# The mathematics behind our calculations

Life tables tell us the fraction of people that are of age 75 that die before before reaching age 76. Call this q(75). We have this information for both males and females. But, we don't have it for a particular individual.

Call p(75) your personal chance of dieing before reaching 76 if you make it to 75. Suppose you are i years of age. Then what we need is to find a way to figure out your p(i), your p(i+1), your p(i+2), etc.

So our problem is to convert this generic number into one appropiate for a particular individual, namely you. There are two basic ways we have to modify it. First if you drive a 100 thousand miles this year, you will have a 1% chance of dieing in an automobile accident. So, your personal chance of death this year, p(i) is q(i) + .01. If you did this next year also, p(i+1) = q(i+1) + .01, etc.

## Multiplicative risks

Other health habits only effect you when you are older. For example, inspite of smoking being very bad for you, your chance of dieing from smoking when you are in your 20s are very slim. This is clear if we think of smoking a pack a day as doubling your chance of death. This is a small difference to a 20 year old (chance of death goes from 1/1000 to 2/1000. So it would be a small chance compaired to the driving risk described above. But, if you are 60 it goes from 1/100 to 2/100, so it is about the same as the driving risk above. But, if you were 80, it would double your 1/20 chance of death to a 1/10 chance of death - - a 5% increase in death rate.

In fact we don't use the above method since it might generate a chance of death greater than one. What we use instead is what is called the Cox proportional hazards model. For small numbers it works out identically to the above. For large probabilities, it makes sure the probability stayes between zero and one. It is easier to state if we work with the chance of staying alive, 1-p(i):

1 - p(i) = (1 - q(i))^(1+c)

where c is a correction. If c is zero, then

1 - p(i) = 1 - q(i)

so the chance of your living through out the year is the same as for a typical person. But, if c = 1,

1 - p(i) = (1 - q(i))^2

If q(i) is small, then p(i) = 2 q(i). So your chance of dieing is doubled. A more accurate way to think about it is that if your c=1, then you have to survive the year twice.

## Combining it all together

Our proceedure then is to first figure out the base accident rate. (Assuming it is mostly due to driving about 10k miles per year). What is left over is health base rate. We then correct the base health rate by using the Cox model. Finally we add in the actual number of miles you drive to get p(i). This is done for each year (assuming your driving and other health habits stay the same.) Finally it is all put together to get your predictions.

ungar@cis.upenn.edu