Questions tagged [independence]

Events (or random variables) are independent when information on some of them tells you nothing about the probability of occurrence (/ distribution) of the others. Please DO NOT use this tag for independent variable use [predictor] instead.

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Are SHAP values potentially misleading when predictors are highly correlated?

Are SHAP (SHapley Additive exPlanations) values potentially misleading when predictors are highly correlated? How and why? If so, is there any guidance on when not to use SHAP? Are there any rules ...
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Independence in gene set enrichment testing

Edit: I apologize if the question is considered to broad. In fact, it is concerning a very specific task in bioinformatic analysis of high throughput data set, and in my opinion the problem presented ...
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Trading signals example from Marginal Revolution blog

From http://marginalrevolution.com/marginalrevolution/2016/09/someone-give-doug-blog.html: Many trading signals reliably predict prices, but not strongly enough to overcome transaction costs (i.e....
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Orthogonal transformations of random vectors and statistical independence

In this old CV post, there is the statement "(...) I have also shown the transformations to preserve the independence, as the transformation matrix is orthogonal." It refers to the $k$-...
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Why does presence of sample correlation violate independence assumption?

Consider the data from this website. They present the attached graph and state "It is clear that children who get more correct in the D0 condition tend to get more correct in the D60 condition. ...
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Confused about multilevel analysis and non independence of observations

I'm still struggling with my understanding of multilevel analysis, wondering if it applies or not to my problem. I'v read here the following (where author gives an example of a multilevel model with ...
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Survival model for an epidemic -- can the observations be treated as independent?

I've been thinking about ways to tackle an epidemic modelling problem I've been working on, and I've come up against a conceptual difficulty over the way survival analysis works. Here's a really ...
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Explicitly Show Conditional Independence In a Mixture Model

Suppose we have the following mixture model: $$\pi \sim Dirichlet(\alpha)$$ $$\theta_1 ,\ldots ,\theta_K \overset{iid}{\sim} N(0,1)$$ $$Z_1 , \ldots , Z_n \mid \pi \overset{iid}{\sim} Categorical(\...
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Understanding the proof of Basu's theorem

In understanding the proof of Basu's thereorem, from wikipedia, I could not understand a few steps. Here's the wiki page. The link to the page. I do not understand the following. We are not making ...
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Independent RVs theorem: rigorous?

I am reproducing here theorem (#3.30) from "All of Statistics" by Larry Wasserman: Let X and Y have joint pdf $f_{X,Y}$ . Then $X\perp Y$ if and only if $f_{X,Y}(x,y)=f_{X}(x)f_{Y}(y)$ for all ...
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is it legitimate to t-test between statistics of a bootstrap?

say I have two samples $x_1$ and $x_2$ and a function which calculates a statistic out of each sample, denotes as $f(x)$. I would like to test the significance of the difference $f(x_1)-f(x_2)$. A ...
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Testing for the existence of dependencies in time series

What methods exist to test for the existence of any sort of dependence in a time series? This is in contrast to something like auto-correlation, which tests for a particular type of dependency. Is ...
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Is chi-squared the right method to compare time periods?

We manufacture foobars. In July, 91% of foobars were defect-free, but in August that figure was 89%. Would chi-squared be the right method to determine if the difference of 2% between July and August ...
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Example of independent observations

I came across an interesting (to me, at least) question. I want to predict the probability that a given boat will win a race. The dataset is something like: Y: winner (0-1) X1: size of the sail X2: ...
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Are statistics calculated from Bootstrapped samples independent?

I was wondering if someone could point me to some reference stating the (in)dependence properties of statistics calculated from bootstrapped samples. In other words, are bootstrapped statistics (mean,...
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Why is assuming Determinacy the same as Independence?

Consider a standard linear model $$y = x\beta_0 + \epsilon$$ or perhaps more generally a non-linear model $$ y = h(x,\beta) + \epsilon$$ I have several times come across the phrase that: To treat ...
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How to Use Chi-Squared Test for Inference about Three-way independence

If I recall correctly, three random variables X, Y, and Z are three-way independent iff these two statements are met: P(X∩Y∩Z) = P(X)P(Y)P(Z) X, Y, and Z are all pairwise independent of each other. ...
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Can two random variables be independent in some basis and dependant in other?

If some random variables forming N dimensions are dependant on each other is it possible that in a different coordinate system they'd be independent? For example if (X, Y) are two dependent RVs is it ...
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LASSO Cox Model after multiple imputation

I want to develop a predictive survival model on a data set with about 8000 subjects and 38 covariates. About 4% of subjects had the event of interest. There are 21 variables with missing values, ...
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Importance of the right-continuity of filtration in definition of strong Markov Property

Taking the definition from wikipedia, With $X = (X_t : t \geq 0) $ as a stochastic process on a probability space $(\Omega, \mathcal{F}, \mathbb{P})$ with natural filtration $\{ \mathcal{F}(t) \}_{t \...
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Expected value of quotient of Poisson distributions

Let $X$ and $Y$ be independent random variables such that $X \sim \text{Poisson}(\lambda \cdot c)$ and $Y \sim \text{Poisson}(\lambda \cdot (1-c))$, where $c$ is a real number in $[0, 1]$. Is there ...
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Imposing independence in multivariate MA process

Consider a 4-dimensional MA(q) process: $$ z_t = A(L) \epsilon_t$$ where $$A(L) = \sum_{j=0}^q A_j L^j$$ is a matrix-polynomial in the first $q$ powers of $L$. What is the best way to impose ...
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Why is conditional independence more important than marginal independence?

Graphical models are based on the idea of representing certain types of conditional independences in a (joint) distribution via a graph, and are an active research area. As argued (correctly I believe ...
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Find $P\{ (A \, \text{or} \, B) \, \text{and} \, (A_1 \, \text{or} \, B_1) \}$ or a lower bound in this specific case

Define $X$, $Y$, $X_1$, $Y_1$, and $Z$ to be some positive random variables, for each of which we know the distribution. Note that these variables are independent of each other. Let $t, a$ two ...
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ANOVA - Independence condition for paired data

Hi I'm reading the book "OpenIntro Statistics" about ANOVA and in section 5.5.4 when explaining the conditions for applying ANOVA under the voice 'Independence' it states: Independence. [...] For ...
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Chi-Square test with very large df

I am trying to use the Chi-Square test for independence of attributes. My dataset has only two columns, but several thousands of rows. Consequently the degree of freedom is also very high. When I use ...
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Consequence of violation of independence assumption on estimates of standard errors

from the first chapter , Introduction to Multilevel Analysis , p.5 of the book , it is written that : Standard statistical tests lean heavily on the assumption of independence of the observations....
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In Berkson's paradox, is $\beta_1 = 0$ or $\ne 0$?

There is "a general phenomenon known as Berkson’s paradox (Berkson, 1946), whereby observations on a common consequence of two independent causes render those causes dependent. For example, the ...
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joint distribution, probability, calculating probabilities under false independent assumption, when the random variables are actually dependent

Suppose I have random variables $X,Y,Z$ and I would like to compute the probability that random variable $X$ is smaller than $Y$ and $Z$: $$ \pi_X \overset{def}{=} Pr(X < Y, X < Z) = \int Pr(x ...
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How to Test Independence of Poisson Variables?

I have a data set of length $10,000$. Each entry, $e_i$, of my data set is associated with two values, $x_i$ and $y_i$. Let $X$ and $Y$ represent the random variables for which my data set is a ...
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Does the independence assumption interact with the generalizability of results in hierarchical designs?

A friend recently posed to me the following problem: Is it possible to do a Chi-square on test results for a group of participants to determine if they made more of type A errors or type B errors? I ...
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How to implement a regression with non-independent observations in the DV?

I have a design where a number of object pairs were rated for association strength. These pairwise association ratings along 5 criteria of interest are expected to be used as IVs in a regression. ...
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Should two equal pseudo-random numbers in a sample count as a "run down" or a "run up" in a runs test?

Taking into consideration the runs test proposed by Knuth given a sample of pseudo-random numbers to test independence, and looking at this example: ...
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2 answers
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How can I estimate the confidence interval of correlations possibly dependend with time?

I have a multivariate problem (with solar data from different meteorological stations) that I am working on my engineering master thesis. I would like to estimate the correlations of different ...
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3 votes
1 answer
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How to test for Pearson correlation when one variable is arbitratily fixed for number of cases?

I have 4 groups of different respondents, each group surveyed on four different dates (points of time). All respondents have answered a psychological questionnaire related to their perception of death ...
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Why should i care about independent triangular arrays?

In Advanced probability class we were building up to proofs of the central limit theorem. We started by stating the theorem of the Classic CLT, then we introduced the independent triangular array and ...
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Independence Testing for Discrete Random Variable

Suppose $X$ and $Y$ are two discrete random variable take values on $\mathcal{X}$ and $\mathcal{Y}$ with their iid observation $\{ X_i\}_{i = 1}^n$ and $\{ Y_i\}_{i = 1}^n$. If both $\mathcal{X}$ and $...
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Joint distribution of sample correlations of variables taken from a multivariate normal distribution

Let us assume multivariate normal vector $(X_1, \cdots, X_n)$ with mean vector $\mu$ and variance-covariance matrix $\Sigma$. A sample correlation will not exactly equal its population parameter, but ...
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some thought about independence and orthogonal, please comment on this if it's wrong

It seems that linearly independent is totally different from independent of random variable concept. Non-zero vectors Orthogonality must imply linearly independence. In Statistics, the relation of ...
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How do I know if my data violate the assumption of independence?

I have data on reaction time (dependent variable, continuous) from participants viewing a visual stimulus that can have 7 different states (-3, -2, -1, 0, 1, 2, 3, independent variable, nominal). I ...
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Density of min(X,Y), max(X,Y) for iid Uniform (Related to other post)

I was looking at Determine density of $\min(X,Y)$ and $\max(X,Y)$ for independently uniform distributed variables There's a very detailed answer, but while I was doing the same exercise by myself I ...
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How can a copula be seen as a characterization of association?

On the post A formal definition of a “measure of association” @kjetil b halvorsen commented the following: A copula could be seen as a characterization of association, so maybe a "measure of ...
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Does independent marginal distribution indicate independent conditional expectation?

Let's say, $z=[z_1,...,z_d] \in R^d$ is a multivariate random variable, and $x\in R^d$ is another multivariate random variable, and $r(x)=[r_1(x),...,r_d(x)]=[\mathbb{E}[z_1|x],...,\mathbb{E}[z_d|x]] =...
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Can we estimate independent parameters when $p > n$?

I am using a ridge regression method to estimate the effect of SNPs (p = 10000) as random effect for a population of n=2000 individuals. I know that when we estimate fixed effects, the number of ...
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If correlation is a measure for linear terms what is a measure for quadratic terms

I was thinking today this question on correlation and dependence and I would like to know how do I measure quadratic terms of dependence say I don't have correlation between two sets. Possible ...
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1 answer
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Determine independence/dependence of random variables

Let $Z_1$ and $Z_2$ be independent standard normal random variables. Let $W = \frac{Z_1 + Z_2}{\sqrt{2}}$ so that $W \sim N(0,1)$. Let $U = Z_1^2 + Z_2^2$ so that $U \sim \chi_2^2$. How can I ...
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2 votes
2 answers
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Dependent and independent events: Students

I am trying to answer this one from Khan Academy. My only question is about the first question. For the first question the answer is 120/300. My question about this is that 120 is just the students ...
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2 votes
1 answer
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How to show two variables are asymptotically independent

Let $X_1,...,X_n$ be iid from $Exp(\theta)$ with density function $f(x) = \frac{1}{\theta}e^{-x/\theta}$. Show that $M_n = X_{n:n} - \theta \ln(n)$ and $T_n = nX_{1:n}$ are asmyptoically independent ...
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Does uneven sampling during 3 time periods violate independence for ANOVA?

A continuous variable is measured at 7 AM, 2 PM, and 7PM, over the course of 2.5 months (summer). An animal is being periodically observed, while the site is monitored. The observation periods are for ...
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Mathematical explanation for this relationship

I have analyzed some data, and noticed that it fits this relationship quite well: $$P(A|B\cap C) = P(A|B)\frac{P(A|C)}{P(A)}$$ I am wondering what the interpretation of this would be mathematically, e....
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