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Introduction. Health services researchers use interaction terms in models with binary dependent variablesExamplesMortality depends on age, gender (and interaction)Readmission depends on nursing turnover rate, CQI program (and interaction)Pre-post treatment control study designDifference-in-diff
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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/