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Determination of Sample Size for Speed Measurement on Urban Arterials
Available online at www.sciencedirect.com
ScienceDirect Transportation Research Procedia 17 (2016) 384 – 390
11th Transportation Planning and Implementation Methodologies for Developing Countries, TPMDC 2014, 10-12 December 2014, Mumbai, India
Determination of Sample Size for Speed Measurement on Urban Arterials Varsha .V.a, Gaurav H. Pandeyb, K. Ramachandra Raob, Bindhu B.K.a* a Department of Civil Engineering, RIT, Kottayam -686516, Kerala, India Department of Civil Engineering and Transportation Research and Injury Prevention Programme, IIT Delhi, Hauz Khas, New Delhi-110016, India
b
Abstract Estimation of speed is an important task for various traffic engineering analyses like crash analysis, establishing speed zones, evaluation of traffic signal locations and so on. Past researchon the sample size estimation was focused on finding the number of GPS based probe vehicles required.However, there are limited studies on the sample size determination of speeds obtained using video-graphic survey. Large samples increases the chances of finding the significant difference but sometimes it leads to wastage of resources. Inadequate sample size may lead to inaccurate results. Hencesample size determination is often an important step in traffic engineering studies. This paper attempts to determine the sample size based on variability and traffic conditions.An evaluation of the sample size determination such as ITE and Hybrid methods were also carried out. From the data analysis, it was found that there was not much variation in speeds for given vehicle type and location. The mean and variance in speeds obtained from first ten speed measurements for a vehicle type are not statistically different from those obtained after one hour of data collection. For some locations, where the proportion of heavy vehicles and flow were low, statistically stable mean speeds of heavy vehicles could not be obtained even after hour of data collection. © 2016 2015 The by Elsevier B.V. B.V. This is an open access article under the CC BY-NC-ND license © TheAuthors.Published Authors. Published by Elsevier Selection and peer-review under responsibility of the Department of Civil Engineering, Indian Institute of Technology Bombay. (http://creativecommons.org/licenses/by-nc-nd/4.0/).
Peer-review under responsibility of the Department of Civil Engineering, Indian Institute of Technology Bombay Keywords:Sample size; Spot speeds; Vehicular types and trajectories;
1. Introduction Spotspeed estimation is an important task in traffic engineering analysis like crash analysis, establishing speed zoning,evaluation of traffic signal locations and so on. Data collection is done by inductance loop, video-graphy, speed gun etc. In some techniques speed measurement is based on measurement of travel time along a section of fixed distance.Even though the inductive loops are said to be the most accurate method of speed data collection, they fail in the overloaded regions where installation and maintenance costs will be much higher. Moreover theymay not be
2352-1465 © 2016 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the Department of Civil Engineering, Indian Institute of Technology Bombay doi:10.1016/j.trpro.2016.11.130
V. Varsha et al. / Transportation Research Procedia 17 (2016) 384 – 390 helpful for collecting data under mixed traffic conditions. Each similar method used for data collection is having its own advantages and limitations. Depending upon the purpose of the study, appropriate mode of data collection can be adopted. For this study, video graphic survey having high resolution cameras is used for collecting the traffic data. Accuracy obtained from video graphic survey mainly depends upon the selection and position of cameras used. Hence special consideration should be given while mounting the cameras.For extraction of the speed data image processing software (TRAZER) was used. Sampling is an important concept which helps in research analysis as it determines the adequate respondents from the total number of target population to be used. Inadequate samplesize may lead to poor results hencesample size determination is often an important step in planning and statistical study.Itis fundamental to traffic engineering analysis.This study was done to estimate the sample size for determining mean speeds for various vehicle categories at different locations. Since video graphic survey was used for the data collectionprocess, randomness in selecting the samples was not considered. Conventional methods like Oppenlander (1963),Hybrid method (Quiroga and Bullock, 1998), ITE (Institute of Transportation Engineering) method (1999), and Modified method (Li et al., 2002) were used to determine the sample size for speed measurements in various vehicular categories A new method considering Regression to Mean (RTM)phenomenon is proposed to find the sample size.The RTM is a statistical phenomenon that occurs when repeated measurements are made on the same subject or unit of observation. The collected speed data from the four locations were used to find their speed variance and mean speed. Regression to mean phenomenon was used to determine the sample size for mean as well as sample size for standard deviation.Statistical tools like two sample t tests were used to check the magnitude of speed change.Speed studies are mainly the performance measures of traffic engineering. Depending upon the accuracy desired data collection methods may vary. Many researchers have carried out studies regarding the sample size estimation of probe vehicles using GPS data. However, there are very few studies on the sample size determination of speeds obtained using video-graphic survey. By using video graphic method bias can be avoided as all vehicles’ data is captured. Further, it also helps in vehicle classification. For the spot speed studies video graphic survey is the best suited method in spite of being time consuming. Accuracy of the method is also up to the expected level. Obtaining the correct number of sample size is crucial for any kind of study. Sample size may vary from study to study depending upon its nature. Hence sample size estimation is fundamental to any traffic engineering analysis.Previous studies in the area of travel time and delay studies used methods like ITE method, Hybrid method and Modified method for the sample size determination. In this paper, an attempt was made to determine the sample size for vehicular speeds obtained by video graphic survey. 2. Literature Review Sample sizes can vary from a fraction of an hour to 24 hours a day, 365 days a year depending on the purpose of the study. Generally for the speed study, peak hours will be included in all samples. Holiday or on the day before or after a holiday is excluded for taking traffic counts. Normally Monday mornings and Friday evenings will show high volumes. Votaw and Levinson (1962) indicate that the sample size is an important factor in the design of sample survey. Oppenlander (1976) developed a procedure for sample size determination, based on the average range of the observed travel speeds. The estimate of standard deviation of the travel speeds can also be used in similar techniques.Quiroga and Bullock (1998) developed a hybrid method for the determination of sample size. 2
n
ª tD u R º « » (1) ¬ dH ¼
n
minimum sample size
tD
normal, two tailed statistics for a confidence interval of 1-D
R
average range
385
386
V. Varsha et al. / Transportation Research Procedia 17 (2016) 384 – 390
d
d H
R
V factor for estimating V from R user-selected allowable error
Li et al. (2002) suggest a modified method to determine the sample size. Use of ܼఈൗଶ values in place of ݐఈൗଶvalues induces some error in sample size, hence proposed a correction factor for calculation of sample size. Revised equation is given below,
ª zD / 2 u V º « H » H n (2) ¬ ¼ 2
n n
minimum sample size
V population standard deviation H user-selected error H n sample size adjustment Barnett et al. (2005) studied regression to mean (RTM) phenomenon and concluded that it is ubiquitous phenomenon in repeated data. It should always be considered as a possible cause of an observed change. Use of better study design and suitable statistical methods can be used to alleviate these effects. Park and Lord (2010) have used graphical methods to illustrate the RTM phenomenon. They used aggregated speed data to show on to reduce RTM bias in before-and-after speed data analysis. From the numerical examples it was found that the estimated magnitude of the mean speed change, due to the introduction of an engineering treatment and the amount of uncertainty which can be measured by the estimated standard error and confidence interval, can be misleading if RTM was not taken into account. Nezamuddin (2010) validated the earlier ITE estimates for moderate traffic conditions on freeways, which was based on standard deviation values for spot speeds for roadways classified by annual average daily traffic (AADT) volumes. This study also found a consistent U-shaped relationship between the standard deviation of speed and traffic volume, which alludes to an inherent speed distribution profile of traffic data. Zhou et al. (2011) determined the small sample size of probe vehicles based on micro-simulation.A simulation-based approach was used to determine the sample size of probe vehicles with consideration of road network coverage and link average speed estimation. The accuracy of estimation increases little and the efficiency decreases significantly with the increase of probe sample size when it reaches a certain level, as proved by microscopic VISSIM simulation. Jung and Gan (2011) have studied sampling and analysis methods for estimation of Average Vehicle Occupancies (AVO). Their paper further describes a detailed process for estimating AVOs at the individual location, facility-type, and county levels. A detailed sampling process designed to select data collection locations and dates on three different facility-types was presented where link-day, was used as the basic sampling unit to determine the minimum sample sizes required to achieve the desired accuracy in AVO estimates.Yang et al.(2011) have studied travel time reliability of probe vehicle system based on minimum sample size analysis.Minimum number of probe vehicles was calculated based on t-statistic, standard deviation and allowable error. Correlation analysis was done between minimum sample size and link length. Linear regression is used to estimate the minimum sample size forecasting model considering all the factors affecting the determination of sample size.Bolbol et al (2012) used journey time data to calculate appropriate sample size for studies that aims to infer the mode of transport from GPS based travel surveys. The sample size required was based on the mode of transport, temporal granularity observed in three zones of London. Maurya and Bokare (2013) observed that the dispersion parameter of speeds not only vary with vehicle type but also with manoeuvres such as acceleration or deceleration. Hence it isconcluded that the sample size (which is a function of dispersion of speed of vehicle) requirement varies with vehicle type and with manoeuvre type.
V. Varsha et al. / Transportation Research Procedia 17 (2016) 384 – 390 3. Methodology 3.1. Data The spot speed data collection was carried out in the city of Ludhiana, Punjab, India,using video-graphic survey on four arterial roads for the peak and non-peak hours. Different types of vehicles ranging from two wheelers to heavy motorized vehicles are analysed. The variation observed in the flow is from moderate to high. The four locations are listed below. 1. Jagraon bridge– Viswakarma Intersection 2. ViswakarmaChowk – Samrala Intersection 3. Jalandhar bypass – Jagraon bridge 4. Jagraon bridge – Bharatnagar Intersection Vehicle detection and analysis is a major problem found in traffic survey mainly due to the lack of lane discipline and seepage of small vehicles like two wheelers. In such cases even the inductive loop detectors fail to classify the vehicles. Usage of laser speed guns was also found to be difficult under mixed traffic conditions as they cannot capture all the vehicles. Macroscopic features like average vehicle speeds, classified vehicle flows, vehicle occupancies and microscopic traffic characteristics like vehicle trajectories, lateral and longitudinal spacing can be obtained from images of video cameras using an offline image processing technique. The sites were selected such that it provides elevation and are as close to centreline of the road as possible: hence only the sites with foot over bridge were selected for mounting camera. Cameras position should be in such a way that it should be able to cover a considerable length of road where vulnerable traffic data falls. This was an important requirement for TRAZER image processing software used for the data extraction. Traffic analyser and enumerators are used for the extraction of prescribed data from the whole traffic data. TRAZER is capable of tracking the vehicles under highly congested conditions Main advantages are; it is nonintrusive, automated, accurate and auditable in nature. The software can process up to four lanes simultaneously. For short time recording 4.5 m tripod is used for holding the cameras and the most common vantage point is the foot over bridge. Using this software (TRAZER) the vehicles were manually marked for higher accuracy. A MATLAB code was developed to analyse the X-Y coordinates given by TRAZER and determine vehicular speed using first and final position of vehicle in its trajectory. The algorithm helps in identifying the portion of scenes that might contain a vehicle from the sequence of images originated from the video cameras. After the identification methodology is carried out, further analysis is done to extract the relevant features. Various methods discussed earlier were used to determine the sample size for the four locations. These methods found sample size with the help of some equations consisting of average range, standard deviation, user selected error, tα and ܼఈൗଶ values.Sample size determination using the methods like Hybrid and Modified methods were dependent on the variation in the standard deviation as well as range. The limitation of the ITE method is that the occurrence of manual error is more in case computing the value of R factor from the x bar chart. Two new methods are proposed in the present study using regression to mean phenomenon. Data points are noted accordingly to the speed data collected. The mean speed and variance were found out from the data and was used for finding the sample size. The occurrence of three consecutively same mean or standard deviation values were used as criteria for finding sample size for mean and standard deviation. Statistical tests were done for the data collected to check whether the data was statistically significant. Prior to t tests, F test was done to check whether the data is having equal or unequal variances. Two sample t tests were carried out according to the F test obtained. Significance of t tests are checked by taking the higher and lower values of the mean speeds with respect to the stabilized one using regression to mean method. 3.2. Results and Discussion Various conventional methods for sample size determination and the proposed methodology are presented in the following tables. The user defined error was taken as 4km/h. F test was done in order to verify the equality of variance.
387
388
V. Varsha et al. / Transportation Research Procedia 17 (2016) 384 – 390 Based on the result of F test, two sample t tests were chosen and are also included in the following tables.From the Table 1, it was clear that for locations 1, 2 and 4, where the data was relatively large, the sample size for mean was higher. The sample size determined using RTM was found to be higher than that obtained using other methods. The method failed to determine the sample size when the proportion of vehicles and flow are low. T tests also indicated that the sample was not significantly varying. Table 1.Sample size estimated by various methods for light motor vehicle in the four locations with the time taken, flow and occupancy
Location
Flow
Occupancy
Location 1 Location 2 Location 3 Location 4
2248 2187 983 2933
0.042 0.020 0.016 0.032
Sample size using RTM for mean 50 40 Not stabilized 60
Sample size using RTM for std. deviation 30 10 Not stabilized 50
ITE Metho d (1999) 13 4 3 7
Hybrid Method (1998)
Modified Method (2002)
t test results
3 11 25 18
6 14 26 20
10 10 10 10
Table 2.Sample size estimated by various methods for heavy motor vehicle in the four locations with the time taken, flow and occupancy
Location
Flow
Occupancy
Location 1 Location 2 Location 3 Location 4
2248 2187 983 2933
0.042 0.020 0.016 0.032
Sample size using RTM for mean 20 Not stabilized Not stabilized Not stabilized
Sample size using RTM for std. deviation 30 Not stabilized Not stabilized Not stabilized
ITE Method (1999)
Hybrid Method (1998)
Modified Method (2002)
t test results
11 4 2 3
3 6 30 16
6 8 29 6
10 9 Not stabilized Not stabilized
From the Table 2,it was observed that the method used for sample size estimation using regression was overestimating the sample size comparing to other methods at location 1.The method failed to determine sample size where the proportion of vehicles and flow were low. T tests also failed at locations where the flow was low. The method for sample size determination using regression to mean failed to estimate the sample size in all the locations other than location 1 due to the low proportion of vehicles and the flow. The three conventional methods were not found to follow any trend. The sample sizes were varying in accordance with the variance in the speed data. Two sample t tests showed that the data was not significantly varying much in case of locations 1 and 2. Table 3.Sample size estimated by various methods for three wheeler vehicle in the four locations with the time taken, flow and occupancy
Location
Flow
Occupancy
Location 1 Location 2 Location 3 Location 4
2248 2187 983 2933
0.042 0.020 0.016 0.032
Sample size using RTM for mean 20 10 Not stabilized 30
Sample size using RTM for std. deviation 10 10 10 10
ITE Method (1999)
Hybrid Method (1998)
Modified Method (2002)
t test results
11 6 4 8
3 4 7 9
6 7 9 12
10 10 10 10
From the Table 3, it was observed that the method used for sample size estimation using regression to mean was overestimating the sample size determined by other methods. The method failed to estimate the sample size for location 3 where the proportion of vehicles and flow are low. Two sample t tests showed that the data was not significantly varying in nature.
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V. Varsha et al. / Transportation Research Procedia 17 (2016) 384 – 390
Table 4.Sample size estimated by various methods for two wheeler vehicle in the four locations with the time taken, flow and occupanc y
Location
Flow
Occupancy
Location 1 Location 2 Location 3 Location 4
2248 2187 983 2933
0.042 0.020 0.016 0.032
Sample size using RTM for mean 30 20 Not stabilized 50
Sample size using RTM for std. deviation 40 10 Not stabilized 20
ITE Method (1999)
Hybrid Method (1998)
Modified Method (2002)
t test results
40 8 2 10
2 7 51 22
5 10 48 25
10 10 10 10
From the Table 4, itcan be seen that the method used for sample size estimation using regression was overestimating the sample size comparing to other methods for location 2 and location 4.The method failed to estimate the sample size for location 3 where the proportion of vehicles and flow was low. Two sample t tests showed the datawas not significantly varying. 4. Conclusions From the data analysis, it was observed that the sample size determined by various methods seemed to be varying with vehicle categories, time of the survey and flow parameters for the spot speed data. Since the sample size and number of locations were less in each case, statistical tests like Shapiro-Wilk’s test and other tests for normality and homogeneity of variance cannot be performed. It may thus be difficult to further analyse the data and develop relationships between the variables. Some of the important conclusions based on the present study are given below. x Sample size determined using regression to mean is overestimating the sample size determined from other conventional methods for all the categories of vehicles excluding the cases where the flow and proportion of vehicles are low. x The sample size determined using regression failed at locations where the flow and proportion of vehicles were low. x Sample size determined from the modified method varies with respect to the change in the standard deviation whereas the sample size determined from hybrid method varies with respect to the change in the average range. x T tests can be used as the best suitable method for the determination of sample sizes for the vehicular mean speeds. x As per the result from t tests, minimum number of speed samples to be taken for the study can be as low as ten (n=10). This is irrespective of the vehicle categories and locations as long as flow and proportion of vehicles were relatively high. x In order to assume that the speeds are normally distributed during analysis, the minimum number of samples should be greater than thirty (n>30). Acknowledgements The authors (GHP and KRR) would like to thank Professor Dinesh Mohan, Guest Professor, IIT Delhi, for triggering ideas on sample size requirements and the need for optimizing data collection efforts in traffic studies. References Barnett, A., G., Van der Pols, J., C., & Dobson, A., J., 2005. Regression to the Mean: What it is and How to Deal With It. International Journal of Epidemiology34, 215–220. Bolbol, A., Cheng, T., Tspakis, & Chow, A., 2012. Sample Size Calculation For Studying Transportation Modes From GPS Data. Procedia Social and Behavioural Sciences Transport Research Arena Europe, 3040 –3050. ITE, 1999.Traffic Engineering Handbook.5th edition, Institution of Transportation Engineers, Washington, D.C. Jung, R.,&Gan, A., 2011.Sampling and Analysis Methods for Estimation of Average Vehicle Occupancies. Journal of Transportation Engineering 137, ASCE, pp.537-546
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V. Varsha et al. / Transportation Research Procedia 17 (2016) 384 – 390 Li, S., K., Zhu, B., H., W., Van Gelder, J., Nagle, & C.,Tuttle., 2002.Reconsideration of SampleSize Requirements for Field Traffic Data Collection Using GPS Devices.Transportation Research Record, Journal of the Transportation Research Board 1804, TRB, National Research Council, Washington, D.C.pp.17–22.. Maurya, A.,K.,andBokare, P., S., 2013.Sample Size Requirements for Vehicle's Speed Data Collection Using Global Positioning System. Procedia Social and Behavioural Sciences Transport Research- Transport Research Group – Transport Research Group 2013 Conference, Agra, India. Nezamuddin, 2010.Statistical Patterns of Traffic Data and Sample Size Estimation.TRB Annual meeting. Oppenlander, J., C.,1963. Sample Size Determination for Spot-Speed Studies at Rural, Intermediate and Urban Locations. Highway Research Record, Journal of the Highway Research Board 35, HRB, Washington, D.C. pp. 78–80. Oppenlander, J., C., 1976. Sample Size Determination for Travel Time and Delay Studies. Traffic Engineering 46.9, Institute of Traffic Engineers, Washington D.C.pp. 25–28. Park, P., Y., & Lord, D., 2010.Investigating Regression to the Mean in Before and After Speed Data Analysis.Submitted for publication at Transportation research board, 89th annual meeting, January 10-14, Washington D.C. Quiroga, C., A.,and Bullock, D., 1998. Determination of Sample Sizes for Travel Time Studies.ITE Journal 68.8, Institute of Transportation Engineers, Washington, D.C. pp. 92–98. Robertson, H., D., 1994. ITE Manual of Transportation Engineering Studies, Prentice Hall, Englewood Cliffs, N.J. Votaw, D., F., and Levinson, H., S., 1962.Elementary sampling for traffic engineers.Eno Foundation for Highway Traffic Control, New Haven, CT. Yang, Y., Yao, E., Qu, D., and Zhang, Y., 2011. Study on Travel Time Reliability of ProbeVehicle System based on Minimum Sample Size Analysis.International Conference of Chinese Transportation Professionals(ICCTP), held in Nanjing, China, August 14–17, ASCE. Zhou, W., Zhao, S., and Liu, K., 2011. Determining the Small Sample Size of Probe Vehicles Based on Micro-Simulation.International Conference of Chinese Transportation Professionals(ICCTP), held in Nanjing, China, August 14–17,ASCE.
ScienceDirect Transportation Research Procedia 17 (2016) 384 – 390
11th Transportation Planning and Implementation Methodologies for Developing Countries, TPMDC 2014, 10-12 December 2014, Mumbai, India
Determination of Sample Size for Speed Measurement on Urban Arterials Varsha .V.a, Gaurav H. Pandeyb, K. Ramachandra Raob, Bindhu B.K.a* a Department of Civil Engineering, RIT, Kottayam -686516, Kerala, India Department of Civil Engineering and Transportation Research and Injury Prevention Programme, IIT Delhi, Hauz Khas, New Delhi-110016, India
b
Abstract Estimation of speed is an important task for various traffic engineering analyses like crash analysis, establishing speed zones, evaluation of traffic signal locations and so on. Past researchon the sample size estimation was focused on finding the number of GPS based probe vehicles required.However, there are limited studies on the sample size determination of speeds obtained using video-graphic survey. Large samples increases the chances of finding the significant difference but sometimes it leads to wastage of resources. Inadequate sample size may lead to inaccurate results. Hencesample size determination is often an important step in traffic engineering studies. This paper attempts to determine the sample size based on variability and traffic conditions.An evaluation of the sample size determination such as ITE and Hybrid methods were also carried out. From the data analysis, it was found that there was not much variation in speeds for given vehicle type and location. The mean and variance in speeds obtained from first ten speed measurements for a vehicle type are not statistically different from those obtained after one hour of data collection. For some locations, where the proportion of heavy vehicles and flow were low, statistically stable mean speeds of heavy vehicles could not be obtained even after hour of data collection. © 2016 2015 The by Elsevier B.V. B.V. This is an open access article under the CC BY-NC-ND license © TheAuthors.Published Authors. Published by Elsevier Selection and peer-review under responsibility of the Department of Civil Engineering, Indian Institute of Technology Bombay. (http://creativecommons.org/licenses/by-nc-nd/4.0/).
Peer-review under responsibility of the Department of Civil Engineering, Indian Institute of Technology Bombay Keywords:Sample size; Spot speeds; Vehicular types and trajectories;
1. Introduction Spotspeed estimation is an important task in traffic engineering analysis like crash analysis, establishing speed zoning,evaluation of traffic signal locations and so on. Data collection is done by inductance loop, video-graphy, speed gun etc. In some techniques speed measurement is based on measurement of travel time along a section of fixed distance.Even though the inductive loops are said to be the most accurate method of speed data collection, they fail in the overloaded regions where installation and maintenance costs will be much higher. Moreover theymay not be
2352-1465 © 2016 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the Department of Civil Engineering, Indian Institute of Technology Bombay doi:10.1016/j.trpro.2016.11.130
V. Varsha et al. / Transportation Research Procedia 17 (2016) 384 – 390 helpful for collecting data under mixed traffic conditions. Each similar method used for data collection is having its own advantages and limitations. Depending upon the purpose of the study, appropriate mode of data collection can be adopted. For this study, video graphic survey having high resolution cameras is used for collecting the traffic data. Accuracy obtained from video graphic survey mainly depends upon the selection and position of cameras used. Hence special consideration should be given while mounting the cameras.For extraction of the speed data image processing software (TRAZER) was used. Sampling is an important concept which helps in research analysis as it determines the adequate respondents from the total number of target population to be used. Inadequate samplesize may lead to poor results hencesample size determination is often an important step in planning and statistical study.Itis fundamental to traffic engineering analysis.This study was done to estimate the sample size for determining mean speeds for various vehicle categories at different locations. Since video graphic survey was used for the data collectionprocess, randomness in selecting the samples was not considered. Conventional methods like Oppenlander (1963),Hybrid method (Quiroga and Bullock, 1998), ITE (Institute of Transportation Engineering) method (1999), and Modified method (Li et al., 2002) were used to determine the sample size for speed measurements in various vehicular categories A new method considering Regression to Mean (RTM)phenomenon is proposed to find the sample size.The RTM is a statistical phenomenon that occurs when repeated measurements are made on the same subject or unit of observation. The collected speed data from the four locations were used to find their speed variance and mean speed. Regression to mean phenomenon was used to determine the sample size for mean as well as sample size for standard deviation.Statistical tools like two sample t tests were used to check the magnitude of speed change.Speed studies are mainly the performance measures of traffic engineering. Depending upon the accuracy desired data collection methods may vary. Many researchers have carried out studies regarding the sample size estimation of probe vehicles using GPS data. However, there are very few studies on the sample size determination of speeds obtained using video-graphic survey. By using video graphic method bias can be avoided as all vehicles’ data is captured. Further, it also helps in vehicle classification. For the spot speed studies video graphic survey is the best suited method in spite of being time consuming. Accuracy of the method is also up to the expected level. Obtaining the correct number of sample size is crucial for any kind of study. Sample size may vary from study to study depending upon its nature. Hence sample size estimation is fundamental to any traffic engineering analysis.Previous studies in the area of travel time and delay studies used methods like ITE method, Hybrid method and Modified method for the sample size determination. In this paper, an attempt was made to determine the sample size for vehicular speeds obtained by video graphic survey. 2. Literature Review Sample sizes can vary from a fraction of an hour to 24 hours a day, 365 days a year depending on the purpose of the study. Generally for the speed study, peak hours will be included in all samples. Holiday or on the day before or after a holiday is excluded for taking traffic counts. Normally Monday mornings and Friday evenings will show high volumes. Votaw and Levinson (1962) indicate that the sample size is an important factor in the design of sample survey. Oppenlander (1976) developed a procedure for sample size determination, based on the average range of the observed travel speeds. The estimate of standard deviation of the travel speeds can also be used in similar techniques.Quiroga and Bullock (1998) developed a hybrid method for the determination of sample size. 2
n
ª tD u R º « » (1) ¬ dH ¼
n
minimum sample size
tD
normal, two tailed statistics for a confidence interval of 1-D
R
average range
385
386
V. Varsha et al. / Transportation Research Procedia 17 (2016) 384 – 390
d
d H
R
V factor for estimating V from R user-selected allowable error
Li et al. (2002) suggest a modified method to determine the sample size. Use of ܼఈൗଶ values in place of ݐఈൗଶvalues induces some error in sample size, hence proposed a correction factor for calculation of sample size. Revised equation is given below,
ª zD / 2 u V º « H » H n (2) ¬ ¼ 2
n n
minimum sample size
V population standard deviation H user-selected error H n sample size adjustment Barnett et al. (2005) studied regression to mean (RTM) phenomenon and concluded that it is ubiquitous phenomenon in repeated data. It should always be considered as a possible cause of an observed change. Use of better study design and suitable statistical methods can be used to alleviate these effects. Park and Lord (2010) have used graphical methods to illustrate the RTM phenomenon. They used aggregated speed data to show on to reduce RTM bias in before-and-after speed data analysis. From the numerical examples it was found that the estimated magnitude of the mean speed change, due to the introduction of an engineering treatment and the amount of uncertainty which can be measured by the estimated standard error and confidence interval, can be misleading if RTM was not taken into account. Nezamuddin (2010) validated the earlier ITE estimates for moderate traffic conditions on freeways, which was based on standard deviation values for spot speeds for roadways classified by annual average daily traffic (AADT) volumes. This study also found a consistent U-shaped relationship between the standard deviation of speed and traffic volume, which alludes to an inherent speed distribution profile of traffic data. Zhou et al. (2011) determined the small sample size of probe vehicles based on micro-simulation.A simulation-based approach was used to determine the sample size of probe vehicles with consideration of road network coverage and link average speed estimation. The accuracy of estimation increases little and the efficiency decreases significantly with the increase of probe sample size when it reaches a certain level, as proved by microscopic VISSIM simulation. Jung and Gan (2011) have studied sampling and analysis methods for estimation of Average Vehicle Occupancies (AVO). Their paper further describes a detailed process for estimating AVOs at the individual location, facility-type, and county levels. A detailed sampling process designed to select data collection locations and dates on three different facility-types was presented where link-day, was used as the basic sampling unit to determine the minimum sample sizes required to achieve the desired accuracy in AVO estimates.Yang et al.(2011) have studied travel time reliability of probe vehicle system based on minimum sample size analysis.Minimum number of probe vehicles was calculated based on t-statistic, standard deviation and allowable error. Correlation analysis was done between minimum sample size and link length. Linear regression is used to estimate the minimum sample size forecasting model considering all the factors affecting the determination of sample size.Bolbol et al (2012) used journey time data to calculate appropriate sample size for studies that aims to infer the mode of transport from GPS based travel surveys. The sample size required was based on the mode of transport, temporal granularity observed in three zones of London. Maurya and Bokare (2013) observed that the dispersion parameter of speeds not only vary with vehicle type but also with manoeuvres such as acceleration or deceleration. Hence it isconcluded that the sample size (which is a function of dispersion of speed of vehicle) requirement varies with vehicle type and with manoeuvre type.
V. Varsha et al. / Transportation Research Procedia 17 (2016) 384 – 390 3. Methodology 3.1. Data The spot speed data collection was carried out in the city of Ludhiana, Punjab, India,using video-graphic survey on four arterial roads for the peak and non-peak hours. Different types of vehicles ranging from two wheelers to heavy motorized vehicles are analysed. The variation observed in the flow is from moderate to high. The four locations are listed below. 1. Jagraon bridge– Viswakarma Intersection 2. ViswakarmaChowk – Samrala Intersection 3. Jalandhar bypass – Jagraon bridge 4. Jagraon bridge – Bharatnagar Intersection Vehicle detection and analysis is a major problem found in traffic survey mainly due to the lack of lane discipline and seepage of small vehicles like two wheelers. In such cases even the inductive loop detectors fail to classify the vehicles. Usage of laser speed guns was also found to be difficult under mixed traffic conditions as they cannot capture all the vehicles. Macroscopic features like average vehicle speeds, classified vehicle flows, vehicle occupancies and microscopic traffic characteristics like vehicle trajectories, lateral and longitudinal spacing can be obtained from images of video cameras using an offline image processing technique. The sites were selected such that it provides elevation and are as close to centreline of the road as possible: hence only the sites with foot over bridge were selected for mounting camera. Cameras position should be in such a way that it should be able to cover a considerable length of road where vulnerable traffic data falls. This was an important requirement for TRAZER image processing software used for the data extraction. Traffic analyser and enumerators are used for the extraction of prescribed data from the whole traffic data. TRAZER is capable of tracking the vehicles under highly congested conditions Main advantages are; it is nonintrusive, automated, accurate and auditable in nature. The software can process up to four lanes simultaneously. For short time recording 4.5 m tripod is used for holding the cameras and the most common vantage point is the foot over bridge. Using this software (TRAZER) the vehicles were manually marked for higher accuracy. A MATLAB code was developed to analyse the X-Y coordinates given by TRAZER and determine vehicular speed using first and final position of vehicle in its trajectory. The algorithm helps in identifying the portion of scenes that might contain a vehicle from the sequence of images originated from the video cameras. After the identification methodology is carried out, further analysis is done to extract the relevant features. Various methods discussed earlier were used to determine the sample size for the four locations. These methods found sample size with the help of some equations consisting of average range, standard deviation, user selected error, tα and ܼఈൗଶ values.Sample size determination using the methods like Hybrid and Modified methods were dependent on the variation in the standard deviation as well as range. The limitation of the ITE method is that the occurrence of manual error is more in case computing the value of R factor from the x bar chart. Two new methods are proposed in the present study using regression to mean phenomenon. Data points are noted accordingly to the speed data collected. The mean speed and variance were found out from the data and was used for finding the sample size. The occurrence of three consecutively same mean or standard deviation values were used as criteria for finding sample size for mean and standard deviation. Statistical tests were done for the data collected to check whether the data was statistically significant. Prior to t tests, F test was done to check whether the data is having equal or unequal variances. Two sample t tests were carried out according to the F test obtained. Significance of t tests are checked by taking the higher and lower values of the mean speeds with respect to the stabilized one using regression to mean method. 3.2. Results and Discussion Various conventional methods for sample size determination and the proposed methodology are presented in the following tables. The user defined error was taken as 4km/h. F test was done in order to verify the equality of variance.
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V. Varsha et al. / Transportation Research Procedia 17 (2016) 384 – 390 Based on the result of F test, two sample t tests were chosen and are also included in the following tables.From the Table 1, it was clear that for locations 1, 2 and 4, where the data was relatively large, the sample size for mean was higher. The sample size determined using RTM was found to be higher than that obtained using other methods. The method failed to determine the sample size when the proportion of vehicles and flow are low. T tests also indicated that the sample was not significantly varying. Table 1.Sample size estimated by various methods for light motor vehicle in the four locations with the time taken, flow and occupancy
Location
Flow
Occupancy
Location 1 Location 2 Location 3 Location 4
2248 2187 983 2933
0.042 0.020 0.016 0.032
Sample size using RTM for mean 50 40 Not stabilized 60
Sample size using RTM for std. deviation 30 10 Not stabilized 50
ITE Metho d (1999) 13 4 3 7
Hybrid Method (1998)
Modified Method (2002)
t test results
3 11 25 18
6 14 26 20
10 10 10 10
Table 2.Sample size estimated by various methods for heavy motor vehicle in the four locations with the time taken, flow and occupancy
Location
Flow
Occupancy
Location 1 Location 2 Location 3 Location 4
2248 2187 983 2933
0.042 0.020 0.016 0.032
Sample size using RTM for mean 20 Not stabilized Not stabilized Not stabilized
Sample size using RTM for std. deviation 30 Not stabilized Not stabilized Not stabilized
ITE Method (1999)
Hybrid Method (1998)
Modified Method (2002)
t test results
11 4 2 3
3 6 30 16
6 8 29 6
10 9 Not stabilized Not stabilized
From the Table 2,it was observed that the method used for sample size estimation using regression was overestimating the sample size comparing to other methods at location 1.The method failed to determine sample size where the proportion of vehicles and flow were low. T tests also failed at locations where the flow was low. The method for sample size determination using regression to mean failed to estimate the sample size in all the locations other than location 1 due to the low proportion of vehicles and the flow. The three conventional methods were not found to follow any trend. The sample sizes were varying in accordance with the variance in the speed data. Two sample t tests showed that the data was not significantly varying much in case of locations 1 and 2. Table 3.Sample size estimated by various methods for three wheeler vehicle in the four locations with the time taken, flow and occupancy
Location
Flow
Occupancy
Location 1 Location 2 Location 3 Location 4
2248 2187 983 2933
0.042 0.020 0.016 0.032
Sample size using RTM for mean 20 10 Not stabilized 30
Sample size using RTM for std. deviation 10 10 10 10
ITE Method (1999)
Hybrid Method (1998)
Modified Method (2002)
t test results
11 6 4 8
3 4 7 9
6 7 9 12
10 10 10 10
From the Table 3, it was observed that the method used for sample size estimation using regression to mean was overestimating the sample size determined by other methods. The method failed to estimate the sample size for location 3 where the proportion of vehicles and flow are low. Two sample t tests showed that the data was not significantly varying in nature.
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Table 4.Sample size estimated by various methods for two wheeler vehicle in the four locations with the time taken, flow and occupanc y
Location
Flow
Occupancy
Location 1 Location 2 Location 3 Location 4
2248 2187 983 2933
0.042 0.020 0.016 0.032
Sample size using RTM for mean 30 20 Not stabilized 50
Sample size using RTM for std. deviation 40 10 Not stabilized 20
ITE Method (1999)
Hybrid Method (1998)
Modified Method (2002)
t test results
40 8 2 10
2 7 51 22
5 10 48 25
10 10 10 10
From the Table 4, itcan be seen that the method used for sample size estimation using regression was overestimating the sample size comparing to other methods for location 2 and location 4.The method failed to estimate the sample size for location 3 where the proportion of vehicles and flow was low. Two sample t tests showed the datawas not significantly varying. 4. Conclusions From the data analysis, it was observed that the sample size determined by various methods seemed to be varying with vehicle categories, time of the survey and flow parameters for the spot speed data. Since the sample size and number of locations were less in each case, statistical tests like Shapiro-Wilk’s test and other tests for normality and homogeneity of variance cannot be performed. It may thus be difficult to further analyse the data and develop relationships between the variables. Some of the important conclusions based on the present study are given below. x Sample size determined using regression to mean is overestimating the sample size determined from other conventional methods for all the categories of vehicles excluding the cases where the flow and proportion of vehicles are low. x The sample size determined using regression failed at locations where the flow and proportion of vehicles were low. x Sample size determined from the modified method varies with respect to the change in the standard deviation whereas the sample size determined from hybrid method varies with respect to the change in the average range. x T tests can be used as the best suitable method for the determination of sample sizes for the vehicular mean speeds. x As per the result from t tests, minimum number of speed samples to be taken for the study can be as low as ten (n=10). This is irrespective of the vehicle categories and locations as long as flow and proportion of vehicles were relatively high. x In order to assume that the speeds are normally distributed during analysis, the minimum number of samples should be greater than thirty (n>30). Acknowledgements The authors (GHP and KRR) would like to thank Professor Dinesh Mohan, Guest Professor, IIT Delhi, for triggering ideas on sample size requirements and the need for optimizing data collection efforts in traffic studies. References Barnett, A., G., Van der Pols, J., C., & Dobson, A., J., 2005. Regression to the Mean: What it is and How to Deal With It. International Journal of Epidemiology34, 215–220. Bolbol, A., Cheng, T., Tspakis, & Chow, A., 2012. Sample Size Calculation For Studying Transportation Modes From GPS Data. Procedia Social and Behavioural Sciences Transport Research Arena Europe, 3040 –3050. ITE, 1999.Traffic Engineering Handbook.5th edition, Institution of Transportation Engineers, Washington, D.C. Jung, R.,&Gan, A., 2011.Sampling and Analysis Methods for Estimation of Average Vehicle Occupancies. Journal of Transportation Engineering 137, ASCE, pp.537-546
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