How Well Do We Understand Probabilities In Medicine?

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Posted on: July 21, 2016

Last Updated: June 25, 2024

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In his bestselling book, Fooled by Randomness (which was named one of the smartest books of all time by Fortune magazine), Nassim Nicholas Taleb discusses a question that was posed to a group of medical doctors.

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A test of a disease presents a rate of 5% false positives. The disease strikes 1/1000 of the population. People are tested at random, regardless of whether they are suspected of having the disease. A patient’s test is positive. What is the probability of the patient being stricken with the disease?

Most doctors answered 95%. Less than one in 5 professionals answered the question correctly.

SOLVING THE PROBLEM

Probability in MedicineIn order to solve Taleb’s question, I will break this problem down into steps using tables. These tables are helpful in understanding a range of similar probability questions.

In solving questions like this, it helps to draw a 2×2 table.

The 2×2 table will be one that is used in appraisal of diagnostic tests.

2 by 2 table for diagnostic test

Key points in the 2×2 table are:

  1. True disease is always on the top
  2. Test is on the left
  3. If comparing with a gold standard test, then gold standard is always on the top

WHAT DOES THE BOX LOOK LIKE FOR THE PROBLEM?

We will now try to fill in the boxes:

  1. Total number of people with the disease is 1
  2. 999 people do not have the disease
  3. Therefore, True Positive = 1 and False Negative = 0
  4. False positives = 0.05×999 = 49.95 = 50 (5% false positives)
  5. TN = 1-FP = 949

The box now looks like this:

Sensitivity, Specificty in medicine

DETERMINING THE PROBABILITY

In order to determine the probability of the patient having the disease, we need to calculate the positive predictive value (PPV) which is different from sensitivity or specificity.

PPV = TP/ TP+FP = 1/51 = 2%

Therefore, there is a 2% probability that the patient has a disease.

TALEB'S DESCRIPTION OF SOLUTION

I will simplify the answer using the frequency approach. Assume no false negatives. Consider that out of 1000 patients who are administered the test, one will be expected to be afflicted with the disease.

Out of a population of the remaining 999 healthy patients, the test will identify about 50 with the disease (it is 95%) accurate.

The correct answer should be that the probability of being afflicted with disease for someone selected at random who presented a positive test is the following ratio: Number of afflicted persons / number of true positives and false positives. Here, 1 in 51 (2%).

IMPLICATIONS

Why is the probability so low, even though the test has only 5% false positives?

The answer is in the low incidence rate of the disease. In rare outcomes, the predictive value of any instrument drops.

Think about applying this principle for suicide prediction where the rate of suicide is even lower. I will highlight the challenges of suicide prediction in another post.

TAKE HOME MESSAGES

  1. Sensitivity and specificity are functions of the test and do not change with prevalence, for example the test in Taleb’s question had a specificity (true negative rate) of 95%
  2. Positive and negative predictive values change with prevalence
  3. PPV and NPV are more important for the patient, as they tell the patient the probability of having or not having a disease

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References

Taleb, N. (2005). Fooled by randomness: The hidden role of chance in life and in the markets (Vol. 1). Random House Incorporated.

How Well Do We Understand Probabilities In Medicine?

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In his bestselling book, Fooled by Randomness (which was named one of the smartest books of all time by Fortune magazine), Nassim Nicholas Taleb discusses a question that was posed to a group of medical doctors.

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content Most doctors answered 95%. Less than one in 5 professionals answered the question correctly.

SOLVING THE PROBLEM

content Probability in MedicineIn order to solve Taleb’s question, I will break this problem down into steps using tables. These tables are helpful in understanding a range of similar probability questions. In solving questions like this, it helps to draw a 2x2 table. The 2x2 table will be one that is used in appraisal of diagnostic tests. 2 by 2 table for diagnostic test Key points in the 2x2 table are:

  1. True disease is always on the top
  2. Test is on the left
  3. If comparing with a gold standard test, then gold standard is always on the top

WHAT DOES THE BOX LOOK LIKE FOR THE PROBLEM?

content We will now try to fill in the boxes:

  1. Total number of people with the disease is 1
  2. 999 people do not have the disease
  3. Therefore, True Positive = 1 and False Negative = 0
  4. False positives = 0.05x999 = 49.95 = 50 (5% false positives)
  5. TN = 1-FP = 949

The box now looks like this: Sensitivity, Specificty in medicine

DETERMINING THE PROBABILITY

content In order to determine the probability of the patient having the disease, we need to calculate the positive predictive value (PPV) which is different from sensitivity or specificity. PPV = TP/ TP+FP = 1/51 = 2% Therefore, there is a 2% probability that the patient has a disease.

TALEB'S DESCRIPTION OF SOLUTION

content

IMPLICATIONS

content Why is the probability so low, even though the test has only 5% false positives? The answer is in the low incidence rate of the disease. In rare outcomes, the predictive value of any instrument drops. Think about applying this principle for suicide prediction where the rate of suicide is even lower. I will highlight the challenges of suicide prediction in another post.

TAKE HOME MESSAGES

content

  1. Sensitivity and specificity are functions of the test and do not change with prevalence, for example the test in Taleb’s question had a specificity (true negative rate) of 95%
  2. Positive and negative predictive values change with prevalence
  3. PPV and NPV are more important for the patient, as they tell the patient the probability of having or not having a disease

Get serious about psychiatry learning

Join Academy by Psych Scene to access over 40 hours of in-depth, high-quality psychiatry courses that earn CPD points/CME credits.

References

Taleb, N. (2005). Fooled by randomness: The hidden role of chance in life and in the markets (Vol. 1). Random House Incorporated.

About the Author

Dr. Sanil Rege

Dr. Sanil Rege is a Consultant Psychiatrist and founder of Psych Scene and Vita Healthcare. He currently practices on the Mornington Peninsula.

About the Author

Dr Sanil Rege

MBBS, MRCPsych, FRANZCP

Dr. Sanil Rege is a Consultant Psychiatrist and founder of Psych Scene and Vita Healthcare. He has dual psychiatry qualifications from the United Kingdom and Australia. He currently practices on the Mornington Peninsula. His focus on combining psychiatry with principles of entrepreneurship has uniquely enabled him to not only contribute to the academic world through his several publications but also add value to the real world by establishing two successful enterprises in a short span of 6 years. He was appointed Associate Professor of Psychiatry at a prestigious Australian University at the age of 32 but left the role to focus on his passion of entrepreneurship in psychiatry. Psych Scene was co-founded to enhance psychiatry education, and Vita Healthcare was to provide the highest quality mental health care to the public. He is passionate about learning from multiple disciplines (Medicine, Psychiatry, Neurosciences, Accounting, Entrepreneurship, Finance and Psychology) with the aim of adding value to the world. By taking on multiple roles of a clinician, entrepreneur, father, educator, investor and MBA student, he recognises that personal development is a journey that needs to touch others lives for the better. He lives by the motto “All the knowledge in the world is not found in one academic discipline” and driven by curiosity. Dr. Sanil Rege is a
 Fellow of the Royal Australian and New Zealand
 College of Psychiatrists and Member of the 
Royal College of Psychiatrists (UK). He has
practiced Psychiatry in the United Kingdom and throughout Australia. He has experience in the assessment
 and management of a broad range of psychiatric disorders, including psychosis, depression, anxiety, post-traumatic stress disorders, personality disorders, neuropsychiatric presentations and consultation-liaison psychiatry.  

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