Questions tagged [leverage]
Leverage is a measure used in regression to highlight observations which are outlying in the space of the predictors.
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Hat matrix and leverages in classical multiple regression
What is Hat matrix and leverages in classical multiple regression? What are their roles? And Why do use them?
Please explain them or give satisfactory book/ article references to understand them.
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Precise meaning of and comparison between influential point, high leverage point, and outlier?
From Wikipedia
Influential observations are those observations that have a relatively large effect on the regression model's predictions.
From Wikipedia
Leverage points are those observations, ...
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Prove the relation between Mahalanobis distance and Leverage?
I have seen formulas on Wikipedia. that relate Mahalanobis distance and Leverage:
Mahalanobis distance is closely related to the leverage statistic, $h$, but has a different scale: $$D^2 = (N - 1)(...
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How to identify outliers and do model diagnostics for an lme4 model?
I need to identify outliers and high leverage points, and perform model diagnostics, in an lme4 model. For outliers and high leverage points, simply making a plot ...
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How to extract/compute leverage and Cook's distances for linear mixed effects models
Does anyone know how to compute (or extract) leverage and Cook's distances for a mer class object (obtained through lme4 package)...
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Diagonal elements of the projection matrix
I am having some problem trying to prove that the diagonal elements of the hat matrix $h_{ii}$ are between $1/n$ and $1$.
Suppose that $\mathrm{Range}(X_{n,k})=K $ the number of columns of our matrix ...
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Cook's distance vs. hat values
What exactly does Cook's distance measure? And how is this different from what hat values measure?
I know hat values measure how distant a point it form its corresponding fitted point. I also know ...
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What is .hat in regression output
The augment() function in the broom package for R creates a dataframe of predicted values from a regression model. Columns created include the fitted values, the ...
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Interpreting case influence statistics (leverage, studentized residuals, and Cook's distance)
I just wanted to clarify some things about leverage, studentized residuals, and Cook's distance:
Does a large (in absolute value) studentized residual mean that a case is an outlier?
Does a large ...
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What are the leverage values for Ridge regression?
In linear least squares the parameter estimates are:
$\hat{\beta} = \left(X^{\top}X\right)^{-1}X^{\top}y$.
In Ridge regression the standardized parameter estimates are given by
$\hat{\beta}_{\Gamma} = ...
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How to handle leverage values?
I have a dataset with 1747 observations. Outcome variable is categorical, while independent variables are continuous, so I decided to use logistic regression for my analysis. I built the model using ...
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Which of these points in this plot has the highest leverage and why?
I am studying the definition of leverage, and I understand it in terms of formulas. However, if I would have a plot like this for instance, how could I see which of these points has the highest ...
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Leverages and effect of leverage points
I just got some question about the hat matrix in linear models.
My first question is:
Why in a balanced one-way layout $(n_1=...=n_c=n_0)$, all leverages $h_{ii}$ have the same value $\frac{1}{n_0}$? ...
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Bounding residual variance with distance from mean
For a linear regression $Y = X\beta + \varepsilon$ with $\varepsilon \sim \mathcal N(0,\sigma^2 I)$, we have $\hat Y = H Y$ for $H = X(X^TX)^{-1}X^T$. This means that $Var(Y - \hat Y) = \sigma^2(I-H)$ ...
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Prove that the leverage converges to 0. Do the residuals in a linear regression really approximate the errors?
In linear regression, $y_i = x_i^T \beta + \epsilon_i$, $i=1,\dots,n$, and $Var(\epsilon_i)=\sigma^2$. It is well known that the residuals $e_i$ have variance $Var(e_i) = \sigma^2 (1-h_{ii})$, where $...
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What insights can be found by using leverage plots?
I'm trying to figure out whether leverage plots can provide valuable information. See example here.
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Proof about the diagonal element of the hat matrix
I'd really appreciate it if you could help me find the proof for the following formula:
$$h_{ii}=1/n + \frac{(x_{i}-\bar{x})^2}{\sum(x_{j}-\bar{x})^2},$$
where $j=1,\ldots,n$.
I don't really know ...
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Leverage Statistic - equation explanation
I came across this equation for the leverage statistic in ISL. Does the denominator translate to the summation of all the squares of $(x_i - mean(x))^2$ excluding the current observation of x? In ...
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R not plotting observations with leverage one
I am building a really simple linear model. I want to test if the frass I got over 3 days from butterfly larvae depend upon the food they ate (diet), the butterfly family (the mother line) and ...
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Effect of leverage on residuals
The variance of residuals in a linear regression is given by :
$$Var(e_i)=(1-h_{ii})\sigma^2$$
This means that residuals have a lower variance than the error terms, and the variance of residuals is ...
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what to do with ridiculous but valid leverage points
So I'm having some difficulty fitting a linear model to the data (see other post here glm model fit - can't find a family/link combination that produces good fit).
In particular, I'm worried ...
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Is there a specific standard error for an individual residual?
On page 97 of Introduction to Statistical Learning book, there is a paragraph on studentized residuals within the context of looking for outliers.
But in practice, it can be difficult to decide how ...
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Individual significance of data points in correlation
In this question on stackoverflow, I asked about how it is possible to find the individual significance of each correlation coefficient of each node. I answered the question myself later stating that ...
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What is the explanation for a regressor losing statistical significance when a high leverage point is dropped?
I'm currently working on an Econometrics project and I've come to a point where I've dropped a high leverage point as identified by cook's distance and a leverage plot (had observations that were ...
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How does the squared residual of a high leverage sample observation differ from the population's variance?
Is the high leverage sample observation considered a subset of the population, forcing its squared residual to be less than or equal to the population variance? Or am I misinterpreting the question? I'...
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Why (or when) can we neglect residuals mutual correlation?
I've found in several books so far only a waving of hands explanation on why one can simply neglect the residuals mutual correlation, in a linear model context, and plot a QQ-graph to do a qualitative ...
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Leverage formula/derivation and Hat matrix
1)
So I know that $h_{ii}$ is just the ith row ith column of $H=X(X^TX)^{-1}X^T$. Intuitively, why is this the case? I understand that H is the projection matrix and leverage is measuring how far away ...
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How to find a wrong predictor value based on other correlated predictors
I have five correlated predictors, ref the following pairs plot:
Now I suspect that sometimes a predictor is wrong, as these come from different sources. In other words, four of the predictors of an ...
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Understanding the ridge leverage scores sampling from an arXiv paper
I give a try to read the arXiv paper Distributed Adaptive Sampling for Kernel Matrix Approximation, Calandriello et al. 2017. I got a code implementation where they compute ridge leverage scores ...
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Understanding Residuals vs Leverage plot in terms of meeting regression assumptions
Can someone help me understand the Residuals vs Leverage plot in terms of meeting the assumption of independence/influence for multiple linear regression models?
My understanding is that the ...
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leverage.plot() R - CAR package
I'm trying to complete a homework regarding added-variables and leverage plots using the CAR package.
In the documentation of the ...
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Multiple Regression Assumptions
This may seem like a basic question, but I'm verifying the assumptions for a multiple regression and have some trouble wrapping my head around homoscedasticity. I have a few questions listed below:
1)...
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Is it possible to derive Leverage figures without a Hat Matrix?
I ran into an impasse while attempting to write code for Cook's Distance: when a regression model reaches only a moderate size, I can't derive a Hat Matrix through my normal matrix math routines ...
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Which observation has the largest variance of the residual?
a) For which of the following observations (obs1, obs2, obs3) is the variance of the residual the largest? Which observation has the highest leverage? And which one the smallest? Explain why.
b) Which ...
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Influential observations and Outliers in Linear Regression model
What is the difference between influential observations and outliers in linear regression model?
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Do leverage values make sense outside of linear models?
I've been looking at leverage plots, but it seems to me that they are always related to linear regression models.
For instance, this explanation of a hat matrix considers a linear regression model: ...
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Logistic regression with rare events, all events have large residuals and are influential points
I am running a logistic regression model where the outcome is relatively rare (250 out of 5000). My main interest is to see if there are differences between age groups, sex, educational levels, income ...
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Leverage and residuals - Show $\frac{e_i^2}{\| (I - H)Y \|^2} \le 1 - h_{ii}$ where $e_i$ is the $i$-th residual and $h_{ii}$ is leverage
Question: Suppose that $\boldsymbol{Y} = \boldsymbol{X}\boldsymbol{\beta} + \boldsymbol{\epsilon}$, and the errors have zero mean, and are uncorrelated with constant variance. Let $\hat{\boldsymbol{\...
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Who originally defined leverage scores to be the diagonal elements of $X(X^TX)^{-1}X^T$?
A nice description of leverage in the sense that I am using it is given here so I will not repeat it.
Who originally defined leverage scores to be the diagonal elements of $X(X^TX)^{-1}X^T$?
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Leverage - influential points
Do we look at the absolute value of the leverage or the relative value?
For instance, based on the chart below, the largest leverage is about 0.023, it is big compared to other data points, but I'm ...
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Simple Linear Regression: Hat-Value $h_i$
I'm trying to finish proving that in simple-regression analysis, $h_i = \frac{1}{n} + \frac{(X_i - \bar{X})^2}{\sum_{j=1}^{n}(X_j - \bar{X})^2}$, where $h_i := h_{ii} = \sum_{j=1}^nh_{ij}^2$, the ...
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leverage() diagnostic test not supported for glmmTMB models in r
I am using a glmmTMB to look at the effect of numerous variables on how far individuals travel (distance).
The example below is ...
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How does centering affect leverage?
For a linear regression model $Y = β_0 + β_1X$, consider the matrix $X$ and $X_c$ with centering the mean.
How do you use algebra to show directly that centering does not change the leverage?
I have ...
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variance decreases when x gets farther from the average x?
I just read the description of Studentized residual on Wikipeida.
I'm confused about what it says about variance, it says that
"the residuals, unlike the errors, do not all have the same variance
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Why divide by 1-leverage?
I'm reading about resampling methods, and specifically leave-one-out cross-validation.
I understood the method, and how to calculate the estimate of the test MSE (Mean squared error):
In the setup ...
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Is it OK if I have some high leverage points in my experimental plots and they weren't outliers?
I've analyzed effects of temperature and sample thickness on drying process via response surface methodology. I've faced to four high leverage points (my four axial runs) that stick together on normal ...
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Using low coverage predictors in models?
The conventional wisdom I run up against is to drop predictors with low coverage without given them much consideration.
By low coverage predictors: I meant predictors whose values are mostly missing. ...
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Multiple regression - High adjusted R^2 but the residuals are high
I used the multiple regression models to derive the outcomes after using AIC pairwise comparison and deleted the outliers, high leverage points.
And it seems good, the adjusted R^2 acheived 0.9543, ...
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How do I go about rectifying violated assumptions when more than one is violated at the same time? [duplicate]
I am currently trying to run a model analyzing the duration of the egg stage of each sex of two species of insect across five different temperatures. All independent variables are categorical. My ...
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