﻿ Multiple Linear Regression

# Multiple Linear Regression

### Statistics > Regression > Regression

Used to investigate and model the relationship between a response variable and one or more predictors.

You can use Multiple Linear Regression to perform simple and multiple regression using least squares. Use this procedure for fitting general least squares models, storing regression statistics and generating prediction and confidence intervals.

### Dialog box items

Response:
Select the column containing the Y, or response variable.

Predictors:
Select the columns containing the X, or predictor variables.

Report:
The display of outputs of VisualStat.

### Data

Enter response and predictor variables in numeric columns of equal length so that each row in your worksheet contains measurements on one observation or subject. VisualStat excludes all observation that contains missing values in the response or in the predictors.

### Statistics

ANOVA Table:
Check to display an analysis of variance table

Regression Coefficients:
Check to display regression coefficients, including value, standard error, beta, t value, p value and confidence interval.

Residuals:
Check to store the residuals.

Fitted values:
Check to store the value the model predicts for the dependent variable.

X'X Inverse:
Check to store the (X'X)-1 matrix.

Variance-Covariance Matrix:
Check to store the variance-covariance matrix.

Leverage values:
Check to store the leverage values.

Cook's distance:
Check to store Cook's distance.

DFITS:
Check to store the DFITS.

### Charts

Displays a histogram, and a scatter plot.

Histogram of residuals:
Check to display a histogram of the residuals.

Residuals vs fits:
Check to plot the residuals versus the fitted values.

### Options

Include constant in equation:
Check to fit a constant term (the y-intercept of the regression line). Uncheck to fit the model without a constant term. VisualStat does not display R2 for this model.

Test Value:
Enter the test mean µ0

Confidence Level: Enter the level of confidence desired. Enter any number between 0 and 100. Entering 90 will result in a 90% confidence interval. The default is 95%.

### Example

Using the Consumer Price Index dataset we can explore the relationship between the prices of various products and commodities. These data represent the average prices of the following items for the months of January between 1981 and 2006.

1.Open the DataBook reg.vstz

2.Select the sheet ConsumerPriceIndex

3.Choose the tab Statistics, the group Regression and the command Regression

4.In Response, select Gasoline. In Predictors, select Electricity, Fuel_Oil and Orange_Juice.

5.Click OK

Report window output

Multiple Linear Regression

The regression equation is

Gasoline = 0.0829 + 0.0132*Electricity + 0.7934*Fuel_Oil - 0.1903*Orange_Juice

Residuals:

Min       1Q  Median      3Q     Max

-0.1973  -0.0357  0.0122  0.0388  0.0875

Coefficients:

Value  Std. Error     Beta  t value  p value  95% LBound  95% UBound

(Intercept)    0.0829      0.1250   0.0000   0.6635   0.5139      0.0750      0.0908

Electricity    0.0132      0.0034   0.2375   3.9359   0.0007      0.0130      0.0134

Fuel_Oil       0.7934      0.0445   0.8890  17.8195   0.0000      0.7905      0.7962

Orange_Juice  -0.1903      0.0867  -0.1228  -2.1953   0.0390     -0.1958     -0.1848

Model Summary:

Multiple R-Squared  0.9581

Adjusted R-Squared  0.9524

Residual Std Error  0.0676

Analysis of Variance

df  Sum of Sq  Mean Sq   F Value   Pr(F)

Regression   3     2.3007   0.7669  167.8123  0.0000

Residual    22     0.1005   0.0046

Total       25     2.4012

### Interpreting the results

Regressing Gasoline on the following three predictor prices: Orange Juice, Fuel and Electricity illustrates significant effects of all these variables as significant explanatory prices (at a = 0.05) for the cost of Gasoline between 1981 and 2006.

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