Top

The goal of regression analysis is to determine the values of parameters for a function that cause the function to best fit a set of data observations that you provide. In linear regression, the function is a linear (straight-line) equation. For example, if we assume the value of an automobile decreases by a constant amount each year after its purchase, and for each mile it is driven, the following linear function would predict its value (the dependent variable on the left side of the equal sign) as a function of the two independent variables which are age and miles:

value = price + depage*age + depmiles*miles

where value, the dependent variable, is the value of the car, age is the age of the car, and miles is the number of miles that the car has been driven. The regression analysis performed by NLREG will determine the best values of the three parameters, price, the estimated value when age is 0 (i.e., when the car was new), depage, the depreciation that takes place each year, and depmiles, the depreciation for each mile driven. The values of depage and depmiles will be negative because the car loses value as age and miles increase.

The least squares regression equation

= b0 + b1 x

is an estimate of the population regression equation

E(Y|X=x) = 0 + 1 x

The response variable, Y, is described by the model

Yi = 0 + 1 Xi + i,

where i is a random error. The usual tests produced by most statisical program packages assume the errors are independent and follow a normal distribution with mean 0 and constant variance. This means that the variability of responses for small X values is the same as the variability of responses for large X values.

This is usually written ~N(0,2)--that is, normally distributed with mean 0 and variance 2--where is a fixed but unknown constant. (The standard error of the estimate estimates .)