When do we know if something is a problem?

I’ve written about this before, both when discussing the implications of an unidentified submarine in the Stockholm waters (described in Should we in Sweden be surprised when we learn that there are foreign submarines in the Stockholm archipelago? Hardly!) and when I try to decide if my bike route to work is dangerous or not (two posts: (i) Risky business on land and (ii) One accident doesn’t mean thing is risky).

Now, a couple of days ago I witnessed the result of one more bike accident at the very spot described before. This time I estimate that the accident was a result of the sharp turns in combination with the curb stones collecting water that had frozen during the night, i.e., the physical layout created a hazardous condition.

So, the question is: I have a record of a very low number of incidents (in the submarine case one incident, and in the bike lane case two incidents), can I judge (calculate) the likelihood of this being evidence of a problem?

Note: In these cases I have a record of the incidents and the number of incidents and conditions for each incident are known. I don’t base it on hearsay or for example on newspaper articles talking about many biking accidents (which could be ten persons describing what could be the same accident). For the submarine incident this means that I trust the description the Swedish Armed Forces gave regarding some basic aspect of the incidents.

Judging if the incidents are a one-time event or evidence of a bigger problem came in the submarine case down to estimating how likely it is that we observe a submarine if it is in the area. In the biking case observing is not a problem, i.e., it is about how often we are on the scene.

For being able to calculate total number of bike accidents we need more information. In short we need to estimate how often I’m on site and what that mean.

Calculation example: Rush-hour accidents (in one intersection)
No of observed incidents: 2

Observation time per day [minutes]: 1

Observations per week [per week]: 9
Average incident observation window [min] (duration of accident): 10

Total period [years]: 1.5

Weeks per year [weeks]: 46
Days per week [days]: 5
Hours per day, rush-hour [hours]: 2

=> Total period [minutes]: 41400

=> Total observation time [minutes]: 6831
=> Observation percentage: 17%

Estimated no of incidents in total (from observation percentage): 12

Bikers per minute (during rush-hour): 2

=> Sample size, calculated [no of bikers]: 13662

=> Incident per observed biker: 0.01%

Statistical error margin (given sample size and a 95% confidence and normal distribution): 0.02%
=> Number of incidents: 0.01% ±0.02%, i.e., between -5* and 29

*) we cannot have negative number of accidents, i.e., assumption about normal is distribution wrong (wrong left tail). Two incidents are observed, i.e., these two could statistically be the only two

=>                                                   

Minimum number of incidents (the observed ones): 2

Expected number of incidents: 12

Maximum number of incidents (given stat. error margin): 29

Per year

Minimum number of incidents: 1

Expected number of incidents: 8

Maximum number of incidents: 19

But if this 1 to 18 accidents per year in one intersection a problem? Is it too often? If we put that question in a safety (risk) perspective we also need to know the consequence of an incident. I estimate that one out 20 incidents lead to serious injuries. The probability for exactly N injuries is then described by the binomial distribution. Given this frequency, the expected number of incidents and the binomial distribution we can calculate the probability for exactly one, two, three… serious accidents per year. Then we can calculate a FN-curve for each of the three number incidents above (1, 8 and 19 per year) where N is the number of persons seriously injured per year in this intersection.

FN-curve describing the probability for N or more serious injured persons per year if there is one accident per year (filled circle); eight accidents per year (circle); and 19 accidents per year (squares). (C) Hans Liwång 2016.

However, to my knowledge there are no FN-criteria for one intersection, so we still do not know if this is a problem or not. I however note that the probability for at least one seriously injured person per year is close to one which sounds high for one insignificant intersection (I pass at least 40 of that size per day).

The most crucial figure in the calculation above is the average duration of an incident. Note also that if my wife, who also twice a day pass the same spot on her bike, tell me that she had an accident on this spot I could not include that information because it is outside my sample.

I promise to get back with a maritime case complementing the FN-curve I did when describing that the risk for refugees in the Mediterranean so high it is intolerable.