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- 1. Regression Analysis CMGT 587AUNIVERSITY OF SOUTHERN CALIFORNIA AL ARIZMENDEZ/CATHRYN LOTTIER
- 2. What is Regression Analysis? The Regression Method, more commonly referred to as Regression Analysis, is the assessment of the relationship of a dependent variable and one or more multiple independent variable(s). It involves techniques for measuring or analyzing multiple variables and their relationship This technique is used to analyze variables with at least one dependent variable (often y) and one or multiple independent variables (often x) to understand a phenomena, make predictions, and/or test hypotheses
- 3. Assumptions Underlying the Method The validity of regression analysis depends on four assumptions: Linearity: where the relationship between dependent and independent variables are directly proportional to each other Independence: an independence of errors with no serial correlation (a random value of Y is assumed to be independent of any other value of Y) Constant variance: having your data values be scattered to the same extent Normality: the random variable of interest is distributed is a normal manner
- 4. When can you use Regression Analysis? Regression Analysis is used to make predictions, so it can virtually be used by anyone Some reasons that you may want to use regression analysis are: To model a phenomena to understand it better in order to make decisions To model a phenomena to understand it better to predict values for that in other places or times (later in these slides, you will see an example of this as we created an example to forecast album sales) To test a hypotheses, but one should note that regression analysis is an estimate or guess, not an accurate data set (we will show an example of this later in the slides with our test of life expectancy vs. literary rates)
- 5. Diving a Little Deeper… Multiple linear regression analysis begins by positing the general form of the relationship in the following model: ϒi = β0 + β1Χi1 + εi More simply put: Outcomei = (b0 + b1xi) + errori Where Y is the dependent variable, β0 is the intercept, β1 is the slope and Χi1 is the independent variable The ε is the residual term, which expresses the composite of all the other types of individual differences that aren’t explicitly identified in the model (a.k.a. random error term)…a reminder that it will never be perfect
- 6. What does that really mean? That equation means that the “outcome” can be predicted from a model and some error associated with that prediction (εi) The outcome variable is represented as yi, which is predicted using a predictor variable (xi) and a parameter (bi) associated with the predictor variable Bi is the line the direction or strength of the relationship or effect B0 tells us what the value of the outcome is when the predictor is 0 (the intercept) The betas tell us what the shape of the model is and what it looks like
- 7. Explanation of R Squared R2 allows one to assess how well the model fits If you square all of the differences, the sum of all the squared differences is known as the total sum of squares (SST ) If an optimal model is fitted to the data, the differences between the observed data points and the values predicted by the regression line can be squared and summed, which is referred to as the sum of squared residuals (SSR) The difference between SST and SSR is the model sum of squares (SSM) R2 is determined by dividing the model sum of squares by the total sum of squares, which is used to describe how well the regression line fits An R2 near 1 indicates that a regression line fits the data well, while an R2 closer to 0 indicates a regression line does not fit the data very well
- 8. Example of Regression Analysis Regression Analysis can be used to forecast the trend of album sales (shown on the y-axis) in relation to the advertising budget (shown on the x-axis)
- 9. Adding Another Variable to the Equation Now, taking it one step further and adding amount of radio play to the equation This turns into multiple regression analysis with more predictors creating a regression plane (or a 3d model) with the line turning into a plane It looks more complicated, but the principles remain the same as linear regression
- 10. Explanation of Multiple Regression Analysis Multiple Regression Analysis Often referred to as OLS (Ordinary Least Squares) regression “multiple regression can establish whether a set of independent variables explains a proportion of the variance in a dependent variable at a significant level (through a significance test of R2)” (Garson, 2012, p. 10) It can also determine the relative predictive importance of the independent variable (by comparing regression weights, also known as beta weights)
- 11. Multiple Regression Analysis While the formula for linear regression analysis looks like this: ϒi = β0 + β1Χ1i + εi Multiple regression analysis looks more like this: ϒi = (β0 + β1Χ1i+ β2Χ2i…+ βnΧni) + εi This shows that the principles are the same aslinear regression, there are just more predictors!
- 12. Talking About the Betas The betas tell the relationship between a particular predictor and the outcome The betas also define the shape of the plane In this instance: the beta 0 is represent where the plane hits the y-axis (value of the outcome when both predictors are zero) b1 represents the slope of the side associated with radio play b2 represents the slope of the side associated with advertising budget This can go on for multiple dimensions with each of the predictors defining the shape
- 13. Simple Linear Regression w/ SPSS Life Expectancy of Females (dependent variable) Literacy of country in percent (independent variable)
- 14. Simple Regression w/ SPSS: Open the Data Set
- 15. Simple Regression w/ SPSS: Scatter It’s always a good idea to do a scatter plot Graphs>Legacy dialogs> Scatter/Dot>Define
- 16. Simple Regression w/SPSS: Scatter Add dependent variable (Life expectancy) on y-axis Add independent variable (Literacy) on x-axis
- 17. Simple Regression : Scatter done, we’re not Strong uphill pattern, expectancy increases with literacy rate, but we need to run a regression line
- 18. Simple Regression w/SPSS: Scatter and Regression Line
- 19. Simple Regression: Plotted, now run itAnalyze> Regression> Linear
- 20. Simple Regression w/SPSS: Output
- 21. Simple Regression w/SPSS: Scatter Coefficients
- 22. Multiple Linear Regression w/SPSSAnalyze>Regression> Linear
- 23. Multiple Linear Regression w/SPSSTop half of output; notice the multiple variables enteredand the single dependent variable (female life expectancy)
- 24. Multiple Linear Regression w/SPSSBottom half of Output:
- 25. Multiple Linear Regression w/SPSS Literacy is one variable, but it is that specific combination of the variables that Multiple Linear Regression tests for makes MLR so powerful

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