Regression Analysis

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Regression Analysis. Scatter plots. Regression analysis requires interval and ratio-level data. To see if your data fits the models of regression, it is wise to conduct a scatter plot analysis. The reason?
Regression AnalysisScatter plots
  • Regression analysis requires interval and ratio-level data.
  • To see if your data fits the models of regression, it is wise to conduct a scatter plot analysis.
  • The reason?
  • Regression analysis assumes a linear relationship. If you have a curvilinear relationship or no relationship, regression analysis is of little use.
  • Types of LinesScatter plot
  • This is a linear relationship
  • It is a positive relationship.
  • As population with BA’s increases so does the personal income per capita.
  • Regression Line
  • Regression line is the best straight line description of the plotted points and use can use it to describe the association between the variables.
  • If all the lines fall exactly on the line then the line is 0 and you have a perfect relationship.
  • Things to remember
  • Regressions are still focuses on association, not causation.
  • Association is a necessary prerequisite for inferring causation, but also:
  • The independent variable must preceded the dependent variable in time.
  • The two variables must be plausibly lined by a theory,
  • Competing independent variables must be eliminated.
  • Regression Table
  • The regression coefficient is not a good indicator for the strength of the relationship.
  • Two scatter plots with very different dispersions could produce the same regression line.
  • Regression coefficient
  • The regression coefficient is the slope of the regression line and tells you what the nature of the relationship between the variables is.
  • How much change in the independent variables is associated with how much change in the dependent variable.
  • The larger the regression coefficient the more change.
  • Pearson’s r
  • To determine strength you look at how closely the dots are clustered around the line. The more tightly the cases are clustered, the stronger the relationship, while the more distant, the weaker.
  • Pearson’s r is given a range of -1 to + 1 with 0 being no linear relationship at all.
  • Reading the tables
  • When you run regression analysis on SPSS you get a 3 tables. Each tells you something about the relationship.
  • The first is the model summary.
  • The R is the Pearson Product Moment Correlation Coefficient.
  • In this case R is .736
  • R is the square root of R-Squared and is the correlation between the observed and predicted values of dependent variable.
  • R-Square
  • R-Square is the proportion of variance in the dependent variable (income per capita) which can be predicted from the independent variable (level of education). 
  • This value indicates that 54.2% of the variance in income can be predicted from the variable education.  Note that this is an overall measure of the strength of association, and does not reflect the extent to which any particular independent variable is associated with the dependent variable. 
  • R-Square is also called the coefficient of determination.
  • Adjusted R-square
  • As predictors are added to the model, each predictor will explain some of the variance in the dependent variable simply due to chance. 
  • One could continue to add predictors to the model which would continue to improve the ability of the predictors to explain the dependent variable, although some of this increase in R-square would be simply due to chance variation in that particular sample. 
  • The adjusted R-square attempts to yield a more honest value to estimate the R-squared for the population.   The value of R-square was .542, while the value of Adjusted R-square was .532. There isn’t much difference because we are dealing with only one variable. 
  • When the number of observations is small and the number of predictors is large, there will be a much greater difference between R-square and adjusted R-square.
  • By contrast, when the number of observations is very large compared to the number of predictors, the value of R-square and adjusted R-square will be much closer.
  • The p-value associated with this F value is very small (0.0000). 
  • These values are used to answer the question "Do the independent variables reliably predict the dependent variable?". 
  • The p-value is compared to your alpha level (typically 0.05) and, if smaller, you can conclude "Yes, the independent variables reliably predict the dependent variable". 
  • If the p-value were greater than 0.05, you would say that the group of independent variables does not show a statistically significant relationship with the dependent variable, or that the group of independent variables does not reliably predict the dependent variable. 
  • Coefficients
  • B - These are the values for the regression equation for predicting the dependent variable from the independent variable. 
  • These are called unstandardized coefficients because they are measured in their natural units.  As such, the coefficients cannot be compared with one another to determine which one is more influential in the model, because they can be measured on different scales. 
  • Coefficients
  • This chart looks at two variables and shows how the different bases affect the B value. That is why you need to look at the standardized Beta to see the differences.
  • Coefficients
  • Beta - The are the standardized coefficients. 
  • These are the coefficients that you would obtain if you standardized all of the variables in the regression, including the dependent and all of the independent variables, and ran the regression. 
  • By standardizing the variables before running the regression, you have put all of the variables on the same scale, and you can compare the magnitude of the coefficients to see which one has more of an effect. 
  • You will also notice that the larger betas are associated with the larger t-values.
  • How to translate a typical tableRegression Analysis Level of Education by Income per capitaPart of the Regression Equation
  • b represents the slope of the line
  • It is calculated by dividing the change in the dependent variable by the change in the independent variable.
  • The difference between the actual value of Y and the calculated amount is called the residual.
  • The represents how much error there is in the prediction of the regression equation for the y value of any individual case as a function of X.
  • Comparing two variables
  • Regression analysis is useful for comparing two variables to see whether controlling for other independent variable affects your model.
  • For the first independent variable, education, the argument is that a more educated populace will have higher-paying jobs, producing a higher level of per capita income in the state.
  • The second independent variable is included because we expect to find better-paying jobs, and therefore more opportunity for state residents to obtain them, in urban rather than rural areas.
  • SingleMultiple RegressionSingle RegressionMultiple RegressionPerceptions of victoryRegression
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