# Multiple regression

## Description

Use Multiple regression to analyse the relationship between a dependent variable and one or more independent variables (predictor variables or explanatory variables).

## Required input

### Dependent variable

The variable whose values you want to predict.

### Independent variables

Select at least one variable you expect to influence or predict the value of the dependent variable. Also called predictor variables or explanatory variables.

### Weights

Optionally select a variable containing relative weights that should be given to each observation (for weighted least-squares regression). Select the dummy variable "*** AutoWeight 1/SD^2 ***" for an automatic weighted regression procedure to correct for heteroscedasticity (Neter et al., 1996). This dummy variable appears as the first item in the drop-down list for Weights.

### Filter

Optionally enter a data filter in order to include only a selected subgroup of cases in the analysis.

### Options

- Method: select the way independent variables are entered into the model.
- Enter: enter all variables in the model in one single step, without checking
- Forward: enter significant variables sequentially
- Backward: first enter all variables into the model and next remove the non-significant variables sequentially
- Stepwise: enter significant variables sequentially; after entering a variable in the model, check and possibly remove variables that became non-significant.

- Enter variable if P< A variable is entered into the model if its associated significance level is less than this P-value.
- Remove variable if P> A variable is removed from the model if its associated significance level is greater than this P-value.
- Report Variance Inflation Factor (VIF): option to show the Variance Inflation Factor in the report. A high Variance Inflation Factor is an indicator of multicollinearity of the independent variables. Multicollinearity refers to a situation in which two or more explanatory variables in a multiple regression model are highly linearly related.
- Zero-order and simple correlation coefficients: option to create a table with correlation coefficients between the dependent variable and all independent variables separately, and between all independent variables.
- Residuals: you can select a Tests for Normal distribution of the residuals.

## Results

The results window for Multiple regression displays:

**Sample size**: the number of (selected) data pairs.

**Coefficient of determination R ^{2}**: this is the proportion of the variation in the dependent variable explained by the regression model. It can range from 0 to 1 and is a measure of the goodness of fit of the model.

**R ^{2}-adjusted**: this is the coefficient for determination adjusted for the number of independent variables in the regression model. Unlike the coefficient of determination, R

^{2}-adjusted may decrease if variables are entered in the model which do not add significantly to the model fit.

**Multiple correlation coefficient**: this coefficient is a measure of how tightly the data points cluster around the regression plane, and is calculated by taking the square root of the coefficient of determination.

**Residual standard deviation**: the standard deviation of the residuals.

**The regression equation**: the different regression coefficients *b _{i}* with standard error

*s*, t-value, P-value, partial and semipartial correlation coefficients r

_{bi}_{partial}and r

_{semipartial}.

- If P is less than the conventional 0.05, the regression coefficient can be considered to be significantly different from 0, and the corresponding variable contributes significantly to the prediction of the dependent variable.
- Partial correlation coefficient r
_{partial}: partial correlation is the correlation between an independent variable and the dependent variable after the linear effects of the other variables have been removed from both the independent variable and the dependent variable (the correlation of the variable with the dependent variable, adjusted for the effect of the other variables in the model). - Semipartial correlation coefficient r
_{semipartial}(in SPSS called*part*correlation): semipartial correlation is the correlation between an independent variable and the dependent variable after the linear effects of the other independent variables have been removed from the independent variable only. The squared semipartial correlation is the proportion of (unique) variance accounted for by the independent variable, relative to the total variance of the dependent variable Y. - Optionally the table includes the Variance Inflation Factor (VIF). A high Variance Inflation Factor is an indicator of multicollinearity of the independent variables. Multicollinearity refers to a situation in which two or more explanatory variables in a multiple regression model are highly linearly related.

**Variables not included in the model**: variables are not included in the model because of 2 possible reasons:

- You have selected a stepwise model and the variable was removed because the P-value of its regression coefficient was above the threshold value.
- The tolerance of the variable was very low (less than 0.0001). The tolerance is the inverse of the Variance Inflation Factor (VIF) and equals 1 minus the squared multiple correlation of this variable with all other independent variables in the regression equation. If the tolerance of a variable in the regression equation is very small then the regression equation cannot be evaluated.

**Analysis of variance**: the analysis of variance table divides the total variation in the dependent variable into two components, one which can be attributed to the regression model (labeled *Regression*) and one which cannot (labeled *Residual*). If the significance level for the F-test is small (less than 0.05), then the hypothesis that there is no (linear) relationship can be rejected, and the multiple correlation coefficient can be called statistically significant.

**Zero-order and simple correlation coefficients**: this optional table shows the correlation coefficients between the dependent variable (Y) and all independent variables X_{i} separately, and between all independent variables.

### Analysis of residuals

## See also

## Link

Go to Multiple regression.