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Simple Linear Regression

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Simple linear regression is a tool for fitting a linear line to a set of data.  It is used when you want to predict the value of the "dependent variable" Y by knowing the value of the "independent variable" X.Figure 1 is an example of a data set with a regression line fit.

The line in the graph can be described as:

where y is the dependent variable (also plotted on the y axis of the graph), x is the independent variable (plotted on the x axis of the graph). The parameters that are estimated are b0 and b1. 

These parameters can be estimated using the following equations:

where X_i and Y_i are the individual observation and n is the number of observations.

The results of a regression are often summarized using an analysis of variance table.
The usual configuration for the table is as follows:

The F test is a test to determine if the regression explains more of the variation than the mean.  Another statistic that is commonly used to describe a regression is the coefficient of determination r2.   This statistic is the proportion of the observed data explained by the regression.  This statistic is a value that ranges from 0 to 1 with 0 being no agreement between the regression and the data and 1 being perfect agreement between the data and the regression.

Another important method of explaining  the results of a regression is to plot the residuals against the independent variable.  This analysis can be used to indicate that the model is mis-specified and  transformation required.

Also See:

Chapter 16 - Simple linear Regression pages 317-330 in:

Zar, J. H. 1999. Biostatistical Analysis. Prentice-Hall, Inc. Englewood Cliffs, New Jersey. 718 pp.