# Weights for multi-round analyses

When data from multiple rounds (and multiple countries) are combined in an analysis, varying design and population weights have to be accounted for. Overall design and perhaps also population weights have to be calculated to allow for valid inference from the combined samples to the population.

The structure of the ESS cumulative data set is illustrated in table 4.2.

ESS Round | Country | Design weight |
---|---|---|

Round I | C1 | DW 1 |

DW 2 | ||

DW 3 | ||

C2 | DW 1 | |

DW 2 | ||

DW 3 | ||

C3 | DW 1 | |

DW 2 | ||

DW 3 | ||

C4 | DW 1 | |

DW 2 | ||

DW 3 | ||

Round II | C1 | DW 1 |

DW 2 | ||

DW 3 | ||

C2 | DW 1 | |

DW 2 | ||

DW 3 | ||

C3 | DW 1 | |

DW 2 | ||

DW 3 | ||

C4 | DW 1 | |

DW 2 | ||

DW 3 | ||

Round III | C1 | DW 1 |

DW 2 | ||

DW 3 | ||

C2 | DW 1 | |

DW 2 | ||

DW 3 | ||

C3 | DW 1 | |

DW 2 | ||

DW 3 | ||

C4 | DW 1 | |

DW 2 | ||

DW 3 |

In line with the above example, in the ESS cumulative data set, multiple respondents (not shown) are nested within multiple weighting classes1 (DW 1 to DW 3) within countries (C1 to C4) within ESS rounds (Round I and II). To simplify presentation here, the number of weighting classes is assumed not to vary between countries and rounds. This is not the case in reality, however.

The problem, then, is how to combine estimators from different rounds and perhaps also different countries. In the following, these two cases will be discussed, namely combining estimators from a) different rounds and the same country, b) from different countries and different rounds.

First of all, some definitions will be useful. Let the design weight of the *w*th weighting class in the *c*th country in round *r* be denoted by *d _{wcr}* and let the population weight be referred to as

*p*.

_{rc}
Let *a* be the vector of length *z* with elements *r*. Of course, a ⊂ l, which is the vector of length *m* with elements *i = {1, ... , m}* indicating ESS rounds where data is available. If we consider rounds one and two, *a* would simply be (1, 2), which, as we can see, is of length *z* = 2. Since, no more than the first two rounds that were chosen are available so far, *l = a*.

Similarly, let *s* be the number of countries for which a combined estimator is to be calculated. The vector *k _{r}* of length

*q*with elements

_{r}*j = {1 , ... , q*indicates the participating countries in the

_{r}}*r*th round.2 The vector

*k*of length

*s*then contains the countries under consideration. To make this easier to illustrate, let us assume that every element (country) of

*k*must be an element of every

*k*, which means that all countries,

_{r}*k*, under consideration must have participated in all rounds under consideration. To illustrate, assume the countries under consideration to be Germany (DE), the United Kingdom (GB), and Portugal (PT). Then

_{c}*k*= (DE, GB, PT).

Let an unbiased estimator, *θ*, that takes into account either population weights, design weights or both be denoted by *θ(p)*, *θ(d)*, and *θ(p,d)*, respectively. Furthermore, let their unbiased combined equivalents be denoted by (p), (d), and (p,d)3.

#### Footnotes

- [1] Most commonly, but not always, the weighting classes correspond to the household size classes.
- [2] Note that, instead of numbers, labels that correspond to international abbreviations are used to identify countries.
- [3] Note that
**combined**is irrespective of the combination involved. Each of the combinations ‘multi-round single-country’, ‘single-round multi-country’ and ‘multi-round multi-country’ is possible.