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Multiple Linear Regression. Relationship between education level, income, and length of time out of school Our new regression equation:. = a + bX 1 + bX 2. Example of Multiple Regression. is the predicted value of the dependent variable (income). X 1.

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example of multiple regression
Relationship between education level, income, and length of time out of school

Our new regression equation:

= a + bX1 + bX2

Example of Multiple Regression

is the predicted value of the dependent variable (income)


is the value of the first predictor variable (education level)


is the value of the second predictor variable (time out of school)

example of multiple regression continued
The new regression equation allows us to:

See whether my two predictor variables, combined, are significantly related to, or predictive of, my dependent variable, and how much of the variance my predictor variables explain in my dependent variable

Test whether each of my predictor variables is significantly related to my dependent variable when controlling for the other predictor variable.

See which of my two predictor variables is the stronger predictor of my dependent variable

Test whether one predictor variable is related to my dependent variable after controlling for the other predictor variable, thus conducting a sort of ANCOVA

Example of Multiple Regression (continued)
what does the table tell us
The data presented in the table on the previous slide reveal that both years of education and years in the workforce are positively correlated with monthly income

Need to answer the following questions:

How much of the variance in income can these two predictor variables explain together?

Will years of education still predict income when we control for the effects of years in the workforce?

Which of these two independent variables will be the stronger predictor of income?

And will each make a unique contribution in explaining variance in income?

To answer these questions, we need to run multiple regression analyses

What Does the Table Tell Us?
the analyses
We begin by computing Pearson correlation coefficients for all three of the variables in the model: The Analyses…
  • These data reveal that level of education and years in the workforce are both correlated with monthly income, and with each other
the analyses continued
In a multiple regression, we’ve got multiple predictor variables trying to explain variance in the dependent variable

In our current example, both of our predictor variables are strongly correlated with our dependent variable, so this condition is met

In addition, for each of our predictor variables to explain a unique, or independent portion of the variance in the dependent variable, our two predictor variables cannot be too strongly related to each other

The Analyses (continued)
shared variance in multiple regression

First independent variable (X1)

Second independent variable (X2)

Shared variance

Unique variance

Dependent variable (Y)

Shared Variance in Multiple Regression
  • Strong correlations among predictor variables is called multicollinearity and can cause problems in multiple regression analysis because it can make it difficult to identify the unique relation between each predictor variable and the dependent variable
  • Use Tolerance, VIF, and Collinearity diagnostics to detect collinearity issues
discussion of model summary
We get an “R” value of 0.946. This is the multiple correlation coefficient(R).

It provides a measure of the correlation between the two predictors combined and the dependent variable. It is also the correlation between the observed value of Y and the predicted value of Y.

We get an “R Square” value (symbolized R2) of 0.896.

This is the coefficient of determination for my combined predictor variables and the dependent variables, and it provides us with a percentage of variance explained. The R2 statistic is the measure of effect size used in multiple regression. Combined, these two predictor variables explain about 90% of the variance in the income variable

The “Adjusted R Square” is 0.866.

Accounts for some of the error associated with multiple predictor variables by taking the number of predictor variables and the sample size into account, and thereby adjusts the R2 value down a little bit

Standard error for the R and R2 value

Discussion of Model Summary
discussion of anova results
F value of 30.095, with a corresponding p value of .000

Reveals that our regression model overall is statistically significant

Total variance of dependent variable is divided into 2 components

Explained variance “regression” compared to unexplained variance “residual”

Mean square = sum squares/df

Discussion of ANOVA Results
discussion of regression coefficients
The unstandardized regression coefficients can be found in the column labeled “B”

It is difficult to compare the size of the unstandardized regression coefficients

In the column labeled “Beta”, the standardized regression coefficients are presented

These regression coefficients have been standardized, thereby converting the unstandardized coefficients into coefficients with the same scale of measurement. In this example, the beta for years of education is a bit larger than the beta for years in the workforce.

In the columns labeled “t value” and “p value” we get measures that allow us to determine whether each predictor variable is statistically significantly related to the dependent variable. In this example, both predictor variables are significant predictors of income.

Discussion of Regression Coefficients
methods of regression analysis
Methods of Regression Analysis
  • How do you decide which variables to include in the final equation?
    • Include only significant correlation coefficients
    • Use a limited number of variables
      • 10:1 rule
    • Enter- usually the default method
    • Forward- variables added one by one if they meet a certain significance value- starts with most significant correlation
    • Backward- variables are removed one by one if they meet a certain significance value- starts with variable that has the smallest partial correlation with the dependent
    • Stepwise- a hybrid of the previous 2- variables are added one at a time however at each step a backward procedure is used to see if any variables should be removed.
issues to consider
Include variables that make sense

predictor variables chosen by selection criteria, need to have logical relationships with the dependent

May force important or logical predictors into the model even if they are not chosen by selection criteria.

Be wary of variable selection results

Issues to Consider