17.3 - Estimating the Probability of Disease

Sensitivity and specificity describe the accuracy of a test. In a clinical setting, we do not know who has the disease and who does not - that is why diagnostic tests are used. We would like to be able to estimate the probability of disease based on the outcome of one or more diagnostic tests. The following measures address this idea.

Prevalence is the probability of having the disease, also called the prior probability of having the disease. It is estimated from the sample as \(\dfrac{\left(a+c\right)}{\left(a+b+c+d\right)}\).

Positive Predictive Value (PV+) is the probability of disease in an individual with a positive test result. It is estimated as \(\dfrac{a}{\left(a+b\right)}\).

Negative Predictive Value (PV - ) is the probability of not having the disease when the test result is negative. It is estimated as as \(\dfrac{d}{\left(c+d\right)}\).

In the FNA study of 114 women with nonpalpable masses and abnormal mammograms,

\(prevalence = \dfrac{15}{114} = 0.13\)

\(PV+ = \dfrac{14}{\left(14+8\right)} = 0.64\)

\(PV - = \dfrac{91}{\left(1+91\right)} = 0.99\)

Thus, a woman's prior probability of having the disease is 0.13 and is modified to 0.64 if she has a positive test result. A women's prior probability of not having the disease is 0.87 and is modified to 0.99 if she has a negative test result.

If the disease under study is rare, the investigator may decide to invoke a case-control design for evaluating the diagnostic test, e.g., recruit 50 patients with the disease and 50 controls. Obviously, prevalence cannot be estimated from a case-control study because it does not represent a random sample from the general population.

Predictive values allow us to determine the usefulness of a test and they vary with the sensitivity and specificity of a test. If all other characteristics held constant, then:

  1. as sensitivity of a test increases, PV - increases and
  2. as specificity of a test increases, PV+ increases.

Predictive values vary with the prevalence of the disease in the population being tested or the pre-test probability of disease in a given individual.

Sensitivity, specificity, and prevalence can be used in a clinical setting to estimate post-test probabilities (predictive values), even though physicians work with one patient at a time, not entire populations of patients. Three pieces of information are necessary prior to performing the test, namely, (1) either the prevalence of the disease or the prior probability of disease, (2) sensitivity, and (3) specificity.

Then, formulae for PV+ and PV- are:

\(PV+ = \dfrac{\text{Prevalence}\times\text{Sensitivity}}{(\text{Prevalence}\times\text{Sensitivity})+\left\{(1-\text{Prevalence})\times (1-\text{Specificity}) \right\}}\)

\(PV- = \dfrac{(1-\text{Prevalence})\times\text{Specificity}}{\left\{(1-\text{Prevalence})\times\text{Specificity})\right\}+\left\{\text{Prevalence}\times (1-\text{Sensitivity}) \right\}}\)

Although PV+ = 14/(14+8) = 0.64 and PV - = 91/(1+91) = 0.99 can be calculated directly from the 2 × 2 data table because the women constituted a random sample, the above formulae yield the same results:

\(PV+ = \dfrac{(0.13)(0.93)}{{(0.13)(0.93) + (0.87)(0.08)}} = 0.64\)

\(PV- = \dfrac{(0.87)(0.92)}{{(0.87)(0.92) + (0.13)(0.07)}} = 0.99\)

The following example is taken from Sackett et al (1985, Clinical Epidemiology ). Suppose a patient with the following characteristics visits a physician:

  • 45-year-old man
  • ambulatory with episodic chest pain
  • no coronary risk factors except smoking one pack of cigarettes per day
  • 3-week history of substernal and precordial pain - stabbing and fleeting
  • physical exam shows a single costochondral junction that is slightly tender, but does not reproduce the patient's pain

From this information, the physician estimates an intermediate pre-test (prior) probability of 60% that this patient has significant coronary artery narrowing.

The physician is not sure whether the patient should undergo an exercise electrocardiogram (ECG). How useful would this test be for this patient?

Suppose it is known from the literature that the sensitivity and specificity of the exercise ECG in coronary artery stenosis (as compared to the gold standard of coronary arteriography) are 60% and 91%, respectively.


\(PV+ = \dfrac{(0.6)(0.6)}{{(0.6)(0.6) + (0.4)(0.09)}} = 0.91\)

\(PV - = \dfrac{(0.4)(0.91)}{{(0.4)(0.91) + (0.6)(0.4)}} = 0.60\)

An additional test characteristic reported in the medical literature is the likelihood ratio, which is the probability of a particular test result (+ or - ) in patients with the disease divided by the probability of the result in patients without the disease. There exists one likelihood ratio for a positive test (LR+) and one for a negative test (LR - ). Likelihood ratios express how many times more (or less) likely the test result is found in diseased versus non-diseased individuals:

\(LR+ = \dfrac{\text{Sensitivity}}{\left(1 - Specificity\right)}\)

\(LR - = \dfrac{\left(1 - \text{Sensitivity}\right)}{\text{Specificity}}\)

From the FNA study in 114 women with nonpalpable masses and abnormal mammograms, LR+ = 0.933/0.081 = 11.52 and LR - = 0.067/0.919 = 0.07. Thus, positive FNA results are 11.52 times more likely in women with cancer as compared to those without, and negative FNA results are .07 times as likely in women with cancer as compared to those without.