1. Hazard Functions for Combination of Causes Farrokh Alemi Ph.D. Professor of Health Administration and Policy College of Health and Human Services, George Mason University 4400 University Drive, Fairfax, Virginia 22030 703 993 1929 falemi@gmu.edu falemi@gmu.edu

2. Purpose Risk faced from one cause over time Risk faced from one cause over time Relative contribution of various causes Relative contribution of various causes Joint effect of causes Joint effect of causes

3. Definitions Function Name FormulaDefinition Related Terms Probability Distribution Function p(X=t) Probability of event occurring at time "t." Cumulative Distribution Function p((X≤t) Probability of the event occurring prior to or at time "t." Sum of PDF for ≤ t Survival Function p(X>t) Probability of the event not occurring prior or at time "t." 1-CDF Hazard Function p(X=t|X≥t) Probability of the event occurring at time t given that it has not occurred prior to this time PDF/SF

4. Definitions Function Name FormulaDefinition Related Terms Probability Distribution Function p(X=t) Probability of event occurring at time "t." Cumulative Distribution Function p((X≤t) Probability of the event occurring prior to or at time "t." Sum of PDF for ≤ t Survival Function p(X>t) Probability of the event not occurring prior or at time "t." 1-CDF Hazard Function p(X=t|X≥t) Probability of the event occurring at time t given that it has not occurred prior to this time PDF/SF

5. Definitions Function Name FormulaDefinition Related Terms Probability Distribution Function p(X=t) Probability of event occurring at time "t." Cumulative Distribution Function p((X≤t) Probability of the event occurring prior to or at time "t." Sum of PDF for ≤ t Survival Function p(X>t) Probability of the event not occurring prior or at time "t." 1-CDF Hazard Function p(X=t|X≥t) Probability of the event occurring at time t given that it has not occurred prior to this time PDF/SF

6. Definitions Function Name FormulaDefinition Related Terms Probability Distribution Function p(X=t) Probability of event occurring at time "t." Cumulative Distribution Function p((X≤t) Probability of the event occurring prior to or at time "t." Sum of PDF for ≤ t Survival Function p(X>t) Probability of the event not occurring prior or at time "t." 1-CDF Hazard Function p(X=t|X≥t) Probability of the event occurring at time t given that it has not occurred prior to this time PDF/SF

7. An Example Year Probability Distribution Function Cumulative Distribution Function Survival Function Hazard Function 10.20 1.000.20 2 0.400.800.25 30.200.60 0.33 40.200.800.400.50 50.201.000.201.00 5+0

8. An Example Year Probability Distribution Function Cumulative Distribution Function Survival Function Hazard Function 10.20 1.000.20 2 0.400.800.25 30.200.60 0.33 40.200.800.400.50 50.201.000.201.00 5+0

9. An Example Year Probability Distribution Function Cumulative Distribution Function Survival Function Hazard Function 10.20 1.000.20 2 0.400.800.25 30.200.60 0.33 40.200.800.400.50 50.201.000.201.00 5+0

10. An Example Year Probability Distribution Function Cumulative Distribution Function Survival Function Hazard Function 10.20 1.000.20 2 0.400.800.25 30.200.60 0.33 40.200.800.400.50 50.201.000.201.00 5+0

11. What Do You Know? “Suppose that a cancer patient is equally likely to die at any time between 0 to 3 years from now. What is the hazard function for this person?” “Suppose that a cancer patient is equally likely to die at any time between 0 to 3 years from now. What is the hazard function for this person?” From Cox LA. Risk Analysis, Foundations, Models and Methods From Cox LA. Risk Analysis, Foundations, Models and Methods

12. Poisson Process of Arrival Constant hazard rates, h Constant hazard rates, h Poisson Distribution Poisson Distribution Arrival of sentinel event Arrival of sentinel event After “t” periods have passed without the event After “t” periods have passed without the event

13. Poisson Process of Arrival Constant hazard function

14. Poisson Process of Arrival Hazard function

15. Hazard Rate & Probability When the sentinel event is rare, the hazard function is essentially the same as the probability distribution function

16. Hazard Function from Different Sources Hazard function from all sources Hazard function from first source Hazard function from source n

17. Hazard Function from Different Sources

18. Attributable Risk Attributable Risk to Source i

19. Attributable Risk Hazard function for source i Attributable Risk to Source i

20. Attributable Risk Hazard function from all sources Hazard function for source i Attributable Risk to Source i

21. Example If the hazard function for medication error caused by a fatigued nurse is 1 in 1000 and the hazard function for medication error caused by illegible prescription order is 2 in 1000, what is the attributable risk to fatigued nurse? If the hazard function for medication error caused by a fatigued nurse is 1 in 1000 and the hazard function for medication error caused by illegible prescription order is 2 in 1000, what is the attributable risk to fatigued nurse?

22. Attributable Risk & Probable Cause Hazard function from all sources Hazard function for fatigued nurse Attributable Risk to fatigued nurse

23. Complete Specification of Combination of Hazard Functions Combinatorial combination Combinatorial combination Not enough data Not enough data Estimate missing effects Estimate missing effects

24. Estimating Missing Hazard Functions 1. Unknown combination of cause X and a binding constraint Y: For example, a cause of medication error is miscalculation by the nurse of the necessary dose. A constraint for this is verification by an independent observer. If a nurse miscalculates the dose but the error is found in verification, then it is unlikely to have a medication error from this cause.

25. Estimating Missing Hazard Functions 2. Unknown hazard of several causes (X, Y, and Z) required to be simultaneously present:

26. Estimating Missing Hazard Functions 2. Unknown hazard of several causes (X, Y, and Z) required to be simultaneously present: For example, for a patient to fall, there must be a slippery floor and some cognitive impairment. The hazard of slippery floor or cognitive impairment by itself is minimal but the combination of these two causes makes falls much more likely.

27. Estimating Missing Hazard Functions 3. Unknown combination of non- interacting, independent causes:

28. Estimating Missing Hazard Functions 3. Unknown combination of non- interacting, independent causes: For example, wrong side surgery might be due to erroneous marking, not following the nurse’s marking or wrong information from the patient. The missing hazard of the combination is the sum of the hazard associated with each cause.

29. Estimating Missing Hazard Functions 4. Unknown combination of interacting, dependent causes:

30. Estimating Missing Hazard Functions 4. Unknown combination of interacting, dependent causes: For example, consider the effect of poor training, fatigue and similar bottles on medication error rates. If fatigue and poor training interact to make things worse, then the hazard of combined poor training and fatigue should be added to the hazard associated with similar bottles.

31. Estimating Missing Hazard Functions 5. Unknown, non-interacting and independent causes

32. Take Home Lesson Hazard Functions Are Key to Understanding Relative Contribution of Multiple Causes to Sentinel Events

33. Minute Evaluations Please use the course web site to ask a question and rate this lecture Please use the course web site to ask a question and rate this lecture