Questions tagged [linear]

For statistical topics which involve the assumption of linearity, for example, linear regression or linear mixed models, or for the discussion of linear algebra as applied to statistics.

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Help factoring matrices out of cross-covariance

I am trying to prove that $\text{Cov}( \boldsymbol{BU}, \boldsymbol B' \boldsymbol W) = \boldsymbol B \text{Cov}(\boldsymbol U, \boldsymbol W) \boldsymbol B'^T$, where $\boldsymbol B$ is $j \times m$...
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Simple Linear Regression Question Confusion

By using this site I found the two linear regression models that the question asked. The equations came out to be: $$US=60.495+18.550x$$ $$China =-2.08+18.296x$$ A follow-up question asked me to find &...
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The shape of line differs fom information coming from linear regression? [closed]

The line which represents linear regression (Using scikit-learn library) is totally different from the information that you can get from the library, e.g. ...
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The second-derivative of the log-likelihood function in logistic regression (page 120) model

In textbook The Elements of Statistical Learning, for logistic regression (page 120) model, the log-likelihood function can be written as $$\ell(\beta)=\sum_{i=1}^N(y_i\beta^Tx_i-\log(1+e^{\beta^Tx_i})...
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Exercises concerning a linear regression model with parameter $\beta$

I'm trying to solve the following exercises. I am unsure of what I'm doing so I appreciate any help. My attempt: (a) We have that \begin{align*} \boldsymbol{\hat \beta} &= (\boldsymbol X^T \...
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How to reconcile these two versions of a "linear model"?

Outline I am taking a course in which the professor, unless I'm badly misunderstanding something, is discussing two varieties of linear models. Version 1 The general linear model is $\boldsymbol Y = \...
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Deriving confidence interval for random-effect model

In R, when using the usual linear regression functions, such as lm(), deriving and plotting confidence bands are easily done with the ...
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73 views

Is this linear model a good fit?

Is this simple linear regression a good fit? Are there any transformations that would improve it? The data is discrete interval count vs discrete interval count (the count of steps walked per time) ...
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Is this formula for the calculation of linear regression confidence intervals correct?

The Matplotlib Documentation on drawing confidence bands on a scatterplot has the following code sample: ...
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Differences in the factor - variable relationship in EFA and component - variable relationship in PCA?

When I read about exploratory factor analyses, I saw equations showing that each variable is a linear composite of different factors - with loadings correspond to the coefficient in front of each ...
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Is there an algorithm to determine what type of prediction function to use in linear regression? [duplicate]

I recently started learning machine-learning and just learned the basics of linear regression. So gradient descent and other optimization algorithms can be used to find the values of θ in the ...
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1answer
74 views

Pearson's Correlation after power transformation of dependent variable

I have a simple model. y ~ x y is continuous (habitat gained per million $) x is continuous ...
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1answer
55 views

PCA - How to show that linear combination has maximum variance

Let $$x_1,..., x_n$$ be random variables with a covariance matrix $\Sigma$. $ a^Tx:=\sum_{i=1}^na_ix_i $ $ \sum a_i^2=1 $ how do I show that the linear combination of 1 and 2 has maximal variance ...
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What are the calculations or maths behind least-squares-minimizing in linear regression used by sklearn [duplicate]

I'm relatively new in the ML field, and this question came up when working with linear regression from sklearn library. After a bit of looking up in the documentation, it states Compute least-squares ...
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Change in coefficient β in log linear model

I'm studying econometrics with Wooldridge's manual. In a problem from Chapter 2, you have to prove that the coefficient β1 does not change when you transform a simple linear regression into a log-...
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Whether it is okay to dichotomise a 4-point Likert-scale outcome item

I'm using an already collected data-set within my research, meaning I was very limited with the outcomes that I could use. One of my two main outcome variables was an item using a 4-point Likert-scale ...
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Help to make sense over increasing validation loss on linear model over BERT embeddings [duplicate]

I am trying to use BERT generated embeddings in a simple linear model with relu and dropouts(0.3) in between two hidden layers of dimensions 256 and 128, respectively. For a binary classification task....
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Are my formulas correct?

This is an incredibly stupid question, but I've found it so hard to find any consistent information on this. I'm trying to calculate a t test for linear regression using R. Here are my formulas: ...
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Linear mixed effects model for longitudinal data and interpretation of effect plot

I've been working on a longitudinal study using a linear mixed-effects model in R. I wonder if there is any way to visualize the results of the linear mixed-effects model. The problem in my situation ...
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Same results with Linear Regression and Lasso

I have fit the data with Linear Regression model, and OLS regression results look like below: Here is how these insignificant betas look like: When I apply Lasso to the same data, I get almost the ...
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22 views

Visualize results of linear mixed effects model of a longitudinal study in R [duplicate]

I've been working on a longitudinal study using a linear mixed-effects model in R. I wonder if there is any way to visualize the results of the linear mixed-effects model. The problem in my situation ...
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1answer
28 views

Interpreting unintuitive fixed effects sizes in linear mixed effects models

We are working on the analysis of an online behavioral experiment in which participants had to solve problems presented in two different linguistic tenses. One of them was ...
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365 views

Why use MSE instead of SSE as cost function in linear regression?

I am studying linear regression and I solved some problems analytically. For that I used the normal and intuitive sum of squared error function. Looking at this function, it makes all sense why it ...
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1answer
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Where would you begin analysing this?

I would really appreciate your help. Despite enjoying stats, this issue has been very much bugging me and I can't get my head around it. Everytime I make a breakthrough I end up back at square one. I ...
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1answer
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Linear regression: multiple points or averages?

I have the following set-up: A material property that is a function of temperature is measured by 3 laboratories. Each laboratory performs two measurements each at 4 different temperatures, i.e. 8 ...
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1answer
34 views

Robust linear regression for complex valued data in R

Are there any existing R packages capable of performing a robust linear regression on complex valued data? I have a set $Y$ of complex valued ($a + b i$) data, that are linearly dependent on another ...
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1answer
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Joint Hypothesis Testing on Multicollinear Regressors?

I was reading the following thread: link when I came across this discussion regarding dropping regressors that demonstrate multicollinearity from a linear regression model: But what if you have ...
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Apply Linear Regression on data with correlation -0.3

X and y look like above. The shape of X is (260, 1), which means one feature with size of 260.If I apply Linear Regression on x and y, R-square is only 0.11 and residuals have varying variance. I'm ...
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Removing variables from a linear regression improves $R^2_{adj.}$

I am working on a linear regression model. The complete model with 11 variables in total has a quite low adjusted R-squared ($R^2_{adj.}$) of 0.11. 4 variables have a significant influence on the DV....
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1answer
58 views

How to determine the best fit slope of a line?

I was going back through linear regression and I found two equations for finding the best fit slope given X and y. This is one: $$m=\frac{\sum_{i=0}^{n}(x_i - \bar{x})(y_i - \bar{y})}{\sum_{i=0}^{n}(...
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1answer
34 views

Effect of increaing number of features on support vector regression with a linear kernel

I am trying to train and optimize a linear kernel support vector regression while analyzing the effect of increasing the number of features used to train the model on the model performance. The number ...
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How to find regression coefficient from summary table [duplicate]

I'm practicing for an exam ,and I wonder how could I solve this type of questions. Find hgb estimate or this question: Find Sum of squares regression (SSreg)
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1answer
17 views

the linear association was different when selecting 3 knots compared to 4 knots in restricted cubic spline based on cox regression model

I applied the restricted cubic spline term of BMI/weight in cox regression to test the linear association between BMI/weight with the outcome. However, the P-value of linear association tested by ...
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1answer
34 views

Eigenvalues in PCA [duplicate]

When I carry out Principal Component Analysis, the outputs are the Eigen-values and Eigen-vectors for each PC. Question: are the Eigen-values directly proportional to the Variance explained by the PC? ...
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1answer
48 views

The correct tool for testing statistically likely source of endpoint?

Background Whilst I have some experience in statistics, I am not trained in the field and so am at somewhat of a loss with respect to what tool I should employ in the following scenario. I have a ...
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1answer
47 views

Residuals in LME models

Good morning everyone! I have implemented the following lme model in r: ...
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31 views

Linear Transformation of a Random Variable with a Laplace Distribution

I have read these two posts ( 1 and 2) about linear transformation of a random variable with a Gaussian distribution. I would like to find the first two moments of a linearly transformed Laplace ...
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33 views

Calculate regression confidence and prediction intervals from the standard errors of the fitted parameters AND the correlation coefficient

In many fields of the natural sciences, it is common practice to report the results of regression analysis as y = a1 + a2 * x. Bad luck, no uncertainties are ...
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1answer
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For linear regression, if the theoretical coefficients and the variance of the error is known, is the theoretical R squared value, F statistic known?

For linear regression, suppose we know the true, theoretical coefficients of the predictors (say, for a simulation) and the standard deviation of the error term (sigma). For instance, suppose we know ...
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127 views

How to determine by what percent the target variable will change if we change a variable by some percent in Linear Regression?

I trained a linear regression model on some data. Now I have the intercept and the other coefficients. How to relate that with percent change in target given some percent change in a feature, keeping ...
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20 views

correlation and simple linear regression [duplicate]

what this sentence means"The correlation squared (r2 or R2) has special meaning in simple linear regression. It represents the proportion of variation in Y explained by X".
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1answer
19 views

Interpreting log multiple linear regression, backtransformations?

I'm investigating adherence to a special diet (that is scored from 0-18) in relation to C-reactive protein level and am in the process of building multiple linear regression models: To achieve a ...
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1answer
61 views

Finding the coefficient of determination from a regression line?

Suppose you are given the following estimated model from a sample of size 1217: $\hat{y} = 1.177663 + 0.0910103x$ and the standard errors of the coefficients are $0.0865446$ and $0.0065643$ ...
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22 views

Establishing relationship between two variables with low sample size(5)

My question is - Is elevation range of plants linked to intraspecific trait variability. I want to explore the relationship between the coefficient of variance (CV) of a certain plant functional ...
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1answer
36 views

Full dataset with few repeated measures

I am dealing with a dataset with the majority of entries with one value per individual but, with three cases with 2 repeated measures each. My first approach would be to pursue linear mixed models, to ...
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36 views

Using variance of y variable as weights for weighted linear regression when both x and y variables contain negative values?

I am currently dealing with a weighted linear regression problem in the context of an instrument calibration in analytical chemistry. Let's assume I have a response variable y and a predictor variable ...
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1answer
35 views

Do I need to transform this data for linear regression? If yes, what type of transformation is best?

The picture below is the dependent variable on the Y and one of the IVs on the X axis. The Dependent variable is range bound between .5 and 12 while the IVs range from 0-over a million depending on ...
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How to calculate parameter estimates from the linear predictor with block effect?

I fit the following linear model with block effect in R: M1 <- lm (RV~EV+Block) And estimated the model means like this: M2 <- lm (RV~EV-1+Block) coef(M2) The output: ...
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33 views

Linear Regression - predicted values constantly below true values

just a short question, which may be easy to solve for most of you. I am just starting with linear regression models in python. Therefore I made a simple multiple linear regression with training and ...
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303 views

Do random forests work better than multinomial logistic regression for prediction of categorical non-binary variables? Why?

I posted another question that was well received. I am posting this new question because it was suggested by other members of Cross Validated. Here is the link of the original question that I posted: ...

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