Multiple Linear Regression Tutorial

SOCR_EduMaterials_AnalysesActivities - Multiple Linear Regression Tutorial

Multiple Linear Regression Tutorial Using LA Neighborhoods Data

Data: We will be using the LA Neighborhoods Data for this tutorial.

Goal: Our goal is to predict the median income using multiple explanatory variables by using SOCR. In this example, we will predict median income using age, proportion of homeowners, and proportion of whites in population.

Step 1: First, we will import the data into the SOCR Simple Regression Analysis Activity. Head to LA Neighborhoods Dataand find the table with the data. Select all of the data, and press Ctrl+C (Command+C on Macs) to copy it.

Step 2: Next, head to http://socr.ucla.edu/htmls/SOCR_Analyses.html, and find the Simple Regression Analysis Activity in the drop-down menu.

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Step 3: Now Click the “PASTE” button under the drop down menu. You should now see the data in the window.

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Step 4: Click on the “MAPPING” tab. This is where we define our dependent and independent variables. The dependent variable is the one we want to make a prediction on, and the independent variables are the ones which we will use to make the prediction. In this example, we add “Income” to the dependent variables list and “Age”, “Homes” and “White” to the independent variables list.

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Step 5: Click “CALCULATE”. You will now be taken to the “RESULTS” tab. Here you can see the regression equation and \(R^2\), among others.

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Step 6: Click “GRAPH”. Here you will see scatterplots of the Income variable against each of the three chosen explanatory variables,

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as well as the residual plots

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and the Normal QQ Plot.

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Step 7: We want to check that the assumptions of linear regression, and make sure that they are met.

Assumption 1: There is a linear relationship between the independent and dependent variables.

• How to check: Make a scatter plot of the variables
• How to fix: Transformations (for example Log(y) vs x), or the relationship is not linear.

Linear model fits the data moderately well

Assumption 2: The variance is constant

• How to check: Look at plot of residuals vs. predicted values. Make sure there is not a pattern, such as the residuals getting larger as the predicted values increase.
• How to fix: Logging of variables, fixing underlying independence or linearity causes.

Slight increase in residuals at the top range of exploratory variables

Assumption 3: Errors are normally distributed.

• How to check: Normal QQ Plot (Should lie close to straight line)
• How to fix: Take out outliers, if applicable. Non-linear transformation may be needed

Assumption met

Conclusions

No major violation of linear regression assumptions, we proceed with our analysis:

We can see from the "Results" tab that the regression equation is:

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Income = -21139.729 +1347.656*Age +49806.135*White +53726.649*Homes + E

The “E” is the error term. “Income” is the predicted value, and “Homes”, “Age”, and “White” are the explanatory variables.

This model states that for every 100 percent increase in homeowner proportion, and everything else held constant, the median household income will increase by $53726.65. For every 1 year increase in median age, and everything else held constant, the median household income will increase by $1,347.66. For every 100 percent increase in the proportion of whites in the population, with everything else held constant, the median household income will increase by $49806.14.

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