Pre- and post-test probability

Pre-test probability and post-test probability are the probabilities of a having condition before and after a diagnostic test. Post-test probability, in turn, can be positive or negative, depending on whether the test falls out as a positive test or a negative test, respectively.

Estimation from prevalence and predictive values

The pre-test probability (the probability of a having a target condition before a diagnostic test) is numerically equal to the prevalence of the condition in the population, assuming that no other risk factor is known about the individual being tested that would result in another pre-test probability than the general population. In such cases, it can be calculated as follows (with dashes removed to avoid mixup with the minus sign):

Pretest probability = (True positive + False negative) / Total sample

Dealing with individuals with other factors is described in separate section below

Also, with the same assumptiopn, the positive post-test probability (the probability of having the target condition if the test falls out positive), is numerically equal to the positive predictive value, and the negative post-test probability (the probability of having the target condition if the test falls out negative) is numerically equal to the negative predictive value, again assuming that the individual being tested does not have any other risk factors than the population used to establish the positive and negative predictive values of the test.

Dealing with individuals with other factors is described in separate section below

Estimation from likelihood ratio

With only pre-test probability and likelihood ratio given, then, the post-test probabilities can be calculated by the following three steps:

• Pretest odds = (Pretest probability / (1 - Pretest probability)
• Posttest odds = Pretest odds * Likelihood ratio

In equation above, positive post-test probability is calculated using the likelihood ratio positive, and the negative post-test probability is calculated using the likelihood ratio negative.

• Posttest probability = Posttest odds / (Posttest odds + 1)

Example

A worked example

A diagnostic test with sensitivity 67% and specificity 91% is applied to 2030 people to look for a disorder with a population prevalence of 1.48%

		Fecal occult blood screen test outcome
	Total population (pop.) = 2030	Test outcome positive	Test outcome negative	Accuracy (ACC) = (TP + TN) / pop. = (20 + 1820) / 2030 ≈ 90.64%	F1 score = 2 × ⁠precision × recall/precision + recall⁠ ≈ 0.174
Patients with bowel cancer (as confirmed on endoscopy)	Actual condition positive (AP) = 30 (2030 × 1.48%)	True positive (TP) = 20 (2030 × 1.48% × 67%)	False negative (FN) = 10 (2030 × 1.48% × (100% − 67%))	True positive rate (TPR), recall, sensitivity = TP / AP = 20 / 30 ≈ 66.7%	False negative rate (FNR), miss rate = FN / AP = 10 / 30 ≈ 33.3%
	Actual condition negative (AN) = 2000 (2030 × (100% − 1.48%))	False positive (FP) = 180 (2030 × (100% − 1.48%) × (100% − 91%))	True negative (TN) = 1820 (2030 × (100% − 1.48%) × 91%)	False positive rate (FPR), fall-out, probability of false alarm = FP / AN = 180 / 2000 = 9.0%	Specificity, selectivity, true negative rate (TNR) = TN / AN = 1820 / 2000 = 91%
	Prevalence = AP / pop. = 30 / 2030 ≈ 1.48%	Positive predictive value (PPV), precision = TP / (TP + FP) = 20 / (20 + 180) = 10%	False omission rate (FOR) = FN / (FN + TN) = 10 / (10 + 1820) ≈ 0.55%	Positive likelihood ratio (LR+) = ⁠TPR/FPR⁠ = (20 / 30) / (180 / 2000) ≈ 7.41	Negative likelihood ratio (LR−) = ⁠FNR/TNR⁠ = (10 / 30) / (1820 / 2000) ≈ 0.366
		False discovery rate (FDR) = FP / (TP + FP) = 180 / (20 + 180) = 90.0%	Negative predictive value (NPV) = TN / (FN + TN) = 1820 / (10 + 1820) ≈ 99.45%	Diagnostic odds ratio (DOR) = ⁠LR+/LR−⁠ ≈ 20.2

Related calculations

• False positive rate (α) = type I error = 1 − specificity = FP / (FP + TN) = 180 / (180 + 1820) = 9%
• False negative rate (β) = type II error = 1 − sensitivity = FN / (TP + FN) = 10 / (20 + 10) ≈ 33%
• Power = sensitivity = 1 − β
• Positive likelihood ratio = sensitivity / (1 − specificity) ≈ 0.67 / (1 − 0.91) ≈ 7.4
• Negative likelihood ratio = (1 − sensitivity) / specificity ≈ (1 − 0.67) / 0.91 ≈ 0.37
• Prevalence threshold = $PT={\frac {{\sqrt {TPR(-TNR+1)}}+TNR-1}{(TPR+TNR-1)}}$ ≈ 0.2686 ≈ 26.9%

This hypothetical screening test (fecal occult blood test) correctly identified two-thirds (66.7%) of patients with colorectal cancer. Unfortunately, factoring in prevalence rates reveals that this hypothetical test has a high false positive rate, and it does not reliably identify colorectal cancer in the overall population of asymptomatic people (PPV = 10%).

On the other hand, this hypothetical test demonstrates very accurate detection of cancer-free individuals (NPV ≈ 99.5%). Therefore, when used for routine colorectal cancer screening with asymptomatic adults, a negative result supplies important data for the patient and doctor, such as reassuring patients worried about developing colorectal cancer. In this example, the positive pre-test probability is calculated as:

• Pretest probability = (2 + 1) / 203 = 0,0148
• Pretest odds = 0,0148 / (1 - 0,0148) =0,015
• Posttest odds = 0,015 * 7,4 = 0,111
• Posttest probability = 0,111 / (0,111 + 1) =0,1 or 10%

As demonstrated, the positive post-test probability is numerically equal to the positive predictive value, equivalent to negative post-test probability being numerically equal to negative predictive value.

Other individual factors

If the individual being tested has other factors that influence the probability of having the target condition of the test, then the prevalence in the population is not completely accurate in representing the pre-test probability, and the predictive value (whether positive or negative) is not completely accurate in representing the positive pre-test probability. Nevertheless, likelihood ratio, being calculated from sensitivity and specificity of the test, does not depend on prevalence in the population, and is less affected by the fact that the individual at hand belongs to a group with increased or decreased prevalence, such as, theoretically, a group of people with exactly the same risk factors as the individual at hand. Thus, post-test probability, as estimated from the likelihood ratio and pre-test probability, is more accurate than the positive predictive value of the test when the tested individual has a different pre-test probability than what is the prevalence of that condition in the population.

However, post-test probability, as estimated from the likelihood ratio and pre-test probability, should still be handled with caution in individuals with other risk factors than the general population, because such factors may also influence the test itself in unpredictive ways, still causing inaccurate results. Preferably, a large group of equivalent individuals should be studied, in order to establish separate positive and negative predictive values for use of the test in such individuals.