1. The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition. Calculating interaction patterns from logit coefficients: Interaction between two categorical independent variables Jane E. Miller, PhD

2. The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition. Overview Logic for calculating overall interaction patterns from coefficients Review: Metric of logit coefficients Two approaches to calculating odds ratio for an interaction Before watching this podcast, watch the podcast on calculating categorical by categorical interaction from OLS coefficients for diagrams and detailed explanation.

3. The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition. Model specification with interactions: race and education Logit(LBW) = f (race, education, race_education) – LBW = low birth weight: <2,500 grams – The log-odds of low birth weight are specified as a function of race, education, and the race-by-education interaction. To specify the model, need ALL of the main effects and interaction term variables related to race and mother’s education Logit(LBW) = f (NHB, <HS, =HS, NHB_<HS, NHB_=HS)

4. The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition. Interaction patterns from logit βs The logic for calculating overall interaction patterns based on logit coefficients is the same as for OLS: – For cases in the reference category for one but not both of the IVs involved in the interaction: One β (the coefficient on a main effect term) – For cases NOT in the reference category for either variable: Three β s (those on two main effects terms and the interaction term)

5. The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition. Example calculation for an interaction from a logit model Suppose we have the following estimated coefficients from a logit model of low birth weight: β NHB = 0.68; β <HS = 0.60; β <HS_NHB = –0.45 Recall that coefficients from logit models are in the metric of ln(relative odds)

6. The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition. Calculating the net effect of the interaction from logit coefficients From the βs, we can calculate the odds ratio for a given group compared to the reference category either by 1. Summing the coefficients on the pertinent main effects and interaction terms, which are in the metric of log-relative odds. Then exponentiating that sum to calculate the odds ratio. or 2. Exponentiating each of the main effects and interaction term coefficients separately. Then calculating the product of the resulting odds ratios.

7. The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition. Exponentiating the sum of the logit βs on main effects and interaction General equation for the odds ratio involving main effects and interaction terms: e (βNHB + β<HS + β[<HS_NHB]) Substituting the estimated βs from the logit model and solving: = e (0.68 + 0.60 + [–0.45]) = e (–0.84) = 2.29 Thus, non-Hispanic black infants born to mothers with HS (the reference category).

8. The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition. Calculating the product of the odds ratios on main effects and interaction terms General equation for the odds ratio involving main effects and interaction terms: e βNHB × e β<HS × e β[<HS_NHB] Substituting the βs from the logit model and solving: e 0.68 × e 0.60 × e (–0.45) = 1.98  1.81  0.64 = 2.29

9. The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition. Both approaches yield same solution Note that the solution is the same regardless of which approach we use to calculate the OR based on the combination of pertinent main effects and interaction coefficients. – Exponentiating the sum of the main effect and interaction βs: e (0.68 + 0.60 + [–0.45]) = e (–0.84) = 2.29 – Multiplying the odds ratios for the main effects and interaction terms: e 0.68 × e 0.60 × e (–0.45) = 1.98  1.81  0.64 = 2.29

10. The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition. Use a spreadsheet to calculate the interaction pattern Spreadsheets can – Store The estimated logit coefficients The input values of the independent variables The correct generalized formula to calculate odds ratios of the outcome for combinations of the IVs involved in the interaction – Graph the overall pattern See spreadsheet template and voice-over explanation

11. The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition. Summary The logic for calculating overall interaction patterns based on logit coefficients is the same as for OLS in terms of number and type of terms involved in the specification. Odds ratios can be computed from the logit coefficients either by – Exponentiating the sum of pertinent βs, – Or calculating the product of the pertinent odds ratios for each of the three terms.

12. The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition. Suggested resources Chapters 9 and 16 of Miller, J.E. 2013. The Chicago Guide to Writing about Multivariate Analysis, 2nd Edition. Podcasts on – Interpreting multivariate coefficients – Calculating interaction patterns from OLS coefficients for 2 categorical independent variables Spreadsheet for calculating the pattern for a categorical by categorical interaction.

13. 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. – Problem set for Chapter 16 – Suggested course extensions for Chapter 16 “Reviewing” exercises.

14. 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