Calculating the shape of a
polynomial from regression
coefficients
Jane E. Miller, PhD
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
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 = β0 + β1X1 + β2X12
– A cubic specification will include linear, square, and
cubic terms:
y = β0 + β1X1 + β2X12 + β3X13
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 incometo-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
The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition.
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.
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 (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.
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  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 β0 because it cancels out when we subtract to
calculate differences across predicted values, in the next step.
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 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.
• Avoid selecting out of range values, since they were
not included in the model that estimated the βs.
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 (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.
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
The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition.
Graphing the polynomial
Predicted birth weight by income-to-poverty ratio
Birth weight (grams)
3,250
3,200
3,150
3,100
3,050
3,000
2,950
0.0
0.5
1.0
1.5
2.0
2.5
3.0
Income-to-poverty ratio (IPR)
3.5
4.0
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 βIPR
– a negative βIPR2
• 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.
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.
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.
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
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.
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
The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition.