7.10 Limited information statistics (Cont’d)

The $M_2$ statistic compares whether the observed and model-predicted first two marginals equal or not. The hypothesis of interest is,

• $H_0$: The model fits data well
• $H_1$: The model does not fit data well

The $M_2$ test statistics is:

$$M_2=N\left(\boldsymbol{p}_2-\hat{\boldsymbol{\pi}}_2\right)^T \hat{\boldsymbol{C}}_2\left(\boldsymbol{p}_2-\hat{\boldsymbol{\pi}}_2\right)$$ The $M_2$ statistic is also $\chi^2$ distributed, with $df$ being the number of elements in $\mathbf{p}_1$ and $\mathbf{p}_2$ minus the number of parameters. The process of obtaining $\hat{\boldsymbol{C}}_2$ is well explained in Maydeu-Olivares & Joe (2005).

References

Maydeu-Olivares, A., & Joe, H. (2005). Limited-and full-information estimation and goodness-of-fit testing in 2 n contingency tables: A unified framework. Journal of the American Statistical Association, 100(471), 1009–1020.