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Epidemiology:
doi: 10.1097/EDE.0000000000000083
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Additive Interaction Between Continuous Risk Factors Using Logistic Regression

Katsoulis, Michail; Bamia, Christina

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Hellenic Health Foundation, Athens, Greece, University of Athens, WHO Collaborating Center for Food and Nutrition Policies, Department of Hygiene, Epidemiology, and Medical Statistics, Athens, Greece, mkatsoulis@hhf-greece.gr

University of Athens, WHO Collaborating Center for Food and Nutrition Policies, Department of Hygiene, Epidemiology, and Medical Statistics, Athens, Greece.

Supported by European Union Seventh Framework Program (FP7/2007–2013) under CHANCES Project (grant agreement no. HEALTH–F3-2010–242244).

Supplemental digital content is available through direct URL citations in the HTML and PDF versions of this article ( www.epidem.com). This content is not peer-reviewed or copy-edited; it is the sole responsibility of the author.

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To the Editor:

Interaction of two risk factors with respect to a health outcome may be better evaluated through deviations from additivity rather than multiplicativity of their effects.1,2 Certain publications, focusing on dichotomous risk factors, have explored additive interaction through Rothman’s indexes.2–7 Continuous risk factors have been investigated for specific cases only8 (eAppendix 1; http://links.lww.com/EDE/A771). We consider here the setting of continuous risk factors X, Y for disease D (0 = absence, 1 = presence), such as the probability that D = 1 increases with higher values of X, Y.7 Let x0x1 and y0y1 denote the arbitrary increments dx = x1x0, dy = y1y0 of X, Y, whereas x0x0 and y0y0, denote that X, Y are fixed at a background level of exposure.

Now, consider the following logistic regression, including X, Y, their product term XY, and covariates Zi, i = 1, 2, …, n:


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(1) p being the probability of D = 1.

We further assume that odds ratios (ORs) as obtained from Equation (1) approximate the relative risks (RRs).4 We show (eAppendix 1; http://links.lww.com/EDE/A771) that measures of additive interaction between X and Y, expressed through Rothman’s indexes, are estimated as shown in the Figure.

FIGURE.
FIGURE.
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Variances and confidence intervals for each of Equations (2) to (7) are calculated using the delta method (eAppendix 2; http://links.lww.com/EDE/A771). Suitable software for estimating the indicated indexes and their variances/confidence intervals can be found in the eAppendix (eAppendix.script; http://links.lww.com/EDE/A771).

For illustration, we used data from 4805 women younger than 45 years, recruited in the Greek EPIC cohort (eAppendix 3; http://links.lww.com/EDE/A771). We hypothesized that increasing body mass index (BMI) (continuous) and increasing nonadherence to the Mediterranean diet (NMD) (ordinal) result in increased prevalence of hypertension. Using Equation (1) with age, height, and education as covariates, we estimated measures of additive interaction through Equations (2) to (7). In these data, the outcome was rare (<10%), and so the above indexes are applicable.5

The association of hypertension with BMI and nonadherence to a Mediterranean diet, estimated through logistic regression, with and without their interaction term, is shown in the Table and eTable 2 ( http://links.lww.com/EDE/A771), respectively. In eTable 2, both factors are shown to be positively associated with prevalence of hypertension (ORBMI = 1.14 and ORNMD = 1.18, respectively). As shown in the Table, we found no evidence of submultiplicativity or supermultiplicativity of these associations (bBMI × NMD = 0.002; test for interaction, P = 0.929). Moreover, there was no indication of deviations from additivity of the effects (Table, lower part) for any combination of BMI and diet. However, deviations are higher for higher BMI and less adherence to Mediterranean diet, as well as for higher increments in both exposures for given background values of BMI and diet.

TABLE.
TABLE.
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In conclusion, we show that measures of additive interaction between continuous risk factors X and Y depend both on the background (x0, y0) and on the “elevated” (x1, y1) levels of exposure. Formal quantification of the influence of background and elevated levels of exposures on the monotonicity of the estimated indexes is, however, complicated and was not further investigated. Our formulae need to be applied with care in case-control studies, given the limitations accompanying estimation of additive interaction when additional covariates are included in logistic regression.6

Notably, Equations (2) to (7) hold for any variables X, Y (continuous, ordered, or dichotomous), given that ordered X, Y is a special case on continuous X, Y, and Equations (2) to (7) reduce to formulae previously proposed for two dichotomous variables.4 The derived formulae add, therefore, to the interpretation of measures of additive interaction, whereas the developed software makes estimation feasible, as shown in the eAppendix (eAppendix.script; http://links.lww.com/EDE/A771). This is important for researchers who wish to consider additive interactions in their data (eg, gene–environmental interactions) but may have been discouraged by the difficulty in estimating and interpreting such measures.

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ACKNOWLEDGMENTS

We thank Antonia Trichopoulou for making available EPIC-Greece data for this study and her useful comments.

Michail Katsoulis
Hellenic Health Foundation
Athens, Greece
University of Athens
WHO Collaborating Center for Food and
Nutrition Policies
Department of Hygiene, Epidemiology, and
Medical Statistics
Athens, Greece
mkatsoulis@hhf-greece.gr

Christina Bamia
University of Athens
WHO Collaborating Center for Food and
Nutrition Policies
Department of Hygiene, Epidemiology, and
Medical Statistics
Athens, Greece.

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REFERENCES

1. Rothman KJ, Greenland S, Walker AM. Concepts of interaction. Am J Epidemiol. 1980; 112:467–470

2. Knol MJ, VanderWeele TJ. Recommendations for presenting analyses of effect modification and interaction. Int J Epidemiol. 2012; 41:514–520

3. Rothman KJ. Modern Epidemiology. 1986; Boston/Toronto Little: Brown and Company

4. Hosmer DW, Lemeshow S. Confidence interval estimation of interaction. Epidemiology. 1992; 3:452–456

5. VanderWeele TJ, Chen Y, Ahsan H. Inference for causal interactions for continuous exposures under dichotomization. Biometrics. 2011; 67:1414–1421

6. Skrondal A. Interaction as departure from additivity in case-control studies: a cautionary note. Am J Epidemiol. 2003; 158:251–258

7. Knol MJ, VanderWeele TJ, Groenwold RH, Klungel OH, Rovers MM, Grobbee DE. Estimating measures of interaction on an additive scale for preventive exposures. Eur J Epidemiol. 2011; 26:433–438

8. Knol MJ, van der Tweel I, Grobbee DE, Numans ME, Geerlings MI. Estimating interaction on an additive scale between continuous determinants in a logistic regression model. Int J Epidemiol. 2007; 36:1111–1118

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