category: literaturenote citekey: aljanahispeedspeedlimitsroad1999 title: "Speed, speed limits and road traffic accidents under free flow conditions" authors: "Aljanahi, A.A.M; Rhodes, A.H; Metcalfe, A.V" year: 1999 date: 1999-00-00 1999 doi: 10.1016/s0001-4575(98)00058-x publication: "Accident Analysis & Prevention" url: "https://doi.org/10.1016/s0001-4575(98)00058-x" zotero_key: CQF3IAWG zotero_storage: 5J8U3XUT collections: transpordi_planeerimine folder: 001_artiklid/Road Safety Indicators firstAuthor: "Aljanahi, A.A.M"

status: converted

Speed, speed limits and road traffic accidents under free flow conditions

A.A.M. Aljanahi ^a , A.H. Rhodes a,1, A.V. Metcalfe b,*

^a *Department of Ci*6il Engineering, *Uni*6ersity of Newcastle upon Tyne, NE¹ ⁷RU, UK ^b Department of Engineering Mathematics, *Uni*6ersity of Newcastle upon Tyne, NE¹ ⁷RU, UK

Received 8 May 1998; received in revised form 8 August 1998; accepted 19 August 1998

Abstract

The relationship between various measures of traffic speed, under free flow conditions, and accident rate is investigated for two groups of sites, one in the Tyne and Wear county of the UK and the other in Bahrain. The effect of speed limits on traffic speed is estimated for both groups of sites. In Bahrain, there is statistically significant evidence of an association between mean speed and accident rate. In Tyne and Wear the statistical evidence is weaker, and points to a stronger relationship between accidents and the variability of traffic speeds. In both areas, there is an apparent decrease in accident rate if the percentage of heavy vehicles increases, with the speed distribution held constant. In both areas the effect of speed limits is to reduce the mean speed of traffic by at least one quarter. Higher speeds are associated with longer trips. © 1998 Elsevier Science Ltd. All rights reserved.

Keywords: Free flow conditions; Traffic speed; Accident rate

1. Introduction

The main benefit of high speed traffic flow is that the cost attributed to travel time is reduced, despite higher fuel and other running costs together with the associated higher environmental costs (Aljanahi, 1995). However, this advantage may be offset by a possible increase in the number of accidents and the fact that injuries sustained are likely to be more severe if accidents occur at higher speeds.

Road traffic accidents are a result of many factors, although excessive speed is often thought to be a major cause. Speed limit signs were introduced as a safety measure; and the relationships between the speed characteristics of traffic, speed limits, and accidents have been debated each time limits have been changed (e.g. Solomon, 1964; Munden, 1967; Brenac, 1990; Finch et al., 1994). Other contributory factors include: road design; traffic volume; weather; reasons for travelling; drivers under the influence of alcohol and drugs; economic pressures on drivers to save time; training of drivers; and road safety publicity campaigns. Fortunately, the frequency of road traffic accidents is decreasing, in the UK at least, despite an increasing number of vehicle registrations (Department of Transport, 1995). There are many explanations for this and any longitudinal investigation of changes in traffic speed characteristics and accident frequencies is likely to give misleading results. Instead, different road links (sites) were compared over the same period. The sites were selected from the County of Tyne and Wear in England over a five year period (1988–1992), and State of Bahrain over a four year period (1987–1990). It was thought that the two areas were sufficiently diverse to allow generalisation of any common trends.

2. Method

².1. Scope of study

The first objective was to investigate any relationship between traffic speed characteristics and personal injury accidents, on high quality road links operating under free-flow traffic conditions. The second objective was to

^* Corresponding author. Tel.: +44-191-222-6233; Fax: +44-191- 222-5498; e-mail: andrew.metcalf@newcastle.ac.uk.

¹ Dr Rhodes, who was Head of Transport Engineering at the University of Newcastle upon Tyne, died in July 1998.

Fig. 1. Hourly mean speed (mph) and flow of traffic (vehicles per h) against time of day, for a Tyne and Wear site with LOSB critical flow of 1680 vehicles per h. Records from three days.

establish a relationship between speed limits and the distribution of traffic speeds. Links were chosen so that features, other than speed limits and the traffic speed characteristics, were as similar as possible. Differences in accident rates could then be compared with the speed characteristics of the traffic, other factors being assumed constant or inconsequential. It would have been preferable to have allocated speed limits to links at random (Spitz, 1984), but such a procedure was quite impractical for this study.

².2. Criteria for site selection

The intention was to select sites with similar physical and environmental characteristics, and then to compare accident records under free-flow traffic conditions. The following criteria were set for the selection of highway links:

1. horizontal curves of less than 4° and longitudinal gradients of up to 3%;
2. link accidents only (no accidents to be considered within 20 metres of a junction);
3. dual-carriageways;
4. good pavement surface conditions;
5. no bus-stops or pedestrian crossings;
6. low land use activity and
7. traffic moving under free-flow conditions throughout the 24-hour day.

Free-flow conditions were defined as traffic flows less than the service flow for Level of Service B (LOSB). This is taken to represent the greatest service flow at which drivers have an unrestricted choice of manoeuvre. The service flow at LOSB was calculated from the following formula (National Research Council, 1985): MSF is the maximum flow rate per lane per hour for passenger cars; N is the number of lanes and the various *f*s are adjustment factors:

service flow = MSF $$\times N \times fw \times f{HV} \times f_p$$

where fw is an adjustment for the effect of restricted lane widths, *f*HV is an adjustment for the number of trucks, buses and recreational vehicles in the traffic stream and fp is an adjustment for the mix of drivers in the population. The National Research Council report provides numerical values for the MSF and the three factors. The free flow criterion was tested by comparing speed, averaged over each hour of the day, with traffic flow during the hour. No statistically significant relationships were found between speed and flow, although the latter occasionally exceeded the critical value at some sites. For example, the LOSB service flow for the Tyne and Wear site for which vehicle flow and speed is plotted in Fig. 1 was calculated as 1680 vehicles per h. Nine suitable dual carriageway sites were identified in Tyne and Wear, and ten in Bahrain.

Table 1 Personal injury accidents (PIA) and traffic characteristics of the sites investigated in Tyne & Wear (TW) and Bahrain (BO)

Sites variables	TW1	TW2	TW3	TW4	TW5	TW6	TW7	TW8	TW9
Mean speed of traffic (km/h)	110.56	107.82	83.52	93.50	78.86	77.46	72.90	72.26	63.89
Standard deviation	14.77	18.84	13.12	17.06	10.91	13.52	10.35	11.75	6.76
Seventh percentile speed	86.18	79.63	64.73	70.33	66.14	59.30	59.66	58.87	53.90
Fifteenth percentile speed	91.68	87.03	69.04	75.83	68.40	63.92	62.83	61.72	58.27
Fiftieth percentile speed	107.90	107.26	81.83	89.27	77.91	77.75	71.63	71.77	63.66
Eighty-fifth percentile speed	125.03	129.15	97.12	114.45	89.14	90.49	83.04	64.68	71.39
Ninety-third percentile speed	131.50	136.65	103.59	123.48	96.59	95.95	88.22	89.30	73.55
5 vehicles/year) Traffic flow (10	39.63	50.00	47.67	46.91	33.33	52.51	44.78	54.21	24.45
Length of the section (km)	44.00	18.00	7.00	0.64	1.16	0.47	0.62	1.05	1.03
Annual mean number of PIA	28.40	18.80	7.40	0.80	0.60	0.60	0.40	0.40	0.20
Speed limit (mph)	70.00	70.00	70.00	50.00	50.00	70.00	40.00	50.00	40.00
% Heavy vehicles (%)	18.40	6.10	4.30	7.70	10.00	11.00	7.00	7.70	10.00
Trip length	4.00	3.00	2.00	3.00	3.00	2.00	3.00	2.00	2.00
Sites Variables	BO1	BO2	BO3	BO4	BO5	BO6	BO7	BO8	BO9	B10
Mean speed of traffic (km/h)	87.80	82.50	80.80	80.04	72.27	61.20	75.50	73.30	72.80	60.10
Standard deviation	15.70	14.80	13.90	14.12	15.28	9.60	16.00	13.80	18.40	10.10
Seventh percentile speed	64.30	61.68	60.48	58.58	52.12	46.68	52.26	52.00	43.97	45.17
Fifteenth percentile speed	71.36	67.32	58.95	63.76	56.78	51.09	59.80	61.00	51.16	49.71
Fiftieth percentile speed	87.21	82.08	80.99	77.08	72.43	61.30	76.16	73.97	73.77	59.57
Eighty-fifth percentile speed	105.23	98.39	95.38	90.52	89.21	71.16	90.30	87.43	91.06	70.60
Ninety-third percentile speed	112.17	106.04	101.09	98.36	96.85	75.31	97.80	92.07	100.04	75.61
5 vehicles/year) Traffic flow (10	64.49	31.72	64.53	69.20	90.54	42.54	53.75	43.13	28.79	11.82
Length of the section (km)	10.67	6.27	3.83	1.52	0.80	0.55	1.90	1.05	2.59	0.82
Annual mean number of PIA	30.25	12.25	6.50	4.25	1.25	1.00	2.25	2.00	1.75	0.50
Speed limit (km/h)	100	100	80	80	80	70	70	70	50	50
% Heavy vehicles (%)	15.70	3.50	15.70	17.70	15.90	21.20	14.00	6.40	7.50	6.40
Lanes	3	2	3	3	3	2	2	2	2	2
Trip length	4.00	3.00	4.00	4.00	3.00	2.00	4.00	3.00	4.00	2.00
Condition	1.00	1.00	1.00	2.00	2.00	4.00	4.00	3.00	3.00	3.00

².3. Data collection

The period over which personal injury accident records were examined had to be long enough to provide sufficiently precise estimates of accident rates, but short enough for current vehicle speeds to be relevant to the comparisons between sites. A five-year period was chosen for Tyne and Wear, but computer records were only available over four years in Bahrain. Some of the sites under observation were closer to urban areas than others, and pedestrian accidents were more likely to occur near urban areas. In the attempt to make comparisons across sites as fair as possible, accidents involving pedestrians have been excluded from the analysis.

The accident data for Tyne and Wear were acquired from the Traffic Accident Data Unit (TADU), Gateshead Metropolitan Borough Council and the Highways Department, Durham County Council. The data were from the five year period, 1988–1992. In Bahrain, the accident data were collected from the Planning and Organisation Division, Traffic and Licensing Department, Ministry of Interior. The data were from the four year period 1987–1990.

No records of the speed characteristics of the sites were available. As a substitute, speed characteristics were estimated at the start of the study using pneumatic sensors laid across the road. Speeds were recorded continuously for several days at each of the sites, and the summary statistics were found to be consistent from day to day. The mean values of these traffic speed characteristics were taken to represent mean values for the time span of the accident records. It was assumed, therefore, that any changes, such as those accounted for by applying a growth rate, in the mean speed of traffic were similar for all the sites.

Traffic counts are much more common than surveys of vehicle speeds. In Tyne and Wear the annual average daily traffic (AADT) counts were available for much the same period as the accident records. The few missing years were interpolated using the local growth rate of traffic. There were no traffic flow records for Bahrain, so traffic counts were made at the same time as the speed recordings. The relevant data are summarised in Table 1 (a and b). Nine or ten sites may seem a small number for a statistical analysis; but the careful matching of sites that could reasonably be assumed to be under free flow conditions, and data acquisition, took a considerable amount of time and effort.

3. Analysis of occurrences of accidents

The total numbers of accidents during the five, or four, year period were initially assumed to have a Poisson distribution, with a mean which depends on the features of the site (e.g. Johansson, 1996).

3.1. Analysis for Tyne and Wear

The following form of model was proposed for the mean (m, accidents per 5 years) of the Poisson distribution:

$$\mu = k L F^a H^b S^c$$

where L is the length of road at the site (km); F is the traffic flow over the site (105 vehicles per year); H is the percentage of traffic classified as heavy vehicles; S is some measure of speed; and a, b, c and k are parameters to be determined from the data. It was assumed that the accident rate was proportional to the length of road being monitored. The statistical software GLIM (NAG, 1998) is a convenient means of estimating the parameters by maximum likelihood. In the logarithmic form of the model,

$$ln(\mu) = ln(k) + ln(L) + a ln(F) + b ln(H) + c ln(S).$$

The parameters a, b, c appear as coefficients of the variables and the logarithm of the parameter k is an intercept. The variable ln(L) has a known coefficient, 1 in this application, and is referred to as an offset in GLIM. The relationship between the rate of accidents and the traffic flow was more difficult to specify. Although the number of vehicles at risk is proportional to traffic flow, the number of accidents does not necessarily increase linearly with traffic flow as, for example, drivers may be more careful in denser traffic. Mountain et al. (1996) estimated that accidents on a sample of road links in England increased in proportion to traffic flow raised to the power of 0.64. They excluded any junction accidents in this part of their analysis and the exponent of traffic flow was significantly less than one (*P*B0.01). However, their investigation was not restricted to links under free flow conditions. A more pessimistic approach is to argue that: each vehicle could be involved in an accidents with each neighbouring vehicle, that the number of accidents is approximately proportional to the number of pairings, and hence that a could be as high as 2. An assumption that accidents are proportional to flow (i.e. a=1) may be reasonable for free flow conditions on dual carriageways (UK Department of Transport, 1981), and the analyses were repeated with the logarithm of F treated as an offset (i.e. a set at 1). Further analyses were carried out using standard deviation, and various measures of speed which combine mean and variability, as the speed variables. The following measures are commonly used in the literature: the upper 85 and 93% quantiles; the sum of the mean and the standard deviation; the square root of the sum of the squares of the mean and the standard deviation, which is the root mean squared speed; the coefficient of upper speed (CUSS) defined as (85−50% quantile)/(50% quantile); and the skewness index (SI) defined as 2×(93−50% quantile)/(93−7% quantile). Results for a selection of speed measures are summarised in Table 2 (a).

The results with the logarithm of length as an offset are qualitatively similar to those with the logarithms of length and flow included as offsets (i.e. a=1). If a is estimated, the evidence against a hypothesis that it equals one is not statistically significant (since the difference between the estimate and 1 is always less than its standard error). The apparent consequence of an increased proportion of heavy vehicles is that the accident rate decreases. This is statistically significant at the 5% level, if a is assumed equal to 1 for analyses using any of the speed measures except the standard deviation and CUSS, because there is no evidence of overdispersion relative to a Poisson variable (the deviance is actually less than its expected value which equals the degrees of freedom) and the estimate of b exceeds twice its standard error (which is based on an assumption of Poisson variability). If the standard deviation is used as a measure of speed the estimate of b is still statistically significantly different from 0 at a 10% level, and if CUSS is used the estimate of b is still negative, though not statistically significantly so. Although the statistical significance of the evidence of an association between an increase in the proportion of heavy vehicles and a reduction in the accident rate may be equivocal for the Tyne and Wear study, considered in isolation, similar findings for the study of sites in Bahrain provide additional evidence for such an association. Possible explanations for the reduction are that heavy vehicles are driven by professional drivers, and that the presence of heavy vehicles increases the concentration of drivers of other vehicles. The influence of the speed measures on the accident rate is not statistically significant at a 5% level but the overall tendency is for increased mean speed, and variability in speed, to be associated with an increase in the accident rate. The estimated coefficients are, generally, similar to their standard errors so there is weak statistical evidence of a reduction in accidents when the speed decreases. Although the estimate of the reduction is imprecise, it is large enough to be of potential practical importance and to justify further studies. For example, the measured upper 93% points (U93) of the speed distribution ranges from 74 to 137

Table 2 Results of GLIM Analyses of Accident Occurrences for Tyne and Wear

| |--------------------------|-- | Speed measure S | Tyne and Wear | | Mean | | a | | b | | c | | Standard deviation | | a | | b | | | | c | | Mean+standard deviation | ln(k) | | a | | b | | c | | 93% quantile | | a | | b | | c | | Root mean square | | a | | b | | c | | CUSS | | | 85–50% | a | 50% | b | | c | | SI | 2(93–50%) | a | (83–7%) | b | | c | | Bahrain | | Mean | | a | | b | | c | | Standard deviation | | a | | b | | c | | Mean+ standard deviation | ln(k) | | a | | | | b | | c | | 93% quantile | | a | | b | | c | | Offset | | | | | | | | | ----------|-------------------------------------|----------|----------------------------------------------------------|---------|----------|--|--|--| | ln(Length) | Parameter estimate (standard error) | | ln(Length × Flow)
Parameter estimate (standard error) | | | | | | | | | | | | | | | | ln(k) | −4.513 | (3.696) | ln(k) | −4.850 | (2.611) | | | | | 0.874 | (1.034) | | | | | | | | −0.307 | (0.251) | b | −0.280 | (0.117) | | | | | 0.713 | (0.708) | c | 0.666 | (0.591) | | | | | Deviance | 3.580 | (5 d.f.) | Deviance | 3.600 | (6 d.f.) | | | | | ln(k) | −1.824 | (4.026) | ln(k) | −3.134 | (1.173) | | | | | 0.600 | (1.187) | | | | | | | | −0.236 | (0.206) | b | −0.176 | (0.105) | | | | | | | | | | | | | | 0.544 | (0.543) | c | 0.419 | (0.390) | | | | | Deviance | 3.620 | (5 d.f.) | Deviance | 3.730 | (6 d.f.) | | | | | −4.342 | (3.654) | ln(k) | −4.775 | (2.568) | | | | | 0.827 | (1.057) | | | | | | | | −0.296 | (0.243) | b | −0.260 | (0.109) | | | | | 0.688 | (0.683) | c | 0.625 | (0.556) | | | | | Deviance | 3.580 | (5 d.f.) | Deviance | 3.610 | (6 d.f.) | | | | | ln(k) | −4.309 | (3.644) | ln(k) | −4.778 | (2.543) | | | | | 0.813 | (1.059) | | | | | | | | −0.283 | (0.233) | b | −0.246 | (0.104) | | | | | 0.677 | (0.662) | c | 0.611 | (0.537) | | | | | Deviance | 3.550 | (5 d.f.) | Deviance | 3.580 | (6 d.f.) | | | | | ln(k) | −4.474 | (3.686) | ln(k) | −4.815 | (2.595) | | | | | 0.867 | (1.038) | | | | | | | | −0.305 | (0.249) | b | −0.276 | (0.115) | | | | | 0.354 | (0.352) | c | 0.329 | (0.292) | | | | | Deviance | 3.580 | (5 d.f.) | Deviance | 3.600 | (6 d.f.) | | | | | ln(k) | 0.320 | (5.468) | ln(k) | | (1.022) | | | | | | | | −0.877 | | | | | | 0.757 | (1.089) | | | | | | | | −0.088 | (0.185) | b | −0.070 | (0.169) | | | | | 0.918 | (0.965) | c | 0.782 | (0.757) | | | | | Deviance | 3.830 | (5 d.f.) | Deviance | 3.870 | (6 d.f.) | | | | | ln(k) | −3.238 | (3.727) | ln(k) | −1.842 | (0.288) | | | | | 1.328 | (0.874) | | | | | | | | −0.161 | (0.195) | b | −0.223 | (0.105) | | | | | −0.451 | (1.658) | c | −0.657 | (1.570) | | | | | Deviance | 4.600 | (5 d.f.) | Deviance | 4.750 | (6 d.f.) | | | | | | | | | | | | | | ln(k) | −13.160 | (4.225) | ln(k) | −12.494 | (4.193) | | | | | 0.531 | (0.492) | | | | | | | | −0.129 | (0.257) | b | −0.353 | (0.108) | | | | | 3.062 | (1.087) | c | 2.609 | (0.960) | | | | | Deviance | 9.500 | (6 d.f.) | Deviance | 10.400 | (7 d.f.) | | | | | ln(k) | −0.562 | (2.063) | ln(k) | −0.201 | (1.855) | | | | | 1.187 | (0.372) | | | | | | | | −0.388 | (0.216) | b | −0.293 | (0.106) | | | | | −0.397 | (0.710) | c | −0.340 | (0.677) | | | | | Deviance | 18.050 | (6 d.f.) | Deviance | 18.310 | (7 d.f.) | | | | | −12.581 | (4.756) | ln(k) | −12.260 | (4.763) | | | | | 0.736 | (0.462) | | | | | | | | | | | | | | | | | −0.214 | (0.248) | b | −0.342 | (0.108) | | | | | 2.690 | (1.128) | c | 2.457 | (1.050) | | | | | Deviance | 12.170 | (6 d.f.) | Deviance | 12.490 | (7 d.f.) | | | | | ln(k) | −10.932 | (4.404) | ln(k) | −10.760 | (4.426) | | | | | 0.800 | (0.443) | | | | | | | | −0.219 | (0.244) | b | −0.318 | (0.107) | | | | | 2.239 | (1.015) | c | 2.082 | (0.955) | | | | | Deviance | 13.200 | (6 d.f.) | Deviance | 13.400 | (7 d.f.) | | | |

Table 2 (Continued)

| Speed measure S
Root mean square | Offset | | | | | | | | | |-------------------------------------|------------|-------------------------------------|----------------------------------------------------------|----------|---------|----------|--|--|--| | | ln(Length) | Parameter estimate (standard error) | ln(Length × Flow)
Parameter estimate (standard error) | | | | | | | | | ln(k)
a | −13.090
0.579 | (4.335)
(0.486) | ln(k) | −12.500 | (4.312) | | | | | | b | −0.150 | (0.255) | b | −0.351 | (0.108) | | | | | | c | 3.003 | (1.100) | c | 2.600 | (0.984) | | | | | | Deviance | 10.090 | (6 d.f.) | Deviance | 10.840 | (7 d.f.) | | | | | CUSS | ln(k) | −1.598 | (1.658) | ln(k) | −1.115 | (1.231) | | | | | 85–50% | a | 1.166 | (0.375) | | | | | | | | 50% | b | −0.378 | (0.219) | b | −0.293 | (0.107) | | | | | | c | −0.012 | (0.766) | c | 0.005 | (0.761) | | | | | | Deviance | 18.360 | (6 d.f.) | Deviance | 18.560 | (7 d.f.) | | | | | SI | ln(k) | −0.573 | (1.013) | ln(k) | −1.364 | (0.282) | | | | | 2(93–50%) | a | 0.695 | (0.377) | | | | | | | | (83–7%) | b | −0.069 | (0.234) | b | −0.236 | (0.107) | | | | | | c | 3.943 | (1.642) | c | 3.360 | (1.457) | | | | | | Deviance | 12.220 | (6 d.f.) | Deviance | 12.830 | (7 d.f.) | | | |

over the nine sites. If the flow is set at its average value of 44×10⁵ vehicles passing over the link in a year, the percentage of heavy vehicles is taken as 10%, and the section length is 1, the estimates of m are 2.8 and 4.2 when the U93 equals 74 and 137, respectively. This corresponds to a reduction in accidents of more than 30%.

3.2. Analysis for Bahrain

The model and analysis were the same as for Tyne and Wear except that m is now measured in accidents per four years. The results are given in Table 2 (b). As for the Tyne and Wear sites, there is no statistically significant evidence against a hypothesis that the mean accident rate is proportioned to the flow, and the conclusions that can be drawn from the study are insensitive to whether a is estimated, or assumed to equal 1. An increase in mean speed is associated with an increase in accidents. The deviance is slightly higher than would be expected with a Poisson distribution (e.g. 9.50) relative to an expected value of 6. Although this is quite common in analyses of road traffic accidents (Nicholson, 1985; Nicholson and Wong, 1993) it is not statistically significant. An approximate 90% confidence interval for c, allowing for possible over dispersion (Maher and Summersgill, 1996), is given by:

$$3.062 \pm t_{6,0.05} \times 1.087 \times \sqrt{(9.50/6)}$$ .

Which gives [0.40, 5.72]. Even the lower limit of 0.40 corresponds to a predicted 9% reduction in the accident rate if the mean speed on the link with the fastest traffic is reduced from 88 to 70 km/h. A value of c equal to 2 predicts a reduction of nearly 40%. Higher values of c are perhaps unrealistic from a practical point of view. The Bahrain accident rate of 105 vehicle per kilometre year is estimated as 0.040, which is twice the estimate for Tyne and Wear (0.018).

4. Analysis of speed limits and speed

Speed limits are the usual means of controlling vehicle speeds under free flow conditions on high speed road links (e.g. Newby, 1970; O'Flaherty, 1986). Their effectiveness depends, partly, on the level of policing and the size of fines. In this section the relationship between speeds, speed limits and other variables is described for the nine Tyne and Wear and ten Bahrain sites in 1992.

⁴.1. Analysis for Tyne and Wear

A trip length variable was assigned to each site. This was based on road-side surveys which are carried out by the local authorities, typically on a four or five year cycle. The value of the trip length variable was set on a scale from 1 up to 5 according to:

1. 25% or less of trips were through traffic;
2. between 25 and 45% of trips were through traffic;
3. between 45 and 55% of trips were through traffic;
4. between 55 and 85% of trips were through traffic;
5. more than 85% of trips were through traffic.

The following multiple regression models were fitted to the mean speed (MS) and upper 85% point of the distribution of speeds (U85).

$$MS = 12.64 + 0.629 \times \text{speed limit in mph} - 0.842$$

×heavy vehicles%+16.0×trip length+1.51
×ln(length of section).

The standard errors of the constant and coefficients

Fig. 2. (a) Residuals from regression of MS against logarithm of length of link for Tyne and Wear. (b) Residuals from regression of U85 against logarithm of length of link for Tyne and Wear. (c) Plot of residuals from regression of MS on Logarithm of length of section for Bahrain. (d) Plot of residuals from regression of U85 on logarithm of length of section for Bahrain.

were 19.82, 0.258, 0.724, 4.89, and 2.39, respectively. The value of R² was 91.1 or 82.1% if adjusted, and the standard deviation of the errors was 6.84. The effect of speedlimit and trip length are statistically significant, beyond the 10% level with a two sided alternative, but all the coefficients have the expected sign. That is, longer section lengths are expected to increase speeds whereas the presence of heavy vehicles is expected to reduce speeds.

U85 = $$6.45 + 0.875 \times \text{speedlimit} - 1.56$$

$\times \text{ heavy vehicles}\% + 2.10 \times \text{trip length} + 0.72$
$\times \text{ ln(length of section)}.$

Standard errors of the constant and coefficients were 31.87, 0.414, 1.164, 7.87 and 3.84, respectively. The value of R² was 85 or 69.9% if adjusted, and the standard deviation of the errors was 10.99. Residuals for both regressions are plotted against the logarithm of the length of the section in Fig. 2 (a and b). The estimates of the coefficients are not changed substantially if the regressions are fitted without constant terms.

⁴.2. Analysis for Bahrain

Two additional variables were coded for the Bahrain links. The number of lanes, which was 2 or 3 (all Tyne and Wear links had two lanes), and the condition of the road surface coded from (1) for excellent to (4) for fair. The following models were fitted.

$$\begin{split} MS = 33.45 + 0.280 \times speedlimit \ in \ km \ per \ h - 0.298 \ \times \ heavy \ vehicles\% + 1.47 \times lanes + 6.06 \ \times \ trip \ length - 0.078 \times condition + 0.841 \ \times \ ln(length \ of \ section). \end{split}$$

Standard errors of the constant and coefficients were

9.12, 0.042, 0.168, 3.15, 1.05, 1.51 and 1.22, respectively. The value of R² was 99.4 or 98.2% if adjusted, and the standard deviation of the errors was 1.19. The effects of speedlimit and trip length are highly statistically signifi-

cant (beyond the $$1\%$$ level).

$$U85 = 42.58 + 0.289 \times \text{speedlimit} - 0.480$$

$$\times \text{ heavy vehicles}\% + 2.08 \times \text{lanes} + 7.45$$

×trip length+1.64×ln(length of section).

Standard errors of the constant and coefficients were 8.24, 0.095, 0.236, 2.91, 1.95 and 2.10, respectively. The value of R² was 96.7 or 92.6% if adjusted, and the standard deviation of the errors was 2.95. Residuals for both regressions are plotted against the logarithm of the length of the section in Fig. 2 (c and d).

5. Conclusions

A feature of road traffic research is that inferences based on statistical information from individual studies are often imprecise. Planners need to take expert judgement and results from similar surveys into account when drawing conclusions. In both areas, the best point estimates for the link between traffic speed and accidents suggest that the number of accidents could be substantially reduced; by either enforcing lower speed limits or at least reducing the spread of vehicle speeds. In Bahrain there is statistically significant evidence of an association between mean speed and accident rate. In Tyne and Wear, where the accident rate was half that of the Bahrain sites, the statistical evidence is weaker, and points to a stronger relationship between accidents and the variability of traffic speeds than the average speed. This is in general agreement with a study, carried out in the UK, by Garber and Gadiraju (1990). However, accidents are bound to be more serious if they occur at higher speeds.

There was no statistically significant evidence against a hypothesis that the accident rate was proportional to the traffic flow. However, for the Tyne and Wear sites the exponent of traffic flow was estimated at about 0.8 for most choices of speed measurement. The main exception was when the skewness index was used, and this gave a noticeably poorer fit to the data. An estimated exponent less than 1 is in broad agreement with the estimate of 0.64 made by Mountain et al. (1996). However, whilst the links they investigated were situated in the UK they did not necessarily satisfy the criteria used for selecting links in this study. The estimated exponents of traffic flow in the best fitting Bahrain models, using the mean or root mean square of speed, were about 0.6. Although the standard errors associated with these estimates were approximately 0.5, so even 90% confidence intervals extend well beyond 1, the point estimates are again close to 0.64. There was some weak evidence of an association between an increase in the proportion of heavy vehicles and a decrease in the accident rate, for both groups of sites. For the Tyne and Wear group of sites the numerical values of the exponent of the proportion of heavy vehicles did not change much if the exponent of traffic flow was estimated simultaneously. In the Bahrain study, the exponent of the proportion of heavy vehicles was reduced in absolute magnitude, for most choices of speed measurement if the exponent of the proportion of heavy vehicles was estimated, but it nevertheless remained negative. It would be worthwhile monitoring the proportion of heavy vehicles in similar investigations.

The relationships between mean speed, and also the upper 85% point of the distribution of speeds, and the explanatory variables: speed limit, percentage of heavy vehicles, trip length, the logarithm of the length of the section, and for Bahrain the number of lanes and the condition of the road surface, were qualitatively as expected. The effects of speed limit and trip length were statistically significant despite the small sample sizes. Lee and Forni (1991) found similar results under free flow conditions although they were not reported as statistically significant. Finch et al. (1994) found, after reviewing cases of imposing speed limits in different countries, that the change in the mean speed of traffic was roughly one quarter of the change in the speed limit. The coefficient of speedlimit in the regression for the Bahrain data is 0.280, and this is in close agreement with their findings. The coefficient in the regression for the Tyne and Wear data is 0.629. But, this is in terms of a mph limit, and if the speedlimit is converted to km/h, the appropriate coefficient becomes 0.391. Although this is higher than the putative one quarter, it is not statistically significantly so. The results suggest that speed limits are a reasonably effective means of reducing traffic speed in free flow conditions. 1994 HMSO: London. .

Acknowledgements

We wish to thanks the following people who offered advice and help during the data collection phase: Mr G. Robinson and his colleagues from the Department of Engineering Services in Gateshead Metropolitan Borough Council; Mr M. Baxter and Mr P. Brown from the Traffic Section, and Mr N. Cowling and his colleagues from the Environment Section of Durham County Council; Mr T. Dimonaco from the Peek Traffic Company; Mr I. Khalaf and his colleagues from the Traffic Section in the Ministry of Works (Bahrain) and Lt Colonel S. Al-Zayani and his colleagues from the Licensing Directorate in the Ministry of Interior (Bahrain).

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