Julius Center.nl Julius Center.nl Health Sciences and Primary Care Estimating additive interaction between continuous determinants M.J. Knol, I. van der

Julius Center.nlJulius 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



Question 1

Which model do you usually use in your research?

a) Linear regression

b) Logistic regression

c) Cox regression

d) Other


Question 2

How do you usually assess interaction in your research?

a) Stratification

b) Product term

c) Never

d) Other


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


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


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

BUAUB U


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


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


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

BUAUB U

27.2% 11.1% 14.7% 4.4%


Example 2x2 table


Young (A-) Old (A+)

Normal BMI (B-) 4.4% 14.7%



• (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


Example 2x2 table

Relative risks (D=hypertension)

Young (A-) Old (A+)

Normal BMI (B-) 1.0 3.3

Overweight (B+) 2.5 6.2

Young (A-) Old (A+)

Normal BMI (B-) 4.4% 14.7%




Example 2x2 table

Relative risks (D=hypertension)

Young (A-) Old (A+)

Normal BMI (B-) 1.0 3.3

Overweight (B+) 2.5 6.2


• 6.2 ~= 3.3 x 2.5 6.2 < 8.4

• Old subjects with overweight have no excess risk for hypertension

• Risk ratio


Example 2x2 table


• Excess risk

• Risk difference


• No excess risk

• Risk ratio

Presence (or direction) of interaction depends on measure of effect


Rothman - Epidemiology: An introduction


Relative excess risk due to interaction

A- A+

B- 1.0 3.3

B+ 2.5 6.2

Additive interaction can also be calculated with relative risks

Formulas to calculate amount of additive interaction

• Absolute risks: (RA+B+-RA-B-) - (RA+B--RA-B-) - (RA-B+-RA-B-)

• Relative risks: (RRA+B+-1) - (RRA+B--1) - (RRA-B+-1)

• Relative excess risk due to interaction (RERI):

RRA+B+ - RRA+B- - RRA-B+ + 1

= 6.2 - 3.3 -2.5 + 1 = 1.4

Note: No additive interaction RERI = 0


0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

A-B- A+B- A-B+ A+B+

Rel

ativ

e ri

sk Interaction effect

Individual effect of B

Individual effect of A

Background risk

1.0

2.5

3.3

6.2


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


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?


Literature

Hosmer & Lemeshow (1992)

• Method additive interaction with logistic regression

• Making one categorical variable: A-B-, A+B-, A-B+, A+B+

• RERI = ORA+B+ - ORA+B- - ORA-B+ + 1 (OR=eβ)

D1 D2 D3

A-B- 0 0 0

A+B- 1 0 0

A-B+ 0 1 0

A+B+ 0 0 1


• Determinants

– Age dichotomous

– BMI dichotomous

• Outcome

– Diastolic blood pressure dichotomous

,000

1,327 ,158 ,000 3,771 2,767 5,140

1,001 ,143 ,000 2,721 2,054 3,605

2,104 ,135 ,000 8,198 6,294 10,678

-3,087 ,115 ,000 ,046

dummy

dummy(1)

dummy(2)

dummy(3)

Constant

B S.E. Sig. Exp(B) Lower Upper

95,0% CI

Example Dummy variables

3 dummy variables


,000

1,327 ,158 ,000 3,771 2,767 5,140

1,001 ,143 ,000 2,721 2,054 3,605

2,104 ,135 ,000 8,198 6,294 10,678

-3,087 ,115 ,000 ,046

dummy

dummy(1)

dummy(2)

dummy(3)

Constant


95,0% CI


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)


,000

1,327 ,158 ,000 3,771 2,767 5,140

1,001 ,143 ,000 2,721 2,054 3,605

2,104 ,135 ,000 8,198 6,294 10,678

-3,087 ,115 ,000 ,046

dummy

dummy(1)

dummy(2)

dummy(3)

Constant


95,0% CI


RERI = ORA+B+ - ORA+B- - 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


However…

• Only for dichotomous determinants, not for continuous ones


Methods and formulas

RERI = ORA+B+ - ORA+B- - ORA-B+ + 1

General formula logistic regression:

ln(odds) = β0 + β1 A + β2 B + β3 AB

- Individual effect of A: ORA+B- = eβ1

- Individual effect of B: ORA-B+ = eβ2

- Combined effect of A and B: ORA+B+ = eβ1+β2 +β3

RERI = eβ1+β2 +β3 - eβ1

- eβ2 + 1

95% CI bootstrap; 2.5th and 97.5th percentile


Example Two dichotomous determinants

• Determinants

– Age dichotomous

– BMI dichotomous

• Outcome


1,327 ,158 ,000 3,771 2,767 5,140

1,001 ,143 ,000 2,721 2,054 3,605

-,225 ,193 ,245 ,799 ,547 1,166

-3,087 ,115 ,000 ,046

age_dich

bmi_dich

age_dich by bmi_dich

Constant


95,0% CI



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

1,327 ,158 ,000 3,771 2,767 5,140

1,001 ,143 ,000 2,721 2,054 3,605

-,225 ,193 ,245 ,799 ,547 1,166

-3,087 ,115 ,000 ,046

age_dich

bmi_dich


Constant


95,0% CI




- eβ2 + 1 = 8.2 – 3.8 – 2.7 + 1 = 8.2 – 5.5 = 2.7

95% CI = (1.3; 4.4)


Combined effect of A and B is 2.7 more than sum of individual effects


1,327 ,158 ,000 3,771 2,767 5,140

1,001 ,143 ,000 2,721 2,054 3,605

-,225 ,193 ,245 ,799 ,547 1,166

-3,087 ,115 ,000 ,046

age_dich

bmi_dich


Constant


95,0% CI


Example Continuous and dichotomous determinant

,255 ,027 ,000 1,290 1,223 1,361

1,398 ,311 ,000 4,049 2,202 7,445

-,063 ,033 ,058 ,939 ,879 1,002

-4,600 ,250 ,000 ,010

age_5

bmi_dich

age_5 by bmi_dich

Constant


95,0% CI

• Determinants

– Age continuous per 5 years

– BMI dichotomous

• Outcome




,255 ,027 ,000 1,290 1,223 1,361

1,398 ,311 ,000 4,049 2,202 7,445

-,063 ,033 ,058 ,939 ,879 1,002

-4,600 ,250 ,000 ,010

age_5

bmi_dich

age_5 by bmi_dich

Constant


95,0% CI

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


,255 ,027 ,000 1,290 1,223 1,361

1,398 ,311 ,000 4,049 2,202 7,445

-,063 ,033 ,058 ,939 ,879 1,002

-4,600 ,250 ,000 ,010

age_5

bmi_dich

age_5 by bmi_dich

Constant


95,0% CI



- eβ2 + 1 = 4.9 - 1.3 - 4.0 + 1 = 4.9 - 4.3 = 0.56

95% CI = (0.27; 1.0)


With each 5 years of increase in age and overweight subjects, relative risk is 0.56 more than if there were no

interaction



0

0.1

0.2

0.3

0.4

0 20 40 60

Age (yrs)

Dia

sto

lic h

yper

ten

sio

n (

%)

BMI=0

BMI=1


0

0.1

0.2

0.3

0.4

0 20 40 60

Age (yrs)

Dia

sto

lic h

yper

ten

sio

n (

%)

BMI=0

BMI=1

additivity


0

0.1

0.2

0.3

0.4

0 20 40 60

Age (yrs)

Dia

sto

lic h

yper

ten

sio

n (

%)

BMI=0

BMI=1

additivity

multiplicativity


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

BARR

RERIAP

11

1

BABA

BA

RRRR

RRS


Conclusion

• Rothman’s theory about biologic interaction as starting point

• Study provides tools to estimate additive interaction and its

uncertainty


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



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

Documents

Julius Center.nl Julius Center.nl Health Sciences and Primary Care Estimating additive interaction between continuous determinants M.J. Knol, I. van der