1. Dr. G. Johnson, www.researchdemystified.org1 Data Analysis: Regression Research Methods for Public Administrators Dr. Gail Johnson

2. Dr. G. Johnson, www.researchdemystified.org2 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

3. Dr. G. Johnson, www.researchdemystified.org3 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

4. Dr. G. Johnson, www.researchdemystified.org4 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.

5. Dr. G. Johnson, www.researchdemystified.org5 Making Sense of Regression The emphasis here is on Understanding the key elements of regression Requirements Application Limitations

6. Dr. G. Johnson, www.researchdemystified.org6 Regression Is A Powerful Analytical Technique Enables researchers to do two things: 1. 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

7. Dr. G. Johnson, www.researchdemystified.org7 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

8. Dr. G. Johnson, www.researchdemystified.org8 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

9. Dr. G. Johnson, www.researchdemystified.org9 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

10. Dr. G. Johnson, www.researchdemystified.org10 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.

11. Dr. G. Johnson, www.researchdemystified.org11 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

12. Dr. G. Johnson, www.researchdemystified.org12 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?

13. Dr. G. Johnson, www.researchdemystified.org13 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

14. Dr. G. Johnson, www.researchdemystified.org14 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

15. Dr. G. Johnson, www.researchdemystified.org15 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.long- range forecasting

16. Dr. G. Johnson, www.researchdemystified.org16 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

17. Dr. G. Johnson, www.researchdemystified.org17 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)

18. Dr. G. Johnson, www.researchdemystified.org18 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

19. Dr. G. Johnson, www.researchdemystified.org19 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

20. Dr. G. Johnson, www.researchdemystified.org20 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

21. Dr. G. Johnson, www.researchdemystified.org21 X Axis: Age of Planes 5 years.......... 20 years Y Axis: Plane Maintenance Costs The dots represent each plane’s age and maintenance cost from prior year Predicted values if perfect relationship $1,000 $500........ 10 years.....

22. Dr. G. Johnson, www.researchdemystified.org22 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

23. Dr. G. Johnson, www.researchdemystified.org23 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)

24. Dr. G. Johnson, www.researchdemystified.org24 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)

25. Dr. G. Johnson, www.researchdemystified.org25 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.

26. Dr. G. Johnson, www.researchdemystified.org26 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

27. Dr. G. Johnson, www.researchdemystified.org27 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

28. Dr. G. Johnson, www.researchdemystified.org28 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.

29. Dr. G. Johnson, www.researchdemystified.org29 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

30. Dr. G. Johnson, www.researchdemystified.org30 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

31. Dr. G. Johnson, www.researchdemystified.org31 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

32. Dr. G. Johnson, www.researchdemystified.org32 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

33. Dr. G. Johnson, www.researchdemystified.org33 What Are the Possible Causes for Urban Decay? 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

34. Dr. G. Johnson, www.researchdemystified.org34 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

35. Dr. G. Johnson, www.researchdemystified.org35 Multiple Regression Equation Y = a + bX 1 + bX 2 + bX 3 + bX 4 + 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

36. Dr. G. Johnson, www.researchdemystified.org36 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

37. Dr. G. Johnson, www.researchdemystified.org37 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 )

38. Dr. G. Johnson, www.researchdemystified.org38 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

39. Dr. G. Johnson, www.researchdemystified.org39 Multiple Regression: Interpretation Results: Y= 6000 + 400X 1 (education) + 200X 2 (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.

40. Dr. G. Johnson, www.researchdemystified.org40 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

41. Dr. G. Johnson, www.researchdemystified.org41 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)

42. Dr. G. Johnson, www.researchdemystified.org42 Multiple Regression Relationship between contributions to political campaigns as a function of age and income. Computer generates this equation: Y = 8 + 2X 1 +.010X 2 (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

43. Dr. G. Johnson, www.researchdemystified.org43 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

44. Dr. G. Johnson, www.researchdemystified.org44 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

45. Dr. G. Johnson, www.researchdemystified.org45 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.

46. Dr. G. Johnson, www.researchdemystified.org46 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.

47. Dr. G. Johnson, www.researchdemystified.org47 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.

48. Dr. G. Johnson, www.researchdemystified.org48 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?

49. Dr. G. Johnson, www.researchdemystified.org49 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

50. Dr. G. Johnson, www.researchdemystified.org50 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.

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