Likelihood Ratios


Menu location: Analysis_Clinical Epidemiology_Likelihood Ratios (2 by k).


This function gives likelihood ratios and their confidence intervals for each of two or more levels of results from a test (Sackett et al., 1983, 1991).


The quality of a diagnostic test can be expressed in terms of sensitivity and specificity. Sensitivity is the ability of the test to pick up what it is testing for and specificity is the ability of the test to reject what it is not testing for.


    Present Absent
TEST: +: a (true +ve) b (false +ve)
-: c (false -ve) d (true -ve)


Sensitivity = a/(a+c)

Specificity = d/(b+d)


Likelihood ratio of a positive test = [a/(a+c)]/[b/(b+d)]

Likelihood ratio of a negative test = [c/(a+c)]/[d/(b+d)]


Likelihood ratios enable you to quantify the effect that a particular test result has on the probability of an outcome (e.g. diagnosis of a disease). Using a simplified form of Bayes' theorem:


posterior odds = prior odds * likelihood ratio



odds = probability/(1-probability)

probability = odds/(odds+1)


These methods can be generalised to more than two possible test outcomes, in which case the data can be arranged into a two by k table (k is the number of test outcomes studied). If one test outcome is called test level j then the likelihood ratio at level j is given by:


likelihood ratio j = p(tj_disease)/p(tj_no disease)


where p(tj_) is the proportion displaying the relevant test result at level j


Technical validation

The confidence intervals for the likelihood ratios are constructed using the iterative method suggested by Gart and Nam (1991).



From Sackett et al. (1991, p. 111).


Initial creatine phosphokinase (CK) levels were related to the subsequent diagnosis of acute myocardial infarction (MI) in a group of patients with suspected MI. Four ranges of CK result were chosen for the study:



  MI No MI
CK ≥ 280: 97 1
CK = 80-279: 118 15
CK = 40-79: 13 26
CK = 1-39: 2 88


To analyse these data in StatsDirect select Likelihood Ratios (2 by K) from the Clinical Epidemiology section of the Analysis menu. Choose the default 95% confidence interval.


For this example:


Result + Feature - Feature Likelihood Ratio 95% CI (Koopman)
1 97 1 54.826087 9.923105 to 311.581703
2 118 15 4.446377 2.772565 to 7.31597
3 13 26 0.282609 0.151799 to 0.524821
4 2 88 0.012846 0.003513 to 0.046227


Here we can say with 95% confidence that CK results of ≥ 280 are at least ten (9.9) times more likely to come from patients who have had an MI than they are to come from those who have not had an MI.


confidence intervals



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