One of the most commonly used strategies for medical decision making mirrors the scientific method of hypothesis generation followed by hypothesis testing. Diagnostic hypotheses are accepted or rejected based on testing.

## Hypothesis Generation

Hypothesis generation involves the identification of the main diagnostic possibilities (differential diagnosis) that might account for the patient’s clinical problem. The patient’s chief complaint (eg, chest pain) and basic demographic data (age, sex, race) are the starting points for the differential diagnosis, which is usually generated by pattern recognition. Each element on the list of possibilities is ideally assigned an estimated probability, or likelihood, of its being the correct diagnosis (pre-test probability—for an example, see table Hypothetical Differential Diagnosis and Pre-Test and Post-Test Probabilities).

Clinicians often use vague terms such as “highly likely,” “improbable,” and “cannot rule out” to describe the likelihood of disease. Both clinicians and patients often misinterpret such semiquantitative terms; explicit statistical terminology should be used instead when available. Mathematical computations assist clinical decision making and, even when exact numbers are unavailable, can better define clinical probabilities and narrow the list of hypothetical diseases further.

### Probability and odds

The **probability** of a disease (or event) occurring in a patient whose clinical information is unknown is the frequency with which that disease or event occurs in a population. Probabilities range from 0.0 (impossible) to 1.0 (certain) and are often expressed as percentages (from 0 to 100). A disease that occurs in 2 of 10 patients has a probability of 2/10 (0.2 or 20%). Rounding very small probabilities to 0, thus excluding all possibility of disease (sometimes done in implicit clinical reasoning), can lead to erroneous conclusions when quantitative methods are used.

**Odds** represent the ratio of affected to unaffected patients (ie, the ratio of disease to no disease). Thus, a disease that occurs in 2 of 10 patients (probability of 2/10) has odds of 2/8 (0.25, often expressed as 1 to 4). Odds (Ω) and probabilities (p) can be converted one to the other, as in Ω= p/(1 − p) or p =Ω/(1 +Ω).

## Hypothesis Testing

The initial differential diagnosis based on chief complaint and demographics is usually very large, so the clinician first tests the hypothetical possibilities during the history and physical examination, asking questions or doing specific examinations that support or refute a suspected diagnosis. For instance, in a patient with chest pain, a history of leg pain and a swollen, tender leg detected during examination increases the probability of pulmonary embolism.

When the history and physical examination form a clear-cut pattern, a presumptive diagnosis is made. Diagnostic testing is used when uncertainties persist after the history and physical examination, particularly when the diseases remaining under consideration are serious or have dangerous or costly treatment. Test results further modify the probabilities of different diagnoses (post-test probability). For example, the table Hypothetical Differential Diagnosis and Pre-Test and Post-Test Probabilities shows how the additional findings that the hypothetical patient had leg pain and swelling and a normal ECG and chest x-ray modify diagnostic probabilities—the probability of acute coronary syndrome, dissecting aneurysm, and pneumothorax decreases, and the probability of pulmonary embolism increases. These changes in probability may lead to additional testing (in this example, probably chest CT angiography) that further modifies post-test probability (see table) and, in some cases, confirms or refutes a diagnosis.

It may seem intuitive that the sum of probabilities of all diagnostic possibilities should equal nearly 100% and that a single diagnosis can be derived from a complex array of symptoms and signs. However, applying this principle that the best explanation for a complex situation involves a single cause (often referred to as Occam's razor) can lead clinicians astray. Rigid application of this principle discounts the possibility that a patient may have more than one active disease. For example, a dyspneic patient with known COPD may be presumed to be having an exacerbation of COPD but actually is also suffering from a pulmonary embolism.

### Hypothetical Differential Diagnosis and Pre-Test and Post-Test Probabilities for a 50-Yr-Old Hypertensive, Diabetic Cigarette Smoker With Chest Pain

## Probability Estimations and the Testing Threshold

Even when diagnosis is uncertain, testing is not always useful. A test should be done only if its results will affect management. When disease pre-test probability is above a certain threshold, treatment is warranted (treatment threshold) and testing is not indicated.

Below the treatment threshold, testing is indicated only when a positive test result would raise the post-test probability *above*the treatment threshold. The lowest pre-test probability at which this can occur depends on test characteristics and is termed the testing threshold. The testing threshold is discussed in greater detail elsewhere.

## Probability Estimations and the Treatment Threshold

The disease probability at and above which treatment is given and no further testing is warranted is termed the treatment threshold (TT).

The above hypothetical example of a patient with chest pain converged on a near-certain diagnosis (98% probability). When diagnosis of a disease is certain, the decision to treat is a straightforward determination of whether there is a benefit of treatment (compared with no treatment, and taking into account adverse effects of treatment). When the diagnosis has some degree of uncertainty, as is almost always the case, the decision to treat also must balance the benefit of treating a sick person against the risk of erroneously treating a well person or a person with a different disorder; benefit and risk encompass both financial and medical consequences. This balance must take into account both the likelihood of disease and the magnitude of the benefit and risk. This balance determines where the clinician sets the treatment threshold.

Conceptually, if the benefit of treatment is very high and the risk is very low (as when giving a safe antibiotic to a patient with diabetes who possibly has a life-threatening infection), clinicians tend to accept high diagnostic uncertainty and might initiate treatment even if probability of infection is fairly low (eg, 30%—see figure Variation of treatment threshold (TT) with risk of treatment). However, when the risk of treatment is very high (as when doing a pneumonectomy for possible lung cancer), clinicians want to be extremely sure of the diagnosis and might recommend treatment only when the probability of cancer is very high, perhaps > 95% (see figure). Note that the treatment threshold does not necessarily correspond to the probability at which a disease might be considered confirmed or ruled in. It is simply the point at which the risk of not treating is greater than the risk of treating.

Quantitatively, the treatment threshold can be described as the point at which probability of disease (p) times benefit of treating a person with disease (B) equals probability of no disease (1 − p) times risk of treating a person without disease (R). Thus, at the treatment threshold

p × B = (1 − p) × R

Solving for p, this equation reduces to

p = R/(B +R)

From this equation, it is apparent that if B (benefit) and R (risk) are the same, the treatment threshold becomes 1/(1 + 1) = 0.5, which means that when the probability of disease is > 50%, clinicians would treat, and when probability is < 50%, clinicians would not treat.

For a clinical example, a patient with chest pain can be considered. How high should the clinical likelihood of acute MI be before thrombolytic therapy should be given, assuming the only risk considered is short-term mortality? If it is postulated (for illustration) that mortality due to intracranial hemorrhage with thrombolytic therapy is 1%, then 1% is R, the fatality rate of mistakenly treating a patient who does not have an MI. If net mortality in patients with MI is decreased by 3% with thrombolytic therapy, then 3% is B. Then, treatment threshold is 1/(3 + 1), or 25%; thus, treatment should be given if the probability of acute MI is > 25%.

Alternatively, the treatment threshold equation can be rearranged to show that the treatment threshold is the point at which the odds of disease p/(1 −p) equal the risk:benefit ratio (R/B). The same numerical result is obtained as in the previously described example, with the treatment threshold occurring at the odds of the risk:benefit ratio (1/3); 1/3 odds corresponds to the previously obtained probability of 25% (see probability and odds).