AI-Enhanced Review

Lexington - IT lexZero Error Numerics uses computer assisted review (CAR) software with active machine learning algorithms. Active machine learning is a type of artificial intelligence (AI). When used in legal search these AI algorithms significantly improve the search, review, and classification of electronically stored information (ESI). For this reason we prefer to call predictive coding by the name AI-enhanced review or AI-enhanced search. For more background on the science involved see

In CARs with AI-enhanced review and search engines, attorneys train a computer to find documents identified by the attorney as a target. The target is typically relevance to a particular lawsuit or legal issue, or some other legal classification, such as privilege. This kind of AI-enhanced review, along with general e-discovery training, are now my primary interests as a lawyer.

Personal Legal Search Background

Ralph and some of his computers at one of his law officesIn 2006 I dropped my civil litigation practice and limited my work to e-discovery. That is also when I started this blog. At that time I could not even imagine specializing more than that. In 2006 I was interested in all aspects of electronic discovery, including computer assisted review. AI-enhanced CARs were still just a dream that I hoped would someday come true.

The use of software in legal practice has always been a compelling interest for me. I have been an avid user of computer software of all kinds since the late 1970s, both legal and entertainment. I even did some game software design and programming work in the early 1980s. My now-grown kids still remember the computer games I made for them.

I carefully followed the legal search and review software scene my whole career, but especially since 2006. It was not until 2011 that I began to be impressed by the new types of predictive coding CAR software entering the market. After I got my hands on the new software, I began to do what had once been unimaginable. I started to limit my legal practice even further. I began to spend more and more of my time on predictive coding types of review work. Since 2012 my work as an e-discovery lawyer and researcher has focused almost exclusively on using predictive coding driven CARs in large document production projects, and on e-discovery training, another passion of mine. In that year one of my cases produced a landmark decision by Judge Andrew Peck that first approved the use of predictive coding, Da Silva Moore. (I do not write about it because it is still ongoing.)

LSS_Gavel_3Attorney Maura R. Grossman and I are among the first attorneys in the world to specialize in predictive coding as an e-discovery sub-niche. Maura is a colleague who is both a practicing attorney and an expert in the new field of Legal Search Science.  We have frequently presented on CLE panels as a kind of technology evangelists for these new methods of legal review. Maura, and her partner, ProfessorGordon Cormack, who is one of the most esteemed information scientists in the field, wrote the seminal scholarly paper on the subject, and more recently an excellent glossary of terms used in CAR (they prefer to call it TAR). Technology-Assisted Review in E-Discovery Can Be More Effective and More Efficient Than Exhaustive Manual Review, Richmond Journal of Law and Technology, Vol. XVII, Issue 3, Article 11 (2011); The Grossman-Cormack Glossary of Technology-Assisted Review, with Foreword by John M. Facciola, U.S. Magistrate Judge2013 Fed. Cts. L. Rev. 7 (January 2013); Evaluation of Machine-Learning Protocols for Technology-Assisted Review in Electronic DiscoverySIGIR’14, July 6–11, 2014.

I recommend your reading of all of their works. I also recommend your study of the website that I put together, and the many references and citations included at Legal Search Science, including the writings of other pioneers in the field, such as  the founders of TREC Legal Track, Jason R. Baron, Doug Oard, and David Lewis, and other key figures in the field, such as information scientists William Webber and EDI’s Herb Roitblat. Also see Baron and Grossman, The Sedona Conference® Best Practices Commentary on the Use of Search and Information Retrieval Methods in E-Discovery (2013).pdf (December 2013).

Advanced CARs Require Completely New Driving Methods

CAR or TAR is more than just new software. It entails a whole new legal method, a new approach to large document reviews. Below is the diagram that I created to show the new workflow I use in a typical CAR project. This is the standard version of the workflow. Further below you will find a variation that uses a slightly more complicated process called continuous active learning (CAL).


For a basic description of the eight steps see the Electronic Discovery Best Practices page on predictive coding.

I have found that driving a CAR properly requires the highest skill levels and is, for me at least, the most challenging activity in electronic discovery. It also shows the promise of being the new tool that we have all been waiting for. When used properly, good predictive coding type software allows attorneys to find the information they need in vast stores of ESI, and to do so in an effective and affordable manner.

In my experience the best software and training methods use what is known as an active learning process in steps four and five in the chart above. My preferred active learning process in the iterative machine learning steps is threefold:

  1. The computer selects documents for review where the software classifier is uncertain of the correct classification. This helps the classifier algorithms to learn by adding diversity to the documents presented for review. This in turn helps to locate outliers of a type your initial judgmental searches in step two and five have missed. This is machine selected sampling, and, according to a basic text in information retrieval engineering, a process is not a bona fide active learning search without this ability. Manning, Raghavan and Schutze, Introduction to Information Retrieval, (Cambridge, 2008) at pg. 309.
  2. Some reasonable percentage of the documents presented for human review in step five are selected at random. This again helps maximize recall and premature focus on the relevant documents initially retrieved.
  3. Other relevant documents that a skilled reviewer can find using a variety of search techniques. This is called judgmental sampling. After the first round of training, aka the seed set, the judgmental sampling by a variety of search methods is used based on the machine selected or random selected documents presented for review, but sometimes the subject matter expert (“SME”) human reviewer follows a new search idea unrelated to the new documents seen.  Any kind of searches can be used for judgmental sampling, which is why I call it a multimodal search. This may include some linear review of selected custodians or dates, parametric Boolean keyword searches, similarity searches of all kinds, concept searches, as well as several unique predictive coding probability searches.

The initial seed set generation, step two in the chart, should also use some random samples, plus judgmental multimodal searches. Steps three and six in the chart always use pure random samples and rely on statistical analysis. For more on the three types of sampling see my blog, Three-Cylinder Multimodal Approach To Predictive Coding.

My insistence on the use of multimodal judgmental sampling in steps two and five to locate relevant documents follows the consensus view of information scientists specializing in information retrieval, but is not followed by several prominent predictive coding vendors. They instead rely entirely on machine selected documents for training, or even worse, rely entirely on random selected documents to train the software.  In my writings I call these processes the Borg approach, after the infamous villains in Star Trek, the Borg, a race half-human robots that assimilates people into machines. (I further differentiate between three types of Borg in Three-Cylinder Multimodal Approach To Predictive Coding.) Like the Borg, these approaches unnecessarily minimize the role of individuals, the SMEs. They exclude other types of search to supplement an active learning process. I advocate the use of all types of search, not just predictive coding.

Professor Cormack and Maura Grossman also performed experiments, which, among other things, tested the efficacy of random only based search.Evaluation of Machine-Learning Protocols for Technology-Assisted Review in Electronic DiscoverySIGIR’14, July 6–11, 2014. They reached the same conclusions that I did, and showed that this random only – Borg approach – is far less effective than even the most simplistic judgmental methods. I reported on this study in full in a series of blogs in the Summer of 2014, Latest Grossman and Cormack Study Proves Folly of Using Random Search for Machine Training, see especially Part One of the series.

The CAL Variation

After study of the 2014 experiments by Professor Cormack and Maura Grossman reported at the SIGIR conference, I created a variation to the predictive coding work flow, which they call CAL, for Continuous Active Learning. Evaluation of Machine-Learning Protocols for Technology-Assisted Review in Electronic DiscoverySIGIR’14, July 6–11, 2014, at pg. 9. Also see Latest Grossman and Cormack Study Proves Folly of Using Random Search for Machine Training – Parts One,  TwoThree and Four. The part that intrigued me about there study was the use of continuous machine training as part of the entire review. This is explained in detail in Part Three of my lengthy blog series on the Cormack Grossman study. I had already known about the ineffectiveness of random only machine training from my own experiments, but had never experimented with the continuous training aspects included in their experiments.

The form of CAL that Cormack and Grossman tested used high probable relevant documents in all but the first training round. (In the first round, the so called seed set, they trained using documents found by keyword search.) This experiment showed that the method of review of the documents with the highest rankings works well, and should be given significant weight in any multimodal approach, especially when the goal is to quickly find as many relevant documents as possible. This is another take-away from this important experiment.

The “continuous” training aspects of the CAL approach means that you keep doing machine training throughout the review project and batch reviews accordingly. This could become a project management issue. But, if you can pull it off within proportionality and requesting party constraints, it just makes common sense to do so. You might as well get as much help from the machine as possible and keep getting its probability predictions for as long as you are still doing reviews and can make last minute batch assignments accordingly.

I have done several reviews in such a continuous training manner without really thinking about the fact that the machine input was continuous, including my first Enron experiment. Predictive Coding Narrative: Searching for Relevance in the Ashes of Enron. But the Cormack Grossman study on the continuous active learning approach caused me to rethink the standard flow chart shown above that I usually use to explain the predictive coding process. The standard work flow that does not use a CAL approach is referred to in the Cormack Grossman report as the simple approach, where you review and train, but then at some point stop training and final review is done. Under the simple approach there is a distinct stop in training after step five, and the review work in step seven is based on the last rankings established in step five.

The continuous work flow is slightly more difficult to show in a diagram, and to implement, but it does make good common sense if you are in a position to pull it off. Below is the revised workflow that illustrates how the training continues throughout the review.


Machine training is still done in steps four and five, but then continues in steps four, five and seven. There are other ways it could be implemented of course, but this is the CAL approach I would use in a review project where such complex batching and continuous training otherwise makes sense. Of course, it is not necessary in any project were the review in steps four and five effectively finds all of the relevant documents required. This is what happened in my Enron experiment. Predictive Coding Narrative: Searching for Relevance in the Ashes of EnronThere was no need to do a proportional final review, step seven, because all the relevant documents had already been reviewed as part of the machine training review in steps four and five. In the Enron experiment I skipped step seven and when right from step six to step eight, production. I have been able to do this is other projects as well.

Hybrid Human Computer Information Retrieval


In further contradistinction to the Borg, or random only approaches, where the machine controls the learning process, I advocate a hybrid approach where Man and Machine work together. In my hybrid CARs the expert reviewer remains in control of the process, and their expertise is leveraged for greater accuracy and speed. The human intelligence of the SME is a key part of the search process. In the scholarly literature of information science this hybrid approach is known as Human–computer information retrieval (HCIR).

The classic text in the area of HCIR, which I endorse, is Information Seeking in Electronic Environments (Cambridge 1995) by Gary Marchionini, Professor and Dean of the School of Information and Library Sciences of U.N.C. at Chapel Hill. Professor Marchionini speaks of three types of expertise needed for a successful information seeker:

  1. Domain Expertise. This is equivalent to what we now call SME, subject matter expertise. It refers to a domain of knowledge. In the context of law the domain would refer to particular types of lawsuits or legal investigations, such as antitrust, patent, ERISA, discrimination, trade-secrets, breach of contract, Qui Tam, etc. The knowledge of the SME on the particular search goal is extrapolated by the software algorithms to guide the search. If the SME also has System Expertise, and Information Seeking Expertise, they can drive the CAR themselves.   Otherwise, they will need a chauffeur with such expertise, one who is capable of learning enough from the SME to recognize the relevant documents.
  2. System Expertise. This refers to expertise in the technology system used for the search. A system expert in predictive coding would have a deep and detailed knowledge of the software they are using, including the ability to customize the software and use all of its features. In computer circles a person with such skills is often called a power-user. Ideally a power-user would have expertise in several different software systems. They would also be an expert in a particular method of search.
  3. Information Seeking Expertise. This is a skill that is often overlooked in legal search. It refers to a general cognitive skill related to information seeking. It is based on both experience and innate talents. For instance, “capabilities such as superior memory and visual scanning abilities interact to support broader and more purposive examination of text.” Professor Marchionini goes on to say that: “One goal of human-computer interaction research is to apply computing power to amplify and augment these human abilities.” Some lawyers seem to have a gift for search, which they refine with experience, broaden with knowledge of different tools, and enhance with technologies. Others do not.

Id. at pgs.66-69, with the quotes from pg. 69.

All three of these skills are required for an attorney to attain expertise in legal search today, which is one reason I find this new area of legal practice so challenging. It is difficult, but not impossible like this Penrose triangle.


It is not enough to be an SME, or a power-user, or have a special knack for search. You have to be able to do it all, and so does your software. However, studies have shown that of the three skill-sets, System Expertise, which in legal search primarily means mastery of the particular software used, is the least important. Id. at 67. The SMEs are more important, those  who have mastered a domain of knowledge. In Professor Marchionini’s words:

Thus, experts in a domain have greater facility and experience related to information-seeking factors specific to the domain and are able to execute the subprocesses of information seeking with speed, confidence, and accuracy.

Id. That is one reason that the Grossman Cormack glossary builds in the role of SMEs as part of their base definition of computer assisted review:

A process for Prioritizing or Coding a Collection of electronic Documents using a computerized system that harnesses human judgments of one or more Subject Matter Expert(s) on a smaller set of Documents and then extrapolates those judgments to the remaining Document Collection.

Glossary at pg. 21 defining TAR.

According to Marchionini, Information Seeking Expertise, much like Subject Matter Expertise, is also more important than specific software mastery. Id. This may seem counterintuitive in the age of Google, where an illusion of simplicity is created by typing in words to find websites. But legal search of user-created data is a completely different type of search task than looking for information from popular websites. In the search for evidence in a litigation, or as part of a legal investigation, special expertise in information seeking is critical, including especially knowledge of multiple search techniques and methods. Again quoting Professor Marchionini:

Expert information seekers possess substantial knowledge related to the factors of information seeking, have developed distinct patterns of searching, and use a variety of strategies, tactics and moves.

Id. at 70.

In the field of law this kind of information seeking expertise includes the ability to understand and clarify what the information need is, in other words, to know what you are looking for, and articulate the need into specific search topics. This important step precedes the actual search, but is an integral part of the process. As one of the basic texts on information retrieval written by Gordon Cormack, et al, explains:

Before conducting a search, a user has an information need, which underlies and drives the search process. We sometimes refer to this information need as a topic …

Buttcher, Clarke & Cormack, Information Retrieval: Implementation and Evaluation of Search Engines (MIT Press, 2010) at pg. 5. The importance of pre-search refining of the information need is stressed in the first step of the above diagram of my methods, ESI Discovery Communications. It seems very basic, but is often under appreciated, or overlooked entirely in the litigation context where information needs are often vague and ill-defined, lost in overly long requests for production and adversarial hostility.

Hybrid Multimodal Bottom Line Driven Review

I have a long descriptive name for what Marchionini calls the variety of strategies, tactics and moves that I have developed for legal search: Hybrid Multimodal AI-Enhanced Review using a Bottom Line Driven Proportional Strategy. See eg. Bottom Line Driven Proportional Review (2013). I refer to it as a multimodal method because, although the predictive coding type of searches predominate (shown on the below diagram as AI-enhanced review – AI), I also  use the other modes of search, including Unsupervised Learning Algorithms (explained in (often called clustering or near-duplication searches), keyword search, and even some traditional linear review (although usually very limited). As described, I do not rely entirely on random documents, or computer selected documents for the AI-enhanced searches, but use a three-cylinder approach that includes human judgment sampling and AI document ranking. The various types of legal search methods used in a multimodal process are shown in this search pyramid.

Multimodal Search Pyramid

Most information scientists I have spoken to agree that it makes sense to use multiple methods in legal search and not just rely on any single method. UCLA Professor Marcia J. Bates first advocated for using multiple search methods back in 1989, which she called it berrypicking. Bates, Marcia J. The Design of Browsing and Berrypicking Techniques for the Online Search Interface, Online Review 13 (October 1989): 407-424. As Professor Bates explained in 2011 in Quora:

An important thing we learned early on is that successful searching requires what I called “berrypicking.” … Berrypicking involves 1) searching many different places/sources, 2) using different search techniques in different places, and 3) changing your search goal as you go along and learn things along the way. This may seem fairly obvious when stated this way, but, in fact, many searchers erroneously think they will find everything they want in just one place, and second, many information systems have been designed to permit only one kind of searching, and inhibit the searcher from using the more effective berrypicking technique.

This berrypicking approach, combined with HCIR, is what I have found from practical experience works best with legal search. They are the Hybrid Multimodal aspects of my AI-Enhanced Bottom Line Driven Review method.

My Battles in Court Over Predictive Coding

In 2012 my case became the first in the country where the use of predictive coding was approved. See Judge Peck’s landmark decision Da Silva Moore v. Publicis, 11 Civ. 1279, _ FRD _, 2012 WL 607412 (SDNY Feb. 24, 2012). In that case my methods of using Recommind’s Axcelerate software were approved. Later in 2012, in another first, an AAA arbitration approved our use of predictive coding in a large document production. In that case I used Kroll Ontrack’s Inview software over the vigorous objections of the plaintiff, which, after hearings, were all rejected. These and other decisions have helped pave the way for the use of predictive coding search methods in litigation.

Scientific Research

In addition to these activities in court I have focused on scientific research on legal search, especially machine learning. I have, for instance, become one of the primary outside reporters on the legal search experiments conducted by TREC Legal Track of the National Institute of Science and Technology. See egAnalysis of the Official Report on the 2011 TREC Legal Track – Part OnePart Two and Part ThreeSecrets of Search: Parts OneTwo, and ThreeAlso see Jason Baron, DESI, Sedona and Barcelona.

After the TREC Legal Track closed down in 2011 the only group participant scientific study to test the efficacy of various predictive coding software, and search methods, is the one sponsored by Oracle, the Electronic Discovery Institute and Stanford. This search of a 1,639,311 document database was conducted in early 2013, with the results reported in Monica Bay’s article, EDI-Oracle Study: Humans Are Still Essential in E-Discovery (LTN Nov., 2013). Here is the below chart published by LTN that summarizes the results.


Monica Bay summaries the findings of the research as follows:

Phase I of the study shows that older lawyers still have e-discovery chops and you don’t want to turn EDD over to robots.

Penrose_triangle_ExpertiseWith respect to my dear friend Monica, I must disagree with her conclusion. The age of the lawyers is irrelevant. The best predictive coding trainers do not have to be old, they just have to be SMEs, power users of good software, and have good search skills. In fact, not all SMEs are old, although many may be. It is the expertise and skills that matter, not age per se. It is true as Monica reports that the lawyer, a team of one, who did better in this experiment than all of the other much larger participant groups, was chronologically old. But that fact is irrelevant. The skill set and small group size, namely one, is what made the difference. SeeLess Is More: When it comes to predictive coding training, the “fewer reviewers the better” – Parts OneTwo, and Three.

Moreover, although Monica is correct to say we do not want to”turn over” review to robots, this assertion misses the point. We certainly do want to turn over review to robot-human teams. We want our predictive coding software, our robots, to hook up with our experienced lawyers. We want our lawyers to enhance their own limited intelligence with artificial intelligence – the Hybrid approach. Robots are the future, but only if and as they work hand-in-hand with our top human trainers. Then they are unbeatable, as the EDI-Oracle study shows.

Secret Shh!For the time being the details of the EDI-Oracle scientific study are still closed, and even though Monica Bay was permitted to publicize the results, and make her own summary and conclusions, participants are prohibited from discussion and public disclosures. For this reason I can say no more on this study, and only assert without facts that Monica’s conclusions are in some respects incorrect, that age is not critical, that the hybrid multimodal method is what is important. I hope and expect that someday soon the gag order for participants will be lifted, the full findings of this most interesting scientific experiment will be released, and a free dialogue will commence. Truth only thrives in the open, and science concealed is merely occult.

Why Predictive Coding Driven CARs Are Important

I continue to focus on this sub-niche area of e-discovery as I am convinced that it is critical to advancement of the law in the 21st Century. Our own intelligence and search skills must be enhanced by the latest AI software. The new search and review methods I have developed allow a skilled attorney using readily available predictive coding type software to review at remarkable rates of speed and cost. The CAR review rates are more than 250-times faster than traditional linear review, and the costs less than a tenth as much. See eg Predictive Coding Narrative: Searching for Relevance in the Ashes of EnronEDI-Oracle Study: Humans Are Still Essential in E-Discovery (LTN Nov., 2013).

My Life as a Limo Driver and Trainer

I have spoken on this subject at many CLEs around the country since 2011. I explain the theory and practice of this new breakthrough technology. I also consult on a hands-on basis to help others learn the new methods. As an old software lover who has been doing legal document reviews since 1980, I also continue to like to do these review projects myself. I like to drive the CARs myself, not just teach others how to drive. I enjoy the interaction and enhancements from the hybrid, human-robot approach. Certainly I need an appreciate the artificial intelligence boosts to my own limited capacities.

I also like to serve as a kind of limo driver for trial lawyers from time to time. The top SMEs in the world (I prefer to work with the best), are almost never also software power-users, nor do they have special skills or talents for information seeking outside of depositions. For that reason they need me to drive the CAR for them. To switch to the robot analogy again, I like and can work with the bots, they cannot.

I can only do my job as a limo driver – robot friend in an effective manner if the SME first teaches me enough of their domain to know where I am going; to know what documents would be relevant or hot or not. That is where decades of legal experience handling a variety of cases is quite helpful. It makes it easer to get a download of the SME’s concept of relevance into my head, and then into the machine. Then I can act as a surrogate SME and do the machine training for them in an accurate and consistent manner.


Working as a driver for an SME presents many special communication challenges. I have had to devise a number of techniques to facilitate a new kind of SME surrogate agency process. Of course, it is easier to do the search when you are also the SME. For instance, in one project I reviewed almost two million documents, by myself, in only two-weeks. That’s right. By myself. (There was no redaction or privilege logging, which are tasks that I always delegate anyway.) A quality assurance test at the end of the review based on random sampling showed a very high accuracy rate was attained. There is no question that it met the reasonability standards required by law and rules of procedure.

It was only possible to do a project of this size so quickly because I happened to be an SME on the legal issues under review, and, just as important, I was a power-user of the software, and have, at this point, mastered my own search and review methods. I also like to think I have a certain knack for information seeking.

Thanks to the new software and methods, what was considered impossible, even absurd, just a few short years ago, namely one attorney accurately reviewing two million documents by him or herself in 14-days, is attainable by many experts. My story is not unique. Maura tells me that she once did a seven-million document review by herself. That is why Maura and Gordon were correct to refer to TAR as a disruptive technology in the Preface to their Glossary. Technology that can empower one skilled lawyer to do the work of hundreds of unskilled attorneys is certainly a big deal, one for which we have Legal Search Science to thank.  It is also why I urge you to study this subject more carefully and learn to drive these new CARs yourself. Either that, or hire a limo driver.

Before you begin to actually carry out a predictive coding project, with or without an expert chauffeur to drive your CAR, you need to plan for it. This is critical to the success of the project. Here is detailed outline of a Form Plan for a Predictive Coding Project that I use as a complete checklist.

My Writings on CAR

A good way to continue your study in this area is to read the articles by Grossman and Cormack, and the over forty or so articles on the subject that I have written since mid-2011. They are listed here in rough chronological order, with the most recent on top. Also see the CAR procedures described on Electronic Discovery Best Practices.

Ralph in the morning reading on his 17 inch MacProI am especially proud of the legal search experiments I have done using AI-enhanced search software provided to me by Kroll Ontrack to review the 699,083 public Enron documents and my reports on these reviews. Comparative Efficacy of Two Predictive Coding Reviews of 699,082 Enron Documents(Part Two); A Modest Contribution to the Science of Search: Report and Analysis of Inconsistent Classifications in Two Predictive Coding Reviews of 699,082 Enron Documents. (Part One). I have been told by scientists that my over 100 hours of search, comprised of two fifty-hour search projects using different methods, is the largest search project by a single reviewer that has ever been undertaken, not only in Legal Search, but in any kind of search. I do not expect this record will last for long, as others begin to understand the importance of Information Science in general, and Legal Search Science in particular. But for now I will enjoy both the record and lessons learned from the hard work involved.

Articles by Ralph Losey on Legal Search

  1. Form Plan of a Predictive Coding Project. Detailed Outline for project planning purposes.
  2. Two-Filter Document CullingPart One and Part Two.
  3. Introducing “ei-Recall” – A New Gold Standard for Recall Calculations in Legal SearchPart One, Part Two and Part Three.
  4. In Legal Search Exact Recall Can Never Be Known.
  5. Visualizing Data in a Predictive Coding ProjectPart One, Part Two and Part Three.
  6. Guest Blog: Talking Turkey by Maura Grossman and Gordon Cormack, edited and published by RCL.
  7. Latest Grossman and Cormack Study Proves Folly of Using Random Search For Machine Training – Part One,  Part Two,  Part Three, and Part Four.
  8. The “If-Only” Vegas Blues: Predictive Coding Rejected in Las Vegas, But Only Because It Was Chosen Too Late. Part One and Part Two.
  9. IT-Lex Discovers a Previously Unknown Predictive Coding Case: “FHFA v. JP Morgan, et al”
  10. Beware of the TAR Pits! Part One and Part Two.
  11. PreSuit: How Corporate Counsel Could Use “Smart Data” to Predict and Prevent Litigation. Also see
  12. Predictive Coding and the Proportionality Doctrine: a Marriage Made in Big Data, 26 Regent U. Law Review 1 (2013-2014).
  13. Less Is More: When it comes to predictive coding training, the “fewer reviewers the better” – Parts One, Two, and Three.
  14. My Basic Plan for Document Reviews: The “Bottom Line Driven” Approach, PDF version suitable for print, or HTML version that combines the blogs published in four parts.
  15. Relevancy Ranking is the Key Feature of Predictive Coding Software.
  16. Why a Receiving Party Would Want to Use Predictive Coding?
  17. Vendor CEOs: Stop Being Empty Suits & Embrace the Hacker Way 
  18. Comparative Efficacy of Two Predictive Coding Reviews of 699,082 Enron Documents(Part Two).
  19. A Modest Contribution to the Science of Search: Report and Analysis of Inconsistent Classifications in Two Predictive Coding Reviews of 699,082 Enron Documents. (Part One).
  20. Introduction to Guest Blog: Quick Peek at the Math Behind the Black Box of Predictive Coding that pertains to the higher-dimensional geometry that makes predictive coding support vector machines possible.
  21. Keywords and Search Methods Should Be Disclosed, But Not Irrelevant Documents.
  22. Reinventing the Wheel: My Discovery of Scientific Support for “Hybrid Multimodal” Search.
  23. There Can Be No Justice Without Truth, And No Truth Without Search (statement of my core values as a lawyer explaining why I think predictive coding is important).
  24. Three-Cylinder Multimodal Approach To Predictive Coding.
  25. Robots From The Not-Too-Distant Future Explain How They Use Random Sampling For Artificial Intelligence Based Evidence Search. Video Animation.
  26. Borg Challenge: Report of my experimental review of 699,082 Enron documents using a semi-automated monomodal methodology (a five-part written and video series comparing two different kinds of predictive coding search methods).
  27. Predictive Coding Narrative: Searching for Relevance in the Ashes of Enron in PDF form for easy distribution and the blog introducing this 82-page narrative, with second blog regarding an update.
  28. Journey into the Borg Hive: a Predictive Coding Narrative in science fiction form.
  29. The Many Types of Legal Search Software in the CAR Market Today.
  30. Georgetown Part One: Most Advanced Students of e-Discovery Want a New CAR for Christmas.
  31. Escape From Babel: The Grossman-Cormack Glossary.
  32. NEWS FLASH: Surprise Ruling by Delaware Judge Orders Both Sides To Use Predictive Coding.
  33. Does Your CAR (“Computer Assisted Review”) Have a Full Tank of Gas?  (and you can also click here for the alternate PDF version for easy distribution).
  34. Analysis of the Official Report on the 2011 TREC Legal Track – Part One.
  35. Analysis of the Official Report on the 2011 TREC Legal Track – Part Two.
  36. Analysis of the Official Report on the 2011 TREC Legal Track – Part Three
  37. An Elusive Dialogue on Legal Search: Part One where the Search Quadrant is Explained.
  38. An Elusive Dialogue on Legal Search: Part Two – Hunger Games and Hybrid Multimodal Quality Controls.
  39. Random Sample Calculations And My Prediction That 300,000 Lawyers Will Be Using Random Sampling By 2022.
  40. Second Ever Order Entered Approving Predictive Coding.
  41. Predictive Coding Based Legal Methods for Search and Review.
  42. New Methods for Legal Search and Review.
  43. Perspective on Legal Search and Document Review.
  44. LegalTech Interview of Dean Gonsowski on Predictive Coding and My Mission to Make Predictive Coding Software More Affordable.
  45. My Impromptu Video Interview at NY LegalTech on Predictive Coding and Some Hopeful Thoughts for the Future.
  46. The Legal Implications of What Science Says About Recall.
  47. Reply to an Information Scientist’s Critique of My “Secrets of Search” Article.
  48. Secrets of Search – Part I.
  49. Secrets of Search – Part II.
  50. Secrets of Search – Part III. (All three parts consolidated into one PDF document.)
  51. Information Scientist William Webber Posts Good Comment on the Secrets of Search Blog.
  52. Judge Peck Calls Upon Lawyers to Use Artificial Intelligence and Jason Baron Warns of a Dark Future of Information Burn-Out If We Don’t.
  53. The Information Explosion and a Great Article by Grossman and Cormack on Legal Search.

Please contact me at if you have any questions.


ei-recallThe backbone of ZEN document review is a  new method for calculating recall in legal search projects using random sampling that we call ei-Recall. This stands for elusion interval recall. We offer this to everyone in the e-discovery community in the hope that it will replace the hodgepodge of methods currently used, most of which are statistically invalid. Our goal is to standardize a new best practice for calculating recall. This lengthy essay will describe the formula in detail, and explain why we think it is the new gold standard. Then we will provide a series of examples as to how ei-Recall works.

We have received feedback on these ideas and experiments from the top two scientists in the world with special expertise in this area, William Webber and Gordon Cormack. Our thanks and gratitude to them both, especially to William, who must have reviewed and responded to a dozen earlier drafts of this blog. He not only corrected initial logic flaws, and there were many, but also typos. As usual any errors remaining are purely our own, and these are our opinions, not theirs.

ei-Recall is preferable to all other commonly used methods of recall calculation, including Herb Roitbalt’s eRecall, for two reasons. First, ei-Recall includes interval based range values, and, unlike eRecall, and other simplistic ratio methods, is not based on point projections. Second, and this is critical, ei-Recall is only calculated at the end of a project, and depends on a known, verified count of True Positives in a production. It is thus unlike eRecall, and all other recall calculation methods that depend on an estimated value for the number of True Positives found.

Yes, this does limit the application of ei-Recall to projects in which great care is taken to bring the precision of the production to near 100%, including second reviews, and many quality control cross-checks. But this is anyway part of the workflow in many Continuous Active Learning (CAL) predictive coding projects today. At least it is in mine, where we take great pains to meet the client’s concern to maintain the confidentiality of their data. See: Step 8 of the EDBP (Electronic Discovery Best Practices), which I call Protections and is the step after first pass review by CAR (computer assisted review, multimodal predictive coding).

Advanced Summary of ei-Recall

Ralph_2015We begin with a high level summary of this method for my more advanced readers. Do not be concerned if this seems fractured and obtuse at first. It will come into clear 3-D focus later as we describe the process in multiple ways and conclude with examples.

ei-Recall calculates recall range with two fractions. The numerator of both fractions is the actual number of True Positives found in the course of the review project and verified as relevant. The denominator of both fractions is based on a random sample of the documents presumed irrelevant that will not be produced, the Negatives. The percentage of False Negatives found in the sample allows for a calculation of a binomial range of the total number of False Negatives in the Negative set. The denominator of the low end recall range fraction is the high end number of the projected range of False Negatives, plus the number of True Positives. The denominator of the high end recall range fraction is the low end number of the projected range of False Negatives, plus the number of True Positives.

Here is the full algebraic explanation of ei-Recall, starting with the definitions for the symbols in the formula.

  • Rl stands for the low end of recall range.
  • Rh stands for high end of recall range
  • TP is the verified total number of relevant documents found in the course of the review project.
  • FNl is the low end of the False Negatives projection range based on the low end of the exact binomial confidence interval.
  • FNh is the high end of the False Negatives projection range based on the high end of the exact binomial confidence interval.

Formula for the low end of the recall range:
Rl = TP / (TP+FNh).

Formula for the high end of the recall range:
Rh = TP / (TP+FNl).

This formula essentially adds the extreme probability ranges to the standard formula for recall, which is: R = TP / (TP+FN).


Quest for the Holy Grail of Recall Calculations

holy.grail.chaliceI have spent the last few months in intense efforts to bring this project to conclusion. I have also spent more time writing and rewriting this blog than any I have ever written in my eight plus years of blogging. I wanted to find the best possible recall calculation method for e-discovery work. I convinced myself that I needed to find a new method in order to take my work as a legal search and review lawyer to the next level. I was not satisfied with my old ways and methods of quality control of large legal search projects. I was not comfortable with my prevalence based recall calculations. I was not satisfied with anyone else’s recall methods either. I had heard the message of Gordon Cormack and Maura Grossman clearly stated right here in their guest blog of September 7, 2014: Talking Turkey. In their conclusion they stated:

We hope that our studies so far—and our approach, as embodied in our TAR Evaluation Toolkit—will inspire others, as we have been inspired, to seek even more effective and more efficient approaches to TAR, and better methods to validate those approaches through scientific inquiry.

I had already been inspired to find better methods of predictive coding, and have uncovered an efficient approach with my multimodal CAL method. But I was still not satisfied with my recall validation approach, I wanted to find a better method to scientifically validate my review work.

Like almost everyone else in legal search, including Cormack and Grossman, I had earlier rejected the so called Direct Method of recall calculation. It is unworkable and very costly, especially in low prevalence collections where it requires sample sizes in the tens of thousands of documents. See Eg. Grossman & Cormack, Comments on ‘The Implications of Rule 26(g) on the Use of Technology-Assisted Review,’ Federal Courts Law Review, Vol. 7, Issue 1 (2014) at 306-307 (“The Direct Method is statistically sound, but is quite burdensome, especially when richness is low.”)

Like Grossman and Cormack, I did not much like any of the other sampling alternatives either. Their excellent Comments articles discusses and rejects Roitblat’s eRecall, and two other methods by Karl Schieneman and Thomas C. Gricks III, which Grossman and and Cormack call the Basic Ratio Method and Global Method. Supra at 307-308.

I was on a quest of sorts for the Holy Grail of recall calculations. I knew there had to be a better way. I wanted a method that used sampling with interval ranges as a tool to assure the quality of a legal search project. I wanted a method that created as accurate an estimate as possible. I also wanted a method that relied on simple fraction calculations and did not depend on advanced math to narrow the binomial ranges, such as William Webber’s favorite recall equation: the Beta-binomial Half formula, shown below.


Webber, W., Approximate Recall Confidence IntervalsACM Transactions on Information Systems, Vol. V, No. N, Article A, Equation 18, at pg. A:13 (October 2012).

Before settling on my much simpler algebraic formula I experimented with many other methods to calculate recall ranges. Most were much more complex and included two or more samples, not just one. I wanted to try to include a sample that I usually take at the beginning of a project to get a rough idea of prevalence with interval ranges. These were the examples shown by my article, In Legal Search Exact Recall Can Never Be Known, and described in the section, Calculating Recall from Prevalence. I wanted to include the first sample, and prevalence based recall calculations based on that first sample, with a second sample of excluded documents taken at the end of the project. Then I wanted to kind of average them somehow, including the confidence interval ranges. Good idea, but bad science. It does not work, statistically or mathematically, especially in low prevalence.

I found a number of other methods, which, at first, looked like the Holy Grail. But I was wrong. They were made of lead, not gold. Some of the one’s that I dreamed up were made of fools gold! A couple of the most promising methods I tried and rejected used multiple samples of various stratas. That is called stratified random sampling as compared to simple sampling.

My questionable, but inspired research method for this very time consuming development work consisted of background reading, aimless pondering, sleepless nights, intuition, trial and error (appropriate I suppose for a former trial lawyer), and many consults with the top experts in the field (another old trial lawyer trick). I ran though many other alternative formulas. I did the math in several standard review project scenarios, only to see the flaws of these other methods in certain circumstances, primarily low prevalence.

Every experiment I tried with added complexity, and added effort of multiple samples, proved to be fruitless. Indeed, most of this work was an exercise in frustration. (It turns out that noted search expert Bill Dimm is right. There is no free lunch in recall.) My experiments, and especially the expert input I received from Webber and Cormack, all showed that the extra complexities were not worth the extra effort, at least not for purposes of recall estimation. Instead, my work confirmed that the best way to channel additional efforts that might be appropriate in larger cases is simply to increase the sample size. This, and my use of confirmed True Positives, are the only sure-fire methods to improve the reliability of recall range estimates. They are the best ways to lower the size of the interval spread that all probability estimates must include.

Finding the New Gold Standard

gold_standard_ei_recallei-Recall meets all of my goals for recall calculation. It maintains mathematical and statistical integrity by including probable ranges in the estimate. The size of the range depends on the size of the sample. It is simple and easy to use, and easy to understand. It can thus be completely transparent and easy to disclose. It is also relatively inexpensive and you control the costs by controlling the sample size (although I would not recommend a sample size of less than 1,500 in any legal search project of significant size and value).

Finally, by using verified True Positives, and basing the recall range calculation on only one random sample, one of the null set, instead of two samples, the chance factor inherent to all random sampling is reduced. I described these chance factors in detail in In Legal Search Exact Recall Can Never Be Known, in the section on Outliers and Luck of Random Draws. The possibility of outlier events is still possible using ei-Recall, but is minimized by limiting the sample to the null set and only estimating a projected range of False Positives. While it is true that the prevalence based recall calculations described in In Legal Search Exact Recall Can Never Be Known, also only use one random sample, that is a sample of the entire document collection to estimate a projected range of relevant documents, True Positives. The number of relevant documents found will (or at least should be in any half-way decent search) be a far larger number than the number of False Negatives. For that reason alone the variability range (interval spread) of the straight elusion recall method should typically be smaller and more reliable.

Focus Your Sampling Efforts on Finding Errors of Omission

ei-recall_SodaThe number of documents presumed irrelevant, the Negatives, or null set, will always be smaller than the total document collection, unless of course you found no relevant documents at all! This means you will always be sampling a smaller dataset when doing an elusion sample, than when doing a prevalence sample of the entire collection. Therefore, if you are trying to find your mistakes, the False Negatives, look for them where they might lie, in the smaller Negative set, the null set. Do not look for them in the larger complete collection, which includes the documents you are going to produce, the Positive set. Your errors of omission, which is what you are trying to measure, could not possibly be there. So why include that set of documents in the random sample? That is why I reject the idea of taking a sample at the end of the entire collection, including the Positives.

The Positives, the documents to be produced, have already been verified enough under my two-pass system. They have been touched multiple times by machines and humans. It is highly unlikely there will be False Positives. Even if there are, the requesting party will not complain about that. Their concern should be on completeness, or recall, especially if any precision errors are minor.

There is no reason to include the Positives in a final recall search in any project with verified True Positives. That just unnecessarily increases the total population size and thereby increases the possibility of an inaccurate sample. Estimates made from a sample of 1,500 documents of a collection of 150,000 documents will always be more accurate, more reliable, than estimates made from a sample of 1,500 documents of a collection of 1,500,000. The only exception is when there is an even distribution of target documents making up half of the total collection – 50% prevalence.

Random_sample_group-of-peopleSample size does not scale perfectly, only roughly, and the lower the prevalence, the more inaccurate it becomes. That is why sampling is not a miracle tool in legal search, and recall measures are range estimates, not certainties. In Legal Search Exact Recall Can Never Be Known. Recall measure when done right, as it is in ei-Recall, is a powerful quality assurance tool, to be sure, but it is not the end-all of quality control measures. It should be part of a larger tool kit that includes several other quality measures and techniques. The other quality control methods should be employed throughout the review, not just at the end like ei-Recall. Maura Grossman and Gordon Cormack agree with me on this. Comments on ‘The Implications of Rule 26(g) on the Use of Technology-Assisted Review,’ supra at 285. They recommend that validation:

consider all available evidence concerning the effectiveness of the end-to-end review process, including prior scientific evaluation of the TAR method, its proper application by qualified individuals, and proportionate post hoc sampling for confirmation purposes.

Ambiguity in the Scope of the Null Set

ei-recall_N_AmbigiousThere is an open-question in my proposal as to exactly how you define the Negatives, the presumed irrelevant documents that you sample. This may be varied somewhat depending on the circumstances of the review project. In my definition above I said the Negatives were the documents presumed to be irrelevant that will not be produced. That was intentionally somewhat ambiguous. I will later state with less ambiguity that Negatives are the documents not produced (or logged for privilege). Still, I think this application should be varied sometimes according to the circumstances.

In some circumstances you could improve the reliability of an elusion search by excluding from the null set all documents coded irrelevant by an attorney, either with or without actual review. The improvement would arise from shrinking the size of the number of documents to be sampled. This would allow you to focus your sample on the documents most likely to have an error.

For example, you could have 50,000 documents out of 900,000 not produced, that have actually been read or skimmed by an attorney, and coded irrelevant. You could have yet another 150,000 that have not been actually been read or skimmed by an attorney, but have been bulked coded irrelevant by an attorney. This would not be uncommon in some projects. So even though you are not producing 900,000 documents, you may have manually coded 200,000 of those, and only 700,000 have been presumed irrelevant on the basis of computer search. Typically in predictive coding driven search that would be because their ranking at the end of the CAL review was too low to warrant further consideration. In a simplistic keyword search they would be documents omitted from attorney review because they did not contain a keyword.

In other circumstances you might want to include the documents attorneys reviewed and coded as irrelevant, for instance, where you were not sure of the accuracy of their coding for one reason or another. Even then you might want to exclude other sets of documents for other grounds. For instance, in predictive coding projects you may want to exclude some bottom strata of the rankings of probable relevance. For example, you could exclude the bottom 25%, or maybe the bottom 10%, or bottom 2%, where it is highly unlikely that any error has been made in predicting irrelevance of those documents.

data-visual_Round_5In the data visualization diagram I explained in Visualizing Data in a Predictive Coding Project – Part Two (shown right) you could exclude some bottom portion of the ranked documents shown in blue. You could, for instance, limit the Negatives searched to those few documents in the 25% to 50% probable relevance range. Of course, whenever you limit the null set, you have to be careful to adjust the projections accordingly. Thus, if you find 1% False Negatives in a sample of a presumably enriched sub-collection of 10,000 out of 100,000 total Negatives, you cannot just project 1% of 100,000 and assume there are a total of 1,000 False Negatives (plus or minus of course). You have to project the 1% from the sample of the size of the sub-collection sampled, and so it would be 1% of 10,000, or 100 False Negatives, not 1,000, again subject to the confidence interval range, a range that varies according to your sample size.

Remember, the idea is to focus your random search to find mistakes on the group of documents that are most likely to have mistakes. There are many possibilities.

In still other scenarios you might want to enlarge the Negatives to include documents that were never included in the review project at all. For instance, if you collected emails from ten custodians, but eliminated three as unlikely to have relevant information as per Step 6 of the EDBP (culling), and only reviewed the email of seven custodians, then you might want to include select documents from the three excluded custodians in the final elusion test.

There are many other variations and issues pertaining to the scope of the Negatives set searched in ei-Recall. There are too many to discuss in this already long article. I just want to point out in this introduction that the makeup and content of the Negatives sampled at the end of the project is not necessarily cut and dry.

Advantage of End Project Sample Reviews

ei-recall_wineBasing recall calculations on a sample made at the end of a review project is always better than relying on a sample made at the beginning. This is because final relevance standards will have been determined and fully articulated by the end of a project. Whereas at the beginning of any review project, the initial relevance standards will be tentative. They will typically change in the course of the review. This is known as relevance shift, where the understanding of relevance changes and matures during the course of the project.

This variance of adjudication between samples can be corrected during and at the end of the project by careful re-review and correction of initial sample relevance adjudications. This also requires correction of changes of all codings made during the review in the same way, not just inconsistencies in sample codings.

The time and effort spent to reconcile the adjudications might be better spent on a larger sample size of the final elusion sample. Except for major changes in relevance, where you would anyway have to go back and make corrections as part of quality control, it may not be worth the effort to remediate the first sample, just so you can still use it again at the end of the project with an elusion sample. That is because of the unfortunate statistical fact of life, that the two recall methods cannot be added to one another to create a third, more reliable number. I know. I tried. The two recall calculations are apples and oranges. Although a comparison between the two range values is interesting, they cannot somehow be stacked together to improve the reliability of either or both of them.

Prevalence Samples May Still Help Guide Search, Even Though They Cannot Be Reliably Used to Calculate Recall

sampleI like to make a prevalence sample at the beginning of a project to get a general idea of the number of relevant documents there might be, and I emphasize general and might, in order to help with my search. I used to make recall calculation from that initial sample too, but no longer (except in small cases under the theory it is better than nothing), because it is simply too unreliable. The prevalence samples can help with search, but not with recall calculations used to test the quality of the search results. For quality testing it is better to sample the null set and calculate recall using the ei-Recall method.

Still, if you are like me, and like to take a sample at the start of a project for search guidance purposes, then you might as well do the math at the end of the project to see what the recall range estimate is using the prevalence method described in In Legal Search Exact Recall Can Never Be Known. It is interesting to compare the two recall ranges, especially if you take the time and trouble to go back and correct the first prevalence sample adjudications to match those of calls made in your second null set sample (that can eliminate the problem of concept drift and reviewer inconsistencies). Still, go with the recall range values of the ei-Recall, not prevalence. It is more reliable. Moreover, do not waste your time, as I did for weeks, trying to somehow average out the results. I traveled down that road and it is a dead-end.

Claim for ei-Recall

Claim_ChartMy claim is that ei-Recall is the most accurate recall range estimate method possible that uses only algebraic math within everyone’s grasp. (This statement is not exactly true because binomial confidence interval calculations are not simple algebra, but we avoid these calculations by use of an online calculator. Many are available.) I also claim that ei-Recall is more reliable, and less prone to error in more situations, than a standard prevalence based recall calculation, even if the prevalence recall includes ranges as I did in In Legal Search Exact Recall Can Never Be Known.

I also claim that my range based method of recall calculation is far more accurate and reliable than any simple point based recall calculations that ignore or hide interval ranges, including the popular eRecall. This later claim is based on what I proved in In Legal Search Exact Recall Can Never Be Knownand is not novel. It has long been known and accepted by all experts in random sampling, that recall projections that do not include high-low ranges are inexact and often worthless and misleading. And yet attorneys and judges are still relying on point projections of recall to certify the reasonableness of search efforts. The legal profession and our courts need to stop relying on such bogus science and turn instead to ei-Recall.

I am happy to concede that scientists who specialize in this area of knowledge like Dr. Webber and Professor Cormack can make slightly more accurate and robust calculations of binomial recall range estimates by using extremely complex calculations such as Webber’s Beta-binomial formula.

Webber_beta-binomial_formula_hyperSuch alternative black box type approaches are, however, disadvantaged by the additional expense from expert consultations and testimony to implement and explain. (Besides, at the present time, neither Webber nor Cormack are available for such consultations.) My approach is based on multiplication and division, and simple logic. It is well within the grasp of any attorney or judge (or anyone else) who takes the time to study it. My relatively simple system thus has the advantage of ease of use, ease of understanding, and transparency. These factors are very important in legal search.


Although the ei-Recall formula may seem complex at first glance, it is really just ratios and proportions. I reject the argument some make that calculations like this are too complex for the average lawyer. Ratios and proportions are part of the Grade 6 Common Core Curriculum. Reducing word problems to ratios and proportions is part of the Grade 7 Common Core, so too is basic statistics and probability.

Overview of How ei-Recall Works

ei-recallei-Recall is designed for use at the end of a search project as a final quality assurance test. A single random sample is taken of the documents that are not marked relevant and so will not be produced or privileged-logged – the Negatives. (As mentioned, definition and scope of the Negatives can be varied depending on project circumstances.) The sample is taken to estimate the total number of False Negatives, documents falsely presumed irrelevant that are in fact relevant. The estimate projects a range of the probable total number of False Negatives using a binomial interval range in accordance with the sample size. A simplistic and illusory point value projection is not used. The high end of the range of probable False Negatives is shown in the formula and graphic as FNh. The low end of the projected range of False Negatives is FNl.

This type of search is generally called an elusion based recall search. As will be discussed here in some detail, well-known software expert and entrepreneur, Herb Rotiblat, who has a PhD in psychology, advocates for the use of a similar elusion based recall calculation that uses only the point projection of the total False Negatives. He has popularized a name for this method: eRecall, and uses it with his company’s software.

I here offer a more accurate alternative that avoids the statistical fallacies of point projections. Rotiblat’s eRecall, and other ratio calculations like it, ignore the interval high and low range range inherent in all sampling. My version includes interval  range, and for this reason an “i” is added to the name: ei-Recall.

ei-Recall is more accurate than eRecall, especially when working with low prevalence datasets, and, unlike eRecall, is not misleading because it shows the total range of recall. It is also more accurate because it uses the exact count of the documents verified as relevant at the end of the project, and does not estimate the True Positives value. I offer ei-Recall to the e-discovery community as a statistically valid alternative, and urge its speedy adoption.

Contingency Table Background

A review some of the basic concepts and terminology used in this article may be helpful before going further. It is also important to remember that ei-Recall is a method for measuring recall, not attaining recall. There is a fundamental difference. Many of my other articles have discussed search and review methods to achieve recall, but this one does not. See eg.

  1. Latest Grossman and Cormack Study Proves Folly of Using Random Search For Machine Training – Part One,  Part Two,  Part Three, and Part Four.
  2. Predictive Coding and the Proportionality Doctrine: a Marriage Made in Big Data, 26 Regent U. Law Review 1 (2013-2014).
  3. Less Is More: When it comes to predictive coding training, the “fewer reviewers the better” – Parts One, Two, and Three.
  4. Three-Cylinder Multimodal Approach To Predictive Coding.

This article is focused on the very different topic of measuring recall as one method among many to assure quality in large-scale document reviews.

Everyone should know that in legal search analysis False Negatives are documents that were falsely predicted to be irrelevant, that are in fact relevant. They are mistakes. Conversely, documents predicted irrelevant, that are in fact irrelevant, are called True Negatives. Documents predicted relevant that are in fact relevant are called True Positives. Documents predicted relevant that are in fact irrelevant are called False Positives.

These terms and formulas derived therefrom are set forth in the Contingency Table, a/k/a Confusion Matrix, a tool widely used in information science. Recall using these terms is the total number of relevant documents found, the True Positives (TP), divided by that same number, plus the total number of relevant documents not found, the False Negatives (FN). Recall is the percentage of total target documents found in any search.


Truly Non-Relevant Truly Relevant
Coded Non-Relevant True Negatives (“TN”) False Negatives (“FN”)
Coded Relevant False Positives (“FP”) True Positives (“TP”)


The standard formula for Recall using contingency table values is: R = TP / (TP+FN).

The standard formula for Prevalence is: P = (TP + FN) / (TP + TN + FP + FN).

The Grossman-Cormack Glossary of Technology Assisted Review. Also see: LingPipe Toolkit class on PrecisionRecallEvaluation.

 General Background on Recall Formulas

Before I get into the examples and math for ei-Recall, I want to provide more general background. In addition, I suggest that you re-read my short description of an elusion test at the end of Part Three of Visualizing Data in a Predictive Coding Project. It provides a brief description of the other quality control applications of the elusion test for False Negatives. If you have not already done so, you should also read my entire article, In Legal Search Exact Recall Can Never Be Known

I also suggest that you read John Tredennick’s excellent article: Measuring Recall in E-Discovery Review: A Tougher Problem Than You Might Realize, especially Part Two of that article. I give a big Amen to John’s tough problem insights.

For the more technical and mathematically minded, I suggest you read the works of William Webber, including his key paper on this topic, Approximate Recall Confidence Intervals (January 2013, Volume 31, Issue 1, pages 2:1–33) (free version in arXiv), and his many less formal and easier to understand blogs on the topic: Why confidence intervals in e-discovery validation? (12/9/12); Why training and review (partly) break control sets, (10/20/14);  Why 95% +/- 2% makes little sense for e-discovery certification, (5/25/13); Stratified sampling in e-discovery evaluation, (4/18/13); What is the maximum recall in re Biomet?, (4/24/13). Special attention should be given to Webber’s recent article on Roitblat’s eRecallConfidence intervals on recall and eRecall (1/4/15), where it is tested and found deficient on several grounds,

voltaire_sketchMy idea for a recall calculation that includes binomial confidence intervals, like most ideas, is not truly original. It is, as our friend Voltaire puts it, a judicious imitation. For instanceI am told that my proposal to use comparative binomial calculations to determine approximate confidence interval ranges follows somewhat the work of an obscure Dutch medical statistician, P. A. R. Koopman, in the 1980s. See: Koopman, Confidence intervals for the ratio of two binomial proportions, Biometrics 40: 513–517 (1984).  Also see: Webber, William, Approximate Recall Confidence IntervalsACM Transactions on Information Systems, Vol. V, No. N, Article A (October 2012); Duolao Wang, Confidence intervals for the ratio of two binomial proportions by Koopman’s methodStata Technical Bulletin, 10-58, 2001.

As mentioned, the recall method I propose here is also similar to that promoted by Herb Roitbalt – eRecall – except that avoids its fundamental defect. I include binomial intervals in the calculations to provide an elusion recall range, and his method does not. Measurement in eDiscovery (2013). Herb’s method relies solely on point projections and disregards the ranges of both the Prevalence and False Negative projections. That is why no statistician will accept Rotibalt’s eRecall, whereas ei-Recall has already been reviewed without objection by two of the leading authorities in the field, William Webber and Gordon Cormack.


ei-Recall is also a superior method because it is based on a specific number of relevant documents found at the end of the project, the True Positives (TP). That is not an estimated number. It is not a projection based on sampling where a confidence interval range and more uncertainty are necessarily created. True Positives in ei-Recall is the number of relevant documents in a legal document production (or privilege log). It is an exact number verified by multiple reviews and other quality control efforts set forth in steps six, seven and eight in Electronic Discovery Best Practices (EDBP), and then produced in step nine (or logged).

In a predictive coding review the True Positives as defined by ei-Recall are the documents predicted relevant, and then confirmed to be relevant in second pass reviews, etc., and produced and logged. (Again see: Step 8 of the EDBP, which I call Protections.) The production is presumed to be a 100% precise production, or at least as close as is humanly possible, and contain no False Positives. For that reason ei-Recall may not be appropriate in all projects. Still, it could also work, if need be, by estimating the True Positives. The fact that ei-Recall includes interval ranges in and of itself make it superior and more accurate that any other ratio method.

ei-Recall_smallIn the usual application of ei-Recall, only the number of relevant documents missed, the False Negatives, is estimated. The actual number of relevant documents found (TP) is divided by the sum of the projected range of False Negatives from the samples of the null set of each strata, both high (FNh) and low (FNl), and the number of relevant documents found (TP). This method is summarized by the following formulas:

Formula for the lowest end of the recall range from the null set sample: Rl = TP / (TP+FNh).

Formula for the highest end of the recall range from the null set sample: Rh = TP / (TP+FNl).

This is a very different from the approach used by Herb Roitblat for eRecall. Herb’s approach is to sample the entire collection to calculate a point projection of the probable total number of relevant documents in the collection, which I will here call P. He then takes a second random sample of the null set to calculate the point projection of the probable total False Negatives contained in the null set (FN). Roitblat’s approach only uses point projections and ignores the interval ranges inherent in each sample. My approach uses one sample and includes its confidence interval range. Also, as mentioned, my approach uses a validated number of True Positives found at the end of a review project, and not a projection of the probable total number of relevant documents found (P). Although Herb never uses a formula per se in his paper, Measurement in eDiscovery, to describe his approach, if we use the above described definitions the formula for eRecall would seem to be: eR = P / (P + FN). (Note there are other speculations as to what Roitblat’s really intends here, as discussed in the comments to Webber’s blog on eRecall. One thing we know for sure, is that although he may change the details to his approach, it never includes a recall range, just a spot projection.)

My approach of making two recall calculations, one for the low end, and another for the high end, is well worth the slight additional time to create a range. Overall the effort and cost of ei-Recall is significantly less than eRecall because only one sample is used in my method, not two. My method significantly improves the reliability of recall estimates and overcomes the defects inherent in ignoring confidence intervals found in eRecall and other methods such as the Basic Ratio Method and Global Method. See Eg: Grossman & Cormack, Comments on ‘The Implications of Rule 26(g) on the Use of Technology-Assisted Review,’ Federal Courts Law Review, Vol. 7, Issue 1 (2014) at 306-310.

The use of range values avoids the trap of using a point projection that may be very inaccurate. The point projections of eRecall may be way off from the true value, as was explained in detail by In Legal Search Exact Recall Can Never Be KnownMoreover, ei-Recall fits in well with the overall work flow of my current two-pass, CAL-based (continuous active learning), hybrid, multimodal search and review method.

Recall Calculation Methods Must Include Range

A fuller explanation of Herb Rotiblat’s eRecall proposal, and other similar point projection based proposals, should help clarify the larger policy issues at play in the proposed alternative ei-Recall approach.

Again, I cannot accept Herb Roitblat’s approach to using an Elusion sample to calculate recall because he uses the point projection of prevalence and elusion only, and does not factor in the recall interval ranges. My reason for opposing this simplification was set out in detail In Legal Search Exact Recall Can Never Be Known. It is scientifically and mathematically wrong to use point projections and not include ranges.

TredennickI note that industry leader John Tredennick also disagrees with Herb’s approach. See his recent article: Measuring Recall in E-Discovery Review: A Tougher Problem Than You Might RealizePart Two. After explaining Herb’s eRecall John says this:

Does this work? Not so far as I can see. The formula relies on the initial point estimate for richness and then a point estimate for elusion.

I agree with John Tredennick in this criticism of Herb’s method. So too does Bill Dimm, who has a PhD in Physics and is the founder and CEO of Hot Neuron. Bill summarizes Herb’s eRecall method in his article, eRecall: No Free LunchHe uses an example to show that eRecall does not work at all in low prevalence situations. Of course, all sampling is challenged by extremely low prevalence, even ei-Recall, but at least my interval approach does not hide the limitations of such recall estimates. There is no free lunch. Recall estimates are just one quality control effort among many.

Maura Grossman and Gordon Cormack also challenge the validity of Herb’s method. They refer to Roitblat’s eRecall as a specious argument. Grossman and Cormack make the same judgment about several other approaches that compare the ratios of point projections and show how they all suffer from a basic mathematical statistical error, which they call the Ratio Method Fallacy. Comments on ‘The Implications of Rule 26(g) on the Use of Technology-Assisted Review,’ supra at 308-309.

Missed_targetIn Grossman & Cormack’s, Guest Blog: Talking Turkey (e-Discovery Team, 2014) they explained an experiment that they did and reported on in the Comments article where they repeatedly used Roitblat’s eRecall, the direct method, and other methods to estimate recall. They used a review known to have achieved 75% recall and 83% precision, from a collection with 1% prevalence. They results showed that in this review “eRecall provides an estimate that is no better than chance.” That means eRecall was a complete failure as a quality assurance measure.

Although my proposed range method is a comparative Ratio Method, it avoids the fallacy of other methods criticized by Grossman and Cormack. It does so because it includes binomial probability ranges in the recall calculations and eschews the errors of point projection reliance. It is true that the range of recall estimates using ei-Recall may be still uncomfortably large in some low yield projects, but at least it will be real and honest, and, unlike eRecall, it is better than nothing.

No Legal Economic Arguments Justify the Errors of Simplified Point Projections 

arrows missing targetThe oversimplified point projection ratio approach can lead to a false belief of certainty for those who do not understand probability ranges inherent in random samples. We presume that Herb Roitblat understands the probability range issues, but he chooses to simplify anyway on the basis of what appears to me to be essentially legal-economic arguments, namely proportionality cost-savings, and the inherent vagaries of legal relevance. Roitblat, The Pendulum Swings: Practical Measurement in eDiscovery.

I disagree strongly with Roitblat’s logic. As one scholar in private correspondence pointed out, Herb appears to fall victim to the classic fallacy of the converse. Herb asserts that “if the point estimate is X, there is a 50% probability that the true value is greater than X.” What *is* true (for an unbiased estimate) is that “if the true value is X, there is a 50% probability that the estimate is greater than X.” Assuming the latter implies the former is classic fallacy of the converse. Think about it. It is a very good point. For a more obvious example of the fallacy of the converse consider this: “Most accidents occur within 25 miles from home; therefore, you are safest when you are far from home.”

Although I disagree with Herb Roitblat’s logic, I do basically agree with many of his non-statistical arguments and observations on document review, including, for instance, the following:

Depending on the prevalence of responsive documents and the desired margin-of-error, the effort needed to measure the accuracy of predictive coding can be more than the effort needed to conduct predictive coding.

Until a few years ago, there was basically no effort expended to measure the efficacy of eDiscovery. As computer-assisted review and other technologies became more widespread, an interest in measurement grew, in large part to convince a skeptical audience that these technologies actually worked. Now, I fear, the pendulum has swung too far in the other direction and it seems that measurement has taken over the agenda.

There is sometimes a feeling that our measurement should be as precise as possible. But when the measure is more precise than the underlying thing we are measuring, that precision gives a false sense of security. Sure, I can measure the length of a road using a yardstick and I can report that length to within a fraction of an inch, but it is dubious whether the measured distance is accurate to within even a half of a yard.

bullseye_arrow_hitAlthough I agree with many of the points of Herb’s legal economic analysis in his article, The Pendulum Swings: Practical Measurement in eDiscoveryI disagree with the conclusion. The quality of the search software, and legal search skills of attorney-users of this software, have both improved significantly in the past few years. It is now possible for relatively high recall levels to be attained, even including ranges, and even without incurring extraordinary efforts and costs as Herb and others suggest. (As a side note, please notice that I am not opining on a specific minimum recall number. That is not helpful because it depends on too many variable factors unique to particular search projects. However, I would point out that in the TREC Legal Track studies in 2008 and 2009 the participants, expert searchers all, attained verified recall levels of only 20% to 70%. See The Legal Implications of What Science Says About Recall. All I am saying is that in my experience our recall efforts have improved and are continually improving as our software and skills improve.)

Further, although relevance and responsiveness can sometimes be vague and elusive as Roitblat points out, and human judgments can be wrong and inconsistent, there are quality control process steps that can be taken to significantly mitigate these problems, including the often overlooked better dialogues with the requesting party. Legal search is not an arbitrary exercise such that it is a complete waste of time to try to accurately measure recall.

I disagree with Herb’s suggestion to the contrary based on his evaluation of legal relevance judgments. He reaches this conclusion based on the very interesting study he did with Anne Kershaw and Patrick Oot on a large-scale document review that Verizon did nearly a decade ago. Document Categorization in Legal Electronic Discovery: Computer Classification vs. Manual ReviewIn that review Verizon employed 225 contract reviewers and a Twentieth Century linear review method wherein low paid contract lawyers sat in isolated cubicles and read one document after another. The study showed, as Herb summarizes it, that the reviewers agree with one another on relevance calls only about 50% of the time.” Measurement in eDiscovery at pg. 6. He takes that finding as support for his contention that consistent legal review is impossible and so there is no need to bother with finer points of recall intervals.

coin_flipI disagree. My experience as an attorney making judgments on the relevancy of documents since 1980 tells me otherwise. It is absurd, even insulting, to call legal judgment a mere matter of coin flipping. Yes, there are well-known issues with consistency in legal review judgments in large-scale reviews, but this just makes the process more challenging, more difficult, not impossible.

Although consistent review may be impossible if large teams of contract lawyers do linear review in isolation using yesterday’s technology, that does not mean consistent legal judgments are impossible. It just means the large team linear review process is deeply flawed. That is why the industry has moved away from the approaches used by the Verizon team review nearly ten years ago. We are now using predictive coding, small teams of SMEs and contract lawyers, and many new innovative quality control procedures, including soon, I hope, ei-Recall. The large team linear review approach of a decade ago, and other quality factors, were the primary causes of the inconsistencies seen in the Verizon approach, not the inherent impossibility of determining legal relevance.

Good Recall Results Are Possible Without Heroic Efforts
But You Do Need Good Software and Good Methods

robot_whispererEven with the consistency and human error challenges inherent in all legal review, and even with the ranges of error inherent in any valid recall calculation, it is, I insist, still possible to attain relatively high recall ranges in most projects. (Again, note that I will not commit to a specific general minimum range.) I am seeing better recall ranges attained in more and more of my projects and I am certainly not a mythical TAR-whisperer, as Grossman and Cormack somewhat tongue in cheek described lawyers who may have extraordinary predictive coding search skills. Comments on ‘The Implications of Rule 26(g) on the Use of Technology-Assisted Review,’ at pg. 298. Any experienced lawyer with technology aptitude can attain impressive results in large-scale document reviews. They just need to use hybrid, multimodal, CAL-type, quality controlled, search and review methods. They also need to use proven, high quality, bona fide predictive coding software. I am able to teach this in practice with bright, motivated, hard-working, technology savvy lawyers.

cesar_millan-Dog_WhispererLegal search is a new legal skill to be sure, just like countless others in e-discovery and other legal fields. I happen to find the search and review challenges more interesting than the large enterprise preservation problems, but they are both equally difficult and complex. TAR-whispering is probably an easier skill to learn than many others required today in the law.  (It is certainly easier than becoming a dog whisperer like Cesar Millan. I know. I’ve tried and failed many times.)

Think of the many arcane choice of law issues U.S. lawyers have faced for over a century in our 50-state, plus federal law system. Those intellectual problems are more difficult than predictive coding. Think of the tax code, securities, M&A, government regulations, class actions. It is all hard. All difficult. But it can all be learned. Like everything else in the law, large-scale document review just requires a little aptitude, hard work and lots of legal practice. It is no different from any other challenge lawyers face. It just happens to require more software skills, sampling, basic math, and AI intuition than any other legal field.

On the other point of bona fide predictive coding software, while I will not name names, as far as I am concerned the only bona fide software on the market today uses active machine learning algorithms. It does not depend instead on some kind of passive learning process (although they too can be quite effective, they are not predictive coding algorithms, and, in my experience, do not provide as powerful a search tool). I am sorry to say that some legal review software on the market today falsely claims to have predictive coding features, when, in fact, it does not. It is only passive learning, more like concept search, than AI-enhanced search. With software like that, or even with good software where the lawyers use poor search and review methods, or do not really know what they are searching for (poor relevance scope), then the efforts required to attain high recall ranges may indeed be very extensive and thus cost prohibitive as Herb Roitblat argues. If your tools and or methods are poor, it takes much longer to reach your goals.

rcl-head_3d_SMALLOne final point regarding Herb’s argument, I do not think sampling really needs to be as cost prohibitive as he and others suggest. As noted before in In Legal Search Exact Recall Can Never Be Known, one good SME and skilled contract review attorney can carefully review a sample of 1,534 documents for between $1,000 and $2,000. In large review projects that is hardly a cost prohibitive barrier. There is no need to be thinking in terms of small 385 document sample sizes, which create a huge margin of error of 5%. This is what Herb Rotiblat and others do when suggesting that all sampling is anyway ineffective, so just ignore intervals and ranges. Any large project can afford a full sample of 1,534 documents to cut the interval in half to a 2.5% margin of error. Many can afford much larger samples to narrow the interval range even further, especially if the tools and methods used allow them to attain their recall range goals in a fast and effective manner.

John Tredennick, who, like me, is an attorney, also disagrees with Herb’s legal-economic analysis in favor of eRecall, but John proposes a solution involving larger sample sizes, wherein the increased cost burden would be shifted onto the requesting party. Recall in E-Discovery Review: A Tougher Problem Than You Might RealizePart Two. I do not disagree with John’s assertions in his article, and cost shifting may be appropriate in some cases. It is not, however, my intention to address the cost-shifting arguments here, or the other good points made in John’s article. Instead, my focus in the remaining section of this article will be to provide a series of examples of ei-Recall in action. For me, and I suspect for many of you, seeing a method in action is the best way to understand it.

Summary of the Five Reasons ei-Recall is the new Gold Standard

ei-Recall_pentagramBefore moving onto the samples, I wanted to summarize what we have covered so far and go over the five main reasons ei-Recall is superior to all other recall methods. First, and most important, is the fact ei-Recall calculates a recall range, and not just one number. As shown by In Legal Search Exact Recall Can Never Be Known, recall statements must include confidence interval range values to be meaningful. Recall should not be based on point projections alone. Therefore any recall calculation method must calculate both a high and low value. The ei-Recall method I offer here is designed for the correct high low interval range calculations. That, in itself, makes it a significant improvement over all point projection recall methods.

The second advantage of ei-Recall is that is only uses one random sample, not two, or more. This avoids the compounding of variables, uncertainties, and outlier events inherent in any system that uses multiple chance events, multiple random samples. The costs are also controlled better in a one sample method like this, especially since the one sample is of reasonable size. This contrasts with the Direct Method, which also uses one sample, but the sample has to be insanely large. That is not only very costly, but also introduces a probability of more human error in inconsistent relevancy adjudications.

The timing of the one sample in ei-Recall is another of its advantages. It is taken at the end of the project when the relevance scope has been fully articulated.

Another key advantage of ei-Recall is that the True Positives used for the calculation are not estimated, and are not projected by random samples. They are documents confirmed to be relevant by multiple quality control measures, including multiple reviews of these documents by humans, or computer, and often both.

Finally, ei-Recall has the advantage of simplicity, and ease of use. It can be carried out by any attorney who knows fractions. The only higher math required, the calculation of binomial confidence intervals, can be done by easily available online calculators. You do not need to hire a statistician to make the recall range calculations using ei-Recall.

To be continued.

First Example of How to Calculate Recall Using the ei-Recall Method

Let us begin with the same simple hypothetical used in In Legal Search Exact Recall Can Never Be Known. Here we assume a review project of 100,000 documents. By the end of the search and review, when we could no longer find any more relevant documents, we decided to stop and run our ei-Recall quality assurance test. We had by then found and verified 8,000 relevant documents, the True Positives. That left 92,000 documents presumed irrelevant that would not be produced, the Negatives.

As a side note, the decision to stop may be somewhat informed by running estimates of possible recall range attained based on early prevalence assumptions from a sample of all documents at or near the beginning of the project. The prevalence based recall range estimate would not, however, be the sole driver of the decision to stop and test. The prevalence based recall estimates alone can be very unreliable as shown In Legal Search Exact Recall Can Never Be Known. That is one of the main reasons for developing the ei-Recall alternative. I explained the thinking behind the decision to stop in Visualizing Data in a Predictive Coding Project – Part Three.

I will not have stopped the review in most projects (proportionality constraints aside), unless I was confident that I had already found all of those (highly relevant) types of documents; already found all types of strong relevant documents, and already found all highly relevant document, even if they are cumulative. I want to find each and every instance of all hot (highly relevant) documents that exists in the entire collection. I will only stop (proportionality constraints aside) when I think the only relevant documents I have not recalled are of an unimportant, cumulative type; the merely relevant. The truth is, most documents found in e-discovery are of this type; they are merely relevant, and of little to no use to anybody except to find the strong relevant, new types of relevant evidence, or highly relevant evidence.

Back to our hypothetical. We take a sample of 1,534 (95%+/-2.5%) documents, creating a 95% confidence level and 2.5% confidence interval, from the 92,000 Negatives. This allows us to estimate how many relevant documents had been missed, the False Negatives.

Assume we found only 5 False Negatives. Conversely, we found that 1,529 of the documents picked at random from the Negatives were in fact irrelevant as expected. They were True Negatives.

The percentage of False Negatives in this sample was thus a low 0.33% (5/1534). Using the Normal, but wrong, Gaussian confidence interval the projected total number of False Negatives in the entire 92,000 Negatives would thus be between 5 and 2,604 documents (0.33%+2.5%= 2.83% * 92,000). Using the binomial interval calculation the range would be from 0.11% to 0.76%. The more accurate binomial calculation eliminates the absurd result of a negative interval on the low recall range (.33% -2.5%= -2.17). The fact that negative recall arises from using the Gaussian normal distribution demonstrates why the binomial interval calculation should always be used, not Gaussian, especially in low prevalence. From this point forward, in accordance with the ei-Recall method, we will only use the more accurate Binomial range calculations. Here the correct range generated by the binomial interval is from between 101 (92,000 * 0.11%) and 699 (92,000 * 0.76%) False Negatives. Thus the FNh value is 699, and FNl is 101.

ei-recall_exampleThe calculation of the lowest end of the recall range is based on the high end of the False Negatives projection: Rl = TP / (TP+FNh) = 8,000 / (8,000 + 699) = 91.96% 

The calculation of the highest end of the recall range is based on the low end of the False Negatives projection: Rh = TP / (TP+FNl) = 8,000 / (8,000 + 101) = 98.75%.

Our final recall range values for this first hypothetical is thus from 92%- 99% recall. It was an unusually good result.


Ex. 1 – 92% – 99%

It is important to note that we could have still failed this quality assurance test, in spite of the high recall range shown, if any of the five False Negatives found was a highly relevant, or unique-strong relevant document. That is in accord with the accept on zero error standard that I always apply to the final elusion sample, a standard having nothing directly to do with ei-Recall. Still, I recommend that the e-discovery community also accept this as a corollary to implement ei-Recall. I have previously explained this zero error quality assurance protocol on this blog several times, most recently in Visualizing Data in a Predictive Coding Project – Part Three where I explained:

I always use what is called an accept on zero error protocol for the elusion test when it comes to highly relevant documents. If any are highly relevant, then the quality assurance test automatically fails. In that case you must go back and search for more documents like the one that eluded you and must train the system some more. I have only had that happen once, and it was easy to see from the document found why it happened. It was a black swan type document. It used odd language. It qualified as a highly relevant under the rules we had developed, but just barely, and it was cumulative. Still, we tried to find more like it and ran another round of training. No more were found, but still we did a third sample of the null set just to be sure. The second time it passed.

Variations of First Example with Higher False Negatives Ranges

I want to provide two variations of this hypothetical where the sample of the null set, Negatives, finds more mistakes, more False Negatives. Variations like this will provide a better idea of the impact of the False Negatives range on the recall calculations. Further, the first example wherein I assumed that only five mistakes were found in a sample of 1,534 is somewhat unusual. A point projection ratio of 0.33% for elusion is on the low side for a typical legal search project. In my experience in most projects a higher rate of False Negatives will be found, say in the 0.5% to 2% range.

Let us assume for the first variation that instead of finding 5 False Negatives, we find 20. That is a quadrupling of the False Negatives. It means that we found 1,514 True Negatives and 20 False Negatives in the sample of 1,534 documents from the 92,000 document discard pile. This creates a point projection of 1.30% (20 / 1534), and a binomial range of 0.8% to 2.01%. This generates a projected range of total False Negatives of from 736 (92,000 * .8%) to 1,849 (92,000 * 2.01%).

Now let’s see how this quadrupling of errors found in the sample impacts the recall range calculation.

ei-recall_example2The calculation of the low end of the recall range is based on the high end of the False Negatives projection: Rl = TP / (TP+FNh) = 8,000 / (8,000 + 1,849) = 81.23% 

The calculation of the high end of the recall range is based on the low end of the False Negatives projection: Rh = TP / (TP+FNl) = 8,000 / (8,000 + 736) = 91.58%.

Our final recall range values for this variation of the first hypothetical is thus 81% – 92%.

In this first variation the quadrupling of the number of False Negatives found at the end of the project, from 5 to 20, caused an approximate 10% decrease in recall values from the first hypothetical where we attained a recall range of 92% to 99%.

Ex. 2

Ex. 2 – 81% – 87%

Let us assume a second variation that instead of finding 5 False Negatives, finds 40. That is eight times the number of False Negatives found in the first hypothetical. It means that we found 1,494 True Negatives and 40 False Negatives in the sample of 1,534 documents from the 92,000 document discard pile. This creates a point projection of 2.61% (40/1534), and a binomial range of 1.87% to 3.53%. This generates a projected range of total False Negatives of from 1,720 (92,000*1.87%) to 3,248 (92,000*3.53%).

ei-recall_example3The calculation of the low end of the recall range is based on the high end of the False Negatives projection: Rl2 = TP / TP+FNh = 8,000 / (8,000 + 3,248) = 71.12% 

The calculation of the high end of the recall range is based on the low end of the False Negatives projection: Rh2 = TP / TP+FNl = 8,000 / (8,000 + 1,720) = 82.30%.

Our recall range values for this variation of the first hypothetical is thus 71% – 82%.

In this second variation the eightfold increase of the number of False Negatives found at the end of the project, from 5 to 20, caused an approximate 20% decrease in recall values from the first hypothetical where we attained a recall range of 92% to 99%.

Ex. 3

Ex. 3 – 71% – 82%

Second Example of How to Calculate Recall Using the ei-Recall Method

We will again go back to the second example used in In Legal Search Exact Recall Can Never Be KnownThe second hypothetical assumes a total collection of 1,000,000 documents and that 210,000 relevant documents were found and verified.

In the random sample of 1,534 documents (95%+/-2.5%) from the 790,000 documents withheld as irrelevant (1,000,000 – 210,000) we assume that only ten mistakes were uncovered, in other words, 10 False Negatives. Conversely, we found that 1,524 of the documents picked at random from the discard pile (another name for the Negatives) were in fact irrelevant as expected; they were True Negatives.

The percentage of False Negatives in this sample was thus 0.65% (10/1534). Using the binomial interval calculation the range would be from 0.31% to 1.2%. The range generated by the binomial interval is from  2,449 (790,000*0.31%) to 9,480 (790,000*1.2%) False Negatives.

ei-recall_example4The calculation of the lowest end of the recall range is based on the high end of the False Negatives projection: Rl2 = TP / TP+FNh = 210,000 / (210,000 + 9,480) = 95.68% 

The calculation of the highest end of the recall range is based on the low end of the False Negatives projection: Rh2 = TP / TP+FNl = 210,000 / (210,000 + 2,449) = 98.85%.

Our recall range for this second hypothetical is thus 96% – 99% recall. This is a highly unusual, truly outstanding result. It is, of course, still subject to the outlier result uncertainty inherent in the confidence level. In that sense my labels on the diagram below of “worst” or “best” case scenario are not correct. It could be better or worse in five times out of one hundred times the sample is drawn in accord with the 95% confidence level. See the discussion near the end of my article In Legal Search Exact Recall Can Never Be Known, regarding the role that luck necessarily plays in any random sample. This could have been a lucky draw, but nevertheless, it is just one quality assurance factor among many, and is still an extremely good recall range achievement.

Ex.4 -

Ex.4 – 96% – 99%

Variations of Second Example with Higher False Negatives Ranges

I now offer three variations of the second hypothetical where each has a higher False Negative rate. These examples should better illustrate the impact of the elusion sample on the overall recall calculation.

Let us first assume that instead of finding 10 False Negatives, we find 20, a doubling of the rate. This means that we found 1,514 True Negatives and 20 False Negatives in the sample of 1,534 documents in the 790,000 document discard pile. This creates a point projection of 1.30% (20/1534), and a binomial range of 0.8% to 2.01%. This generates a projected range of total False Negatives of from 6,320 (790,000*.8%) to 15,879 (790,000*2.01%).


Now let us see how this doubling of errors in the second sample impacts the recall range calculation.

The calculation of the low end of the recall range is: Rl = TP / (TP+FNh) = 210,000 / (210,000 + 15,879) = 92.97% 

The calculation of the high end of the recall range is: Rh = TP / (TP+FNl) = 210,000 / (210,000 + 6,320) = 97.08%.

Our recall range for this first variation of the second hypothetical is thus 93% – 97%

The doubling of the number of False Negatives from 10 to 20, caused an approximate 2.5% decrease in recall values from the second hypothetical where we attained a recall range of 96% to 99%.

Ex. 5 -

Ex. 5 – 93% – 97%

Let us assume a second variation where instead of finding 10 False Negatives at the end of the project, we find 40. That is a quadrupling of the number of False Negatives found in the first hypothetical. It means that we found 1,494 True Negatives and 40 False Negatives in the sample of 1,534 documents from the 790,000 document discard pile. This creates a point projection of 2.61% (40/1534), and a binomial range of 1.87% to 3.53%. This generates a projected range of total False Negatives of from 14,773 (790,000*1.87%) to 27,887 (790,000*3.53%).

ei-recall_example6The calculation of the low end of the recall range is now: Rl = TP / (TP+FNh) = 210,000 / (210,000 + 27,887) = 88.28% 

The calculation of the high end of the recall range is now: Rh = TP / (TP+FNl) = 210,000 / (210,000 + 14,773) = 93.43%.

Our recall range for this second variation of second hypothetical is thus 88% – 93%.

The quadrupling of the number of False Negatives from 10 to 40, caused an approximate 7% decrease in recall values from the original where we attained a recall range of 96% to 99%.

Ex. 6 – 88% – 93%

If we do a third variation and increase the number of False Positives found by eight-times, from 10 to 80, this changes the point projection to 5.22% (80/1534), with a binomial range of 4.16% to 6.45%.  This generates a projected range of total False Negatives of from 32,864 (790,000*4.16%) to 50,955 (790,000*6.45%).

ei-recall_example7The calculation of the low end of the recall range is: Rl = TP / (TP+FNh) = 210,000 / (210,000 + 50,955) = 80.47%. 

The calculation of the high end of the recall range is: Rh = TP / (TP+FNl) = 210,000 / (210,000 + 32,864) = 86.47%.

Our recall range for this third variation of the second hypothetical is thus 80% – 86%.

The eightfold increase of the number of False Negatives, from 10 to 80, caused an approximate 15% decrease in recall values from the second hypothetical where we attained a recall range of 96% to 99%.

Ex. 7 - 80% - 86%

Ex. 7 – 80% – 86%

By now you should have a pretty good idea of how the ei-Recall calculation works, and a feel for how the number of False Negatives found impacts the overall recall range.

Third Example of How to Calculate Recall Using the ei-Recall Method where there is Very Low Prevalence

A criticism of many recall calculation methods is that they fail and become completely useless in very low prevalence situations, say 1%, or sometimes even less. Such low prevalence is considered by many to be common in legal search projects.

upside_down_plane_stampObviously it is much harder to find things that are very rare, such as the famous, and very valuable, Inverted Jenny postage stamp with the upside down plane. These stamps exist, but not many. Still, it is at least possible to find them (or buy them), as opposed to a search for a Unicorn or other complete fiction. (Please, Unicorn lovers, no hate mail!) These creatures cannot be found no matter how many searches and samples you take because they do not exist. There is absolute zero prevalence.

unicornThis circumstance sometimes happens in legal search, where one side claims that mythical documents must exist because they want them to. They have a strong suspicion of their existence, but no proof. More like hope, or wishful thinking. No matter how hard you look for such smoking guns, you cannot find them. You cannot find something that does not exist. All you can do is show that you made reasonable, good faith efforts to find the Unicorn documents, and they did not appear. Recall calculations make no sense in crazy situations like that because there is nothing to recall. Fortunately that does not happen too often, but it does happen, especially in the wonderful world of employment litigation.

We are not going to talk further about a search for something that does not exist, like a Unicorn, the zero prevalence. We will not even talk about the extremely, extremely rare, like the Inverted Jenny. Instead we are going to talk about prevalence of about 1%, which is still very low.

In many cases, but not all, very low prevalence like 1%, or less, can be avoided, or at least mitigated, by intelligent culling. This certainly does not mean filtering out all documents that do not have certain keywords. There are other, more reliable methods than simple keywords to eliminate superfluous irrelevant documents, including elimination by file type, date ranges, custodians, and email domains, among other things.

When there is a very low prevalence of relevant documents, this necessarily means that there will be a very large Negatives pool, thus diluting the sampling. There are ways to address the large Negatives sample pool, as I discussed previously. The most promising method is to cull out the low end of the probability rankings where relevant documents should anyway be non-existent.

Even with the smartest culling possible, low prevalence is often still a problem in legal search. For that reason, and because it is the hardest test for any recall calculation method, I will end this series of examples with a completely new hypothetical that considers a very low prevalence situation of only 1%. This means that there will be a large size Negatives pool: 99% of the total collection.

We will again assume a 1,000,000 document collection, and again assume sample sizes using 95% +/-2.5% confidence level and interval parameters. An initial sample of all documents taken at the beginning of the project to give us a rough sense of prevalence for search guidance purposes (not recall calculations), projected a range of relevant documents of from 5,500 to 16,100.

The lawyers in this hypothetical legal search project plodded away for a couple of weeks and found and confirmed 9,000 relevant documents, True Positives all. At this point they are finding it very difficult and time consuming to find more relevant documents. What they do find is just more of the same. They are sophisticated lawyers who read my blog and have a good grasp of the nuances of sampling. So they know better than to simply rely on a point projection of prevalence to calculate recall, especially one based on a relatively small sample of a million documents taken at the beginning of the project. See In Legal Search Exact Recall Can Never Be KnownThey know that their recall level could be only a 56% recall 9,000/16,100 (or perhaps far less, in the event the one sample they took was a confidence level outlier event, or there was more concept drift than they thought). It could also be near perfect, 100% recall, when they consider the binomial interval range going the other way. The 9,000 documents they had found was way more than the low range of 5,500. But they did not really consider that too likely.

They decide to stop the search and take a second 1,534 document sample, but this time of the 991,000 null set (1,000,000 – 9,000). They want to follow the ei-Recall method, and they also want to test for any highly relevant or unique strong relevant documents by following the accept on zero error quality assurance test. They find -1- relevant document in that sample. It is just a more of the same type merely relevant document. They had seen many like it before. Finding a document like that meant that they passed the quality assurance test they had set up for themselves. It also meant that using the binomial intervals for 1/1534, which is from 0.00% and 0.36%, there is a projected range of False Negatives of from between -0- and 3,568 documents (991,000*0.36%). (Actually, a binomial calculator that shows more decimal places than any I have found on the web (hopefully we can fix that soon) will not show zero percent, but some very small percentage less than one hundredth of a percent, and thus some documents, not -0- documents, and thus something slightly less than 100% recall.)

ei-recall_example8They then took out the ei-Recall formula and plugged in the values to see what recall range they ended up with. They were hoping it was tighter, and more reliable, than the 56% to 100% recall level they calculated from the first sample alone based on prevalence.

Calculation for the low end of the recall range: Rl = TP / (TP+FNh) = 9,000 / (9,000 + 3,568) = 71.61%.  

Calculation for the high end of the recall range: Rh = TP / (TP+FNl) = 9,000 / (9,000 + 0) = 100%.

The recall range using ei-Recall was 72% – 100%.

Ex. 8 - 72% - 100%

Ex. 8 – 72% – 100%

The attorneys’ hopes in this extremely low prevalence hypothetical were met. The 72%-100% estimated recall range was much tighter than the original 56%-100%. It was also more reliable because it was based on a sample taken at the end of the project when relevance was well defined. Although this sample did not, of and by itself, prove that a reasonable legal effort had been made, it did strongly support that position. When considering all of the many other quality control efforts they could report, if challenged, they were comfortable with the results. Assuming that they did not miss a highly relevant document that later turns up in discovery, it is very unlikely they will ever have to redo, or even continue, this particular legal search and review project.

Would the result have been much different if they had doubled the sample size, and thus doubled the cost of this quality control effort? Let us do the math and find out, assuming that everything else was the same.

ei-recall_example9This time the sample is 3,068 documents from the 991,000 null set. They find two relevant documents, False Negatives, of a kind they had seen many times before. This created a binomial range of 0.01%  to 0.24%, projecting a range of False Negatives from 99 to 2,378 (991,000 * 0.01% — 991,000 * 0.24%). That creates a recall range of 79% – 99%.

Rl = TP / (TP+FNh) = 9,000 / (9,000 + 2,378) = 79.1%.  

Rh = TP / (TP+FNl) = 9,000 / (9,000 + 99) = 98.91%.

Ex. 9 - 79% - 99%

Ex. 9 – 79% – 99%

In this situation by doubling the sample size the attorneys were able to narrow the recall range from 72% – 100% to 79% – 99%. But was it worth the effort and doubling of  cost? I do not think so, at least not in most cases. But perhaps in larger cases, it would be worth the expense to tighten the range somewhat and so increase somewhat the defensibility of your efforts. After all, we are assuming in this hypothetical that the same proportional results would turn up in a sample size double that of the original. The results could have been much worse, or much better. Either way, your results would be more reliable than an estimate based on a sample half that size, and would have produced a tighter range. Also, you may sometimes want to take a second sample of the same size, if you suspect the first was an outlier.

Let is consider one more example, this time of an even smaller prevalence and larger document collection. This is the hardest challenge of all, a near Inverted Jenny puzzler. Assume a document collection of 2,000,000 and a prevalence based on a first random sample for search-help purposes, where again only one relevant was found in the sample of 1,534 sample. This suggested there could be as many as 7,200 relevant documents (0.36% * 2,000,000). So in this second hypothetical we are talking about a dataset where the prevalence may be far less than one percent.

ei-recall_extreme_low_prevalenceAssume next that only 5,000 relevant documents were found, True Positives. A sample 1,534 of the remaining 1,995,000 documents found -3- relevant, False Negatives. The binomial intervals for 3/1534, is from 0.04% to 0.57%, producing  a projected range of False Negatives of from between 798 and 11,372 documents (1,995,000 * .04% — 1,995,000 * 0.57%). Under ei-Recall the recall range measured is 31% – 86%.

Rl = TP / (TP+FNh) = 5,000 / (5,000 + 11,372) = 30.54%.  

Rh = TP / (TP+FNl) = 5,000 / (5,000 + 798) = 86.24%.

31% – 86% is a big range. Most would think too big, but remember, it is just one quality assurance indicator among many.

Ex. 10 - 31% - 86%

Ex. 10 – 31% – 86%

The size of the range could be narrowed by a larger sample. (It is also possible to take two samples, and, with some adjustment, add them together as one sample. This is not mathematically perfect, but fairly close, if you adjust for any overlaps, which anyway would be unlikely.) Assume the same proportions where we sample 3,068 documents from 1,995,000 Negatives, and find -6- relevant, False Negatives. The binomial range is 0.07% – 0.43%. The projected number of False Negatives is 1,397 – 8,579 (1,995,000*.07% – 1,995,000*.43%). Under ei-Recall the range is 37% – 78%.

Rl = TP / (TP+FNh) = 5,000 / (5,000 + 8,579) = 36.82%.  

Rh = TP / (TP+FNl) = 5,000 / (5,000 + 1,397) = 78.16%.

Ex. 11 - 37% - 78%

Ex. 11 – 37% – 78%

The range has been narrowed, but is still very large. In situations like this, where there is a very large Negatives set, I would suggest taking a different approach. As discussed in Part One, you may want to consider a rational culling down of the Negatives. The idea is similar to that behind stratified sampling. You create a subset or strata of the entire collection of Negatives that has a higher, hopefully much higher prevalence of False Negatives than the entire set. See eg. William Webber, Control samples in e-discovery (2013) at pg. 3

CULLING.filters_MULTIMODALAlthough Webber’s paper only uses keywords as an example of an easy way to create a strata, in reality in modern legal search today there are a number of methods that could be used to create the stratas, only one of which is keywords. I use a combination of many methods that varies in accordance with the data set and other factors. I call that a multimodal method. In most cases (but not all), this is not too hard to do, even if you are doing the stratification before active machine learning begins. The non-AI based culling methods that I use, typically before active machine learning begins, include parametric Boolean keywords, concept, key player, key time, similarity, file type, file size, domains, etc.

After the predictive coding begins and ranking matures, you can also use probable relevance ranking as a method of dividing documents into strata. It is actually the most powerful of the culling methods, especially when it comes to predicting irrelevant documents. The second filter level is performed at or near the end of a search and review project. (This is all shown in the two-filter diagram above, which I may explain in greater detail in a future blog.) The second AI based filter can be especially effective in limiting the Negatives size for the ei-Recall quality assurance test. The last example will show how this works in practice.

Low_prevalence_exampleWe will begin this example as before, assuming again 2,000,000 documents where the search finds only 5,000. But this time before we take a sample of the Negatives we divide them into two strata. Assume, as we did in the example we considered in Part One, that the predictive coding resulted in a well defined distribution of ranked documents. Assume that all 5,000 documents found were in the 50%, or higher, probable relevance ranking (shown in red in the diagram). Assume that all of the 1,995,000 presumed irrelevant documents are ranked 49.9%, or less, probable relevant (shown in blue in the diagram). Finally assume that 1,900,000 of these documents are ranked 10% or less probable relevant. Thus leaving 95,000 documents ranked between 10.1% and 49.9%.

Assume also that we have good reason to believe based on our experience with the software tool used, and the document collection itself, that all, or almost all, False Negatives are contained in the 95,000 group. We therefore limit our random sample of 1,534 documents to the 95,000 lower midsection of the Negatives. Finally, assume we now find -30- relevant, False Negatives, none of them important.

ei-recall_ex_StrataThe binomial range is 0.80% – 2.01%, but this time the projected number of False Negatives is 1,254 – 2,641 (95,000*1.32%  — 95,000*2.78%). Under ei-Recall the range is 72.37% – 80.06%.

Rl = TP / (TP+FNh) = 5,000 / (5,000 + 2,641) = 72.37%.  

Rh = TP / (TP+FNl) = 5,000 / (5,000 + 1,245) = 80.06%.

We see that culling down the Negative set of documents in a defensible manner can lead to a much tighter recall range. Assuming we did the culling correctly, the resulting recall range would also be more accurate. On the other hand, if the culling was wrong, based on incorrect presumptions, then the resulting recall range would be less accurate.

Ex. 12 - 72% - 80%

Ex. 12 – 72% – 80%

The fact is, no random sampling techniques can provide completely reliable results in very low prevalence data sets. There is no free lunch, but, at least with ei-Recall the bill for your lunch is honest because it includes ranges. Moreover, with intelligent culling to increase the probable prevalence of False Negatives, you are more likely to get a good meal.


ei-Recall_pentagramThere are five basic advantages of ei-Recall over other recall calculation techniques:

  1. Interval Range values are calculated, not just a deceptive point value. As shown by In Legal Search Exact Recall Can Never Be Known, recall statements must include confidence interval range values to be meaningful.
  2. One Sample only is used, not two, or more. This limits the uncertainties inherent in multiple random samples.
  3. End of Project is when the sample of the Negatives is taken for the calculation. At that time the relevance scope has been fully developed.
  4. Confirmed Relevant documents that have been verified as relevant by iterative reviews, machine and human, are used for the True Positives. This eliminates another variable in the calculation.
  5. Simplicity is maintained in the formula by reliance on basic fractions and common binomial confidence interval calculators. You do not need an expert to use it.

I suggest you try ei-Recall. It has been checked out by multiple information scientists and will no doubt be subject to more peer review here and elsewhere. Be cautious in evaluating any criticisms you may read of ei-Recall from persons with a vested monetary interest in the defense of a competitive formula, especially vendors, or experts hired by vendors. Their views may be colored by their monetary interests. I have no skin in the game. I offer no products that include this method. My only goal is to provide a better method to validate large legal search projects, and so, in some small way, to improve the quality of our system of justice. The law has given me much over the years. This method, and my other writings, are my personal payback.

I offer ei-Recall to anyone and everyone, no strings attached, no payments required. Vendors, you are encouraged to include it in your future product offerings. I do not want royalties, nor even insist on credit (although you can do so if you wish, assuming you do not make it seem like I endorse your product). ei-Recall is all part of the public domain now. I have no product to sell here, nor do I want one. Although I do hope to create an online calculator soon for ei-Recall. When I do, that too will be a give away.

eLeetMy time and services as a lawyer to implement ei-Recall are not required. Simplicity is one of its strengths, although it helps if you are part of the eLeet. I think I have fully explained how it works in this lengthy article. Still, if you have any non-legal technical questions about its application, send me an email, and I will try to help you out. Gratis of course. Just realize that I cannot by law provide you with any legal advice. All articles in my blog, including this one, are purely for educational services, and are not legal advice, nor in any way a solicitation for legal services. Show this article to your own lawyer or e-discovery vendor. You do not have to be 1337 to figure it out (although it helps).