I have two equations with this format: $$Ds= A+A^2+\alpha_1\tag{1}$$ and $$Ds= M+M^2+\alpha_2 \tag{2}$$
Knowing that $(1)$ explains 72% of $Ds$ and $(2)$ 20%. I want to combine these two equations into one and know how much this equation explains. Something like:
$$Ds= A+A^2+\alpha_1+M+M^2+\alpha_2$$ (I know it cannot be a sum, but I don't know how to combine this).
Thank you for your answers.
SOSA
P.D: I don't have good notion in mathematics, and I'm not sure about the tag for this question.
Sorry for my bad explanation.
The facts are: I have two factors Age (A) and matrilineal link (M) I have a parameter the David score (Ds, hierarchy rank) I made a quadratic regression with the factor A, and I found that this regression explains 72% of Ds. I did the same thing with the factor M and I found that this regression explains 20% of Ds. So now I want to combine these two equations to explain the parameter Ds with these two factors.
Thank you again for your help
I have this idea: Ds=[(Ds= A+A^2+alpha1) + (Ds= M+M^2+alpha2)]/2
Is it correct? But How can I say how much they explain Ds?
SOSA
1 Answer
I do not think you have enough information at this point. The reason is the you want to fit the surface
$$ Ds(A,M) = \begin{pmatrix}1\\A\\A^2\end{pmatrix}^\top \begin{vmatrix} K_{11} & K_{12} & K_{13} \\ K_{21} & K_{22} & K_{23} \\ K_{31} & K_{32} & K_{33} \end{vmatrix} \begin{pmatrix}1\\M\\M^2\end{pmatrix}$$
$$ Ds(A,M) = K_{11} + K_{12} M + K_{13} M^2 + K_{21} A + K_{22} A M + K_{23} A M^2 + K_{31} A^2 + K_{32} A^2 M + K_{33} A^2 M^2 $$
and you have only sampled a constant $M=M_\star$ varying $A$ giving you
$$Ds = (K_{11}+K_{12}M_\star+K_{13} M_\star^2) \\ + (K_{21}+K_{22}M_\star+K_{23} M_\star^2) A \\ + (K_{31}+K_{32}M_\star+K_{33} M_\star^2) A^2 \\ = \alpha_1 + \beta_1 A + \gamma_1 A^2$$
and a constant $A=A_\star$ varying $M$ giving you
$$Ds = (K_{11}+K_{21}A_\star+K_{31} A_\star^2) \\ + (K_{12}+K_{22}A_\star+K_{32} A_\star^2) M \\ + (K_{13}+K_{23}A_\star+K_{33} A_\star^2) M^2 \\ = \alpha_2 + \beta_2 M + \gamma_2 M^2$$
Equating your regression coefficients $\alpha_1$, $\beta_1$, $\gamma_1$ and $\alpha_2$, $\beta_2$, $\gamma_2$ to the coefficients $K_{ij}$ of the 3x3 matrix you have 6 equations and 9 unknowns.
You can assume some of the coefficients are zero like $K_{32}=K_{23}=K_{33}=0$ (by ignoring the cross effects of $A$ and $M$) to give you.
$$Ds = (K_{11}+K_{12}M_\star+K_{13} M_\star^2) + (K_{21}+K_{22}M_\star) A + (K_{31}) A^2 \\ = \alpha_1 + \beta_1 A + \gamma_1 A^2$$
and
$$Ds = (K_{11}+K_{21}A_\star+K_{31} A_\star^2) + (K_{12}+K_{22}A_\star) M + (K_{13}) M^2 \\ = \alpha_2 + \beta_2 M + \gamma_2 M^2$$
