Operating Speed Prediction Models for Horizontal Curves on Rural Four-Lane Highways Huafeng Gong and Nikiforos Stamatiadis icant influence on the driving task. More complex environments can increase a driver’s mental workload, which can cause errors and possibly lead to a crash (1). The lack of consistency in roadway geometric design has also been identified as an apparent potential cause of increasing driver’s mental workload, which can lead to driver error (2). It has been found that driver error is one leading contributor to crashes (3). A requirement for roadway design is to meet driver expectations by creating a consistent roadway design. Studies have examined the relationships between design speed and operating speed on rural two-lane highways (4–8). Most studies have concluded that operating speeds and design speeds are often not in agreement, indicating that roadway design does not always meet driver expectations. Those studies also indicated that design inconsistency exists on those roads. It is therefore reasonable to assume that similar inconsistencies could be found on rural four-lane highways. Currently, the two speed-based approaches—design speed and operating speed—are mainly used for evaluating design consistency. The design speed-based method is used by AASHTO for evaluating design consistency involving selection of a design speed. It has been documented that the design speed-based method does not guarantee consistency (9). The operating speed-based method is widely used in Australia, Canada, and countries in Europe and has also been proposed for the United States. The core of the operating speed-based method is the ability to predict operating speeds. Most of the work completed to date is for rural two-lane facilities, and little is known for rural four-lane highways. Several studies have indicated that horizontal curvature is highly related to crashes. A past study documented that the average crash rate for highway segments that include horizontal curves is about three times the average crash rate for tangent segments (10). Crash rates on horizontal curves are 1.5 to 4.0 times greater than on tangents on rural two-lane highways (11). Data from the 2005 Kentucky Traffic Collision Facts Report showed that the percentage of fatality crashes on curves was 1.37% and 0.5% on tangents (12). This finding indicates that in Kentucky, the percentage of fatality crashes on curves is approximately 2.7 times as high as on tangents. Examination of fatal crashes by highway type showed that rural two-lane highways had the highest rates. Data from Kentucky traffic crash analysis showed that the fatal rate in recent 5 years (2001 to 2005) on rural undivided four-lane highways was the second highest rate (1.6/100 MVM) (13). Moreover, a comparison of crash rates by the number of lanes showed that the crash rate on rural undivided four-lane highways was the highest among the crash rates on all rural two-way highways, which in turn was slightly greater than on rural two-lane two-way highways. Other studies have observed that
Previous studies documented that a uniform design speed does not necessarily guarantee design consistency on rural two-lane facilities. Since a similar process is also followed for four-lane rural highways, it is reasonable to assume that similar inconsistencies could be found on such roadways. The operating speed-based method has been extensively used in other countries as the primary method to examine design consistency. Numerous studies have been completed on rural two-lane highways for predicting operating speeds and evaluating design consistency. However, few studies have considered these issues for rural four-lane highways. Therefore, prediction models for rural four-lane highways are needed. This study aims to develop models to predict operating speeds on horizontal curves of rural four-lane highways. A parallel study documented that speeds on inside and outside lanes are different; therefore, two multiple linear regression models are developed. For the inside lane, the significant factors are shoulder type, median type, pavement type, approaching section grade, and horizontal curve length. For the outside lane, factors include shoulder type, median type, approaching section grade, presence of approaching curve, and curve radius and length. The factors in the two models indicate that the curve itself mainly influences a driver’s speed choice. The models were validated by using the datasplitting approach, and validation shows that there are no statistical differences between the predicted and field-observed operating speeds.
Traditionally, design speed has been selected to determine the radii of horizontal curves for roadway design. One significant weakness of the design speed concept is that it uses the design speed of the most restrictive geometric element within the roadway section, usually a horizontal or vertical curve of the alignment, as the design speed of the entire road. Therefore, the speeds that motorists travel on tangents are not explicitly considered following this design approach. The result could be potential inconsistencies among successive sections of a road. These inconsistencies might result in a sudden change in three aspects of the roadway environment: characteristic of the roadway, driver workload, and driver operating speed. A sudden change in the roadway characteristic might surprise motorists, and such sudden changes might violate driver’s expectancy. Driver’s expectancy is formed by driving experience and has signifH. Gong, 216 Raymond Building, and N. Stamatiadis, 265 Raymond Building, Department of Civil Engineering, University of Kentucky, Lexington, KY 40506-0281. Corresponding author: N. Stamatiadis,
[email protected]. Transportation Research Record: Journal of the Transportation Research Board, No. 2075, Transportation Research Board of the National Academies, Washington, D.C., 2008, pp. 1–7. DOI: 10.3141/2075-01
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four-lane highways have higher crash rates than two-lane highways as well (14, 15). Numerous studies have been completed on rural two-lane highways for predicting operating speeds and evaluating design consistency. However, few studies have considered these issues for rural four-lane highways. Therefore, prediction models for rural four-lane highways are needed. The objective of this study is to develop models to predict operating speeds for horizontal curves on rural four-lane highways.
PREVIOUS MODELS FOR OPERATING SPEED PREDICTION Operating speed models have been developed for rural two-lane highways in numerous studies since the 1950s. Of the models developed, most were speed predictions for passenger-car vehicles while few were developed for heavy or light trucks. Most of the studies used the 85th percentile speed to represent the operating speed. Linear and nonlinear regression models were mainly developed. The methodologies used for developing models are primarily statistical methods. The statistical models mainly include simple linear regression, multiple linear regression, ordinary least-squares model for panel data, two-step, and so forth. Most of the existing models are two-dimensional (2-D) models, which only considered horizontal curve and vertical curve. According to a study intended to develop three-dimensional (3-D) (cross-section, horizontal curve, and vertical curve) models for operating speed predicting, the maximum differences between the observed and predicted speed using 3-D model and 2-D model on some sites reached 35% (16). The 3-D models have significant higher values of coefficient of determination due to the cross section considered in the model. Most of the studies developed regression models based on the data collected, but without any validation. Also, almost all studies provided the measurement of fit of the models, like R2 and R 2adj, but did not assess the quality of prediction. Early studies mostly used curve radius (radius or degree of curvature) as the predictor (17, 18). Later studies used additional predictors that mainly consisted of geometric features (5, 19–20), while in some models, traffic and pavement information also was used (21, 22). Variables identified as significantly relating to operating speed include radius of the curve, length of the curve, length of the preceding and successive tangents, grades, superelevation, annual average daily traffic (AADT) volume, pavement condition, approach speed, and speed limit. The most frequently used predictors are radius, length of curve, length of tangent, grade, superelevation, and lane width. A large number of studies, 35, used radius as an explanatory variable for operating speed prediction, while only 6 used length of curve (23). Length of tangent and grade were also used as explanatory variables in a few studies (16, 24). The data collection device used to record vehicle speed mainly includes radar gun, light detection and ranging gun, following vehicle, stop watch, and detector. In most cases, manually operated radar guns were used. The use of a radar gun is usually accompanied by human error and cosine error, since at least two people are needed in operating the device and reading angle. In some cases, the presence of speed collectors might influence drivers’ behavior. Also, in some studies, the number of observations per site is fewer than 100, and there are some sites recording the speed of only 25 vehicles. In studies with under 100 observations, few assessed the quality of their samples. Therefore, the accuracy of these models might be questionable.
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Previous studies did not consider the effects of characteristics of drivers and vehicles on operating speed. Due to difficulties in collecting driver information and vehicle characteristics in the field, most studies potentially assumed that driver and vehicle characteristics do not influence operating speeds.
DATA COLLECTION A three-step process was followed to define the curves to be used in this study. The first step was to use the Kentucky Highway Performance Monitoring System (HPMS) to select sections that conform to the site selection criteria based on rural four-lane nonfreeway or nonparkway segments. The 2005 HPMS database was used to identify these sections, and 371 were identified as four-lane rural nonfreeway or parkway sections. The second step was to identify the possible curves from the sections selected in the first step. The 371 sections consist of straight and curved sections. To identify the sections with curves, all sections were imported to the ArcMap based on the Statewide_m database. The curves of each section selected were identified one by one in the ArcMap environment. The general criterion used in this step was that a curve is not within or in proximity to any intersection. A total of 283 curves were identified this way, and approximately one-half of them (121) were randomly selected for on-site verification. The third step was to use on-site visits to determine the accuracy of the information. A number of sites were eliminated based on the site visits for a variety of reasons, including proximity to traffic signals; actual location within developed (nonrural) areas; presence of intersections with local roads (that were not included in the state database); and current reconstruction of the curve. Of note is that the initial distinction used for rural areas was obtained from HPMS and the functional classification. However, the site visits verified the rural character of the area. This approach yielded a total of 63 horizontal curves on 33 rural four-lane highways for inclusion in this study. Among the 63 sites, 13 were randomly selected to measure speeds in both directions of travel, resulting in a total of 76 curves. These 13 curves were used in the model development, since some of the roadway features were not exactly the same in both directions. For example, there were differences in the two directions of traffic on the right shoulder width, shoulder type, clear zone, presence of the approaching curve, and grade of the approaching tangent. The Kentucky HPMS and the Kentucky Highway Information System (HIS) were used as the primary data sources to extract the required geometric data, which were verified with on-site visits. One of the most important operating speed predictors in the literature review is the curve radius. In HPMS, a road has been divided into segments with the same geometric characteristics. The horizontal geometric data recorded in HPMS do not identify the actual curve radius but rather indicate the range that includes the radius. In HIS, the degree of curve has been provided. However, due to the questionable accuracy of the data, these curve radii are not used; therefore, an estimation procedure was used. In this study, ArcGIS and AutoCAD are used to measure the horizontal curve radii and lengths. The statewide ArcGIS accuracy is 2 m, considered adequate to properly locate the curve and estimate the radius with the procedure described here (standards from Division of Planning, Kentucky Transportation Cabinet, 2004). In addition, Global Positioning System devices were used to identify locations of the curve where the measurements were to be taken, and they were cross referenced with the HIS mile points
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and computed radius of the curve. The procedure for estimating the curve radius is as follows: 1. Extract location information from HPMS to develop a database. 2. Add the mile points obtained from the Map Click software to the database. 3. Import the database and the shape file of the measured statewide roads to ArcGIS. 4. Mark the sites selected in ArcGIS. 5. Export the marked sites and these roadway sections to AutoCAD. 6. Draw horizontal curves to simulate the real curve. 7. Measure the curve radii and the lengths of the curves using AutoCAD tools. Speed data were collected with radar guns from May to December in 2006 during daylight, off-peak periods, and under good weather conditions. The data collectors were required to be located where they could see the measurement point, whereas drivers could not see them, to avoid influencing the drivers’ operating speeds. The direction of the radar guns was also checked to ensure that the positions would eliminate or at least minimize any potential cosine errors. Originally, automated devices were to be used, but this was not feasible due to lack of distinction among vehicle types (only automobiles were used here) and time and personnel requirements for setting these devices up (closing of road by state employees or police, to both install and uninstall the devices). Vehicle type was identified on site by observation. Only passenger cars were used. The time headway was required at least 5 s between consecutive vehicles to collect truly free-flow speeds (25). Initially, for the first 52 curves, speeds were measured at the beginning, middle, and ending points on both the inside and outside lanes along a curve. Thus, speeds were measured at six spots at each of these curves. An initial analysis showed that the speeds for these sites at the three locations were statistically the same, so it was determined to continue the collection process by measuring speeds only at the middle of the curve. At the remaining 24 curves, speeds were measured at the two middle points on both the inside and outside lanes along a curve. Therefore, speeds were measured at a total of 360 spots. On the basis of previous speed collection experience, initially at least 100 observations were to be taken at each site. However, some segments had low AADT. Also, on some sites, most vehicles travel on the outside lane (close to right shoulder), and few vehicles travel on the inside lane. Therefore, fewer observations were typically taken at sites with low AADT, as well as on some inside lanes. The quality of these speeds was examined in data reduction.
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each site. Moreover, verifying the assumption was more important for the sites where few spot speeds were obtained than at other places. Insufficient spot speed samples cannot represent the real population, and thus they will likely produce meaningless results. Therefore, 183 spots with fewer than 100 spot speeds were examined. The total number of observations for these sites ranged from 30 to 97. After using the Kolmogorov-Smirnov test and the normal probability plots, 11 of the 183 spots were identified as lacking normality. Therefore, the 85th percentile speeds measured on 337 spots were available for further data analysis.
METHODOLOGY Model Development Procedure Some studies on rural two-lane highways used simple linear regression method for developing operating speed-prediction models. In this study, both simple linear and multiple regression methods were used. The purpose was to obtain the best model for the prediction by comparing the simple linear regression models and the multiple linear regression models. The model development procedure is shown in Figure 1. Scatter plots were used to identify possible relationships between the independent variables and the 85th percentile speed. Using the available variables, possible regression models were then developed. The statistic Cp, the coefficient of determination R2, and the adjusted coefficient of determination R 2adj were used to select candidate variables. At the same time, multicollinearity among the candidate variables based on the regression models was examined for reducing potential bias. The variance inflation factor was used to test multicollinearity. The models with high R 2adj (using R2 in simple linear models) and appropriate Cp were then chosen.
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Extreme or unreasonable data usually refer to outliers in statistics. An outlier is an observation that lies outside the overall pattern of a distribution and is distant from the rest of the data (26). The presence of an outlier indicates some possible problem. When using the SPSS software to obtain the box plots of the 85 percentile speeds, it was found that there were only two outlier sites. Outliers were then considered those sites where such differences were outside of the box plots. The speeds collected at these two curves were slightly influenced by downstream construction, so the two curves were eliminated. Therefore, 74 curves were qualified for this study. A basic assumption for speeds is that the observations obtained are from a normal distribution. This assumption must be verified for
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Procedure of model development.
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Data Splitting The R2 or R 2adj of a regression is a measure of the fit of the regression to the sample data. However, they are not considered adequate measures of the regression model’s ability to assess the quality of prediction (27). For assessing the prediction quality, a validation process is necessary, and the data-splitting method is used here. In this method, the data set is partitioned into two groups. One group of N1 samples is used to develop the prediction models. The remaining N2 samples are used to assess the predictive ability of the models. The mean square forecast error and the mean absolute forecast error are two commonly computed measures for model validation. Usually, when the sample size is huge, the data set is partitioned evenly. When it is small, some statisticians have suggested using much more samples to develop models and the remaining to validate (28). With use of a macroprogram, 50 of the 74 sample curves were randomly selected to develop the models in this study. The remaining 24 sample curves were used to validate the models.
where V85 = 85 percentile speed (mph), ST = shoulder type index (if type is surfaced, ST = 1; else, ST = 0), MT = median type index (if type is positive barrier, MT = 0; no barrier, MT = 1), PT = pavement type index (if type is bituminous, PT = 0; concrete, PT = 1), AG = approaching section grade index (if absolute grade ≥ 0.5%, AG = 1; else, AG = 0), and LC = length of curve (ft). Also, R2 = .6836, R 2adj = .6477, and mean error = 1.38 (mph). The grade of the approaching section is the grade of the approaching section directly connected with the curve, regardless of whether the section is a tangent or another curve. Outside Lane The final model for the outside lane is as follows
Model Development Preceding statistical analyses showed that on rural four-lane highways, the operating speeds on inside and outside lanes were significantly different, and on each of the two lanes, speeds at the beginning, middle, and ending points were statistically the same (24). Therefore, two separate models were developed for each lane, that is, inside and outside lane. In this study, the estimated 85th percentile speeds refer to the operating speeds at the middle points of the curves. Each model was developed based upon the model development procedure presented in Figure 1. The best variables capable of predicting operating speed were selected among all possible variables. These variables included the AADT, shoulder type, right shoulder width, left shoulder width, pavement type, median type, median width, left clear zone, right clear zone, approaching tangent length, approaching tangent grade, presence of approaching curve, radius of curve, length of curve, and road width. A model was developed for each variable alone as well as combinations of variables. Each model was evaluated, and its ability to predict operating speeds was determined. The most appropriate model was then selected as the best prediction. Of note is that none of the curves selected had spirals; they were simply circular curves. Moreover, the shoulder width at these sites is for the entire shoulder, without any distinction as to the paved and unpaved widths.
SPEED MODELS Using the procedure described, models were first developed for each lane and then validated. These models are presented in the next section.
Inside Lane The final multiple linear regression model for the inside lane is as follows: V85 = 51.520 + 1.567ST − 2.795MT − 4.001PT − 2.150 AG + 2.221 ln ( LC )
V85 = 60.779 + 1.804ST − 2.521MT − 1.071AG − 1.519FC ⎡ ( LC ) ⎤ + 0.00047 R + 2.408 ⎢ ⎥ ⎣ R ⎦ where FC is the front curve index (if approaching section is a curve, FC = 1; else, FC = 0) and R is the curve radius (ft). Also, R2 = .5015; R 2adj = .4320; and mean error = 1.47 (mph). Discussion of Model The variables included in the models and their accompanying signs provide some mixed results. In general, the median type is considered a means for reducing speeds; therefore, the negative sign in both models is appropriate. This variable is coded as binary, that is, presence or no, where a positive median is considered to be a concrete or guardrail barrier. Another factor that could have an influence on, and possibly a combined effect on, the operating speed is the median width. Even though the median width was a variable included in the database, it was not a statistically significant contributor and it was not part of the prediction model. It could therefore be hypothesized that the correlation of median width and type may have an influence, and this should be examined in the future. The shoulder type showed also consistent results for both models. In both models, it has a positive sign, indicating that paved shoulders have a positive effect on operating speeds, that is, increasing them. This result is intuitive, assuming that the presence of paved surface encourages drivers to travel at higher speeds. Of note is that for both median and shoulder type, the size of the coefficients was similar in both models, indicating a somewhat equal effect on operating speeds. The effect of the approaching grade also showed similar results in the two models. In this case, roadway segments with some grade (absolute value greater than 0.5%) showed a reduction in speeds. This result is also generally intuitive where lower speeds are expected on roads with some grade than those without any. It is possible that there may be a different effect of this variable based on whether the grade was positive or negative. However, it was not possible to validate this assumption because the grades were given as absolute values in the HPMS, and the site visits did not allow for an adequate definition of the sign of the grade.
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The pavement type is a variable included in the inside lane model. The sign indicated that concrete pavements result in lower speeds when compared with results on bituminous pavements. This trend could be considered intuitive, assuming that the noise level generated by each pavement type has an effect on operating speed. It is therefore possible that bituminous pavements with the lower noise result in higher operating speeds. However, the magnitude of the coefficient may be of concern, since it seems to have a larger influence than any of the other variables do. This may be an area of additional work in the future, to either validate this impact or estimate a moderated effect of this variable. The type of segment preceding the curve (front curve) was also a significant variable for the outside lane model. In this case, if the segment was another curve, then the speed was reduced. This result is also intuitive, and it assumes that drivers entering the curve in question from a preceding curve will drive at lower speeds. Of interest is the impact of the curve itself on operating speeds. The model developed for inside lane shows that the impact of the curve on operating speeds is clearly intuitive, since only the curve length is included in the model. In this case, the increase of curve length results in an increase of operating speed. However, the model for the outside lane shows that the impact of the curve is not as clearly intuitive because both horizontal curve radius and length of the curve are in the model. For assessing the impact on operating speed, a 3-D figure is helpful (Figure 2). This figure shows the operating speed changes due to the changes of curve radius and deflection angles (it is equal to the value of curve length divided by radius). It clearly shows that, generally, operating speed increased as curve radius or deflection angles, or both, increased.
data for validation (24 sample curves) are those remaining after the random selection.
Inside Lane The differences between the measured and predicted speeds (measured minus predicted) ranged from −4.50 to 4.79 mph (Figure 3) (forecast errors were sorted by ascending order). The mean of the differences was only 0.20 mph, while the mean of the absolute differences was 1.88 mph. The mean squared error reached 4.89 mph2. The mean absolute percent difference was only 2.89%, which indicated that the prediction error rate was very low. These statistics indicated that the prediction error was low. The paired t-test was used to statistically examine the differences. Using the Kolmogorov-Smirnov test, it was found that the null hypothesis that the distribution of the data was normal could not be rejected at the 95% confidence level (p-value = .793). The normal probability plot of the data also showed that the data are normally distributed. The assumption diagnostic indicated that the paired t-test was suitable for the data. Results of the paired t-test showed that, at the 95% confidence level, the null hypothesis—that there was no difference between the means of the measured and predicted operating speeds—could not be rejected (p-value = .663). It could be further concluded that there was no statistical difference between the measured and predicted operating speeds.
Outside Lane The range of the speed differences (measured minus predicted) on the outside lane was from 3.54 to 5.65 mph. The mean of the differences was only −0.28 mph, while the mean of the absolute differences was 1.84 mph. The mean squared error was 5.08 mph2, and the mean absolute percent difference was 2.93%. These statistics are very close to the statistics calculated for the inside lane and indicated that the prediction error was low. Using the Kolmogorov-Smirnov test, it was found that the null hypothesis that the distribution of the data was normal could not be
Validation of Model The objective of validation is to evaluate the accuracy with the speeds predicted by using the models developed. Each of the two models for inside and outside lanes is validated separately. As previously noted, the two models were developed using 50 sample curves randomly selected from the 74 sample curves of the database. The
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rejected at the 95% confidence level ( p-value = .614). The normal probability plot of the data also indicated that the data were normally distributed. The assumption assessment indicated that the paired t-test was suitable for the data. Results of the paired t-test showed that the null hypothesis—that there was no difference between the means of the two speeds—could not be rejected at the 95% confidence level ( p-value = .553). It could be further concluded that there was statistically no difference between the measured and predicted operating speeds.
Limitations of Model As for any model developed, the models presented here have certain limitations. Included are the range of radius (538 to 7,704 ft); range of lengths of horizontal curves (775 to 5,780 ft); range of AADT (5,220 to 26,900 passenger cars per day); and range of design speeds (40 to 70 mph). It should be emphasized that the models are applicable for curve sections, and the use of these models for roadway sections outside of the ranges noted is not recommended without any additional validation.
DISCUSSION OF RESULTS AND CONCLUSIONS This study is the first comprehensive one on predicting operating speed models focusing on horizontal curves of rural four-lane highways (nonfreeway or nonparkway). As noted, most of the previous studies focused on rural two-lane highways mainly due to the extent of the two-lane rural network and the high fatality rates on these roadways. Two multiple linear regression models were developed for operating speed prediction, since the operating speeds on inside and outside lanes were different. The two models focused on horizontal curves of rural four-lane highways. For the inside lane, the significant factors were shoulder type, median type, pavement type, approaching section grade, and horizontal curve length. For the outside lane, the foregoing factors except pavement type were included, as well as curve radius and presence of approaching curve. In comparing the significant factors in the two models, it could be concluded that there were some common factors, including shoulder type, approaching section grade, and curve length. All these factors showed intuitive results, indicating that paved shoulders contribute to higher speeds, the absence of median barriers results in lower operating speeds,
and the presence of grades produces a decrease in the speeds. For both models, the magnitude of the impact was similar, indicating an agreement between the models. In both models, curve length was found to be a significant predictor, indicating that the curve radius did not solely determine operating speeds on rural four-lane highways. In contrast, some models for rural two-lane highways used only curve radius as the explanatory variable. Validation for the models indicated that the models are appropriate for application and the explanatory variables are reliable. However, the limitations of use of these models should be considered when used in design. The site selection was based upon geographic information system (GIS) technology. This technology is able to convert between data and graphics, such as Excel database, to ArcGIS shape file. In this way, site information can be queried as numeric data, and site surroundings can be visualized. By contrast, most previous studies selected study sites based on highway databases, maps, or on-site visits. GIS technology facilitates and improves efficiency and reduces cost for site selection. Not all of the geometric elements examined were included in the two models. This does not mean that the elements excluded from the models did not significantly affect operating speed. The included elements meant only that these elements were significant to operating speed, and their combination could mostly explain the operating speed. It is possible that future studies will include more of the variables used here. Therefore, additional work in this area is needed to explore this potential. The first step in the model development process was to examine the influence of each variable on the model and its ability to predict operating speeds. Results from this step showed that there were some elements that did not significantly affect operating speeds. This finding could be attributed to either the range of the data or the sites examined. Moreover, when models including more variables were examined, the same elements were found significant. This indicated that certain interactions among the geometric elements were examined. Results further showed that the multiple linear regression model was much more appropriate than another model for rural four-lane highways. This finding also supports the need for further study of these variables and the collection of additional data to further validate and possibly revise these models. The work presented here is a first step toward development of speed prediction models for rural four-lane highways. The models developed for horizontal curves address this issue, but additional work is needed for tangent sections, as well as transition sections
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between tangents and curves. As noted, the magnitude of some of the coefficients presented may be questionable. The finding that bituminous pavements may result in higher speeds could be considered acceptable, but the large difference between bituminous and concrete pavements may be relative high. Another area of focus could be the combined effect of the median width and type. The interaction of these variables was not examined, due to lack of data, but this may well be an area that could influence operating speeds on multilane roads. A third area for future concentration is the effect of the grade on the speeds and especially the type of grade (up or down). The available data did not allow for any such distinction. Also, even though grade itself was significant, the effect of positive or negative grades was not evaluated. Completion of such models will allow for development of design consistency models similar to those developed for two-lane rural roads, aiming to improve highway design practices.
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