Multiple Regression Extension of Simple Linear Regression –

1. Multiple Regression Extension of Simple Linear Regression – using multiple predictors each predictor could help predict or explain additional variability in the response/criterion variable However: What should be the effect of using any additional predictors?

2. Multiple Regression What should be the effect of using additional predictors? Logically, unless correlation with DV is 0, each predictor will improve prediction (explain additional variance in DV) So just adding variables as predictors at random will usually “improve” model Creates potential for misuse of the strategy

3. IDEALLY each predictor should be - correlated with the DV - uncorrelated with other predictors (r over .8 undesirable) each predictor should explain some unique variability in DV each predictor should make sense!

4. Best situation CLEAR THEORY or LOGIC determines the predictors selected Examples Relationship Commitment satisfaction with outcomes (+) investments in relationship (+) attractiveness of available alternatives (-) Job Satisfaction salary physical conditions social conditions

5. Simple linear regression Yp = a + bX (+ residuals) Multiple regression Yp = a + b1X1 + b2X2 (+ residuals) a is value of Y when all X = 0 (regression constant) b’s are ‘partial regression coefficients’ slope for each predictor when other predictors held constant

6. Graph of relationship when two predictors are used Now try to fit a plane rather than a line – to minimize the errors of prediction

7. Multiple regression Yp = a + b1X1 + b2X2 + b3X3 (+ residuals) Commitment = a + b1 (satisfaction) + b2 (investments) + - b3 (alternatives) (+ residuals) A weighted linear combination of predictors comparison to ANOVA – main effects only model

8. Let’s return to the question of predicting Exam 2 grades – using multiple predictors • Undergraduate GPA (0-4 scale) • GRE Verbal (200-800 scale) • GRE Quantitative (200-800 scale) • Exam 1 grade (0-100 scale) • Mean Homework grade (0-10 scale) Note variety of scales for predictors, weights (partial regression coefficients) will be variable to take those into account

9. Ideally, all predictors are related to the criterion, and are unrelated to each other

10. Using just Exam 1 score, the correlation between Exam 1 and Exam 2 was r = .637, r2 = .406 Now the R, between the set of predictors and Exam 2 is .735, and R2 = .540 Since gpatot, grev, greq were all not significant, should they be excluded from the equation? Exam 2 (Pred) = 16.06 + .19 (gpa) - .00(grev) + .00(greq) +.44(exam1) +3.67(homework)

11. Assumptions - never likely to satisfy all Essentially same as for r, but at a multivariate level Independent Observations Interval/Ratio Data – or at least pretend Normality – all Predictors (X’s) and Response (Y) - errors of prediction are normally distributed Linearity all X’s have linear relationship with Y - errors of prediction/predicted scores are linear Equality of Variances (Homoscedasticity) - variability of errors of Y are same at all values of X

12. Assumptions can be evaluated within SPSS at the multivariate level. In the Regression window, choose Plots and request zresid (Y) and zpred (X). The tables at the right demonstrate the patterns that would indicate each violation. Although deciding when there is ‘enough’ discrepancy is still subjective. From Tabachnick & Fidell (2007). Using multivariate statistics (5th). Boston: Allyn & Bacon. Example to follow

13. Predicting Rated Distress : (1) none to Extreme (9) - when partner is emotionally unfaithful using Age and Rated Distress over Sexual Infidelity as predictors. All 3 variables are skewed.

14. Other Considerations in Multiple Regression -- Truncated Range – same as with r, can lead to poor assessment of ‘real’ R -- Outliers due to multivariate deviation Discrepancy (distance) –outlier on criterion Leverage – outlier on predictors Influence – combines D & L to assess influence on solution (change in regression coefficients if case deleted)

15. How these would appear in a simple linear regression situation From Tabachnick & Fidell (2007). Using multivariate statistics (5th). Boston: Allyn & Bacon.

16. Other Considerations in Multiple Regression -- Outliers due to multivariate deviation Simple diagnostic for Influence is to request Cook’s Distance statistic in Regression window, Save option. Values over 1 would suggest potentially strong influence.

17. Line with outlier Cook’s distance = 92.6 Note that residual for the outlier is not great, but it has strong influence on solution Line without outlier

18. Other Considerations in Multiple Regression • Sample Size – if too small may get good, but meaningless prediction – too little variability Minimum sample sizes recommended (to detect moderate effect sizes, 13% with power of approximately .80) (Green, S. B. (1991) How many subjects does it take to do a regression analysis? Multivariate Behavioral Research, 26, 499-510.) • For test of a model n= 50 +8p • For test of individual predictors in model 104 + p • p = number of predictors Can also conduct a power analysis based on the effect size you desire to select your sample size

19. Other Considerations in Multiple Regression • Multicollinearity or Singularity • Singularity – when one predictor is a combination of other predictors included • Multicollinearity - when other predictors can account for a high degree of variability in a predictor

20. Other Considerations in Multiple Regression • Diagnostics for Multicollinearity or Singularity Tolerance is used as diagnostic statistic If other predictors used to predict a predictor, what variance is shared? But reported as 1-R2,so closer to 1 is better - less than .2 indicates a problem Variance Inflation Factor (VIF) also used. It is the reciprocal of Tolerance, so can range from 1 up. Reflects degree to which standard error of b is increased due to correlations among predictors. value of 4 cause for some concern value of 10 serious problem

21. Assessing the Outcome Testing the Overall Model as a single outcome How well do the set of predictors (X’s) predict the criterion (Y) Ho: all b’s are = 0, all partial regression coefficients = 0 Or Ho: R = 0, the Multiple Correlation Coefficient = 0 R = correlation of actual Y with weighted linear combination of predictors (X’s) Or – since weighted linear combination leads to predicted scores R = correlation of actual Y with predicted Yp

22. Reminder: Partitioning the Variability in Y SSTotal = Sum (Y - Mean Y)2 variability of Y scores from the mean Separated into SSregression = Sum (Yp – Mean Y) 2 Improvement in predictions when using X (variability in Y explained by X), rather than assuming everyone gets the Mean SSresidual = Sum (Y - Yp) 2 Degree to which predictions do not match the actual scores (prediction errors that have been minimized)

23. Example from Simple Linear Regression Mean IQ = 105 This would be your best ‘guess’ for every person if you had no useful predictor Improvement in Prediction using GPA Residual – distance from the prediction line Residual much greater here Mean GPA = 3.06

24. Test using F – similar to simple linear regression Partition SST into • SSregression (explained by weighted combination) • SSresidual (unexplained) F = SSregression /df regressiondf = (p + a) - 1 SSresidual / df residual df = n – p - 1 F = MSregression = explained(systematic+unsystematic) MSresidual unexplained (unsystematic) Was R reliably different from 0 ? Yes, if F is significant Recall: Standard Error of the Estimate = SQRT (MS residual) Number of parameters in model (predictors + intercept) – df often indicated as p, since always only one a (df = p +1 -1)

25. R2 = SSregression = explained variability SST total variability % of variance accounted for by the model (see next slide for ANOVA example) Adjusted R2 for better estimate for population, adjusted based on number of predictors and sample size Adjusted R2 = 1- ((1-R2) (n-1/n-p-1))so lower if small sample, but many predictors Can use R2 for describing a sample

26. Partial Test of model in which there are 3 predictors used to predict the rating on “Sensitive”, the DV Example from Handout Packet, Page approx. 47

27. In some cases, the purpose of the regression analysis is simply to see if the Model “works”. Does it explain variance in the criterion? Can it be used to make predictions? Thus, the overall test of the model is all you need, and you can interpret the R2 or R2adj and the SEE, if plan predictions In other cases, you might want to know how the individual predictors contributed to the overall model.

28. Assessing the contribution of individual predictors Dependent upon the set of predictors included! Partial regression coefficient– can test to see if b = 0 Is b = 0 (slope = 0) when other predictors are held constant Tested using a t test with df = n – p – 1 Beta – partial regression coefficient when all variables are standardized (standardized slope) If b is significant, so is beta Test of Partial Regression Coefficient is like a typical statistical test of significance - it is or is not significant, and is influenced by sample size

29. Can also evaluate predictors based on “effect size” measures (practical significance) These would be “significant” if b is significant Partial correlation (pr) – as described in simple covariation section correlation of predictor (X1) with DV (Y) after removing the variance in both explained by the other predictors So both X1 and Y are adjusted before correlation is calculated All other X’s are ‘partialed’ out of X1 and Y pr2 – shared variance within context – what % of variability in Y does X1 explain after other variables’ contributions to explaining both are removed there is less than 100% of variability of Y left for X1 explain

30. Semi-partial (part) correlation (sr) – correlation of predictor (X1) with DV (Y) after removing the variance of X1 shared with the other predictors So X1 is adjusted by removing variance shared with other X’s But all variability in Y is left to be explained Assesses ‘unique’ contribution of X1 to explaining Y There is 100% of variability in Y to explain for each X in model sr2 is considered best measure of individual predictor importance (practical significance) R2 will be lowered by sr2 for predictor when it is removed from model (BOTH pr and sr ARE STILL DEPENDENT ON THE MODEL USED) WHY?

31. Shared variability DV and X1 IV 1 a Variability of DV (Y) Shared variability IV 1 and X2 b d c IV 2 Shared variability DV and X2 Partial correlation (X1) = a/(a + d) Semi-partial correlation (X1) = a/(a + b + c + d)

32. Types of Multiple Regression Standard – all predictors entered together contribution of each depends on others in the group Assumes other variables would usually be there and/or are relevant Four Humor Styles Investment Model Variables Big Five Personality Dimensions

33. Hierarchical Regression - enter in planned sequence Can enter individual predictors one at a time Or enter groups of variables at separate steps As new predictors are added, each one can only explain variability that is left Assess change in R2 at each step (increase significantly) and overall model when done Predicting adult IQ Parental IQ Prenatal experience Early infant experience Education

34. Statistical Methods – Let the data determine inclusion in the model, not based on a logical or theoretical ‘plan’ Assess each step by evaluating change in R or R2 Usually an exploratory tool in possible model building Requires a larger sample to have confidence (40 cases per predictor)

35. Stepwise Begins with single best predictor Adds next best, and assesses if model is better At each step, each variable is reassessed, and might be kept or removed Stops when adding additional variables does not significantly improve model (R)

36. Forward inclusion Begins with single best predictor Adds next best, and assesses improvement Once in, stay in, but only stay if improved model (R) Backward exclusion Begins with full model Removes weakest contributor and assesses loss Keeps removing unless significant drop in R

37. Research questions using Multiple Regression Assess Overall Model Assess individual predictors Effects of adding or changing predictors on overall model on other individual predictors Predictions in new sample

38. Other Multiple Regression issues/applications Suppressor variables – variables that improve the model due to correlations with other predictors, not criterion. They ‘suppress’ variance in another predictor that is ‘noise’ evident if simple r with criterion very low but contributes to model (sr is higher) can also produce a change in sign from r to b (i.e. positive r but negative b)

39. Other issues/applications Mediation Models Relationship of X to Y is mediated by some other variable C Positive use of Humor for self Perceived Stress Positive Personality (optimistic, hopeful, happy) Positive use of Humor for self Perceived Stress (High self-enhancing/Low self-defeating) a b C’ Humor use (H) predicts Perceived Stress (c) - the direct path Humor use predicts Positive Personality (PP) (a) Positive Personality predicts Perceived Stress, with Humor in model (b) In Hierarchical Model, enter PP first, then H, if PP mediates H, H no longer ‘contributes’ to the model – the c’ path, indirect, not significant

40. Other issues/applications Moderator Models – relationship of predictor with the criterion depends upon some other variable (just like an interaction in ANOVA) Yp = a + b1x1 + b2x2 + b3(x1x2) + residuals Often requires some modifications of the data prior to the analysis - centering variables to avoid multicollinearity (if predictors do not have true 0 scores) Main effects Interaction term added to equation

41. Best situation CLEAR THEORY to be tested Relationship Commitment (low 8 – 72 high) satisfaction with outcomes (+) (low 3 – 21 high) investments in relationship (+) attractiveness of available alternatives (-) ( low 6 – high 48) (Subj low 6 – 54 high) (Obj none 0 - ?? Lots)

42. In Handout Packet Begin by examining the individual variables for normality and outliers etc. Can request Cook’s D to assess for outlier influence Can check assumptions using plot from regression analysis

43. Then look at the simple correlations (r)Expect predictors to correlate with criterion, but not a lot with each other

44. Check to see how well the model worked R and R2, and test of significance Standard error of the estimate To describe sample To generalize to population Typical residual

45. Now can look at individual predictors check collinearity see which predictors are individually significant look at individual contributions (semi-partial or part r2) 95

46. Go through example in SPSS • Look at G*Power • Stepwise example in Handouts