Blog > BAYESIAN STATISTICS, LIKELIHOOD RATIO AND PROGNOSTIC FACTOR RESEARCH: In search of a unique scientific identity for Homoeopathy

When a doctor states that an allopathic (conventional) medicine will work, he refers to a considerable amount of certainty that the medicine works better than a placebo in the average patient, not excluded from randomized controlled trials (RCTs) and provided his/her diagnosis is correct.2If a doctor with adequate training prescribes a Homoeopathic medicine he/she can give an estimate of the chance the medicine will work for the patient in front of him, based on the symptoms this individual presents. This may be no more than a chance of, say, 60% with a considerable amount of uncertainty, but it is relevant for the individual patient. It is also not possible to know if this cure is due to the effect of the prescribed Homoeopathic medicine or other factors such as spontaneous recovery or placebo‑effect.3

There are differences between diagnostic tests like ultrasonography in Allopathy which are considered gold standards and the methodology used in homeopathy. The most important problem is that our ‘gold standard’ is not easy to define. There is a somewhat vague general understanding about the meaning of ‘cure’. One of the main principles in our method is the relation between a symptom and the curative effect of a medicine. The symptom indicates that a medicine is more likely to have an effect than we could expect it by mere coincidence.

In conventional medicine we are used to assessing diagnostic tests. To assess a diagnostic test we need a gold standard to compare the test with. The gold standard is regarded as the best approximation of the truth. For instance, to assess ultrasonography for diagnosis of appendicitis the best standard is the result of laparotomy (and histology). After laparotomy we can divide patients in four groups, according to outcome4:

This can be depicted in a 2x2 table (Table 1):

i

test positive a b a+b

test negative c d c+d

a+c b+d a+b+c+d

We will use the notation a-d to indicate the possible results of a test.

The likelihood ratio is a constant that indicates the relation between prior-odds (the odds before the test) and posterior-odds (the odds after the test). This relation is given by the formula:

The LR can indicate the change if the test is positive (defined as LR+) and if the test is negative (defined as LR-).

The mathematical formulas regarding the parameters in the 2x2 table are:

LR+ = (a/(a+c)) / (b/(b+d))

LR- = (c/(a+c)) / (d/(b+d))

If LR = 1 nothing changes. The higher the LR (+) the better the test if the result is positive. For a negative result the test is better if the LR (-) is closer to zero. As an example we take the ultrasonography for the diagnosis of appendicitis. Literature shows that the LR (+) = 7.6 and the LR (-) = 0.27. Now we can calculate the posterior-chance for all prior-chances using the formula mentioned above.

It shows how the probability of appendicitis changes after positive (LR+) and after negative (LR-) ultrasonography. If our suspicion of the existence of appendicitis was 33%, the probability after the test rises to 79%, a negative ultrasonography would lower the probability to 12%.

Validation of Homoeopathic medicines is about validating effectiveness in individual cases. Homoeopathic practitioners base their expectation that a medicine will work on the experience that specific symptoms of the patient indicate specific medicines. The prevalence of such symptoms is higher in a population responding well to a specific medicine than in the remainder of the population. This principle has a solid mathematical foundation in Bayes’ theorem which identifies Homoeopathic symptoms as prognostic factors, and offers an interesting perspective of individualized research.3

In prognostic factor research, the probability that a medicine will be effective depends upon the presence of certain factors, i.e. specific symptoms. Prognostic factor research in homeopathy can be assessed by applying Bayes’ theorem, which tells us how to use practical experience gathered from the past for prescribing in new situations. It is based on the mathematical conditional probability. Like a diagnosis, the probability that a homeopathic medicine will work (prognosis) increases if the patient has a specific symptom indicating this medicine. Adding other symptoms indicating the same medicine stepwise increases the chance that the medicine will be effective. The essence of Bayes theorem in this context states that if a symptom has a higher prevalence in the ‘population responding to a specific medicine’ than the prevalence in the remainder population’, the probability of cure increases.5,6

The 18th century reverent and mathematician Thomas Bayes described a more pragmatic design for the search for truth. The conviction that something is true is built up gradually by subsequent observations. In medicine many facts are neither true, nor false, but probable. So is the result of most medical treatments. Bayes reformed the formula for conditional probability and thus described how our conviction of the truth of a certain fact increases or decreases by subsequent observations. In epidemiology this is expressed by Likelihood Ratio (LR) and odds. LR+ stands for increase in likelihood if a symptom is present; LR- stands for decrease in likelihood if the symptom is absent. Bayes\\\' formula is, after two centuries of struggle, accepted all over the world and present in many computer programs.4

Symptoms and personal characteristics are prognostic factors for a favourable response to a specific medicine , in Homoeopathy. An important challenge in this research is establishing causality between medicine and improved health.

One of the main principles in our method is the relation between a symptom and the curative effect of a medicine. The symptom indicates that a medicine is more likely to have an effect than we could expect it by mere coincidence. The ‘gold standard’ in homeopathy is the fact that the medicine worked. Instead of the diagnostic value of a test we measure the prognostic value of a symptom.

symptom present a b a+b

symptom absent c d c+d

a+c b+d a+b+c+d

a+c = all the patients that got the medicine with a positive effect

b+d = all the other patients, including the ones that got the medicine without positive effect

We take the symptom as the diagnostic test and the medicine as the illness to be diagnosed. If we take a closer look at the formula for the LR(+) we see:

a/(a+c) is for the prevalence of the symptom in the population that responds to a medicine.

b/(b+d) is for the prevalence of the symptom in the population that does not respond to that medicine.

If the symptom is more frequently present where the medicine was successful than in the rest of the population the LR(+)>1. In other words the more the symptom is confined to the medicine (and not to the rest) the higher the likelihood ratio(+)4.

The importance of a symptom in relation to a certain medicine is represented by the typefaces in homeopathic repertories. Bold type or bold and underlined, represent the most important symptoms of medicines. There are, however, some inconsistencies in the repertory. One is the representation of rare medicines. Changing typefaces on the basis of LR and power of the argument could correct this shortcoming.

The meaning of the typefaces in the repertory is not very clear. Kent gives no explanation in the preface to his repertory; he merely explains what kind of symptoms is more valuable. We find some explanation in one of the sources of the repertory; Hering’s ‘Guiding symptoms’. Hering gives indications for: ‘Symptoms occasionally confirmed’, ‘Symptoms more frequently confirmed’, ‘Symptoms verified by cures’ and ‘Symptoms repeatedly verified’. When a symptom is frequently confirmed in the treatment with a certain medicine its becomes more important, especially when the symptom is rare.

Rare medicines are medicines with little data. If there is little experience with a medicine, its symptoms are not frequently confirmed. This means that there is no emphasis for these symptoms in the repertory, even if the symptoms are characteristic for the medicine. In Kent\\\'s original repertory, Latrodectus mactans is not mentioned in the rubric \\\'fear of death\\\', despite the fact that the symptom is very prominent in the materia medica of Latrodectus mactans. The medicines Natrum muriaticum and Sulphur, however, are present in this rubric. These medicines are used very frequently and we might wonder if \\\'fear of death\\\' is really more frequently present in patients responding well to these medicines than in the remainder of the population. The materia medica does not give us that information because the materia medica does not compare all symptoms with other medicines.

The comparison of each symptom with other medicines can be provided by the repertory. At least, it should do so, but – as explained elsewhere – many repertory rubrics are seriously flawed because medicine entries are based on absolute occurrence while they should be based on relative occurrence (prevalence). A symptom is an indication for a specific medicine only if it occurs more frequently in patient responding well to that medicine than in other patients. In statistical terms, LR should be greater than one. To assess the LR for each medicine for the symptom, we have to count prevalence of the symptom in populations responding well to specific medicines and in the whole population. This research process is called PFR.

In Homoeopathy, the population responding well to any specific medicine is just a small part of the whole population, so the remainder of the population is nearly the whole population. The smaller the denominator in this formula (prevalence in the remainder of the population), the higher the LR 4.

At present it is not possible to test the usefulness of the LR in homeopathy, which is mainly because we have no data about the prevalence of our symptoms in our patient population, and we are just starting to gather data about successful cases. The prevalence of a symptom in patients responding to a particular homeopathic medicine will be determined by collecting a sufficient number of cases that showed a curative effect from that medicine. But even then we must be careful. Homeopathic symptoms are not easy to define, and often vague. Our gold standard, the cure is perhaps even more difficult to define. Also, present materia medica and repertory are not very clear about the exact meaning of many symptoms.

Our first goal is to investigate the possibility of assessing the LR of homeopathic symptoms. Our long-term goal is to update our materia medica and repertory by means of statistical instruments that match the homeopathic methodology.

Data of study by Commissie Methode en Validering of the Dutch society of homeopathic physicians (VHAN)4-

The first prospective assessment of homeopathic symptoms started June 2004, ten practitioners participated. Six symptoms were assessed: Diarrhoea from anticipation, fear of death, grinding teeth at night, herpes lips, sensitivity to injustice and loquacity. This is a mix of vague and less vague symptoms. Several computer programs were adapted to record and export the presence of the symptoms in each patient and all medicines and their results prescribed to each patient. The ten participating doctors were already trained in assessing results during consensus meetings that were organised since 1997. During these meetings doctors presented their best cases regarding two medicines and discussed results; what is the score according to the Glasgow homeopathic outcome scale (GHHOS) and was it due to the medicine?

During the prospective assessment of LR two consensus meetings each year were held to define symptoms and to discuss intermediate results. These meetings revealed differences between doctors in interpreting results and difficulties in interpreting vague symptoms.

In March 2007, 3367 patients were included and 3246 prescriptions evaluated. Some results regarding the symptom \\\'Fear of death\\\' are shown in table 3. There were 131 patients with fear of death in the total population of 3367 patients (3.9%). Patients reacting well were defined by GHHS results between 2-4, i.e. not only the presented complaint was better, but also constitutional effects were visible.

Fear of death n=131

LR+ 95% CI

Aconitum 6.5 1.9 to 21.9Anacardium 12.1 6.2 to 23.7

Arsenicum album 6.4 3.1 to 13.2

Conium 3.7 1.0 to 13.6

Veratrum album 10.4 3.5 to 30.9

Sulphur 0.35 0.05 to 2.5

The amount of data permitted to assess only a small number of LRs with significant values. Many values, like for Calcarea carbonica (Calc.) (LR= 1.2), Cimiciguga (Cimic.) (LR=4.3), Lac caninum (Lac-c.) (LR=4.3), Nitricum acidum (Nit-ac.) (LR=1.7) and Phosphorus (Phos.) (LR=1.4) had 1 in their 95% confidence interval. If we look at the underlying figures, however, we can see that their data are more reliable then the data of the original repertory. If only 3 of 64 patients responding well to Calcarea carbonica have a fear of death we can hardly imagine that this symptom strongly indicates Calcarea, as suggested by the repertory. The symptoms is repeatedly seen in patients responding well to this medicine, but also prescribed many times. But there were 5 out of 11 Anacardium patients with fear of death.

We conclude that the bold type entries in this repertory-rubric of Calc. and Phos are incorrect, they are just due to the frequent use of these medicines. Nit-ac. should probably not be mentioned in bold type, and Sulphur should not be in this rubric. On the other hand, Anacardium is a surprising outcome; this medicine is strongly related to fear of death.

Of course, we must realise that this is one group of doctors, with their training and experience. There could be differences in other groups in other countries. But if we constitute repertory-rubrics this way we are much more informed than by the existing repertory.

This first Dutch study led to three conclusions4 :

1. LR research is feasible when using the proper software in daily practice.

2. It is possible to gather a large amount of data without interfering with daily practice.

3. There are many mistakes in the repertory. Most of them concern unjust entries of frequently used medicines, but there is no general rule that permits us to discard all frequently used medicines from a rubric.

By making these estimations more explicit and by performing prospective studies assessing LR of homeopathic symptoms, practicioner’s can achieve significant improvement in their method in the following ways-

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