National Academies Press: OpenBook

Recent Decline in Public Transportation Ridership: Analysis, Causes, and Responses (2022)

Chapter: Chapter 4 - Bus Ridership and Frequency Trends by Time of Day in Four Cities

« Previous: Chapter 3 - Multicity Evaluation
Page 33
Suggested Citation:"Chapter 4 - Bus Ridership and Frequency Trends by Time of Day in Four Cities." National Academies of Sciences, Engineering, and Medicine. 2022. Recent Decline in Public Transportation Ridership: Analysis, Causes, and Responses. Washington, DC: The National Academies Press. doi: 10.17226/26320.
×
Page 33
Page 34
Suggested Citation:"Chapter 4 - Bus Ridership and Frequency Trends by Time of Day in Four Cities." National Academies of Sciences, Engineering, and Medicine. 2022. Recent Decline in Public Transportation Ridership: Analysis, Causes, and Responses. Washington, DC: The National Academies Press. doi: 10.17226/26320.
×
Page 34
Page 35
Suggested Citation:"Chapter 4 - Bus Ridership and Frequency Trends by Time of Day in Four Cities." National Academies of Sciences, Engineering, and Medicine. 2022. Recent Decline in Public Transportation Ridership: Analysis, Causes, and Responses. Washington, DC: The National Academies Press. doi: 10.17226/26320.
×
Page 35
Page 36
Suggested Citation:"Chapter 4 - Bus Ridership and Frequency Trends by Time of Day in Four Cities." National Academies of Sciences, Engineering, and Medicine. 2022. Recent Decline in Public Transportation Ridership: Analysis, Causes, and Responses. Washington, DC: The National Academies Press. doi: 10.17226/26320.
×
Page 36
Page 37
Suggested Citation:"Chapter 4 - Bus Ridership and Frequency Trends by Time of Day in Four Cities." National Academies of Sciences, Engineering, and Medicine. 2022. Recent Decline in Public Transportation Ridership: Analysis, Causes, and Responses. Washington, DC: The National Academies Press. doi: 10.17226/26320.
×
Page 37
Page 38
Suggested Citation:"Chapter 4 - Bus Ridership and Frequency Trends by Time of Day in Four Cities." National Academies of Sciences, Engineering, and Medicine. 2022. Recent Decline in Public Transportation Ridership: Analysis, Causes, and Responses. Washington, DC: The National Academies Press. doi: 10.17226/26320.
×
Page 38
Page 39
Suggested Citation:"Chapter 4 - Bus Ridership and Frequency Trends by Time of Day in Four Cities." National Academies of Sciences, Engineering, and Medicine. 2022. Recent Decline in Public Transportation Ridership: Analysis, Causes, and Responses. Washington, DC: The National Academies Press. doi: 10.17226/26320.
×
Page 39
Page 40
Suggested Citation:"Chapter 4 - Bus Ridership and Frequency Trends by Time of Day in Four Cities." National Academies of Sciences, Engineering, and Medicine. 2022. Recent Decline in Public Transportation Ridership: Analysis, Causes, and Responses. Washington, DC: The National Academies Press. doi: 10.17226/26320.
×
Page 40
Page 41
Suggested Citation:"Chapter 4 - Bus Ridership and Frequency Trends by Time of Day in Four Cities." National Academies of Sciences, Engineering, and Medicine. 2022. Recent Decline in Public Transportation Ridership: Analysis, Causes, and Responses. Washington, DC: The National Academies Press. doi: 10.17226/26320.
×
Page 41
Page 42
Suggested Citation:"Chapter 4 - Bus Ridership and Frequency Trends by Time of Day in Four Cities." National Academies of Sciences, Engineering, and Medicine. 2022. Recent Decline in Public Transportation Ridership: Analysis, Causes, and Responses. Washington, DC: The National Academies Press. doi: 10.17226/26320.
×
Page 42
Page 43
Suggested Citation:"Chapter 4 - Bus Ridership and Frequency Trends by Time of Day in Four Cities." National Academies of Sciences, Engineering, and Medicine. 2022. Recent Decline in Public Transportation Ridership: Analysis, Causes, and Responses. Washington, DC: The National Academies Press. doi: 10.17226/26320.
×
Page 43

Below is the uncorrected machine-read text of this chapter, intended to provide our own search engines and external engines with highly rich, chapter-representative searchable text of each book. Because it is UNCORRECTED material, please consider the following text as a useful but insufficient proxy for the authoritative book pages.

33   The analysis conducted in Chapter 3 uses service and ridership information by month at the transit agency level, which was provided by those agencies to NTD. While these data have been used to delineate the broad trends in ridership, they don’t help answer the “when” and “where” questions, which are instrumental to identify the causes of ridership decline and define strategies to reverse the trend. Yet, understanding the underlying dynamics of bus ridership is integral for the effective management of bus transit systems. Data collected from automatic passenger counters (APCs) can provide a unique window into the distribution of ridership by time of day and day of the week. This chapter analyzes transit ridership trends on a hyper-local level: segments of seven consecutive stops on the same route and direction. Combined with standardized schedule data from the General Transit Feed Speci- fication (GTFS), ridership can be compared with service levels. The quantity of service, which is measured in terms of frequency, is fundamental to understanding the distribution of demand, both temporally and spatially. Service frequency determines the feasibility and reliability of transit trips. Therefore, in the following section, trends in bus ridership and service quantity over time are explored together in four transit agencies: • TriMet in Portland, Oregon • Miami-Dade Transit in Miami, Florida • Metro Transit in Minneapolis/St. Paul, Minnesota • Metropolitan Atlanta Rapid Transit Authority (MARTA) in Atlanta, Georgia These four transit agencies have all experienced losses in bus ridership, following the national trend. Serving passengers in different regions of the United States, these four bus systems are dif- ferent in key ways that make them representative of other large to mid-sized transit agencies as a group. The four agencies operate in differently sized cities with varying density profiles. At the same time, they are similar enough in size to be compared to each other. They all serve between 50 million and 57 million bus trips per year. Most importantly, they were all early adopters of passenger counting technology. Over multiple years, they have maintained the high standard of data collection, cleaning, and processing required for this analysis. 4.1 Bus Ridership Trends Identifying the temporal dimension of the recent bus ridership decline is crucial to under- standing the underlying causes and finding solutions to reverse the trend. There is a common narrative that bus ridership has been especially declining during off-peak and weekends (Bliss, 2017b). According to a report from UCLA (University of California, Los Angeles) and the analysis in Chapter 5, the Bay Area Rapid Transit (BART) authority has experienced a concentration of demand during peak hours at the expense of night and weekend service (Blumenberg et al., C H A P T E R 4 Bus Ridership and Frequency Trends by Time of Day in Four Cities

34 Recent Decline in Public Transportation Ridership: Analysis, Causes, and Responses 2020). Likewise, rail ridership in New York has been declining at a faster rate during nights and on the weekends (Fitzsimmons, 2017). But has bus ridership declined at a faster rate during off-peak and weekends? A report from MBTA shows that between 2014 and 2018, off-peak bus ridership declined by 10% while peak ridership only dropped by 8% (Thistle and Zimmer, 2019). There is still little evidence, however, as to whether the trend in Boston is representative of what is happening in other bus systems. Therefore, more research is needed to identify the temporal dynamics of bus ridership decline. APC data can be used to track ridership trends by time of day and day of the week over time. Tracking these trends helps determine whether some time periods are driving the decline while others are bucking the trend, or whether all time periods are affected equally. The relative dif- ference in ridership change between time periods may be explained by changes in service levels. Therefore, tracking when transit agencies have modified their service over the last several years is necessary to contextualize the trends. The graphical representation of both the supply and demand for bus service over time can then support a more refined analysis of their causal relationship. Another temporal element affecting bus ridership is the seasonality. Especially in cities expe- riencing wide fluctuations in weather throughout the year, cold or hot temperatures may affect mode choice, route choice, times of travel, and the rate of telecommuting. Seasonality also impacts the schedules of universities and school-age children and thus vacations and commuting hours. While the seasonality of ridership is well established in the literature, an understanding of how its impacts are distributed by time of day and day of the week is still lacking (Kashfi et al., 2015; Stover and McCormack, 2012). Understanding how pronounced these fluctuations are may inform the service allocation process. It is also interesting to see how transit agencies respond to the fluctua- tions in demand with their service. Finally, comparing the seasonality of both service and ridership by time period can help shed light on the yearly fluctuations in travel demand. Figure 4-1 shows the relative change in bus ridership over time in Portland, Miami, Minneapolis/ St. Paul, and Atlanta. As shown in the legend at the bottom, weekday time periods are represented by colored solid lines, whereas the weekend time periods are black and grey dashed lines. The time- table on the right shows how time periods are defined. There is a clear downward trend in most cities across all time periods. Weekend night rider- ship (light grey) is declining at a steeper rate than the other time periods in all four cities. While weeknights (light purple) seem to be following weekend trends, middays (dark purple) are much closer to a.m. and p.m. peaks (blue and red, respectively), which experienced the slowest decline across the four MSAs. In sum, although no time period has escaped the downward ridership trend, weeknights and weekend nights are particularly affected. In order to understand better the ridership trends displayed in Figure 4-1, it is impor- tant to take into consideration how service has changed over the years. Figure 4-2 shows the relative change in bus service provided, measured in total vehicle trips, in Portland, Miami, Minneapolis/St. Paul, and Atlanta. Service quantity seems to have some slight seasonality in Minneapolis and no perceptible seasonality in Portland. While the ridership trends explored previously are consistent across cities, there are some important differences in terms of service allocation policies: • In Portland, bus service quantity has increased in all time periods, especially off-peak. • In the Twin Cities, bus service increased at night and over the weekends but remained con- stant during the weekdays. • In Miami and Atlanta, bus service during peak hours has been reduced. Miami also cut bus service during nights and weekends, while Atlanta increased bus service in those time periods. • Overall, night and weekend service consistently increased at a faster rate or declined at a slower rate in all agencies except for Miami, where it fluctuated.

Figure 4-1. Relative change in bus ridership by time period in four MSAs.

Figure 4-2. Relative change in total vehicle trips by time period in four MSAs.

Bus Ridership and Frequency Trends by Time of Day in Four Cities 37   Seasonality In addition to ridership trends over the years, Figure 4-1 captures the seasonality of bus ridership. In Miami and Atlanta, ridership barely fluctuates throughout the year. While both cities have a warmer climate in the winter months, they also experience heat waves over the summer. The lack of transit ridership seasonality in Atlanta and Miami may reflect the lack of alternative options for bus riders in both cities. While the Portland and Minneapolis transit systems serve 4.5% and 6.1% of commuting trips, respectively, Atlanta and Miami only have a 3% to 3.1% transit mode share. Therefore, the smaller pool of bus riders in these two cities may be relying on the bus regardless of the weather condition. Portland and Minneapolis/St. Paul exhibit strong seasonality. Interestingly, week- day ridership is out of phase with nighttime and weekend ridership, which tends to peak in the summer. Weekday ridership, on the other hand, peaks in the spring and fall. Interviews with transit planners revealed that weekday ridership is heavily influenced by students, whose travel demand is diminished over the summer. In the Twin Cities, all public high school students get a transit pass; in Portland, students have access to special fares. Night and weekend ridership is highest during the summer due to the prevalence of leisure activities such as the Minnesota State Fair and various sporting events. The changes in service quantity in Figure 4-2 do not explain the ridership declines observed in Figure 4-1. Although only Atlanta cut service during the peaks and only Miami cut service across all time periods, bus ridership has been declining for all transit agencies and all time periods, even when service was increased. Paradoxically, in Portland, Minneapolis, and Atlanta, weekdays and weekend nights have experienced the greatest bus ridership losses while also gaining the most or losing the least service. In Miami, night and weekend service was reduced at approximately the same rate as during weekdays. These results indicate that the change in bus service quantity does not correspond to the variation in ridership change between time periods in each transit agency and overall between transit agencies. Therefore, the common causes of bus ridership decline are likely to be external. Among the external factors that may particularly affect evening and weekend ridership is ride-hailing, which started operating in cities throughout the United States in 2012. The analysis in Chapter 3 shows that the entrance of ride-hailing in a metropolitan area is correlated with bus ridership decline. Ride-hailing has been found to be far more prevalent during nighttime and weekends, when bus ridership tends to flatten (Tirachini et al., 2019; Feigon and Murphy, 2018; Dias et al., 2019; Li, 2019). 4.2 Comparing Bus Ridership and Productivity by Time Period Although the changes in frequency cannot explain why bus ridership has been declining since 2012, they may be key to reversing the trend. While service planners strive to match the supply to the times and places that have the greatest concentration of demand, they must also provide suf- ficient service throughout the day and across the network to ensure wide accessibility. However, research has shown that demand varies throughout the day beyond what temporal accessibility can explain (Legrain et al., 2015). Therefore, to understand how transit service planners can

38 Recent Decline in Public Transportation Ridership: Analysis, Causes, and Responses allocate frequency to maximize ridership, it is important to ask how and why the demand for bus service varies throughout the day. In this section, the relative distribution of demand and supply for buses by time period is compared. The research team used the bus ridership data from Figure 4-1 and the bus service allocation data in Figure 4-2 to assess the bus service productivity in the four cities. Figure 4-3 shows this bus service productivity in terms of passenger boardings and alightings per trip per stop by time period over the years in Portland, Miami, Minneapolis/St. Paul, and Atlanta. The productivity is normalized by the number of stops to compare routes of different lengths. This is intuitively equivalent to the average number of passenger boardings each time a bus passes a bus stop. All four transit agencies have similar productivities across time periods, although TriMet has slightly greater productivities overall. Across all four agencies, a common thread is that nights— both during the week and over the weekends—are the least productive time periods, whereas peak periods and midday are the most productive. Weekday time periods are the most produc- tive. Interestingly, midday productivity is at the same level as—or in the case of Atlanta and Miami, even higher than—a.m. peak. 4.3 Bus Ridership Elasticity to Frequency Evaluating the causal relationship between bus service quantity and ridership can help support important policy decisions. Transit service planners are tasked with deciding when and where to add or remove service. The amount of service provided is a key lever available for transit agencies to affect bus ridership. While there are other important factors participating in this decision process—such as equitable access, connection to places of strategic importance, and reliability— the anticipated impact on ridership is the initial consideration. Ridership not only is a measure of impact on local mobility but also it determines revenue through fares, which are instrumental for a transit agency’s capacity to deliver service. Fortunately, unlike external factors, which tend to impact travel demand and mode choice continuously over time, changes in frequency typically have immediate effects on ridership that can therefore be measured directly. The concept of elasticity is closely related to productivity, introduced in the previous section. Elasticity informs the rate of change in productivity. Since productivity measures the number of passengers per trip, an elasticity > 1 means that adding service results in increased productivity. On the other hand, if the elasticity is between 0 and 1, then each additional vehicle trip will generate ridership but not as much as the average vehicle already in operation. Since productivity varies by route and by time period, comparing the absolute change in rider- ship would not be overly insightful. If, for example, ridership declined by 5% uniformly across Productivity A metric of comparison is productivity, which measures passengers per vehicle trip. Analyzing the data temporally allows us to identify which time periods are the most productive. This is particularly important since research has shown that providing service during peak hours is more expensive due to the directionality of demand, short or split operator shifts, and the need for high vehicle capacity. By looking at four diverse and large to mid-sized transit agencies, a comparison can be established between agencies to determine how time periods rank among each other across transit agencies.

Figure 4-3. Productivity in passenger boardings and alightings by time period.

40 Recent Decline in Public Transportation Ridership: Analysis, Causes, and Responses the population and trip purposes, the routes and vehicle trips with the greatest ridership would experience the greatest loss. Since elasticity is a relative measure, it allows for comparison of times and places that have different levels of demand and supply to begin with. When combined, elasticity and productivity make it possible to anticipate the impact of service changes on rider- ship. For example, the change in ridership—denoted Δ Ridership—that results from adding one more vehicle on the route (i.e., Δ Frequency = 1) is the elasticity multiplied by the productivity, as shown in the equations below: = ∆ ∆ = ∆      ∆      ∆ = ∆ =   % % 0 0 0 0 Elasticity Ridership Frequency Ridership Ridership Frequency Frequency Ridership Elasticity Ridership Frequency Ridership Elasticity Productivity p p The last equation shows that the expected change in ridership generated by adding one vehicle trip is the productivity multiplied by elasticity. Therefore, estimating elasticities and productivities allows for comparisons that help anticipate the impact of service changes. The times and places that have high productivity and high elasticity have the greatest potential growth in ridership and could be part of a strategy to reverse the decline. 4.4 Modeling Elasticity for Bus Services Previous research conducted by the authors in Portland, Miami, Minneapolis/St. Paul, and Atlanta found that at point-in-time, the most frequent routes generally have the most produc- tivity (Berrebi and Watkins, 2020). The research revealed that over time, ridership was inelastic to frequency. In every city besides Minneapolis/St. Paul, this was particularly true on the most Elasticity The relationship between transit ridership and service is quantified as elasticity. Elasticity measures the sensitivity of demand to marginal changes in an explana- tory variable. The concept of elasticity is widely used in economics and business to estimate how demand changes in response to changes in price—or in the case of transit, how demand changes with regard to transit fares. In this analysis, how- ever, the research team investigated the impact of service quantity, measured as the total number of vehicle trips. Service elasticity gives the relative change in ridership resulting from the relative change in service. In other words, elasticity is the percentage change in transit ridership assuming that service increased by 1%. • When elasticity < 1, a 1% increase in service yields a less than 1% increase in ridership. • When elasticity = 1, a 1% increase in service yields a 1% increase in ridership. • When elasticity > 1, a 1% increase in service yields a more than 1% increase in ridership.

Bus Ridership and Frequency Trends by Time of Day in Four Cities 41   frequent routes. Nonetheless, an analysis investigating these dynamics at the time-period level is still lacking. This is important since transit agencies have adopted policies to adjust service differently by time of day and day of the week. Passengers traveling to their destination will typically use the same set of modes and routes to return to their origin. On a deeper level, perceived service reliability is needed the most when it is the weakest (i.e., when and where frequency is the lowest). These places and time periods have an outsize effect on travel behavior. Being able to compare elasticities to service quantity may pro- vide strategic tools to help transit agencies set frequencies to maximize ridership throughout the day. With four transit agencies serving as case studies, common threads in hyper-local dynamics can be identified to shed light on the broader trends among large to mid-sized bus agencies. This analysis explores the connection between bus ridership and service frequency at the route- segment level. Figure 4-4 shows a typical map of route-segments in Portland. Using data from the Census Bureau’s Longitudinal Employer-Household Dynamics (LEHD) program, the research team collected population and jobs data within a 0.25-mile radius of each route-segment. Some route-segments have bus stops spaced out so that the 0.25-mile buffers do not overlap (see, for example, the blue segment). Others, such as the purple, orange, and green segments, are much Figure 4-4. Map illustrating route-segment buffers in Portland.

42 Recent Decline in Public Transportation Ridership: Analysis, Causes, and Responses more compact. For those segments, population and jobs are only counted once, even if they fall in the overlap of several bus stops within the same route-segment. The model results are presented in this section. Modeling steps and assumptions are described in Appendix G of TCRP Web-Only Document 74. The objective of this section is to evaluate the elasticity of ridership to frequency by time period, which is defined in the same way here as in Figures 4-1, 4-2, and 4-3. Table 4-1 shows the results of this analysis for Portland, Miami, Minneapolis, and Atlanta. The top half of the table shows the estimated linear time trend, and the bottom half shows the estimates of ridership elasticity to frequency. The trends observed in Figure 4-1 are also seen in the model of Table 4-1. As shown by the negative values for the time-trend, there is a downward Route-Segment Route-segments are defined in this report as clusters of seven adjacent stops on the same route and direction. Population and jobs from the LEHD program are assigned to route-segments within 0.25 miles. This distance corresponds to the typical walking distance to access a bus stop, according to the Transit Capacity and Quantity of Service Manual (Kittelson & Associates et al., 2013). Portland Miami Minneapolis Atlanta Weekday Time-trend Peak (a.m.) 0 -0.05 -0.01 -0.06 Midday -0.02 -0.05 -0.03 -0.07 Peak (p.m.) 0 -0.05 -0.02 -0.06 Night -0.01 -0.07 -0.03 -0.08 Weekend Time-trend Day -0.01 -0.05 -0.03 -0.06 Night -0.02 -0.1 -0.05 -0.08 Weekday Frequency Elasticity Peak (a.m.) 0.61 0.68 0.77 0.58 Midday 0.55 0.68 0.73 0.65 Peak (p.m.) 0.52 0.38 0.83 0.6 Night 0.72 0.94 0.78 0.76 Weekend Frequency Elasticity Day 0.42 0.84 0.71 0.5 Night 0.39 0.94 0.92 0.77 Population and Jobs Not Significant (p > 0.05) Positive Effect Note: Peak (a.m.) = 6–9 a.m., Peak (p.m.) = 4–7 p.m., Weekend Day = 6 a.m.–7 p.m. Table 4-1. Results for ridership elasticity to frequency by time period.

Bus Ridership and Frequency Trends by Time of Day in Four Cities 43   trend in all agencies and time periods except for Portland in a.m. and p.m. peaks, which both have a time-trend value of 0. As shown by the magnitude of this time-trend, Miami and Atlanta experienced greater declines in ridership (declines ranging from 5% to 10%) compared to Port- land and Minneapolis (declines ranging from 0% to 5%) across all time periods. In general, peak periods have incurred less of a decline (0% to 6%) than other periods, with nighttime declining the most (1% to 10%). It is also important to compare ridership elasticity to frequency by time period. As shown in Table 4-1, weeknights are the closest to 1.0, meaning they are among the most elastic time periods across agencies. Except for Portland, weekend nights are also close to 1.0, meaning they are among the most elastic time periods. These elasticities close to 1.0 indicate that investing in nighttime service should not lead to substantial losses in productivity. While nighttime service is unlikely to reach current daytime levels of productivity, it could be part of a broader policy of service expansion. If, for example, a transit agency increased bus service by 10% across all time periods, nighttime productivity would remain lower than daytime but by a smaller margin. 4.5 Conclusion Automatic passenger count data were used to examine the trends in bus ridership on a hyper- local level by time of day and day of the week. Based on schedule data, the research team investi- gated the sensitivity of ridership to changes in service frequency. This relationship was explored graphically by looking at the relative changes in ridership, service quantity, and productivity over time. The researchers then evaluated the impact of service changes on ridership while con- trolling for linear time trends. They found that while bus ridership is declining across time periods, weekday and weekend nights differ substantially from weekday time periods. In Figure 4-1, Figure 4-2, and Table 4-1, it was found that nighttime bus ridership has declined the most. These factors are unlikely to be internal since service frequencies have increased more during weekday and weekend nights than during weekday periods in all cities except for Miami, where service was cut throughout the week. Despite the decline in bus ridership at night, there is good reason to be hopeful. While these time periods are generally the least productive, they also have the highest elasticity of ridership to frequency. Unlike other time periods, increasing service at night does not lead to a rapid decline in productivity. The variable cost of increasing service has been shown to be lower than the average operating cost of existing service across time periods, especially for midday and weeknight periods (Taylor et al., 2000). This study found not only that ridership is affected by service levels in each specific time period but also that all-day frequency impacts ridership independently. Ultimately, measuring how ridership responds to changes in service is one key to reversing the recent decline in bus ridership. Bus ridership appears to be “peaking,” with ridership declining most at night despite having the most increase or least decrease in service. This indicates that some external factor must be at play.

Next: Chapter 5 - Examining the Peaking Phenomenon in Bay Area Rapid Transit Ridership »
Recent Decline in Public Transportation Ridership: Analysis, Causes, and Responses Get This Book
×
 Recent Decline in Public Transportation Ridership: Analysis, Causes, and Responses
MyNAP members save 10% online.
Login or Register to save!
Download Free PDF

Rethinking mission and service delivery, rethinking fare policy, giving transit priority, careful partnering with shared-use mobility providers, and encouraging transit-oriented density are among the strategies transit agencies can employ to increase ridership and mitigate or stem declines in ridership that started years before the COVID-19 pandemic.

The TRB Transit Cooperative Research Program's TCRP Research Report 231: Recent Decline in Public Transportation Ridership: Analysis, Causes, and Responses provides a deep-dive exploration of the ridership losses already being experienced by transit systems prior to the COVID-19 pandemic and explores strategies that appear to be key as we move to the new normal of a post-pandemic world.

Supplemental to the report are TCRP Web-Only Document 74: Recent Decline in Public Transportation Ridership: Hypotheses, Methodologies, and Detailed City-by-City Results and an overview presentation.

READ FREE ONLINE

  1. ×

    Welcome to OpenBook!

    You're looking at OpenBook, NAP.edu's online reading room since 1999. Based on feedback from you, our users, we've made some improvements that make it easier than ever to read thousands of publications on our website.

    Do you want to take a quick tour of the OpenBook's features?

    No Thanks Take a Tour »
  2. ×

    Show this book's table of contents, where you can jump to any chapter by name.

    « Back Next »
  3. ×

    ...or use these buttons to go back to the previous chapter or skip to the next one.

    « Back Next »
  4. ×

    Jump up to the previous page or down to the next one. Also, you can type in a page number and press Enter to go directly to that page in the book.

    « Back Next »
  5. ×

    To search the entire text of this book, type in your search term here and press Enter.

    « Back Next »
  6. ×

    Share a link to this book page on your preferred social network or via email.

    « Back Next »
  7. ×

    View our suggested citation for this chapter.

    « Back Next »
  8. ×

    Ready to take your reading offline? Click here to buy this book in print or download it as a free PDF, if available.

    « Back Next »
Stay Connected!