The Probability of Disease. William J. Long. Cambridge, MA hospital admitting door (or doctors oce, or appropriate

Transcription

1 Repinted fom Poceedings of the Fifteenth Annual Symposium on Compute Applications in Medical Cae, pp , 1991 The Pobability of Disease William J. Long MIT Laboatoy fo Compute Science Cambidge, MA 2139 This pape addesses the natue of the pio pobabilities of diseases fo pobabilistic diagnostic easoning. Because diseases die in thei chonicity, occuence, eoccuence, and likelihood of becoming pat of the patient population, easoning in tems of the fequency of disease episodes is necessay to captue the impotant distinctions. Even with these complexities, it is possible to fomulate a easonably accuate, computationally tactable, fequency estimation method fo combinations of diseases. This method also suggests ways in which the needed numbes can be estimated fom patient data. 1 Intoduction Many medical diagnosis pogams use pobabilistic infeence to eason about the attibution of ndings (fo example [1, 2, 3]). The usual paadigm is that diseases have some pio pobability of occuing and poduce ndings with some pobability, possibly though some intemediate states. Thus, the view of the diagnosis poblem coesponds well to easoning about Bayesian pobability netwoks. A fundamental assumption is that the pimay causes, the diseases, ae independent. If the diseases ae known not to be independent, etiologies can be added to the netwok as pimay causes to eliminate the dependencies. This epesentation of the diagnostic poblem has geat intuitive appeal, but anumbe of diculties aise when applying it to eal medical domains. The poblem addessed by this papeiswhy one often nds patients with seveal seemingly independent diseases when the poduct of the pobabilities would pedict that this would be a ae event. Conside the situation whee disease A has a pio pobability of.1 and disease B has a pio pobability of.5. Does that mean that the next patient in the doo has pobabilities of.1 of having A,.5 of having B, and.5 of having both? The answe This eseach was suppoted by National Institutes of Health Gant No. R1 HL3341 fom the National Heat, Lung, and Blood Institute and No. R1 LM4493 fom the National Libay of Medicine. is no fo seveal easons. 1) People aely come into a hospital admitting doo (o doctos oce, o appopiate context fo the medical domain) unless they think they have something that needs attention. Thus, the pobability of some disease in the patient is high and cetainly much highe than in the man on the steet. 2) A o B may not always cause the patient togoto the hospital. 3) The pobability of the combination of the diseases, assuming that they ae causally independent, still depends on how long each one pesists. If both A and B have acute onset and do not emain long (like appendicitis, fo example), it is vey unlikely that they will occu togethe. If one o both is a chonic disease, it is much moe likely that the patient who comes into the hospital will have both. 2 The Natue of the Poblem The basic poblem is the meaning of the pobability of disease. That is, the meaning of the numbes, often called pios, that ae assigned to the pimay disease entities. To have a pobability thee must be a dened population ove which it is detemined. A simple answe might be to assume the population is that seved by the hospital (o cae unit). This model seves well fo a pogam such asdedombal's acute abdomen pogam[4], whee the diseases in question ae acute events, typically occu once in a lifetime, and always cause the patient to go to the hospital. Vaiations fom these chaacteistics cause poblems. If one consides the pobability of inuenza, it becomes clea that the issue is not the numbe of patients with the disease, but the fequency of the disease. Since each peson typically has inuenza seveal times duing his o he lifetime, it does not make sense to talk about the pobability of the disease. It makes moe sense to say that the aveage fequency of inuenza is once in ten yeas. Othe diseases ae chonic and pesent anothe set of issues. If a patient becomes diabetic, they emain diabetic fo the est of thei lifetime. It makes sense to say that thee is a cetain pobability that a peson will be diabetic by the end of thei life, but that obscues the poblem since the patient can

2 ente the hospital many times as a diabetic o befoe they become diabetic. The context in which theoc- cuence of disease is of concen is the hospital visit. Theefoe, a typical peson with diabetes will account fo seveal \patients", that is, seveal hospital visits. Thus, the numbe of concen is the expected fequency of hospital visits fo a paticula disease. The example of inuenza also aises the issue of whethe the peson with a disease will become a patient. We could say that we ae only inteested in diseases sevee enough to equie a hospital visit, but that is not sucient. If the patient is othewise healthy, in- uenza is unlikely to equie a hospital visit, but if the patient hasachonic disease that saps thei stength, it is much moe likely to equie a hospital visit. Thus, even though the two diseases may have independent etiologies, the pobability that a peson becomes pat of the patient population may be dependent on othe diseases. Accounting fo this phenomenon is paticulaly impotant if the domain is limited to a paticula specialty. Fo example, the fequency of patients with pneumonia alone in a cadiology clinic is small, because those patients ae teated elsewhee. Howeve, the fequency of pneumonia on top of an existing cadiac poblem is high, because pneumonia tends to decompensate the cadiac poblems. 3 Example The basic poposal is to view the poblem in tems of the fequency of disease events in the patient. To see the pactical implications, conside a population of 1 people and two possible diseases A and B. Disease A is a chonic incuable disease that occus in one out of a hunded people. Once a peson has it, they aveage a hospital visit fo a \ae-up" evey ve yeas and they have an aveage emaining lifespan of twentyyeas. Disease B is an acute disease that always sends a peson to the hospital but with pope hospital teatment is quickly esolved. It occus in the aveage peson once evey foty yeas. Conside the expected hospitalizations ove the lifetimes of all of the people in the population, assuming that the aveage lifetime in the population is 8 yeas. Fo disease A, the 1 people that contact it will each aveage fou hospitalizations fo a total of 4 hospitalizations caused by disease A. Fo disease B, the aveage peson will have it twice fo a total of 2 hospitalizations. Thee ae also 2 peson yeas when people have disease A and could also contact disease B. Thus, thee ae 5 expected hospitalizations caused by disease B in which disease A is pesent. If \ae-ups" of A and incidents of B ae consideed to have no time duation, the possibility of both happening togethe can be ignoed. Thus, thee ae 45 hospitalizations of patients with disease A, 11% of which ae caused by disease B. Conside that fo disease B the chances of being hospitalized fo the disease by itself is.1, but if it occus when disease A is pesent, the patient is always hospitalized. Now thee ae 2 hospitalizations fo B, but thee ae still 5 fo the combination of A and B. 4 Fequency of Patients with a Disease in a Setting Given that the numbe to calculate is the expected fequency of patients with the disease in the paticula cae setting, thee ae seveal factos that go into that calculation. 1) The pobability that a patient will contact the disease duing a given time peiod. This numbe may be dependent on age, sex o othe demogaphic factos, but should be independent ofany othe disease except whee the dependency is explicitly epesented. 2) The time couse of the disease. In the domain of cadiovascula disodes that we have been studying, thee ae two pimay types of time couse, diseases that ae acute, equiing a single admission, and ones that ae pemanent unless sugically coected. Thee ae some othe time couses in othe domains, but these cove the main issues. Acute diseases can usually be consideed to be events at a point in time. As such, the pobability oftwo such diseases happening at the same time in the patient isvanishingly small unless thee is a causal elation between them. The pactical implications of this assumption needs to be caefully consideed fo each domain, othewise impotant situations may be uled out. Fo example, the causal elation between diseases may be vague inuenza can cause a myocadial infaction by putting stess on a heat with peexisting coonay atey disease. Also, the hospital stay fo an acute disease may be long enough fo the patient to contact a second acute disease, such as pneumonia. This is paticulaly impotant when it is the second disease that bings the patient into the hospital. Chonic diseases can be consideed always pesent. 3) Fo chonic diseases, it is necessay to know the expected fequency of signicant episodes and the eect of the disease on the patient's life expectancy. That is, if a peson contacts a chonic disease at age 4 with yealy episodes equiing hospitalization but with no change in life expectancy, thee will be many admissions with that disease. If howeve, the aveage life expectancy afte contacting the disease is a yea, thee will be few admissions. The fequency of episodes of a disease may be dependent on many factos, but it is dicult to get sucient data on paticula diseases to chaacteize these elations in much detail. 4) The nal facto is the likelihood that the patient with a disease episode will become pat of the cohot of inteest. This is usu-

3 ally dependent ontheoveall state of the patient. If these factos can be specied fo the diseases of concen, the next step is to compute the expected fequency of disease. Conside st the single disease situation fo a patient ofagey. Fo an acute disease contacted on aveage q(y) times pe patient-yea and equiing admission h faction of the time, the expected fequency is q(y) h. Fo a chonic disease, the calculation is moe complicated. The patient contacted the disease at some time, suvived anothe a yeas and had an episode at age y. Since the fequency is elative to all patients who attained age y, the suvival needs to be elative to nomal suvival. Since all such patients ae included, the function is summed ove all ages less than y. An appoximationofthechange in suvival that is easonably accuate fo a wide ange of diseases and is computationally tactable is the declining exponential[5]. That is, the size of the population of patients who contacted the chonic disease changes elative to the population of patients without the disease ove the a yeas at e ;a. Anothe way to think about this equation is in tems of the numbe of yeas it would take to kill half of the patients that would have emained alive without the disease. If a is that numbe of yeas, = :693=a. In the simplest case whee the incidence of disease q is constant fo age, the pobability of hospitalization h is constant, and the fequency of episodes f is independent of both age and length of time the disease has been pesent, the expected fequency of episodes is obtained by summing ove a fom to age y: 1 ; e ;a e ;y da = The faction epesents the aveage numbe of yeas the patient would have had the disease. Fo example, if the patient is 5yeas old and is.5, eecting a ve yea diminished suvival of 78%, the expected length of time the patient had the disease is 18 yeas. If thee is an episode on aveage evey 5 yeas, this will be (on aveage) the thid o fouth episode. If the yealy isk of contacting the disease is.1 and an episode always causes a hospital admission, the expected fequency of hospital admissions is :1 :2 1: 18:4 =:37 pe peson pe yea. Fo simple dependencies of q on age, it is possible to genealize this elation. Fo example, if thee is no isk of the disease until a cetain age b and then the yealy isk is q, the fequency becomes 1 ; e ;(y;b) : If thee is a change in the isk at some age fom q to kq, the fequency is (1 ; k ; 1 k e ;(y;b) ; 1 k e ;y ): Often, the fequency of episodes of a chonic disease incease ove time. One waytomodelthisbehav- io is with the fequency as fe ka. That is, initially the fequency is f but that doubles in log 2=k yeas. The expected fequency of episodes would be 1 ; e qfe ka he ;a da = (k;)y : k ; Cae must be taken in epesenting a disease this way because the fequency of episodes neve gets above about half a dozen times a yea befoe the episodes mege into a single hospital stay o the patient succumbs to the disease. It is possible fo the k to be lage than the exponent. Fo example, fo a disease that has an excess motality of 2% each veyeas, the is about.5. If the fequency of hospital episodes inceases fom one evey fou yeas to fou a yea ove thity yeas with the disease, the k is.9. This simply means that the 22% of the patients (adjusting fo nomal life expectancy) emaining afte 3 yeas with the disease ae esponsible fo moe episodes than the initial goup when they contacted it. Thus, this scheme is able to appoximate the kind of chaacteistics that mightbeknown about a disease and accounts fo the fundamental dieences between acute and chonic diseases. 5 Fequency of Two Diseases The poblem we stated with is how to estimate the expected fequency of multiple diseases in the patient and we now have the machiney to addess that poblem. The two cases of concen ae an acute disease and a chonic one, and two chonic diseases. As discussed above, the pobability that two acute diseases happen togethe is zeo unless thee is a causal elationship between them. If this is a poblem, one could assign time extents to them. Fo example, just multiplying the fequencies togethe is the fequency that they happen in the same yea. An altenative solution would be to include a nonspecic causal mechanism that can cause cetain acute diseases when the patient is aleady \sick" with an acute disease. In each domain the knowledge enginee needs to ascetain the signicance and appopiate model fo acute diseases that happen togethe. 5.1 An Acute and a Chonic Disease When the patient has both an acute and a chonic disease, the acute disease stuck duing the tenue of

4 the chonic disease. Thee is not a coincidence of an episode of the chonic disease with the acute disease, fo the same easons that acute diseases do not happen togethe. Thus, the patient has contacted the chonic disease c, had it fo some peiod of time, contacted the acute disease a, and has been hospitalized by the combination. The fequency with which that occus is q c 1 ; e ;c y c q a h ajc : The only pat of this equation that is unknown is the pobability h ajc of being hospitalized with the acute disease given the chonic one. Unde almost all situations it will be highe than the pobability ofbe- ing hospitalized with the acute disease alone, but it may have nothing to do with the pobability ofbeing hospitalized with an episode of the chonic disease, since thee is no episode. A plausible mechanism fo estimating the pobability would be to have anaddi- tional facto fo chonic diseases indicating the degee to which the disease in geneal \compomises" the patient tobecombined with the pobability of hospitalization fo the acute disease. Howeve, one can think of situations whee this would not be satisfactoy and moe empiical expeience is needed to develop an appopiate mechanism fo handling this in geneal. 5.2 Two Chonic Diseases When the patient has two chonic diseases, one is undegoing an episode, but not both. Fotunately, pat of the diagnosis is which disease is having an episode. The situation is that the patient contacted chonic disease 1, some time passed, contacted chonic disease 2, some time passed, is now expeiencing an episode of 1, and is hospitalized. Altenatively, disease 2 peceded disease 1. The st expected fequency of the st situation is q1e ;1(b;a) q2e ;(1+2)(y;b) f1h 1j2 : The fequency of this happening fo all times a and b is Z b q1e ;1(b;a) q2e ;(1+2)(y;b) f1h 1j2 da db: Integating and adding the coesponding tem fo disease 2 peceding disease 1 gives a fomula fo the expected fequency of episodes of disease 1: 1 ; e ;1 y 1 ; e ;2 y q1 1 q2 2 f1h 1j2 : This expession can be gouped into fou pats, the expected yeas of diseases 1 and 2, the fequency of episodes and the pobability of hospitalization. The pobability of hospitalization depends on the episode of disease 1 in the context of both diseases. If thee is some way to chaacteize the degee of compomise of the two diseases, this might be the pobability implied by an episode of disease 1 combined with the compomise of disease 2. If the fequencies of episodes of the diseases ae modeled as inceasing, the expected fequency is 1 ; e (k1;1)y 1 ; e ;2 y q1 k1 ; 1 q2 2 f1h 1j2 : 6 Application to Medical Domains Given that this epesentation of patient hospitalizations in tems of fequency of disease episodes makes it possible to povide appopiate estimates of the expected fequency of dieent disease combinations, the emaining question is how to get the needed numbes. Thee ae seveal kinds of infomation available in hospital ecods about the occuence of diseases. Fo each admission thee is infomation about the pimay poblem and a list of the othe poblems that the patient had. In addition, one can look back in the patient ecod o statements of the patient's histoy and get a petty good indication of how many times and when the patient was admitted in the past fo the same poblems. This infomation is less eliable because thee can be omissions fo a numbe of easons. If one has the numbe of past admissions fo a paticula disease and the numbe of yeas the patient has had that disease, the fequency of disease episodes can be estimated diectly (with appopiate coections fo estimating the beginning of the disease fom the st episode). The est of the numbes can be estimated fom disease combinations. If we have data coveing a peiod of time, the numbe of episodes c of a paticula acute disease is an estimate of nph whee n is the size of the patient population. If the statistics cove all of the instances of the disease, h can be assumed to be one. The population size is the same fo all of the diseases. Fo chonic diseases, let p = q( 1 ; e;y Then, the count of episodes of the disease is an estimate of nf p (assuming h = 1). Since p is a function of the age of the patient, age anges need to be consideed. The p's can be estimated by looking at disease combinations. The expected numbe ofacute disease 1 in patients with chonic disease 2 is np1p2. Thus, the atio of the counts of the combination to the count of the acute disease c 1j2 =c1 estimates p2 (of the ):

5 chonic disease). Similaly, the count of admissions fo episodes of chonic disease 1 when chonic disease 2 is also pesent is an estimation of nf1p1p2, so the atio of counts c 1j2 =c1 estimates p2. In geneal, all episodes of othe diseases estimate the p i ofachonic disease, theefoe P p i c j jji P j c j whee all of the diseases j ae ones in which the faction admitted is independent of disease i (ie, h jji = h j ). Once the p's of chonic diseases have been estimated, the population size can be estimated and the pobabilities of acute diseases chosen accodingly. The p's of chonic diseases also give anothe way of estimating the fequency of episodes if that fequency is independent of the length of time the patient has had the disease. If not, a moe thoough analysis of the pattens of episode fequency must be done. 7 Summay This pape pesents a solution to the poblem of estimating the likelihood of combinations of diseases in the patient. The poblem is ecast in tems of the expected fequency of the disease episodes that would be encounteed in a paticula setting. This claies the natue of the calculation and makes it possible to account fo the impotantchaacteistics of the poblem. The solution is pactical because the factos needed to do the estimated fequency calculations can be estimated fom the kind of data nomally available about diseases. Refeences [1] William Long, \Medical Diagnosis Using a Pobabilistic Causal Netwok," Applied Aticial Intelligence, 3: , [2] K. G. Olesen, U. Kjaeul, et al, \A Munin Netwok fo the Median Netwok A Case Study on Loops," Applied Aticial Intelligence, 3: , [3] M. Shwe, B. Middleton, et al, \A Pobabilistic Refomulation of the Quick Medical Refeence System," 14th SCAMC, Washington, DC, pp79-794, 199. [4] F. T. dedombal, D. J. Leape, et al, \Compute- Aided Diagnosis of Acute Abdominal Pain," Bitish Medical Jounal, 2: 9-13, [5] J. Robet Beck, Jeome P. Kassie, and Stephen G. Pauke, \A Convenient Appoximation of Life Expectancy (The `DEALE')," The Ameican Jounal of Medicine, 73: , 1982.