Chi-square and likelihood ratio statistics

Perhaps the state of the system at a certain time is completely independent from what happened previously. Or may be it does depend on the past state. How do investigators separate these two cases?

A well-known tool to investigate whether the present state of the system depends on the previous state is the chi-square statistic [Bishop, Fienberg HollandBishop 1975,EverittEveritt1992]. This statistic is usually employed on structural contingency tables in order to determine if there is a dependence between 2 (or many) categorical variables. As was shown by scientists, this statistic may well be applied on transitional frequency matrices, a special case of contingency tables [Bishop, Fienberg HollandBishop 1975,Gottman Kumar RoyGottman Kumar Roy1990,Bakeman GottmanBakeman Gottman1986].

The chi-square statistic tests whether transitions are a first-order Markov process, that is, if the state of the system at time $t$ depends on its state at time $t-1$, or if state at time $t+1$ depends on state at time $t$. Higher order Markov processes refer to a dependence at higher lags ($t-2$, $t-3$, and so on; they are reviewed in chapter 8).

Researchers can test four different types of hypothesis using chi-square statistics [CastellanCastellan1979]:

1. fit of frequencies to specified probabilities
2. fit of marginal frequencies to specified probabilities
3. conditional fit of row (or column) frequencies to specified probabilities
4. variables independence

We will discuss the fourth type of hypothesis and let readers refer to Castellan for more details about the three others, since they are less frequent.