7.8 Full information statistics (Cont’d)

Two full-information statistics can be defined as $${\chi }^2=N\sum _y\frac{(p_y-{\hat{\pi }}_y{)}^2}{{\hat{\pi }}_y}$$ and $$G^2=2N\sum _y{p}_y\frac{p_y}{{\hat{\pi }}_y}$$

Both statistics are ${\chi }^2$ distributed. The associated degrees of freedom is equal to the number of response patterns - the number of parameters - 1.

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

Exercise: write your own function to calculate these two statistics.