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Jane E. Miller, PhD

Calculating interaction patterns from logit coefficients: Interaction between two categorical independent variables. Jane E. Miller, PhD. Overview. Logic for calculating overall interaction patterns from coefficients Review: Metric of logit coefficients

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Jane E. Miller, PhD

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  1. Calculating interaction patterns from logit coefficients: Interaction between two categorical independent variables Jane E. Miller, PhD

  2. 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. 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. 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. Example calculation for an interaction from a logit model • Suppose we have the following estimated coefficients from a logit model of low birth weight: βNHB= 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. 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 coefficientsseparately. • Then calculating the product of the resulting odds ratios.

  7. 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 have 2.29 times the odds of LBW as those born to non-Hispanic white women with >HS (the reference category).

  8. 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: eβNHB× e β<HS× e β[<HS_NHB] • Substituting the βs from the logit model and solving: e0.68× e0.60 × e(–0.45) = 1.98  1.81  0.64 = 2.29

  9. 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: e0.68× e0.60 × e(–0.45) = 1.98  1.81  0.64 = 2.29

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

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