Introduction to Decision Analysis

Transcription

1 Introduction to Decision Analysis M.Sc. (Tech) Yrjänä Hynninen Dept of Mathematics and Systems Analysis Analytics and Data Science seminar, October 16, 2017

2 Learning objectives Develop an understanding for: What is decision analysis How can decisions be structured with decision trees What is the value of information in supporting decisions MS-E2 courses in the data analysis minor MS-E2134 Problem Solving and Decision Making MS-E2177 Seminar on Case Studies in Operations Research Other relevant MS-E2 courses MS-E2192 Systems research seminar 2

3 Data analytics in supporting decisions The scientific process of transforming data into insight for making better decisions. source: Lisa Kart: Advancing Analytics (Gartner).

4 Operations Research Science of better Operations research is the attack of modern science on complex problems arising in the direction and management of large systems of men, machines, materials and money in industry, business, government and defense. Its distinctive approach is to develop a scientific model of the system, incorporating measurements of factors such as change and risk, with which to predict and compare the outcomes of alternative decisions, strategies or controls. The purpose is to help management determine its policy and actions scientifically. OR Quarterly 3(3): 282,

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6 Operations Research in Aalto Systems Analysis Laboratory Group in the Department of Mathematics and Systems Analysis 6

7 SAL Professors Harri Ehtamo Optimization models Game theory Fabricio Oliveira Stochastic optimization Production planning Ahti Salo Risk and decision analysis Invesment theory Kai Virtanen (PoP) Simulation OR in defence forces Raimo Hämäläinen (prof. emeritus) Environmental decision making Systems intelligence Risto Lahdelma (double affiliation) Linear programming Energy models

8 Basic elements of decision problems Decision context What to study in Aalto? Decision alternatives Which program? Bachelor, Master or Doctoral studies? Decision outcomes Expected future job salary, time to complete studies, study workload, Uncertainty about the outcomes We cannot a priori be sure about future job salary, study time,... Values, objectives, and attributes Value is expression of what the decision maker (DM) cares about ( Have a good life ) Objectives set direction for how value can be realized ( Be wealthy ) Attributes measure the attainment of objectives ( Monthly salary ) 8

9 Dimensions of unknowing Certainty DM knows for sure what the state of nature will be Decisions can still be hard due to the presence of large number of alternatives or multiple criteria Multiattribute value theory (MAVT) Ignorance DM knows all possible states of nature, but does not know their probabilities Risk DM knows all possible states of nature and can assign a probability to each state 9

10 Maximizing Expected Monetary Value (EMV) Expected Monetary Value (EMV) = Probability * Outcome Calculate the average outcomes when the future includes the scenarios that may or may not happen Select the alternative that has the highest EMV Justification by the Law of Large Numbers The average of the outcomes converges to the EMV as the number of repetitions approaches infinity In the long run, better of taking alternatives with highest EMVs 10

11 St. Petersburg paradox Consider the following game: 1. Toss a coin until first heads comes up 2. Receive 2 n euros where n is the number of tosses you made How much would you pay to participate in this game? Tosses The Expected Monetary Value of this game is infinite! n 2 n EMV

12 Risk attitudes Certainty Equivalent (CE) is a certain outcome that is equally preferred to a simple game with two outcomes Lottery Probability Outcome heads tails EMV 50 CE? What is the highest price CE you would accept to buy the lottery? Risk Premium = Expected Monetary Value - Certainty Equivalent RP = EMV - CE 12

13 Example: Assess your risk attitude Which lottery would you prefer? Why? Lottery 1 Lottery 2 heads tails EMV What are your certainty equivalents of the lotteries? Are they consistent with your preferences? E.g. If you prefer lottery 1, then check if CE(lottery 1) > CE(lottery 2) 13

14 Utility function Utility function captures the DM s risk preferences Risk-averse: RP > 0 concave utility function Risk-neutral: RP = 0 linear utility function Risk-seeking: RP < 0 convex utility function A risk-averse DM would accept a sure compensation CE (< EMV) instead of the lottery. A risk-neutral DM is indifferent between receiving EMV for sure and the lottery. A risk-seeking DM would enjoy the thrill of the game. 14

15 Problem structuring with decision trees Represent problems as chains of consecutive decisions and chance events Squares for decisions Circles for chance events Uncertainties associated with chance events are modelled by probabilities. Decision alternative 1 p 1 Event 1 Utility 1 Decision alternative 2 Decision node p 2 Event 2 Chance node Utility 2 Decision outcomes (leaves of the trees) 15

16 A simple decision tree Should you take an umbrella with you in the morning? These are the utilities Take the umbrella p It rains 0.4 Protected from rain 1-p It doesn t rain 0.9 Burden of carrying p It rains 0 Get soaked Do not take the umbrella It doesn t rain 1-p 1 Wonderful! 16

17 Modelling with decision trees Decision trees can analyse a variety of contexts Real options, value of information Continuous variables can be discretized Common mistakes: Decision and chance nodes are in wrong order Only chance nodes whose results are known can precede a decision node Incorrect derivation of probabilities Probabilities depend on earlier decisions and event outcomes 17

18 Solving decision trees General principle: Determine the path with the maximum expected utility or EMV (Expected Monetary Value) Prerequisite The utility (or monetary value) at each end node The probabilities for each chance event Process Proceed from right to left starting from the end Compute the expected utility (or EMV) for each chance node At each decision node, choose the alternative for which the expected utility (or EMV) is highest 18

19 Example: Value of inspection (1/4) Your brother is going to buy a cottage out of two alternatives 1. A new cottage for An old cottage for The old cottage may be moldy, which is hard to ascertain. Your brother estimates that there is a 15 % probability for this defect If the old cottage is defective, he will have to buy a new cottage and receives only for the old one Before buying, he can have the cottage inspected at a cost of If the cottage is OK, the inspector will confirm this without exception If the cottage is defective, there is a 20 % chance that the defect will go unnoticed 19

20 Example: Value of inspection (2/4) A decision tree presentation of this problem Inspect p G 1-p G Do not inspect Good Bad New Old New Old New Old p D G 1-p D G p D B 1-p D B p D 1-p D Defect No defect Defect No defect Defect No defect We need to derive the probabilities p G, p D G, p D B and p D before the tree can be solved. 20

21 Example: Solving a decision tree (3/4) The following probabilities are given p D = P(Defect) = 0.15 (your brother s prior estimate) P( Good No defect) = p G N = 1.0 P( Good Defect) = p G D = 0.20 p G, p D G and p D B derived as follows p p p G D G D B P(" Good") P(" Good" No defect) P( No defect) P(" Good" Defect) P( Defect) P( Defect P( Defect P(" Good" Defect) P( Defect) " Good") P(" Good") P(" Bad" Defect) P( Defect) " Bad") P(" Bad")

22 Example: Solving a decision tree (4/4) No inspection & Old: Inspect Do not inspect > *( )+0.966*( ) = Good Bad New Old New New Old To maximize EMV, buy the old cottage without inspection Inspect if it costs ( ) = less Old Defect No defect Defect Defect No defect

23 Case study: Value of genetic testing Primary source: Yrjänä Hynninen, Miika Linna, Eeva Vilkkumaa, Value of genetic testing in the prevention of cardiovascular events, Manuscript.

24 Value of genetic testing in preventing CVD CVD = Cardiovascular disease Traditional tests cannot reliably detect risk groups More than half of all heart disease events involve individuals with estimated risk at low or average levels Genetic testing offers possibility to improve reliability More accurate but more costly Too expensive to test everyone Resources for testing are away from resources for treating What strategy should one use for testing? Individual: I want all possible tests Society: Test only those whose treatment is likely to change 24

25 Evaluation of outcomes Health outcomes measured as quality-adjusted lifeyears (QALYs) One year of living in perfect health equals 1 QALY Costs as euros discounted to year 2015 Trade-off between health benefits and costs determined through the societal willingness-to-pay λ (WTP; /QALY) How many euros are we (as a society) at most willing to pay for a treatment that extends the life of a person by 1 QALY? Net health benefit is NHB = Health outcomes Costs λ 25

26 Parameters of the study Health outcomes Estimate Source CVD free (QOL) 0.90 [24] Disutility due to non-fatal CVD event [25 27] (QALY) Probability of death in case of event 22 % [28] Expected time of CVD event 5.76 years [18] Risk reduction if statin treatment -25 % [29] Annual side effect of statin treatment [30,31] (QOL) Discount rate of health outcomes 3 % [23] Costs Costs of obtaining Framingham Risk 173 [33] factors incl. blood panel, doctor and nurse visits ( ) Genetic testing ( ) 200 Assumption Annual statin costs ( /person) 53 [34,35] Annual monitoring of a patient receiving 173 [33] statins (in primary prevention) Annual secondary prevention 451 [36] Non-fatal CVD event (undiscounted) National Discharge Register Annual statin medication (undiscounted) 226 National Discharge Register Fatal CVD event (undiscounted) [35,36] Productivity loss due to non-fatal CVD [38,39] Willingness-to-pay threshold Assumption Discount rate of costs 3 % [23] CVD, cardiovascular disease; QOL, quality of life; QALY, quality-adjusted life-year. 26

27 Value of treatment Example: Statin medication as prevention Decreases risk of heart disease event on average by 25 % Annual costs Negative side effects Treating everyone is not economical nor good treatment practice Expected NBH for treating 15 % Prior risk 20 % 85 % 20 % 80 % Expected NBH for not treating 27

28 Value of a single test 1/3 This example is not from the case study Tests not fully accurate Patient healthy Patient sick Test says healthy True negative Pr = 0.9 False negative Pr = 0.2 Test says sick False positive Pr = 0.1 True positive Pr =0.8 Treat Impact of treatment Not treat NHB scaled from -100 to 0 Patient healthy Treatment cost Risk of complications NHB=-10 No cost NHB=0 Patient sick Treatment cost Healthy benefit NHB=-60 Emergency treatment cost Risk of death or severe complications NHB=

29 Value of a single test 2/3 This example is not from the case study Prior risk updated using Bayes theorem to reflect the tests P test says sick sick P sick P sick test says sick = P test says sick P healthy test says sick = 1 P sick test says sick P sick test says healthy = P healthy test says healthy P test says healthy sick P sick P test says healthy = 1 P sick test says healthy Let P sick = 0.2, other probabilities on previous slide P sick test says sick = P sick test says healthy =

30 Value of a single test 3/3 This example is not from the case study Test says sick p = = 0.24 Not treat Treat Test says healthy 1 p = = 0.76 Treat Not treat Solve tree to get expected NHB when testing Sick Healthy Sick Healthy Sick Healthy Sick Healthy

31 Alternative testing strategies No treatment Do not test or treat any patient Treatment (without testing, no delay in starting treatment) Use prior risk only to determine whether to treat with statin medication or not FRS (some delay in starting treatment) Carry out Framingham Risk Score (FRS) test to determine whether to treat or not FRS & GRS simultaneously (some delay) Carry out FRS and Genetic Risk Score (GRS) tests simultaneously to determine whether to treat or not FRS & GRS optionally (most delay in starting treatment) Carry out FRS and, based on its results, optionally GRS to determine whether to treat or not 31

32 Decision tree for testing strategies 32

33 Optimal testing and treatment strategy Solving the decision tree for prior probabilities 0, 0.01, 0.02,..., 1 gives optimal treatment strategy Knowledge: Prior Prior+FRS Prior+FRS+GRS 33

34 Sensitivity analysis WTP ( /QALY) ### ### ### ### ### ### ### ### SW SW SW SW ### domi ## ### ### ### domisw SW SW SW SW domidomi### ### ### domidomi### ### ### ### domi### ### ### ### ### ### ### ### ### domi domdomi### ### ### ### domidomidomi### ### SW SW SW domi### SW SW SW ### ### domidomi### ### ### ### domidomidomisw SW SW SW domisw domisw Cost of GRS ( ) Two-way sensitivity analysis of WTP threshold and the cost of GRS. The gray area represents the WTP-cost combinations for which GRS is a part of the optimal testing strategy. 34

35 Summary and conclusions Operations Research is a growing field Data Analytics provides new opportunities to support decision making Data valuable only if it helps make better decisions! Paying for more accurate estimates beneficial only if it is likely enough that the decision with these estimates leads to another decision than what would otherwise have been taken Simple decision tree models can help identify where additional data is most valuable What is the expected value of data analytics? Does it cover the expenses? 35

36 Reading materials Overviews M.J.Mortenson, N.F. Doherty, S. Robinson (2014) Operational Research from Taylorism to Terabytes: A Research Agenda for the Analytics Age, European Journal of Operational Research ( Continued readings RL Keeney (1996). Value-Focused Thinking: A Path to Creative Decisionmaking. Harvard University Press, Cambridge MA. S French, J Maule and N Papamichail (2009). Decision Behaviour, Analysis and Support. Cambridge University Press, Cambridge. A Salo, J Keisler, A Morton (2011). Portfolio Decision Analysis: Improved Methods for Resource Allocation, Springer, New York. 36