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Appendix 6. The Iterative Proportional Fitting procedure

A6.1. The Iterative Proportional Fitting (IPF) procedure is used when reliable estimates for a desired cross-classification cannot be obtained directly, but estimates of the variables of interest, and possibly some related variables, are available at a higher level of aggregation. An additional requirement for the use of the IPF procedure is that information on the relationship between the variables is available at the desired level of cross-classification.

A6.2. In the IPF procedure, these two sources of information appear as two distinct classes of inputs known as the association structure and the allocation structure. The association structure, representing the relationship between available estimates, is typically a two-dimensional table of estimates, and the allocation structure consists of estimates of various 'marginals' of the table. (A 'marginal' of a table is the set of quantities obtained by adding across all categories of any one or more of the cross-classifying variables in the table).

A6.3. The IPF procedure produces new estimates for each cell in the table by adjusting the initial estimates (the association structure) to agree with the marginal constraints provided by the allocation structure, in an iterative fashion. For illustration, take the case where the association structure is a two-dimensional table, with two one-dimensional marginals. First the elements of each row of the table would be prorated so that their total equals the corresponding marginal estimate, and then the elements of each column would be prorated so that their total equals the corresponding estimate in the other marginal. After this step, the estimates in the table would no longer add across the rows to agree with the first marginal, and so the steps would be repeated until the procedure converged to the unique solution which adds to the marginals while preserving the relationships as specified by the association structure.

A6.4. As population estimates and components of population change deal with whole numbers of persons, after convergence a rounding process that maintains the marginal totals is employed.

A6.5. When an IPF procedure needs to be applied to a distribution with positive and negative values in either the association structure or the allocation structure, plus-minus proration (see Appendix 7) is substituted for the standard method of proration.

A6.6. For a more detailed description of the IPF procedure, see Purcell, N.J., and Kish L. (1979) Estimations for small domains, Biometrics, 35, pp 365-384.