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 === Joint Method of Agreement and Difference === If a group of situations has only one antecedent in common and all exhibit the same result, and if another group of similar situations lacks that antecedent and fails to exhibit the result, then that antecedent causes the result. Symbolically, abc⇒ZYX, ade⇒ZWV, and afg⇒ZUT; bdf⇒YWU and bceg⇒XVT, ∴a⇒Z.

This method is very similar to the methods of agreement and of difference, but it lacks the simple, simultaneous pairing of presence or absence between one antecedent and a corresponding result. Effectively, this method treats each ‘situation’ or experiment as one sample in a broader experiment demonstrating that whenever a is present, Z results, and whenever a is absent, Z is absent. The method makes the seemingly unreasonable assumption of ‘all other things being equal’; yet this assumption is valid if the experiment is undertaken with adequate randomization.

Method of Concomitant Variations
If variation in an antecedent variable is associated systematically with variation in a consequent variable, then that antecedent causes the observed variations in the result. Symbolically, abc⇒Z, ab∆c⇒∆Z, ∴c⇒Z; or abc⇒WXYZ, ab∆c⇒WXY∆Z, ∴c⇒Z.

The method of concomitant variations is like a combination of the methods of agreement and difference, but it is more powerful than either. Whereas the methods of agreement or difference merely establish an association, the method of concomitant variations quantitatively determines the relationship between causal and resultant variables. Thus the agreement and difference methods treat antecedents and consequents as attributes: either present or absent. The method of concomitant variations treats them as variables.

Usually one wants to know whether a relationship is present, and if so, what that relationship is. This method simultaneously addresses both questions. Furthermore, nonlinear relationships may fail the method of difference but be identified by the method of concomitant variation. For example, a method-of-difference test of the efficacy of a medication might find no difference between medicated and unmedicated subjects, because the medicine is only useful at higher dosages.

A quantitative relationship between antecedent and result, as revealed by the method of concomitant variation, may provide insight into the nature of that relationship. It also permits comparison of the relative importance of various causal parameters. This technique, however, is not immune to two limitations of the two previous methods:
 * determination that a significant relationship exists does not prove causality; and
 * other variables must be prevented from confounding the result. If they cannot be kept constant, then their potential biasing effect must be circumvented via randomization.

The correlation techniques described earlier in this chapter exploit the method of concomitant variations.

Method of Residues
If one or more antecedents are already known to cause part of a complex effect, then the other (residual) antecedents cause the residual part of the effect. Symbolically, abc⇒WXYZ, ab⇒WXY, ∴c⇒Z.