Titel van de Slide

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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 Health Sciences and Primary Care
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Question 1

• Which model do you usually use in your research?
• Linear regression
• Logistic regression
• Cox regression
• Other

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Question 2

• How do you usually assess interaction in your
  research?
• Stratification
• Product term
• Never
• Other

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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

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Background

• Synonyms
• Interaction
• Effect (measure) modification
• Synergy
• Interaction is present when effect of A is
  different across strata of B
• (or vice versa)

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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

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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

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Example dataset

• Utrecht Health Project
• Baseline data
• N4897
• 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

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Example 2x2 table
Absolute risks (Dhypertension)
27.2
11.1
14.7
4.4
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Example 2x2 table
Absolute risks (Dhypertension)

• Interaction as departure from additivity
• (27.2 - 4.4) (14.7 - 4.4) (11.1 - 4.4) ?
  22.8 gt 17.0
• Old subjects with overweight have excess risk for
  hypertension
• Risk difference

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Example 2x2 table
Absolute risks (Dhypertension)
Relative risks (Dhypertension)
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Example 2x2 table

• Interaction as departure from multiplicativity
• 6.2 3.3 x 2.5 ? 6.2 lt 8.4
• Old subjects with overweight have no excess risk
  for hypertension
• Risk ratio

Relative risks (Dhypertension)
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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

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Rothman - Epidemiology An introduction
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Relative excess risk due to interaction

• Additive interaction can also be calculated with
  relative risks
• Formulas to calculate amount of additive
  interaction
• Absolute risks (RAB-RA-B-) - (RAB--RA-B-) -
  (RA-B-RA-B-)
• Relative risks (RRAB-1) - (RRAB--1) -
  (RRA-B-1)
• Relative excess risk due to interaction (RERI)
• RRAB - RRAB- - RRA-B 1
• 6.2 - 3.3 -2.5 1 1.4
• Note No additive interaction ? RERI 0

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6.2
3.3
2.5
1.0
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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

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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?

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Literature

• Hosmer Lemeshow (1992)
• Method additive interaction with logistic
  regression
• Making one categorical variable A-B-, AB-,
  A-B, AB
• RERI ORAB - ORAB- - ORA-B 1 (OReß)

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Example Dummy variables

• Determinants
• Age ? dichotomous
• BMI ? dichotomous
• Outcome
• Diastolic blood pressure ? dichotomous

3 dummy variables
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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)
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Example Dummy variables
RERI ORAB - ORAB- - ORA-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
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• However
• Only for dichotomous determinants, not for
  continuous ones

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Methods and formulas

• RERI ORAB - ORAB- - ORA-B 1
• General formula logistic regression
• ln(odds) ß0 ß1 A ß2 B ß3 AB
• Individual effect of A ORAB- eß1
• Individual effect of B ORA-B eß2
• Combined effect of A and B ORAB eß1ß2 ß3
• RERI eß1ß2 ß3 - eß1 - eß2 1
• 95 CI ? bootstrap 2.5th and 97.5th percentile

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Example Two dichotomous determinants

• Determinants
• Age ? dichotomous
• BMI ? dichotomous
• Outcome
• Diastolic blood pressure ? dichotomous

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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
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Example Two dichotomous determinants
RERI eß1ß2 ß3 - eß1 - eß2 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
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Example Continuous and dichotomous determinant

• Determinants
• Age ? continuous per 5 years
• BMI ? dichotomous
• Outcome
• Diastolic blood pressure ? dichotomous

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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
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Example Continuous and dichotomous determinant
RERI eß1ß2 ß3 - eß1 - eß2 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
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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

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Conclusion

• Rothmans theory about biologic interaction as
  starting point
• Study provides tools to estimate additive
  interaction and its uncertainty

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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 Health Sciences and Primary Care
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• Nagelkerke R2 is measure for model fit
• 2 dichotomous determinants 0.12
• 1 dichotomous and 1 continuous determinant 0.12
• 2 continuous determinants 0.14