<p>Factor separation is widely used in the analysis of numerical simulations. &#160;It allows changes in properties of a system to be attributed to changes in multiple variables associated with that system. &#160;There are many possible factor separation methods; here we discuss three previously-proposed methods that have been applied in the field of climate modelling: the linear factor separation, the Stein and Alpert (1993) factor separation, and the Lunt et al (2012) factor separation. &#160;We show that, when more than two variables are being considered, none of these three methods possess all four properties of 'uniqueness', 'symmetry', 'completeness', and 'purity'. &#160;Here, we extend each of these methods so that they do possess these properties for any number of variables, resulting in three factor separation methods -- the 'linear-sum' , the 'shared-interaction', and the 'scaled-total'. &#160;We show that the linear-sum method and the shared-interaction method reduce to be identical in the case of four or fewer variables, and we conjecture that this holds for any number of variables. &#160;We present the results of the factor separations in the context of studies that used the previously-proposed methods. &#160;This reveals that only the linear-sum/shared-interaction factor separation method possesses a fifth property -- `boundedness', and as such we recommend the use of this method in applications for which these properties are desirable.&#160; &#160;The work described here is in review in Geoscientific Model Development - see https://gmd.copernicus.org/preprints/gmd-2020-69 .</p>