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.