1. Multiple Regression 4 Sociology 5811 Lecture 25 Copyright © 2005 by Evan Schofer Do not copy or distribute without permission

2. Announcements Schedule: –Today: Multiple regression hypothesis tests, assumptions, and problems –Next Class: More diagnostics Including “outliers”, which you should address for the final paper. Don’t miss class! Reminder: Final paper deadline coming up soon! Questions about the paper?

3. Review: Interaction Terms Interaction Terms: Effect of a variable changes within groups or levels of a third Example: Effect of income on happiness may be different for women and men If men are more materialistic, each dollar has a bigger effect Issue isn’t men = “more” or “less” than women –Rather, the slope of a variable coefficient (for income) differs across groups Essentially: different regression line (slope) for each group.

4. Review: Interaction Terms Visually: Women = blue, Men = red INCOME 100000800006000040000200000 HAPPY 10 9 8 7 6 5 4 3 2 1 0 Overall slope for all data points Note: Here, the slope for men and women differs. The effect of income on happiness (X1 on Y) varies with gender (X2). This is called an “interaction effect”

5. Review: Interaction Terms Examples of interaction: –Effect of education on income may interact with type of school attended (public vs. private) Private schooling has bigger effect on income –Effect of aspirations on educational attainment interacts with poverty Aspirations matter less if you don’t have money to pay for college.

6. Review: Interaction Terms Interaction effects: Differences in the relationship (slope) between two variables for each category of a third variable Option #1: Analyze each group separately Look for different sized slope in each group Option #2: Multiply the two variables of interest: (DFEMALE, INCOME) to create a new variable –Called: DFEMALE*INCOME –Add that variable to the multiple regression model.

7. Review: Interaction Terms Example: Interaction of Race and Education affecting Job Prestige: DBLACK*EDUC has a negative effect (nearly significant). Coefficient of -.576 indicates that the slope of education and job prestige is.576 points lower for Blacks than for non-blacks.

8. Continuous Interaction Terms Two continuous variables can also interact Example: Effect of education and income on happiness –Perhaps highly educated people are less materialistic –As education increases, the slope between between income and happiness would decrease Simply multiply Education and Income to create the interaction term “EDUCATION*INCOME” And add it to the model.

9. Interpreting Interaction Terms How do you interpret continuous variable interactions? Example: EDUCATION*INCOME: Coefficient = 2.0 Answer: For each unit change in education, the slope of income vs. happiness increases by 2 –Note: coefficient is symmetrical: For each unit change in income, education slope increases by 2 Dummy interactions effectively estimate 2 slopes: one for each group Continuous interactions result in many slopes: Each value of education*income yields a different slope.

10. Interpreting Interaction Terms Interaction terms alters the interpretation of “main effect” coefficients Including “EDUC*INCOME changes the interpretation of EDUC and of INCOME See Allison p. 166-9 –Specifically, coefficient for EDUC represents slope of EDUC when INCOME = 0 Likewise, INCOME shows slope when EDUC=0 –Thus, main effects are like “baseline” slopes And, the interaction effect coefficient shows how the slope grows (or shrinks) for a given unit change.

11. Dummy Interactions It is also possible to construct interaction terms based on two dummy variables –Instead of a “slope” interaction, dummy interactions show difference in constants Constant (not slope) differs across values of a third variable –Example: Effect of of race on school success varies by gender African Americans do less well in school; but the difference is much larger for black males.

12. Dummy Interactions Strategy for dummy interaction is the same: Multiply both variables –Example: Multiply DBLACK, DMALE to create DBLACK*DMALE Then, include all 3 variables in the model –Effect of DBLACK*DMALE reflects difference in constant (level) for black males, compared to white males and black females You would observe a negative coefficient, indicating that black males fare worse in schools than black females or white males.

13. Interaction Terms: Remarks 1. If you make an interaction you should also include the component variables in the model: –A model with “DFEMALE * INCOME” should also include DFEMALE and INCOME There are rare exceptions. But when in doubt, include them 2. Sometimes interaction terms are highly correlated with its components That can cause problems (multicollinearity – which we’ll discuss more soon)

14. Interaction Terms: Remarks 3. Make sure you have enough cases in each group for your interaction terms –Interaction terms involve estimating slopes for sub- groups (e.g., black females vs black males). If you there are hardly any black females in the dataset, you can have problems 4. “Three-way” interactions are also possible! An interaction effect that varies across categories of yet another variable –Ex: DMale*DBlack interaction may vary across class They are mainly used in experimental research settings with large sample sizes… but they are possible.

15. Multiple Regression Hypothesis Tests Hypothesis tests can be conducted independently for all slopes (b) of X variables For X 1, X 2 …X k, we can test hypotheses for b 1, b 2 …b k Null/Alternative hypotheses are the same: H0:  k = 0 H1:  k  0; Or, one-tailed tests: H1:  k > 0, H1:  k < 0 Hypothesis tests are about the slope controlling for other variables in the model Sometimes people explicitly mention this in hypotheses NOTE: Results with “controls” may differ from bivariate hypothesis tests!

16. Multiple Regression Hypothesis Tests Formula for MV hypothesis tests: Where b is a slope, s b is a standard error k represents the kth independent variable K = total number of independent variables T-test degrees of freedom depends on N and number of independent variables Compare observed t-value to critical t; or p to 

17. Multiple Regression Estimation Calculating b’s involves solving a set of equations to minimize squared error Analogous to bivariate, but math is more complex The optimal estimator has minimum variance and is referred to as “BLUE”: Best Linear, Unbiased Estimate The BLUE Multiple Regression has more assumptions than bivariate.

18. Multiple Regression Assumptions As discussed in Knoke, p. 256 Note: Allison refers to error (e) as disturbance (U); And uses slightly different language… but ideas are the same! 1. a. Linearity: The relationship between dependent and independent variables is linear Just like bivariate regression Points don’t all have to fall exactly on the line; but error (disturbance) must not have a pattern –Check scatterplots of X’s and error (residual) Watch out for non-linear trends: error is systematically negative (or positive) for certain ranges of X There are strategies to cope with non-linearity, such as including X and X-squared to model curved relationship.

19. Multiple Regression Assumptions 1. b. And, the model is properly specified: –No extra variables are included in the model, and no important variables are omitted. This is HARD! Correct model specification is critical If an important variable is left out of the model, results are biased (“omitted variable bias”) –Example: If we model job prestige as a function of family wealth, but do not include education Coefficient estimate for wealth would be biased –Use theory and previous research to decide what critical variables must be included in your model.

20. Multiple Regression Assumptions 2. All variables are measured without error Unfortunately, error is common in measures –Survey questions can be biased –People give erroneous responses (or lie) –Aggregate statistics (e.g., GDP) can be inaccurate This assumption is often violated to some extent –We do the best we can: Design surveys well, use best available data And, there are advanced methods for dealing with measurement error.

21. Multiple Regression Assumptions 3. The error term (e i ) has certain properties Recall: error is a cases deviation from the regression line –Not the same as measurement error! After you run a regression, SPSS can tell you the error value for any or all cases (called the “residual”) 3. a. Error is conditionally normal –For bivariate, we looked to see if Y was conditionally normal. For multivariate regression, we look to see if error is conditionally normal Examine “residuals” (e i ) for normality at different values of X variables.

22. Multiple Regression Assumptions 3. b. The error term (e i ) has a mean of 0 –This affects the estimate of the constant (Not a huge problem) This is not a critical assumption to test. 3. c. The error term (e i ) is homoskedastic (has constant variance) Note: This affects standard error estimates, hypothesis tests –Look at residuals, to see if they spread out with changing values of X Or plot standardized residuals vs. standardized predicted values.

23. Multiple Regression Assumptions 3. d. Predictors (X i s) are uncorrelated with error –This most often happens when we leave out an important variable that is correlated with another X i –Example: Predicting job prestige with family wealth, but not including education –Omission of education will affect error term. Those with lots of education will have large positive errors. Since wealth is correlated with education, it will be correlated with that error! –Result: coefficient for family wealth will be biased (vastly overestimated).

24. Multiple Regression Assumptions 4. In systems of equations, error terms of equations are uncorrelated Knoke, p. 256 –This is not a concern for us in this class Worry about that later!

25. Multiple Regression Assumptions 5. Sample is independent, errors are random Technically, part of 3.c. –Not only should errors not increase with X (heteroskedasticity), there should be no pattern at all! Things that cause patterns in error (autocorrelation): –Measuring data over long periods of time (e.g., every year). Error from nearby years may be correlated. Called: “Serial correlation”.

26. Multiple Regression Assumptions More things that cause patterns in error (autocorrelation): –Measuring data in families. All members are similar, will have correlated error –Measuring data in geographic space. Example: data on 50 US states. States in a similar region have correlated error Called “spatial autocorrelation” There are variations of regression models to address each kind of correlated error.

27. Multiple Regression Assumptions Regression assumptions and final projects: 1. Check all your assumptions… but present results for only 1 or 2 X variables. 2. Multivariate assumption checks involve plots of e (“error” or “residual”) to test linearity, heteroskedasticity This contrasts bivariate, where you plotted X vs. Y Don’t forget to focus on “e”! 3. Also, you should check for outliers To be discussed soon!

28. Regression: Outliers Note: Even if regression assumptions are met, slope estimates can have problems Example: Outliers -- cases with extreme values that differ greatly from the rest of your sample More formally: “influential cases” Outliers can result from: Errors in coding or data entry Highly unusual cases Or, sometimes they reflect important “real” variation Even a few outliers can dramatically change estimates of the slope, especially if N is small.

29. Regression: Outliers Outlier Example: -4 -2 0 2 4 4 2 -2 -4 Extreme case that pulls regression line up Regression line with extreme case removed from sample

30. Regression: Outliers Strategy for identifying outliers: 1. Look at scatterplots or regression partial plots for extreme values Easiest. A minimum for final projects 2. Ask SPSS to compute outlier diagnostic statistics –Examples: “Leverage”, Cook’s D, DFBETA, residuals, standardized residuals.

31. Regression: Outliers SPSS Outlier strategy: Go to Regression – Save –Choose “influence” and “distance” statistics such as Cook’s Distance, DFFIT, standardized residual –Result: SPSS will create new variables with values of Cook’s D, DFFIT for each case –High values signal potential outliers –Note: This is less useful if you have a VERY large dataset, because you have to look at each case value.

32. Scatterplots Example: Study time and student achievement. –X variable: Average # hours spent studying per day –Y variable: Score on reading test CaseXY 12.628 21.413 3.6517 44.131 5.258 61.916 73.56 Y axis X axis 0 1 2 3 4 30 20 10 0

33. Outliers Results with outlier:

34. Outlier Diagnostics Residuals: The numerical value of the error –Error = distance that points falls from the line –Cases with unusually large error may be outliers –Note: residuals have many other uses! Standardized residuals –Z-score of residuals… converts to a neutral unit –Often, standardized residuals larger than 3 are considered worthy of scrutiny But, it isn’t the best outlier diagnostic.

35. Outlier Diagnostics Cook’s D: Identifies cases that are strongly influencing the regression line –SPSS calculates a value for each case Go to “Save” menu, click on Cook’s D How large of a Cook’s D is a problem? –Rule of thumb: Values greater than: 4 / (n – k – 1) –Example: N=7, K = 1: Cut-off = 4/5 =.80 –Cases with higher values should be examined.

36. Outlier Diagnostics Example: Outlier/Influential Case Statistics HoursScoreResidStd ResidCook’s D 2.60289.321.01.124 1.4013-1.97-.215.006.65174.33.473.070 4.10317.70.841.640.258-3.43-.374.082 1.9016-.515-.056.0003 3.506-15.4-1.68.941

37. Outliers Results with outlier removed:

38. Regression: Outliers Question: What should you do if you find outliers? Drop outlier cases from the analysis? Or leave them in? –Obviously, you should drop cases that are incorrectly coded or erroneous –But, generally speaking, you should be cautious about throwing out cases If you throw out enough cases, you can produce any result that you want! So, be judicious when destroying data.

39. Regression: Outliers Circumstances where it can be good to drop outlier cases: 1. Coding errors 2. Single extreme outliers that radically change results Your results should reflect the dataset, not one case! 3. If there is a theoretical reason to drop cases –Example: In analysis of economic activity, communist countries may be outliers If the study is about “capitalism”, they should be dropped.

40. Regression: Outliers Circumstances when it is good to keep outliers 1. If they form meaningful cluster –Often suggests an important subgroup in your data Example: Asian-Americans in a dataset on education In such a case, consider adding a dummy variable for them –Unless, of course, research design is not interested in that sub-group… then drop them! 2. If there are many –Maybe they reflect a “real” pattern in your data.

41. Regression: Outliers When in doubt: Present results both with and without outliers Or present one set of results, but mention how results differ depending on how outliers were handled For final projects: Check for outliers! At least with scatterplots But, a better strategy is to use partialplots and Cooks D (or similar statistics) –In the text: Mention if there were outliers, how you handled them, and the effect it had on results.

44. Extra Slides

45. Review Types of regression variables, and interpretation of coefficients: 1. Normal variable coefficient: Reflect slope of line relating one variable to the dependent var The effect of a 1-point change in X on Y 2. Dummy variable: Reflects difference in the constant for a group compared to omitted group Here, the effect is the difference in constant (level) of Y for different groups.

46. Review 3. Interaction term: Dummy * Continuous: Indicates differences in slope for different groups Example: DFEMALE*Education affecting income –Coefficient indicates difference in slope for dummy group compared to slope of reference group 4. Interaction term: Dummy * Dummy: Indicate differences in the constant Example: DFEMALE*DBLACK –Coefficient indicates difference in constant between black females and black males (and white females).

47. Review 5. Interaction term: Continuous * Continuous: Indicates differences in slope for different values of other variable Example: ParentsWealth*Education affecting income –Coefficient indicates difference in slope for each unit change in other continuous variable.

48. Log Transformations 1. Linearity and log transformations: When should you log your variables? There are two common reasons: –1. To reduce extreme skewness (which often leads to non-linearity –2. For variables where the social meaning is clearly non-linear.

49. Log Transformations Example: Country GDP per capita –Highly skewed –Also, a shift from $1000 to $2000 is much more socially significant than shift from $30,000 to 31,000 Other example: wages (interval) Log transformations should be used judiciously Don’t log all variables to achieve a modest improvement in linearity.

50. Multiple Regression Problems Another common regression problem: Multicollinearity Definition: collinear = highly correlated –Multicollinearity = inclusion of highly correlated independent variables in a single regression model Recall: High correlation of X variables causes problems for estimation of slopes (b’s) –Recall: variable denominators approach zero, coefficients may wrong/too large.

51. Multiple Regression Problems Multicollinearity symptoms: –Addition of a new variable to the model causes other variables to change wildly Note: occasionally a major change is expected (e.g., if a key variable is added, or for continuous interaction terms) –If a variable typically has a small effect; but when paired with another variable, BOTH have big effects in opposite directions.

52. Multiple Regression Problems Diagnosing multicollinearity: 1. Look at correlations of all independent vars –Watch out for variables with correlations above.7 –Correlations of over.9 are really bad 2. Use advanced tools: –Tolerances, VIF (Variance Inflation Factor) 3. Watch out for symptoms mentioned previously.

53. Multiple Regression Problems Solutions to multcollinearity –It can be difficult if a fully specified model requires several collinear variables 1. Drop unnecessary variables 2. If two collinear variables are really measuring the same thing, drop one or make an index –Example: Attitudes toward recycling; attitude toward pollution. Perhaps they reflect “environmental views” 3. Advanced techniques: e.g., Quantile regression, Ridge regression.

54. Entering Variables Into Regressions Question: For final papers, how should you enter variables into a regression? Forward, backward, stepwise, or all at once? –I recommend entering variables all at once, rather than using an automated procedure Automated procedures are more useful for advanced models –It is often interesting to present more than one model Example: Show how coefficients change with the addition of new variables.