PHLT 8500 Week 2 Model Build Example

A Reason for Every Term: Building a First Multivariable Model of Hemoglobin A1c on Log-Scaled Well Arsenic in a Composite Private-Well Cohort, With Each Covariate Justified Before Its Coefficient Appears

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Walden University

PHLT 8500: Advanced Biostatistics

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Month Day, Year

A Reason for Every Term: Building a First Multivariable Model of Hemoglobin A1c on Log-Scaled Well Arsenic in a Composite Private-Well Cohort, With Each Covariate Justified Before Its Coefficient Appears

Purpose of the Model

This paper builds the first multivariable model for the term's question: among adults without diabetes who drink from private wells, how is the arsenic concentration in their well water associated with hemoglobin A1c, holding constant factors that could otherwise distort that association? The data are the composite well-testing cohort introduced in Week 1, 1,612 adults aged 35 to 74. The model is built for estimation, not prediction. That purpose governs every choice below: a term enters because it is needed to estimate the arsenic association without bias or because it sharpens the estimate, never because it raises R-squared.

A Row for Every Term

Table 1

Terms in the Model and the Reason for Each

Note. Composite cohort. Every term was chosen and coded before the adjusted model was fit.

Arsenic on a Log Scale

Last week's simple model scaled arsenic in steps of ten micrograms, and the residual plot showed that a few wells above 100 micrograms per liter were pulling the fitted line toward themselves. Arsenic concentrations are roughly lognormal, so the model uses the base-2 logarithm. The coefficient then describes the change in A1c for each doubling of well arsenic, a unit that treats a rise from 2 to 4 micrograms per liter as comparable to a rise from 20 to 40. Centering at the cohort median of 6 means the intercept refers to a typical well. The choice assumes the association is linear in log arsenic, an assumption the Week 9 comparison will test against a more flexible curve.

Age, Centered

Age is centered at 55 so that the intercept describes a 55-year-old rather than a newborn. Age enters for two reasons: A1c climbs with age, and older households are more often served by deep bedrock wells, where arsenic concentrations are higher. It is therefore associated with both exposure and outcome without lying on the path between them, the pattern that makes a variable a confounder (Greenland et al., 1999).

A Reference Group Chosen on Purpose

Reference categories are chosen so that comparisons are stable and meaningful. For education, the reference is a college degree, the largest group at 41% of the cohort, which keeps the standard errors of the other categories small. For smoking, the reference is never smokers, because comparisons with people who have never smoked are the ones readers expect. For sex, women are the reference simply because they are the majority of participants. None of these choices changes the arsenic coefficient; they change only how the other coefficients read.

Rice as a Competing Exposure

Rice accumulates inorganic arsenic from soil and irrigation water, so it is a second route of the same exposure. Rice intake is not caused by well arsenic, but if it is associated with A1c through diet and with arsenic dose through its own content, leaving it out would blur the comparison. Including it also reduces unexplained variation in A1c, which narrows the confidence interval for the well arsenic term.

Crude Beside Adjusted

The crude coefficient fell by about a quarter, from 0.024 to 0.018, when age was added, close to the Week 1 forecast of a quarter to a third. The remaining covariates moved it only slightly further, to 0.017. That pattern says age was the main confounder among those measured. The fully adjusted intercept, 5.58%, describes a 55-year-old never-smoking woman with a college degree, a body mass index of 28 and a well at the median arsenic level and who eats no rice.

Showing the crude estimate next to the adjusted one lets readers see how much the covariates mattered. Westreich and Greenland (2013) recommend presenting the exposure coefficient as the focus of the table, and the other coefficients are reported in an appendix without causal interpretation.

Table 2

Arsenic Coefficient Before and After Adjustment (Composite Figures)

Note. Composite figures; coefficients are in A1c percentage points.

Held Back, With a Reason

Two variables were deliberately left out. Recent seafood intake raises urinary arsenic mainly through arsenobetaine, a form considered of low toxicity (Navas-Acien et al., 2011). It matters when exposure is measured in urine, but this model measures arsenic in well water, which seafood cannot change, so seafood is neither a confounder nor a competing exposure here. Home water treatment was also held back. Many households install treatment after receiving a high arsenic result, so treatment is partly a consequence of the exposure, and conditioning on it would strip away some of the very association under study. Both decisions will be revisited in Week 3, when the causal diagram is drawn formally.

Limitations

The model is cross-sectional: arsenic and A1c were measured at the same visit, so it cannot show which came first, although well water arsenic is stable over years. Residual confounding is possible from unmeasured factors such as diet quality. And well water arsenic is not personal dose, since people drink different amounts and some drink bottled water.

Summary

In the first multivariable model, A1c sits about 0.017 percentage points higher for every doubling of well arsenic once the six covariates are held fixed. Each term has a written reason, two candidates were held back with reasons and the crude and adjusted estimates appear side by side.

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

Navas-Acien, A., Francesconi, K. A., Silbergeld, E. K., & Guallar, E. (2011). Seafood intake and urine concentrations of total arsenic, dimethylarsinate and arsenobetaine in the US population. Environmental Research, 111(1), 110-118. https://doi.org/10.1016/j.envres.2010.10.009

Westreich, D., & Greenland, S. (2013). The Table 2 fallacy: Presenting and interpreting confounder and modifier coefficients. American Journal of Epidemiology, 177(4), 292-298. https://doi.org/10.1093/aje/kws412