase epsilon}; lowercase eta subscript {lowercase t} is normal and identically distributed (mean equals 0, variance equals lowercase sigma superscript {2} subscript {lowercase eta}; lowercase omega subscript {lowercase t} is normal and identically distributed (mean equals 0, variance equals lowercase sigma superscript {2} subscript {lowercase omega}

and

covariance (lowercase epsilon subscript {lowercase t}, lowercase eta subscript {lowercase t}) equals covariance (lowercase epsilon subscript {lowercase t}, lowercase omega subscript {lowercase t}) equals covariance (lowercase eta subscript {lowercase t}, lowercase omega subscript {lowercase t}) equals 0 for all lowercase t.

We refer to zt as the intervention variable, defined as:

lowercase z subscript {lowercase t} equals 0 when lowercase t is less than lowercase tau or 1 when lowercase t is greater than or equal to lowercase tau

Thus, zt takes on a value of zero up until time lowercase tau, the month and year of the known speed limit change. The overall fit of the model might be improved by searching for possible interventions rather than pre-specifying their timing. Indeed, there may be a time lag before drivers adapted to the new limits. We decided to retain the more conservative strategy of using the timing of the legal changes and considering only pure level shifts at those times. With regard to potential time lags, the variables defined by equation (5) would differ for only one or two months. If a substantial effect exists, it would still be detected. As for the host of other potential interventions, we preferred to focus solely on the impact of speed-limit changes and to avoid concerns about mining the data. When the component parameters of the structural model are estimated, the intervention parameter lowercase lambda can be used to assess the impact of the speed-limit change. The value 100 times (expontential of (lowercase lambda) minus 1) approximates the percentage increase in the number of fatal crashes after the speed limit was exposed. The exact value is more complex as a result of using the transformation ln(xt + 1) rather than ln(xt); the differences are slight unless the mean level is very low when percentage changes are rather unreliable anyway. The computer package STAMP 5.0, developed by Harvey and his associates, was used to perform the analyses presented in this study.

Example: Rural Arizona

As an example of this method of analysis, consider rural interstates in Arizona. The speed limit was changed in April 1987 and December 1995. Thus, an intervention variable was specified for each of these months, defined as in equation (5). The original time series is shown in figure 1(a).

The series is decomposed into level, seasonal, and irregular components represented graphically in figure 1, panels a, b, and c, after transformation back to the original units. We see a significant increase in the level around 1987 but none around 1995. This indicates that around 1987 the average number of fatal crashes significantly increased, but not so elsewhere. This increase occurs at the same time as a speed-limit increase. Statistically, it is estimated that the 1987 speed-limit increase resulted in a 41% increase in rural interstate crashes in Arizona (see table 1). There is no statistical evidence that the 1995 speed-limit increase had any additional effect on the number of fatal crashes. This may change as more observations become available, better defining the impact of the policy change.

Next, we see from the seasonal component that there appears to be a strong monthly effect on the number of fatal crashes. For this example, there are considerably more crashes in June, July, and August compared with the other months. Such seasonal patterns exist for most states and reflect the higher traffic levels in summer months. The irregular component is simply what is left over after the level and seasonal components are taken into account.

The Structural Time Series Modeling approach tells us that there is a strong seasonal effect on the number of fatal crashes and that there is a significant increase in the number of such crashes around the time the speed limit was changed. For this particular series, the plot of the level component suggests that, after the initial jolt of the speed-limit change, the trend gradually moves back to its original level. This phenomenon was observed for a number of states, but not all. Such a movement would be consistent with the slight overall decline in fatal accidents nationally over this time period, as shown in figure 2. This observation is made tentatively, since partial adjustment effects were neither modeled nor tested. The picture is further complicated as state laws were enacted at different times. Nevertheless, we view this as a question worthy of further exploration since several distinct hypotheses exist, with quite different policy implications. Such hypotheses include 1) drivers adjusted to driving at higher speeds, 2) states increased enforcement of driving laws, and 3) automobile safety was improved. However, we stress that our analysis was not designed to examine these questions; rather, they are important issues for further investigation.

Statistical Analysis

Each state's rural and urban interstates were analyzed using the structural modeling approach with deterministic step intervention variables at the time(s) of the speed-limit increases. Rural interstates are subject to 1987 and 1996 changes, while urban interstates were only changed around 1996. We will refer to the changes around 1987 as the FIRST speed-limit increases and those made around 1996 as the SECOND speed-limit increases.

Results for Individual States

We can summarize the findings as follows:

  • 19 of 40 states experienced a significant increase in fatal crashes along with the FIRST speed-limit increases on rural interstates (figure 3).
  • 10 of 36 states experienced a significant increase in fatal crashes along with SECOND speed-limit increases on rural interstates (figure 4).
  • 6 of 31 states experienced a significant increase in fatal crashes along with the speed-limit increases on urban interstates (figure 5).

Table 1 shows the states with significant changes on rural interstates, the estimated monthly percentage impact of the speed-limit change, and the numbers of fatal crashes in 1986 to 1988. From this table, we can see the monthly percentage increase in the number of fatal crashes attributable to the speed-limit changes. The numbers of total fatal crashes for 1986 to 1988 are included for two reasons: 1) to interpret the percentages in terms of real numbers and 2) to see if the number of fatal crashes increases in the year after the speed-limit change. The purpose behind the first reason is to see, without minimizing the value of human life, what the significant increase translates to in terms of actual number of crashes. For example, suppose a state averages 36 crashes per year, or 3 per month, and the estimated monthly increase of fatal crashes is about 33%. The expected increase in the number of crashes is about one per month. Although statistically significant, such an increase is small in absolute numbers and may be attributable to other factors. The purpose behind the second reason is to assess whether drivers gradually adjust to new driving conditions. For example, Arizona, as graphically displayed in figure 1, had an increase in the number of crashes the year of the speed-limit change but a decrease from that level in subsequent years. This suggests that drivers in Arizona may have learned how to drive safely at the new limit. Such patterns are not consistent across states, and this issue requires further investigation.

Table 2 shows the same information for the urban interstates and also includes 1998 data but includes only states that experienced a statistically significant increase in the number of fatal crashes. During the 1996 set of changes, some states encountered a negative impact, a decline in the number of fatal crashes after the speed-limit increase. While this effect may be real, it is difficult to attribute it to the increase in speed limits. Therefore, the results are not included in table 2.

In order to get an idea of how many fatal crashes are associated with a particular speed-limit increase, we first remove from the fitted model the term relating to the increase for those states that had a significant increase in fatal crashes. We then analyze the difference between the modified expected and actual numbers. We approximate the predicted number of fatal crashes had the speed limit not increased by dividing the observed number of fatal crashes by one plus the percent change. Tables 3 and 4 show this information separated by rural and urban interstates. We see that the estimated overall percentage increases are of the same order as the individual increases, resulting in approximately an additional 200 rural and 80 urban fatal crashes per year. It is important to note that these numbers only represent a crude approximation of the effect of the speed-limit increase.

Seasonality

One of the powerful benefits of using structural modeling is that instead of removing seasonality, the effect of a specific month is directly modeled. The strength of the seasonal pattern was one of the most surprising aspects of this analysis. Figures 6 and 7 show the following:

  • 29 states exhibited seasonality at the 0.05 level of significance on rural interstates (figure 6)
  • 18 states exhibited seasonality at the 0.05 level of significance on urban interstates (figure 7).

The extent of seasonality varies by state. Most states typically have a higher number of fatal crashes in August. Some states have different patterns with interpretations unique to that state. For instance, Florida tends to have more fatal crashes in March on its urban interstates. One possible interpretation of this could be the increase of traffic from college students traveling to Florida on spring break. In general, seasonal peaks appear to coincide with peak holiday seasons. Most states do not produce monthly data on vehicle-miles traveled, so we cannot adjust the data in a consistent manner for such effects.

Aggregate Analysis

Though the analysis is by state, it is of interest to generalize the effect of speed limit increases to the nation as a whole. To answer this question, we use a "Super t-Test." We first record the t-values of the intervention variables for all states. Positive t-values indicate a positive impact (increase) of the number of fatal crashes. Of fatal crashes, they determine the significance of the individual impact of the policy change. To answer the question whether or not fatal crashes increase along with speed-limit increases, we then perform a one-sided t-test to determine whether the mean of the t-values of all of the intervention variables is significantly greater than zero. If we reject the null hypothesis, we can conclude that there is indeed an increase in the number of fatal crashes. It does not tell us, however, how large this increase is, only if, on average, an effect exists.

The Super t-Test for Rural Interstates resulted in a t-statistic of 10.6 with 39 degrees of freedom (one-tailed p-value ≈ 0.000) for the FIRST set of speed-limit changes and a t-statistic of 4.0 with 36 degrees of freedom (one-tailed p-value = 0.0002) for the SECOND set of speed-limit changes. For urban interstates, the Super t-test gave a t-statistic of 1.373 with 30 degrees of freedom (one-tailed p-value = 0.090). We see from the Super t-Tests that rural interstates appear to be affected by speed-limit increases, while the effect for urban interstates is weak.

Conclusion and Future Work

The purpose of the study is to investigate the relationship between speed limits and traffic-related fatalities. Specifically, we sought to discover if an increase in the speed limit results in a higher incidence of fatal crashes.

We carried out the data analysis using a time-series technique known as structural modeling. This approach enables us to partition a series into its level, trend, seasonal, and irregular (or residual) components and to evaluate the impact of major interventions such as speed-limit changes. Based on a review of the past literature, we formulated the impact of a speed-limit change as a one-time percentage increase in the number of accidents, after which the seasonal and trend patterns in the series would be expected to remain similar to those of past years. The analysis was performed for each state, separately for urban and rural interstates. Although the results are statistically significant as noted above, the numbers in some states may be small. The seasonal patterns probably reflect changes in the number of vehicle-miles traveled (VMT), with peaks occurring during holiday seasons. Seasonal analysis is critical to understanding any changes in pattern since unadjusted comparisons for a few months immediately before and after a change could be seriously in error. Our analysis allows comparisons to be made after proper adjustment for seasonal effects. Overall, increases were seen in some states following speed-limit changes. These increases were predominantly on rural rather than urban interstates.

Acknowledgments

We wish to acknowledge the sponsorship of the Consumers Union for this research. The views expressed in this paper are those of the authors and are not to be attributed in any way to the Consumers Union. We also wish to thank Tom Wassel of AstraZeneca, Eric Falk of Ernst & Young, and Eric Rosenberg of Consumers Union for data collection and analysis assistance.

References

DeLurgio, S.A. 1998. Forecasting Principles and Applications. Boston: Irwin McGraw-Hill.

Farmer, C.M., R.A. Retting, and A.K. Lund. 1997. Effect of 1996 Speed-Limit Changes on Motor Vehicle Occupant Fatalities. Insurance Institute for Highway Safety, Washington, DC.

Harvey, A. and J. Durbin. 1986. The Effects of Seat Belt Legislation on British Road Casualties. Journal of the Royal Statistical Society, Series A 149:187-227.

Kendall, M. and J.K. Ord. 1990. Time Series. London: Edward Arnold.

Ledolter, J. and K.S. Chan. 1996. Evaluating the Impact of the 65-mph Maximum Speed Limit on Iowa Rural Interstates. The American Statistician 50, 1.

U.S. Department of Transportation (USDOT), National Highway Traffic Safety Administration (NHTSA). 1992. Effects of the 65-mph Speed Limit Throughout 1990. Report to Congress.

_____. 1998. The Effect of Increased Speed Limits in the Post-NMSL Era. Report to Congress.

Address for Correspondence and End Notes

J. Keith Ord, Georgetown University, McDonough School of Business, 320 Old North, Washington, DC 20057. Email: ordK@msb.edu.

1 Specifically, only states with increases between December 8, 1995 and April 1, 1996 are considered.
2 Further information regarding the data and their collection is available from the author.
3http://www-fars.nhtsa.dot.gov/
4 See chapter 13 of Kendall and Ord (1990) or chapter 12 of DeLurgio (1998) for a description.