Questions tagged [variance]
The expected squared deviation of a random variable from its mean; or, the average squared deviation of data about their mean.
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mean and variance
Let X be a discrete random variable such that X = 0 with probability 0.5 and X = 1 with probability 0.5. Let Y be a discrete random variable such that Y = 1 when X = 1 and Y = 0 when X = 0. What is ...
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Why is the variance smaller for the same coefficient in a reduced regression model vs. full regression model?
Let's say we have two estimators for $\beta$.
$\beta$ denotes all a full set of coefficients, one for each covariate in a dataframe.
$\beta$ can be split into $\beta_p$ and $\beta_r$, where $p$ ...
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Expectation and Variance of two sets
In genomics, you have an input control (I) and a treatment (T) where then you determine the ratio T/I. You perform multiple replicates for each but the number of replicates is not always the same. ...
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Variance in set of different random variables
Imagine we have sample of values a_1, a_2,...a_n, with a_i value originating from a normal distribution with mean mu_i_a and variance var_i_a, thus each value is single realization of different random ...
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Unbiased estimator of $\sigma^4$
In the post [here], the user asked the question
$\{X_i\}_1^n$ is random sample from $N(\mu, \sigma^2)$ with unknown parameters. Find an unbiased estimator of $\sigma^4$.
The solution uses a property ...
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Which type of t-test is best to use for my data
I'm looking for some guidance on which type of t-test is most suitable to use for my data.
I want to compare the means of the variable 'Rate' between two groups in my data. I have 6 years of data and ...
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Misunderstanding on the use of Popoviciu and von Szokefalvi Nagy's inequalities on the variance of a unbiased estimator
Let $X_1,\cdots,X_n$ be (discrete in my case) i.i.d. and bounded between $m$ and $M$. I'm interested in bounding the variance of an unbiased estimator:
$$\mathbb{V}\left[\frac1n\sum_{i=1}^nX_i\right]$$...
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Understanding assumptions of equivalence of random effects variances; what to do when violated?
The random effects model is stated as:
$Y_{ij} = \mu + \tau_i+\epsilon_{ij}$
Where,
$\tau_i \overset{iid}{\sim} \mathcal{N} (0, \sigma^2_{\mu} ) \\ $
$\epsilon_{ij} \overset{iid}{\sim} \mathcal{N} (0, ...
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Standard deviation of standard deviation under non-normality
In this post, an unbiased estimator for the standard deviation of the standard deviation under normality is provided.
I would be interested in such an estimator without the normality assumption, i.e., ...
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Resample random variable to fit different variance
Suppose I have samples drawn from a random variable, and I want to multiply that random variable with a scalar constant. How should I transform the samples such that they would have been drawn from ...
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How to choose appropriate variance parameters when simulating data for a linear mixed model
I am in the process of fitting a linear mixed model to artificial data generated using simr::makeLmer(), with the goal of conducting power analysis for a study ...
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Difference between F-test and confidence intervals on variance estimates
Given n samples from a normally-distributed variable X, we estimate variance as $s^2=\frac{1}{n-1}\sum{(x_i - \bar{x})^2}$. We can also get a confidence interval for such a variance estimate as:
$$...
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Hypothesis testing of normal distribution, unknown mean unknown variance
Suppose that we make measurements of an effect, and we know that the values that we obtain follow a normal distribution. But we don't know the mean nor the variance. The hypothesis is that 95% of the ...
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Propagation of confidence interval through subtraction of variance estimates
Consider a biochemical method which is sensitive to the concentration of reagent A.
The method produces a signal with variance var(method) when reagent A is repeatedly sampled from the same stock (no ...
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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.
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Covariance matrix for data
Assume $n*p$ data matrix $X$, where n is the number of observations and p is the number of features. We are interested in the covariance among features.
I have seen notations where covariance matrix ...
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R: nlme: can we use varIdent or varFixed to model known variances?
Can anyone familiar with nlme kindly explain how does the varIdent, with option fixed actually work?
Documentation says:
fixed.....
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Derivation of the formula for E(MSE) in Recursive Partitioning for Heterogeneous Causal Effects (Athey and Imben 2015)
This post answers a question from the Recursive Partitioning for Heterogeneous Causal Effects paper by Athey and Imbens.
However, I am blocked at an even earlier stage from the previous post. I don't ...
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Recover Variances From GLS Model (Phylogenetic Least Squares on Evolutionary Tree)
I don't know how to calculate the variance of a variable when all of its observations have an arbitrary correlation structure.
I am simulating the evolution of animals as they branch apart into ...
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Using the standard error of the difference between means (s.e.d.m.) as opposed to the standard deviation
I'm trying to teach myself an intuitive understanding of basic statistics and have encountered the following problem in a textbook:
Q: Seven observers were shown five dishes containing mustard seeds. ...
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Cross sectional variance of random walks
Suppose that there are $J$ markets, and prices in every market $j$ follow a random walk. That is, for any time $t$, the price $p_t$ is the sum of shocks up to $t$:
$$p_t = \sum_{i=1}^t \epsilon_i^j$$
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How does the heteroskedasticity-robust SE equal the conservative estimator of SE for sample average treatment effect?
I have been told that the heteroskedasticity-robust standard error of $\hat{\beta}_1$ from an OLS regression with a binary $X_i$: $Y_i = \beta_0 + \beta_1 X_i + u$ should be the same as the ...
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Variance of Fourth Sample Central Moment [closed]
I am trying to derive a formula for the variance of the fourth sample central moment $m_4=\frac{1}{n}\sum_{i=1}^n (X_i-\bar{X})^4$ (where $X_i$ is the $i$th realization of a random variable, $\bar{X}$ ...
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Variance partitioning with crossed and nested factors: A simulation study
I'm working on a variance partitioning study, where I've simulated a dataset with crossed and nested factors. I have three predictor variables and one response variable. The first predictor is crossed ...
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Conjugate prior for univariate normal with same mean and unknown sum of two variances
I have a Bayesian inference problem where the likelihood function is conditioned on two unknown variances.
$$\log\mathcal{L}(d\mid \sigma_n,\sigma_s) = -\frac{1}{2} \log (\sigma_n^2 + \sigma_s^2) -\...
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Calculating variance of a dependent variable with multiple fixed effects
I am kind of a stats noob but I figure someone here may have some insight. I am running a linear mixed effects model in R, reminiscent of what's below:
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Calculate F test when restricting intercept term
Restrict the intercept term, i.e.
In this case, I want to calculate F-test.
The formula is as follows:
How can I calculate SSR for restricted version? What is its formula? Thank you.
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Lower bounding weighted sample variance
Let us assume that we draw a sample $\{X_i\}_{i=1}^N$ from a random variable $X$ and we have a discrete probability distribution $q_{ij}$, i.e. $0 \leq q_{ij}\leq 1$ and $\sum_{ij} q_{ij} =1$ (the $...
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How to solve MME [closed]
I have spent an entire day trying to figure out how to solve these mixed model equations without any luck. I am therefore reaching out to this community. The assignment is as follows:
Using the ...
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Sample size to achieve an error E on mean estimation, for a normal distribution with unknown variance
Disclaimer: This is actually related to a previous question. I decided to post this as a new question as I feel it specifies the problem enough to be considered a separate question. If mods believe ...
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Recentered influence function and OLS interpretation
I am working with Recentered Influence Functions (RIF) to estimate regressions in distribution. We have the following regression
$RIF(F_y, \nu (F_y)) = \beta_0 + \beta_1X + \varepsilon$
where $\nu(F_y)...
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Why do small samples have big variances?
I am trying to understand why they say that small samples have big variances. For example, suppose the population variance of some variable is 100. I take 5 samples and the variance is 156 ... in this ...
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Additive property of the regression coefficients (slopes)
If $y_1$, $y_2$, and $y_3$ are time series such that:
$$y_1=y_2+y_3$$
Suppose all those variables were regressed against index $x$, so we get coefficients $r_{y_1}$, $r_{y_2}$, and $r_{y_3}$. In ...
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Conditional variance notation
Why does the formula $\operatorname{Var} (Y\mid X)= E \left (\left(Y- E (Y\mid X)\right )^2\mid X\right )$ condition $EY$, but not $Y$, on $X$?
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Triplets, Sampling Criteria and Variance?
Say I have a population size of 1000 students from schools in different towns (assume they are in the same class year, of mixed gender, same country/state). The mean, variance and standard deviation ...
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When A/B testing a two sample hypothesis test of means, should we always use the welch t-test? [duplicate]
The Welch t-test is best used when we cannot make an equal variance assumption between our treatment and control groups (our two samples).
However, in A/B testing, it's not clear to me how we could ...
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Does a variance decomposition make sense with a non-linear link function?
I am doing a variance decomposition, with a hierarchal random intercept model like the one below (BRMS R Code):
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Can a sequence with increasing variance satisfy weak stationarity
In Section 3 of Page 3, the notes that one of the conditions for weak stationarity is that $\gamma_{X}(h)=Cov(X_t, X_{t+h})$, essentially that the covariance of $X_s$ and $X_t$ depend only on $t-s$ ...
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Estimating variance based on parameters
I have a list of measurements $y_1,\ y_2,\ ...,\ y_n$ of quantity $Y$ and a list of parameters associated with each measurement $(A,\ B,\ ...)_j,\ j=1...n$. The distribution of $Y$ is symmetric, but ...
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Unbiased estimators and moment of moments
Following section 7.4 of Rose and Smith "Mathematical Statistics with Mathematica" (book available online here), I'm trying to use the Fundamental Expectation Result (eq 7.15) and other ...
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What is the distribution of the estimate of residual variance in linear regression?
As the question says, what is the distribution of the estimate of residual variance in a standard gaussian linear regression?
I'm confused because I know in theory the observed $y$ subtract the ...
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Comparing scree plots or explained variance of two groups with different number of features after PCA
I want to define the dimensionality of a group as the number of PC features that can explain 80% of the variance in the group dataset. This intuition seems to work for a single group, however, if I ...
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Given two rvs $X$ and $Y$, if $X Y = Z$, is it possible to change the mean and sd of $X$ without changing the mean and sd of $Y$ and $Z$
I have two lognormal rvs $X$ and $Y$, and a third rv $Z$ which is the product of the former two. I know the mean and standard deviation of the three.
Is it possible to calculate an alternative pair of ...
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Bounding the distance of empirical average from its expected value
Suppose we have three sequences of random variables, $(Y_n)_n$, $(W_n)_n$, and $(X_n)_n$ such that:
If $Y_n=a$, then $X_n=b$. If $X_n=b$, then $W_n=c$. That is
$$
1_{[Y_n=a]}\leq 1_{[X_n=b]}\leq 1_{[...
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Variance of Influence Functions, Cross-fitting, and the Propensity Score
Following example 2 in this paper, suppose I wanted to estimate $\psi = E[E[Y|X,A=a]] $ and I had an influence function follows:
$$
IF(\psi) = \frac{A}{\pi(X)}\{Y-\mu(X)\} - \psi
$$
where $\pi(X)$ is ...
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How can I calculate the proportion of influence 2 factors have on a continuous variable?
I have a dependent continuos, but not normal variable y and want to know how much of the variance of y is explained by x_1 and x_2. Both x variables are nominal. The Levene test y~x_1+x_2 led to the ...
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Shouldn't we consider the difference in variance between population and a sample while calculating confidence intervals?
To comprehend the concept of confidence intervals, I came up with an example. I want to share it here for your better understanding what my question is all about.
Suppose, we want to figure out what ...
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Significance testing between two set of SNPs
After performing GWAS, I calculated the percentage of phenotypic variance (SNP-based heritability) for top SNPs and random SNPs using GREML (GCTA).
The variance of random SNPs was calculated for 3 ...
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Total generalized variance for Box-Cox transformed components
I have a couple Gaussian mixture models where each component comes from (component-wise) Box-Cox transformed data. These models do not describe the same data: the individual components are selected ...
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Metric for run-to-run consistency of time series data
If I run $n$ samples of a physical experiment, I expect to see roughly similar time vs. position plots but with slight variations run-to-run. What are good statistical metrics to quantify the ...
