
Note: This beta test page is for demonstration only, not for
official or clinical use.

Calculations
with Bayes' Theorem
in Decision Trees. 
Consideration
of Utilities for Treat and NoTreat Options 

Section
1. 
Problem:
You need to know probability of disease given a test result.
Instead, you know probability of disease in the target population prior
to the test, and probability of test results given various diseases
(and no disease). In Section 1, cells for entering values contain the
available information. Tree in the middle of this page is generated
information that is used in the second decision tree in Section 2. That
tree on the bottom of this page shows the use of utilities and Bayes
calculations in a Treat, Test, or Do Nothing decision analysis.
In the theoretical baseline numbers used here, the most favored action with the
highest expected utility is "do the test".

Yellow
boxes are where you can enter new data.
Default data relates to hypothetical prostate test.











Enter names for what
can happen: 

Patient have disease: 






Patient be well: 






Test be positive: 






Test be negative: 










What are the known,
general probabilities?














Probability 



Prior probability of disease, p(Disease):






Prob of positive test given
disease, Sensitivity, p(T+/D): 










Prob of negative test given
no disease, Specificity, p(T/noD): 













Step
1: Information Input From Above: 

Step
2. Tree "inverted" by Bayes' Theorem: 










State of world 

Test result 


Test result 

State of world 












Test positive 




Disease 





















Probability 




Probability 

Disease 

 


Test positive 

 


Chance 
 



Chance 




 




 

Probability 

Test negative 


Probability 

No disease 

 




 



 




 



 

Probability 


 

Probability 
Chance 




Chance 
 



 

Test positive 


 

Disease 

 




 



 




 



No Disease 

Probability 


Test negative 

Probability 



 




 


Chance 
 



Chance 
 

Probability 

 


Probability 

 



Test negative 




No disease 





















Probability 




Probability 


















Probability of positive test
result: 
















p(T+) = P(T+/D) x p(D) +
p(T+/NoD) x p(NoD) 












Probability of disease state
given test result: 






Bayes' Theorem: 

















p(T+/D) x p(D) 




p(D/T+) = 
 





p(T+/D) x p(D) + p(T+/NoD) x p(NoD)





Section
2. 
Bayes
Decision Tree Using Outcomes Utilities
for the Treat, Test, or Do Nothing Decision Analysis. 









Fill in
this information (yellow boxes) 














Name
of treatment: 




Name of test: 


Name of "do nothing": 
















Outcomes











Name 


Utility 


Treat disease 







Treat no disease 







Not treat disease 







Not treat no disease


















Action


Test results 


Events
(disease) 

Outcomes
(Utilities) 















Disease 

Treat disease 
























Probability 

Utility 

Treat 




 






Chance 
 








 



Expected utility 




No disease 

Treat no disease 

 








 








 




Probability 

Utility 

 








 




Disease 

Treat disease 

 








 








 

Test positive 


Probability 

Utility 

 




 



 



Chance 
 



 

Probability 


 



 

 


No disease 

Treat no disease 
Choice 
 

 






 

 






 

 


Probability 

Utility 

Test 

 







Chance 



Disease 

Not treat dis 



 






Expected utility 

 






 

Test negative 


Probability 

Utility 

 




 



 



Chance 
 



 

Probability 


 



 




No disease 

Not treat no dis 

 








 








 




Probability 

Utility 

 








 




Disease 

Not treat disease 

 








 








Do Nothing 




Probability 

Utility 






 






Chance 
 



Expected utility 




 








No disease 

Not treat No dis 
























Probability 

Utility 
