Diagnostic Reasoning: Likelihood Ratios in Every Day Life

I was sitting in morning report the other day listening to a case of a patient that presented with fatigue, generalized weakness, and weight loss over the period of six months. The work-up was pointing to the disease sarcoidosis. Sarcoidosis is a rare inflammatory disease characterized by diffuse granulomas. Granulomas are clusters of active immune system cells that cause dysfunction in whatever tissue they invade.

Besides having the common symptoms, the patient had several other key findings as well. There was bilateral hilar lymphadenopathy on chest CT, his calcium was high, and an angiotensin converting enzyme level was elevated.  All common in sarcoidosis.

Sarcoidosis is confirmed by biopsy of an enlarged lymph node. The issue that arose was the specialist did not want to commit to treatment until confirmation via biopsy. This would require an invasive procedure known as a bronchoscopy. The patient was to be discharged to follow-up as an outpatient to schedule the procedure, which could take weeks, without a confident diagnosis or treatment plan.

This reminded me of the concepts testing thresholds, treatment thresholds, and the use of likelihood ratios in diagnostic uncertainty.

The concept of testing thresholds and treatment thresholds can be understood as follows:

·      In every case, you start with a 0% degree of certainty.

·      As a physician interviews and examines a patient, a potential diagnosis begins to reveal itself.

·      When a threshold for a specific diagnosis is passed, the physician will order tests to help rule in or rule out the diagnosis. This is the testing threshold.

·      As data from tests come in, certainty either increases or decreases, when there is enough certainty, a treatment is chosen, this is crossing the treatment threshold.

Concept depicted in figure below.

Pre-test probability is essentially the prevalence of the disease in the general population, or the prevalence of the disease in patients with certain symptoms.

In practice, testing and treatment thresholds are often a gut feeling. However, calculation of likelihood ratios and their effect on probability help quantify certainty.

We’ll use this case as an example.

Sarcoidosis is present in 0.07% of the population. It’s a rare disease. The prevalence of 0.07% can be used as your pretest probability (starting line). However, this patient had symptoms of generalized weakness, fatigue, and weight loss. The percentage of patients with those complaints that have sarcoidosis we’ll estimate around 5%.  I’ll use this pre-test probability for simplicity.  

A likelihood ratio is a measurement that either amplifies or dampens probability. If we order a test, and the test is positive, that increases the likelihood of the disease. If the test is negative, it decreases the likelihood. We can use historical population data to estimate these ratios.

This patient had several key findings: 1) An elevated angiotensive converting enzyme level (ACE), 2) hilar lymphadenopathy, and 3) hypercalcemia. (Technicality- In the case above the patient had hypercalcemia, for sarcoidosis you would check for elevated calcium levels in the urine.)

A high angiotensive converting enzyme has a positive likelihood ratio 3.5-5.0 for sarcoidosis . We can use a likelihood ratio nomogram to get our posttest probability- 10%. The positive test doesn’t help us much. It moves our pre-test probability of 5% to a posttest probability of 10%.  (see 1st nomogram)

Moving on to the 2^nd finding or hilar lymphadenopathy.  Hilar lymphadenopathy has a positive likelihood ratio that is high. It is specific for the disease meaning a high percentage of patients with the disease will have this finding. Although no exact likelihood ratio is known, since it is so specific, we’ll estimate the ratio to be > 10. Starting at our new pre-test probability of 10% we move to a post probability of 50%. (see 2^nd nomogram.)

Moving on to the third finding of hypercalcemia.  Hypercalcemia is found in 20-30% of patients with sarcoidosis. Not as common as hilar lymphadenopathy but associated with the disease. Hypercalciuria has a positive likelihood ratio of between 4-8.  For simplicity, we’ll call it 5. Starting at our new pre-test probability of 50% we move to a posttest probability of 90%. (See third nomogram)

You have a diagnostic certainty of 90%. The treatment for sarcoidosis is prednisone. This is a corticosteroid. It is low cost, and in the short-term carries moderate, modifiable, risks. Long term, medications like prednisone can be problematic. Here you are, a high diagnostic certainty and a treatment that is low cost, easy to administer, and in the short term, tolerable.

My term for this is probability stacking.

Yet, the specialist still wants a biopsy? What are you to do?

There are many reasons why the specialist may want a biopsy. Maybe they don’t believe what they see.  Maybe they were trained to treat only after biopsy confirmation? Maybe they’re worried your treatment will lower the diagnostic yield of the biopsy? It could be for a number of reasons. I would argue, you don’t need it. You’ve excluded other potential causes, and you’re carrying a high degree of certainty. Go for it. Help this poor patient out.

You can use probability stacking in everyday life. Imagine walking into the living room to a beautiful mural of purple and blue scribbles on the wall. There are markers with caps off at it’s base. You find you’re six-year-old sipping on some juice watching TV. Oddly, there are blue and purple marks all over his hands.  Do you need to investigate further? Probably not.

That’s likelihood ratios in everyday life.