Linked Questions

5
votes
3answers
546 views

Why are the ends of the prediction interval wider in the regression? [duplicate]

Usually, the prediction interval has this shape in the image. I don't know why the end of the interval is wider than the center.
0
votes
1answer
751 views

Finding the the Confidence Interval with Linear Extrapolation [duplicate]

Suppose I have some slightly messy data with an r squared of 0.9, and so fits a line pretty well. If I were to extrapolate based on the slope and intercept of the fit, I would expect my y values to ...
0
votes
0answers
48 views

Standard error of linear regression fit [duplicate]

I have a naive question about regression. How does R function predict.lm compute the 95% confidence interval of the fitted line? In particular why is this not a straight line? ...
0
votes
0answers
26 views

computing confidence curves around the fitting line [duplicate]

I am reading a book about linear regression in which there is a plot showing the regression line and two curves they call a confidence band (also can be seen here). As I learn something last week that ...
1
vote
0answers
22 views

How to intuitively understand the shapes of confidence and prediction intervals in simple linear regression [duplicate]

Assume the linear regression $Y=b_{0}+b_{1}X$ The $100(1-a)$% prediction interval for $x=x_{*}$ is given by whereas the The $100(1-a)$% confidence interval for $E(Y|X=x_{*})$ is given by What I don'...
38
votes
2answers
31k views

Understanding shape and calculation of confidence bands in linear regression

I am trying to understand the origin of the curved shaped of confidence bands associated with an OLS linear regression and how it relates to the confidence intervals of the regression parameters (...
14
votes
1answer
15k views

Why does the standard error of the intercept increase the further $\bar x$ is from 0?

The standard error of the intercept term ($\hat{\beta}_0$) in $y=\beta_1x+\beta_0+\varepsilon$ is given by $$SE(\hat{\beta}_0)^2 = \sigma^2\left[\frac{1}{n}+\frac{\bar{x}^2}{\sum_{i=1}^n(x_i-\bar{x})^...
12
votes
2answers
1k views

Understanding the shape of confidence interval for polynomial regression (MLR)

I have difficulties to grasp the shape of the confidence interval of a polynomial regression. Here is an artificial example, $\hat{Y}=a+b\cdot X+c\cdot X^2$. The left figure depicts the UPV (unscaled ...
6
votes
1answer
2k views

How to calculate the specific Standard Error relevant for a specific point estimate within a linear regression?

My objective is to actually get to the Standard Error for every point estimate of a linear regression. I originally thought that the Standard Error of a linear regression model was constant. And, ...
4
votes
2answers
3k views

Confidence interval for first order linear regression

I implemented first order deming regression on an array of x and y values. I tried to calculate the confidence interval, which ...
0
votes
0answers
4k views

How to calculate the standard error of a predictive linear model using each of the coefficients' standard errors?

Using R, I have built a linear model from the fluorescence produced by a set of genetic sequences under certain experimental conditions. The final linear model provides a number of different metrics ...
2
votes
2answers
2k views

Equation for Confidence Interval of Linear Regression

I've done a multivariate linear regression. The results specify each parameter and the 95% confidence interval for each parameter. I did this using Python and StatsModels (not that it matters), and ...
1
vote
0answers
1k views

Why is a prediction interval wider than a confidence interval for values of the predictor? [duplicate]

I have this graph that depicts PI being larger than CI for values of 14.5 and 24 for the independent/predictor variable. CI take into account regression coefficients, which are estimates. The PI has ...
4
votes
1answer
415 views

Interpretation of conditional variance of estimator of intercept in linear regression

$Y_i=a+bX_i+e_i$. $Y_i$ and $X_i$ are scalar r.v. We have, $$ V(\hat b|X)=\frac{\sigma^2}{n\left(\bar{X^2}-\left[\bar{X}\right]^2\right)} $$ and, $$ V(\hat a|X)=\frac{\sigma^2 \bar{X^2}}{n\left(\bar{X^...
1
vote
1answer
260 views

Is always true that in the simple linear regression the width of prediction interval corresponding to new observationx=xo increases linearly with xo

Is it always true that in the simple linear regression model the width of the prediction interval corresponding to a new observation x=xo increases linearly with xo? Thanks in advance

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