correlation may be due to $x$ and $y$ being strongly correlated, but it may also be due to $n$ having a high variance. The higher the variance of $n$, the higher the size-dependent correlation. In fact, if $n$ is distributed according to a log-normal distribution with a very large variance, the size-dependent correlation will be close to $1$, regardless of the extent to which $x$ and $y$ are correlated. The strength of the size-dependent correlation then mainly reflects the variance of the size of institutions.

In our analysis, we consider both a size-dependent and a size-independent perspective. We calculate the proportion of publications that belong to the top $10\%$ most highly cited publications in their field and year, which we call the PP(top $10\%$). In addition, we use PP($4^*$) to denote the proportion of publications with a $4^*$ rating in the REF. The PP(top $10\%$) and PP($4^*$) are similar in spirit.[^10] They aim to identify whether publications have a high impact or are of high quality (“world leading”), respectively. Other citation metrics, such as those based on average normalised citation counts, are more difficult to translate into a $4^*$ rating system. Both the PP($4^*$) and the PP(top $10\%$) are clearly 