1. The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition. Calculating the shape of a polynomial from regression coefficients Jane E. Miller, PhD

2. The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition. Overview Functional form of a polynomial Solving a polynomial for values of the independent variable Illustrative example – Calculations – Chart Spreadsheet to perform calculations

3. The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition. Specifying a model with a polynomial To specify a polynomial function of an independent variable (IV), include linear and higher-order terms for that IV in the model. E.g., – A quadratic specification will include linear and square terms (variables): Y = β 0 + β 1 X 1 + β 2 X 1 2 – A cubic specification will include linear, square, and cubic terms: y = β 0 + β 1 X 1 + β 2 X 1 2 + β 3 X 1 3

4. The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition. Example: Birth weight as a quadratic function of IPR If a birth weight model includes both income- to-poverty ratio (IPR) and IPR 2 as independent variables, it yields the following quadratic specification: Birth weight = β 0 + (β IPR × IPR) + (β IPR 2 × IPR 2 ) – β IPR is the coefficient on the linear term – β IPR 2 is the coefficient on the square term – IPR is the value of the income-to-poverty ratio variable for each case – IPR 2 is IPR-squared for each case

5. The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition. Solving a polynomial based on βs Birth weight (grams) = β 0 + (β IPR × IPR) + (β IPR 2 × IPR 2 ) Substituting the estimated βs into the general equation gives: Birth weight (grams) = 3,317.8 + (80.5  IPR) + (–9.9  IPR 2 ) Which can be solved for specific values of IPR Estimated coefficients are shown in table 9.1 of The Chicago Guide to Writing about Multivariate Analysis, 2nd Edition.

6. The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition. Calculating the shape of the polynomial, holding constant all other IVs To calculate the effect of the IV in the polynomial holding constant all other independent variables in the model – The intercept (β 0 ) and the β i s related to other IVs in the model will cancel out when you subtract to calculate the difference between values. – So you don’t have to include those coefficients in these calculations.

7. The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition. Calculating predicted values of the dependent variable (DV) for different values of the IV Solve the equation (80.5  IPR) + (–9.9  IPR 2 ) for each of several selected values of X 1. E.g., – At IPR = 1.0 Birth weight = (80.5  1.0) + (–9.9  1.0 2 ) = 70.6 grams – At IPR = 2.0 Birth weight = (80.5  2.0) + (–9.9  2.0 2 ) = 121.4 grams – At IPR = 3.0 Birth weight = (80.5  3.0) + (–9.9  3.0 2 ) = 151.8 grams β IPR = 80.5; β IPR 2 = –9.9. We ignore β 0 because it cancels out when we subtract to calculate differences across predicted values, in the next step.

8. The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition. Selecting values of the IV for which to calculate the predicted DV Select values of the IV to solve for by picking 5 or 6 values that span the observed range of X i in your data. E.g., – The income-to-poverty ratio (IPR) ranges from 0 to 5, so plug in values at 1-unit increments across that range. – Mother’s age ranges from 15 to 49, so if your model specifies a polynomial function of age, solve it for values at 5-year increments across that range. Avoid selecting out of range values, since they were not included in the model that estimated the βs.

9. The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition. Calculating the effect of changes in the IV on the DV Compute the difference in predicted value of the dependent variable (DV, Y) by subtracting predicted values of Y for different values of the IV (X i ). – E.g., predicted Y (X i = 2) – predicted Y (X i = 1) Important to do this for several pairs of values of the IV because when the association is specified with a polynomial, by definition, X i will not have a constant marginal effect on Y.

10. The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition. Example: Birth weight as a quadratic function of IPR As we saw earlier: – At IPR = 1.0, predicted birth weight = 70.6 grams – At IPR = 2.0, predicted birth weight = 121.4 grams – At IPR = 3.0, predicted birth weight = 151.8 grams Thus the marginal effect of moving from IPR = 1.0 to IPR = 2.0 is 121.4 – 70.6 = 50.8 grams IPR = 2.0 to IPR = 3.0 is 151.8 – 131.4 = 30.4 grams

11. The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition. Graphing the polynomial

12. The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition. Coefficients and shape of the quadratic In this example, the decreasing marginal positive effect of IPR on birth weight is due to the combination of – a positive β IPR – a negative β IPR 2 Recall: β IPR = 80.5; β IPR 2 = –9.9 Other combinations of positive and negative signs on the linear and squared terms will generate different shapes of the quadratic function.

13. The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition. Using a spreadsheet to calculate pattern of a polynomial from βs Spreadsheets are well-suited to conducting repetitive, multistep calculations. Type in: – Estimated coefficients on the polynomial terms, – Selected values of the independent variable, – Formulas to calculate predicted value of the dependent variable from the βs and values of the independent variable (IV). Generalize the formulas to apply to all values of the IV. Create a chart to portray the association between the IV and DV across the observed range of the IV.

14. The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition. Summary A regression model involving a polynomial will include separate variables for each term in the polynomial. The overall shape of the pattern can be calculated by solving the polynomial for values of: – The estimated coefficients on the polynomial terms, and – Selected values of the independent variable across its observed range in the data. A spreadsheet is an efficient way to calculate and graph the polynomial.

15. The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition. Suggested resources Miller, J. E. 2013. The Chicago Guide to Writing about Multivariate Analysis, 2nd Edition. – Chapter 10, section on polynomials – Appendix D, using a spreadsheet for calculations Podcast on – Interpreting regression coefficients Spreadsheet templates – Spreadsheet basics – Solving for a quadratic

16. The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition. Suggested practice exercises Study guide to The Chicago Guide to Writing about Multivariate Analysis, 2nd Edition. – Suggested course extensions for chapter 10 “Applying statistics and writing” question #7. “Revising” questions #6 and 9.

17. The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition. Contact information Jane E. Miller, PhD jmiller@ifh.rutgers.edu Online materials available at http://press.uchicago.edu/books/miller/multivariate/index.html