Friday, March 26, 2010

Solving previous problem

If we analyze the previous problem, we could consider that there are three decisions that the decision maker could take:
  1. No treatment
  2. Treatment with aspirin
  3. Treatment with warfarin
So, we should draw a decision three with these three decisions at first branch of tree.
There are three different uncertainties that could appear for treatments. Tow of these three is applied also for no treatment.
If we draw the decision three it should looks like the following pictures. (I just provide two pictures, because the tree for treatment with warfarin is the same as the aspirin, just different valuse)

If we calculate the Expected Value (EV, which clarifies the best decision that we could make) it will be clear that patients should be treated with warfarin.

A decision tree of the problem is given at the end of this document.
No treatment & CVA & affected = ‐158,000 SEK
No treatment & CVA & unaffected = ‐76,600 SEK
No treatment & No CVA = 0
Aspirin/Warfarin & side‐effect & CVA & affected = ‐(15,000+158,000) = ‐173,000 SEK
Aspirin/Warfarin & side‐effect & CVA & unaffected = ‐(15,000+76,600) = ‐91,600 SEK
Aspirin/Warfarin & side‐effect & haemorrhage = ‐(15,000+32,000) = ‐47,000 SEK
Aspirin/Warfarin & no side‐effect & CVA & affected = ‐(15,000+158,000) = ‐173,000 SEK
Aspirin/Warfarin & no side‐effect & CVA & unaffected = ‐(15,000+76,600) = ‐91,600 SEK
Aspirin/Warfarin & no side‐effect & No CVA = ‐15,000 SEK
Folding back the tree:
EMV(no treatment) =( 0.25*(‐158,000) + 0.75*(‐76,600))*0.8 = ‐77,560
EMV(aspirin treatment) =(((( 0.25*(‐173,000)+0.75*(‐91,600))*0.644)+0.356*(‐47,000))*0.048) + (((0.25*(‐173,000)+0.75*(‐91,600))*0.5)+0.5*(‐15,000))*0.952 = ‐64,692
EMV(warfarin treatment) = (((0.25*(‐173,000)+0.75*(‐91,600))*0.5)+0.5*(‐47,000))*0.072 + (((0.25*(‐173,000)+0.75*(‐91,600))*0.1)+0.9*(‐15,000))*0.928 = ‐28,639
Thus, if we merely look at the costs for the hospital, patients should be treated with warfarin.
These kind of analysis are called subjective analysis. If you are interested to deal with these kind of problem, I recommend the Making Hard Decisions: An Introduction to Decision Analysis book by Robert T. Clemen.

Tuesday, March 23, 2010

Decision Tree

I hade a course previous quarter. It was about how to use decision tree for taking correct decisions. It was very amazing and I want to rewrite one of the question of final exam for showing the types of question that decision tree could help us to solve. This quarter I have the second and advance version of that course called Decision and Risk Analysis, second course.

You are working as a medical advisor at a big hospital and your department is specialized within cardiology. Your manager has asked you to formally analyze the medical condition of atrial fibrillation. In the analysis you will merely look at costs for the hospital and not consider other criteria. Atrial fibrillation is a common condition, which carries with it a significant risk for stroke (80%) if left untreated. Treatment with the medicines warfarin or aspirin significantly reduces this risk, but there are side-effects to both treatments. For the average patient (at moderate risk), treatment with aspirin has a slightly smaller risk for side-effects than warfarin, 4.8% as opposed to 7.2%, but aspirin reduces the risk for stroke less effectively than warfarin. The serious side-effects are either cerebrovascular accident, CVA during the treatment or haemorrhage. Of patients treated with aspirin and affected by side-effects, 64.4% get during their treatment, whereas this number is only 50% for patients treated with warfarin and affected by side-effects.

For the patients who do not suffer from immediate side-effects during the aspirin treatment, the risk for CVA is still 50%, whereas the risk for CVA is only 10% for patients treated by warfarin and not suffering from side-effects during the period of treatment. If an average patient (moderate risk) with atrial fibrillation gets a stroke (CVA), regardless of treatment or no treatment, he or she is classified as affected or unaffected. Out of all CVA:s that occur, 25% are affected and 75% are unaffected. Transition costs(those that happen just once) for the treatment of a patient with CVA is established to 76,600 SEK (unaffected patients), whereas the state costs (those that remain patients for their Iifetime) for treatment of a patient with CVA is estimated to 158,000SEK (affected patients). The treatment of a haemorrhage is estimated to 32,000 SEK, and the cost of a medical treatment (either aspirin or warfarin) is 15,000 SEK.

What would be your recommendation to the hospital in the handling of moderate risk patients with atrial fibrillation and why?

But how could we solve it?

If you draw the decision tree, you will see that the warfarin is the best medical treatment.