Calculating the shape of a polynomial from regression coefficients

1. Calculating the shape of a polynomial from regression coefficients Jane E. Miller, PhD

2. Overview • Functional form of a polynomial • Solving a polynomial for values of the independent variable • Illustrative example • Calculations • Chart • Spreadsheet to perform calculations

3. 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 = β0 + β1X1+ β2X12 • A cubic specification will include linear, square, and cubic terms: y = β0 + β1X1+ β2X12+ β3X13

4. Example: Birth weight as a quadratic function of IPR • If a birth weight model includes both income-to-poverty ratio (IPR) and IPR2 as independent variables, it yields the following quadratic specification: Birth weight = β0 + (βIPR × IPR) + (βIPR2 × IPR2) • βIPR is the coefficient on the linear term • βIPR2 is the coefficient on the square term • IPR is the value of the income-to-poverty ratio variable for each case • IPR2 is IPR-squared for each case

5. Solving a polynomial based on βs • Birth weight (grams) = β0+ (βIPR× IPR) + (βIPR2 × IPR2) • Substituting the estimated βs into the general equation gives: Birth weight (grams) = 3,317.8 + (80.5IPR) + (–9.9 IPR2) • 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. 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 (β0) and the βis 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. Calculating predicted values of the dependent variable (DV) for different values of the IV • Solve the equation (80.5IPR) + (–9.9 IPR2) for each of several selected values of X1. E.g., • At IPR = 1.0 • Birth weight = (80.5 1.0) + (–9.9 1.02) = 70.6 grams • At IPR = 2.0 • Birth weight = (80.5  2.0) + (–9.9 2.02) = 121.4 grams • At IPR = 3.0 • Birth weight = (80.5 3.0) + (–9.9 3.02) = 151.8 grams βIPR = 80.5; βIPR2 = –9.9. We ignore β0because it cancels out when we subtract to calculate differences across predicted values, in the next step.

8. 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 Xi 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. • Avoidselectingout of range values, since they were not included in the model that estimated the βs.

9. 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 (Xi). • E.g., predicted Y (Xi = 2) – predicted Y (Xi = 1) • Important to do this for several pairs of values of the IV because when the association is specified with a polynomial, by definition, Xi will not have a constant marginal effect on Y.

10. 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. Graphing the polynomial

12. 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 βIPR • a negativeβIPR2 • Recall: βIPR = 80.5; βIPR2 = –9.9 • Other combinations of positive and negative signs on the linear and squared terms will generate different shapes of the quadratic function.

13. 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. 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. 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. 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. Contact information Jane E. Miller, PhD jmiller@ifh.rutgers.edu Online materials available at http://press.uchicago.edu/books/miller/multivariate/index.html