Interaction Terms in Logit and Probit models

1. Interaction Terms in Logit and Probit models Edward C. Norton UNC at Chapel Hill AcademyHealth 2004

2. Introduction Health services researchers use interaction terms in models with binary dependent variables Examples Mortality depends on age, gender (and interaction) Readmission depends on nursing turnover rate, CQI program (and interaction) Pre-post treatment control study design Difference-in-differences models

3. Linear Models Coefficient on the interaction term gives the magnitude and sign of interaction effect Use t-test for statistical significance Easy to compute marginal effects (for continuous variables) or incremental effects (for dummy variables)

4. Nonlinear Models Magnitude does not equal coefficient on interaction term (or usual marginal effect) Sign may be different (!) Statistical significance is not z-statistic on interaction term Conditional on the independent variables (like marginal effect of one variable)

5. Literature Reviewed 72 articles published in 13 top economics journals on JSTOR between 1980 and 2000 All used interaction term in a nonlinear model (e.g., logit, probit, tobit, neg. bin) None interpreted interaction effect correctly

6. Outline Review interaction terms in OLS Review marginal and incremental effects in logit and probit for single variable Interaction effects in nonlinear models Show examples Intuition, formulas, graph, Stata code Papers and Stata code available on web

7. Difference-in-Differences Model Experiment Rats� weight depends on drug OLS regression

8. Interpretation of DD Models

9. Continuous and Continuous Crop yield depends on rainfall and fertilizer OLS regression Marginal effect is ?12 If all estimated coefficients are positive: More rain makes fertilizer more effective at increasing crop yield

10. OLS Summary Look at coefficient on interaction term Significance is t-statistic on interaction term Interaction effect is a double difference, or a double derivative

11. Logit and Probit Marginal Effect Probabilities are scale of interest Not log odds No one has ever given me a clear interpretation of log odds Whatever interpretation you understand, it will not work for interaction terms Ultimately, we care about probabilities Paraphrase Emmett Keeler

12. Marginal Effect of Single Variable in Nonlinear Models More complicated in nonlinear models Not constant Vary with covariates Smaller when the overall probability is small Policy interpretation makes sense ME = ?x?pdf ME = ?x?F ?(1-F) if logit ME = ?x?? if probit

13. Marginal Effect Graphs CDF and pdf (slope of CDF) for logit

14. Interaction Effects in Nonlinear Models General principles Compute double difference or double derivative (or one of each) Expect values to differ for each observation More complicated if squared terms Stata command mfx (not recommended) Does not know if variables interacted Does not show distribution

15. General Formula Interaction effect is double derivative or double difference

16. Example: Probit with DD Interaction effect is double difference

17. Example: Logit with CC Interaction effect is double derivative

18. Standard Errors Equation for standard errors � Delta method Depends on whether continuous or dummy variables interacted Is ugly Provides no intuition, no point in deriving here See paper (Ai & Norton, 2003) for details

19. Example Mello, Stearns, and Norton (HE, 2002) Predict whether elderly person in HMO 1993?1996 MCBS data Logit model Interact age with health status and market penetration (all continuous) N = 38,185

20. inteff Command logit y x1 x2 x1x2 $x inteff y x1 x2 x1x2 $x, savedata(intdata.dta,replace) savegraph1(intg1.gph,replace) savegraph2(intg2.gph,replace)

21. Logit Results

22. Model 1 Interaction Effects

23. Model 1 Interaction Effects

24. Model 1 z-statistics

25. Model 1 z-statistics

26. Model 2 Interaction Effects

27. Model 2 Interaction Effects

28. Model 2 z-statistics

29. Model 2 z-statistics

30. Extensions Triple interactions (including DDD models) Follow same logic, take triple derivatives or differences Squared or higher-order terms Follow same logic, take full derivatives or differences Box-Cox and log transformation Above formulas only apply to normally distributed error We have derived formulas for general case

31. Odds Ratio Odds ratio interpretation does not work for interaction terms Need ratio of odds ratios (which means ??)

32. Linear Probability Model LPM is OLS with dummy dependent variable Interaction effects are as simple as in OLS Problems with LPM Predictions may be outside [0,1] interval Assumes constant marginal effect Heteroskedasticity (fixable) May prefer LPM if model has fixed effects

33. Predictnl Our Stata command inteff only works For logit and probit models If exactly two variables interacted If no squared or other non-linear terms E.g., age and age squared and age cubed Predictnl is general Stata command Works for all nonlinear models Works for all interactions, squared terms Hard to use

34. Conclusions Interaction effects are more complicated in nonlinear models than in OLS Only looking at coefficient on the interaction term is wrong: Wrong magnitude Wrong sign Wrong statistical significance Our paper supplies formulas and examples

35. References Ai & Norton. 2003. Interaction terms in logit and probit models. Economics Letters 80(1):123?129. Norton, Wang, & Ai. 2004. Computing interaction effects in logit and probit models. The Stata Journal 4(2):103?116. For copies of papers and Stata 8 programs to compute interaction effects and standard errors edward_norton@unc.edu www.unc.edu/~enorton/