# What implies correlation?

• weetabixharry
In summary: Many thanks for trying to answer this question yourself and for providing some helpful feedback. Unfortunately, I don't think this is a question that can be adequately answered with just the information you've given.

#### weetabixharry

If I have three scalar random variables: $a$, $b$ and $c$, which are each zero-mean and have some nonzero variances, and I know:

1) The correlation between $a$ and $b$ is nonzero.

2) The correlation between $b$ and $c$ is nonzero.

Does this imply that the correlation between $a$ and $c$ is nonzero?

I feel like the answer must be yes, but I don't have any sound mathematical reasoning for it. Any advice would be greatly appreciated!

Simple counterexample: Suppose that a and c are uncorrelated random variables each with zero mean and nonzero variance and suppose that bac, with α and γ non-zero constants. By construction, b is correlated with each of a and c, but a and c are (by construction) uncorrelated.

D H said:
Simple counterexample: Suppose that a and c are uncorrelated random variables each with zero mean and nonzero variance and suppose that bac, with α and γ non-zero constants. By construction, b is correlated with each of a and c, but a and c are (by construction) uncorrelated.

Many thanks for clearing that up so elegantly. It's easy when you know how!

Perhaps I can develop my understanding of a similar problem here without starting a new topic:

If, again, I have three scalar random variables $a$, $b$ and $c$ which are each zero-mean and have some nonzero variances... and in this case $a$ and $b$ are uncorrelated:

$\mathcal{E} \left\{ ab^*\right\} = 0$

where $\mathcal{E}\left\{\right\}$ denotes expectation and $*$ denotes complex conjugate (although the variables probably need not be complex for this example).

What I'd like to know is whether, in general, we can find a $c$ which can sort of 'recorrelate' $a$ and $b$:

$\mathcal{E}\left\{ cab^*\right\} > 0$

I can't seem to find such a case using numerical examples in Matlab, but I'd really like to figure out a proper mathematical approach to this. Any advice or insights would be very much appreciated!

Correlation refers to the degree of linear relationship between two variables. In other words, it measures how closely the values of two variables are related to each other. A nonzero correlation implies that there is a relationship between the two variables, and this relationship may be positive or negative.

In the given scenario, we have three variables - a, b, and c, and we know that the correlation between a and b is nonzero, and the correlation between b and c is nonzero. This means that there is a relationship between a and b, and between b and c.

Based on this information, it is reasonable to assume that there may also be a relationship between a and c. However, it is important to note that correlation does not necessarily imply causation. Just because there is a correlation between two variables does not mean that one variable causes the other.

To determine whether the correlation between a and c is nonzero, we would need to calculate the correlation coefficient between these two variables. If the correlation coefficient is nonzero, then we can say that there is a relationship between a and c. If the correlation coefficient is zero, then we cannot conclude that there is a relationship between a and c, even though there is a relationship between a and b and between b and c.

In conclusion, the given information suggests that there may be a relationship between a and c, but it cannot be confirmed without calculating the correlation coefficient between these two variables. It is important to always consider the limitations of correlation and to use caution when interpreting its results.

## 1. What is correlation and how is it measured?

Correlation is a statistical measure that shows the relationship between two variables. It is measured using a correlation coefficient, which ranges from -1 to +1. A positive correlation coefficient indicates a positive relationship between the variables, while a negative correlation coefficient indicates a negative relationship.

## 2. What does a high or low correlation coefficient mean?

A high correlation coefficient (close to +1 or -1) indicates a strong relationship between the variables, meaning that changes in one variable are highly associated with changes in the other variable. On the other hand, a low correlation coefficient (close to 0) indicates a weak relationship between the variables, meaning that changes in one variable do not necessarily affect the other variable.

## 3. Can correlation imply causation?

No, correlation does not imply causation. Just because two variables are correlated does not mean that one causes the other. There may be other factors at play that influence both variables, or the relationship may be purely coincidental.

## 4. How do you determine if a correlation is significant?

To determine if a correlation is significant, you can calculate the p-value, which indicates the probability of obtaining a correlation coefficient as extreme as the observed value by chance. A p-value less than 0.05 is typically considered statistically significant.

## 5. What are some limitations of correlation analysis?

One limitation of correlation analysis is that it only measures the strength and direction of a relationship between two variables, but it does not provide information about the underlying cause of the relationship. Additionally, correlation does not account for other variables that may affect the relationship, and it cannot be used to make predictions or determine causality.