<!DOCTYPE article PUBLIC "-//NLM//DTD JATS (Z39.96) Journal Archiving and Interchange DTD v1.0 20120330//EN" "JATS-archivearticle1.dtd">
<article xmlns:xlink="http://www.w3.org/1999/xlink">
  <front>
    <journal-meta>
      <journal-title-group>
        <journal-title>P. Goel)</journal-title>
      </journal-title-group>
    </journal-meta>
    <article-meta>
      <title-group>
        <article-title>Investigating the duality of CBR: Performance and Interpretability</article-title>
      </title-group>
      <contrib-group>
        <contrib contrib-type="author">
          <string-name>Prateek Goel</string-name>
          <xref ref-type="aff" rid="aff0">0</xref>
        </contrib>
        <aff id="aff0">
          <label>0</label>
          <institution>Dept. of Computer and Information Sciences, Drexel University</institution>
          ,
          <addr-line>3675 Market St, Philadelphia, 19104</addr-line>
          ,
          <country country="US">USA</country>
        </aff>
      </contrib-group>
      <pub-date>
        <year>2023</year>
      </pub-date>
      <volume>000</volume>
      <fpage>0</fpage>
      <lpage>0003</lpage>
      <abstract>
        <p>Case-based reasoning (CBR) is an artificial intelligence (AI) methodology with applications across diferent domains. Its reasoning is rooted in human intuition making this methodology universally accepted as interpretable. In contrast, black box models like neural networks have a widely accepted notion with high-performance accuracy and low interpretability. Despite low interpretability, there is a ready acceptance of black box models in academia and industry for automated decision-making. The focus of this research is investigating the relationship between performance and interpretability for CBR by exploring a variety of ways to note the impact of enhancing performance on CBR interpretability.</p>
      </abstract>
      <kwd-group>
        <kwd>eol&gt;Interpretability</kwd>
        <kwd>Case-Based Reasoning</kwd>
        <kwd>Feature Weighting</kwd>
        <kwd>Statistical Relevance</kwd>
      </kwd-group>
    </article-meta>
  </front>
  <body>
    <sec id="sec-1">
      <title>1. Introduction</title>
      <p>diferent ways to improve CBR performance (while maintaining its interpretability) would detail
impact factors that can serve as a reference to future researchers and provide for the ready
acceptance of CBR in high-stakes decision-making as an interpretable alternative to black-box
methods.</p>
    </sec>
    <sec id="sec-2">
      <title>2. Literature Review</title>
      <p>
        Case-based Reasoning Case-Based Reasoning (CBR) is an AI methodology that uses
knowledge from past experiences to solve new problems. What distinguishes it from other reasoning
methodologies is its ability to identify the most “similar” experiences for solving a problem.
Since the solution of the experience(s) may not be fully applicable to the current problem, it
allows the methodology to tweak (or adapt) the past experience to solve the problem [
        <xref ref-type="bibr" rid="ref6 ref7">6, 7</xref>
        ].
      </p>
      <p>
        CBR has two foundational perspectives of understanding. The first perspective is as a process
cycle (called CBR cycle[
        <xref ref-type="bibr" rid="ref6">6</xref>
        ]) where a problem is formulated using details from encountered
experience, followed by identification of the similar past problem(s) capable of providing a
solution. The solutions to the identified similar problems are adapted to form a new solution.
An alternative perspective is with respect to the overall knowledge of the CBR system stored in
separate knowledge containers. Each container possesses a diferent aspect of knowledge
contributing to the overall knowledge of the CBR cycle[
        <xref ref-type="bibr" rid="ref6 ref7">6, 7</xref>
        ]. This study focuses on the knowledge
of similarity contained within the similarity container for the CBR cycle[
        <xref ref-type="bibr" rid="ref6">6</xref>
        ].
      </p>
      <p>
        A set of approaches pertaining to the similarity container are reviewed. Wetterscherek and
Aha [
        <xref ref-type="bibr" rid="ref8">8</xref>
        ] list a variety of similarity-based weight learning approaches that allocate similarity
weights per feature of a case problem to obtain the most similar cases to an encountered case
problem. Maximini et al. [
        <xref ref-type="bibr" rid="ref9">9</xref>
        ] suggest using the concept of ’generalizing’ cases to form cases
with a feature range for the selection of appropriate cases. Jalali and Leake [
        <xref ref-type="bibr" rid="ref10">10</xref>
        ] suggest defining
rules guided by the adaptation step within the retrieval step.
      </p>
      <p>
        Interpretability Interpretability in A.I. stems from diferent notions inspired to gather a
holistic understanding of a machine learning model’s operability. The field has gained immense
popularity in the past decade primarily due to the increased usage of black box AI models
in high-stakes decision-making. Despite the stakes, the field lacks a unifying definition to
bind all afiliated notions [
        <xref ref-type="bibr" rid="ref11 ref2">2, 11</xref>
        ]. Numerous sources have pointed out the subjective nature of
interpretability listing it as a domain-specific notion with no unifying definition [
        <xref ref-type="bibr" rid="ref2 ref4 ref5">4, 2, 5</xref>
        ].
      </p>
      <p>
        Interpretability, as it stands today, is an amalgamation of diferent notions. Rudin [
        <xref ref-type="bibr" rid="ref2">2</xref>
        ] details
an interpretable model that has either usability or some structural knowledge of the domain
it is being applied to. Arrieta [
        <xref ref-type="bibr" rid="ref4">4</xref>
        ] refers to it as a passive character of the model such that the
model makes sense to the human observer. Doshi-Velez and Kim [
        <xref ref-type="bibr" rid="ref12">12</xref>
        ] relate it to explainability
and suggest an interpretable model can explain its reasoning in understandable terms.
      </p>
      <p>
        An important subject of discussion is the relationship between performance and
interpretability for diferent AI methods. The DARPA XAI Program [
        <xref ref-type="bibr" rid="ref3">3</xref>
        ], showcased a notion of
the performance-explainability tradeof. This relation is showcased as inversely proportional
where the performance of a model increases at the cost of its explainability. This trade-of
has been a subject of discussion where either the trade-of has been rejected completely [
        <xref ref-type="bibr" rid="ref2">2</xref>
        ] or
suggested to hold true under specific circumstances [
        <xref ref-type="bibr" rid="ref4">4</xref>
        ].
      </p>
      <p>
        CBR methodology claims to have “natural” transparency (or inherent interpretability) [
        <xref ref-type="bibr" rid="ref1 ref2">2, 1</xref>
        ].
Due to the methodology’s foundations lying in psychological plausibility, there is universal
consensus on the interpretability of CBR to the point where the methodology itself is used to
provide post-hoc explanations for other black box models. Keane and Kenny [
        <xref ref-type="bibr" rid="ref1">1</xref>
        ] use the “twin
systems” approach by creating an ANN-CBR twin with shared weights where simultaneous
learning of feature weights provides for interpretations using CBR. Caruana et al. [
        <xref ref-type="bibr" rid="ref13">13</xref>
        ] generate
case-based explanations for artificial neural networks and decision trees.
      </p>
      <p>
        Rudin [
        <xref ref-type="bibr" rid="ref2">2</xref>
        ] and Miller [
        <xref ref-type="bibr" rid="ref14">14</xref>
        ] identify the limitation of humans processing information by listing
how humans are limited to processing 7 (+-2) cognitive entities at one time. Using this notion,
Lage et al. [
        <xref ref-type="bibr" rid="ref15">15</xref>
        ] list diferent factors (called “cognitive chunks") that make models interpretable
by building decision sets and conducting human-subject experiments to analyze the impact on
performance. The conclusions are used to seed the investigation of CBR interpretability in this
research and analyze how various CBR methods accordingly evaluate.
      </p>
    </sec>
    <sec id="sec-3">
      <title>3. Research Plan</title>
      <p>This research aims to investigate methods to improve CBR performance. In addition, the impact
of these methods on interpretability will be noted. Finally, the structure of the relationship
between performance and interpretability will be analyzed and the level of sensitivity is observed
in case a relationship is verified.</p>
      <sec id="sec-3-1">
        <title>3.1. Research Objectives</title>
        <p>The first objective of this research is to improve CBR performance. The aim is to investigate a
variety of methods and approaches to enhance CBR performance, where the approaches are
limited to the retrieval step of the CBR cycle (or similarity container). The second objective is to
note the impact of CBR approaches considered in the first research objective on interpretability.
This objective would entail identifying the factors to govern CBR interpretability objectively
and using these factors to witness changes in the approaches considered in the first research
objective. Additionally, this objective aims to describe the structure of the relationship between
performance and interpretability for the approaches considered in research objectives one and
two. The aim is to identify the level of sensitivity between the two characteristics for approaches
when an impact is noted.</p>
      </sec>
      <sec id="sec-3-2">
        <title>3.2. Approaches and Methodologies</title>
        <p>The entirety of this research is divided into two phases. Phase 1 studies examine the literature
on what has been proposed for performance in CBR and note increased performance as a direct
consequence of a proposed alternative to similarity and retrieval-based approach.</p>
        <p>This notion is explored via three studies. The first study incrementally increases the number
of weights per feature to describe the relative relevance of features describing the cases. This
includes methods involving clustering and weight-learning approaches. The second study uses
the proposed methods in the literature for enhancing CBR performance. This comparison
study would be comparing existing approaches in the literature by identifying datasets used
across reviewed literature. The third study compares weight methodologies and their impact
on performance. Where the focus is on increasing the number of weights in the first study, this
study focuses on how weights are learned while keeping the number of weights constant. An
additional aim of this phase is coming up with an approach built using approaches considered
within the studies, that mimic the performance behavior of a neural network model.</p>
        <p>
          Phase 2 investigates CBR interpretability notions to obtain factors that influence the
interpretability of CBR objectively. Consequently, this phase analyzes the relationships between the
performance and interpretability of CBR approaches and notes the sensitivity of the impact
one has on the other. Using the suggested concept of “cognitive chunks" discussed in Lage et al.
[
          <xref ref-type="bibr" rid="ref15">15</xref>
          ] as information units, this phase studies the relationship, with respect to information units,
between performance and interpretability by analyzing diferent approaches from Phase 1. By
associating these information units with a number of weights per feature, the goal is to gain
some visibility in their relationship. Additionally, this phase hypothesizes the existence of “soft
spots" in the performance-interpretability relationship for diferent methods. These spots are
points at which the limited increase in performance comes at a greater cost to interpretability.
To draw out the relationships for every method listing their respective soft spots, a quantitative
evaluation would be done detailing the performance-interpretability relationship. Post this
quantitative evaluation, a user study will be conducted to get validation from humans to confirm
the hypothesis and selection of the “soft spots" and note the level of correctness of the identified
performance-interpretability relationship.
        </p>
      </sec>
    </sec>
    <sec id="sec-4">
      <title>4. Progress Summary</title>
      <p>
        The first study for Phase 1 is ongoing. The baseline approach of gradient descent-based weight
learning (1 weight per feature) is altered by clustering cases according to some distance metric.
Once cases are clustered, each cluster has its own sets of weights (using gradient descent) to
separate the cases within. This gives every feature within a case represented by two weights
instead of one. One weight suggests the cluster the instance belongs to and the other weight
corresponds to the gradient descent method for the instance feature. This methodology was
evaluated using a 5-fold cross-validation split using the Credit Default Binary classification
Dataset [
        <xref ref-type="bibr" rid="ref16">16</xref>
        ], comprised of 30,000 instances with 23 features in each instance. The performance
was evaluated using accuracy as a metric to compare against baseline retrieval using gradient
descent-based weight learning. The noted average accuracy was marginally higher than the
baseline model with performance accuracy noted to be 81.4% for this method and 79.8% for the
baseline. The optimum number of clusters was 3 and calculated using the elbow method. To
confirm the significance of these results McNemar’s test[
        <xref ref-type="bibr" rid="ref17">17</xref>
        ] was performed. The null hypothesis,
suggesting the same performance accuracy across both models, is rejected with a 95% confidence
level With a p-value of 2.016e-12.
      </p>
      <p>The design for the second study is ongoing where the methods discussed in the reviewed
literature, presented in the Related Works section, will be applied to note enhancements in
performance.</p>
    </sec>
    <sec id="sec-5">
      <title>5. Conclusion</title>
      <p>This research is in its preliminary phases and no conclusions have been drawn yet. For
limitations, the scope considers only the classification task. The investigations are, also, limited to
the retrieval step and the similarity container.</p>
    </sec>
  </body>
  <back>
    <ref-list>
      <ref id="ref1">
        <mixed-citation>
          [1]
          <string-name>
            <given-names>M. T.</given-names>
            <surname>Keane</surname>
          </string-name>
          ,
          <string-name>
            <given-names>E. M.</given-names>
            <surname>Kenny</surname>
          </string-name>
          ,
          <article-title>How case-based reasoning explains neural networks: A theoretical analysis of xai using post-hoc explanation-by-example from a survey of ann-cbr twin-systems</article-title>
          , in: K. Bach,
          <string-name>
            <surname>C.</surname>
          </string-name>
          Marling (Eds.),
          <source>Case-Based Reasoning Research and Development</source>
          , Springer International Publishing, Cham,
          <year>2019</year>
          , pp.
          <fpage>155</fpage>
          -
          <lpage>171</lpage>
          .
        </mixed-citation>
      </ref>
      <ref id="ref2">
        <mixed-citation>
          [2]
          <string-name>
            <given-names>C.</given-names>
            <surname>Rudin</surname>
          </string-name>
          ,
          <article-title>Stop explaining black box machine learning models for high stakes decisions and use interpretable models instead</article-title>
          ,
          <source>Nature Machine Intelligence</source>
          <volume>1</volume>
          (
          <year>2019</year>
          )
          <fpage>206</fpage>
          -
          <lpage>215</lpage>
          . URL: https://doi.org/10.1038/s42256-019-0048-x. doi:
          <volume>10</volume>
          .1038/s42256-019-0048-x.
        </mixed-citation>
      </ref>
      <ref id="ref3">
        <mixed-citation>
          [3]
          <string-name>
            <given-names>D.</given-names>
            <surname>Gunning</surname>
          </string-name>
          , D. Aha,
          <article-title>Darpa's explainable artificial intelligence (xai) program</article-title>
          ,
          <source>AI</source>
          Magazine
          <volume>40</volume>
          (
          <year>2019</year>
          )
          <fpage>44</fpage>
          -
          <lpage>58</lpage>
          . URL: https://ojs.aaai.org/aimagazine/index.php/aimagazine/article/view/ 2850. doi:
          <volume>10</volume>
          .1609/aimag.v40i2.
          <fpage>2850</fpage>
          .
        </mixed-citation>
      </ref>
      <ref id="ref4">
        <mixed-citation>
          [4]
          <string-name>
            <given-names>A.</given-names>
            <surname>Barredo Arrieta</surname>
          </string-name>
          ,
          <string-name>
            <given-names>N.</given-names>
            <surname>Díaz-Rodríguez</surname>
          </string-name>
          ,
          <string-name>
            <given-names>J. Del</given-names>
            <surname>Ser</surname>
          </string-name>
          ,
          <string-name>
            <given-names>A.</given-names>
            <surname>Bennetot</surname>
          </string-name>
          ,
          <string-name>
            <given-names>S.</given-names>
            <surname>Tabik</surname>
          </string-name>
          ,
          <string-name>
            <given-names>A.</given-names>
            <surname>Barbado</surname>
          </string-name>
          ,
          <string-name>
            <given-names>S.</given-names>
            <surname>Garcia</surname>
          </string-name>
          ,
          <string-name>
            <given-names>S.</given-names>
            <surname>Gil-Lopez</surname>
          </string-name>
          ,
          <string-name>
            <given-names>D.</given-names>
            <surname>Molina</surname>
          </string-name>
          ,
          <string-name>
            <given-names>R.</given-names>
            <surname>Benjamins</surname>
          </string-name>
          ,
          <string-name>
            <given-names>R.</given-names>
            <surname>Chatila</surname>
          </string-name>
          ,
          <string-name>
            <given-names>F.</given-names>
            <surname>Herrera</surname>
          </string-name>
          ,
          <article-title>Explainable artificial intelligence (xai): Concepts, taxonomies, opportunities and challenges toward responsible ai</article-title>
          ,
          <source>Information Fusion</source>
          <volume>58</volume>
          (
          <year>2020</year>
          )
          <fpage>82</fpage>
          -
          <lpage>115</lpage>
          . URL: https://www.sciencedirect.com/science/ article/pii/S1566253519308103. doi:https://doi.org/10.1016/j.inffus.
          <year>2019</year>
          .
          <volume>12</volume>
          . 012.
        </mixed-citation>
      </ref>
      <ref id="ref5">
        <mixed-citation>
          [5]
          <string-name>
            <given-names>G. K.</given-names>
            <surname>Dziugaite</surname>
          </string-name>
          ,
          <string-name>
            <given-names>S.</given-names>
            <surname>Ben-David</surname>
          </string-name>
          ,
          <string-name>
            <given-names>D. M.</given-names>
            <surname>Roy</surname>
          </string-name>
          ,
          <article-title>Enforcing interpretability and its statistical impacts: Trade-ofs between accuracy and interpretability</article-title>
          ,
          <year>2020</year>
          . arXiv:
          <year>2010</year>
          .13764.
        </mixed-citation>
      </ref>
      <ref id="ref6">
        <mixed-citation>
          [6]
          <string-name>
            <given-names>M. M.</given-names>
            <surname>Richter</surname>
          </string-name>
          ,
          <string-name>
            <given-names>R. O.</given-names>
            <surname>Weber</surname>
          </string-name>
          ,
          <source>Case-based reasoning: a Textbook</source>
          ,
          <volume>1</volume>
          <fpage>ed</fpage>
          ., Springer Berlin, Heidelberg,
          <year>2013</year>
          . doi:https://doi.org/10.1007/978-3-
          <fpage>642</fpage>
          -40167-1.
        </mixed-citation>
      </ref>
      <ref id="ref7">
        <mixed-citation>
          [7]
          <string-name>
            <given-names>A.</given-names>
            <surname>Aamodt</surname>
          </string-name>
          ,
          <string-name>
            <given-names>E.</given-names>
            <surname>Plaza</surname>
          </string-name>
          ,
          <article-title>Case-based reasoning: Foundational issues, methodological variations, and system approaches</article-title>
          ,
          <source>AI</source>
          Communications
          <volume>7</volume>
          (
          <year>1994</year>
          )
          <fpage>39</fpage>
          -
          <lpage>59</lpage>
          . URL: https://doi.org/10.3233/ AIC-1994-7104. doi:
          <volume>10</volume>
          .3233/AIC-1994-
          <volume>7104</volume>
          ,
          <fpage>1</fpage>
          .
        </mixed-citation>
      </ref>
      <ref id="ref8">
        <mixed-citation>
          [8]
          <string-name>
            <given-names>D.</given-names>
            <surname>Wettschereck</surname>
          </string-name>
          , D. W. Aha, Weighting features,
          <source>in: Proceedings of the First International Conference on Case-Based Reasoning Research and Development</source>
          , ICCBR '95,
          <string-name>
            <surname>SpringerVerlag</surname>
          </string-name>
          , Berlin, Heidelberg,
          <year>1995</year>
          , p.
          <fpage>347</fpage>
          -
          <lpage>358</lpage>
          .
        </mixed-citation>
      </ref>
      <ref id="ref9">
        <mixed-citation>
          [9]
          <string-name>
            <given-names>K.</given-names>
            <surname>Maximini</surname>
          </string-name>
          ,
          <string-name>
            <given-names>R.</given-names>
            <surname>Maximini</surname>
          </string-name>
          ,
          <string-name>
            <given-names>R.</given-names>
            <surname>Bergmann</surname>
          </string-name>
          ,
          <article-title>An investigation of generalized cases</article-title>
          , in: K. D.
          <string-name>
            <surname>Ashley</surname>
          </string-name>
          , D. G. Bridge (Eds.),
          <source>Case-Based Reasoning Research and Development</source>
          , Springer Berlin Heidelberg, Berlin, Heidelberg,
          <year>2003</year>
          , pp.
          <fpage>261</fpage>
          -
          <lpage>275</lpage>
          .
        </mixed-citation>
      </ref>
      <ref id="ref10">
        <mixed-citation>
          [10]
          <string-name>
            <given-names>V.</given-names>
            <surname>Jalali</surname>
          </string-name>
          ,
          <string-name>
            <given-names>D. B.</given-names>
            <surname>Leake</surname>
          </string-name>
          ,
          <article-title>Adaptation-guided case base maintenance</article-title>
          ,
          <source>in: AAAI Conference on Artificial Intelligence</source>
          ,
          <year>2014</year>
          .
        </mixed-citation>
      </ref>
      <ref id="ref11">
        <mixed-citation>
          [11]
          <string-name>
            <given-names>Z. C.</given-names>
            <surname>Lipton</surname>
          </string-name>
          ,
          <article-title>The mythos of model interpretability: In machine learning, the concept of interpretability is both important and slippery</article-title>
          .,
          <source>Queue</source>
          <volume>16</volume>
          (
          <year>2018</year>
          )
          <fpage>31</fpage>
          -
          <lpage>57</lpage>
          . URL: https: //doi.org/10.1145/3236386.3241340. doi:
          <volume>10</volume>
          .1145/3236386.3241340.
        </mixed-citation>
      </ref>
      <ref id="ref12">
        <mixed-citation>
          [12]
          <string-name>
            <given-names>F.</given-names>
            <surname>Doshi-Velez</surname>
          </string-name>
          ,
          <string-name>
            <given-names>B.</given-names>
            <surname>Kim</surname>
          </string-name>
          ,
          <article-title>Towards a rigorous science of interpretable machine learning</article-title>
          ,
          <year>2017</year>
          . arXiv:
          <volume>1702</volume>
          .
          <fpage>08608</fpage>
          .
        </mixed-citation>
      </ref>
      <ref id="ref13">
        <mixed-citation>
          [13]
          <string-name>
            <given-names>R.</given-names>
            <surname>Caruana</surname>
          </string-name>
          ,
          <string-name>
            <given-names>H.</given-names>
            <surname>Kangarloo</surname>
          </string-name>
          ,
          <string-name>
            <given-names>J. D.</given-names>
            <surname>Dionisio</surname>
          </string-name>
          ,
          <string-name>
            <given-names>U.</given-names>
            <surname>Sinha</surname>
          </string-name>
          ,
          <string-name>
            <given-names>D.</given-names>
            <surname>Johnson</surname>
          </string-name>
          ,
          <article-title>Case-based explanation of non-case-based learning methods</article-title>
          ,
          <source>Proc AMIA Symp</source>
          (
          <year>1999</year>
          )
          <fpage>212</fpage>
          -
          <lpage>215</lpage>
          .
        </mixed-citation>
      </ref>
      <ref id="ref14">
        <mixed-citation>
          [14]
          <string-name>
            <given-names>G. A.</given-names>
            <surname>Miller</surname>
          </string-name>
          ,
          <article-title>The magical number seven plus or minus two: some limits on our capacity for processing information</article-title>
          ,
          <source>Psychol Rev</source>
          <volume>63</volume>
          (
          <year>1956</year>
          )
          <fpage>81</fpage>
          -
          <lpage>97</lpage>
          .
        </mixed-citation>
      </ref>
      <ref id="ref15">
        <mixed-citation>
          [15]
          <string-name>
            <given-names>I.</given-names>
            <surname>Lage</surname>
          </string-name>
          ,
          <string-name>
            <given-names>E.</given-names>
            <surname>Chen</surname>
          </string-name>
          ,
          <string-name>
            <given-names>J.</given-names>
            <surname>He</surname>
          </string-name>
          ,
          <string-name>
            <given-names>M.</given-names>
            <surname>Narayanan</surname>
          </string-name>
          ,
          <string-name>
            <given-names>B.</given-names>
            <surname>Kim</surname>
          </string-name>
          ,
          <string-name>
            <given-names>S. J.</given-names>
            <surname>Gershman</surname>
          </string-name>
          ,
          <string-name>
            <given-names>F.</given-names>
            <surname>Doshi-Velez</surname>
          </string-name>
          ,
          <article-title>Human evaluation of models built for interpretability</article-title>
          ,
          <source>Proceedings of the AAAI Conference on Human Computation and Crowdsourcing</source>
          <volume>7</volume>
          (
          <year>2019</year>
          )
          <fpage>59</fpage>
          -
          <lpage>67</lpage>
          . URL: https://ojs.aaai.org/index. php/HCOMP/article/view/5280. doi:
          <volume>10</volume>
          .1609/hcomp.v7i1.
          <fpage>5280</fpage>
          .
        </mixed-citation>
      </ref>
      <ref id="ref16">
        <mixed-citation>
          [16]
          <string-name>
            <surname>I.-C. Yeh</surname>
          </string-name>
          , C.
          <article-title>-h.</article-title>
          <string-name>
            <surname>Lien</surname>
          </string-name>
          ,
          <article-title>The comparisons of data mining techniques for the predictive accuracy of probability of default of credit card clients</article-title>
          ,
          <source>Expert Syst. Appl</source>
          .
          <volume>36</volume>
          (
          <year>2009</year>
          )
          <fpage>2473</fpage>
          -
          <lpage>2480</lpage>
          . URL: https://doi.org/10.1016/j.eswa.
          <year>2007</year>
          .
          <volume>12</volume>
          .020. doi:
          <volume>10</volume>
          .1016/j.eswa.
          <year>2007</year>
          .
          <volume>12</volume>
          .020.
        </mixed-citation>
      </ref>
      <ref id="ref17">
        <mixed-citation>
          [17]
          <string-name>
            <given-names>Q.</given-names>
            <surname>McNEMAR</surname>
          </string-name>
          ,
          <article-title>Note on the sampling error of the diference between correlated proportions or percentages</article-title>
          ,
          <source>Psychometrika</source>
          <volume>12</volume>
          (
          <year>1947</year>
          )
          <fpage>153</fpage>
          -
          <lpage>157</lpage>
          .
        </mixed-citation>
      </ref>
    </ref-list>
  </back>
</article>