PHLT 8500 Week 5 Outcome Type Brief Example

An Event Is Not a Measurement: A Brief on Modeling Diabetes by the Six-Year Visit With Logistic Regression Instead of the Linear Model Built for Hemoglobin A1c

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Walden University

PHLT 8500: Advanced Biostatistics

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Month Day, Year

An Event Is Not a Measurement: A Brief on Modeling Diabetes by the Six-Year Visit With Logistic Regression Instead of the Linear Model Built for Hemoglobin A1c

The New Outcome

The term's models so far have treated hemoglobin A1c as a continuous measurement. The cohort's main public health question, however, is whether people develop diabetes. Among the 1,612 enrollees free of diabetes at baseline, 1,420 completed the six-year visit, and 176 of them, 12.4%, had developed diabetes by then, counting either a new physician diagnosis or a follow-up A1c at the 6.5% threshold. The outcome is now binary: each person either developed diabetes or did not. This brief explains why the linear model no longer fits and what replaces it. Participants who missed the six-year visit are set aside here and handled properly in Week 10.

Predictions Below Zero

A linear model fit to a 0-or-1 outcome, called a linear probability model, estimates the probability of diabetes as a straight-line function of the predictors. Fit with the Week 3 adjustment set, it predicts a probability below zero for 38 participants, mostly young, lean, never-smoking adults with low-arsenic wells, and above 0.5 for a handful at the other extreme. A negative probability has no meaning. The model's errors also cannot be normally distributed or equally variable, because each residual can take only two values for any set of predictors.

The Logit Link in Two Sentences

Logistic regression models the log of the odds of diabetes, rather than the probability itself, as a linear function of the predictors. Because the log odds can take any value while the probability it implies always lies between 0 and 1, predictions stay within the possible range.

Results

Per doubling of well arsenic, the odds of developing diabetes by the six-year visit rise by 14%, holding the other terms constant. The model has 10 parameters and 176 events, about 18 events per parameter, comfortably above the levels Vittinghoff and McCulloch (2007) found to produce acceptable bias and coverage.

Table 1

Logistic Regression for Diabetes by the Six-Year Visit (Composite Figures, n = 1,420, 176 cases)

Note. Composite figures. Also adjusted for sex, smoking, education and rice intake; covariate odds ratios are shown for context only and are not causal estimates.

Odds Are Not Risks

An odds ratio of 1.14 is often reported as "14% higher risk," which is not quite right. Odds and risks diverge as an outcome becomes common, and the odds ratio then exaggerates the risk ratio. Zhang and Yu (1998) give a correction using the baseline risk. With a risk of about 12% in the reference group, the odds ratio of 1.14 corresponds to a risk ratio of roughly 1.12, a 12% higher risk per doubling. The difference here is small because diabetes is moderately rare by six years, but stating risk ratios or predicted risks avoids the error entirely. For a reader, the clearest summary is a pair of predicted risks: about 11.0% for a typical participant with a 3 microgram per liter well and 13.8% for the same person with a 12 microgram per liter well, two doublings higher.

Fit, With Its Caveats

The Hosmer-Lemeshow test gives p = .41, meaning no evidence of poor calibration across ten risk groups. The test has known weaknesses: its result depends on how many groups are used, and it has little power in modest samples while flagging trivial misfit in very large ones. A calibration plot of predicted against observed risk by decile is more informative and shows predictions within two percentage points of observed risk in every decile. Discrimination, the area under the ROC curve, is 0.72, but discrimination matters more for prediction models than for estimating one association, so it is reported without weight.

One Family per Outcome Type

The change from A1c to diabetes is a change of outcome type, and each type has its model family. A continuous measurement suits linear regression; a binary event suits logistic regression, or a log-binomial or modified Poisson model when risk ratios are wanted directly; a count suits Poisson or negative binomial regression; and time until an event, with some people followed for less than the full period, suits survival models. The choice follows from the outcome, not from the software default. Because this analysis ignored when diabetes occurred and dropped those lost to follow-up, it answers a narrower question than a survival model would.

Consistency With Prior Evidence

A positive association between arsenic and incident diabetes is consistent with prospective findings in other populations (Kuo et al., 2015), although exposure there was measured in urine and the populations differ.

References

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

Vittinghoff, E., & McCulloch, C. E. (2007). Relaxing the rule of ten events per variable in logistic and Cox regression. American Journal of Epidemiology, 165(6), 710-718. https://doi.org/10.1093/aje/kwk052

Zhang, J., & Yu, K. F. (1998). What's the relative risk? JAMA, 280(19), 1690-1691. https://doi.org/10.1001/jama.280.19.1690