Letter 4: PROBABILITIES AND ODDS IN DIAGNOSIS

Definitions

"Probability" is defined as the likelihood of an Event or Index outcome expressed as a percentage of all outcomes. It includes the Index outcome itself plus the other Events or outcomes that may occur. The Event or Index outcome is counted twice and included in the numerator as well as in the denominator.

For example: In tossing a coin 100 times, the head may come out 50 times as well as the tail. The probability of the tail coming out is 50 out of 100 or 50%. The 100 times the coin is tossed includes both the Index outcome (head) itself as well as the other outcomes (tail).

"Odds" is another means of expressing likelihood. It compares the Index outcome on one side and all other outcomes (excluding the Index outcome) on the other side to come up with a ratio of the two. In this situation, odds count the Index outcome only once. Going back to the coin tossing exercise, odds would compare the 50 tails on one side with the 50 heads on the other side for a ratio of 50:50, or 1.

Using the concept of "odds" in medical decision analysis is important because most mathematical calculations are simpler when expressed in “odds form.” Using odds allows clinicians to do faster mental calculations, which would be difficult using probabilities because of its complex formulas.

To compare odds and probability: odds of 1 are the same as 50% probability. Any probability less than 50% has odds between 0 and 1. Probabilities higher than 50% can go from odds just over 1 up to infinity. Odds of 0.5 is the same as probability of around 33%, and odds of 1.5 is the same as probability of 60%. The odds of 99 are roughly the same as 99% probability. Above odds of 99 or probability of 99% is not worth considering because there are no odds equivalent to 100% probability.

Using Odds in Diagnosis

In the practice of medicine, it makes more sense to frame the thinking process in favor of "likelihood" rather than "certainty." To pursue certainty is impractical, extremely costly, and oftentimes dangerous to patients. Medical practice is a field of uncertainty as opposed to engineering or accounting where outcomes are accurately predicted. In medicine, outcomes are not assured -- the best way to predict an outcome is in terms of odds.

Here's an illustration of how odds and probabilities can be applied in diagnosis.

Let us mix 100 green and red marbles in a bucket. A win is if you pick a green marble. There are 51 green and 49 red marbles. In this scenario, you should bet on green marbles for an overall win, knowing that you will not win all the time. If there were 85 green and 15 red, you would still bet on green as the winner, of course, but you do not need to know beyond 51:49 to bet on green to win. So, it does not make sense to put more effort or cost to find out if it was 51:49 versus 85:15 in order to decide on which color to bet on.

This concept can be applied when getting a clinical history. One clinician can ask 10 questions to arrive at a reasonable diagnostic probability, but these 10 questions may not give him a 0.9 probability of hitting the mark. If another clinician could reduce the questions to five and arrive at a reasonable probability of just over 0.5, he would be as good as, if not better than, the clinician who asks 10 questions.

When asking questions it makes sense to have a structure for a particular chief complaint (CC).  For this particular CC, the same five or more questions can be asked every time.

Let us take the example of a 10-year-old boy with a CC of “Coughing.”

These are the structured questions that I ask all the time:

1. Is the cough longer than two weeks?

2. Is there no fever?

3. Is the coughing more after playing, running, exercise, or laughing?

4. Did he ever have bronchitis or pneumonia before?

5. Is there asthma, allergy, or hay fever in the family?

If there are at least three "Yes" answers to these five questions, I put asthma as the first probability. These five questions are the green marbles. The chance of being right is high and only five questions were asked. More questions can be asked, but the likelihood of asthma as the true diagnosis will not increase much.

Once a diagnostic likelihood of over 0.5 is made, after just a few questions, asking more questions will reduce the efficiency and effectiveness of the clinician. It is difficult to attain certainty with clinical history. As long as more than a likelihood of 0.5 in the clinical diagnosis is achieved, the clinician can be satisfied and treat depending on the severity of the target disease, the values of the patients or parents, or the “standard of practice” in the community.

Using Odds in Diagnostic Tests

Once the clinician has an estimate of the most likely diagnosis based on clinical history and physical findings, he decides on whether to treat or not. This decision depends on his estimate of the severity of the target condition or disease, the values of the patient or parent, and the standard of practice in the community.

If the clinician is considering an illness that is very serious like meningitis or a malignancy, he is obligated to do some laboratory tests before starting any form of treatment. However, if, from his estimate, the target condition is not serious at all and there is enough time to wait, he has two decisions to choose from: to observe or to do more laboratory tests. 

In most instances, observing and doing no tests is the better approach since the patient might get better while the clinician is observing. Or the patient may develop more symptoms that the clinician can use to request less or narrowed-down tests to arrive at a definite diagnosis.

Let us take this example of a 10-year-old girl who had belly pain for about six weeks. The clinician had done his history and physical examination and concluded that there were no serious conditions that require immediate attention. About two weeks later, the mother called and said that there is blood in the stool. This was the first time this symptom was noticed.

The clinician requested the mother to bring her daughter in again for follow up. After doing the routine history and pertinent physical examination, including a rectal exam, the clinician considered Crohn’s Disease as one of the top three possibilities in his differential diagnosis. He requested for CBC, ESR, and some blood tests that are sensitive to diagnose Crohn’s Disease. If any of these were positive, then the clinician would consider endoscopy for biopsy.

After the decision to observe the child after the initial visit, new symptoms appeared that helped the clinician limit the list of possibilities. Suppose that during the initial visit, he requested for a battery of tests, there would have been a good chance that all of the tests would be normal, which would not help at all. This would probably even complicate the problem if false positive tests, like a positive occult blood in the stool, or a WBC of 18,000 with normal hemoglobin and platelet, appeared in the results.

Now let's say that the patient or the parent wants the clinician to do some tests after he decided to just observe. There are two ways the clinician can deal with it. 

One is, he can explain to the parent, “At this time, with all the information from the story of your daughter and the physical examination, my best bet is there is no serious condition that we should worry about. But if the belly pain persists, gets worse, or there are new symptoms like blood in the stools, you can come back any time and we will immediately do some tests to figure out the cause.”

Another approach is to say: “We can do some tests now. However, sometimes certain tests complicate the problem. We might get a false positive result, which will make me pursue or order more unnecessary tests that could hurt your child even more, as compared to if we first observe for more symptoms. There's also the possibility that your child will get better while we are observing.”

End Letter 4 of 6