PHLT 8500 Week 3 Variable Selection Example

Drawing the Map Before Choosing the Variables: A Causal Diagram for Well Arsenic and Hemoglobin A1c That Decides Which Covariates to Adjust For, Which to Leave Alone and What Stepwise Selection Would Have Done Instead

Student Name

Walden University

PHLT 8500: Advanced Biostatistics

Instructor Name

Month Day, Year

Drawing the Map Before Choosing the Variables: A Causal Diagram for Well Arsenic and Hemoglobin A1c That Decides Which Covariates to Adjust For, Which to Leave Alone and What Stepwise Selection Would Have Done Instead

The Effect Named Before the Variables

Variable selection only makes sense relative to a target. The target here is the total effect of well water arsenic on hemoglobin A1c among adults without diabetes in the composite private-well cohort: how A1c would differ if the same people drank water with twice as much arsenic. Naming that effect first matters because a variable that should be adjusted for when estimating one effect may need to be left alone when estimating another. A model for predicting A1c, for example, would happily include every strong predictor, but a model for estimating an effect must avoid adjusting for consequences of the exposure.

Arrows With Reasons

A directed acyclic graph represents assumptions about which variables cause which (Greenland et al., 1999). Each arrow below is a claim with a reason, and each missing arrow is a claim too: that one variable does not directly affect another.

The diagram does not prove anything; it makes assumptions visible so readers can dispute a particular arrow instead of the whole model.

Table 1

Arrows in the Diagram and the Reason for Each

Note. Composite diagram. Body mass index and sex enter as causes of A1c only.

Upstream, Adjusted

A variable needs adjustment if it opens a backdoor path, a route from exposure to outcome that begins with an arrow pointing into the exposure. Age and education each open one: well arsenic ← age → A1c and well arsenic ← education → A1c. Adjusting for both closes those paths. Smoking lies on a path through education, so once education is adjusted, smoking is not needed to remove bias, but it is a strong cause of A1c and including it improves precision without opening any new path. VanderWeele and Shpitser (2011) propose adjusting for every pre-exposure covariate that causes the exposure, the outcome or both, a practical rule that leads to the same adjustment set here: age, education and smoking, with sex, body mass index and rice intake added for precision because they cause the outcome or its dose.

Downstream, Left Alone

Three variables sit downstream of the exposure, and the diagram says not to adjust for any of them.

Home Treatment

Treatment lies on the path from well arsenic to dose. Adjusting for it would remove the part of the association that runs through the fact that untreated households drink more arsenic, which is most of the effect of interest. Schisterman et al. (2009) call this overadjustment and show that it biases estimates toward the null.

Urinary Arsenic

Urinary arsenic is a marker of dose. Adjusting for it would compare people with the same urinary arsenic but different well arsenic, a comparison with no clear meaning.

Fasting Insulin

Insulin resistance may be a mechanism by which arsenic affects glucose. Adjusting for it would estimate only the part of the effect not running through insulin, which is a different question from the total effect.

A Variable That Is Easy to Misplace

Body mass index could be drawn either way. If arsenic affected weight, body mass index would be a mediator and should be left out. The literature on arsenic and adiposity is mixed, and in this cohort body mass index does not differ across arsenic levels. The diagram treats it as a cause of A1c only and includes it for precision. A sensitivity model without body mass index changes the arsenic coefficient from 0.017 to 0.018, so the decision does not affect the conclusion.

What Stepwise Would Have Done

For comparison, a backward stepwise procedure starting from all available variables and removing those with p-values above .10 was run on the same data. It kept urinary arsenic, fasting insulin, body mass index and age and dropped education and rice intake. Because urinary arsenic is a marker of dose and strongly correlated with well arsenic, its inclusion pulled the well arsenic coefficient down to 0.004, near zero. Stepwise selection chose variables by their association with A1c, not by their position relative to the exposure, so it adjusted for two downstream variables and dropped an upstream confounder. Greenland (1989) warned that significance-based selection can produce biased effect estimates and misleading confidence intervals, and this example shows the mechanism: a near-null estimate created by adjusting away the exposure's own pathway.

The Adjustment Set

Table 2

Final Adjustment Decisions

Note. Composite decisions consistent with the Week 2 model.

Limits of the Diagram

The diagram reflects current knowledge and could be wrong. An unmeasured factor, such as overall diet quality, might affect both where people live and their A1c, leaving a backdoor path open. The analysis cannot rule that out; it can only state the assumption, so readers can judge which unmeasured pattern would be needed to bias the estimate.

Summary

The causal diagram identifies age and education as confounders, smoking, sex, body mass index and rice intake as precision variables and home treatment, urinary arsenic and fasting insulin as downstream variables to leave alone. Stepwise selection, by contrast, adjusted for the exposure's own markers and nearly erased the estimate.

References

Greenland, S. (1989). Modeling and variable selection in epidemiologic analysis. American Journal of Public Health, 79(3), 340-349. https://doi.org/10.2105/AJPH.79.3.340

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

Schisterman, E. F., Cole, S. R., & Platt, R. W. (2009). Overadjustment bias and unnecessary adjustment in epidemiologic studies. Epidemiology, 20(4), 488-495. https://doi.org/10.1097/EDE.0b013e3181a819a1

VanderWeele, T. J., & Shpitser, I. (2011). A new criterion for confounder selection. Biometrics, 67(4), 1406-1413. https://doi.org/10.1111/j.1541-0420.2011.01619.x