DCluster (version 0.2-7)

achisq: Another Implementation of Pearson's Chi-square Statistic

Description

Another implementation of Pearson's Chi-square has been written to fit the needs in package DCLuster.

achisq.stat is the function that calculates the value of the statistic for the data.

achisq.boot is used when performing a non-parametric bootstrap.

achisq.pboot is used when performing a parametric bootstrap.

Details

This statistic can be used to detect whether observed data depart (over or above) expected number of cases significantly. The test considered stands for relative risks among areas to be equal to an (unknown) constant $\lambda$, while the alternative hypotheses is that not all relative risks are equal.

The actual value of the statistic depends on null hypotheses. If we consider that all the relative risks are equal to 1, the value is

T=$$\sum _i\frac{(O_i-E_i{)}^2}{E_i}$$

and the degrees of freedom are equal to the number of regions.

On the other hand, if we just consider relative risks to be equal, without specifying their value (i.e., $\lambda$ is unknown), $E_i$ must be substituted by $E_i\frac{O_{+}}{E_{+}}$ and the number of degrees of freedom is the number of regions minus one.

When internal standardization is used, null hypotheses must be all relative risks equal to 1 and the number of degrees of freedom is the number of regions minus one. This is due to the fact that, in this case, $O_{+}=E_{+}$.

References

Potthoff, R. F. and Whittinghill, M.(1966). Testing for Homogeneity: I. The Binomial and Multinomial Distributions. Biometrika 53, 167-182.

Potthoff, R. F. and Whittinghill, M.(1966). Testing for Homogeneity: The Poisson Distribution. Biometrika 53, 183-190.

See Also

DCluster, achiq.stat, achisq.boot, achisq.pboot