PHLT 8500 Week 7 Collinearity Diagnostic Example

When Two Covariates Climb Together: A Diagnostic Check of Collinearity Between Age and Years at the Current Residence in the Well Arsenic Model, and Why the Exposure Estimate Survives It

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Walden University

PHLT 8500: Advanced Biostatistics

Instructor Name

Month Day, Year

When Two Covariates Climb Together: A Diagnostic Check of Collinearity Between Age and Years at the Current Residence in the Well Arsenic Model, and Why the Exposure Estimate Survives It

Why This Check

A reviewer of the Week 4 model asked whether years lived at the current residence should be added, since it approximates how long each participant has drunk from the tested well. The suggestion is reasonable, but years at residence rises with age, and in the composite private-well cohort the two are strongly correlated. This paper measures the overlap, shows what it does to the coefficients, checks whether it affects the arsenic estimate and weighs remedies against the model's purpose. A brief review of residuals and influential points follows.

Overlap, Measured

The correlation between age and years at the current residence is 0.72. Variance inflation factors quantify how much each coefficient's variance is inflated because a predictor can be partly predicted from the others.

A variance inflation factor of 2.1 means the variance of the age coefficient is roughly doubled, so its standard error is about 1.46 times larger. Common rules of thumb flag values above 5 or 10, and by those rules nothing here is a problem. O'Brien (2007) argues that such thresholds are arbitrary and that whether inflation matters depends on the sample size and the precision a question requires. The useful question is not whether a number crosses a line but what happens to the estimates that matter.

Table 1

Variance Inflation Factors, With and Without Years at Residence (Composite Figures)

Note. Composite figures.

Coefficients That Swing

When years at residence enters, the age coefficient halves and its interval widens to include zero, while years at residence takes up part of what age had carried. This is the classic signature of collinearity: the model can estimate the combined contribution of the two variables well but struggles to divide it between them, because few participants are old with few years at their address or young with many. Neither coefficient is wrong; each is imprecise about a quantity the data barely inform.

Table 2

Selected Coefficients, With and Without Years at Residence (Composite Figures)

Note. Composite figures, expressed in A1c percentage points.

The Exposure, Untouched

The arsenic coefficient moves from 0.017 to 0.016, with an interval of nearly identical width. Its variance inflation factor rises only from 1.08 to 1.11. Collinearity among covariates inflates the variances of those covariates, not of a predictor that is only weakly correlated with them. Since the model exists to estimate the arsenic association, the collinearity between age and years at residence does not threaten its purpose.

Remedies Weighed Against Purpose

Keep Both

Keeping both terms costs nothing for the arsenic estimate and adjusts for any confounding by residence duration, though the age and residence coefficients become hard to read. Westreich and Greenland (2013) caution against interpreting covariate coefficients anyway, so that loss matters little.

Drop Years at Residence

Dropping it returns to the Week 4 model. That is defensible if residence duration does not open a separate backdoor path. Duration plausibly affects A1c only through age and cumulative exposure, so it is not clearly a confounder.

Rethink What Duration Means

The reviewer's real interest may be cumulative exposure, not a covariate. If duration matters because longer residence means more years of arsenic, the right response is a different exposure, such as arsenic concentration multiplied by years of use, analyzed as its own question rather than added to this model.

Decision

The model keeps the Week 4 specification and reports the model with years at residence as a sensitivity analysis showing that the arsenic estimate is stable. Greenland (1989) recommends choosing covariates according to the effect being estimated, and years at residence does not change the estimate of the effect this model targets.

What the Reviewer Gains

The reviewer's suggestion was not wasted. The expanded model shows that the arsenic estimate does not depend on whether residence duration is in the model, which answers a question readers would otherwise raise: could people who have lived longer at an address differ in ways that explain both higher arsenic and higher A1c? The near-identical estimates say no, at least through that route. The exercise also surfaced a more interesting question, whether cumulative exposure predicts A1c better than current concentration, which could not be answered by adding a covariate and deserves its own analysis with an exposure defined as concentration multiplied by years of use. Recording both points in the paper turns a diagnostic chore into a contribution to the study's agenda.

Residuals, Briefly

Plotting residuals over fitted values reveals no curvature, only a slight widening at higher fitted values, consistent with A1c being mildly right-skewed. Refitting with heteroskedasticity-robust standard errors changes the arsenic interval only slightly, to 0.003 to 0.031. A normal quantile plot of residuals shows a modestly heavy right tail, which matters little for inference with 1,612 participants. The largest Cook's distance is 0.02, and refitting without the ten most influential observations gives an arsenic coefficient of 0.016. No single participant drives the result, and no observation was removed from the final model on the strength of these checks.

Summary

Age and years at residence overlap strongly enough to make their separate coefficients unstable, but the overlap leaves the arsenic estimate essentially unchanged. The Week 4 model is kept, with the expanded model reported as a sensitivity check and cumulative exposure noted as a question for separate analysis. Residual and influence diagnostics show no problem that would change the conclusion.

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

O'Brien, R. M. (2007). A caution regarding rules of thumb for variance inflation factors. Quality & Quantity, 41(5), 673-690. https://doi.org/10.1007/s11135-006-9018-6

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