View source for Likelihood ratios in diagnostic testing

{{Distinguish|Likelihood-ratio test}}
{{Use dmy dates|date=July 2013}}

In [[evidence-based medicine]], [[likelihood ratio]]s 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 rule]]s was made at a symposium on information theory in 1954.<ref>{{cite journal | doi = 10.1126/science.182.4116.990 | author = Swets JA. | year = 1973 | title = The relative operating characteristic in Psychology | url = | journal =Science | volume = 182 | issue = 14116 | pages = 990–1000 | pmid = 17833780 }}</ref> In medicine, likelihood ratios were introduced between 1975 and 1980.<ref>{{cite journal | doi = 10.1056/NEJM197507312930505 |vauthors=Pauker SG, Kassirer JP | year = 1975 | title = Therapeutic Decision Making: A Cost-Benefit Analysis | url = | journal =NEJM | volume = 293 | issue = 5 | pages = 229–34 | pmid = 1143303 }}</ref><ref>{{cite journal | doi = 10.1148/114.3.561 |vauthors=Thornbury JR, Fryback DG, Edwards W | year = 1975 | title = Likelihood ratios as a measure of the diagnostic usefulness of excretory urogram information. | url = | journal =Radiology | volume = 114 | issue = 3 | pages = 561–5 | pmid = 1118556 }}</ref><ref>{{cite journal | doi = |vauthors=van der Helm HJ, Hische EA | year = 1979 | title = Application of Bayes's theorem to results of quantitative clinical chemical determinations. | url = http://www.clinchem.org/content/25/6/985.long | journal =Clin Chem | volume = 25 | issue = 6 | pages = 985–8 | pmid = 445835 }}</ref>

==Calculation==

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

The positive likelihood ratio is calculated as

:<math> LR+ = \frac{\text{sensitivity}}{1 - \text{specificity}} </math>

which is equivalent to

:<math> LR+ = \frac{\Pr({T+}|D+)}{\Pr({T+}|D-)} </math>

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&minus;" denote that the result of the test is positive or negative, respectively. Likewise, "D+" or "D&minus;" 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&minus;).

The negative likelihood ratio is calculated as<ref name=altman>{{cite book |author1=Gardner, M. |author2=Altman, Douglas G. |title=Statistics with confidence: confidence intervals and statistical guidelines |publisher=BMJ Books |location=London |year=2000 |pages= |isbn=978-0-7279-1375-3 |oclc= |doi= |accessdate=}}</ref>

:<math> LR- = \frac{1 - \text{sensitivity}}{\text{specificity}} </math>

which is equivalent to<ref name=altman/>

:<math> LR- = \frac{\Pr({T-}|D+)}{\Pr({T-}|D-)} </math>

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 variable|dichotomous]] outcomes; a separate likelihood ratio is simply calculated for every level of test result and is called interval or stratum specific likelihood ratios.<ref>{{cite journal | doi = 10.1067/mem.2003.274 |vauthors=Brown MD, Reeves MJ | year = 2003 | title = Evidence-based emergency medicine/skills for evidence-based emergency care. Interval likelihood ratios: another advantage for the evidence-based diagnostician | url = | journal =Ann Emerg Med | volume = 42 | issue = 2| pages = 292–297 | pmid = 12883521 }}</ref>

The [[Pre- and post-test probability#Pre-test probability|pretest odds]] of a particular diagnosis, multiplied by the likelihood ratio, determines the [[Pre- and post-test probability|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.<ref>{{cite journal | doi = 10.1001/jama.247.18.2543 |vauthors=Harrell F, Califf R, Pryor D, Lee K, Rosati R | year = 1982 | title = Evaluating the Yield of Medical Tests | url = | journal = JAMA | volume = 247 | issue = 18| pages = 2543–2546 | pmid = 7069920 }}</ref>

Research suggests that physicians rarely make these calculations in practice, however,<ref name="pmid9576412">{{cite journal |vauthors=Reid MC, Lane DA, Feinstein AR |title=Academic calculations versus clinical judgments: practicing physicians' use of quantitative measures of test accuracy |journal=Am. J. Med. |volume=104 |issue=4 |pages=374–80 |year=1998 |pmid=9576412| doi = 10.1016/S0002-9343(98)00054-0}}</ref> and when they do, they often make errors.<ref name="pmid11934776">{{cite journal |vauthors=Steurer J, Fischer JE, Bachmann LM, Koller M, ter Riet G |title=Communicating accuracy of tests to general practitioners: a controlled study |journal=BMJ |volume=324 |issue=7341 |pages=824–6 |year=2002 |pmid=11934776 |pmc=100792| doi = 10.1136/bmj.324.7341.824}}</ref> A [[randomized controlled trial]] compared how well physicians interpreted diagnostic tests that were presented as either [[sensitivity (tests)|sensitivity]] and [[specificity (tests)|specificity]], a likelihood ratio, or an inexact graphic of the likelihood ratio, found no difference between the three modes in interpretation of test results.<ref name="pmid16061916">{{cite journal |vauthors=Puhan MA, Steurer J, Bachmann LM, ter Riet G |title=A randomized trial of ways to describe test accuracy: the effect on physicians' post-test probability estimates |journal=Ann. Intern. Med. |volume=143 |issue=3 |pages=184–9 |year=2005 |pmid=16061916 |doi=10.7326/0003-4819-143-3-200508020-00004}}</ref>

== Estimation Table ==
This table provide examples of how changes in the likelihood ratio affects post-test probability of disease.
{| class="wikitable"
!Likelihood Ratio
!Approximate* Change 
in Probability<ref>{{cite journal |title=Simplifying likelihood ratios |journal=Journal of General Internal Medicine |date=2002-08-01 |issn=0884-8734 |pmc=1495095 |pages=647–650 |volume=17 |issue=8 |doi=10.1046/j.1525-1497.2002.10750.x |language=en |first=Steven |last=McGee}}</ref>
!Effect on Posttest
Probability of disease<ref>{{cite book |last1=Henderson |first1=Mark C. |last2=Tierney |first2=Lawrence M. |last3=Smetana |first3=Gerald W. |title=The Patient History |edition=2nd |publisher=McGraw-Hill |year=2012 |isbn=978-0-07-162494-7 |location= |page=30}}</ref>
|-
!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
|}
<nowiki>*</nowiki>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 ===
# Pre-test probability: For example, if about 2 out of every 5 patients with abdominal distension have ascites, then the pretest probability is 40%.
# Likelihood Ratio: An example "test" is that the physical exam finding of bulging flanks has a positive likelihood ratio of 2.0 for ascites.
# Estimated change in probability: Based on table above, a likelihood ratio of 2.0 corresponds to an approximately + 15% increase in probability. 
# 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&minus;.<ref name="urlLikelihood ratios">{{cite web |url=http://www.poems.msu.edu/InfoMastery/Diagnosis/likelihood_ratios.htm |title=Likelihood ratios |accessdate=2009-04-04 |url-status=dead |archiveurl=https://archive.is/20020820071706/http://www.poems.msu.edu/InfoMastery/Diagnosis/likelihood_ratios.htm |archivedate=20 August 2002 |df=dmy-all }}</ref> A worked example is shown below.
{{SensSpecPPVNPV}}

[[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%).<ref>[http://www.medcalc.org/calc/diagnostic_test.php Online calculator of confidence intervals for predictive parameters]</ref>

==Estimation of pre- and post-test probability==
{{Further|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:<ref>[http://www.cebm.net/index.aspx?o=1043 Likelihood Ratios], from CEBM (Centre for Evidence-Based Medicine). Page last edited: 1 February 2009</ref>

*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''.

Odds are converted to probabilities as follows:<ref>[http://www.abs.gov.au/AUSSTATS/abs@.nsf/Lookup/4441.0.55.002Explanatory+Notes5Jun+2012] 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 </ref>


:<math>\begin{align}(1)\ Odds = Probability/(1-Probability)   
\end{align}</math>      


multiply equation (1) by (1 - Probability) 


:<math>\begin{align}(2)\ Probability & = Odds \times (1 - Probability) \\
& = Odds - Probability \times Odds \\
\end{align}</math>


add (Probability x Odds) to equation (2)



:<math>\begin{align}(3)\ Probability + Probability \times Odds & = Odds \\
Probability \times (1 + Odds) & = Odds \\
\end{align}</math>


divide equation (3) by (1 + Odds)


:<math>\begin{align}(4)\ Probability = Odds/(1 + Odds)
\end{align}</math>


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==
{{reflist}}

==External links==
;Medical Likelihood Ratio Repositories
* [http://lrdatabase.com The Likelihood Ratio Database]
* [http://www.getthediagnosis.org GetTheDiagnosis.org: A Database of Sensitivity and Specificity]
* [http://www.thennt.com/home-lr/ The NNT: LR Home]

{{Medical research studies}}

[[Category:Medical statistics]]
[[Category:Evidence-based medicine]]
[[Category:Summary statistics for contingency tables]]