of an institution in a certain UoA. As such, a reputational yardstick is *size-independent*: it concerns an average or percentage, not a total, and it does not depend on the size of an institution. In the REF, funding is related to the total scientific quality of the publications of an institution in a certain UoA. As such, funding is *size-dependent*: institutions with more output or staff generally receive more funding. Of course, quality also affects funding: institutions that do well receive more funding than equally sized institutions that do less well. Both the size-dependent and size-independent perspectives are relevant to the REF. We therefore believe that both perspectives are important in deciding whether metrics can replace peer review.

Many studies of the REF and its predecessors have analysed either size-dependent or size-independent correlations. Size-dependent correlations are typically much higher than size-independent correlations. For example, @Mryglod2013 [@Mryglod2013a] found size-dependent correlations on the order of $0.9$ but much lower correlations for size-independent metrics. Similarly, @Harzing2017 found a very high size-dependent correlation.

Size-dependent correlations can be expected to be larger in general. Let us make this a bit more explicit. Suppose we have two size-independent metrics $x$ and $y$ (e.g. metrics and peer 