relationship between education level, income, and length of time out of school our new regression...
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Relationship between education level, income, and length of time out of school
Our new regression equation:
is the predicted value of the dependent variableis the predicted value of the dependent variable (income) (income)Yis the value of the first predictor variable (education level)is the value of the first predictor variable (education level)
is the value of the second predictor variable (time out of school)is the value of the second predictor variable (time out of school)
X1
X2
= a + bX1 + bX2 Y
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
Education Level Education Level (X1) in years(X1) in years
Years Working Years Working (X(X2)2)
Monthly Income Monthly Income (Y) in thousands(Y) in thousands
Case 1Case 1 66 1010 11
Case 2Case 2 88 1414 1.51.5
Case 3Case 3 1111 88 11
Case 4Case 4 1212 77 22
Case 5Case 5 1212 2020 44
Case 6Case 6 1313 1515 2.52.5
Case 7Case 7 1414 1717 55
Case 8Case 8 1616 2222 66
Case 9Case 9 1616 3030 1010
Case 10Case 10 2121 1010 88
MeanMean 12.912.9 1515 4.14.1
Standard DeviationStandard Deviation 4.254.25 7.27.2 3.123.12
Correlation Correlation CoefficientCoefficient
rr = 0.83 = 0.83 rr = 0.70 = 0.70
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
We begin by computing Pearson correlation coefficients for all three of the variables in the model:
These data reveal that level of education and years in the workforce are both correlated with monthly income, and with each other
Years of Years of EducationEducation
Years in Years in WorkforceWorkforce
Monthly Monthly IncomeIncome
Years of Years of EducationEducation
1.001.00
Years in Years in WorkforceWorkforce
0.3100.310 1.001.00
Monthly IncomeMonthly Income 0.8260.826 0.6950.695 1.001.00
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
Strong correlations among predictor variables is called Strong correlations among predictor variables is called multicollinearitymulticollinearity and can cause problems in multiple and can cause problems in multiple regression analysis because it can make it difficult to regression analysis because it can make it difficult to identify the unique relation between each predictor identify the unique relation between each predictor variable and the dependent variablevariable and the dependent variable
Use Tolerance, VIF, and Collinearity diagnostics to detect Use Tolerance, VIF, and Collinearity diagnostics to detect collinearity issuescollinearity issues
First independent First independent variable (variable (XX11))
Unique varianceUnique variance
Dependent variable (Dependent variable (YY))
Second independent Second independent variable (variable (XX22))
Shared varianceShared variance
Variance ExplainedVariance Explained
RR R SquareR Square Adjusted R Adjusted R SquareSquare
Std. Error Std. Error of the of the
EstimateEstimate
0.9460.946 0.8960.896 0.8660.866 1.14051.1405
ANOVA ResultsANOVA Results
Sum of Sum of SquaresSquares
dfdf Mean SquareMean Square F ValueF Value P ValueP Value
RegressionRegression 78.29578.295 22 39.14739.147 30.09530.095 0.0000.000
ResidualResidual 9.1059.105 77 1.3011.301
TotalTotal 87.40087.400 99
Regression CoefficientsRegression Coefficients
Un-Un-standardized standardized CoefficientsCoefficients
Standardized Standardized CoefficientsCoefficients
BB Std. ErrorStd. Error BetaBeta t Valuet Value p Valuep Value
InterceptIntercept -5.504-5.504 1.2981.298 -4.421-4.421 0.0040.004
Years Edu.Years Edu. 0.4950.495 0.0940.094 0.6760.676 5.2705.270 0.0010.001
Years WorkYears Work 0.2100.210 0.0560.056 0.4850.485 3.7833.783 00070007
Model Summary
ANOVA
Coefficients
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
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
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.
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.
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