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

Blog > BAYESIAN STATISTICS, LIKELIHOOD RATIO AND PROGNOSTIC FACTOR RESEARCH: In search of a unique scientific identity for Homoeopathy
DR.SUHANA.P.AZIS, MD Research Officer (H), Scientist-1, Central Council for Research in Homoeopathy, Ministry of AYUSH, Government of India, New Delhi Published: '. October 17, 2018

How do we value homeopathic symptoms?

There is of course Hahnemann’s famous § 153, stating that peculiar, rare symptoms are most valuable. But we know that each remedy has its own characteristics, also called ‘keynotes’. These symptoms are not so rare, but observed more frequently in relation to that remedy than the rest of our population. So Homoeopaths prescribe on symptom value. But how sure can we be about the value of the symptom. Clinical experience shows that a Homeopathic medicine cannot be prescribed on indication alone. Recently, conventional medicine is adapting this centuries old homeopathic insight in personalised medicine. Researching the influence of multiple variables on outcomes (confounders) requires a suitable strategy.1
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. 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.4


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:

a: the test (ultrasonography) is positive and the illness (appendicitis) is present: true positive
b: the test is positive and the illness is absent: false positive
c: the test is negative and the illness is present: false negative
d: the test is negative and the illness is absent: true negative

This can be depicted in a 2x2 table (Table 1):
                        illness present    illness absent     
test positive                a                      b                 a+b
test negative               c                      d                 c+d
                                 a+c                  b+d           a+b+c+d

Table 1: 2x2 table showing relationships between the results of diagnostic tests and the presence of illness
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:

Posterior odds (+) = LR(+) x prior-odds.

The transformations between odds and chance are as follows:
Odds = chance / (1-chance)
Chance = odds / (1+odds)

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%.


Prognosis research in conventional setup seeks to understand and improve future outcomes in people with a given disease or health condition. More people now live with disease and conditions that impair health than at any other time in history; prognosis research provides crucial evidence for translating findings from the laboratory to humans, and from clinical research to clinical practice. A prognostic factor is any measure that, among people with a given start point (such as diagnosis of disease), is associated with a subsequent endpoint (such recovery).1
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

Thus, LR = (Prevalence in the target population)/ (Prevalence in the remainder of the population)

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. Prognostic factor research could become one of the main pillars of Homoeopathy’s scientific identity.3
In evidence based medicine we are looking for proof of efficacy of treatments for specific diagnoses. Due to over-emphasis of RCT the value of clinical experience has become under-estimated.7 Each separate case gives some qualitative information and collections of similar cases allow quantitative analysis.


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.

The 2x2 diagram for a homeopathic symptom is shown below:
                               medicine worked       rest     
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

Assessing questions

In homeopathy we ask questions to get ideas about possible medicines or to confirm medicines that might be applicable. Some questions seem more effective than others. In this respect homeopathy does not differ from conventional medicine. Can we assess questions the same way we assess diagnostic tests?
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.
So the LR(+) = (a/(a+c))) / (b/(b+d)) = (prevalence of the symptom with the medicine) / (prevalence of the symptom with all others).

Or in other words: The likelihood ratio (+) of a symptom compares the presence of that symptom in the successful prescriptions of that medicine, with the frequency of this symptom in the unsuccessful prescription of the same medicine and all prescriptions of other medicines4.
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.


LR gives us the opportunity to assess Homeopathic symptoms scientifically. The best way for this is prospective research. If we want to perform prospective research on LR we will have to evaluate an enormous amount of symptoms. This requires a method that interferes with daily practice as little as possible.
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 21.9
     Anacardium             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

LR results of the symptom \\\'Fear of death\\\' after 3246 evaluated prescriptions, with 95% confidence intervals (95% CI)
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-
•    Physician can be more certain about the value of the symptoms and able to estimate the certainty that his/her medicine will be effective.
•    Symptoms are better described and results become reproducible.
•    Structural mistakes of the repertory are solved.
•    Homoeopathy can develop its own scientific identity.

1.    Riley RD, Hayden JA, Steyerberg EW, Moons KG, Abrams K et al. Prognosis research strategy (PROGRESS) 2: Prognosic Factor Research. PloS Med 2013; 10(2): e1001380. doi: 10.1371/journal.pmed.1001380 
2.    Smulders Y. Evidence based: To be or not to be? Neth J Med 2011; 69: 53‑4.
3.    Rutten L. Prognostic factor research in Homoeopathy. Indian J Res Homoeopathy 2016; 10: 59-65.
4.    Rutten L, Stolper E, Lugten R, Barthels R. Likelihood Ratio: A modern approach for classical homeopathy. Commissie Methode en Validering of the Dutch society of homeopathic physicians (VHAN).  (accessed on 10.07.2018)
5.    Rutten ALB, Stolper CF, Lugten RFG, Barthels RWJM. Assessing likelihood ratio     of clinical symptoms: handling vagueness. Homeopathy.2003;92:182-186
6.    Rutten ALB. Bayesian homeopathy: talking normal again. Homeopathy 2007; 96: 120-124.
7.    Vandenbroucke JP. In defense of case reports and case series. An n Intern Med. 2001; 134:330-334.