DF chi-square: q_x from std table

In the χ2 test in X 4.11 (also displayed in CMP upgrade 2008-09), expected deaths are calculated using qx from AM92 table. There are 16 groups from age 30 through 45, both inclusive. The solution tests the statistic with 16 degrees of freedom i.e. equal to the number of groups.

In the core reading of Chapter 13, it is stated that we lose one degree of freedom for each parameter that we estimate, and we lose a further indeterminate number of degrees of freedom because of the constraints imposed by the choice of standard table.

In X 4.11, I believe choosing AM92 to determine the expected deaths at each age amounts to a ‘constraint’. If so, the degrees of freedom should have been reduced. But that is not so. Pointers?

Thanks.

 

I don't have the question here, but may I know what is the test statistic calculated?

I agree with you that they should test for less than 16 d.o.f, but sometimes the solution will just test the max d.o.f. If your statistic is greater than this critical value, then no matter how many degree of freedom you lose, you will reach the same conclusion which is to reject the fit.

 

Chapter 12, Section 7.1 proves quite helpful here. We're told that the Chi-square test...

In X4.11, we're interested in the first case. So ignore Chapter 13 for now - that's all about graduation, and not appropriate here.

Carry on reading 12.7.1, and you'll see that, when we've got m age groups, the test statistic should have m degrees of freedom. There are no constraints that prevent any x-year old from joining the "died aged x" category (any of the lives at risk could die), so no degrees of freedom are lost.

 

Thanks Jensen and Mark.

Mark: the difference you pointed out is relevant in context. The default thought 'test for graduation' needs a re-set in my mind! You are right: one is not testing for graduation here, so no degrees of freedom lost for standard table. Thanks again.