Multiple regression

Example of the multiple regression equation

For the multiple regression Y = 2 + 3A + 4B + 5C, what is the predicted value of Y when A = 1, B = 3 and C = 2 Y = 27. In this example, variable C holds the highest regression weight (B), so can be considered to contribute MOST to the model.

What is the aim of multiple regression in regards to the regression weight for each predictor?

It calculates the best regression weights (B) with which to multiply the predictor to produce the best prediction of the criterion variable.

What is the partial regression weight?

The relation between Y and one of the predictor variables after partialling out the the relationship of that predictor variable with other predictor variables.

What are the different ways of entering variables into a regression equation?

1) Enter 2) Hierarchical 3) Stepwise 4) Backwards

Backwards method of entering in variables into regression equation

All predictors entered and then removed until the prediction gets worse.

What is the 'enter' method of regression?

All variables are entered at once,

Stepwise method of entering variables into regression equation.

Best predictor entered in first, then second best etc...

Standardised beta in multiple regression

Beta score derived from Z-scores, produces a value with no intercept. In linear regression with only one predictor, this would be equal to the correlation co-efficient. However, this is not the case with multiple regression. It is useful for assessing the relative importance of variables.

How do you perform a hierarchical regression in SPSS?

Enter in the IV as you would in a linear regression, but make sure to click 'next' when you have finished entering in the variables for a certain block. Then look at the models in order to ascertain how much of the variance in the DV is accounted for by each model.

What is R-squared (effect size)

How well the predicted regression line approximates the actual data points. It is the amount of variance in criterion explained by the regression equation within the current sample.

If one of the predictors in a multiple regression is not significant, what can be done?

If a predictor is not significant, this means its partial regression weight is not significantly different from zero. In this case, it can be removed from the equation to see if the predictive value is improved.

How do you interpret the output to see the regression weights in order to calculate the multiple regression?

In order to obtain the values, you need to look at the SPSS output. Interpret this in a similar way you would a linear regression output, except with more variables.

What is an important thing to note about A (intercept) in multiple regression?

It can also be referred to as B0.

What is the purpose of multiple regression?

Multiple regression extends the principles of linear regression by using more than one variable as a predictor. It shows the relative importance of the predictors (if one predicts a higher amount of variance), and whether a dependent variable is best predicted by a combination of variables rather than one.

What are the assumptions needed to perform a multiple regression?

Normally distributed data Linear relationship between variables Homoscedasticity: criterion has an equal level of variability for each value of the predictor which can be checked by visual examination of the residuals.

More detail on the hierarchical method of entering variables into regression equation

Predictor variables are entered into the equation in blocks dependent on some theoretical basis e.g. 'age' may be entered in first in order to control for its effects on the DV, before entering in another block of variables.

What is multicollinearity?

This is when predictor variables strongly correlate with each other (correlation of .8 or higher). In this case only one variable is assigned predictive value, and they cancel each other out.

What is AR-squared?

This refers to the amount of variance in the criterion explained by the regression equation within the population. It adjusts R-squared down with every predictor added to the model and is generally lower than R-squared.

What is variance inflation factor?

This shows how much variance of a regression coefficient is increased due to collinearity. Guidelines on acceptable levels of this is mixed, though generally 8 - 10

What is the formula for multiple regression

Y' = A + B1X1 + B2X2 + B3X3.... A= the intercept B1 refers to the regression weight of the first variable (X1), B2 refers to the regression weight of the second variable (X2) and so on... From this, you can determine the relative importance of each predictor in determining the criterion.