thekingofkings
commented
8 years ago How to derive the p-value of coefficients in standard LR?
The T-test is used in LR to get the coefficients p-value.
According to Wiki page T-test
t = sqrt(p) Z / s,
where Z and s are functions of data. Z = (Xbar - u) / (sigma / sqrt(n))
Assumptions:
The following assumptions are made in T-test:
- X follows a normal distribution with mean μ and variance σ2
- s^2 follows a χ2 distribution with p degrees of freedom under the null hypothesis, where p is a positive constant
- Z and s are independent
Test slope of a regression line
t_score = ( beta^hat - beta_0 ) / SE_beta, where
beta^hat is learned coefficient, beta_0 is the null hypothesis beta = 0, and SE_beta is the standard error of least-squares estimates.
To apply T-test, then the SE_beta^2 should follow a chi^2 distribution, which implies that sum of squares of residuals has a something to do with Gaussian.
Therefore, the residuals should be independent Gaussian variable.
More reference
An explicit requirements for T-test on linear regression is available here.
- The dependent variable Y has a linear relationship to the independent variable X.
- For each value of X, the probability distribution of Y has the same standard deviation σ.
- For any given value of X,
- The Y values are independent.
- The Y values are roughly normally distributed (i.e., symmetric and unimodal). A little skewness is ok if the sample size is large.
In standard regression
How to derive the confidence interval and p-value of coefficients?
The Gaussian distribution assumptions is made. But in more detail, what follows the Gaussian distribution. And why?
In spatial auto-correlation
The spatial auto-correlation has co-linear issue in constructing predictor variables. This violates some assumptions, so it is harder to calculate the p-value and confidence interval.