We are interested in seeing if counting the number of MCM cells less than 46 or 46 or more is related to the chance of a positive biopsy. The results of a trial are given below from which we can a high sensitivity but a low positive predictive value (PPV).
|
|
True |
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|
- Biopsy |
+ Biopsy |
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Pred |
Cell count < 46 (-) |
27 |
0 |
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|
Cell count 46 or more (+) |
4 |
8 |
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The PPV = P(a positive biopsy given you have a MCM Cell Count of 46 or more) = 8 /(8+4) = 0.67
The Sensitivity = P(a MCM Cell Count of 46 or more given you have a positive biopsy) = 8/(8+0) = 1.00
This is explained by all positive biopsies having MCM cell counts of 46 or more but not all MCM cell counts of 46 or more having positive biopsies (only 2/3rds do).
Another way of seeing this related to the graphical method of Gigerenzer (2003) for explaining conditional probabilities is
Positive Biopsies have the likelihood of having 46 or more cells (Predicted positive equal to number in cell A) or less than 46 cells (Predicted negative equal to number in cell B )
Negative Biopsies have the likelihood of having 46 or more cells (Predicted positive equal to number in cell C) or less than 46 cells (Predicted negative equal to number in cell D)
PPV = P(Pred positive | a positive Biopsy) / [P(Pred positive | a positive Biopsy) + (Pred positive | a negative Biopsy)] = A / (A+C) = P(True positive (positive Biopsy) | a Pred positive (46 or more cells))
Sensitivity = P(Pred positive | a positive Biopsy) / [P(Pred positive | a positive Biopsy) + P(Pred negative | a positive Biopsy)] = A / (A+B) = P(Pred positive (46 or more cells) | True positive (positive Biopsy))
Reference
Gigerenzer G (2003) Reckoning with Risk: Learning to Live with Uncertainty. Penguin.