Method

For a sequence of $m$ modalities, there are $m \times m = m^2$ first-order transitional frequencies, as represented by the general transitional frequency matrix#3571#> (cf table 3.1). These observed transitional frequencies are compared with what would be expected if the state of the system at time $t+1$ was independent from its previous state. The best estimates are given by the maximum likelihood estimates [Bishop, Fienberg HollandBishop 1975,EverittEveritt1992]:

$$E_{ij} = N \hat{p}_{i+} \hat{p}_{+j} = N \frac{n_{i+}}{N} \frac{n_{+j}}{N}$$ (5.9)

$$= \frac{n_{i+} n_{+j}}{N}$$ (5.10)

These expected frequencies are then replaced in the well known $X^2$ formula:

$$\chi^2 = \sum_{i,j} \frac{(n_{ij} - E_{ij})^2}{E_{ij}}$$ (5.11)

A less known statistic, the likelihood ratio $G^2$, may also be used for testing independence between categorical variables:

$$G^2 = 2 \sum_{ij} n_{ij} \log \frac{n_{ij}}{E_{ij}}$$ (5.12)

with $(m-1)^2$ degrees of freedom.

The two statistics $X^2$ and $G^2$ have asymptotically the same behavior. Should they differ too greatly, such as with sparse tables, researcher is advised to be cautious about the validity of the results.