The Good Doctor

Logic and reason

Patrick J. Hurley, A Concise Introduction to Logic, 12th Edn.

Gestalt, Odds, Nuance, and Rhyme

Statistical measures

The Normal Distribution

Natural frequencies aid in the communication of probabilities based on decision-tree analysis, particularly in matters of health and disease (Girgerenzer, 45).¹

“M” = mammography, “U” = ultrasound; “+” and “−” denote positive and negative test results, respectively [4]

Outcomes are rarely (if ever) truly mutually exclusive. This understanding leads to conditional probabilities. Conditional probabilities can be calculated using Bayes Theorem, in which odds are the ratio of chance of conditional events. Odds is a good way to numerically represent conditional probabilities, because it intrinsically reflects that there is an inherent relationship (ratio) between apparently separate events.

The numerator (or total population) in risk is inclusive of the index case but that is not so in calculating the odds, which instead directly reflects this ratio of case: non-case (rather than case: total population). The ratio of 1:9 above, for instance, intrinsically reflects this inherent relationship between case and non-case.

One-in-2 (one-half) events: 1:1 or 50/50 chance

An odds ratio of 1 (1:1) means that the odds in favour of an event = 1, or the probability of success is 50%; and the odds against = 1, or the probability of failure also is 50%.

An infinitesimally low odds ratio of 0:1 describes a situation where the odds in favour of an event = 0, or a 0% probability of success is 0%; while the odds against are infinite, and the probability of failure is 100%. Conversely, an infinite odds ratio of 1:0 describes an event in which the odds in favour are infinite, i.e. the probability of success is 100%; and the odds against = 0; or a 0% probability of failure.

Two-thirds and one-third events: 2:1 and 1:2 odds

An odds ratio of 2:1 describes an event where the odds in favour are 2, i.e. the probability of success is 67%; and the odds against = 0.5; the probability of failure is 33%. Conversely, an odds ratio of 1:2 implies that the odds in favour of en event is 0.5, i.e. a 33% probability of success; while the odds against is 2, a 67% probability of failure.

Three-quarter and one-quarter events: 3:1 and 1:3 odds

An odds ratio of 3:1 described the situation where the odds in favour = 3, or there is a 75% (3/4) probability of success; while the odds against = 0.33, or there is a 25% probability of failure. An odds ratio of 1:3, on the other hand, implies that the odds in favour are 0.33, i.e. the probability of success is 25%; and the odds against = 3, or the probability of failure = 75%.

Four-fifth and one-fifth events: 4:1 and 1:4 odds

An odds ratio of 4:1 describes the situation where the odds in favour = 4, or the probability of success is 80%. An odds against = 0.25, predicts a probability of failure of 20%. Conversely, an odds ratio of 1:4 describes a situation where the odds in favour are 0.25, i.e. the probability of success is only 20%; and the odds against = 4, or the probability of failure of 80%.

Likewise, calculations for the following odds ratios (OR):

OR 9:1 means: odds in favour = 9 (probability of success = 90%); odds against = 0.1 (probability of failure = 10%)
OR 10:1 means: odds in favour = 10 (probability of success = 90.90%); odds against = 0.1 (probability of failure = 9.09%)
OR 99:1 means: odds in favour = 99 (probability of success = 99%); odds against = 0.01 (probability of failure = 1%)
OR 100:1 means: odds in favour = 100 (probability of success = 99.0099%); odds against = 0.01 (probability of failure = 0.90%).

The Directed Approach: Process of Elimination

A useful guide to think about the process of elimination often used by experienced clinicians to arrive at a workable diagnosis by directed questioning within the confines of limited interview time is the CoreIM podcast, A 45F with Persistent Bleeding: 20Qs Core IM Team vs. Expert

References

Gigerenzer, G. Reckoning with risk: Learning to live with uncertainty. London: Penguin Books Ltd., 2002.
Hoffrage U, Krauss S, Martignon L, Gergerenzer G. “Natural frequencies improve Bayesian reasoning in simple and complex inference tasks.” Frontiers in Psychology. http://journal.frontiersin.org/article/10.3389/fpsyg.2015.01473/full.

Medical Decision Making

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