# Applying the principle of logical elimination to probabilistic diagnosis

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<ul><li><p>MEI). INFOHRI. (1981), VOL. 6, NO. 1, 25-32 </p><p>Applying the principle of logical elimination to probabilistic diagnosis </p><p>D . E. H. LLEWE1,YN Department of Medicine, Kings College Hospital Medical School, Denmark Hill, London SE5 8RX, UK </p><p>(Received Apri l 1980) </p><p>Kevwords : Diagnosis, Substnntiating cotmeiition, Probability thorerrr </p><p>.A probability theorem is described which identifies the factors to be considered when applying the principle of logical elimination to probabilistic diagnosis. The proof is based only on the multiplication, addition and universal bound axioms of probability. When new tests are being designed from knowledge of disease mechanisms, criteria based on the theorem can be used to assess their suitability for interpretation in a logical manner. A simple convention is adopted to allow probabilities to be substantiated in terms of observed frequencies. </p><p>Larticie decrit un theoreme de probabilitt qui identifie les facteurs a considerer lorsquon applique le principe de Iklimination logique au diagnostic probabiliste. La preuve est basee uniquement sur la multiplication, Iaddition et les axiomes universellement lies a la probabilitt. Lorsque de nouvelles epreuves sont creees a &amp;me les connaissances des mechanismes pathologiques, des critires bases sur le theoreme peuvent servir a &amp;valuer leur convenance a une interpretation logique. Une simple convention est adoptee qui permet les probabilites diitre etablies en terme des frkquences observees. </p><p>1. Introduction Mathematical probability may be regarded as any measure on a set which lies in </p><p>the interval 0, 1 . The way in which such a probability is calculated can vary providing that certain rules of consistency are satisfied . However, a probability is usually based on observed frequencies ; this relationship cor- responds to the one which exists between particular and universal propositions in logic . Thus, consider the following argument which is a simplified example of a familiar type of medical reasoning. The argument is first stated in terms of universal propositions, and then in terms of particular propositions. </p><p>All patients with localized right lower quadrant abdominal pain have either appendicitis or non-specific abdominal pain. Some patients with appendicitis have guarding, but none of those with non-specific abdominal pain have guarding. Therefore, all patients with localized right lower quadrant pain and guarding have appendicitis. If we choose one of the patients referred to in this argument, then we may deduce the corresponding particular propositions: this patient with localized right lower quadrant abdominal pain must (i.e. with a probability of one) have either appendicitis or non-specific abdominal pain. A patient with appendicitis may (i.e. with a probability greater than zero) have guarding, but one with non-specific abdominal pain cannot (i.e. the probability is zero) have guarding. Therefore, a </p><p>Info</p><p>rm H</p><p>ealth</p><p> Soc</p><p> Car</p><p>e D</p><p>ownl</p><p>oade</p><p>d fr</p><p>om in</p><p>form</p><p>ahea</p><p>lthca</p><p>re.c</p><p>om b</p><p>y U</p><p>B K</p><p>iel o</p><p>n 10</p><p>/26/</p><p>14Fo</p><p>r pe</p><p>rson</p><p>al u</p><p>se o</p><p>nly.</p></li><li><p>patient \vith localized right loiver quadrant abdominal pain and guarding must ( i .e. with a probability of one) ha\-e appendicitis .' A probabilistic version of this reasoning using real data has been described in a preliminary communication [ S ] . </p><p>We shall explore the extent to Lvhich logical arguments by elimination can be performed in terms of frequencies and their corresponding probabilities when these may be less than one (or 'all') and greater than zero (or 'none'). The analysis will be onl&gt;- based on axioms which are consistent \vith all interpretations ofprobability. No other assumptions Lvill be made, for example statistical independence or mutual exclusion between attributes . </p><p>2. Theorem Let us select an element, e, from a set [ Ul which is defined as all those and only </p><p>those elements mith the attribute U(for example selecting a patient from the set of all those and only those registered with a particular general practitioner). We then obser\.e a list o f attributes (symptoms, signs and test results for example) designated S'T, Sy . . . Sf (ST being the value of the variable S,, so that ST is the result of doing the test S, ) . We \vish to know the probability of subsequently observing that one or more of another series of hidden attributes (for example those which define diagnostic categories) designated D,, D , . . . D, will also be manifested by the element (the patient) which has been selected. </p><p>Let S* represent a combination ofattributes such that S" = ST A S% . . . A Sf, and theset :S*: = fSt :n [Sf l . . . nlS:f. I,et ST beany memberofthelistofattr ibutes ST, Sf . . . S,*, so that tS*j &amp; {ST). Let D j be any member of the list o f possible attributes to be inferred, which are D,, D , . . . D,. Let D=D, v Dz v . . . v D,, and the set f D ) = ( D , f u [ D , j . . . u[D, f . Let D, be aparticular Dj. </p><p>We will regard p(D,!S*) as the realization of a probability measure on a selected element, e , such that </p><p>when No( ( A ; ) means the number of elements in the set {A). 'l'his can be written more succinctly as p(D,iS")= Fr(D,!S+) i.e. the probability of D,, given the presence of S*, is equal to the frequency of those with D, in those with Sx. </p><p>2.1 . Theorein </p><p>2 . 2 . Proof By symmetry of the multiplication rule [ 3 ] : </p><p>Info</p><p>rm H</p><p>ealth</p><p> Soc</p><p> Car</p><p>e D</p><p>ownl</p><p>oade</p><p>d fr</p><p>om in</p><p>form</p><p>ahea</p><p>lthca</p><p>re.c</p><p>om b</p><p>y U</p><p>B K</p><p>iel o</p><p>n 10</p><p>/26/</p><p>14Fo</p><p>r pe</p><p>rson</p><p>al u</p><p>se o</p><p>nly.</p></li><li><p>Logical elimination in probabilistic diagnosis </p><p>Now </p><p>Thus </p><p>p(Sf/D,) - n + 1 1 i = 1 </p><p>27 </p><p>(3) </p><p>Again, </p><p>m </p><p>&lt; 1 P(D,IS*). P(S") +P(WS*) . P ( S 9 </p><p>= Cp(S*/Dj) .P(Dj) +P(S*/') .P(') </p><p>j # x </p><p>m </p><p>j + x </p><p>m </p><p>&lt; 1 any P(STIDj) . P(Dj) + any P ( V / D ) .P(D&gt; </p><p>= C any p(ST/Dj) .p(Dj) + any p(B/ST) .p(Sf) j f X l &lt; ; &lt; ) I 1 d i d n </p><p>m </p><p>(4) </p><p>Thus by substituting equations (3) and (4) in equation (2) we obtain equation (1). This inequality defines the factors to be considered if logical principles are to be </p><p>applied reliably to probabilistic diagnosis. These factors are particularly relevant to those attempting to design powerful tests whose results they intend to be interpreted by logical exclusion. </p><p>j + x l &lt; i d n 1 &lt; idu </p><p>3. The postulated diagnosis The term D, corresponds to a postulated diagnosis which is represented by </p><p>appendicitis in the introductory example. (The presence of guarding would thus have been sought in an attempt to confirm the suspected presence of appendicitis.) The postulated diagnosis is chosen from D,, D, . . . D,,I in a Popperian spirit. This means that there is complete freedom of choice, and that this step is analogous to the creative way in which a scientist will choose a hypothesis to test by experiment [ 7 ] . However, this choice will be influenced by the probability of each D j given the initial evidence and also the relative seriousness of each D j in case it is important to avoid delay by choosing a serious diagnosis earlier. The factors to be considered when applying the theorem will now be discussed. </p><p>4. The differential diagnoses The list D,, D, . . . Dm represents the differential diagnoses associated with the </p><p>attribute Sy chosen to act as the 'initiating evidence' or 'lead'. (French's-Index of </p><p>Info</p><p>rm H</p><p>ealth</p><p> Soc</p><p> Car</p><p>e D</p><p>ownl</p><p>oade</p><p>d fr</p><p>om in</p><p>form</p><p>ahea</p><p>lthca</p><p>re.c</p><p>om b</p><p>y U</p><p>B K</p><p>iel o</p><p>n 10</p><p>/26/</p><p>14Fo</p><p>r pe</p><p>rson</p><p>al u</p><p>se o</p><p>nly.</p></li><li><p>28 D. E . H . Llezcelj*ri </p><p>Differential Diagirosis  is a familiar test \vhich describes the differential diagnoses o f traditional 'leads'.) This 'lead' will initially consist of the presenting complaint (localized right lower quadrant abdominal pain for example) but any ST can be used as a lead subsequently (like guarding), especially if it is more frequently accounted for by fewer differential diagnoses. The term represents the set of patients with all those other diagnoses which are not included as one of the rn listed diagnoses. It may be verified from the inequality that the stronger probability bounds (i.e. the higher guaranteed minimum probabilities) Mould be obtained by using the SiyF where the value ofp(B/Sf), r n and eachp(D,) is lolvest. Thus i f a new test was being designed to act as a 'lead' from ivhich any D j could be chosen as a postulated diagnosis, then m should be small, and p(D/ST) should also be small. For reasons explained below, each p(SJID,) should be as high as possible. </p><p>In the example given in the introduction there were only two differential diagnoses (i.e. M = 2), all other possibilities being said never to occur with localized right lower quadrant pain. Thus p(D/Sp) \vas zero, so that the value of p(S f ) was immaterial. Howel.er, according to the data shown in table 1, based on patients admitted to a surgical unit with an acute abdominal pain , p(Siw;) (i.e. the unconditional probability of localized right lower quadrant pain) would have been 0.36, and p(D/ST) (i.e. the probability of some other conditions given localized right lower quadrant pain) lvould have been 1 -098=0.02. </p><p>5. Unconditional probabilities The terms p(D,) and p(Sp) are unconditional probabilities. The latter can be </p><p>written more full! asp(Sr) =p(S,+IU) = Fr(Sr/U). Such a probability is thus equal to the frequency with which this piece ofe\idence occurs in the set of patients on which all the frequencies were observed. This is quite different, for example, from the 'Bayesian prior probability' Lvhich is represented by the same notation . In the set of patients represented by table 1, the unconditional probability of appendicitis (i.e. ~(4,) =p(D,) ) would be 029 , and that of non-specific abdominal pain (p[D,]) would be 0.5. </p><p>6. Discriminating between differential diagnoses I n order to continue to guarantee the highest possible probability of the </p><p>postulated diagnosis, D,, an observation Sp needs to be selected for each differential diagnosis D, which occurs as rarely as possible in D j and as commonly as possible in </p><p>'1';iklc. I . Frequent! data on paticnrs admitted to a surgical unit \vith acute non-specific abdominal pain (SS.AI' is non-specitic abdominal pain) </p><p>Frequency of Frequency of symptom or sign appendicitis or </p><p>in disease Frequent! of NSAP in ( ' ' , , I symptrirns o r sign i n those with </p><p>~ l l p:\tirnts 5tudird symptom or sign ~ ~~~ Appendicitis SSXP ( K P ) ( ' ' , , ) </p><p>~ .~ - ________.___~ ~ ~~ </p><p>]tight loner quadrant pain 73 28 3 6 0 x (;liarding 83 9 35 82 l</p></li><li><p>Logical elimination in probabilistic diagnosis 29 </p><p>4,. In an absolutely logical situation, guarding would, for example, be said never to occur in non-specific abdominal pain, so thatp(ST/Dj) would be zero. Since this was the only differential diagnosis in the introductory example, and since p(D/S?) is also zero, the entire numerator sums to zero. In this case, the value of p ( D j ) was immaterial since p(Dj) . (p(ST/D,) = O . </p><p>Such a situation is, of course, unrealistic, but such reasoning could occur by drawing exaggerated conclusions from knowledge of physiological mechanisms. The technique of eliciting guarding is based on the observation that patients with appendicitis have an inflamed abdominal peritoneum which results in the charac- teristic reflex response [lo]. This should not occur in non-specific abdominal pain, hence the temptation to infer that guarding would be absent in the latter. As a result there would be an unwarranted certainty in the final diagnosis. A similar conclusion might result from the suggestion that, from anatomical considerations, only appendicitis and non-specific abdominal pain could cause localized right lower quadrant pain [lo]. However, these hypotheses should be tested by assessing the frequency of the findings in those with the diagnostic categories in question. </p><p>In practice, we see from table 1 that the frequency of guarding in non-specific abdominal pain would be about 0.09, and guarding in appendicitis 0.83. Since the frequency of some other diagnoses in those with localized right lower quadrant abdominal pain was 0.02, then numerator would sum to: </p><p>p ( D 2 ) . (p(ST/DJ +p(Sf) .p(D/Sf) =(0.50)(0.09 + (0.36) (002) =0.0.52. </p><p>7. Subdividing diagnoses The way in which the numerator sums to zero in absolutely logical elimination </p><p>obscures the factors represented by the denominator, since the value of the latter becomes immaterial providing that p(S*/D,) &gt; 0. When the numerator is greater than zero, the denominator factors become very important. Now </p><p>r n 1 </p><p>gives a fail-safe lower-bound forp(S*/D,). The latter is equal to the frequency with which the combination S* occurs in the postulated diagnosis D,, for example the frequency with which localized right lower quadrant pain with guarding occurs in those with appendicitis. However, </p><p>n c P(ST/D,) - n + 1 i = l </p><p>may be equal to or less than zero, especially if n is large and manyp(ST/D,) are low. In these circumstances all that can be said is that p(S*/D,)&gt;O and as a result, p(D,/S*) 2 0 (which is obvious from axioms anyway). If this occurs then is raises the question of whether there are two (or more) possible subdivisions of the postulated diagnosis, and that neither alone may be capable of causing all the observed attributes at the same time. For example, localized right lower quadrant abdominal pain may be caused by appendicitis in 73i0 of cases where the appendix is touching the abdominal wall. However, right-sided rectal tenderness may occur in the 24% of patients where the appendix has turned down into the pelvis where it cannot cause </p><p>Info</p><p>rm H</p><p>ealth</p><p> Soc</p><p> Car</p><p>e D</p><p>ownl</p><p>oade</p><p>d fr</p><p>om in</p><p>form</p><p>ahea</p><p>lthca</p><p>re.c</p><p>om b</p><p>y U</p><p>B K</p><p>iel o</p><p>n 10</p><p>/26/</p><p>14Fo</p><p>r pe</p><p>rson</p><p>al u</p><p>se o</p><p>nly.</p></li><li><p>localized right lower quadrant pain as \veil. In this case the upper bound ofp(S*/D,) Ivould be mas (0, 0.73 + 0 2 3 - 2 + 1 =max (0, -0.03) = O . Thus if a new test were being designed to support a diagnosis, and the frequency of the test result was low in those \vith that diagnosis, then the reason for this should be carefully considered in terms of disease mechanisms. I f necessary, the diagnosis should be subdivided so that separate tests can be designed in order that the frequency of their values in each condition is a s near to one as possible, for example, divided into D,, (pelvic appendicitis) and D , (abdominal appendicitis). </p><p>8. Probabilistic inference by elimination The frequency of localized right lower quadrant pain in those with appendicitis </p><p>n-as 0.73, and the frecluenc&gt;- of guarding i n appendicitis Ivas 0.83. Since PI= 2 in the example in the introduction, then the \ d u e of the denominator is equal to max (0, 0.73 + 0.83 - 1 + 1) (0.29) = (0.56) (0.29) = 0,162. Thus a lower bound of the prob- ability of appendicitis, gii.en localized right lower quadrant abdominal pain and guarding \\ith a numerator value of 0.052 and a denominator of 0.162, would be </p><p>The frequency of appendicitis in those with localized right lower quadrant pain and guarding \\-as 0....</p></li></ul>