Tools: Major ways to access Twitter data

Twitter’s standard APIs. Twitter’s standard application programming interfaces (APIs) are the most common entry points to Twitter data. Twitter provides a series of publicly available APIs that offer free but restricted access to data. The Streaming API and the Search API are two of them. Drawing from Twitter’s official guideFootnote ¹ and other research (Driscoll & Walker, Reference Driscoll and Walker2014; Kim et al., Reference Kim, Nordgren and Emery2020), we summarize the main features of and differences between these two APIs for research.

The Streaming API is designed to return tweets in a real-time stream. Two options are currently available: filtered stream and sampled stream. The filtered stream enables researchers to screen millions of new tweets in any given second and extract only those matching a specific set of filter rules (e.g., up to 400 key words, 5,000 user identities [IDs], and 25 locations). By default, the filtered stream returns 1% of tweets per hour. If the researcher wants to use the filtered Streaming API to collect all tweets with the hashtag #TwitterAPI, for example, and there are fewer matched tweets than the allowed cap, the researcher will obtain all tweets containing that hashtag; otherwise, the researcher will receive a sample of the #TwitterAPI tweets.

In comparison, the sampled stream is designed to return a likely random selection of all newly posted tweets (Pfeffer et al., Reference Pfeffer, Mayer and Morstatter2018) in real time that are free from filtering constraints. Researchers who have specific topics to collect are better served using the filtered stream. When researchers do not have a specific research topic in mind and are interested in taking the temperature of all conversations occurring on Twitter for general monitoring purposes, the sampled stream is a better tool to use.

The Search API is another popular entry point for accessing Twitter data. It was designed to return historical tweets collected by matching rules defined by users. The standard Search API enables researchers to access at no cost a sample of tweets published in the past seven days. For researchers who only need the past seven days of Twitter data or who are willing to collect tweets every seven days to trace historical data, the standard Search API may be a good choice. If the research project needs historical tweets posted over a longer period and/or beyond the free-tier sampling rate, one would need to use paid premium plans of Twitter’s Search API. As discussed later, two premium-level Search API products allow backtracking of tweets for a longer period (e.g., past 30 days) or tracing historical data back as far as 2006.

Third-party platforms. Researchers interested in other more user-friendly but costly access points can use third-party platforms. The market for social media data vendors has been growing rapidly, offering an alternative way for researchers to access social media data, including Twitter (see Figure 1). Examples such as Salesforce Social Studio, Crimson Hexagon, and Synthesio differ considerably in their pricing structure, data access type and cap, dashboard and interface, availability of historic search, and data output. A typical Twitter data output from these third-party platforms (see Figure 2) consists of several data fields, such as time stamp, tweet URLs, publisher names and handles, tweet content, location, language, and even a basic sentiment score calculated by the platform.

Figure 1.

Example of a dashboard layout of a third-party platform.

Figure 2.

Example of data output from a third-party platform (not limited to the selected columns).

Note: other third-party platforms might give a different output.

Compared with Twitter’s APIs, these third-party platforms provide a user-friendly dashboard to organize search term setup, data filtering and download, and summary reports, hence requiring a less steep learning curve, especially for those unfamiliar with the API language. Some of these platforms also provide access to other popular social media sites, including public accounts on Facebook, Instagram, and YouTube, as well as online news websites and discussion forums. Typically, researchers can filter on language, time period, location, and source, although the availability of such parameters varies from platform to platform. Importantly, it is rare for third-party platforms to offer unlimited access. Some may place a cap on the maximum return of mentions per query without limiting the total daily queries allowed to run (e.g., 50,000 mentions per search), others may impose a monthly cap (e.g., three million), and still others may restrict how far back in time researchers can search (e.g., one to three years). These varying restrictions impact data quality and should be scrutinized before subscribing.

Shared tweet IDs. Using tweet IDs shared by other researchers is another way to collect Twitter data besides using APIs or purchasing third-party services. Researchers sometimes share metadata in the spirit of open science (Dienlin et al., Reference Dienlin, Johannes, Bowman, Masur, Engesser, Kümpel, Lukito, Bier, Zhang, Johnson, Huskey, Schneider, Breuer, Parry, Vermeulen, Fisher, Banks, Weber, Ellis and de Vreese2021) or for other purposes. One of the data sets used in the empirical analysis here (Jiang et al., Reference Jiang, Chen, Yan, Lerman and Ferrara2020) was collected using this method. Publicly released Twitter data sets that comply with Twitter’s terms of service can be found at specialized websites such as DocNow Catalog (Reference Catalogn.d.). Using tweet IDs, one can relatively easily “rehydrate” back into full tweets using tools such as rehydratoR (Coakley & Steinert-Threlkeld, Reference Coakley and Steinert-Threlkeldn.d.) or packages such as rtweet (Kearney et al., Reference Kearney, Heiss and Briatten.d.).

Financial costs

Twitter’s standard APIs. Standard APIs under version 1.1 (V1.1) allow researchers to collect a small sample of tweets and associated metadata for free. They are good choices for pilot-testing initial research ideas. Researchers can search tweets for the past seven days with desired filtering and sampling and interact with the Twitter platform at no cost. If a researcher is primarily interested in gathering a random sample of topic-matched or general Twitter discourses, the standard APIs are probably sufficient as long as the researcher has the skills to set up the API connection to cover the required time period. However, researchers who need longer historical coverage or higher sampling density beyond the free standard APIs will need to switch to a premium subscription, which allows researchers to collect full-archive tweet data with a current monthly cap of 1.25 million. The current pricing for premium Search Twitter APIs ranges from $149 to $2,499 per month, depending on the types of features and the level of access needed (see Table 1).

Table 1.

Financial costs of different data collection circumstances.

Third-party access. Based on our own research on several popular third-party platforms on the market, the subscription costs of third-party platforms vary considerably, ranging from $7,000 to $50,000 for an annual contract based on quotes received in 2019. It is important to compare multiple platforms and closely examine data availability and cost structure, including, but not limited to, how the maximum cap is placed, whether the number of topics is restricted, historical data access, coverage of social media sources, user access authorization, and data output. For example, some platforms might not be able to provide a tweet’s original ID, making it hard for researchers to further extract profile information and other metadata about a matched tweet. Or a platform might claim that overall data access is unlimited but still impose a daily cap on the total number of mentions allowed to output. Some platforms charge by the number of topics to be examined, offering access to three years of historical data and allowing separate dashboards for different users on the same research team. Other platforms charge by a monthly maximum, going back to when Twitter was founded and requiring team members to share the same dashboard. Most platforms cover data from the big four social media websites (Facebook, Twitter, Instagram, and YouTube) plus some online news websites. Some also cover foreign social media sites such as China’s Weibo. It is useful to request a demonstration from a sales representative, download a sample data output, and examine whether all the required information is available.

Skill sets

Twitter’s standard APIs. As indicated in Table 2, several technical skills are necessary for using the standard APIs for data collection, storage, and analysis. To collect tweets through standard APIs, researchers need some basic knowledge about a data science programming language such as R or Python. That fundamental knowledge will allow researchers to employ several useful open-source packages available in R or Python to access Twitter’s APIs, such as twitteR (Gentry, Reference Gentryn.d.), python-twitter (Python-Twitter Developers, n.d.), and tweepy (Roesslein, Reference Roessleinn.d.). There are many useful online tutorials that teach researchers how to use these tools, though it can take a few weeks of intensive study to grasp these off-the-shelf packages without diving into the technical details of these programming languages. Additionally, researchers need to understand Boolean operation, which is necessary to build search queries. Once the search query is developed and activated to start data collection, researchers also must be able to debug their code to address error messages. For instance, errors such as a broken connection or reaching an API quota might occur. Researchers need to understand these messages or patiently access the resources available to understand them in order to write code to fix the errors. Pilot tests are often an effective way to identify where and when the codes might break.

Table 2.

Skill sets required for using various Twitter tools.

Regarding data storage and data analysis, researchers can often process up to several million tweets on a single updated laptop as long as the analytical strategy allows processing tweets sequentially in batches. Researchers can store their raw tweets spread across several files based on day or hour. While the data can be slowly analyzed in this basic form, to analyze these files more quickly, researchers can use “embarrassingly parallel”Footnote ² programming in R or Python to examine each file separately if the job of each file is independent of the others (i.e., the result of one file does not need to be finished in order to serve as the input to the next file).

Third-party access. Third-party platforms with well-organized query structures and interactive interfaces can relieve researchers from massive programming work but still require some basic technical skills. Resources for researchers regarding platform functionality are generally available through reviewing official instruction manuals, watching video tutorials, and consulting technical support teams.

Researchers should be aware of platform policies and limits while designing a project. Policies refers to the platform-imposed rules that specify ways to extract matched tweets, available Boolean operations to support query construction, and format and size to standardize data export. It is common for platforms to implement their own policies to keep services consistent across customers. Common caps include data rates, data volume per request, request cap for a period, and request length. All of these can affect data quality as well as researchers’ time and computational resources requirements. It is worth noting that automation scripts (e.g., Selenium) can reduce time and resource burdens.

Specialized access: Full-archive access and researcher access

While many research questions can be addressed using conventional access points through Twitter APIs, third-party platforms, and shared tweet IDs, certain research questions require full-archive access. For example, projects aiming to analyze message diffusion structure and dynamics typically need to reconstruct the complete diffusion chains for tweets of interest such as fake news (Vosoughi et al., Reference Vosoughi, Roy and Aral2018) and health messages (Meng et al., Reference Meng, Peng, Tan, Liu, Cheng and Bae2018). In these situations, only the complete collection of tweets matched with filtering criteria can offer the granularity, resolution, and time stamps necessary to accurately infer who retweeted whom and when along the diffusion chain. Relatedly, research projects involving networks, such as the follower-followee network or the network formed by quoting, replying, or retweeting, are better served by obtaining full-archive access, because random samples of tweets remain prone to systematic biases in recovering key network-level attributes (González-Bailón et al., Reference González-Bailón, Wang, Rivero, Borge-Holthoefer and Moreno2014; Lee et al., Reference Lee, Kim and Jeong2006). For these reasons, we briefly introduce methods for access, financial cost, and skill sets relevant to the full-archive access to Twitter data.

Twitter’s enterprise-level APIs enable users to access its full archive dating back to the first tweet in March 2006. Compared with the standard version, enterprise APIs remove a lot of usage restrictions, but they are financially costly for individual researchers and graduate students. In addition, users must request enterprise-level access from and negotiate related terms with Twitter. One of the enterprise packages is the full-archive Search API. According to Twitter, this API provides complete and instant access to the full corpus of Twitter data. It is query-based access to the tweet archive with minute granularity. Using this approach, users define the filter rules, and tweets matching those queries become available from the Search API about 30 seconds after being published.

Often, these data are accessed via Twitter’s PowerTrack APIs, commonly known as the Firehose endpoints. PowerTrack API, which also provides access to the full Twitter archive, is the enterprise-level package based on the standard filtered Streaming API. Full-archive access, previously operated by data providers such as Gnip and now available directly from Twitter, offers 100% data coverage and allows complex search filters to match tweets. By applying the PowerTrack filtering language (e.g., geo-location and key words), users can filter the real-time stream of tweets. Decahose is another enterprise-level API based on the sampled Streaming API. It delivers a 10% random sample of the real-time Twitter Firehose. Table 3 summarizes two major types of Twitter APIs, Search APIs and Streaming APIs, and their three subscription levels: standard, premium, and enterprise.

Table 3.

Twitter’s standard, premium, and enterprise levels of Search and Streaming APIs.

Around mid-2020, Twitter made available to invited institutions and research groups the streaming endpoint as part of its PowerTrack API, to provide free Firehose access to COVID-19-related tweets. In October 2020, Twitter tested the new Academic Research product track in a private beta program and publicly launched it in January 2021. In particular, Twitter released a new version of its APIs (V2) that is built on new functionalities, making it easier to collect and analyze the public conversation and simpler to scale up or down without changing APIs, as well as making it friendlier to academic researchers.Footnote ³ Previously, full-archive data were inaccessible to most researchers.

Enterprise APIs, such as Firehose via the Historical PowerTrack API, offer the highest level of access and reliability. Twitter currently does not have a fixed pricing structure. Accessing this type of APIs needs to be negotiated directly with Twitter, and users are charged on a case-by-case basis by assessing the nature and volume of requested data.

Given the large volume of data output from Firehose access, multiple workstations/servers can considerably enhance efficiency in handling data storage and processing. For example, in the analysis below of the COVID-19 Firehose corpus, a mere week’s volume in June easily reached more than 30 million tweets, and we had to find dedicated servers to store the entire corpus covering the second half of 2020, with the size measured in terabytes. This volume of data can overwhelm the most powerful laptops on the market. Raw tweets usually come in JSON files, which are better reorganized using distributed processing frameworks such as Hadoop or Spark to enable more efficient storage, filtering, and retrieval (e.g., via SQL).

Analytically, while analytical strategies such as dictionary-based scaling can be carried out in batches, almost independently of each other, other approaches such as topic modeling might require more advanced distributed processing to update parameter estimates globally across the entire corpus. Also, one needs to pay attention to whether the preferred analytical strategy is scalable and friendly to distributed processing. For example, although structural topic modeling is a preferred method to examine the linkage between metadata and topic structures (Roberts et al., Reference Roberts, Stewart and Tingley2019), it is typically less scalable and slower to run than the latent Dirichlet allocation algorithms adapted to distributed processing on large-scale data sets.

To bypass the technical complexities of setting up the distributed data storage and analytical infrastructure (e.g., Hadoop, distributed processing), one feasible option is to seek collaboration with computer and data scientists. We should caution that not all such interdisciplinary collaborations bear out. It is critically important to understand and resolve disciplinary differences such as incentives, research goals (e.g., explanatory theory building versus prediction improvement), publication processes (e.g., slower journal papers versus faster conference proceedings), and authorship norms. Our experiences suggest that grand challenges of common interest to multiple disciplines, such as the COVID-19 crisis, and opportunities for grant applications might create strong enough incentives to initiate and sustain such collaborations.

The pros and cons of Twitter as organic data

Twitter data differ in nature from the data that researchers collect from traditional quantitative methods such as surveys or experiments. Survey data are researcher controlled and designed, whereas social media data can be viewed as more organic (Groves, Reference Groves2011). We highlight several nuances (i.e., advantages coexist with challenges) about data quality that researchers should consider due to this organic nature. The first nuance is data quality versus data newness. Data quality is more likely ensured in surveys and experiments, as researchers have more control of which participants to recruit and what questions to ask. Nevertheless, the emergent nature of social media discussions may offer researchers opportunities to identify new, previously unidentified perspectives and frames (Klašnja et al., Reference Klašnja, Barberá, Beauchamp, Nagler, Tucker, Atkeson and Alvarez2017, p. 17). “Newness” is a strength of social media data and is especially useful for studying emerging life sciences issues where the right questions to ask are elusive. However, data newness comes with a data quality challenge that requires researchers to develop methods to indirectly evaluate user characteristics such as user identity and motivations (Chen & Tomblin, Reference Chen and Tomblinin press).

The second nuance is control and cost. Researchers have more control of the data generation process in survey and experiments. Yet, it is costly to collect survey data, especially if a large sample of panel data is needed. For using Twitter to collect social media data, researchers cannot control sample composition, but they can assess sample representativeness of targeted populations through the limited profile information provided by the users. If researchers have domain knowledge about social media platform users, then social media is potentially much cheaper for collecting large and time-series data sets. Researchers also need to be aware that the data generation from social media can suffer problems such as algorithmic bias, polarization, and segregation (Baeza-Yates, Reference Baeza-Yates2018; Colleoni et al., Reference Colleoni, Rozza and Arvidsson2014; Lawrence et al., Reference Lawrence, Sides and Farrell2010). In fact, social media companies regularly conduct A/B testing (i.e., a simple controlled experiment with two variants) to understand their users. Yet, researchers know little about what A/B testing these companies are doing and how it may confound research results (Freiling et al., Reference Freiling, Krause, Scheufele and Chenin press). For instance, suppose the platform is experimenting with misinformation correction messages and the treatment is effective at changing users’ opinions. If the researcher collects data on users’ opinions during the experimental time frame, he or she might reach biased conclusions because the researcher does not know about this misinformation correction intervention.

Finally, there is also a theoretical consideration of whether the 1% random sample from the Streaming API is indeed random. As Twitter points out,Footnote ⁵ its standard API (V1.1) focuses on relevance and not completeness. Research also suggests that the universality of the 1% ceiling with millisecond filtering is unclear, and therefore there is no assurance of a random sample (Kim et al., Reference Kim, Hansen, Murphy, Richards, Duke and Allen2013; Pfeffer et al., Reference Pfeffer, Mayer and Morstatter2018). We provide more details on how we can assess the randomness later.

Bot intervention

Bot accounts are prevalent and evolving (Cresci, Reference Cresci2020). Bots account for 37.2% (Imperva, Reference Roberts2020) of all internet traffic. Existing literature shows that at least 39 countries, including the United States, the United Kingdom, Canada, Russia, and China, have reported cases of political manipulation involving social bots during political events (e.g., elections) or in normal times (Bastos & Mercea, Reference Bastos and Mercea2019; Bolsover & Howard, Reference Bolsover and Howard2019; Cresci, Reference Cresci2020; Luceri et al., Reference Luceri, Badawy, Deb and Ferrara2019; Stukal et al., Reference Stukal, Sanovich, Bonneau and Tucker2017).

It is necessary to assess whether bots warrant concern about the reliability and representativeness of social media data, as bots and humans have become increasingly difficult to distinguish from each other. The answer depends upon the nature of the research question. For example, when the interest is the discursive strategies of legitimate users such as political elites, bot detection might not be necessary, since politicians are seldom willing to hand over their social media accounts to algorithms. However, when the goal is to depict online discourses or topical trends in public opinion, naively treating social media data as exclusively human traces without considering the existence of bots may lead to misjudgment (Duan et al., Reference Duan, Li, Josephine, Yang, Chen, Shah and Yangn.d.).

Methods have been developed for detecting social bots. One off-the-shelf bot detection tool is Botometer (Observatory on Social Media, Reference Social Median.d.), a machine learning-based system which evaluates the activity of a Twitter account and generates a bot-like score (Davis et al., Reference Davis, Varol, Ferrara, Flammini and Menczer2016). Besides the machine learning approach, other approaches are graph-based (Hurtado et al., Reference Hurtado, Ray and Marculescu2019), crowdsourcing-based (Wang et al., Reference Wang, Mohanlal, Wilson, Wang, Metzger, Zheng and Zhao2012), and anomaly-based (Echeverria & Zhou, Reference Echeverria and Zhou2017; Orabi et al., Reference Orabi, Mouheb, Al Aghbari and Kamel2020).

Sample representativeness

For big data research, the “bigness” of data size can never guarantee representativeness of the target population, a critical challenge facing scholars who use social media data. Despite its popularity as a source of accessible digital trace data, Twitter attracted a mere 22% of Americans as self-reported ever users in 2019 (Perrin & Anderson, Reference Perrin and Anderson2019). Even for Facebook, the most popular social networking site to date, still around 30% of Americans had never used its service as of 2019 (Perrin & Anderson, Reference Perrin and Anderson2019). The sample representativeness of the general population of Twitter data is still debated, but it appears to be highly contingent on the research question. For certain topics, research has shown that Twitter is a credible source to reproduce real-world outcomes such as protest and violence (Muchlinski et al., Reference Muchlinski, Yang, Birch, Macdonald and Ounis2021; Sobolev et al., Reference Sobolev, Chen, Joo and Steinert-Threlkeld2020; Steinert-Threlkeld, Reference Steinert-Threlkeld2018; Zhang & Pan, Reference Zhang and Pan2019). However, for other research questions, such as studying public opinion across the United States, we align with other scholars who express caution about extrapolating Twitter users’ opinions to other Americans, such as about people’s happiness level (Jensen, Reference Jensen2017). Twitter users are younger and are more likely to be Democrats compared with the general public, and 10% of users created 80% tweets (Wojcik & Hughes, Reference Wojcik and Hughes2019). As Barberá & Rivero (Reference Barberá and Rivero2015) suggested, Twitter provides researchers with the opportunity to study public opinion, yet the validity of generalizations needs to be assessed along with the biases within political discussions on Twitter.

Other sources of data bias

Two additional sources of potential biases may further threaten the quality of obtainable big social media data: (1) the quality of search terms for data retrieval and (2) the black box of APIs and third-party platforms especially their sampling logic. There have been extensive discussions about search term development and evaluation (Kim et al., Reference Kim, Huang and Emery2016; King et al., Reference King, Lam and Roberts2017), but little is known about the sampling quality of APIs and third-party platforms, which is our focus in the empirical assessment.

Researchers relying upon APIs and third-party platforms rarely have the opportunity to verify how sampling is handled on the back end. For example, according to Twitter documentation, compared with the latest API (V2), its free-access standard API (V1.1) is less likely to return true randomly sampled data, because it is designed to improve relevance for consumer-use cases. Although this potential deviation from true random sampling has been corrected in API (V2), such biases might have permeated numerous third-party platforms that rely upon Twitter’s APIs.

To evaluate the extent to which such biases in sampling of full-access data might exist in Twitter’s standard API (1% data coverage) and third-party platforms, we next use the same list of search terms related to COVID-19 to compare the distributions of one specific type of content attribute—moral appeals—in three collections of COVID-19 tweets for a two-week period of data (July 1–July14, 2020, Coordinated Universal Time): Twitter’s standard API (1% coverage), a third-party platform that claims to provide 10% random sampling, and Twitter’s full-archive access (the ground truth).

Third-party platform and Twitter API representativeness of full-archive data

Over the past decade, interest has grown in applying moral foundations theory (MFT) (Graham et al., Reference Graham, Haidt, Koleva, Motyl, Iyer, Wojcik and Ditto2013; Haidt, Reference Haidt2012) to investigate the roles of moral appeals and moral values in political ideology, public opinion, and individuals’ processing of information related to controversial medical and scientific issues (Clifford & Jerit, Reference Clifford and Jerit2013; Skitka & Morgan, Reference Skitka and Morgan2014; Wolsko et al., Reference Wolsko, Ariceaga and Seiden2016; Yang et al., Reference Yang, Maloney, Tan and Cappella2018). MFT has become a useful theoretical framework for scholars studying politics and life science communication. In this analysis, we use the most recent extended Moral Foundations Dictionary (Hopp et al., Reference Hopp, Fisher, Cornell, Huskey and Weber2020) to measure each tweet’s mentioning of the five types of moral appeals (care, fairness, loyalty, authority, and sanctity). We focus on relevance scoring and do not distinguish moral virtues (e.g., a tweet praising mask-wearing as caring for others) from vices (e.g., a tweet condemning violation of social distancing that harms others). Since many social science questions concern extracting theoretical constructs similar to moral appeals from social media data, this empirical assessment is intended to provide insights into the prevalence (or the lack of) of potential sampling biases likely to generalize to other research contexts.

If Twitter’s standard API and third-party platforms indeed return random samples of matched tweets under identical screening criteria (e.g., language, time period, and key words), we should expect to see similar distributions of any content feature—including moral appeals—across the three data sets. Figure 3 presents the density plots comparing the three data sources, stratified by moral appeal type. Similar to previous studies (Hopp et al., Reference Hopp, Fisher, Cornell, Huskey and Weber2020), the empirical distributions largely resemble a normal distribution. The overall shape, central tendency, and spread of these distributions appear comparable from visual inspection, which is consistent with similar estimated means and standard deviations across the three data sets, per moral appeal category (see Table 4 for details on basic descriptive statistics).

Figure 3.

Comparisons of score distributions of five moral appeals.

Table 4.

Comparison of five basic statistics across three data sources.

Further, we perform a series of Kolmogorov-Smirnov (K-S) tests to further investigate the statistical comparability across the three data sources. In the two-sample case, the K-S test statistic is an indicator of the maximum distance between the empirical cumulative distribution functions of two samples (Massey, Reference Massey1951). In our study, two steps were taken: first, we repeatedly sampled 10,000 tweets without replacement from the full-archive Firehose 1,000 times, and then we calculated the K-S statistics between each sample against the entire Firehose corpus to derive an empirical distribution of the K-S test statistics under the null hypothesis, respectively for each moral appeal category. Next, for each moral appeal type, we estimated the K-S statistics comparing the entire third-party corpus and also the entire standard API corpus with the Firehose, respectively (see the two dashed vertical lines in Figure 4).

Figure 4.

K-S statistics distribution comparison.

As Figure 4 indicates, for most moral appeal categories, the two observed K-S test statistics lay beyond the 97.5% quantile cut-off points (two-tailed tests) in the empirical K-S distributions under the null, suggesting statistically significant discrepancies in the shape of most moral appeal distributions between the Firehose and the other two data sets. This pattern casts doubt on the assumption that the other two corpora could be treated as random samples from the Firehose. With that said, we should note that K-S tests are sensitive to sample size and can easily pick up noise in local discrepancies. Given our large sample sizes, we would like to emphasize the lack of substantial discrepancies between the three data sets in terms of the descriptive statistics reported in Table 4 and the highly comparable density plots in Figure 3.