Hi there, a couple of questions regarding statistical tests for graduation: 1. For the chi square test, Section 7.1 gives the formula for the test statistic as the sum of the [z_x]^2, where [z_x] are the standardised deviations. But the accompanying example in the notes calculates the test statistic as the sum of (A-E)^2 / E, as is usual for chi square tests. While these two formulae are the same for the Poisson and multiple state models, they differ for binomial models. Which is correct? 2. For the cumulative deviations test, the solution to Question 12.19 applies a continuity correction to the number of deaths [d_x] in the calculation of the test statistic, whereas the solution to Question 12.13 does not. Which is correct?
1. for binomial model, assumption is made that under H0: qx = 0, hence variance of Dx is N*qx*(1-qx) = N*qx as (1-qx) = 1 under H0. Hence , Zx^2 and the usual formuale (A-E)^2 / E become same.
1. If you're doing a chi-square test on a graduation, you do the sum of (zx)^2. This is fine for either the binomial or poisson models. What manish_rex says is nearly correct... For the binomial model, we say that since qx is small, it's approximately equal to 0 (rather than qx is equal to zero). So, under the binomial model: Dx (the number of deaths aged x) ~ bin(Ex, qx) Using the normal approximation to the binomial distribution (for large samples): Dx ~ N(Ex*qx, Ex*qx*(1-qx)) And since 1-qx is approximately 1, the variance simplifies to Ex*qx. This means the standardised deviation is of the form: zx= dx - Ex*qx/sqrt(Ex*qx) and (zx)^2 = (dx -Ex*qx)^2/Ex*qx = (Actual deaths - Expected deaths)^2/Expected deaths As you'd wish for a chi-squared test. 2. Looking at recent examiners reports in which cumulative deviations tests have been carried out, the continuity correction is not applied. So don't worry about it.
