Some thoughts on the use of statistical
sampling in legal research
by Carlos N. BOUZAHERRERA, Professor
at the Faculty of Mathematic of University of Havana, Cuba.
Much of legal research is based on discovering
facts through analyzing a lot of papers. Electronically Stored Information
(ESI) poses issues on using data stored electronically. With the increase of
data volumes, a need of reducing costs, without violating the accepted
assumptions poses urgently mid changes in the law firms. The reduction of costs
should not be solved by using “low‐cost lawyers”.
This paper discusses on the use of Technology
Assisted Review and Statistical Sampling for retrieving information and some
examples are discussed for illustrating.
A broad definition of legal research is that: it
is a process which looks for identifying and retrieving what is needed for
supporting legal decisionmaking. Hence, we may consider that it starts with
the analysis of the facts on a particular and ends with the binomial
applicationcommunication of the results of the investigation.
Nowadays statistical evidence, sustained by
probabilistic reasoning, plays an important role in common life. It is
expanding its area of applications to criminal investigations, prosecutions and
trials. Particularly, forensic scientific evidence, including DNA, produced by
expert witnesses, is one of the emerging areas for statistical applications.
That sustains that if you are involved in criminal adjudication, having a
comprehension of the basics of probability and statistics is needed. In other
legal researches, a similar situation is present: data must be retrieved and
analyzed. Misunderstandings of what statistical information at hand are to be
processed and interpreted, as well as of the role of the involved
probabilities, have contributed towards serious miscarriages of justice. These
facts suggest including in the education for lawyers a training on statistical
thinking on how it should be used in legal research.
Actually, some processes use statistical
sampling for providing evidence at the court yard. The
correctness of the statistical procedures used, are being taken into account in
the allegation of decisions by the court. Hence, having a good statistic
advisor is one of the actual needs of the law firms.
Another problem is related with the need of
dealing with Bigdata. They are being used in different legal issues at least in
the past 20 years. The presence of Bigdata poses to the investigators to deal
with responding to:
– How much data they have?
– Which is the structure of the data
(structured, unstructured, textbased, internal and external)?
– Is it possible to analyze the existing data in
real time for instantaneous decisionmaking?
– Are the data reliable?
Much of legal research is based on discovering
facts through analyzing a lot of papers. Electronically Stored Information
(ESI) poses issues on using data stored electronically. With the increase of
data volumes, a need of reducing costs, without violating the accepted
assumptions poses urgently mid changes in the law firms. The reduction of costs
should not be solved by using “low‐cost lawyers”.
To give a modern response when dealing with Bigdata
an emerging technology for the retrieval of document information is connected
with Technology Assisted Review (TAR) and Statistical Sampling. They are
occupying a distinguished place as a tool for the research of law firms, as it
reduces risks and improves productivity in eDiscovery processes. TAR is based
on statistical models and it is being accepted as some kind of standard
statistical tool for analyzing Bigdata problems posed by the existence of ESI.
Actually, many U.S. courts are endorsing the use
of predictivecoding technologies. Consequently U.S. lawfirms are encouraging
structuring task groups for improving Bigdata practice.
§ 1 – Some uses of statistical sampling
The analysis of data always has posed a
complicated task to law firms. Nowadays the available data overcomes the
capacity of the attorneys if some modern technique is not used for sampling and
providing relevant information. Consider the use of applying Statistical
Sampling to discovery of relevant and responsive documents. Though it is not a
common practice, it is increasing its role in legal research. The reasons are
the usual in statistical research. Its use is cost‐effective in many tests as
its behavior has been reasonably effective in finding relevant and responsive
documents.
Sampling is currently used in many areas of the
Social Sciences. In particular, sociology studies use sampling models for
obtaining information. The theoretical frame uses the fact that the study deals
with finite population of well identified units (U={u_{1}
,…,u_{N}}). Selecting a sub set of
them generates a sample (sÌU). Using
judgmental sample was initially the approach up to the general acceptation that
probability sampling is the only way of obtaining “representative samples”.
Statistical sampling is of use in many aspects
of the administration of justice. Providing facts coming from well supported
statistical research is a source of evidence. The court analyzes the results of
statistical research but it must be aware of what is scientifically correct and
how some models may be used for manipulating the results. Then, statistical
experts are to be contracted by law firms for designing their needs of
developing statistical inquires. On the other hand, the court must have an
adequate counterpart for giving support to the righteous of the conclusions
derived by the research.
The use of statistical evidence has proved to be
of considerable support in the court. In some areas, they are currently used.
Statistical sampling is accepted for estimating
Medicare overpayments. Unfortunately, there are not wellestablished guidelines
for sampling methodologies, as in other areas. Hence, there is a basis for
considering whether a statistical principle, or method, is to be preferred to
another one. There is a need of establishing some standards for considering
when a statistical study is valid or not. In USA, the programs of Medicare have
established that a statistical sampling evidence should be considered as acceptable,
only if it uses a probability sampling design. That is observing any sample s
must have a probability P(s)
Î [0,1].
The importance of modeling adequately is
exemplified by some trials as the following ones:
– Transyd Enterprises
LLC D/B/A Transpro Medical Transport (Appellant) vs
(Beneficiaries) Trailblazer Health Enterprises LLC (Contractor), Claim for
Part B. Benefits, 2009 WL 5764287 (Sept. 15, 2009). MAC rejected the
appellant’s argument “PSC’s sampling methodology is invalid” because the PSC failed
to document that its statisticians possessed at least a master’s degree in
statistics or the equivalent.
– Robert D. Lesser, M.D. & Assocs. (Appellant) vs
(Beneficiaries) Pinnacle Business Solutions,
Inc. (Contractor), Claim for Part B Benefits, 2011 WL 5263619, Docket No.
M11358 (Feb. 18, 2011). The Council noted that ALJ relied on the 60day
timeline in the MPIM, which applies to prepayment and post payment review for
MR (Medical Review) purposes. The case arose from a statistical sampling review
by the Benefit Integrity unit of the ZPIC.
– The
MMPIM General Medicine, P.C. (Appellant) (Beneficiaries) Palmetto, GBA
(Contractor), Claim for Part A Benefits, 2010 WL 7232825, Docket No.
M101933 (Nov. 24, 2010): The Council found appellant’s case was based on
unsupported speculations and conjectures. It addresses claimed that
stratification should have been used, stating that the statistical sampling
guidelines did not require stratification of every sample in order to make the
sampling valid.
§ 2 – Technology
Assisted Review (TAR) and Statistical Sampling
A particularly important task in legal work is
text classification. Different studies suggest that machine learning techniques
outperforms the classic manual document review developed by lawyers. They
support that Technology Assisted Review (TAR) and Statistical Sampling increase
both productivity and accuracy at a lower cost. Empirical evidence sustains
that the use of TAR reduces the review time in a 75% of the time and the cost
is only 30% of the classic methods.
Those are the reasons why one of the more
accepted sampling procedures is using TAR. There are not many publications on
its theoretical properties but the comparison of the cost reduction, due to its
use has increased its popularity in legal research. Many law firms are
considering how unassisted document review performs in comparison with TAR,
which is validated by statistical sampling models.
TAR uses the expertise of attorneys and the
methods of machinelearning to automatize the prioritization of documents to be
reviewed. The ranking uses a measure of the responsiveness of document to a
particular matter. By using it for dealing with Big data the firms reduce costs
and key documents are obtained faster.
Some recent documented evidences of the
usefulness of using TAR are:
– Da Silva Moore vs Publicis Groupe8. Andrew Peck (US Magistrate
Judge) gave his opinions the validity of judgmental and statistical sampling
for validating the results of predictive coding. (The Case for Statistical
Sampling in e‐Discovery7).
– Kleen Products vs
Packaging Corporation of America9. Nan Nolan (US Magistrate Judge) heard
the testimony, for sustaining the validity of the sampling process, used by
defense. The validity of testing the results based on research terms, instead
of predictive coding, in finding relevant documents was on trial. the parties
had to determine with sampling procedure was acceptable for them. Once an
agreement was obtained on the keyword to be looked for using sampling the
research and discussion went forward.
– Global Aerospace vs Landow Aviation. The court stated that
predictive coding (aka TAR) including a statistical model for validating the
protocol was adequate for locating and retrieving documents for production.
The reduction of the costs is important but in
addition the consistency of using TAR and sampling is considerably larger than
the so called “linear review”. Linear reviews are developed by the reviews,
performed by attorneys of the documents. The inconsistency of the reviews due
to human error in not measured. Commonly no statistical sampling is used and
hence, the reliability of such reviews is not possible. Therefore, the
inconsistency of reviewers is unknown. TAR is validated with statistical
sampling and it is highly consistent, and hence more reliable, compared with results
of unassisted reviews performed by attorneys. Therefore, using it the lawyers
assure that the process achieves a large level of success in identifying the relevant
and responsive documents.
Well known sampling models as stratification
allows improving the quality of the review process. For example, if a ranking
of the importance of the documents is made previously, the consistency may be
improved by using an unequal probability sampling or ranked set sampling. Such
approaches save time as they avoid expecting for “first‐level” reviews. For example, documents ranked first receive
a preferential treatment in terms of the probability of being selected.
Corporate law departments deal with large
amounts of data from invoices, and need to determine the factors influencing rates
for negotiating better, deals based on that data. a free mobile application
that aggregates data from thousands of law firm invoices is TyMetrix
Legal Analytics. TyMetrix RateDriver™
mobile application uses the statistical model from Real Rate Report™. It is a
statistical analysis of legal invoices.
§ 3 – A
study
Less documented is its use in providing evidence
on reclamations on the contamination due to enterprises. A question is: are the
levels of contamination acceptable? The enterprise produces reports to the
governmental agencies. On doubts on the accuracy of the reports
environmentalists supported claims of farmers that the water used for
agriculture was being contaminated. Their claim is based on the observed
behavior of the production of the land.
The case was Farmers,
F. (Appellant) vs (Beneficiaries) Chemical enterprise, CE (Contractor), Claim
for Part A contamination of the water is affecting the fertility of lands: The
appellant considered that the reported data which supported that the
contamination levels were within the accepted interval were not correct. The
arguments of the enterprise were unsupported speculations and the conjectures
cannot be proved without a statistical study. The statisticians supporting the
appellants claimed that the measurements of the sensors at the factory output
were providing not accurate information. They selected some points in the
course of the water source and obtained their own measurements. A sample of
them were compared with the ones made at the output of the factory by the
sensor of the enterprise owners.
The
set of measures of the outputs were considered as binary (0, 1) indicating whether they coincided with the
ones of the other sensors (correctly classified=1, incorrectly classified=0). The
results of N measurements are summarized in the Table 1.
Considering that the classification
is equivalent to a doubleblind method that is they are made independently.
Each measurement generates a value
Summarizing
is obtained the next table
Table 1. Classification of N measurements of 2 sensors

Correct 
Incorrect 
Total 
Correct 
_{} 


Incorrect 



Total 



Different agreement indexes were
considered. They are function of
p_{ij} = q_{2}
= q_{1}
= where p_{i+}
=
Were evaluated the following indexes
Dice
A value close to zero means that the sensors have a small “agreement”.
Correlation coefficient
Note that we are dealing with attributes (categorical variables). In this case, the correlation coefficient of Pearson may be rewritten, in terms of Table 1 as
The values of will be in the interval
[1, 1]. If r»1, the sensors behave
similarly, r<0 means that they highly disagree
and they are “independent” if r»0.
Measure of Differences
An increase of D means that they largely disagree
Kappa
k =
A large value of it means the existence of a high level of agreement.
3 sensors were placed and data were collected during a month. The values of the indexes were computed for each one and compared with the reports of the enterprise. Each one was evaluated considering the belonging to the accepted levels of contamination fixed by the law. They are reported in the next table
Table 2. Values of the indexes of 3 sensors
Sensor 
Dice 
Correlation coefficient 
Measure of Differences 
Kappa 
1 
0,801 
0,128 
0,333 
0,302 
2 
0,823 
0.001 
0,301 
0,010 
3 
0,774 
0,300 
0,352 
0,364 
Then, it was documented that the lectures of the enterprise had a low agreement when classifying the violating of the accepted level of contaminator with the other sensors.
The court fixed a fine to the enterprise for avoiding their responsibilities with the environment and a calculation of the damage to the farmers is in progress. The statisticians of the enterprise alleged that they assumed that the measurements were normally distributed but the appellant´s proved that this probability assumption was incorrect and that categorical data analysis must have been used by then for controlling.
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[Accessed:
september, 2016]
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