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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]
The numerator of this ratio is less than the denominator; so, the likelihood ratio is between 0 and 1. Low values of the likelihood ratio mean that the observed result was much less likely to occur under the null hypothesis as compared to the alternative.
From this, the likelihood ratios of the test can be established: [2] Likelihood ratio positive = sensitivity / (1 − specificity) = 66.67% / (1 − 91%) = 7.4; Likelihood ratio negative = (1 − sensitivity) / specificity = (1 − 66.67%) / 91% = 0.37; Pretest probability (in this example) = 0.03; Pretest odds = 0.03 / (1 - 0.03) = 0.0309
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 = P T = T P R ( − T N R + 1 ) + T N R − 1 ( T P R + T N R − 1 ) {\displaystyle PT={\frac {{\sqrt {TPR(-TNR+1)}}+TNR-1}{(TPR+TNR-1 ...
The likelihood ratio is central to likelihoodist statistics: the law of likelihood states that the degree to which data (considered as evidence) supports one parameter value versus another is measured by the likelihood ratio. In frequentist inference, the likelihood ratio is the basis for a test statistic, the so-called likelihood-ratio test.
The Karlin–Rubin theorem can be regarded as an extension of the Neyman–Pearson lemma for composite hypotheses. [1] Consider a scalar measurement having a probability density function parameterized by a scalar parameter θ, and define the likelihood ratio () = / ().
The positive predictive value (PPV), or precision, is defined as = + = where a "true positive" is the event that the test makes a positive prediction, and the subject has a positive result under the gold standard, and a "false positive" is the event that the test makes a positive prediction, and the subject has a negative result under the gold standard.
If this ratio is 1, the evidence is indifferent; if greater than 1, the evidence supports the value a against b; or if less, then vice versa. In Bayesian statistics, this ratio is known as the Bayes factor, and Bayes' rule can be seen as the application of the law of likelihood to inference.