# Differential Diagnosis: Bayesian view of diagnostic accuracy

Before delving further into the causal structure of functional movement systems I feel compelled to complete the 5 part series promised on the benefits of using DAGs to learn about and develop reasoning on differential diagnosis. The series is promised at the end of this post - here.

It promised 5 posts - this one is #4: demonstrate how Bayes theorem is related to our measures of diagnostic accuracy (sensitivity / specificity; likelihood ratios). The goal is to bridge the gap between these commonly applied and considered metrics, and what DAGs (as Bayesian Networks) have to offer us clinically. The 5th post will be to extend the understanding of Bayes theorem by introducing “Bayesian Networks” as another name for / form of graphical causal models (at this point we will have come full circle).

Let’s start with the general idea of “Disease” and “Sign or Symptom”. Of course, disease could be condition, dysfunction, or generalized as any causal factor. Sign or symptoms could be results of a test, or generalized as any effect. Diagnostic testing is abduction. It is observing effects and making inferences to the cause. Therefore, underlying all diagnostic testing is a causal association with a causal structure. The simplest causal structure as we have seen before is one with two nodes (variables) and one edge (connection):

Here D represents disease and S represents sign/symptom.

As previously explained in several posts by now, the probability of S is conditional to the occurrence of it’s causal factor D. We denote this conditional probability as: P(S | D) which means the probability of S given D. To facilitate the connection to our established concepts of sensitivity, specificity and likelihood ratios we will consider D and S as binary variables (yes / no). |

The first thing to point out is that the notation P(S | D) generally stands for 2^n probabilities with n being the number of variables (instantiations of the state of S and D in this case). For example: |

P(S | D) includes: P(+S | +D), P(-S | -D), P(+S | -D), P(-S | +D) |

The first connection to make is that P(+S | +D) is the sensitivity - the proportion of True Positives |

The second connection is that P(-S | -D) is specificity - the proportion of True Negatives |

P(-S | +D) is : 1 - Sensitivity which is the proportion of False Negatives; and P(+S | -D) is: 1 - Specificity which is the proportion of False Positives (I can provide a worksheet with all the algebra if anyone is interested ;) |

You may recall that the equations for likelihood ratios are:

+LR = Sensitivity / (1 - Specificity) and -LR = (1 - Sensitivity) / Specificity

Therefore: +LR = P(+S | +D) / P(+S | -D) and -LR = P(-S | +D) / P(-S | -D) |

Which is just saying that the +LR is the ratio of True Positives to False Positives; and the -LR is the ratio of False Negatives to True Negatives; and explains why the greater the +LR the better the Sign/Symptom for ruling in a Disease (or condition) and the lower (closer to 0) the -LR the better the Sign/Symptom is for ruling out a Disease (or condition).

If you recall Bayes equation, then you notice that Sensitivity is the conditional probability necessary to calculate P(+D | +S), the probability of Disease conditional on the presence of a Sign/Symptom - the equation is reviewed in this post. And you notice that Sensitivity is the conditional probability necessary to calculate P(-D | -S), the probability of not Disease conditional on the absence of a Sign/Symptom. |

Using Bayes and spelling it all out:

Probability of the Disease given the presence of the Sign/Symptom is equal to:

Sensitivity * Probability of Disease / Probability of Sign/Symptom

Notation is much easier once you can follow it:

P(+D | +S) = P(+S | +D) * P(D) / P(S) |

There is a difference between Bayes and the metrics of diagnostic accuracy that must be kept in mind and is what keeps us from using the diagnostic metrics from making claims such as: “Because the sensitive is .9, the probability of having the Disease when having the Sign/Symptom is 90%.” (not true!) It is that Sensitivity in it’s raw form: P(+S | +D) is not giving is the probability of Disease (cause) with the observation of the Sign/Symptom (effect). It is giving us the Probability of the Sign/Symptom (effect) when we KNOW someone has the Disease (cause). To convert from a Cause to Effect to an inverse probability of observe Effect and infer Cause (abduction) we must multiply by the probability of the disease, and divide by the probability of the symptom (based on our prior knowledge). The Likelihood ratios are better for sure, but they still do not tell you the probability of the disease given the symptom. |

Going back to the the simple DAG:

This DAG denotes: P(S | D)(recall that P(S | D) refers to all possible instantiations) and has partly to do with the underlying mechanisms, whether D is a definite cause or a possible cause. But it also has to do with whether there are other factors (variables) involved, for example, exacerbating factors, definite eliminating factors, or possible eliminating factors. Keeping things general, we can modify the causal structure to include other possible factors (sets of possible factors are depicted as “U”: |

What we learn from the graphical model, that is still an abstraction of reality but perhaps a more realistic model than the original, is that the true: P(S | D) is really: P(S | D, U1, U3, U4), and that U1 is independent of U3 and U4; but that U3 and U4 are not independent of one another based on this graphic as U3 can cause D, which can cause U4. In other words, the graphic reveals that the situation is MUCH MORE COMPLEX than the original assumptions of the traditional diagnostic metric approach (using sensitivity, specificity and likelihood ratios) just do not capture. Not only is the complexity not captured, but I believe that is it not hard to imagine how a Bayesian Network (DAG, graphical model) approach is much more helpful to the overall goal of differential diagnosis. Don’t worry - we can still calculate probability distributions and have a metric for our diagnostic accuracy - it would just be based on a more accurate model of reality - that is it’s causal structure. As I quoted in the post on Models: “All models are wrong, some models are useful” George Box |