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Probabilistic-Causal Analysis

From these premises, it is evident that clinical diagnosis employs the so-called hypothetical-deductive method, known as DN^[1] (deductive-nomological model^[2]). However, this is not realistic, as the medical knowledge used in clinical decision-making rarely contains causal deterministic laws that allow for causal explanations and, consequently, the formulation of clinical diagnoses, among other things in the specialist context. Let us re-examine the case of Mary Poppins, this time adopting a probabilistic-causal approach.

Consider a number Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} of individuals, including those who report Orofacial Pain, who generally have bone degeneration of the Temporomandibular Joint. However, there might also be other apparently unrelated causes. We must mathematically translate the 'relevance' of these causal uncertainties in determining a diagnosis.

The casual relevance

To do this we consider the degree of causal relevance Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (cr)} of an event Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E_1} with respect to an event Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E_2} where:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E_1} = patients with bone degeneration of the temporomandibular joint.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E_2} = patients reporting orofacial pain.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E_3} = patients without bone degeneration of the temporomandibular joint.

We will use the conditional probability Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P(A \mid B)} , that is the probability that the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} event occurs only after the event Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B} has already occurred.

With these premises the causal relevance Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle cr} of the sample Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} of patients is:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle cr=P(E_2 \mid E_1)- P(E_2 \mid E_3)}

where

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P(E_2 \mid E_1)} indicates the probability that some people (among Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} taken into consideration) suffer from Orofacial Pain caused by bone degeneration of the Temporomandibular Joint,

while

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P(E_2 \mid E_3)} indicates the probability that other people (always among Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} taken into consideration) suffer from Orofacial Pain conditioned by something other than bone degeneration of the Temporomandibular Joint.

Since all probability suggest that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P(A \mid B)} is a value between Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0 } and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1 } , the parameter Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (cr)} will be a number that is between Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -1 } and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1 } .

The meanings that we can give to this number are as follows:

we have the extreme cases (which in reality never occur) which are:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle cr=1} indicating that the only cause of orofacial pain is bone degeneration of the TMJ,
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle cr=-1} which indicates that the cause of orofacial pain is never bone degeneration of the TMJ but is something else,
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle cr=0} indicating that the probability that orofacial pain is caused by bone degeneration of the TMJ or otherwise is exactly the same,

and the intermediate cases (which are the realistic ones)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle cr>0} indicating that the cause of orofacial pain is more likely to be bone degeneration of the TMJ,
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle cr<0} which indicates that the cause of orofacial pain is more likely not bone degeneration of the TMJ.

Second Clinical Approach

(hover over the images)

Figure 1: Patient reporting "Orofacial pain in the right hemilateral"
Figure 2: Patient's TMJ Stratigraphy showing signs of condylar flattening and osteophyte
Figure 3: Computed Tomography of the TMJ
Figure 4: Axiography of the patient showing a flattening of the chewing pattern on the right condyle
Figure 5: EMG Interferential Pattern. Overlapping upper traces corresponding to the right masseter, lower to the left masseter.

So be it then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P(D)} the probability of finding, in the sample of our Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} people, individuals who present the elements belonging to the aforementioned set Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle D=\{\delta_1,\delta_2,...,\delta_n\}}

In order to take advantage of the information provided by this dataset, the concept of partition of causal relevance is introduced:

The partition of causal relevance

Always be Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} the number of people we have to conduct the analyses upon, if we divide (based on certain conditions as explained below) this group into Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k} subsets Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C_i} with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i=1,2,\dots,k} , a cluster is created that is called a "partition set" Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pi} :

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pi = \{C_1, C_2,\dots,C_k \} \qquad \qquad \text{with} \qquad \qquad C_i \subset n , }

where with the symbolism Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C_i \subset n } it indicates that the subclass Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C_i} is contained in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} .

The partition Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pi} , in order for it to be defined as a partition of causal relevance, must have these properties:

For each subclass Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C_i} the condition must apply Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle rc=P(D \mid C_i)- P(D )\neq 0, } ie the probability of finding in the subgroup Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C_i} a person who has the symptoms, clinical signs and elements belonging to the set Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle D=\{\delta_1,\delta_2,...,\delta_n\}} . A causally relevant partition of this type is said to be homogeneous.
Each subset Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C_i} must be 'elementary', i.e. it must not be further divided into other subsets, because if these existed they would have no causal relevance.

Now let us assume, for example, that the population sample Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} , to which our good patient Mary Poppins belongs, is a category of subjects aged 20 to 70. We also assume that in this population we have those who present the elements belonging to the data set Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle D=\{\delta_1,.....\delta_n\}} which correspond to the laboratory tests mentioned above and precisa in 'The logic of classical language'.

Let us suppose that in a sample of 10,000 subjects from 20 to 70 we will have an incidence of 30 subjects Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p(D)=0.003} showing clinical signs Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta_1} and $\delta _{4}$ . We preferred to use these reports for the demonstration of the probabilistic process because in the literature the data regarding clinical signs and symptoms for Temporomandibular Disorders have too wide a variation as well as too high an incidence in our opinion.^[3]^[4]^[5]^[6]^[7]^[8]

An example of a partition with presumed probability in which TMJ degeneration (Deg.TMJ) occurs in conjunction with Temporomandibular Disorders (TMDs) would be the following:


$P(D\|Deg.TMJ\cap TMDs)=0.95\qquad \qquad \;$	where	$\mathbb {C} _{1}\equiv Deg.TMJ\cap TMDs$
$P(D\|Deg.TMJ\cap noTMDs)=0.3\qquad \qquad \quad$	where	$C_{2}\equiv Deg.TMJ\cap noTMDs$
$P(D\|noDeg.TMJ\cap TMDs)=0.199\qquad \qquad \;$	where	$C_{3}\equiv noDeg.TMJ\cap TMDs$
$P(D\|noDeg.TMJ\cap noTMDs)=0.001\qquad \qquad \;$	where	$C_{4}\equiv noDeg.TMJ\cap noTMDs$

«A homogeneous partition provides what we are used to calling Differential Diagnosis.»

Clinical situations

These conditional probabilities demonstrate that each of the partition's four subclasses is causally relevant to patient data $D=\{\delta _{1},.....\delta _{n}\}$ in the population sample $PO$ . Given the aforementioned partition of the reference class, we have the following clinical situations:

Mary Poppins $\in$ degeneration of the temporomandibular joint $\cap$ Temporomandibular Disorders

Mary Poppins $\in$ degeneration of the temporomandibular joint $\cap$ no Temporomandibular Disorders

Mary Poppins $\in$ no degeneration of the temporomandibular joint $\cap$ Temporomandibular Disorders

Mary Poppins $\in$ no degeneration of the temporomandibular joint $\cap$ no Temporomandibular Disorders

To arrive at the final diagnosis above, we conducted a probabilistic-causal analysis of Mary Poppins' health status whose initial data were $D=\{\delta _{1},.....\delta _{n}\}$ .

In general, we can refer to a logical process in which we examine the following elements:

an individual: $a$
its initial data set $D=\{\delta _{1},.....\delta _{n}\}$
a population sample $n$ to which it belongs,
a base probability $P(D)=0,003$

At this point we should introduce too specialized arguments that would take the reader off the topic but that have an high epistemic importance for which we will try to extract the most described logical thread of the Analysandum/Analysans concept.

The probabilistic-causal analysis of $D=\{\delta _{1},.....\delta _{n}\}$ is then a couple of the following logical forms (Analysandum / Analysans^[9]):

Analysandum $=\{P(D),a\}$ : is a logical form that contains two parameters: probability $P(D)$ to select a person who has the symptoms and elements belonging to the set $D=\{\delta _{1},\delta _{2},...,\delta _{n}\}$ , and the generic individual $a$ who is prone to those symptoms.

Analysan $=\{\pi ,a,KB\}$ : is a logical form that contains three parameters: the partition $\pi$ , the generic individual $a$ belonging to the population sample $n$ and $KB$ (Knowledge Base) which includes a set of $n>1$ statements of conditioned probability.

For example, it can be concluded that the definitive diagnosis is the following:

$P(D|Deg.TMJ\cap TMDs)=0.95$ - this means that our Mary Poppins is 95% affected by TMDs, since she has a degeneration of the Temporomandibular Joint in addition to the positive data $D=\{\delta _{1},.....\delta _{n}\}$

↑ Sarkar S, «Nagel on Reduction», in Stud Hist Philos Sci, 2015».
PMID:26386529
DOI:10.1016/j.shpsa.2015.05.006
↑ DN model of scientific explanation, also known as Hempel's model, Hempel–Oppenheim model, Popper–Hempel model, or covering law model
↑ Pantoja LLQ, De Toledo IP, Pupo YM, Porporatti AL, De Luca Canto G, Zwir LF, Guerra ENS, «Prevalence of degenerative joint disease of the temporomandibular joint: a systematic review», in Clin Oral Investig, 2019».
PMID:30311063
DOI:10.1007/s00784-018-2664-y
↑ De Toledo IP, Stefani FM, Porporatti AL, Mezzomo LA, Peres MA, Flores-Mir C, De Luca Canto G, «Prevalence of otologic signs and symptoms in adult patients with temporomandibular disorders: a systematic review and meta-analysis», in Clin Oral Investig, 2017».
PMID:27511214
DOI:10.1007/s00784-016-1926-9
↑ Bonotto D, Penteado CA, Namba EL, Cunali PA, Rached RN, Azevedo-Alanis LR, «Prevalence of temporomandibular disorders in rugby players», in Gen Dent».
PMID:31355769
↑ da Silva CG, Pachêco-Pereira C, Porporatti AL, Savi MG, Peres MA, Flores-Mir C, De Luca Canto G, «Prevalence of clinical signs of intra-articular temporomandibular disorders in children and adolescents: A systematic review and meta-analysis», in Am Dent Assoc, 2016». - PMCID:26552334
DOI:10.1016/j.adaj.2015.07.017
↑ Gauer RL, Semidey MJ, «Diagnosis and treatment of temporomandibular disorders», in Am Fam Physician, 2015».
PMID:25822556
↑ Kohlmann T, «Epidemiology of orofacial pain», in Schmerz, 2002».
PMID:12235497
DOI:10.1007/s004820200000
↑ Westmeyer H, «The diagnostic process as a statistical-causal analysis», in APA, 1975».
DOI:10.1007/BF00139821

This is an Open Access resource!

[:1-1] Sarkar S, «Nagel on Reduction», in Stud Hist Philos Sci, 2015».
PMID:26386529
DOI:10.1016/j.shpsa.2015.05.006

[2] DN model of scientific explanation, also known as Hempel's model, Hempel–Oppenheim model, Popper–Hempel model, or covering law model

[:2-3] Pantoja LLQ, De Toledo IP, Pupo YM, Porporatti AL, De Luca Canto G, Zwir LF, Guerra ENS, «Prevalence of degenerative joint disease of the temporomandibular joint: a systematic review», in Clin Oral Investig, 2019».
PMID:30311063
DOI:10.1007/s00784-018-2664-y

[:3-4] De Toledo IP, Stefani FM, Porporatti AL, Mezzomo LA, Peres MA, Flores-Mir C, De Luca Canto G, «Prevalence of otologic signs and symptoms in adult patients with temporomandibular disorders: a systematic review and meta-analysis», in Clin Oral Investig, 2017».
PMID:27511214
DOI:10.1007/s00784-016-1926-9

[:4-5] Bonotto D, Penteado CA, Namba EL, Cunali PA, Rached RN, Azevedo-Alanis LR, «Prevalence of temporomandibular disorders in rugby players», in Gen Dent».
PMID:31355769

[:5-6] Silva CG, Pachêco-Pereira C, Porporatti AL, Savi MG, Peres MA, Flores-Mir C, De Luca Canto G, «Prevalence of clinical signs of intra-articular temporomandibular disorders in children and adolescents: A systematic review and meta-analysis», in Am Dent Assoc, 2016». - PMCID:26552334
DOI:10.1016/j.adaj.2015.07.017

[:6-7] Gauer RL, Semidey MJ, «Diagnosis and treatment of temporomandibular disorders», in Am Fam Physician, 2015».
PMID:25822556

[:7-8] Kohlmann T, «Epidemiology of orofacial pain», in Schmerz, 2002».
PMID:12235497
DOI:10.1007/s004820200000

[9] Westmeyer H, «The diagnostic process as a statistical-causal analysis», in APA, 1975».
DOI:10.1007/BF00139821

This is an Open Access resource!

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