All Questions
Tagged with variance correlation
131 questions
43
votes
2
answers
75k
views
Variance of product of dependent variables
What is the formula for variance of product of dependent variables?
In the case of independent variables the formula is simple:
$$ \operatorname{var}(XY) = E(X^2Y^2) - E(XY)^2 = \operatorname{var}(X) \...
- 133
25
votes
3
answers
14k
views
Coefficient of Determination ($r^2$): I have never fully grasped the interpretation
I want to fully grasp the notion of $r^2$ describing the amount of variation between variables. Every web explanation is a bit mechanical and obtuse. I want to "get" the concept, not just mechanically ...
- 3,017
14
votes
1
answer
7k
views
Does transformation of r into Fisher z benefit a meta-analysis?
Usually $r$ is transformed into Fisher $z$ to test difference between two $r$ values. But, when a meta-analysis is to be performed, why we should take such a step? Does it correct for measurement ...
user10619
14
votes
1
answer
7k
views
variance of the mean of correlated and uncorrelated data
I read this paragraph in James et al, Introduction to Statistical Learning, p183-184 [1]:
Since the mean of many highly correlated quantities has higher
variance than does the mean of many ...
- 275
13
votes
3
answers
999
views
What does it mean for observations to be uncorrelated and have constant variance?
I am learning about linear regression from the textbook Elements of Statistical Learning by Friedman, Tibshirani, and Hastie. In this section they suppose we have a set of training data $(x_1, y_1), \...
- 343
12
votes
1
answer
3k
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Understanding $\operatorname{Cov}(X,X) = \operatorname{Var}(X)$ intuitively
I just saw this question and the wonderful accepted answer in this forum. I was then triggered to try understanding intuitively why division of $S_xS_y$ is normalizing the covariance:
$$\frac{\...
- 223
11
votes
2
answers
17k
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Correlation between sine and cosine
Suppose $X$ is uniformly distributed on $[0, 2\pi]$. Let $Y = \sin X$ and $Z = \cos X$. Show that the correlation between $Y$ and $Z$ is zero.
It seems I would need to know the standard deviation of ...
- 113
11
votes
2
answers
2k
views
How to estimate the variance of correlated observations?
Assume we have n observations $x_i$ (i from 1 to n), each from the a normal distribution with mean 0 and some variance component: $X_i \sim N(0, \sigma^2)$. The random variables $X_i$s have some (let'...
- 21.9k
11
votes
1
answer
10k
views
Expected value and variance of sample correlation
I've been looking for an expression for the expected value and variance of the sample correlation coefficient. Most of the sources I've found say
$$
Var(Cor(X, Y)) \approx \frac{(1-\rho^2)^2}{n-1},...
- 1,563
10
votes
2
answers
37k
views
Covariance of a variable and a linear combination of other variables
Let $X,A,B,C,D$ be time-series variables and the covariance between any two pairs of these are known.
Suppose we want to find $\textrm{cov}(X,aA + bB + cC + dD)$, where $a,b,c,d$ are constants.
Is ...
user14281
9
votes
4
answers
5k
views
Why highly correlated means higher variance?
I am reading the book Introduction to Statistical Learning and on page 183, the book states that
Since the mean of many highly correlated quantities has higher variance than does the mean of many ...
- 93
9
votes
2
answers
814
views
Variance of sample autocorrelation (Ljung-Box)
The Ljung-Box and Box-Pierce tests make use of the sample autocorrelation
$$ r_k = \frac {\sum_{t=k+1}^n a_ta_{t-k}} {\sum_{t=1}^n a_t^2}$$
and the Ljung-Box test exploits the result that
$$Var(r_k) = ...
- 34.8k
9
votes
1
answer
4k
views
Why does $r^2$ between two variables represent proportion of shared variance?
Firstly, I appreciate that discussions about $r^2$ generally provoke explanations about $R^2$ (i.e., the coefficient of determination in regression). The problem I'm seeking to answer is generalizing ...
- 759
8
votes
3
answers
13k
views
Covariance greater than Variance?
It is straightforward to verify that for two random variables $X$ and $Y$ with variances $\sigma^2_X \neq \sigma^2_Y$, we have that
$$\Big|{\rm Cov}(X, Y)\Big| \leq \max\{\sigma^2_X,\, \sigma^2_Y\}$$
...
8
votes
1
answer
3k
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How to combine standard errors for correlated variables
What is the formula for calculating the standard error of a quantity (A) that is the ratio of 2 quantities (A = B/C) if B and C are correlated?
According to page 2 of http://www.met.rdg.ac.uk/~...
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8
votes
2
answers
2k
views
Sample mean and variance independence in the case of correlated observations
Are the sample mean and sample variance of correlated normal observations independent?
The classic theory relies on the independence of observations, but this is not the case. So as an example I ...
- 449
8
votes
1
answer
10k
views
What's the relationship between covariance, shared variance, and common variance?
I've generally assumed that shared variance and common variance were the same thing. However, here it is written that "Common variance is the realm of total collinearity. On the other hand, the term "...
7
votes
1
answer
5k
views
Variance of sum of dependent random variables
Can you guys help me prove the following:
$$
\operatorname{Var}\left[\frac{1}{m}\sum_{i=1}^my_i\right]=\frac{1}{m}(1-\rho)\sigma^2+\rho\sigma^2
$$
where the sampled predictions ($y_is$) have ...
- 133
6
votes
2
answers
46k
views
Standard error from correlation coefficient
Many studies only report the relationship between two variables (e.g. linear or logistic equation), $n$, and $r^2$. I want to use these reported statistics to reproduce this relationship with its ...
- 61
6
votes
2
answers
3k
views
Linear Mixed Effects Model Variances
Consider the following model:
\begin{equation}
Y_i = X_i\beta + Z_ib_i + \varepsilon_i,
\end{equation}
where $b_i \sim N(0, D)$, and $\varepsilon_i \sim N(0, R_i(\gamma))$.
The variance of $Y_i$ ...
- 843
6
votes
1
answer
3k
views
Why would we ever use Covariance over Correlation and Variance over Standard Deviation?
I am unable to understand the practical use of Covariance and Variance.
In my understanding, Covariance and Correlation are both measures of how one variable changes with respect to another. The only ...
- 163
6
votes
1
answer
5k
views
What can be inferred from "covariance matrix of residuals" and "correlation matrix of residuals" after VAR?
I have this VAR:
summary(VAR(V6CADModelSt45obs1D.df[,c(5,3,2,6,1,4)], p=5, type="none", ic="SC"))
The following is the result of this VAR:
<...
- 549
6
votes
1
answer
2k
views
How to assess similarity of two sets of Principal Component Analysis loadings
A predictive model that I currently use relies on PCA with varimax rotation to reduce the dimensionality of the data (whether this is appropriate is a separate question).
The dataset consists of ...
- 91
5
votes
2
answers
2k
views
Can one have two random variables, perfectly correlated, but with different variances (as percent of their mean)?
I'm thinking of the difference between two random variables, e.g. the spread between two stock prices.
- 679
5
votes
1
answer
622
views
If three random variables have the same variance, what will the co-variances look like?
I'm curious to know, if three random variables have the same variance, what will the co-variances look like? Can somebody help me to figure it out?
- 2,743
5
votes
2
answers
1k
views
Finding correlation coefficient of $X$ and $XY$
Let $X$ and $Y$ be independent random variables with nonzero variances. I'm looking to find the correlation coefficient $\rho$ of $Z=XY$ and $X$ in terms of the means and variances of $X$ and $Y$, i.e....
- 221
5
votes
1
answer
2k
views
distribution of sample variance of correlated observations
It is well known that if we have n i.i.d. observations of a normal random variable, then Cochran's theorem tells us that:
$\frac{(n-1)s^2}{\sigma^2} \widetilde{} χ^2_{n-1}$
But what if the samples ...
- 449
5
votes
1
answer
701
views
correlation coefficient in linear regression
My interest is to develop a relation of the correlation coefficient when the data (both the dependent and independent variables) have measurement errors.
Intro
The measured values are related to the ...
- 460
5
votes
0
answers
288
views
Would a large variance in only one variable also yield a larger correlation coefficient? [closed]
I'm still learning statistics and am trying to understand the relationship between variance and correlations to generate accurate hypotheses for a study. After reading around, I get that the larger ...
- 51
4
votes
1
answer
663
views
Variance of the Sum of Correlated Variables in R
I'm looking to compute $n\text{Var}\left(\frac{1}{n}\sum_{i=1}^nX^{(i)}\right) = \frac{1}{n}\sum_{i=1}^n\sum_{j=1}^n\text{Cov}\left(X^{(i)},X^{(j)}\right)$ in R. Assuming the X's not to be iid, we get
...
- 43
4
votes
2
answers
985
views
Intuitive understanding of variance of sum vs variance of difference
$\newcommand{\Var}{\operatorname{Var}}\newcommand{Cov}{\operatorname{Cov}}$Mathematically,
$\Var(X + Y) = \Var(X) + \Var(Y) + 2\Cov(X,Y)$ and
$\Var(X - Y) = \Var(X) + \Var(Y) - 2\Cov(X,Y)$
This ...
- 529
4
votes
1
answer
5k
views
Covariance of product
A question has been asked regarding the Variance of product of dependent variables. I am interested in the case in which $X$ is a vector and $Y$ is a scalar and the two variables are independent.
...
- 3,263
4
votes
2
answers
2k
views
Correlation not significant because there is not enough variance?
I have a question about correlations again.
I have a dichotomous variable that I want to correlate with the another one (metric) by using the point-biserial correlation coefficient. I get a non-...
- 1,350
4
votes
1
answer
174
views
How can population variance be estimated from a bivariate sample?
Let's assume a bivariate population with a correlation $\rho$ and a common $\sigma$ so that $\Sigma = \sigma^2 \begin{pmatrix}1 & \rho \\ \rho & 1\end{pmatrix}$.
I would like to know the ...
- 361
4
votes
2
answers
675
views
If two predictors are uncorrelated, is the variance explained by multiple regression the sum of variance explained by both linear regressions?
Pretty much what it says in the title. I don't know too much about statistics and I worry I'm getting this wrong.
There are variables $X$ and $Y$, they are uncorrelated by design, because one has ...
4
votes
2
answers
521
views
Why do we use $R^2$ instead of $R$ in linear regression?
$R^2$ equals the "amount of variance explained by the model".
However, we rarely use variance in descriptive statistics. We say a sample's weight is 78 ± 13 kg, which is $\bar x$ ± $\sigma$ (...
- 51
4
votes
2
answers
809
views
Source for claim that 2 measures that correlate at .70+ measure the same construct?
I am trying to locate a source/sources for this claim (from a reviewer):
I (and other measurement experts) believe that a correlation of .70 or
higher indicates that two constructs are very much ...
- 41
3
votes
1
answer
4k
views
When is the variance of the sum of random variables greater than the sum of the variances?
My professor asked my class to 'qualitatively' analyze the two scenarios with the assumption that there is no previous knowledge held in the concept of covariance (as we have not covered that chapter ...
- 101
3
votes
1
answer
78
views
Let $X, Y$ be independent RVs given the variances and no means what is correlation coefficient of $X$ and $Z=2X+Y$?
Let $X, Y$ be independent RV given the variance and no means what is correlation coefficient of $X$ and $Z=2X+Y$?
Given $var(X)=3, var(Y)=4$ and $\mathbf{E}[X]$ and $\mathbf{E}[Y]$ are not known, let $...
- 730
3
votes
1
answer
457
views
Minimum variance of the mean for $n$ correlated random variables
If $X_1,\cdots,X_n$ all have the same variance equal to 1, then $0\leq \mbox{Var}[\bar{X}]\leq 1$ where $\bar{X}=(X_1 + \cdots + X_n)/n$. The upper bound is attained if $\mbox{Cov}[X_k,X_l]=1$ for all ...
3
votes
1
answer
189
views
How to compute correlation of random slopes for X between two Conditions with (X*Condition|subject) model in lme4?
We have the following model:
Y ~ X*Condition + (X*Condition|subject)
Y = dichotomous variable; values 0,1
X = continous variable; values ca. 0-3000
Condition ...
- 1,782
3
votes
1
answer
2k
views
Variance of sample correlation coefficient
On wiki page about Fisher transformation I read that variance of sample correlation coefficient becomes smaller as population correlation coefficient (in absolute value) gets closer to 1. Could ...
- 451
3
votes
1
answer
3k
views
Variance of Z for Z = X + Y, when X and Y correlated
So I'm trying to show that ${\rm Var}(Z) \le 2({\rm Var}(X)+{\rm Var}(Y))$ for $Z = X + Y$. This seems to be pretty easy to show given that $X$ and $Y$ are uncorrelated. But I'm running into trouble ...
- 167
3
votes
2
answers
452
views
Adjusted R-Squared in terms of variance
Say that I am performing a multiple linear regression with 3 variables. If I want to say that two of these variables account for some percentage of the observed variance in the third variable, should ...
3
votes
1
answer
71
views
How to calculate the correlation coefficient from minimal distributional assumptions?
Let random variables $X_1,X_2,\ldots,X_n$ satisfy $$(X_i,X_j)\stackrel{d}{=}(X_1,X_2)\quad \forall i, j$$
(that is, these variables are identically distributed and all their bivariate marginal ...
- 109
3
votes
5
answers
5k
views
Estimating correlation with DCC GARCH
I have used a DCC Garch model to estimate the co-movement between 2 indices using the following command in Stata:
...
- 63
3
votes
1
answer
103
views
Determining the relationship between salesperson and products sold?
Context:
I have a series of figures for car sales that show me (a) the usual number of car sales for particular models and (b) the number of car sales by a particular car salesperson for each model.
...
- 131
3
votes
0
answers
585
views
Is Fisher's z transformation necessary when comparing the variances of correlation coefficients?
I am working on functional connectivity between brain regions, where functional connectivity is represented by the Pearson correlation coefficient between the time series (fMRI) of brain regions. I am ...
- 31
2
votes
2
answers
126
views
"Dependency" definition
Origin Lab has in their fitting parameter's statistics "Dependency". Each parameter has a dependency. It's not like the covariance between 2 parameters. I thought it could be defined from ...
- 133
2
votes
2
answers
428
views
Outlier and correlation
Hi, I have a question.
The scatter plot doesn't show any type of correlation and there is an outlier.
If the outlier was to be removed, would the correlation:
Increase dramatically
Increase ...
- 21