the following examples. If a split-off household is not selected into tracking, the parent household is followed with certainty. The probability of selection (p2 in the equation below) for this household is one. Similarly, if the household is selected into tracking but did not split, the parent household is followed with certainty. The probability of selection for this household is also one.

The situation becomes more complex when the household selected into tracking has split. In that case, the probabilities of selection must be pooled and averaged. It may be more intuitive to think of this in terms of expansion factors, which are the reciprocal of the probability of selection. If a household from an EA with 10 households is selected into tracking and one member splits from the original household, the parent household would just represent itself, and therefore have an expansion factor of one. Since there are only two original households selected out of the ten possible, any split-offs from the selected household would represent the potential split-offs of five households in the EA. Since the definition of parent and split-off is arbitrary, however, the expansion factors (one and five respectively) are averaged over two households, giving each a value of three. The probability of selection would then be the reciprocal or<sup>1</sup> 3 .

Similarly, if a household from an EA with 10 households is selected into tracking and splits into two additional households, the parent household would still have an expansion factor of one and each split-off five. In this case, it would be averaged over three households, giving each a value of<sup>11</sup> 3 . The probability would then be 11<sup>3</sup> for each household. Additionally, the calculations remain the same regardless of whether the split-off household is found and interviewed or if it becomes part of the attrition calculations, although in the case of attrition, it is necessary to make assumptions about the number of households formed by the lost members.

The generalized formula for the probability of selection at this step is: where _z_ is the number of new households entering the survey from a given parent household (as multiple split-off members can form either a single new or multiple new households) and _q_ is the number of households in the EA in round 1.

It may seem somewhat counterintuitive that the sum of the conditional probabilities of selection into tracking for the parent and split-off households is not equal to one, since they stemmed from the same initial household. Since these probabilities total less than one, when the inverse is taken, the weight will increase by some number greater than one. When this step is incorporated into the final weight calculations, it will increase the total sum of the weights in such a way as to mirror the natural population growth of new households created between survey rounds.

# _4) Fair share correction_

Since the follow-up rules for the panel survey include interviewing split-off members in their new households along with all the current members of that household, the weights must allow for the incorporation of those who now live with original sample members. For example, if a young man living with his parents in round 1 has formed a new household in round 2 and resides with his wife and newborn child, the wife and infant will be incorporated into the survey and thus require a probability of selection. Such corrections, known as fair share corrections, are routinely used to distribute weight to new sample 5