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Introduction to Regression Procedures

Introductory Example

Regression analysis is the analysis of the relationship between one variable and another set of variables. The relationship is expressed as an equation that predicts a response variable (also called a dependent variable or criterion) from a function of regressor variables (also called independent variables, predictors, explanatory variables, factors, or carriers) and parameters. The parameters are adjusted so that a measure of fit is optimized. For example, the equation for the ith observation might be

y_i = \beta_0 + \beta_1 x_i + \epsilon_i

where yi is the response variable, xi is a regressor variable, \beta_0 and \beta_1 are unknown parameters to be estimated, and \epsilon_i is an error term.

You might use regression analysis to find out how well you can predict a child's weight if you know that child's height. Suppose you collect your data by measuring heights and weights of 19 school children. You want to estimate the intercept \beta_0 and the slope \beta_1 of a line described by the equation

{Weight} = \beta_0 + \beta_1 {Height} + \epsilon

where

Weight
is the response variable.

\beta_0, \beta_1
are the unknown parameters.

Height
is the regressor variable.

\epsilon
is the unknown error.

The data are included in the following program. The results are displayed in Figure 3.1 and Figure 3.2.

   data class;
      input Name $ Height Weight Age;
      datalines;
   Alfred  69.0 112.5 14
   Alice   56.5  84.0 13
   Barbara 65.3  98.0 13
   Carol   62.8 102.5 14
   Henry   63.5 102.5 14
   James   57.3  83.0 12
   Jane    59.8  84.5 12
   Janet   62.5 112.5 15
   Jeffrey 62.5  84.0 13
   John    59.0  99.5 12
   Joyce   51.3  50.5 11
   Judy    64.3  90.0 14
   Louise  56.3  77.0 12
   Mary    66.5 112.0 15
   Philip  72.0 150.0 16
   Robert  64.8 128.0 12
   Ronald  67.0 133.0 15
   Thomas  57.5  85.0 11
   William 66.5 112.0 15
   ;
   symbol1 v=dot c=blue height=3.5pct;
   proc reg;
      model Weight=Height;
      plot Weight*Height/cframe=ligr;
   run;


The REG Procedure
Model: MODEL1
Dependent Variable: Weight

Analysis of Variance
Source DF Sum of
Squares
Mean
Square
F Value Pr > F
Model 1 7193.24912 7193.24912 57.08 <.0001
Error 17 2142.48772 126.02869    
Corrected Total 18 9335.73684      

Root MSE 11.22625 R-Square 0.7705
Dependent Mean 100.02632 Adj R-Sq 0.7570
Coeff Var 11.22330    

Parameter Estimates
Variable DF Parameter
Estimate
Standard
Error
t Value Pr > |t|
Intercept 1 -143.02692 32.27459 -4.43 0.0004
Height 1 3.89903 0.51609 7.55 <.0001

Figure 3.1: Regression for Weight and Height Data

intreg2.gif (4266 bytes)

Figure 3.2: Regression for Weight and Height Data

Estimates of \beta_0 and \beta_1 for these data are b0=-143.0 and b1=3.9, so the line is described by the equation

Weight = -143.0 + 3.9* Height

Regression is often used in an exploratory fashion to look for empirical relationships, such as the relationship between Height and Weight. In this example, Height is not the cause of Weight. You would need a controlled experiment to confirm scientifically the relationship. See the "Comments on Interpreting Regression Statistics" section for more information.

The method most commonly used to estimate the parameters is to minimize the sum of squares of the differences between the actual response value and the value predicted by the equation. The estimates are called least-squares estimates, and the criterion value is called the error sum of squares

{SSE} = \sum_{i=1}^n ( y_i - b_0 - b_1 x_i )^2

where b0 and b1 are the estimates of \beta_0 and \beta_1 that minimize SSE.

For a general discussion of the theory of least-squares estimation of linear models and its application to regression and analysis of variance, refer to one of the applied regression texts, including Draper and Smith (1981), Daniel and Wood (1980), Johnston (1972), and Weisberg (1985).

SAS/STAT regression procedures produce the following information for a typical regression analysis:

SAS/STAT regression procedures can produce many other specialized diagnostic statistics, including

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