Now we can calculate the Hazard function. This is the function that calculates the probability of the event occurring this year if it has not occurred in the prior years. This is calculated as the ratio of the probability function and the survival function. Note the hazard function increases each year as the more the prosthesis does not fail the more likely that it will fail in the remaining years. So now we can answer the question asked earlier regarding what is the probability of the failure in year 3 if the hip replacement has been successful in years 1 and 2. This probability is 33%, significantly higher than the probability of failing in year 3 which was calculated to be 20%. 

Arrival Process

Processes with constant hazard rate have a Poisson distribution. Even when the hazard rate is not constant over time, it is a constant value at any particular time and therefore the Poisson distribution can be used to measure it. In these circumstance, any sentinel event can be thought of to have a Poisson distribution and arriving for the first time after �t� periods of time. In this sense, the Poisson distribution allows us to estimate the probability of a sentinel event occurring in the next time period given that it has not occurred so far.

The Poisson formula calculates the relationship between the probability of a sentinel event occurring in the next time period based on the event�s constant hazard rate �h�. Note that e to the power of the hazard rate is the probability that the event will not occur during the next time period. One minus this calculation is the probability that it will occur.

p(Surviving next period) = e^h

p(Sentinal event next period)=1-e^h

The Poisson formula can be turned around to calculate the hazard rate. Solving for h, we obtain a way of calculating the hazard function from the probability of occurrence of the sentinel event. This formula is commonly used in many risk analyses because the Poisson process allows the interpretation of hazard rate as the count of arrivals of a random event. Since the effect of multiple causes can be counted and added together, then this interpretation allows us to add and subtract hazard rates to estimate the effects of multiple causes and constraints.

h=-log(1-p)      

Where p is the probability of sentinel event occurring the next time period.

Different Source of Hazards

If we can assume that various causes are independent of each other, then the hazard function from all sources can be calculated as the sum of hazard function from each source. This is a very helpful concept as it allows us to calculate the total impact of multiple causes.  If h(t) shows the combined hazard function and h_i(t) shows the hazard associated with cause "i", then the hazard function of the combination can be calculated as:

h=h₁+h₂+...+h_n

For example, under assumption of independence, the hazard function for a smoker exposed to asbestos is said to be the sum of hazard due to smoking plus the hazard due to asbestos.

h=h_Smoking+h_asbestos

The combined hazard risk can also be used to calculate the attributable risk to a specific cause.  The attributable risk due to a cause is calculated as the ratio of its source specific hazard rate divided by the hazard function from all sources.

AR_i= h_i/ h

An example can help demonstrate the concept of attributable risk. 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 the nurse and to the physician?  Using the formula for attributable risk we calculate this as the ratio of nurse�s hazard rate over the total hazard rate. In this case it is 33%. In contrast, 66% of the medication error risk is attributable to the physician�s poor prescription writing skills. The most probable cause of the event is the physician's ineligible writing. In this fashion, relative attribution can be made to separate risks. Correct attribution, of course, is important to priorities one sets for remedies to the problem. If 66% of risk is due to ineligible prescriptions, then it might be best to resolve this problem first before addressing the problem of fatigued nurses. Of course, the best action is to address both causes but if we cannot and need to set a priority, the analysis shows how to focus on key causes that are more responsible for the observed sentinel events.

Missing Combination of Hazards

It is often not possible to collect sufficient data to estimate the possible combinatorial combination of impact of multiple causes of a sentinel event. For example, if there are 5 causes for a sentinel event and each cause maybe present or absent, then there are 2 to the power of 5 possible combination of causes � some of which are quite rare and therefore unlikely to occur even in long streams of data. In these circumstances, the hazard associated with combination of causes can be estimated from the available hazard functions.  The following provides a set of heuristic for how to estimate the hazard of missing combination of causes:

1. The first step is to set all hazard functions for combination of causes and a binding constraint to zero � as in these circumstances the constraint will nullify the effect of various causes.
   
         h_x&y=0             If x is a cause but y is a binding constraint on the cause
         h_x&y=h_x-h_y        If x is a cause but y is a non-binding constraint on the cause
   
   By the word "binding" constraint, we mean that when the constraint is present, the probability of the sentinel event is zero. For example, a cause of medication error is miscalculation of the necessary dose. A constraint for this cause is verification by an independent observer. If a nurse miscalculates the dose but the error is found in verification, then there will not be any medication errors due to miscalculation. Of course, one might argue that in very rare situations the verification maybe in error too, in which case verification is no longer a binding constraint. It reduces the hazard of miscalculation but not to zero, to a small number near zero.
2. In the second step, we estimate the hazard functions for causes that only have an effect when they simultaneously occur together. Usually, the variable p, the conditional probability of the sentinel event from the combined causes can be estimated from a registry of sentinel events. From this value, the hazard function for the combined causes can be estimated:
   
         h_x&y&z=-log(1-p_x&y&z)
   
   The individual cause hazard functions are not available and can be estimated to be zero or a minimal value because the causes will have no effect or a minimal effect unless all of them are present:
   
         h_x= h_x&y= h_x&z= h_y= h_y&z=0 or minimal value
   
   For example, for a patient to fall, there must be a slippery floor and some cognitive impairment. Both are causes of falls but neither is likely to have an effect by itself. In these circumstances, the hazard of slippery floor or cognitive impairment by themselves is minimal but the combination of these two causes makes falls much more likely.
3. In the third step, we estimate the missing hazard for combination of non-interacting independent causes.  The combined hazard is the sum of the available hazard functions for each of the causes.
   
          h_x&y&z= h_x+ h_y+ h_z             If X, Y and Z are 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 combination of these three causes might be considered non-interacting and independent. Under this assumption, the missing hazard of the combination is the sum of the hazard associated with each cause.
4. In the fourth step, we estimate the missing hazard of combination of interacting dependent causes. To do this, one of course needs an estimate of the interaction between the causes. When two causes interact, their joint hazard is different from the sum of the two.  In calculating the combined effect of a set of causes, it is important to use pair wise joint hazard rate instead of the sum of the hazards of each cause.  If the interaction increases the hazard, then the maximum of various sum of pair wise combination of the hazards is selected.  If the interaction decreases the hazard, then the minimum of various sum of pair wise combination of hazards is calculated. For example, assume that the three causes X, Y and Z interact to increase the risk.  The combined hazard of these three risks is calculated as the maximum of sum of pair wise combinations of causes.
   
           h_x&y&z= Maximum {h_x&y+ h_z, h_x&z+ h_y, h_y&z+ h_x}    
   
   For example, if poor training and fatigue interact with each other to increase medication error rates, then the combined hazard should be used in estimating the combination of these two causes and other causes. So if we want to estimate the effect of poor training, fatigue and similar bottles, then the combined hazard is equal to the hazard of poor training and fatigue plus the hazard associated with similar bottles.
5. In the last step, we estimate the value of hazard of various causes when only the hazard of combination of causes is available.  In the absence of any other information about the relative impact of each cause, the hazard associated with each cause could be estimated to be equal:
   
           h_x = h_y= h_z  = h_x&y&z/3    

The point of this lecture has been that hazard functions are key to understanding the combination of various causes.  We presented heuristics that can be used to estimate the hazards associated with missing combination of causes.  More details on an algorithm to estimate missing combination of causes is available in the section titled:  Analytical Relapse.

What Do You Know?

1. This section is under development

Presentations

To assist you in reviewing the material in this lecture, please see the following resources:

1. See the slides for the lecture

2. Listen to narrated lecture on calculating hazard functions and multiple causes.  See narrated lecture in three parts:
   
   Part 1:  Calculation of Hazard Function from Distributions
    
   
   Part 2:  Risk Attributed to Different Causes
    
   
   Part 3:  Risk Attributed to Different Cases (Continued)
    

Narrated lectures require use of Flash.

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