0
$\begingroup$

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 $\sqrt i$.

$\endgroup$
3
  • 1
    $\begingroup$ 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. $\endgroup$
    – whuber
    Commented Nov 26, 2018 at 14:06
  • 1
    $\begingroup$ 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.. $\endgroup$
    – P db
    Commented Nov 26, 2018 at 16:38
  • $\begingroup$ 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. $\endgroup$
    – whuber
    Commented Nov 26, 2018 at 17:28

1 Answer 1

0
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.