Data Analysis: Regression

1. Data Analysis: Regression Research Methods for Public Administrators Dr. Gail Johnson Dr. G. Johnson, www.researchdemystified.org

2. Making Sense of Regression • Regression analysis is an advanced analytical technique—with the ability to consider many different variables that might explain something like differences in income or declining crime rates Dr. G. Johnson, www.researchdemystified.org

3. Making Sense of Regression • Why include in an introductory research methods textbook? • Because regression results are often reported in the news • Because regression is not hard to understand conceptually-building on what we know about relationships and measures of association –even if the actual equations are intimidating and unclear because so many symbols are used Dr. G. Johnson, www.researchdemystified.org

4. Back to the Premise of Demystifying Statistics • When advocates of particular policies try to persuade, they often use statistics. • The fancier statistics might be appropriate but can also bedazzle or intimidate. • Having an insider’s view about measuring relationships using quantitative data may demystify these statistical techniques. Dr. G. Johnson, www.researchdemystified.org

5. Making Sense of Regression • The emphasis here is on • Understanding the key elements of regression • Requirements • Application • Limitations Dr. G. Johnson, www.researchdemystified.org

6. Regression Is A Powerful Analytical Technique Enables researchers to do two things: • Determine the strength of the relationship • The r-squared value • Small “r” for regression with only one independent variable • Capital “R” for regression with more than one independent variable Dr. G. Johnson, www.researchdemystified.org

7. Regression Is A Powerful Analytical Technique 2. Determine the impact of the independent variable(s) on the dependent variable • The regression coefficient is the predicted change in the dependent variable for every one unit of change in the independent variable • Collectively, the regression coefficients enable the researchers to make estimates of how the dependent variable will change using different scenarios for the independent variables Dr. G. Johnson, www.researchdemystified.org

8. 1. R-square And Its Companions • r = correlation coefficient (overall fit or measure of association, which is also called r, Pearson’s r, Pearson Product Moment Correlation coefficient, or zero-order coefficient). • We’ve seen this in prior chapter • r-square = proportion of the explained variance the dependent variable (also called the coefficient of determination) • 1 minus r-square = proportion of unexplained variance in the dependent variable Dr. G. Johnson, www.researchdemystified.org

9. Interpreting R-Square Is Easy • Or at least as easy as any measure of association • Fake Example: Researchers look at GRE scores and academic performance in graduate school as measured by grade point average • The hypothesis is that people who have high GRE scores will also have high GPAs • From an admission’s committee perspective: the belief that GRE scores are a good predictor of future academic success and are, therefore, a good criteria for admission decisions • The researchers report an r-squared of .2 Dr. G. Johnson, www.researchdemystified.org

10. Interpreting R-Square Is Easy • R-square is similar to a measure of association: • It varies from 0 to 1: zero indicating no relationship, 1 indicating a perfect relationship • Except that it gives more information—it gives an estimate of how much change in the dependent variable (in this case, GPAs) are explained by GRE scores. • Interpretation of prior slide: GRE’s explain 20 percent of the change in GPAs • This means that 80 percent of the changes in GPA are explained by other factors. Dr. G. Johnson, www.researchdemystified.org

11. Discussion • If you were making a recommendation to the admissions committee, how much emphasis should they give GRE scores in admission decisions? • Explain/defend your reasoning Dr. G. Johnson, www.researchdemystified.org

12. A Different R-Squared, A Different Decision? • Suppose the researchers found an r-squared of .65? • What would you recommend? Why? • What other factors might be important in predicting academic success in graduate school? Dr. G. Johnson, www.researchdemystified.org

13. Paradox of High R-squares • Researchers want to obtain results with a high R-square • They want to build models that explain as much as possible about what affects the dependent variable • That is, they want to discover good predictive models • But sophisticated users should be suspicious of results with a high R-squared Dr. G. Johnson, www.researchdemystified.org

14. Generating High R-squares • Problem of multi-collinearity • This means using independent variables that are highly correlated with each other • Including median income and poverty rates for example • They will throw off the mathematics that may give a falsely high r-squared • Aggregating data in ways that reduce sample size can generate high r-squares Dr. G. Johnson, www.researchdemystified.org

15. Generating High R-squares • Researchers might decide to get rid of “outliers”—the data points that are really, really far away from the bulk of the data • If the data point is truly incorrect—clearly someone typed it I wrong, it can be deleted. • Otherwise, researchers should accept the outliers as part of the way things are • For more information, see Taken from J. Scott Armstrong, 1985, long-range forecasting, 2nd ed., P. 487. Dr. G. Johnson, www.researchdemystified.org

16. 2. Regression Wizardry: Predicting Change • Regression follows the same concepts of relationships, then takes it to the next level • It allows researchers to predict the change in the dependent variable based on every unit change in the independent variable • This is the regression coefficient (or partial regression coefficient in multiple regression analysis) • If the regression coefficient = .05, it means that for every one unit change in the GRE score, there will be a .05 increase in the GPA score • Assuming, of course, that there is a strong relationship Dr. G. Johnson, www.researchdemystified.org

17. Other Examples of the Regression Coefficient • For every one unit change in years of education, there is a $2,000 change in yearly individual income. • For every one unit change in the age of a plane, there is a $500 change in maintenance costs. • For every one unit change in age, there is a .3 percent decrease in memory test scores among adults. • (note: these are all fake data) Dr. G. Johnson, www.researchdemystified.org

18. Regression Requirements • Requirements: • Assumes a linear relationship • Uses random sample or census data • Works with interval/ratio level data • It is possible to convert a nominal variable into a “dummy variable”—which means that it only has two variables: 0 and 1—to use as an independent variable • For example: Gender: female 0, male 1 Dr. G. Johnson, www.researchdemystified.org

19. Ordinary Least Squares Regresion • There are many types of regression tools • For our purposes, I am sticking with what they call “ordinary least squares” (OLS) that can only be used with interval/ratio level data (i.e. real numbers) • There are other types to handle other data situations • For example, logistic regression is use with nominal dependent variable with only 2 categories • For example: Drug Use: yes or no Dr. G. Johnson, www.researchdemystified.org

20. The Concept of “Least Squares” • Regression analysis used here is based on the idea of “least squares” • The computer creates an imaginary "best" straight line through a set of data, such that for any value of X, the value of Y can be predicted Dr. G. Johnson, www.researchdemystified.org

21. The dots represent each plane’s age and maintenance cost from prior year . Y Axis: Plane Maintenance Costs . . . . Predicted values if perfect relationship . . . . . . . . . $1,000 . . . . . . . $500 . . 20 years 5 years 10 years X Axis: Age of Planes Dr. G. Johnson, www.researchdemystified.org

22. The Concept of “Least Squares” • This line is selected because it yields the smallest total distance between every data point and this perfect line. • The distances are squared as part of the calculation—hence the name, “least squares” • The line is useful to the extent that the difference between the predicted line and the actual data points is small Dr. G. Johnson, www.researchdemystified.org

23. Simple Regression Equation • Y = a + bX + e • Where: • Y = predicted value of the dependant variable • a = the constant or Y intercept (where the imaginary line crosses the Y access) • b = the regression coefficient • X = the independent variable • e = error (the computer will estimate the likely error) Dr. G. Johnson, www.researchdemystified.org

24. Applying Simple Regression • Researchers are asked to estimate maintenance costs for next year’s budget • This large state that has a fleet of planes used by public officials to make it easy to visit all parts of the state • Analysts believe that there is a relationship between maintenance costs and use of the planes (measured by the miles flown) • Y= plane maintenance costs measured in dollars (the dependent variable) • X = miles flown (the independent variable) Dr. G. Johnson, www.researchdemystified.org

25. How It Is Applied • Analysts collect data over the past two years and crunch it. The computer gives these results: Y = 100 and .020X • The constant is 100: • If they do not fly at all, the computer estimates there is still a cost of $100 • The .020 is the regression coefficient: • This gets interpreted as: for every mile flown, there is $.02 change in maintenance costs. Dr. G. Johnson, www.researchdemystified.org

26. Simple Regression Y = 100 and .020X • Interpreting the regression coefficient: • For every mile flown, the maintenance costs goes up by 2 cents. • For every 100 miles flown, costs are $2 • For every 1,000 miles, the costs are $20 • For every 100,000 miles, the costs are $20,000 Dr. G. Johnson, www.researchdemystified.org

27. Making Maintenance Cost Estimates • They can then solve the equation: • Assuming 100,000 miles will be flown, how much will they need to budget for maintenance? • 100,000 multiplied by .020 = $20,000 • Y= 100 + $20,000 + error • The estimate maintenance will cost: $20,100 + error Dr. G. Johnson, www.researchdemystified.org

28. Yes, but… • How strong is the relationship between miles flown and maintenance costs? • Before we put too much faith in these budget estimates, we will want to look at the r-squared • Like any measure of association, there is some choice about what is “good enough”, since it would be exceedingly rare to get an r-squared close to a perfect 1. Dr. G. Johnson, www.researchdemystified.org

29. Simple Regression: Another Example • Hypothesis: If schools have a higher percentage of poor children, then they will have lower test scores. • A regression analysis shows: • A regression coefficient of -.04 • An r-squared value of .25 Dr. G. Johnson, www.researchdemystified.org

30. Simple Regression • Interpretation? • Regression coefficient: For every increase in the percent of children in poverty within a school, the average test score goes down by .04 • R-squared: 25% of the test scores are explained by the percent of children in poverty in the school • Researchers will ask: what other factors might explain differences in test scores in the schools? • They will want to build a bigger model that will include more factors Dr. G. Johnson, www.researchdemystified.org

31. Life More Complex • Rarely will any one single variable cause big changes in another variable, especially complex phenomena • Warning bells should sound when anyone states that a single variable caused a complex problem • The economic collapse is due to consumer debt • The economic collapse is due to corporate greed Dr. G. Johnson, www.researchdemystified.org

32. Discussion: Complexity of Public Policy Issues • What are the possible causes the 2008 economic downturn? • What are the possible explanations for the declining crime rate from 1991 to 2004? • In 1991, the national violent crime rate was: • 1991: 753 per 100,000 population • 2004: 463 per 100,000 population Dr. G. Johnson, www.researchdemystified.org

33. Lack of jobs High % of absentee landlords Low % of homeowners Poor quality of schools Increased concentration of poor Increase in drugs, crime Aging housing stock Flight of middle class to suburbs Corruption Aging infrastructure Business flight to suburbs What Are the Possible Causes for Urban Decay? Dr. G. Johnson, www.researchdemystified.org

34. Multiple Regression: Added Power • Multiple regression does four things: • Provides the an overall measure of the predictive strength of the model: the R-square • Predict the dependent variable based on the summed contributions of the independent variables. • Determines the impact of each independent variable on the dependent variable while controlling for the other variables (these are the partial regression coefficients) • Determines the relative strength of each of the independent variable using the beta weights Dr. G. Johnson, www.researchdemystified.org

35. Multiple Regression Equation Y = a + bX1 + bX2 + bX3 + bX4 + e. Y = dependent variable X1 = independent variable 1, controlling for X2, X3, X4 X2 = independent variable 2 controlling for X1, X3, X4 X3 = independent variable 3 controlling for X1, X2, X4 X4= independent variable 4 controlling for X1, X2, X3 Dr. G. Johnson, www.researchdemystified.org

36. Multiple Regression Equation • It has the same basic structure of simple regression • Y is still the dependent variable • There is still a constant (a) and some amount of error (e) that the computer calculates • But there are more Xs to represent the multiple independent variables Dr. G. Johnson, www.researchdemystified.org

37. Multiple Regression Equation • The b in front of the Xs will be the Partial Regression Coefficients • The separate impact on dependent variable controlling for all the other independent variables (sometimes called “holding them constant) Dr. G. Johnson, www.researchdemystified.org

38. Multiple Regression: An Example Hypothesis: Income is a function of education and seniority? We suggest that income (the dependent variable) will increase as both education and seniority increases (two independent variables) Y (Income) = a + education + seniority+ error based on Lewis-Beck example Dr. G. Johnson, www.researchdemystified.org

39. Multiple Regression: Interpretation Results: Y= 6000 + 400X1 (education) + 200X2 (seniority) R square = .67 • First look at the R-Square: This shows a strong relationship—so analysis can continue • Partial regression coefficients: • For every year of education, holding seniority constant, income increases by $400. • For every year of seniority, holding education constant, income increases by $200. Dr. G. Johnson, www.researchdemystified.org

40. Multiple Regression: Application Estimate the income of someone who has 10 years of education and 5 years of seniority We solve the regression equation: • Multiply the 10 years of education by the regression coefficient of 400: equals 4,000 • Multiply 5 years of senior by the regression coefficient of 200: equals 1,000 • Put it together with the constant and you have • Y=6000 + 400(10) + 200(5) + error Y= $ 11,000 + error Dr. G. Johnson, www.researchdemystified.org

41. Multiple Regression: Beta Weights Relationship between contributions to political campaigns as a function of age and income? Y= campaign contribution (dollars) X1 = age (years) X2 = income (dollars) Dr. G. Johnson, www.researchdemystified.org

42. Multiple Regression Relationship between contributions to political campaigns as a function of age and income. Computer generates this equation: Y = 8 + 2X1 + .010X2 (age) (income) Interpreting the partial regression coefficients: • For every one year increase in age, contributions go up by $2. • For every dollar increase in income, contributions go up .01 dollars Dr. G. Johnson, www.researchdemystified.org

43. Multiple Regression: Beta Weights • But which is stronger? • We cannot tell because age and income are measured differently (years versus dollars) • Need to look at the Beta Weights • Beta Weights are Standardized--thus making all variables comparable • But they have a very limited application Dr. G. Johnson, www.researchdemystified.org

44. Beta Weights • Returning to age and income as predictors of campaign contributions, the computer gives us these beta weights Age = .15 Income = .45 • Which is the strongest of the two? • Income is the highest, therefore the stronger of the two Dr. G. Johnson, www.researchdemystified.org

45. Takeaway Lesson • When reading research results about relationships, my best advice is to exercise healthy skepticism and ask the tough questions before asserting—or believing—that research results are irrefutable facts merely because of sophisticated mathematics. Dr. G. Johnson, www.researchdemystified.org

46. Takeaway Lesson • Knowing how difficult it is to demonstrate causality or program impacts, be mindful when people present research asserting they have found a cause-effect relationship. • Be especially cautious when people claim they have a found a single cause for a complex phenomenon even when they use advanced statistical techniques. Dr. G. Johnson, www.researchdemystified.org

47. Takeaway Lesson • At the same time, be cautious in believing variables are not connected or that programs do not have an impact based on data from one study. • “More research is needed” is not a self-employment program for researchers • It is also important to know when statistics are just too frail to give a clear answer. Dr. G. Johnson, www.researchdemystified.org

48. Ask the Tough Questions • Are they using data that is likely to be unknown or difficult to measure? • Do the proxy measures they use make sense? • Do they state all of their assumptions in constructing their measures used in their calculations? • Is the analysis appropriate to the situation? • Do they provide measures of association and are they strong enough? Dr. G. Johnson, www.researchdemystified.org

49. Ask the Tough Questions • Is there design strong enough to rule out possible rival explanations? • Even with fancy statistics, the basic principles of good research design still must be met—especially when attempting to answer cause-effect questions • I might show a high r-square between stock market activity and sunspot activity—but I still need a good theory to explain why they are connected Dr. G. Johnson, www.researchdemystified.org

50. Remember: It Is OK To Ask For Help • It is also important to recognize that statistics can be so technical that it necessary to bring in experts to make sense of complex and confusing research results. • No one expects you to know it all from one required research methods course—or remember it 10 years later • My point: remember that it really is OK to bring in the experts to make sense of research that focuses on issues that matter. Dr. G. Johnson, www.researchdemystified.org