Zero cells

• Zero cells arise for two reasons: either that combination is impossible (structural zero) or it is rare and happens not to occur in the sample.

• The first table shown in this course () had a structural zero: no men on maternity leave. If you were to tabulate academics' jobs at both ends of a five-year span, one triangle of the table would be empty: no one gets demoted. Nearly half the table would be occupied by structural zeros.

• Structural zeros are not a problem: weight them out of the analysis (using /CSTRUCTURE in SPSS, for instance). We lose a degree of freedom for each cell weighted out, but they do not contribute any information, so this is correct.

• Sampling zeros are a different problem: they are an exacerbated case of sparseness. Depending on their prevalence and location in a table they may not be a problem.

• But if a model contains a term relating to a marginal that contains a zero total (e.g., if there is an entire row of sampling zeros) the true value of some corresponding parameters will be infinite: the computer will probably estimate a large parameter with a ridiculously large standard error, and the estimate will be very unstable.

• Sampling zeros can also cause practical problems for the program, causing it to have difficulty in converging (this is closely related to parameters being infinite).

• An easy solution to this problem is to add a small constant to the zero (or all) cells: Agresti suggests $10^{-8}$; the algorithm has no difficulty with very small numbers but true zeros confuse it.

• Agresti (section 7.4) and Lindsey (Ch. 5) are good on zero cells. Lindsey (in an earlier book) provides a macro which detects which zero cells are causing problems in a model, and refits the model excluding these cells. This can be regarded as a conservative test, as it excludes cells, reducing the degrees of freedom, and is more likely to affect complex models, making it more likely that they will be rejected.

• This automates a proposal Agresti makes, of excluding problem cells and refitting. Lindsey's macro is written in GLIM, but I have a Stata version which I can make available to anyone interested.