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I have a question on testing mortality table. Suppose I am given a simple mortality table:

age  | prob of dying  | prob of surviving
---------------------------------------
20   |  0.01          |   0.99
21   |  0.02          |   0.98
22   |  0.03          |   0.97
23   |  0.04          |   0.96
...

I want to test whether the table fit my observed data, i.e., whether the Observed values

age  | actual dead    | actual survivors
---------------------------------------
20   | 0              |  397
21   | 1              |  189
22   | 0              |  136
23   | 2              |  100
...

fit the expected values

age  | actaul dead       | actual survivors
---------------------------------------
20   | 3.97 (397*0,01)   |  393.03
21   | 3.8   (190*0.02)  |  186.2
22   | 4.08  (136*0.03)  |  131.92
23   | 4.08 (102*0.04)   |  97.92
...

How do I do that? What method should I use? Should I use Chi-Square, although my expected data is quite small? or there is another method?

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The expecteds aren't all that low, the chi-squared should work fine.

There are other alternatives.

You could do an "exact" test, akin to a Fisher Exact test, but based off the binomial.

You could even do a G-test, another large sample test based off the likelihood-ratio test.

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  • $\begingroup$ Thank you very much for the answer. But can I use Fisher's Exact test in this situation, I thought it is used as a test of independence? in my example the expected values are not calculated by the formula $\endgroup$ – naninko Apr 16 '13 at 12:30
  • $\begingroup$ Hence the use of the phrase "akin to"; it's not actually the exact test that Fisher did. I made some changes to what I wrote; the 'condition on both margins' didn't make sense. $\endgroup$ – Glen_b -Reinstate Monica Apr 16 '13 at 12:40
  • $\begingroup$ Thank you very much for the answer. I alreeady used Chi Square and G test. I am not sure if that is the right apporach, since I already have predetermined probabilities, so my expectations are calculated by the formula $$ E(X_{20})=p_{20}*n_{20} $$ and not by the formula used in exact tests for contingcency tables (row*column/total_number) $$ E(X_{20})=\frac{n_{20}*d}{n} $$ $p_{20}$ -prob a 20 year old to die $n_{20}$ -total number of 20 year olds $d$- total number of dead $Could I still use a test of independence, like Fisher's or chi square, or should I consider a different approach? $\endgroup$ – naninko Apr 16 '13 at 13:04
  • $\begingroup$ Check the first paragraph here, and then this section $\endgroup$ – Glen_b -Reinstate Monica Apr 16 '13 at 13:40

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