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# The complicated answer to “How long do I have to live?”

February 18, 2015 | 1 Comment

Physicians say that once a patient hears the word “cancer”, their brain shuts down and they don’t hear anything else. To be fair to the patients, understanding survival statistics isn’t always simple.

Let’s take just one example:

The three-year survival rate is different from the third-year survival rate. If you have been told that the three-year survival rate is 50% and now it is the third year since your diagnosis, your probability of surviving the year is likely to be much higher than 50%

Let’s take a look at this example, with the number of patients diagnosed each year and how many were alive the 1st, 2nd and 3rd year after diagnosis

Year | N | 1st | 2nd | 3rd

2012 | 75 | 60 _ | 56 _| 48

2013 | 63 | 55 _| 31 _|___

2014 | 42 | 37 _| ___|___

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The probability of survival year 1 = 152/180 = .84

The probability of survival year 2 = 87/115 = .76

The probability of survival year 3 = 48/56 =.86

To find the probability of survival in the THIRD YEAR you divide the number of people alive at the end of three years, which is 48, by the number of people alive at the beginning of the third year, which is 56. (The number of people who survived the second year is the same as the number of people who were alive at the beginning of the third year.)

48/56 = 86% probability of survival the third year.

So, IF YOU HAVE SURVIVED TO THE BEGINNING OF THE THIRD YEAR, your probability of survival in that year is 86%.

However, if you asked me on day 1 what your probability of living three years is, I would say 55% (actually, 54.9024% if you want to be precise).

How can your three-year survival be lower than third-year survival? Here’s how:

*We can only measure third-year survival on people who survived the first two years … *

We followed (75+63 +42) = 180 people for one year. At the end of that year, we had 152 survivors (60 +55 + 37).

So, first year survival rate = 152/180 = 84%

Of those 84%, only 76% survived the second year. Of the people who survived the second year, 86% survived the third. So, what percent survived all three years?

.84 x .76 x .86 = .549024 or, 54.9%

Sometimes people will look at three-year survival rate and think, WRONGLY,

The three-year survival rate is only a little better than 50% and I have already lived to the third year, I must have a 50-50 chance of dying this year.

Actually, that is not correct. As the example shows, your chance of surviving the third-year may be substantially greater than the three-year survival rate.

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Want to exercise your brain while having fun? Play Fish Lake, canoe down rapids, escape your enemies and review fractions. If you are already smart enough, consider donating a copy to a low-income school or after-school program.

# Comments

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## Blogroll

- Andrew Gelman's statistics blog - is far more interesting than the name
- Biological research made interesting
- Interesting economics blog
- Love Stats Blog - How can you not love a market research blog with a name like that?
- Me, twitter - Thoughts on stats
- SAS Blog for the rest of us - Not as funny as some, but twice as smart. If this is for the rest of us, who are those other people?
- Simply Statistics, simply interesting
- Tech News that Doesn’t Suck
- The Endeavor -John D Cook - Another statistics blog

I think you forgot to write something you meant to write.

“The three-year survival rate is different from the third-year survival rate. If you have been told that

Let’s take a look at this example…”