1. Julius Center.nl Julius Center.nl Health Sciences and Primary Care Estimating additive interaction between continuous determinants M.J. Knol, I. van der Tweel, D.E. Grobbee, M.E. Numans, M.I. Geerlings Julius Center, University Medical Center Utrecht Center for Biostatistics, Utrecht University The Netherlands Julius Center.nl Julius Center.nl Health Sciences and Primary Care

2. Question 1 Which model do you usually use in your research? a)Linear regression b)Logistic regression c)Cox regression d)Other

3. Julius Center.nl Julius Center.nl Health Sciences and Primary Care Question 2 How do you usually assess interaction in your research? a)Stratification b)Product term c)Never d)Other

4. Julius Center.nl Julius Center.nl Health Sciences and Primary Care Overview Background - interaction Example dataset Calculation of additive and multiplicative interaction Interaction in regression analysis Additive interaction in logistic regression –Example Additive interaction between continuous determinants –Formulas and example Application

5. Julius Center.nl Julius Center.nl Health Sciences and Primary Care Background Synonyms: Interaction Effect (measure) modification Synergy Interaction is present when effect of A is different across strata of B (or vice versa) A B D

6. Julius Center.nl Julius Center.nl Health Sciences and Primary Care Background Rothman discerns two types of interaction Statistical interaction –Departure from the underlying statistical model Biologic interaction –Two causes are needed to produce disease –Four classes involving determinants A and B: A U B UAUBU

7. Julius Center.nl Julius Center.nl Health Sciences and Primary Care Background Interaction as departure from additivity: combined effect of determinants A and B is larger (or smaller) than sum of individual effects of A and B Interaction as departure from multiplicativity: combined effect of determinants A and B is larger (or smaller) than product of individual effects of A and B Rothman: biologic interaction = interaction as departure from additivity

8. Julius Center.nl Julius Center.nl Health Sciences and Primary Care Example dataset Utrecht Health Project Baseline data N=4897 44.9% male Mean age (sd) = 39.3 (12.5) years Determinants –Age (A)  cut off at 40 years –BMI (B)  cut off at 25 kg/m2 Outcome –Diastolic blood pressure  cut off at 90 mm Hg

9. Julius Center.nl Julius Center.nl Health Sciences and Primary Care Example 2x2 table Absolute risks (D=hypertension) Young (A-)Old (A+) Normal BMI (B-)4.4%14.7% Overweight (B+)11.1%27.2% A U B UAUBU 11.1%14.7%4.4%

10. Julius Center.nl Julius Center.nl Health Sciences and Primary Care Example 2x2 table Absolute risks (D=hypertension) Young (A-)Old (A+) Normal BMI (B-)4.4%14.7% Overweight (B+)11.1%27.2% Interaction as departure from additivity: (27.2 - 4.4) ~= (14.7 - 4.4) + (11.1 - 4.4)  22.8 > 17.0 Old subjects with overweight have excess risk for hypertension Risk difference

11. Julius Center.nl Julius Center.nl Health Sciences and Primary Care Example 2x2 table Relative risks (D=hypertension) Young (A-)Old (A+) Normal BMI (B-)1.03.3 Overweight (B+)2.56.2 Young (A-)Old (A+) Normal BMI (B-)4.4%14.7% Overweight (B+)11.1%27.2% Absolute risks (D=hypertension)

12. Julius Center.nl Julius Center.nl Health Sciences and Primary Care Example 2x2 table Relative risks (D=hypertension) Young (A-)Old (A+) Normal BMI (B-)1.03.3 Overweight (B+)2.56.2 Interaction as departure from multiplicativity: 6.2 ~= 3.3 x 2.5  6.2 < 8.4 Old subjects with overweight have no excess risk for hypertension Risk ratio

13. Julius Center.nl Julius Center.nl Health Sciences and Primary Care Example 2x2 table Interaction as departure from additivity: Excess risk Risk difference Interaction as departure from multiplicativity: No excess risk Risk ratio Presence (or direction) of interaction depends on measure of effect

14. Julius Center.nl Julius Center.nl Health Sciences and Primary Care Rothman - Epidemiology: An introduction

15. Julius Center.nl Julius Center.nl Health Sciences and Primary Care Relative excess risk due to interaction A-A+ B-1.03.3 B+2.56.2 Additive interaction can also be calculated with relative risks Formulas to calculate amount of additive interaction Absolute risks: (R A+B+ -R A-B- ) - (R A+B- -R A-B- ) - (R A-B+ -R A-B- ) Relative risks: (RR A+B+ -1) - (RR A+B- -1) - (RR A-B+ -1) Relative excess risk due to interaction (RERI): RR A+B+ - RR A+B- - RR A-B+ + 1 = 6.2 - 3.3 -2.5 + 1 = 1.4 Note: No additive interaction  RERI = 0

16. Julius Center.nl Julius Center.nl Health Sciences and Primary Care 1.0 2.5 3.3 6.2

17. Julius Center.nl Julius Center.nl Health Sciences and Primary Care Short summary Difference between additive and multiplicative interaction Interaction depends on measure of effect However, it is possible to assess additive interaction when using relative rather than absolute risks Rothman: biologic interaction  additive interaction

18. Julius Center.nl Julius Center.nl Health Sciences and Primary Care Interaction in regression analysis Product term in regression model Linear regression model  additive interaction Logistic regression model  multiplicative interaction What if you want to asses additive interaction but you have a logistic regression model?

19. Julius Center.nl Julius Center.nl Health Sciences and Primary Care Literature Hosmer & Lemeshow (1992) Method additive interaction with logistic regression Making one categorical variable: A-B-, A+B-, A-B+, A+B+ RERI = OR A+B+ - OR A+B- - OR A-B+ + 1 (OR=e β ) D1D2D3 A-B-000 A+B-100 A-B+010 A+B+001

20. Julius Center.nl Julius Center.nl Health Sciences and Primary Care Determinants –Age  dichotomous –BMI  dichotomous Outcome –Diastolic blood pressure  dichotomous Example Dummy variables 3 dummy variables

21. Julius Center.nl Julius Center.nl Health Sciences and Primary Care Example Dummy variables Age: OR = 3.8 (2.8-5.1) BMI: OR = 2.7 (2.1-3.6) Age and BMI: OR = 8.2 (6.3-10.7)

22. Julius Center.nl Julius Center.nl Health Sciences and Primary Care Example Dummy variables RERI = OR A+B+ - OR A+B- - OR A-B+ + 1 = 8.2 – 3.8 – 2.7 + 1 = 2.7 Excess risk due to interaction is 2.7 Combined effect of A and B is 2.7 more than sum of individual effects  Significant ‘positive’ interaction on additive scale

23. Julius Center.nl Julius Center.nl Health Sciences and Primary Care However… Only for dichotomous determinants, not for continuous ones

24. Julius Center.nl Julius Center.nl Health Sciences and Primary Care Methods and formulas RERI = OR A+B+ - OR A+B- - OR A-B+ + 1 General formula logistic regression: ln(odds) = β 0 + β 1 A + β 2 B + β 3 AB -Individual effect of A: OR A+B- = e β1 -Individual effect of B: OR A-B+ = e β2 -Combined effect of A and B: OR A+B+ = e β1+β2 +β3 RERI = e β1+β2 +β3 - e β1 - e β2 + 1 95% CI  bootstrap; 2.5th and 97.5th percentile

25. Julius Center.nl Julius Center.nl Health Sciences and Primary Care Example Two dichotomous determinants Determinants –Age  dichotomous –BMI  dichotomous Outcome –Diastolic blood pressure  dichotomous

26. Julius Center.nl Julius Center.nl Health Sciences and Primary Care Example Two dichotomous determinants Age: OR = 3.8 (2.8-5.1) BMI: OR = 2.7 (2.1-3.6) Product term age and BMI: OR = 0.80 (0.55-1.17) Combined effect of A and B is 0.80 times less than product of individual effects  No significant interaction on multiplicative scale

27. Julius Center.nl Julius Center.nl Health Sciences and Primary Care Example Two dichotomous determinants RERI = e β1+β2 +β3 - e β1 - e β2 + 1 = 8.2 – 3.8 – 2.7 + 1 = 8.2 – 5.5 = 2.7 95% CI = (1.3; 4.4) Excess risk due to interaction is 2.7 Combined effect of A and B is 2.7 more than sum of individual effects  Significant ‘positive’ interaction on additive scale

28. Julius Center.nl Julius Center.nl Health Sciences and Primary Care Example Continuous and dichotomous determinant Determinants –Age  continuous per 5 years –BMI  dichotomous Outcome –Diastolic blood pressure  dichotomous

29. Julius Center.nl Julius Center.nl Health Sciences and Primary Care Example Continuous and dichotomous determinant Age: OR = 1.3 (1.2-1.4) BMI: OR = 4.0 (2.2-7.4) Product term age and BMI: OR = 0.94 (0.88-1.00)  No significant interaction on multiplicative scale

30. Julius Center.nl Julius Center.nl Health Sciences and Primary Care Example Continuous and dichotomous determinant RERI = e β1+β2 +β3 - e β1 - e β2 + 1 = 4.9 - 1.3 - 4.0 + 1 = 4.9 - 4.3 = 0.56 95% CI = (0.27; 1.0) Excess risk due to interaction is 0.56 With each 5 years of increase in age and overweight subjects, relative risk is 0.56 more than if there were no interaction  Significant ‘positive’ interaction on additive scale

31. Julius Center.nl Julius Center.nl Health Sciences and Primary Care

34. Application of methods Other measures of additive interaction Proportion attributable to interaction (AP) Synergy index (S) Spreadsheet on www.juliuscenter.nl Regression coefficients RERI, AP, S Bootstrap script S-PLUS

35. Julius Center.nl Julius Center.nl Health Sciences and Primary Care Conclusion Rothman’s theory about biologic interaction as starting point Study provides tools to estimate additive interaction and its uncertainty

36. Julius Center.nl Julius Center.nl Health Sciences and Primary Care Estimating additive interaction between continuous determinants M.J. Knol, I. van der Tweel, D.E. Grobbee, M.I. Geerlings Julius Center for Health Sciences and Primary Care University Medical Center Utrecht The Netherlands Julius Center.nl Julius Center.nl Health Sciences and Primary Care

37. Nagelkerke R 2 is measure for model fit 2 dichotomous determinants: 0.12 1 dichotomous and 1 continuous determinant: 0.12 2 continuous determinants: 0.14