is postulated, in which formula_21 is the "i"th observation of the dependent variable (hypothesized to be the caused variable), formula_22 for "j"=1...,"k" is the "i"th observation on the "j"th independent variable (hypothesized to be a causative variable), and formula_23 is the error term for the "i"th observation (containing the combined effects of all other causative variables, which must be uncorrelated with the included independent variables). If there is reason to believe that none of the formula_24s is caused by "y", then estimates of the coefficients formula_25 are obtained. If the null hypothesis that formula_26 is rejected, then the alternative hypothesis that formula_27 and equivalently that formula_24 causes "y" cannot be rejected. On the other hand, if the null hypothesis that formula_26 cannot be rejected, then equivalently the hypothesis of no causal effect of formula_24 on "y" cannot be rejected. Here the notion of causality is one of contributory causality as discussed above: If the true value formula_31, then a change in formula_24 will result in a change in "y" "unless" some other causative variable(s), either included in the regression or implicit in the error term, change in such a way as to exactly offset its effect; thus a change in formula_24 is "not sufficient" to change "y". Likewise, a change in formula_24 is "not necessary" to change "y", because a change in "y" could be caused by something implicit in the error term (or by some other causative explanatory variable included in the model).