Methods and formulas for coefficients in Fit Regression Model

Select the method or formula of your choice.

Coefficient (Coef)

The formula for the coefficient or slope in simple linear regression is:

The formula for the intercept (b0) is:

In matrix terms, the formula that calculates the vector of coefficients in multiple regression is:

b = (X'X)-1X'y

Notation

TermDescription
yiith observed response value
mean response
xiith predictor value
mean predictor
Xdesign matrix
yresponse matrix

Standard error of the coefficient (SE Coef)

For simple linear regression, the standard error of the coefficient is:

The standard errors of the coefficients for multiple regression are the square roots of the diagonal elements of this matrix:

Notation

TermDescription
xiith predictor value
mean of the predictor
Xdesign matrix
X'transpose of the design matrix
s2mean square error

T-value

Notation

TermDescription
test statistic for the coefficient
estimated coefficient
standard error of the estimated coefficient

P-value – Coefficients table

The two-sided p-value for the null hypothesis that a regression coefficient equals 0 is:

The degrees of freedom are the degrees of freedom for error, as follows:

np – 1

Notation

TermDescription
The cumulative distribution function of the t distribution with degrees of freedom equal to the degrees of freedom for error.
tjThe t statistic for the jth coefficient.
nThe number of observations in the data set.
pThe sum of the degrees of freedom for the terms. The terms do not include the constant.
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