Multiple Linear Regression Tutorial - Revision history

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JayZzz at 02:39, 29 July 2011

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JayZzz: Created page with '==SOCR_EduMaterials_AnalysesActivities - Multiple Linear Regression Tutorial== '''Multiple Linear Regression Tutorial Using LA Neighborhoods Data''' '''Data:''' We will be …'

2011-07-29T02:38:48Z

Created page with '==SOCR_EduMaterials_AnalysesActivities - Multiple Linear Regression Tutorial== '''Multiple Linear Regression Tutorial Using LA Neighborhoods Data''' '''Data:''' We will be …'

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==[[SOCR_EduMaterials_AnalysesActivities]] - Multiple Linear Regression Tutorial==

'''Multiple Linear Regression Tutorial Using LA Neighborhoods Data'''

'''Data:''' We will be using the [http://wiki.stat.ucla.edu/socr/index.php/SOCR_Data_LA_Neighborhoods_Data#Data_Source 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 white in population.

''Step 1:'' First, we will import the data into the SOCR Simple Regression Analysis Activity. Head to [http://wiki.stat.ucla.edu/socr/index.php/SOCR_Data_LA_Neighborhoods_Data#Data_Source LA Neighborhoods Data]and 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. [[File:MReg1.png|center|800px]]

''Step 3:'' Now Click the “PASTE” button under the drop down menu. You should now see the data in the window. [[File:SReg3.png|center|800px]]

''Step 4:'' You should now see the data in the window. 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.[[File:MReg2.png|center|800px]]

''Step 5:'' Click “CALCULATE”. You will now be taken to the “RESULTS” tab. Here you can see the regression equation and <math>R^2</math>, among others. [[File:MReg3.png|center|800px]]

''Step 6:'' Click “GRAPH”. Here you will see scatterplots of the Income variable against each of the three chosen explanatory variables, [[File:MReg4.png|center|800px]] as well as the residual plots [[File:MReg5.png|center|800px]] and the Normal QQ Plot. [[File:MReg6.png|center|800px]]

''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 (age) and dependent variable (income).
* How to check: Make a scatter plot of income and age
* 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
'''''Assumptions 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:
[[File:MReg7.png|center|800px]]

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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@@ Line 21: / Line 21: @@
 ''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 (age) and dependent variable (income).
+Assumption 1: There is a linear relationship between the independent and dependent variables.
-* How to check: Make a scatter plot of income and age
+* 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 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'''''

@@ Line 34: / Line 34: @@
 * 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
-'''''Assumptions met'''''
+'''''Assumption met'''''
 '''Conclusions'''

@@ Line 13: / Line 13: @@
 ''Step 3:'' Now Click the “PASTE” button under the drop down menu. You should now see the data in the window. [[File:SReg3.png|center|800px]]
-''Step 4:'' You should now see the data in the window. 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.[[File:MReg2.png|center|800px]]
+''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.[[File:MReg2.png|center|800px]]
 ''Step 5:'' Click “CALCULATE”. You will now be taken to the “RESULTS” tab. Here you can see the regression equation and <math>R^2</math>, among others. [[File:MReg3.png|center|800px]]

@@ Line 5: / Line 5: @@
 '''Data:''' We will be using the [http://wiki.stat.ucla.edu/socr/index.php/SOCR_Data_LA_Neighborhoods_Data#Data_Source 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 white in population.
+'''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 [http://wiki.stat.ucla.edu/socr/index.php/SOCR_Data_LA_Neighborhoods_Data#Data_Source LA Neighborhoods Data]and find the table with the data. Select all of the data, and press Ctrl+C (Command+C on Macs) to copy it.