# Linear Regression

Linear Regression estimates the coefficients of the linear equation, involving one or more independent variables, that best predict the value of the dependent variable. For example, you can try to predict a salesperson's total yearly sales (the dependent variable) from independent variables such as age, education, and years of experience.

**Example.** Is the number of games won by a basketball team
in a season related to the average number of points the team scores
per game? A scatterplot indicates that these variables are linearly
related. The number of games won and the average number of points
scored by the opponent are also linearly related. These variables
have a negative relationship. As the number of games won increases,
the average number of points scored by the opponent decreases. With
linear regression, you can model the relationship of these variables.
A good model can be used to predict how many games teams will win.

**Statistics.** For each variable: number of valid cases, mean,
and standard deviation. For each model: regression coefficients, correlation
matrix, part and partial correlations, multiple *R*, *R* ^{2},
adjusted *R* ^{2}, change in *R* ^{2}, standard
error of the estimate, analysis-of-variance table, predicted values,
and residuals. Also, 95%-confidence intervals for each regression
coefficient, variance-covariance matrix, variance inflation factor,
tolerance, Durbin-Watson test, distance measures (Mahalanobis, Cook,
and leverage values), DfBeta, DfFit, prediction intervals, and casewise
diagnostic information. Plots: scatterplots, partial plots, histograms,
and normal probability plots.

Linear Regression Data Considerations

**Data. **The dependent and independent variables should be
quantitative. Categorical variables, such as religion, major field
of study, or region of residence, need to be recoded to binary (dummy)
variables or other types of contrast variables.

**Assumptions.** For each value of the independent variable,
the distribution of the dependent variable must be normal. The variance
of the distribution of the dependent variable should be constant for
all values of the independent variable. The relationship between the
dependent variable and each independent variable should be linear,
and all observations should be independent.

To Obtain a Linear Regression Analysis

This feature requires the Statistics Base option.

- From the menus choose:
- In the Linear Regression dialog box, select a numeric dependent variable.
- Select one or more numeric independent variables.

Optionally, you can:

- Group independent variables into blocks and specify different entry methods for different subsets of variables.
- Choose a selection variable to limit the analysis to a subset of cases having a particular value(s) for this variable.
- Select a case identification variable for identifying points on plots.
- Select a numeric WLS Weight variable for a weighted least squares analysis.

WLS. Allows you to obtain a weighted least-squares model. Data points are weighted by the reciprocal of their variances. This means that observations with large variances have less impact on the analysis than observations associated with small variances. If the value of the weighting variable is zero, negative, or missing, the case is excluded from the analysis.

This procedure pastes REGRESSION command syntax.