Probability Statistics Question I have this formula for determining $x$ and $y$'s effect on
$$a\mapsto\frac{(xy/z)}{(xy/z)+ (1-x)(1-y)/(1-z)}$$
If this formula assumes x and y have equal affect on a (say 50% each), how would i modify this formula to reflect if x had 60% effect and y had 40% effect on a? I 'm not sure how to handle the denominator bc there is a (1-x)*(1-y)/(1-z) term.
I had initially assumed if x and y are equally weighted then that would mean 50% for each of x and y. So, if x was 60% and y was 40% weight, than I'd have 60%/50% and 40%/50% as weights for x and y. In other words, I thought the formula might be a = (1.2x*.8y)/{(1.2x*.8y/z)+1.2(1-x).8(1-y)/(1-z)} but I wasn't sure if that was right. Or, should I rewrite the formula as such: a = (1.2x*.8y)/{(1.2x*.8y/z)+(1-1.2x)(1-.8y)/(1-z)}
 A: As the comments correctly pointed out, a definite answer would strictly require some clarification about the exact meaning of "60% effect" and "40% effect". It is clear that talking about 50% and 50% effects means that $x$ and $y$ have an equal impact on $a$, but for different magnitudes of effects it is often not so simple to weight variables. The situation is rather simple for linear models, as in these cases weighting reduces to applying appropriate linear coefficients: for instance, in a model $a=\alpha x+\beta y+\epsilon$ we can weight variables setting  $a=2(j\alpha x+k\beta y)+\epsilon$, where $j$ and $k$ are weighting coefficients whose sum equals 1. For percentages of 60% and 40%, this would lead to $a=2(0.6\alpha x+0.4\beta y)+\epsilon=1.2\alpha x+0.8\beta y+\epsilon$, where the coefficients $1.2$ and $0.8$ are those cited in the question.
However, weighting becomes more complex for nonlinear models. For example, considering the main part of the numerator of the formula reported in the opening problem, i.e. $xy$, we cannot weight variables using simple linear coefficients (the weighting $jx \,ky$ clearly cannot work: simply note that it would yield the same results of the opposite weight $kx\,jy$). A possible approach in this case could be to use the weighted geometric mean by the formula $(x^jy^k)^\frac{1}{j+k}$, where $j$ and $k$ are exponents that can be easily calculated from the percentages. For example, to weight variables according to a 60% and 40% scheme, we could write $(x^3y^2)^\frac{2}{5}$. However, it should be highlighted that this is not the only possible approach for weighting such a nonlinear model.
A similar procedure can be used for the denominator. In particular, the weighted geometric mean can also be applied  to the product $(1-x)(1-y)$. However, it should again be noted that this is only one of the possible methods to achieve a 60/40% weighting.
