Elsevier

Mayo Clinic Proceedings

Volume 74, Issue 11, November 1999, Pages 1061-1071
Mayo Clinic Proceedings

A Perspective on Standardizing the Predictive Power of Noninvasive Cardiovascular Tests by Likelihood Ratio Computation: 1. Mathematical Principles

https://doi.org/10.4065/74.11.1061Get rights and content

The current practice of reporting positive and negative predictive value (PV), sensitivity (Se), and specificity (Sp) as measures of the power of noninvasive cardiovascular tests has significant limitations. A test result's PV and its comparison with other test results are highly dependent on the pretest disease prevalence at which it is determined; the citation of sensitivity and specificity provides no succinct or explicit quantitation of the rule-in and rule-out power of a test. This article presents a rationale for the use of an alternative standard for expressing predictive power in the form of positive and negative likelihood ratios, (+)LR and (-)LR. The likelihood ratios are composite expressions of test power, which incorporate the Se and Sp and their respective complements [(1 - Se) and (1 - Sp)], thus yielding single unambiguous measures of positive and negative predictive power. The likelihood ratios are calculated as follows: (+)LR = Se(l- Sp) and (-)LR = Sp/(I- Se). On analysis of the predictive value equations, the likelihood ratios equal the quotients of the posttest predictive value odds to the pretest prevalence odds for disease and no disease, respectively, as follows: (+)LR = (+)PVOd/POD and (-)LR = (-)PVOn/PON, where (+)PVO d is positive predictive value odds for disease, POD is prevalence odds for disease, (-)PVOn is negative predictive value odds for no disease, and PON is prevalence odds for no disease. Thus, the likelihood ratios are measures of the odds advantage in posttest probability of disease or no disease relative to pretest probability, independent of disease prevalence in the tested population. The quotients of the (+)LR or the (-)LR among test results studied in a common population are direct expressions of their relative predictive power in that population, The likelihood ratio principle is applicable to the evaluation of the predictive power of multiple tests performed in a common population and to estimating predictive power at multiple test thresholds.

Section snippets

Defining Predictive Value In Terms Of Posttest Characteristics

By current conventions, the predictive power of a noninvasive cardiovascular test result is determined by analy­sis of a representative patient group that is referred for definitive diagnostic evaluation termed an index population2,3,4 The occurrence of disease in the index population is determined by an independently established diagnostic standard such as coronary angiography. The prevalence of disease (PD) is the proportion of the index population expressed as the percentage that has

Defining Predictive Value In Terms Of Sensitivity, Specificity, And Disease Prevalence

The sensitivity and specificity of a test and their complements are termed test performance indices and can be defined on the basis of the posttest characteristics (Figure 2, column 1). The sensitivity (Se) of a test is the quotient of true positives divided by the sum of true positives and false negatives. The complement of sensitivity (1 - Se) is the quotient of false negatives divided by the sum of true positives and false negatives. The specificity (Sp) of a test is the quotient of true

Limitations In Predictive Value Estimates

(3a), (3b) provide a background for analysis of the limitations of the (+)PVd and (-)PVn as measures of the relative power of 2 or more diagnostic tests. Such an analysis will be applied to a comparison of the predictive power of 2 commonly applied noninvasive diagnostic tests for coronary artery disease (CAD), the exercise electrocardiogram (Ex ECG) and the planar exercise thallium scintigram (Ex TI). The sensitivity and specificity levels for this illustration are derived from a meta-analysis

Limitations In Citation Of Sensitivity And Specificity As Measures Of Test Power

In considering the factors in equations (3a) and (3b), it is evident that the predictive power of a positive test result (rule-in power), as expressed in the relationship of (+)PV d to PD, resides in the Se and (1 - Sp), while the predictive power of a negative test result (rule-out power), as expressed in the relationship (-)PVn to PN, resides in the Sp and (1 - Se). Hence, any estimate of positive test power based on the test performance indices must include the Se and (1 - Sp), while any

The Likelihood Ratio As A Composite Measure Of The Predictive Power Of Positive And Negative Test Results

A simplified and easily comprehended expression of the positive and negative predictive power of a noninvasive test is uncovered when sensitivity, specificity, and their complements in equations (3a) and (3b) are consolidated, yielding the following equations:

(+)PVd=PDPD+[PN/(Se/1-Sp)]×100

(-)PVn=PNPN+[PD/(Sp/1-Se)]×100

Considering that the PN term in (4a) is equivalent to (100 - PD) and that the PD term in (4b) is equivalent to (100 - PN), it is apparent that the quotient (Se/1 - Sp)

The Likelihood Ratio As A Probability Odds Measure

A fundamental quality of the likelihood ratio as a measure of predictive power is elucidated when the units of the predictive value equations (6a) and (6b) are converted from a percentage to an odds form (Appendix 3). Conversion to odds: 1 is accomplished by dividing a given percentage in the range 0.0 to <100% by its 100% complement as follows: Odds: 1 = (Percent/100-Percent).)

For a decimal in the range 0.0 to <1.0, conversion to odds: 1 is accomplished by dividing the decimal by its

Estimating The Predictive Power For Combined Test Results

The likelihood ratios provide a convenient means for estimating the power of multiple tests, applied to a common index population. Such an estimate of test power for combined test results is referred to as an effective likelihood ratio (ELR).

The effective likelihood ratios can be estimated by 2 general approaches: (1) calculation of the predicted ELR for disease (predicted ELRd) or for no disease (predicted ELRn) and (2) calculation of the observed ELRd for disease (observed ELRd) or for no

Reconciling Differences In Reported Likelihood Ratio Definition

Weinstein and Fineberg2 define a test likelihood ratio as the ratio of the frequency of a particular test result in those with disease to its frequency in those without disease. In translating this definition in terms of the sensitivity, specificity, and their complements, Sackett7 and Boyko8 have defined the likelihood ratios in the following relationships:

LRfor(+)Test=Se(1-Sp)

LRfor(-)Test=(1-Se)Sp

The reader will notice that the LR for a negative test result defined in (b) is

Acknowledgments

The author expresses his appreciation to Professor Fred B. Weissler, professor of mathematics, University of Paris XIII, for his valuable review of a draft of the manuscript and to Julienne M. Montgomery for her able assistance in its preparation.

References (9)

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