Five methods of inductive reasoning laid out by John Stuart Mill in his A System of Logic (1848). Specifically, they guide in establishing evidence of causal relations. The summations below are taken from A System of Logic's 1859 edition.
1. The Method of Agreement
If two or more instances of the phenomenon under investigation have only one circumstance in common, the circumstance in which alone all the instances agree, is the cause (or effect) of the given phenomenon. (Mill 224)
2. The Method of Difference
If an instance in which the phenomenon under investigation occurs, and an, instance in which it does not occur, have every circumstance save one in common, that one occurring only in the former; the circumstance in which alone the two instances differ, is the effect, or cause, or a necessary part of the cause, of the phenomenon. (Mill 225)
3. The Joint Method of Agreement and Difference
If two or more instances in which the phenomenon occur have only one circumstance in common, while two or more instances in which it does not occur have nothing in common save the absence of that circumstance; the circumstance in which alone the two sets of instances differ, is the effect, or cause, or a necessary part of the cause, of the phenomenon. (Mill 229)
4. The Method of Residues
Subduct from any phenomenon such part as is known by previous inductions to be the eject of certain antecedents, and the residue of the phenomenon is the effect of the remaining antecedents. (Mill 230)
5. The Method of Concomitant Variations
Whatever phenomenon varies in any manner whenever another phenomenon varies in some particular manner, is either a cause or an effect of that phenomenon, or it connected with it through some fact of causation. (Mill 233)