Diagnostic odds ratio

The diagnostic odds ratio is defined mathematically as:

${\displaystyle {\text{Diagnostic odds ratio, DOR}}={\frac {TP/FN}{FP/TN}}={\frac {TP/FP}{FN/TN}}={\frac {TP\cdot TN}{FP\cdot FN}}}$[2][3]

where ${\displaystyle TP}$, ${\displaystyle FN}$, ${\displaystyle FP}$ and ${\displaystyle TN}$ are the number of true positives, false negatives, false positives and true negatives respectively.[1]

The diagnostic odds ratio ranges from zero to infinity, although for useful tests it is greater than one, and higher diagnostic odds ratios are indicative of better test performance.[1] Diagnostic odds ratios less than one indicate that the test can be improved by simply inverting the outcome of the test – the test is in the wrong direction, while a diagnostic odds ratio of exactly one means that the test is equally likely to predict a positive outcome whatever the true condition – the test gives no information.

The diagnostic odds ratio may be expressed in terms of the sensitivity and specificity of the test:[1]

${\displaystyle {\text{DOR}}={\frac {{\text{sensitivity}}\times {\text{specificity}}}{\left(1-{\text{sensitivity}}\right)\times \left(1-{\text{specificity}}\right)}}}$

It may also be expressed in terms of the Positive predictive value (PPV) and Negative predictive value (NPV):[1]

${\displaystyle {\text{DOR}}={\frac {{\text{PPV}}\times {\text{NPV}}}{\left(1-{\text{PPV}}\right)\times \left(1-{\text{NPV}}\right)}}}$

It is also related to the likelihood ratios, ${\displaystyle LR+}$ and ${\displaystyle LR-}$:[1]

${\displaystyle {\text{DOR}}={\frac {LR+}{LR-}}}$

The log diagnostic odds ratio is sometimes used in meta-analyses of diagnostic test accuracy studies due to its simplicity (being approximately normally distributed).[4]

Traditional meta-analytic techniques such as inverse-variance weighting can be used to combine log diagnostic odds ratios computed from a number of data sources to produce an overall diagnostic odds ratio for the test in question.

The log diagnostic odds ratio can also be used to study the trade-off between sensitivity and specificity[5][6] by expressing the log diagnostic odds ratio in terms of the logit of the true positive rate (sensitivity) and false positive rate (1 − specificity), and by additionally constructing a measure, ${\displaystyle S}$:

${\displaystyle D=\log {\text{DOR}}=\log {\left[{\frac {TPR}{(1-TPR)}}\times {\frac {(1-FPR)}{FPR}}\right]}=\operatorname {logit} (TPR)-\operatorname {logit} (FPR)}$

${\displaystyle S=\operatorname {logit} (TPR)+\operatorname {logit} (FPR)}$

It is then possible to fit a straight line, ${\displaystyle D=a+bS}$. If b ≠ 0 then there is a trend in diagnostic performance with threshold beyond the simple trade-off of sensitivity and specificity. The value a can be used to plot a summary ROC (SROC) curve.[5][6]

Consider a test with the following 2×2 confusion matrix:

We calculate the diagnostic odds ratio as:

${\displaystyle {\text{DOR}}={\frac {TP/FP}{FN/TN}}={\frac {26/12}{3/48}}=34.666\ldots \approx 35}$

This diagnostic odds ratio is greater than one, so we know that the test is discriminating correctly. We compute the confidence interval for the diagnostic odds ratio of this test as [9, 134].

The diagnostic odds ratio is undefined when the number of false negatives or false positives is zero – if both false negatives and false positives are zero, then the test is perfect, but if only one is, this ratio does not give a usable measure. The typical response to such a scenario is to add 0.5 to all cells in the contingency table,[1][7] although this should not be seen as a correction as it introduces a bias to results.[5] It is suggested that the adjustment is made to all contingency tables, even if there are no cells with zero entries.[5]

• Sensitivity and specificity
• Binary classification
• Positive predictive value and negative predictive value
• Odds ratio