4. Let the line $y=k x+m$ passing through any point $P$ on the ellipse $\frac{x^{2}}{4}+y^{2}=1$ intersect the ellipse $\frac{x^{2}}{16}+\frac{y^{2}}{4}=1$ at points $A$ and $B$, and let the ray $P O$ intersect the ellipse $\frac{x^{2}}{16}+\frac{y^{2}}{4}=1$ at point $Q$. Then the value of $S_{\triangle A B Q}: S_{\triangle A B O}$ is $\qquad$
4.3.

Let $d_{(X, Y Z)}$ denote the distance from point $X$ to line $Y Z$.
Notice that, $\frac{S_{\triangle A B Q}}{S_{\triangle A B O}}=\frac{d_{(Q, A B)}}{d_{(O, A B)}}=\frac{|P Q|}{|O P|}$.
Let $\frac{|O Q|}{|O P|}=\lambda$. Then $\overrightarrow{O Q}=-\lambda \overrightarrow{O P}$.
Let $P\left(x_{0}, y_{0}\right)$. Then $Q\left(-\lambda x_{0},-\lambda y_{0}\right)$.
Thus, $\frac{x_{0}^{2}}{4}+y_{0}^{2}=1$,
$\frac{\left(-\lambda x_{0}\right)^{2}}{16}+\frac{\left(-\lambda y_{0}\right)^{2}}{4}=1$.
Therefore, $\lambda=2$. Hence, $\frac{|P Q|}{|O P|}=3$.
Thus, the required value is 3.