# Regression

## Contents

## Introduction

Regression analysis helps understand how the dependent variable changes when any of the independent variables is varied, while any other independent variables are held fixed.

## Inputs

**Regression Name:** Specify a name for your regression – if you are intending on undertaking more than one, a unique name relating to the variables being processed is advised

**Regression Dataset Input: **The dataset containing the variables to be tested.

**Regression Dependent Variable: **The response (dependent) variable to be used in the regression.

**Regression Independent Variable(s): **The explanatory (independent) variable(s) to be used in the regression.

## Outputs

Output includes regression coefficients and correlation coefficients.

**Regression Coefficients:**

- Intercept
- Estimate(s)
- Std. Error
- t value (the estimated value divided by its estimated standard error)
- Pr(>|t|) (the probability for testing the hypothesis)
- sigma (standard deviation)
- r.squared : R
^{2}Co-efficient of determination (the amount of variation in the dependent variable explained by variation in the independent variables) - adj.r.squared: Adjusted R
^{2} - fstatistic : F (the ratio of two measures of variability)
- fstatistic : DFR (degrees of freedom for regression )
- fstatistic : DFE (degree of freedom for error)

**Correlation Coefficients:** a measurement of how two variables are related.

## Advanced

The following formula is used in the implementation of linear regression.

\(ŷ = a + b_i x_i\)where:

\(y\) is dependent variables

\(x\) are the independent variable(s)

\(ŷ\) is the vector of fitted values

\(a\) is the y intercept

\(b\) is the estimate(s) of the slope

The residual vector is \(y-ŷ\).

## References

- Buechler, S.(2007) Statistical Models in R: Some Examples, Department of Mathematics, University of Notre Dame.
- Ferrari, D. and T. Head (2010) Regression in R Part I: Simple Liner Regression, Statistical Consulting Center, UCLA Department of Statistics.
- http://en.wikipedia.org/wiki/Residual_sum_of_squares.