# Combining estimators from multiple rounds and the same country

It is often of interest to combine data from two ESS waves conducted in the same country in order to estimate a mean or proportion in that country. This may be the case if, for example, many respondents refuse to answer a certain item. Combining two or more waves can thus help to increase the sample size. The implicit assumption, however, is that the samples are taken from the same population. This should always be kept in mind.

Our combined multi-round single-country estimator is a weighted average of combinations of the respective single-round unbiased estimators. Weights are chosen in relation to the effective sample sizes, namely

wrc = nrc/deffrc

where nrc is the net sample size and deffrc the design effect (see Chapter 5 for more details about design effects) of the variable under study1 in round r and country c, respectively.

The combined multi-round single-country estimator can then be expressed as

The construction of the combined estimator as a weighted average of the single-round estimators takes two basic principles into account: firstly, an estimator from a wave with many respondents is trusted more than one from a wave with fewer respondents. Secondly, for a given sample size, more trust is placed in the estimator that has a lower design effect.

In the following example, the combined estimator of the STFLIFE variable is constructed for Germany, the United Kingdom and Portugal, respectively. In this context, the estimators for the first and second round of ESS are considered.

### Example 3

Consider a case involving combining the estimators of the variable ‘satisfaction with life’ (STFLIFE) from the first and second rounds of the ESS in Germany, the United Kingdom and Portugal.

First of all, the weighted estimators have to be calculated. These are shown in the following table.

Table 4.3. Satisfaction with life (STFLIFE), weighted mean
Country Round I Round II
DE6.96 7.17
GB7.07 7.40
PT5.91 6.15

When combining the estimators for Germany, the weighted average of the round I and round II estimators has to be calculated. We thus need the design effect and the net sample size for the STFLIFE variable in Germany in both rounds. Together with those of the United Kingdom and Portugal, they are shown in the following table:

Table 4.4. Design effect and net sample size for Satisfaction with life (STFLIFE)
Design effect Net sample size
Country Round I Round II Round I Round II
DE6.104.9529162870
GB2.322.3720451897
PT3.723.3914982052

The combined estimator for overall satisfaction with life in Germany is

The combined estimators for rounds one and two in the United Kingdom and Portugal, calculated in the same way, are 7.22 and 6.06, respectively.

We can see that, in Germany, slightly more weight is given to the estimator from the second round - mainly due to the smaller design effect. In the UK, on the other hand, the estimator in the first wave is trusted slightly more, mainly because of the larger sample size (the design effects of both waves are almost equal). In Portugal, however, much more weight is given to the estimator in the second round, as indicated by both the net sample size and the design effect.

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#### Footnotes

• [1] Where design effects are not available for a certain study variable, an overall design effect may serve this purpose.