- test for
**Independence**between (nominal or ordinal) variables containing frequencies - test whether the occurrence of one variable (e.g. eye color
{blue, green}) is independent of another (e.g., sex {male, female})
using a
**Contingency table** - test for
**Goodness of fit** - identify those cells that are the main contributors for any
overall effect using
**Cell-wise examination**

- calculate
**expected frequencies**under the null hypothesis (Note: use**Goodman's Test for quasi-independence**if the diagonal or other cells are structurally zero) - calculate overall significance of a contingency table with
**X**Statistic: Σ(Observed frequency^{2}- Expected frequency_{i })_{i}^{2}**Likelihood-Ratio Statistic**(also known as Likelihood Ratio Chi-square, LRX^{2}, G^{2},**G**-statistic, negative log likelihood, nLL): - 2Σ*f*ln (_{i}*f*/_{i}*f*_{i}_{exp})

- compare these values to a Χ
^{2}distribution - These analyses can be performed using Java DataGrinder applets
- If sample sizes are small you can use a
**Randomization Test**(i.e., Monte Carlo Simulation)

- make sure that
**no expected frequencies are < 3**. lump cells with adjacent classes - X
^{2}and G are sample statistics; Χ^{2}is a theoretical frequency distribution. - G statistic is preferrable to X
^{2}:- The distribution of G fits a Χ
^{2}distribution more closely than does the distribution of X^{2} - G-values are additive while X
^{2}- values are not. Thus, total G can be partitioned into two components, the G for heterogeneity and the pooled G - in more complex designs G is easier to compute than X
^{2}as it is calculated from the observed frequencies only

- The distribution of G fits a Χ
- perform
**William's correction**for estimationg actual type 1 error.**Yate's**correction is too conservative - perform a cell-wise examination of the matrix using
**Freeman-Tukey**deviates (o - observed cell frequency; e - expected cell frequency) as:

F-T = Math.sqrt(o)+Math.sqrt(o+1)-Math.sqrt((4*e)+1)); compare to: Math.sqrt(df*c ^{2}_{a[1]}/nCells);

last modified: 3/18/08