Table of Contents
Preface, xvii
Acknowledgments, xxiii
About the Authors, xxv
How to Contact the Authors and Access the Data Sets, xxvii
Chapter 1 ▪ Overview: The Role of Statistics in
Transportation Engineering 1
1.1 What Is Engineeri ng? 2
1.2 What Is Transport atio n Engineeri ng? 5
1.3 Go al of the Textboo k 7
1.4 Over view of the Textboo k 7
1.5 Who Is the Audie nce for This Textboo k? 8
1.6 Relax—Ever ything Is Fi ne 8
Refere nces 9
Chapter 2 ▪ Graphical Methods for Displaying Data 11
2.1 Intro duction 11
2.2 Histo gram 13
2.3 Bo x and Whis ker Plot 16
2.4 Quanti le Plot 18
2.5 Scatter Plot 19
2.6 Parallel Plot 22
2.7 Time Series Plot 23
viii ◾ Table of Contents
2.8 Qualit y Co ntro l PL OTS 24
2.9 Co ncluding Rem arks 26
Homewor k Pro blems 26
Refere nces 27
Chapter 3 ▪ Numerical Summary Measures 29
3.1 Intro duction 29
3.2 Measures of Ce ntr al Tendency 30
3.2.1 Mean 30
3.2.2 Median 32
3.2.3 Trimmed Mean 33
3.2.4 Mode 33
3.2.5 Proportion 34
3.3 Measures of Relati ve Standing 35
3.3.1 Percentile 35
3.3.2 Quartile 36
3.4 Measures of Vari abilit y 36
3.4.1 Range 36
3.4.2 Variance 37
3.4.3 Standard Deviation 37
3.4.4 Interquartile Range 38
3.4.5 Coefficient of Variation 38
3.5 Measures of Asso ciatio n 39
3.5.1 Pearson’s Sample Correlation Coefficient 40
3.5.2 Spearman’s Rank Correlation Coefficient 41
3.6 Co ncluding Rem arks 43
Homewor k Pro blems 44
Refere nces 46
Chapter 4 ▪ Probability and Random Variables 47
4.1 Intro duction 47
4.2 Sample Spaces and Events 47
4.3 Inter pret atio n of Pro babilit y 48
Table of Contents ◾ ix
4.4 Random Vari able 50
4.4.1 Functions of a Random Variable 52
4.5 Expectatio ns of Random Vari ables 53
4.5.1 Expected Values Are Linear 54
4.6 Co vari ances and Corre latio n of Random
Vari ables 54
4.7 Com puti ng Expecte d Values of Functio ns
of Random Vari ables 55
4.8 Co nditio nal Pro babilit y 57
4.9 Bayes ’ Theorem 59
4.10 Co ncluding Rem arks 60
Homewor k Pro blems 61
Refere nces 63
Chapter 5 ▪ Common Probability Distributions 65
5.1 Intro duction 65
5.2 Dis crete Distri butio ns 66
5.2.1 Binomial Distribution 66
5.2.2 Poisson Distribution 69
5.2.3 Negative Binomial Distribution 71
5.3 Co nti nuous Distri butio ns 73
5.3.1 Normal Distribution 74
5.3.2 t-Distribution 78
5.3.3 Lognormal Distribution 79
5.3.4 Exponential Distribution 80
5.3.5 Gamma Distribution 81
5.3.6 Chi-Square Distribution 82
5.3.7 F-Distribution 84
5.3.8 Beta Distribution 85
5.4 Co ncluding Rem arks 85
Appendix: Table of the Most Po pular
Distri butio ns in Transport atio n Engineeri ng 87
x ◾ Table of Contents
Homewor k Pro blems 87
Refere nces 88
Chapter 6 ▪ Sampling Distributions 91
6.1 Intro duction 91
6.2 Random Sampling 92
6.3 Sampling Distri butio n of a Sample Mean 94
6.4 Sampling Distri bution of a Sample
Vari ance 103
6.5 Sampling Distri butio n of a Sample
Pro portio n 103
6.6 Co ncluding Rem arks 106
Homewor k Pro blems 107
Refere nces 109
Chapter 7 ▪ Inferences: Hypothesis Testing and
Interval Estimation 111
7.1 Intro duction 111
7.2 Fundame ntals of Hypot hesis Testi ng 112
7.3 Infere nces on a Single Po pulatio n Mean 114
7.3.1 Hypothesis Tests about a Population Mean 114
7.3.1.1 z-Test 116
7.3.1.2 t-Test 117
7.3.2 Interval Estimates for a Population Mean 119
7.3.2.1 Bias-Adjusted Confidence Intervals 120
7.4 Infere nces about Two Po pulatio n
Means 123
7.4.1 Hypothesis Tests about Equality of Two
Population Means 123
7.4.1.1 Paired t-Test 123
7.4.1.2 Pooled t-Test 125
7.4.1.3 Unequal Variance t-Test 125
Table of Contents ◾ xi
7.4.2 Interval Estimates and Bias Adjustment for
Difference of Two Population Means 127
7.5 Infere nces about One Po pulatio n
Vari ance 131
7.6 Infere nces about Two Po pulatio n
Vari ances 133
7.6.1 Confidence Intervals for the Ratio of
Two Variances 134
7.6.2 Bias-Corrected Confidence Interval for the Ratio
of Two Variances 134
7.6.3 One-Sided Tests 135
7.7 Co ncluding Rem arks 135
Appendix: Welch (1938) De grees of Free dom for
the Uneq ual Vari ance t-Test 136
Homewor k Pro blems 136
Refere nces 138
Chapter 8 ▪ Other Inferential Procedures: ANOVA and
Distribution-Free Tests 139
8.1 Intro duction 139
8.2 Com pariso ns of More than Two
Po pulatio n Means 140
8.3 Multi ple Com pariso ns 143
8.4 One- and Multiw ay AN OVA 144
8.5 Ass umptio ns for AN OVA 149
8.6 Distri butio n-Free Tests 151
8.6.1 The Kolmogorov–Smirnov Goodness-of-
Fit Test 152
8.6.2 The Kruskal–Wallis Approach to ANOVA 153
8.7 Co nclusio ns 154
Homewor k Pro blems 154
Refere nces 155
xii ◾ Table of Contents
Chapter 9 ▪ Inferences Concerning Categorical Data 157
9.1 Intro duction 157
9.2 Tests and Co nfidence Inter vals for a
Single Pro portio n 157
9.3 Tests and Co nfidence Inter vals for Two
Pro portio ns 161
9.4 Chi-Square Tests Co ncer ning More Than
Two Po pulatio n Pro portio ns 163
9.4.1 Chi-Square Test for Univariate Categorical Data 163
9.4.2 Tests for Independence of Two Categorical
Variables 166
9.4.3 Tests for Homogeneity of Two or More
Populations 171
9.5 The Chi-Square Goo dness -of-Fit Test for
Checking Distri butio nal Ass umptio ns 174
9.6 Co nclusio ns 175
Homewor k Pro blems 176
Refere nces 177
Chapter 10 ▪ Linear Regression 179
10.1 Intro duction 179
10.2 Sim ple Li near Regressio n 180
10.2.1 Correlation Coefficient 181
10.2.2 Fitting a Straight Line 183
10.2.2.1 Estimating Slope and Intercept 185
10.2.2.2 Inferences on Slope and Intercept 186
10.2.3 Prediction 189
10.3 Transform ations 191
10.4 Underst anding and Calculati ng R2 192
10.5 Veri fying the Main Ass umptio ns in Li near
Regressio n 193
10.6 Com pari ng Two Regressio n Li nes at a
Poi nt and Com pari ng Two Regressio n
Parameters 194
Table of Contents ◾ xiii
10.7 The Regressio n Dis conti nuit y Desi gn (RDD ) 195
10.8 Multi ple Li near Regressio n 195
10.8.1 Confidence Intervals on Regression Parameters 199
10.8.1.1 Extrapolation 199
10.9 Vari able Selectio n for Regressio n Models 199
10.10 Additio nal Co llinearit y Iss ues 201
10.11 Co ncluding Rem arks 202
Homewor k Pro blems 203
Refere nces 204
Chapter 11 ▪ Regression Models for Count Data 205
11.1 Intro duction 205
11.2 Poisso n Regressio n Model 206
11.3 Over dis persio n 208
11.4 Assessi ng Goo dness of Fit of Poisso n
Regressio n Models 209
11.5 Ne gati ve Bi nomi al Regressio n Model 215
11.6 Co ncluding Rem arks 226
Appendix: Maxim um Li kelihoo d Estim atio n 226
Homewor k Pro blems 228
Refere nces 229
Chapter 12 ▪ Experimental Design 231
12.1 Intro duction 231
12.2 Com pariso n of Dire ct Obser vatio n and
Desi gned Experime nts 231
12.3 Moti vatio n for Experime ntatio n 233
12.3.1 The Good Experiment 233
12.3.2 The Poor Experiment 233
12.4 A Three -Factor , Two Le vels per Factor
Experime nt 235
12.4.1 A Common Fallacy: Changing One Factor
at a Time Experiments 236
xiv ◾ Table of Contents
12.5 Factori al Experime nts 237
12.6 Fr actio nal Factori al Experime nts 239
12.7 Scree ning Desi gns 241
12.8 D-Optim al and I-Optim al Desi gns 242
12.9 Sample Size Determi natio n 245
12.10 Fie ld and Quasi -Experime nts 248
12.11 Co ncluding Rem arks 252
Appendix: Choi ce Modeling of Experime nts 253
Homewor k Pro blems 254
Refere nces 255
Chapter 13 ▪ Cross-Validation, Jackknife, and Bootstrap
Methods for Obtaining Standard Errors 257
13.1 Intro duction 257
13.2 Met hods for Standard Error Estim atio n
When a Close d-Form Form ula Is Not
Available 258
13.3 Cross -Validation 259
13.4 The Jackknife Met hod for Obtaining
Standard Errors 260
13.5 Bootstr apping 262
13.6 Co ncluding Rem arks 265
Homewor k Pro blems 265
Refere nces 266
Chapter 14 ▪ Bayesian Approaches to Transportation
Data A nalysis 267
14.1 Intro duction 267
14.2 Fundame ntals of Bayesi an Statisti cs 268
14.3 Bayesi an Infere nce 270
14.3.1 Conjugate Priors 271
14.3.1.1 For Poisson Likelihood 271
14.3.1.2 For Binomial Likelihood 272
14.3.1.3 For Normal Likelihood 273
Table of Contents ◾ xv
14.3.2 Point Estimation 277
14.3.3 Uncertainty Estimation 279
14.3.3.1 Posterior Standard Deviation 279
14.3.3.2 Credible Intervals 281
14.4 Co ncluding Rem arks 284
Homewor k Pro blems 284
Refere nces 286
Chapter 15 ▪ Microsimulation 287
15.1 Intro duction 287
15.2 Over view of Traffic Microsim ulatio n
Models 287
15.3 Analyzing Microsim ulatio n Output 294
15.3.1 Model Calibration Validation and Verification 294
15.3.2 Aggregate Goodness-of-Fit Measurements 295
15.3.3 Statistical Analysis of Microsimulation Output 297
15.3.4 Identifying the Statistically Optimal Number of
Simulation Runs 301
15.3.5 Microsimulation Calibration 302
15.3.6 Computer-Based Microsimulation Calibration 304
15.3.7 Absolute vs. Relative Model Accuracy 306
15.4 Per form ance Measures 307
15.4.1 Overview of Performance Measures 308
15.4.2 Common Performance Measures Used in
Simulation Analysis 310
15.4.3 Travel Time 311
15.4.4 Total Delay 313
15.4.5 Queue Length (Oversaturation) 316
15.4.6 Number of Vehicles Completing Simulation 317
15.4.7 Percentage of Corridor (OD Movements)
Congested (PCC) Measure 317
15.4.8 Travel Time Variability 319
15.4.9 Level of Service 320
xvi ◾ Table of Contents
15.4.10 Travel Rate 321
15.4.11 Performance Indices 321
15.5 Co ncluding Rem arks 321
Homewor k Pro blems 322
Refere nces 322
Appendix: Soft Modeling and Nonparametric
Model Building, 325
INDEX, 331
Preface
The basic concept of transportation—the movement of goods and people
over time and space—has changed little since the Romans developed their
transportation system over two thousand years ago. Today we have far
more extensive transportation infrastructure systems that include roads,
waterways, railways, and air transport options, and much more sophisticated
technology than in the past, but the overall objective of transportation
engineers remains the same—to plan, design, construct, and maintain
the various transportation modal systems in the safest and most efficient
manner possible. To achieve this goal, transportation professionals have to
be able to answer fairly sophisticated questions, such as:
Which pavement is most economical for a given situation?
What roadway geometry is safer?
What traffic control device works best?
Where should we invest our limited resources to produce the most
favorable outcome?
To answer these types of questions, the engineers and planners identify
a clear hypothesis, collect relevant data (either through experiment
or observation), and develop reasonable conclusions from the data, all of
which will require the transportation professional to have a solid grounding
in statistics. This text is designed to provide the necessary background
knowledge to make informed transportation-related decisions.
With transportation accounting for between 10% and 20% of the U.S.
economy, the types of questions listed above are asked thousands of times
per day across the country. Unfortunately, textbooks that relate specifically
to transportation statistics, to which a transportation professional
can turn for help, are very few. Though there are many general engineering
statistics books that can provide the necessary background material, these
books do not address statistics from the unique perspective of transportation.
This textbook helps to fill that gap by discussing statistical concepts
in the context of transportation planning and operations.
Our anticipated audience is comprised of transportation professionals,
both planners and engineers, who are looking for more sophisticated
information than is found in a general undergraduate statistics course,
frequently required by a professional program. This textbook would be
ideal for an introductory graduate class in transportation statistics that
could be taken by first-year students specializing in pavements engineering,
transportation systems engineering, or urban planning. In addition,
the book would be useful for working professionals who would like to
learn more about some of the concepts to which they are exposed during
their transportation careers.
While much of what planners and engineers do has been codified
(e.g., take twenty random pavement samples and test to failure, etc.),
this book will help explain the why behind the standard methods. Lastly,
we assume the reader has a basic knowledge of introductory probability
and statistics as well as a strong working knowledge of basic transportation
concepts.
We, the authors, have over fifty combined years of using statistical
techniques for transportation research and teaching. Two factors initiated
our collaborative development of this textbook: (1) a graduate
statistical course established by the authors at Texas A&M University
approximately ten years ago, and (2) over ten years of collaboration on
various transportation research projects that involved many of the statistical
techniques discussed in this book. Based on these experiences,
we made two fundamental, yet dichotomous, observations that motivated
us to write this text. The first is that many transportation professionals
suffer from statistical anxiety—not because the concepts are so
difficult, but because they have not been presented in a way that could be
easily understood by practicing engineers and planners. We felt the best
way to reduce this fear was to use specific transportation examples and
problems, all based on real situations using real data, to illustrate the key
concepts in the text. The second is that there is also, ironically, a danger
of overconfidence. That is, many transportation professions assume
that their tried and true statistical methods, which they have become
accustomed to using in their professional lives, would continue to be
appropriate, even though the underlying assumptions of these statistical
techniques are no longer valid. Equally important are the numerous new
techniques that could be used in everyday practice but have not been
well explained in the available literature. That is, the transportation professionals
did not know what they did not know. We hope this textbook
fulfills this need.
We have made a concerted effort to define explicitly the underlying
assumptions and to provide references and insight so the readers will
know when they need to seek outside help from practicing statisticians.
In addition, terms have been carefully defined from both a statistical and
a transportation perspective. In our experience, the jargon adopted by the
transportation and statistics profession often works at cross-purposes; we
are hopeful that this text will help make the conversations between transportation
professionals and statisticians smoother and more productive
for both parties.
