PHLT 8500 Week 8 Interaction Analysis Example

Does Arsenic Matter More for People With Obesity? An Interaction Analysis of Well Arsenic and Body Mass Index on Diabetes by the Six-Year Visit, Reported on Both the Multiplicative and the Additive Scale

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Walden University

PHLT 8500: Advanced Biostatistics

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Month Day, Year

Does Arsenic Matter More for People With Obesity? An Interaction Analysis of Well Arsenic and Body Mass Index on Diabetes by the Six-Year Visit, Reported on Both the Multiplicative and the Additive Scale

A Reason to Look

Interaction should be examined when there is a reason to expect it, not searched for across every pair of variables. Here there is a reason. Experimental and population studies suggest that arsenic may impair insulin secretion, while obesity raises insulin resistance, so the two could act together: a person whose insulin demands are already high might be less able to absorb an added insult to secretion. Prospective work has also linked arsenic exposure to incident diabetes (Kuo et al., 2015), so the main association is plausible before any interaction is considered. The question is whether the association between well arsenic and diabetes by the six-year visit differs between participants with and without obesity in the composite private-well cohort, 1,420 adults with 176 new cases. The question was stated in the analysis plan before any stratified result was examined.

Coding for Clarity

To keep the presentation readable, arsenic is split at 10 micrograms per liter, the federal maximum for public water, with wells at or above that value treated as high, and body mass index as 30 or higher versus lower. The continuous versions are analyzed in an appendix with the same conclusions. The model is logistic, adjusted for age, sex, smoking, education and rice intake, with a product term between high arsenic and obesity.

Risks in Four Groups

Knol and VanderWeele (2012) recommend presenting each combination against a single reference group, as here, so readers can see the joint effect alongside each factor's separate effect.

Table 1

Diabetes by the Six-Year Visit by Arsenic and Obesity (Composite Figures)

Note. Composite figures, crude risks. Adjusted estimates in Table 2 are similar.

Main Effects, Redefined

Once a product term is in the model, the coefficient for high arsenic no longer describes an average association. It describes the association among participants without obesity, the reference level of the other variable. Likewise, the obesity coefficient describes obesity's association among participants with low-arsenic wells. The product term measures how much the arsenic association, on the log odds scale, differs between the two body mass index groups. Readers who interpret the arsenic coefficient as a general effect after adding an interaction are misreading the model.

Two Scales, Two Answers

On the multiplicative scale, there is essentially no interaction: the odds ratio for high arsenic is about 1.3 in both groups, and their ratio is 1.04. On the additive scale, the picture changes. High arsenic is associated with 1.9 more cases per 100 people among those without obesity but 4.8 more per 100 among those with obesity. The relative excess risk due to interaction, 0.43, summarizes this departure from additivity.

Both answers are correct; they answer different questions. A constant ratio applied to a higher baseline risk produces a larger absolute difference. Because people with obesity start at higher risk, the same relative increase from arsenic adds more cases among them.

Table 2

Interaction on Two Scales (Composite Figures, Adjusted)

Note. Composite figures.

Which Scale Matters for Public Health

For deciding where to act, the additive scale is usually the relevant one. If a county can afford to provide arsenic treatment systems to only some households, the number of cases prevented per household treated is what matters, and that number is larger among residents with obesity. Knol and VanderWeele (2012) recommend reporting both scales for this reason, so readers can judge both the relative strength of association and its absolute consequences.

Precision

The intervals are wide. The relative excess risk interval runs from -0.37 to 1.23, so the data are compatible with no additive interaction as well as with a substantial one. With only 150 participants in the highest-risk group, the analysis is underpowered for interaction, as most interaction analyses are. Detecting an interaction of this size reliably would take several times as many participants with both exposures. The appropriate conclusion is that the absolute burden associated with high arsenic appears larger among people with obesity, consistent with their higher baseline risk, and that the data cannot establish whether the two exposures act together beyond that.

What the Analysis Does Not Show

Even a clear additive interaction would not show biological synergy on its own. An interaction in a model is a pattern in the data, and reading it as a mechanism requires the same causal assumptions as any effect estimate, now for both exposures: no unmeasured confounding of arsenic and no unmeasured confounding of obesity (Greenland et al., 1999). Obesity's confounders, such as physical activity and diet, were never the target of this model's adjustment set, so a causal reading of the joint effect would go beyond what the model was built to support.

Summary

High well arsenic is associated with roughly 30% higher odds of diabetes by the six-year visit regardless of obesity, so there is no multiplicative interaction. Because people with obesity have higher baseline risk, the same relative increase corresponds to more than twice as many additional cases per 100 among them, a departure from additivity whose interval includes zero. Both scales are reported, and the additive scale is the one that informs who should be offered treatment systems first.

References

Greenland, S., Pearl, J., & Robins, J. M. (1999). Causal diagrams for epidemiologic research. Epidemiology, 10(1), 37-48. https://doi.org/10.1097/00001648-199901000-00008

Knol, M. J., & VanderWeele, T. J. (2012). Recommendations for presenting analyses of effect modification and interaction. International Journal of Epidemiology, 41(2), 514-520. https://doi.org/10.1093/ije/dyr218

Kuo, C.-C., Howard, B. V., Umans, J. G., Gribble, M. O., Best, L. G., Francesconi, K. A., Goessler, W., Lee, E., Guallar, E., & Navas-Acien, A. (2015). Arsenic exposure, arsenic metabolism, and incident diabetes in the Strong Heart Study. Diabetes Care, 38(4), 620-627. https://doi.org/10.2337/dc14-1641