1. What Are the Odds? The Annals of Thoracic Surgery
Gary L. Grunkemeier, PhD, YingXing Wu, MD 
The Annals of Thoracic Surgery 
Volume 83, Issue 4, Pages (April 2007)
DOI: /j.athoracsur
Copyright © 2007 The Society of Thoracic Surgeons Terms and Conditions

2. Fig 1
The relationship between odds and probabilities (solid line). Probabilities range from 0 to 1 (0% to 100%). Odds range from 0 to infinity and are always larger than probabilities. The dashed line is the line of identity. For low-risk events, odds and probabilities are similar; for example, 0.25 and 0.20 (20%), respectively, shown by the black circle. For high-risk events, odds are much larger; for example, 4 and 0.80 (80%), shown by the gray circle.
The Annals of Thoracic Surgery  , DOI: ( /j.athoracsur )
Copyright © 2007 The Society of Thoracic Surgeons Terms and Conditions

3. Fig 2
The relationship between a patient’s expected mortality (horizontal axis) and that patient’s mortality (vertical axis) at the hands of providers with various observed-to-expected (O/E) ratios (dashed lines) and odds ratios (ORs) (solid lines). At a provider with O/E = 1 and OR = 1, the patient’s mortality risk is exactly as expected. At providers with O/E greater than 1, a statistical impossibility can arise because the mortality risk can exceed 100%, revealing a defect in the O/E measure itself. Unlike O/E ratios, all combinations of patient risk and provider ORs result in valid, that is, less than 100%, risks. The patient with an invalid risk of 120% based on O/E = 2 (gray circle) has a valid risk of 75% based on OR = 2 (black circle).
The Annals of Thoracic Surgery  , DOI: ( /j.athoracsur )
Copyright © 2007 The Society of Thoracic Surgeons Terms and Conditions

4. Fig 3
Risk-adjusted mortality after percutaneous coronary intervention (PCI) for the six Provident Health System hospitals (Table 1), as measured by observed-to-expected (O/E) ratios (horizontal bars) and odds ratios (ORs; solid circles). Confidence intervals are computed by three methods for the O/E ratios: normal approximation, bootstrap percentile, and bootstrap accelerated bias-corrected (BCa), and by two methods for the ORs: the likelihood-based confidence limits (vertical lines) and normal approximation limits applied to the logarithm of the OR (short horizontal bars).
The Annals of Thoracic Surgery  , DOI: ( /j.athoracsur )
Copyright © 2007 The Society of Thoracic Surgeons Terms and Conditions

5. Fig 4
The relationship between odds ratios (ORs) and observed-to-expected (O/E) ratios depends on the expected event rate. For ratios less than 1, they are very close (ORs are slightly smaller). For ratios greater than 1, they are fairly close when the expected event rates are 5% or less. The greater the expected mortality, the greater the difference. An OR of 4 is equivalent to an O/E ratio of 3.1 when the expected risk is 10% (black circle), and to an O/E ratio of 2.5 when the expected risk is 20% (gray circle).
The Annals of Thoracic Surgery  , DOI: ( /j.athoracsur )
Copyright © 2007 The Society of Thoracic Surgeons Terms and Conditions