Raking

The Two-Variable Raking Procedure

In two-variable (or two-dimensional) raking, population totals are known for the strata of two distinct variables. This situation can be represented by a rectangular array of cell estimates based on the initial sample weights with the “true” population row and column totals, called the row and column control totals, known. One objective is to adjust the cell entries so that both the row and column sums of the cell entries add up to the control totals.

The initial sample weights reflect the probabilities with which the sample units were selected and may incorporate a nonresponse adjustment as well. To preserve the original sample design and possible non-response adjustment, one objective of the raking procedure is to change the initial cell estimates as little as possible, subject to their adding up to the control totals in both dimensions.

The steps in raking are as follows. For the first variable (the row variable in a two-dimensional cross-tabulation), multiply the first row by the control total for row 1 divided by the sum of the cell entries for row 1. This adjustment will now make the cell entries for row 1 sum to the control total for that row. Then do the corresponding adjustment (multiplying the appropriate raking ratio for row 2 by each row 2 entry) to row 2. Continue in this way until all rows have been adjusted with each row summing to its respective control total.

For the second variable, multiply the first column by the control total for column 1 divided by the sum of the cell entries for column 1. At this point column 1 adds up to the control total for column 1, but the rows may no longer add up to their respective control totals. Continue multiplying each column by its respective raking ratio until all columns add up to their respective control totals. Then repeat the raking ratio adjustments on the rows, then the columns in an iterative fashion. (The appropriate raking ratios for the rows and columns will likely change in each iteration until the process converges.) Raking stops when all the rows and columns are within a pre-specified degree of tolerance or if the process is not converging. If raking converges, the new sampling weight for a sampling unit in a particular cell is the initial sampling weight times the raking-adjusted cell estimate divided by the initial cell estimate.

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