All Questions
Tagged with variance correlation
131 questions
1
vote
0
answers
24
views
Variance of weighted average of 𝑛 correlated random variables
This answer explains how to calculate the variance of an average of n correlated random variables. How can I do it for a weighted average of n correlated random variables? My random variables are ...
- 11
12
votes
1
answer
3k
views
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:
$$\frac{\...
- 10.3k
0
votes
1
answer
22
views
Intraclass correlation -- which one?
I have data collected from an employee survey, in which employees are asked to rate various aspects of their work experience (like engagement, collaboration, and teamwork). Each row is a record of an ...
- 5,294
9
votes
2
answers
814
views
Variance of sample autocorrelation (Ljung-Box)
The Ljung-Box and Box-Pierce tests make use of the sample autocorrelation
$$ 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) = ...
- 21.2k
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 ...
- 133k
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 $(x_1, y_1), \...
- 14.3k
0
votes
0
answers
114
views
Derive the expectation and variance of squared sample correlation: delta-method or else?
I would like to obtain the expectation and variance of the squared Pearson sample correlation ($\operatorname{E}(R_{lk}^2)$ and $V(R_{lk}^2)$) between two random variables $l$ and $k$ following a ...
- 215
0
votes
0
answers
55
views
Consequences of ignoring correlation on standard error
I want to know the mathematical reason why between-individual standard error is under-estimated, and conversely, why within-individual standard error is overestimated if we fail to take correlation ...
- 58.8k
0
votes
0
answers
29
views
Why does the ICC differ when comparing multilevel models with using log transformations of the outcome variable in R using lme4?
I'm currently working on a multilevel modeling project in R utilizing the lme4 package. The primary aim of the research is to assess the relative importance of between-family and within-family ...
- 82.8k
0
votes
0
answers
11
views
How to use ICC with given data
I have data with columns like this:
ENTITY Avg_Score_1 N_obs_1 Avg_Score_2 N_obs_2
With sample values:
001 | 0.997 | 900 | 1.13 | 905
002 | 0.890 | 250 | 0.96 | 251
For about 1000 unique ENTITY values,...
Jbell
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:
$$ \operatorname{var}(XY) = E(X^2Y^2) - E(XY)^2 = \operatorname{var}(X) \...
1
vote
1
answer
76
views
How to quantify the similarity between three sets of complex numbers? [closed]
I have multiple groups of measurements, each containing three sets of complex numbers (impedances of the same thing measured under three conditions). The Nyquist plots belows shows two of such groups.
...
- 121
5
votes
1
answer
701
views
correlation coefficient in linear regression
My interest is to develop a relation of the correlation coefficient when the data (both the dependent and independent variables) have measurement errors.
Intro
The measured values are related to the ...
CommunityBot
- 1
7
votes
1
answer
5k
views
Variance of sum of dependent random variables
Can you guys help me prove the following:
$$
\operatorname{Var}\left[\frac{1}{m}\sum_{i=1}^my_i\right]=\frac{1}{m}(1-\rho)\sigma^2+\rho\sigma^2
$$
where the sampled predictions ($y_is$) have ...
- 11.5k
1
vote
0
answers
43
views
How to compute the variance for this process?
I have a sequences of random iid non-correlated positive integers sampled from some (unknown) distribution $\boldsymbol X=[n_1,n_2,...,n_N]$.
From it, I built another sequence in the following way:
$$ ...
- 10.3k
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:
<...
- 103
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 $\bar x$ ± $\sigma$ (...
- 67.1k
1
vote
2
answers
899
views
Defining Diversity in Ensemble Learning
I have a few questions regarding on how diversity is defined since I've seen differing definitions in different papers.
In the paper "Measures of Diversity in Classifier Ensembles and their ...
- 46
0
votes
0
answers
30
views
Equivalence between two expressions for autocorrelation
Have that
$$
\text{Corr}(X_t,X_{t+h}) = \frac{\text{Cov}(X_t,X_{t+h})}{\sqrt{\text{Var}(X_t)\text{Var}(X_{t+h})}}
$$
and
$$
\rho(h) = \frac{\gamma_X(h)}{\gamma_X(0)}.
$$
Those are both ways to express ...
- 69.5k
1
vote
0
answers
17
views
How can I reduce correlation between two independents variable?
Edvard, the evaluator in sample B, does not know Richard, the target subject in sample A. However, the two, independently, give the same answer/Likert value (1-5) to 30% of the questionnaire items. It ...
- 11
0
votes
0
answers
140
views
How to extract the unique variance of a variable
Is there any procedure to quickly obtain the variance that is unique to a variable among a correlated group of variables?
For example a star's temperature mass and volume are related (3 variables), ...
- 334k
0
votes
0
answers
45
views
How do you express the variogram $\gamma(h)$ in terms of correlation taking into account also the nugget
let's say that I have a spatial random field $z(\textbf{x})$ with $\textbf{x}$ the spatial coordinate. I can define the semi-variogram as:
$\gamma(h)=\frac{1}{2}E[(z(\textbf{x+h})-z(\textbf{x}))^2]$
...
- 111
0
votes
1
answer
186
views
Should we standardize the numerator or the denominator of a rate ratio if their variance are different?
I have a disease rate calculated for people aged <50 (early-onset) and aged 50+ (late-onset), for every US county. The rate ratio is the early-onset rate divided by the late-onset rate for every ...
- 101k
0
votes
0
answers
33
views
Deriving a standard deviation of a random variable using correlated other random variables
Consider three random variables: $$u_1, u_2, u_3 \;\; with \;\; E[u_j]=0 \;\; and\;\; Var[u_j]=\sigma^2_j\;\;for\;\;j=1,2,3$$
Here, we know the values $\sigma_1,\;\sigma_2$ and $\rho_{13},\;\rho_{23}$...
- 441
1
vote
0
answers
98
views
When would correlation between two variables not exist?
If we have two random variables $X$ and $Y$, then $\text{corr}(X,Y)=\dfrac{
\text{cov}(X,Y)
}{
\sqrt{
\text{var}(X)\text{var}(Y)
}
}$.
This correlation will not be defined if either variable has an ...
- 67.1k
1
vote
1
answer
174
views
Variance of average of 𝑛 correlated random variable where 𝑛 is random variable also
I have a sum of n correlated variables $\sum_{i=1}^n X_i$ and would like to estimate the variance of the sum. I do it with two different methods.
Firstly, I can decompose sum as multiplication of the ...
- 3,416
0
votes
2
answers
186
views
How do I calculate the weighted variance, $\sigma^2$, of a set of $N$ random variables considering their correlation $\rho$? [duplicate]
In a finance textbook of mine, there is an equation for calculating the variance $\sigma^2$ of a portfolio of two risky assets (i.e. random variables) $X$ and $Y$ by considering the correlation $\rho$ ...
- 1,389
2
votes
1
answer
121
views
Meaning of "average correlation" in Var$(\bar X)$
From some process I got a series of values. I want to compute the variance of the mean from this series. The series is built with contiguous sub-series. In each sub-series the values are correlated. ...
- 86.5k
0
votes
0
answers
104
views
Asymptotic variance of identically distributed but non-independent random variables
I have a question in computing the asymptotic variance of a sequence of random variables that is identically distributed but are not independent. Suppose we have a sequence of i.i.d. random variables $...
- 153
0
votes
0
answers
26
views
Power calculation to check if multiple predictive models are correlated
Say there is a classification setting so that $$ { \{(x_1,y_1),(x_2,y_2), ..., (x_n,y_n)\} } $$ is a set of $n$ observations, and then there is a set of $m$ estimators (models) that take an individual ...
- 223
2
votes
1
answer
355
views
R function to compute variance of average of correlated random variables
I want to calculate the variance of the average of n correlated variables. I found a formula for that in Borenstein et al. (2009) Introduction to Meta-Analysis.
$$\operatorname{Var}\left(\frac{1}{m}\...
- 151
0
votes
0
answers
55
views
Variance of the sum of N correlated random variables with equal variance [duplicate]
According to this Wikipedia article,
In general, the variance of the sum of $n$ variables is the sum of their covariances.
So if the variables have equal variance ${\sigma}^2$ and the average ...
- 101
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 ...
- 11.7k
0
votes
0
answers
28
views
Proving non-correlation with very disperse distributions
I'm fairly new to statistics and came up with a problem.
I have a sample with a variation coefficient CV = 0.517 for variable x, and I want to prove this variable is not correlated with a second ...
- 235
2
votes
1
answer
369
views
Correlation matrix for 2d normal variables with components constrained by $y_1 + y_2 = x_1 + x_2$
I'm following this tutorial on Canonical Correlation Analysis, and had a question about an example from that tutorial.
Question:
On Page 4 of that document, the following example is given:
"...
- 334k
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 ...
- 1,592
1
vote
0
answers
52
views
Correlation Based Models vs Covariance Based Models
I am trying to better understand why some models are "covariance based" vs. why some other models are "correlation based".
1) For example, a Multivariate Normal Distribution ...
- 314
1
vote
1
answer
88
views
What is the new sample size of a within study summary effect for outcomes with different sample sizes?
I am performing a meta-analysis and want to compute a summary effect (i.e. weighted mean) for studies that report data on more than one effect size. The same participants are involved, however the ...
- 17.9k
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 ...
- 9,660
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'...
- 21.2k
1
vote
0
answers
38
views
Comparing variances of multiple correlated variables
I am trying to find a way to compare the variances of multiple variables in the same sample. More specifically, I have a sample of 118 participants who completed eight items (on the same 0-100 ...
- 95
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 $...
- 2,937
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},...
- 107
8
votes
1
answer
3k
views
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/~...
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 ...
- 2,130
1
vote
0
answers
33
views
Correlation of Subsets - When the population correlation is known
Suppose I have a population of N pairs of (X,Y). I know the correlation of the population is Z. I now break the population into two unequal sets (n1 + n2 = N and n1 <> n2). If I calculate the ...
2
votes
1
answer
196
views
Variance of the Product of Correlated Random Variables?
I would like to multiply two correlated random variables, but I'm getting a negative variance. Please point out where I'm wrong.
Variable1 and ...
- 419
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....
- 813
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 ...
- 54.3k
0
votes
0
answers
34
views
Best model for various bad situations
"What type of predictive model would best handle a wide variety of data issues.
Extreme heteroscedasticity
bi or tri modal distribution’s
heavily correlated predictors…
Saw this as an Amazon ...
- 503