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 No-Treat 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-/no-D):    
       

 
  
   

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+/No-D) x p(No-D)      
                 
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+/No-D) x p(No-D)      
 
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
 

Automatic recalculation    

Health Decision Strategies, LLC.

 

Adapted from R. Hamm's spreadsheet calculators.