Appendix 3: Little and Su Method

Formulae

The Little and Su method was implemented as follows:

(a) Column (wave) effects of the form

where

were computed for each wave j = 1, …, m, and for each age group h = 1, …, c, where Y_hj is the sample mean of variable Y for wave j, age group h based on complete cases and Y_h is the global mean of variable Y for age group h based on complete cases.

(b) Row (person) effects of the form

were computed for both complete and incomplete cases. Here the summation is over recorded waves for case i; m_i is the number of recorded waves; Y_hij is the variable of interest for case i, wave j, age group h; and c_hj is the simple wave correction from (a).

(c) Cases were ordered by Y_h⁽ⁱ⁾, and incomplete case i is matched to the closest complete case, say I within age group h.

(d) Missing value Y_hij was imputed by

where the three terms in square parentheses represent the row, column, and residual effects, the first two terms estimate the predicted mean, and the last term is the stochastic component of the imputation from the matched case.

Example

Suppose we have the following small sample of fictitious responses to current wages and salaries.

All cases

From this example, we see that observation 1 did not respond to the current wages and salaries question in wave 1, but provided responses in subsequent waves. Observations 5 and 6 also partially responded and wages and salaries information are not provided in all 3 waves.

The first step in the Little and Su method is to calculate the column effects based on complete cases only. Complete cases were defined as individuals that were interviewed in all 3 waves and responded in all 3 waves for the variable of interest. In this example, the complete cases are:

Complete cases

The column effects are calculated using formula (a) above and are computed to be:

Column effects

The Little and Su method incorporates trend information into the imputed amounts via the column effects. In this example, the wave 1 column effect of 0.70 indicates that the mean current wages and salaries in wave 1 is 30% lower than the overall mean current wages and salaries, and the means in waves 2 and 3 are 6% and 24% higher than the overall mean, respectively.

Next, the row effects are calculated using formula (b) above and are computed to be:

Row effects

The sample is then ordered by the row effects, and the closest donor is identified.

Sorted by row effects

Once the closest donor has been identified, the missing value is imputed by multiplying the actual value for the variable of interest of the donor with the row effect of the recipient divided by the row effect of the donor.

In this example, the imputed current wages and salary amounts using the Little and Su method are highlighted below.

Impute missing values