# Questions tagged [linear-model]

Refers to any model where a random variable is related to one or more random variables by a function that is linear in a finite number of parameters.

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### Interaction term is significant WITHOUT main affects… AND main effects are significant WITHOUT interaction term?

I am trying to determine the effect of a person's weight and the incline that they are running over on their running speed. I'm just using a simple linear model in R, but I get a weird situation where ...
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### Doesn't adding a quadratic term in Linear Reg Model violate independence of predictors (Multicollinearity)?

I was going through an example of Polynomial regression and could not understand why adding quadratic term doesn't violate Linear Model assumption of multi-collinearity as we are just squaring the ...
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### Relationship between two linear regression models

This is a homework question from my statistics course about regression, dummy variables. It provides the information below, Firstly, a dummy variable is defined ...
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### Meaning of $\hat{\beta}$ of the linear regression model [closed]

In the simple linear regression model, $\hat{\beta}$ is the sum of independent normally distributed random variables. Is it false because in linear regression there is $\beta$ and not $\hat{\beta}$?
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### Can least-squares linear regression ever produce no solution at all?

Is it ever possible for least-squares linear regression (linear in both features and weights) NOT to produce a solution? That is, after we set each partial derivative to zero, can the resulting system ...
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### Is the assumption of normality of the error term needed to use p-value?

I have been thinking lately about the following: 1. Is the normality assumption of the error term really needed in order to make use of p-values for linear regression models? A previous CV post (...
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### Resources suggestion about linear model

I was wondering if you could tell me about some self-learning resources for linear model theories. My professor has been using "A First Course in Linear Model Theory, Ravishanker and Dey. Publisher: ...
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### Alternative to plug-in estimation for log-tranformed linear model

I want to estimate a relationship of the form: $$y=ax^b\times\epsilon$$ If I log this model i get: $$\log(y)=\log(a)+b\log(x)+ \log(\epsilon)$$ If I then proceed and estimate this model using a ...
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### Interpretation of related lower-order interaction when higher-order interaction is significant

How do you investigate lower-order interactions which include one factor that is part of a significant higher-order interaction? Let's assume a research design with 4 factors (A, B, C, D), each with ...
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### In a linear model, how is correlation among the independent variables related to uncertainty in the model coefficients? [closed]

Suppose I have a linear model Y=AX, and I tune A based on observed data. I know that correlation among my independent variables, X, will increase the uncertainty in my model coefficients, A. How do I ...
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### Unbiasedness and Variance of Predictions

Here is the problem I'm working on: I'm not quite sure if I'm showing either unbiasedness property right, and am stuck on finding the expressions for the variances. Here's what I've done so far. (a) ...
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### What is considered as parameters/coefficients/weights?

I am doing lesson 2 of the fast-ai course, and I find myself with a doubt about the course. Are the weights/parameters only the slope and the cut of axis y of a simple linear function $y = ax+b$? ...
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### Linear regression for multi-class classification

Linear regression can be used for binary classification where it competes with logistic regression. While the fitted values from linear regression are not restricted to lie between 0 and 1, unlike ...
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### Linear Dependency

In the context of matrices, how would one explain the notion of 'linear dependency between two or more variables' in an intuitive way? Would that imply near-perfect or perfect correlation amongst ...
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### Could data be described by a straight line when Pearson Correlation Coefficient has the highest absolute values?

Suppose that there is a dataset of 2D points $(x_i, y_i)$, consider the following statement: "When the Pearson Correlation Coefficient(PCC) between $x$ and $y$ is equal to -1 or 1 (highest absolute ...
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### Why does the linear regression algorithm assume the input residuals (errors) to be normal distributed? [duplicate]

I am trying to know the assumptions of linear regression (LR). I understand linear regression needs the relationship between the independent and dependent variables to be linear, but LR also assumes ...
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### Using a subset of parameters in joint confidence region of a linear model

For a standard linear model of the form $y = X\beta + \epsilon$, where $\beta$ is a vector of parameters. we can calculate an individual confidence interval for each parameter (of 1-$\alpha$ quartile)....
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### Two different ways to write variance of OLS beta

I know that for OLS, we can write $var(\hat{\beta}) = \sigma^2 (X^{T}X)^{-1}$. Then for a the last variable $p$, we have $var(\hat{\beta}_p) = \frac{\sigma^2}{\langle x_p, x_p \rangle}$. However, we ...
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### What is the variance of the estimator in ordinary least squares with correlated residuals

If we assumed that $y \sim N(X\beta,S)$ where S= \$\sigma^2\begin{bmatrix} 1 & \rho & \rho &...\\ \rho & 1 & \rho &...\\ \rho & \rho & 1 &...\\ \rho & \rho &...
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### Decreasing trend in residual vs fitted value plot

Here is my residual vs fitted value plot. It shows a decreasing trend. Can someone explain to me what could cause this to happen and how do I correct my model to produce a better fit? I am fitting ...