Apply Bayes Theorem to Update the Repertory

The homeopathic repertory is a very important instrument, but still an instrument. It is not possible to prescribe successfully without knowledge or consulting of the materia medica. The Repertory is about knowledge from the past, but the patient before you is another person than the numerous anonymous patients responsible for the data in the repertory. There are so many things that we don’t know about the Repertory, like: what is the meaning of ‘Herpes about lips’? Is this once a year, or ten times a year? It will make a difference if it is ten times a year. And, if you made the repertorisation with, say, eight symptoms, do you prescribe the first remedy? Many times you won’t, and maybe you take the last remedy, because seeing this suggestion you suddenly perceive that the patient perfectly suits that medicine for reasons that were not in the repertorisation.

We might look at a repertorisation like a weather forecast; there are numerous other variables that influence what you are going to do the next day. But still, you want the weather forecast to be correct. Sadly enough, the repertory is not correct. Over two centuries thousands of cases have been reported and the homeopathic database has grown correspondingly. The data were recorded in the homeopathic materia medica and the homeopathic repertory. As the database grew we lost sight and grip on the prerequisites for entering information into the materia medica and the repertory. As a general rule a symptom or characteristic is entered in the materia medica if it is seen in a patient responding well to a specific medicine, and the medicine is entered in the corresponding repertory-rubric. Therefore, the entries are based on absolute occurrence. Some medicines are used frequently, others seldom. If a medicine is frequently used any symptom will come up eventually in a patient responding well to that medicine, especially if the symptom is also frequently occurring. This is due to mere chance.

One of the well-known problems of the homeopathic repertory, is that especially larger rubrics are unreliable and that especially frequently used medicines are over-represented. As said, this is due to chance. In this respect the computer becomes a major threat to the reliability of the homeopathic repertory and materia medica. Updating these sources has become very easy and all manufacturers of homeopathic programs for repertories and materia medica advertise the completeness and vast number of data they comprise. As yet there are no generally accepted rules for ascertaining the reliability of our data. Such rules should be clear, unambiguous and reproducible. First we need a sound theoretical ground and then a methodology based on this ground. The most elegant and widely used theory is Bayes’ theorem. If we apply Bayes’ theorem the computer becomes our most valuable companion. We need large amounts of data and systematic gathering of these data.

Bayes theorem

Reverend Thomas Bayes (1702-1761) based his theorem on the law of conditional probability and it is explicitly or implicitly used to update prior beliefs in a particular hypothesis after observations or experiments.[4] The founder of homeopathy, Hahnemann (1755-1843), already based his use of the homeopathic medicine Rhus toxicodendron on his observation that ‘Amelioration from motion’ was a symptom that occurred more frequently in patients responding well to that medicine than in other patients. Based on this experience homeopathic physicians will prefer Rhus toxicodendron (and other medicines related to the same symptom) if the complaints are better from motion. We refer to patients who respond well to a specific medicine as ‘medicine population’, the patients that respond well to Rhus toxicodendron constitute the ‘Rhus toxicodendron population’.

The addition ‘more than in other patients’ is a crucial element in finding ways to select reliable entries for the homeopathic materia medica and repertory. This can be translated into Bayes theorem. This theorem has several expressions, one of them is:

Posterior odds = LR * prior odds

LR = Likelihood Ratio = prevalence in target population / prevalence in remainder of the population

Odds = chance / (1-chance); chance = odds / (1+odds)

The Likelihood Ratio (LR) is always larger than zero. If LR>1 the posterior odds increases; if LR<1 (>0) the posterior odds decreases.

Our specification of LR is derived from diagnostic research. In diagnostic research we test for a diagnosis with a certain test and compare the outcome with a reference (gold) standard.

This process gives an outcome like table 1.

Table 1: 2×2 contingency table for assessing diagnostic tests

Bayes formula can be applied repeatedly. After the first positive test, the chance that the diagnosis is correct increases. This posterior chance becomes the prior chance for the next test. This process is called sequential updating.

Algorithm for homeopathy

Bayes’ theorem enables us to formulate an algorithm for homeopathy; experience from the past tells us what to do in new cases. People who respond well on a specific homeopathic medicine, have specific symptoms that distinguish them from other homeopathic medicines. In other words: the prevalence of certain symptoms in the ‘medicine population’ is larger than in the remainder of the population.

To assess the LR of a homeopathic symptom we use the model of diagnostic tests with specification as in table 2.

Table 2: 2×2 contingency table for assessing relation between symptom and effect

With Bayes’ formula and sequential updating we can try to describe the decision process in homeopathy. Suppose we have a patient with joint pain and several symptoms and characteristics. The most simple model, is that the belief that one specific medicine could work is updated after a number of sequential informations. We assume that the prior chance that a medicine will work, without any information, is 5%.

Table 3: homeopathic diagnostic model 1; simple sequential updating of chance of effect, regarding only one medicine

Table 3 represents a simplified hypothetical model; we consider one medicine and all information increases sequentially the probability of effect from medicine A. In homeopathic practice all kinds of information could be relevant: complaint, relations to food, modalities (influences on the complaint) and personal characteristics. We assume that all information is mutually independent. Real practice, of course is more complicated, several possible medicines must be compared.

Repertory and LR

To make a comparison with the existing entries of Kent’s repertory we have to translate type (expressing importance of the symptom related to that medicine) into numbers. Such a translation is arbitrary, A cut-off value like LR>1.5 for plain type means that we regard a medicine as indicated if the prevalence of the symptom in the medicine-population is at least 1.5 times larger than in the rest-population. A possible translation from type into LR could be like Table 4:

Table 4: Repertory entries translated into LR values

The choice of LR>6 for bold type could be motivated by the opinion of many homeopathic doctors that three good symptoms are enough to prescribe a medicine. Three good symptoms with LR=6 render a combined LR=6*6*6=216. With this combined LR a prior chance of 1% would climb to 69%. With LR=3 we would need 5 symptoms to get the same result. Therefore, LR=3 or entries in italics stand for 5 necessary symptoms to prescribe with confidence. The lower value of LR=1.5 for plain type is rather arbitrary, lower values wouldn’t make much difference for the chance that a medicine could work.

The Committee for Methods and Validation of the Dutch homeopathic doctors association performed prospective research to assess the Likelihood Ratio (LR) of six symptoms regarding homeopathic medicines.

Methods

From June 2004 until December 2007 we conducted an observational study including all consecutive new patients older than two years. Observers were 10 experienced Dutch homeopathic doctors, working as consultants especially for chronic cases. There were no limitations as to disease, the only limitations were the use of single homeopathic medicines and the possibility to evaluate effect. Practices were divided over the Netherlands. Candidates were selected among participants in our materia medica validation project and received a questionnaire in advance, see appendix 1. In the first consensus meeting with the participants the symptoms were defined. Six symptoms were assessed: 1. ‘Diarrhoea from anticipation’, 2. ‘Fear of death’, 3. ‘Grinding teeth during sleep’, 4 ‘Recurrent herpes lips’, 5. ‘Sensitive to injustice’ and 6. ‘Loquacity’. These six symptoms were checked in all patients. Results were recorded for each prescribed medicine, after evaluation according to the Glasgow Homeopathic Hospital Outcome Scale (GHHOS). The use of the GHHOS scale was already trained in the consensus meeting.

Results

In the end 4094 patients were included; 1314 (32.1%) male, 2752 (67.2%) female, 28 (0.7%) missing values. Mean age was 39.62, standard deviation 20.952, range 3 to 95. The male/female ratio by age is shown in figure 5. Females were over-represented between age 20 and 60.

Despite the consensus meeting to define symptoms and feedback on differences between doctors, differences in prevalence of symptoms remained. These differences were most pronounced for the vaguest symptoms, sensitive to injustice and loquacity. See Figure 1.

Figure 1: inter-observer variation of prevalence of symptoms

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