Likelihood ratios in diagnostic testing

In evidence-based medicine, likelihood ratios are used for assessing the value of performing a diagnostic test. They use the sensitivity and specificity of the test to determine whether a test result usefully changes the probability that a condition (such as a disease state) exists. The first description of the use of likelihood ratios for decision rules was made at a symposium on information theory in 1954.[1] In medicine, likelihood ratios were introduced between 1975 and 1980.[2][3][4]

Not to be confused with Likelihood-ratio test.

Calculation

Two versions of the likelihood ratio exist, one for positive and one for negative test results. Respectively, they are known as the positive likelihood ratio (LR+, likelihood ratio positive, likelihood ratio for positive results) and negative likelihood ratio (LR–, likelihood ratio negative, likelihood ratio for negative results).

The positive likelihood ratio is calculated as

${\displaystyle {\text{LR}}+={\frac {\text{sensitivity}}{1-{\text{specificity}}}}}$

which is equivalent to

${\displaystyle {\text{LR}}+={\frac {\Pr({T+}\mid D+)}{\Pr({T+}\mid D-)}}}$

or "the probability of a person who has the disease testing positive divided by the probability of a person who does not have the disease testing positive." Here "T+" or "T−" denote that the result of the test is positive or negative, respectively. Likewise, "D+" or "D−" denote that the disease is present or absent, respectively. So "true positives" are those that test positive (T+) and have the disease (D+), and "false positives" are those that test positive (T+) but do not have the disease (D−).

The greater the value of the LR+ for a particular test, the more likely a positive test result is a true positive. On the other hand, an LR+ < 1 would imply that non-diseased individuals are more likely than diseased individuals to receive positive test results.

The negative likelihood ratio is calculated as[5]

${\displaystyle {\text{LR}}-={\frac {1-{\text{sensitivity}}}{\text{specificity}}}}$

which is equivalent to[5]

${\displaystyle {\text{LR}}-={\frac {\Pr({T-}\mid D+)}{\Pr({T-}\mid D-)}}}$

or "the probability of a person who has the disease testing negative divided by the probability of a person who does not have the disease testing negative."

The calculation of likelihood ratios for tests with continuous values or more than two outcomes is similar to the calculation for dichotomous outcomes; a separate likelihood ratio is simply calculated for every level of test result and is called interval or stratum specific likelihood ratios.[6]

The pretest odds of a particular diagnosis, multiplied by the likelihood ratio, determines the post-test odds. This calculation is based on Bayes' theorem. (Note that odds can be calculated from, and then converted to, probability.)

Application to medicine

A likelihood ratio of greater than 1 indicates the test result is associated with the disease. A likelihood ratio less than 1 indicates that the result is associated with absence of the disease. Tests where the likelihood ratios lie close to 1 have little practical significance as the post-test probability (odds) is little different from the pre-test probability. In summary, the pre-test probability refers to the chance that an individual has a disorder or condition prior to the use of a diagnostic test. It allows the clinician to better interpret the results of the diagnostic test and helps to predict the likelihood of a true positive (T+) result.[7][8]

Research suggests that physicians rarely make these calculations in practice, however,[9] and when they do, they often make errors.[10] A randomized controlled trial compared how well physicians interpreted diagnostic tests that were presented as either sensitivity and specificity, a likelihood ratio, or an inexact graphic of the likelihood ratio, found no difference between the three modes in interpretation of test results.[11]

Estimation table

This table provide examples of how changes in the likelihood ratio affects post-test probability of disease.

Likelihood ratio	Approximate* change in probability[12]	Effect on posttest Probability of disease[13]
Values between 0 and 1 decrease the probability of disease (-LR)
0.1	−45%	Large decrease
0.2	−30%	Moderate decrease
0.5	−15%	Slight decrease
1	−0%	None
Values greater than 1 increase the probability of disease (+LR)
1	+0%	None
2	+15%	Slight increase
5	+30%	Moderate increase
10	+45%	Large increase

*These estimates are accurate to within 10% of the calculated answer for all pre-test probabilities between 10% and 90%. The average error is only 4%. For polar extremes of pre-test probability >90% and <10%, see "Estimation of pre- and post-test probability" section below.

Estimation example

1. Pre-test probability: For example, if about 2 out of every 5 patients with abdominal distension have ascites, then the pretest probability is 40%.
2. Likelihood Ratio: An example "test" is that the physical exam finding of bulging flanks has a positive likelihood ratio of 2.0 for ascites.
3. Estimated change in probability: Based on table above, a likelihood ratio of 2.0 corresponds to an approximately +15% increase in probability.
4. Final (post-test) probability: Therefore, bulging flanks increases the probability of ascites from 40% to about 55% (i.e., 40% + 15% = 55%, which is within 2% off the exact probability of 57%).

Calculation example

A medical example is the likelihood that a given test result would be expected in a patient with a certain disorder compared to the likelihood that same result would occur in a patient without the target disorder.

Some sources distinguish between LR+ and LR−.[14] A worked example is shown below.

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%

		Patients with bowel cancer (as confirmed on endoscopy)
		Condition positive	Condition negative	Prevalence = (TP+FN)/Total_Population = (20+10)/2030 ≈1.48%	Accuracy (ACC) = (TP+TN)/Total_Population = (20+1820)/2030 ≈90.64%
Fecal occult blood screen test outcome	Test outcome positive	True positive (TP) = 20 (2030 x 1.48% x 67%)	False positive (FP) = 180 (2030 x (100 - 1.48%) x (100 - 91%))	Positive predictive value (PPV), Precision = TP / (TP + FP) = 20 / (20 + 180) = 10%	False discovery rate (FDR) = FP/(TP+FP) = 180/(20+180) = 90.0%
	Test outcome negative	False negative (FN) = 10 (2030 x 1.48% x (100 - 67%))	True negative (TN) = 1820 (2030 x (100 -1.48%) x 91%)	False omission rate (FOR) = FN / (FN + TN) = 10 / (10 + 1820) ≈ 0.55%	Negative predictive value (NPV) = TN / (FN + TN) = 1820 / (10 + 1820) ≈ 99.45%
		TPR, Recall, Sensitivity = TP / (TP + FN) = 20 / (20 + 10) ≈ 66.7%	False positive rate (FPR),Fall-out, probability of false alarm = FP/(FP+TN) = 180/(180+1820) =9.0%	Positive likelihood ratio (LR+) = TPR/FPR = (20/30)/(180/2000) ≈7.41	Diagnostic odds ratio (DOR) = LR+/LR− ≈20.2	F1 score = 2 · Precision · Recall/Precision + Recall ≈0.174
		False negative rate (FNR), Miss rate = FN/(TP+FN) = 10/(20+10) ≈ 33.3%	Specificity, Selectivity, True negative rate (TNR) = TN / (FP + TN) = 1820 / (180 + 1820) = 91%	Negative likelihood ratio (LR−) = FNR/TNR = (10/30)/(1820/2000) ≈0.366

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
• Likelihood ratio positive = sensitivity / (1 − specificity) = 0.67 / (1 − 0.91) = 7.4
• Likelihood ratio negative = (1 − sensitivity) / specificity = (1 − 0.67) / 0.91 = 0.37
• Prevalence threshold = ${\displaystyle PT={\frac {{\sqrt {TPR(-TNR+1)}}+TNR-1}{(TPR+TNR-1)}}}$ = 0.19 => 19.1%

Hence with large numbers of false positives and few false negatives, a positive screen test is in itself poor at confirming the disorder (PPV = 10%) and further investigations must be undertaken; it did, however, correctly identify 66.7% of all cases (the sensitivity). However as a screening test, a negative result is very good at reassuring that a patient does not have the disorder (NPV = 99.5%) and at this initial screen correctly identifies 91% of those who do not have cancer (the specificity). Likewise, below a disease prevalence of 19.1%, the PPV for a screening test with these sensitivities and specificities drops significantly and is therefore more unreliable.

Confidence intervals for all the predictive parameters involved can be calculated, giving the range of values within which the true value lies at a given confidence level (e.g. 95%).[15]

Estimation of pre- and post-test probability

Further information: Pre- and post-test probability

The likelihood ratio of a test provides a way to estimate the pre- and post-test probabilities of having a condition.

With pre-test probability and likelihood ratio given, then, the post-test probabilities can be calculated by the following three steps:[16]

${\displaystyle {\text{pretest odds}}={\frac {\text{pretest probability}}{1-{\text{pretest probability}}}}}$

${\displaystyle {\text{posttest odds}}={\text{pretest odds}}\times {\text{likelihood ratio}}}$

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.

Odds are converted to probabilities as follows:[17]

${\displaystyle {\begin{aligned}(1)\ {\text{ odds}}={\frac {\text{probability}}{1-{\text{probability}}}}\end{aligned}}}$

multiply equation (1) by (1 − probability)

${\displaystyle {\begin{aligned}(2)\ {\text{ probability}}&={\text{odds}}\times (1-{\text{probability}})\\&={\text{odds}}-{\text{probability}}\times {\text{odds}}\end{aligned}}}$

add (probability × odds) to equation (2)

${\displaystyle {\begin{aligned}(3)\ {\text{ probability}}+{\text{probability}}\times {\text{odds}}&={\text{odds}}\\{\text{probability}}\times (1+{\text{odds}})&={\text{odds}}\end{aligned}}}$

divide equation (3) by (1 + odds)

${\displaystyle {\begin{aligned}(4)\ {\text{ probability}}={\frac {\text{odds}}{1+{\text{odds}}}}\end{aligned}}}$

hence

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

Alternatively, post-test probability can be calculated directly from the pre-test probability and the likelihood ratio using the equation:

• P' = P0 × LR/(1 − P0 + P0×LR), where P0 is the pre-test probability, P' is the post-test probability, and LR is the likelihood ratio. This formula can be calculated algebraically by combining the steps in the preceding description.

In fact, post-test probability, as estimated from the likelihood ratio and pre-test probability, is generally more accurate than if estimated from the positive predictive value of the test, if the tested individual has a different pre-test probability than what is the prevalence of that condition in the population.

Example

Taking the medical example from above (20 true positives, 10 false negatives, and 2030 total patients), the positive pre-test probability is calculated as:

• Pretest probability = (20 + 10) / 2030 = 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; the negative post-test probability is numerically equal to (1 − negative predictive value).

References

1. Swets JA. (1973). "The relative operating characteristic in Psychology". Science. 182 (14116): 990–1000. Bibcode:1973Sci...182..990S. doi:10.1126/science.182.4116.990. PMID 17833780.
2. Pauker SG, Kassirer JP (1975). "Therapeutic Decision Making: A Cost-Benefit Analysis". NEJM. 293 (5): 229–34. doi:10.1056/NEJM197507312930505. PMID 1143303.
3. Thornbury JR, Fryback DG, Edwards W (1975). "Likelihood ratios as a measure of the diagnostic usefulness of excretory urogram information". Radiology. 114 (3): 561–5. doi:10.1148/114.3.561. PMID 1118556.
4. van der Helm HJ, Hische EA (1979). "Application of Bayes's theorem to results of quantitative clinical chemical determinations". Clin Chem. 25 (6): 985–8. PMID 445835.
5. Gardner, M.; Altman, Douglas G. (2000). Statistics with confidence: confidence intervals and statistical guidelines. London: BMJ Books. ISBN 978-0-7279-1375-3.
6. Brown MD, Reeves MJ (2003). "Evidence-based emergency medicine/skills for evidence-based emergency care. Interval likelihood ratios: another advantage for the evidence-based diagnostician". Ann Emerg Med. 42 (2): 292–297. doi:10.1067/mem.2003.274. PMID 12883521.
7. Habibzadeh, Farrokh; Habibzadeh, Parham (15 June 2019). "The likelihood ratio and its graphical representation". Biochemia Medica. 29 (2): 193–199. doi:10.11613/BM.2019.020101. PMC 6457916. PMID 31015780.
8. Harrell F, Califf R, Pryor D, Lee K, Rosati R (1982). "Evaluating the Yield of Medical Tests". JAMA. 247 (18): 2543–2546. doi:10.1001/jama.247.18.2543. PMID 7069920.
9. Reid MC, Lane DA, Feinstein AR (1998). "Academic calculations versus clinical judgments: practicing physicians' use of quantitative measures of test accuracy". Am. J. Med. 104 (4): 374–80. doi:10.1016/S0002-9343(98)00054-0. PMID 9576412.
10. Steurer J, Fischer JE, Bachmann LM, Koller M, ter Riet G (2002). "Communicating accuracy of tests to general practitioners: a controlled study". BMJ. 324 (7341): 824–6. doi:10.1136/bmj.324.7341.824. PMC 100792. PMID 11934776.
11. Puhan MA, Steurer J, Bachmann LM, ter Riet G (2005). "A randomized trial of ways to describe test accuracy: the effect on physicians' post-test probability estimates". Ann. Intern. Med. 143 (3): 184–9. doi:10.7326/0003-4819-143-3-200508020-00004. PMID 16061916.
12. McGee, Steven (1 August 2002). "Simplifying likelihood ratios". Journal of General Internal Medicine. 17 (8): 647–650. doi:10.1046/j.1525-1497.2002.10750.x. ISSN 0884-8734. PMC 1495095. PMID 12213147.
13. Henderson, Mark C.; Tierney, Lawrence M.; Smetana, Gerald W. (2012). The Patient History (2nd ed.). McGraw-Hill. p. 30. ISBN 978-0-07-162494-7.
14. "Likelihood ratios". Archived from the original on 20 August 2002. Retrieved 4 April 2009.
15. Online calculator of confidence intervals for predictive parameters
16. Likelihood Ratios Archived 22 December 2010 at the Wayback Machine, from CEBM (Centre for Evidence-Based Medicine). Page last edited: 1 February 2009
17. from Australian Bureau of Statistics: A Comparison of Volunteering Rates from the 2006 Census of Population and Housing and the 2006 General Social Survey, Jun 2012, Latest ISSUE Released at 11:30 AM (CANBERRA TIME) 08/06/2012

External links

Medical likelihood ratio repositories

• The Likelihood Ratio Database
• GetTheDiagnosis.org: A Database of Sensitivity and Specificity
• The NNT: LR Home