# A question about the change in value of multiple correlation coefficient on multiplying each value with a variable quantity

Question:

Let $$X=(X_1,X_2,\ldots,X_p)'$$ be a random vector. What can you say about the change of multiple correlation coefficient of $$X_1$$ on $$X_2,X_3,\ldots,X_p$$ if each $$X_i$$ is multiplied by $$\sqrt i$$ ($$i=1,2,\ldots,p$$)? Justify your answer.

I know the formula of multiple correlation coefficient is;

$$\rho_{1\cdot23\ldots p}=\sqrt{1-\frac{|R|}{R_{11}}}\,,$$

where $$|R|$$ is the determinant of the correlation matrix $$R$$ and $$R_{11}$$ is the cofactor of the $$(1,1)$$ element of $$R$$.

But I really cannot figure out, how to deal with this variable quantity $$\sqrt i$$.

• For each $X_i,$ $\sqrt{i}$ is not variable: it's constant. If that doesn't make the answer immediately clear, consider the case $p=2$ in detail.
– whuber
Nov 26 '18 at 14:06
• Thank you..As per your direction, I have calculated and found out that such a change, leaves Correlaion(Xi, Xj) the same as after multiplying sqrt(i)..So the multiple correlation coefficient seems to remain the same as before..
– P db
Nov 26 '18 at 16:38
• Right. Identify why that is--how you do it depends on your definition of correlation and/or the formulas you use--and see whether you can generalize your observation.
– whuber
Nov 26 '18 at 17:28

Let $$Y_i=\sqrt iX_i$$ for all $$i$$, which directly gives you $$\operatorname{Corr}(Y_i,Y_j)=\frac{\sqrt i\sqrt j\operatorname{Cov}(X_i,X_j)}{\sqrt{i\operatorname{Var}(X_i)}\sqrt{j\operatorname{Var}(X_j)}}=\operatorname{Corr}(X_i,X_j)\quad\forall\,i,j$$
This means the correlation matrix of $$(Y_1,\ldots,Y_p)$$ is identical to that of $$(X_1,\ldots,X_p)$$, whence the multiple correlation coefficient of $$Y_1$$ on $$Y_2,\ldots,Yp$$ is also the same as that of $$X_1$$ on $$X_2,\ldots,X_p$$ using the formula in your post.
We can also possibly justify this by recalling that the multiple correlation coefficient of $$X_1$$ on $$X_2,\ldots,X_p$$ is the maximum possible correlation between $$X_1$$ and any linear function of $$X_2,\ldots,X_p$$. Intuitively, multiplying each $$X_i$$ by a constant should not change this correlation.