1 G Lect 5W Regression assumptions Inferences about regression Predicting Day 29 Anxiety Analysis of sets of variables: partitioning the sums of squares Predicting Day 29 Anger with two Day 28 support measures, after adjusting for Day 28 Mood G Multiple Regression Week 5 (Wednesday)

2 G Lect 5W Usual OLS Regression Assumptions Needed for unbiased estimates »Model is properly specified Linear model? Selection characteristics included as IVs? Reliable IVs? Needed for efficient estimates »Independent observations »Homoscedastic residuals Needed for inference »Independent residuals »Homoscedastic residuals »Normally distributed residuals

3 G Lect 5W Inferences about regression Suppose we believe that reaction time, Y, is inverse-linearly related to amount of cereal subjects eat. »Y=A+BX+e, where B<0 We collect data from 20 students on the grams(x10) of cereal they ate, and we measure their reaction time in identifying ambiguous stimuli. Suppose we obtain estimates of Â=453 and B=-.6 Question: Is there really evidence that X and Y are related? »Can we reject H 0 : B=0? (In this case H 0 is credible!) ˆ

4 G Lect 5W B estimates are random variables Even if B is truly zero, it is unlikely that B will be zero »The least squares criterion guarantees that B will fit even chance association between Y and X. »Especially for small samples, chance associations can be striking. Example of chance results: ˆ ˆ

5 G Lect 5W The compelling nature of Random patterns Formal statistical inference methods tell us how often to expect such striking patterns by chance alone. Two approaches »Wald test (ratio of B to sd B ) »ANOVA test ˆ ˆ

6 G Lect 5W Multiple Regression Inference: Single variables Y = B 0 + B 1 X 1 + B 2 X B q X q + e Formal question: What can be said about an individual coefficient, B q in the context of the full model (i.e. “adjusting for X 1, X 2,..., X q-1 ”) »Test null hypothesis, H 0 : B q = 0 »Compute 95% CI, (L q,U q ) around B q »How much variance in Y does X q account, given that some variance is already fitted by X 1, X 2,..., X q-1 ? Example from CCWA:Does gender add to the prediction of salary when experience and productivity are included in the model?

7 G Lect 5W Example: Predicting Depressed mood day 29 In bar exam study, let's revisit the prediction of depression on day 29 as a function of depression and anxiety on day 28. What can we say about »The relation of anxiety28 to depression29 when depression 28 is adjusted? »The residual distribution? »Homoscedasticity? »Adequacy of the linear model? »Alternative scaling of depression?

8 G Lect 5W Multiple Regression Inference: Fit of whole equation Example: Suppose that outcome is productivity of workgroups in a corporation and X’s are characteristics of work setting, such as space/employee, ambient noise level, distance to restrooms, etc. Y = B 0 + B 1 X 1 + B 2 X B q X q + e What can be said about the whole set of variables (i.e., X 1, X 2,..., X q ) in relation to Y? »Test the null hypothesis, H 0 : B 1 = B 2 =... =B q =0 »Alternative formulation, H 0 : R 2 =0

9 G Lect 5W Decomposition of Regression and Residual Variance Step 1: Estimate regression coefficients using OLS and compute predicted (fitted) values of Y (Y). Step 2: Estimate Regression Sums of Squares as  (Y-Y) 2, MSR=SSR/df Step 3: Estimate Residual Sums of Squares as  e 2, MSE=SSE/df Under H 0, MSR/MSE is distributed as central F on (q,n-q-1) df SourcedfSSMS Regression q  (Y-Y) 2 SSR/q Residualn-q-1  e 2 SSE /(n-q-1) ^ ^ ^ ^ ^ ¯ ¯

10 G Lect 5W Test of Incremental R 2 due to X q Hierarchical Regression »Fit reference model with X 1, X 2,...,X q-1 Determine Regression Sums of Squares This determines R 2 of reference model »Fit expanded model with X q added to reference model Determine increase in Regression Sums of Squares (SS q ) »on 1 df for single predictor X q Determines R 2 increment »“semipartial squared correlation” Determine Sums of Squares & Mean Squares for residual from expanded model »MSE is mean square for residual »on (n-q-1) degrees of freedom »Under null hypothesis, H 0 :B q =0 MS q is simply fitted random variation MS q /MSE ~ F[1, (n-q-1)]

11 G Lect 5W Example: Predicting Anger on Day 29 with Day 28 Measures Does Anger on day 28 improve the fit of Anger on day 29 after four other moods have been included in the model? Do two emotional support variables on day 28 improve the fit of Anger 29 after five moods have been included?

12 G Lect 5W Numerical Results