Sparseness and sampling zeros

• Sparseness occurs when the number of cells in a table is relatively large in comparison with the sample size. It may also occur in parts of a table which exhibits strong association: if many cases are on the diagonal, the bottom left and top right corners may be sparsely populated.
• Sparseness is a general problem in that the properties of $G^2$ and $X^2$ that we depend on for assessing fit are asymptotic large sample properties: we can trust these measures less when there is sparseness. For instance, where there are several cells with fitted values less than about five.
• General sparseness can be dealt with by collapsing the table, or collapsing variable categories (good for localised sparseness), or by getting more data.
• Some localised sparseness is not a disaster: models can usually cope with parts of a table which have a low probability.