A Spline Dose-Response Analysis of Air Pollution

Why Allow a Dose-Response Curve to Bend?

A straight-line regression describes a constant change in the average outcome for each unit increase in exposure. That can be a useful summary, but it may not describe every setting well: the association between air pollution and lung function might be weaker over one exposure range and stronger over another. A piecewise-linear spline allows the fitted line to change slope at a selected exposure value, called a knot, while remaining continuous there.

The aim is not to make a curve look complicated. It is to describe a plausible change in the exposure-response pattern, compare that description with a simpler linear fit, and report what the data support. A spline does not by itself establish that pollution causes a change in lung function.

Model and Interpretation

Suppose \(Y_i\) is lung function, measured as forced expiratory volume in one second (FEV1) in liters, for person \(i\). Let \(x_i\) be that person's air-pollution exposure, measured in micrograms per cubic meter. For a knot \(k\), define the hinge term \((x_i-k)_+=\max(0,x_i-k)\). Below or at the knot it equals zero; above the knot it equals the amount by which exposure exceeds the knot.

This specification gives one straight segment on each side of the knot. It does not create a jump at the knot: the hinge term is zero there, and the two segments meet at the same fitted value. The model's slopes describe adjusted mean associations: they compare expected FEV1 at different exposures while holding the listed adjustment variables fixed.

Why the Spline Is Continuous

The hinge term determines the change in slope without adding a discontinuity in the fitted mean. To see this, temporarily hold age and smoking status fixed and write the exposure part of the mean as a function of \(x\).

At \(x=k\), both expressions equal \(\beta_0\) plus the same fixed adjustment terms, so the fitted mean is continuous. Below the knot, the derivative with respect to exposure is \(\beta_1/10\) liters per microgram per cubic meter; above it, the derivative is \((\beta_1+\beta_2)/10\). Because the coefficients here are defined per 10 exposure units, it is often clearest to report the slopes directly in liters per 10 micrograms per cubic meter.

Assumptions and the Comparison with a Line

Before using coefficient tests or confidence intervals, specify the regression model and its assumptions. For the model above, the usual conditions are that the observations are independent, the conditional mean is described adequately by the included terms, and the errors have mean zero and constant variance at fixed predictor values. Normal errors are additionally used for exact small-sample \(t\) and \(F\) inference. In an application, these conditions cannot be guaranteed from a formula alone: the study design and residual diagnostics provide evidence about whether they are reasonable.

To compare the spline with a straight line, fit both models to the same analysis sample, using the same adjustment variables. The linear model is the spline model with \(H_0:\beta_2=0\). Under this null hypothesis, the slope does not change at the knot. A test of this one coefficient therefore assesses evidence for a slope change at that knot, conditional on the other terms. It does not test every possible nonlinear curve, nor does failure to reject prove that the true relationship is linear.

For a single coefficient, the usual test statistic is \(t=\hat{\beta}_2/SE(\hat{\beta}_2)\), with residual degrees of freedom \(df\). Its equivalent partial \(F\)-statistic is \(F=t^2\), with 1 numerator degree of freedom and the same residual degrees of freedom. This comparison relies on the regression assumptions stated above; in particular, the usual finite-sample reference distributions rely on normal errors.

Worked Comparison with a Linear Fit

The spline-versus-line comparison focuses on the added hinge term. It should not be confused with asking whether exposure is associated with the outcome at all: a test of \(\beta_2\) concerns whether the slope changes, not whether both segment slopes are zero.

Uncertainty for Coefficients and Fitted Differences

A coefficient estimate is not exact. A confidence interval for \(\beta_2\) describes uncertainty about the slope change, while a confidence interval for a fitted difference describes uncertainty about a particular contrast in adjusted mean FEV1. These intervals answer different questions. A pointwise interval for the mean at one exposure also does not, by itself, give a simultaneous confidence band for the entire curve.

For an ordinary two-sided \(t\)-based interval, the estimate plus or minus the appropriate critical value times its standard error gives the interval. This calculation requires the model and inference assumptions to be reasonable; interpreting a fitted contrast also requires that the compared covariate values make sense for the data.

Inspecting Residuals and the Exposure Range

A residual is an observed outcome minus the fitted outcome: \(e_i=y_i-\hat{y}_i\). Residuals plotted against fitted values and exposure can reveal patterns that suggest the mean structure misses curvature, the spread changes with exposure, or a few observations deserve closer examination. A curved pattern in residuals may indicate that the chosen two-segment shape is inadequate; a fan-shaped pattern may suggest nonconstant variance. A plot without obvious patterns is reassuring but does not establish that the model is correct.

The hinge construction guarantees continuity, not good fit. The knot should be chosen and reported transparently, and a spline with one knot can still be too simple or unnecessarily flexible for a particular data set. Diagnostic findings should be described with their consequences: for example, unexplained curvature weakens confidence in the reported slopes, while a few extreme residuals may make estimates sensitive to individual observations.

Finally, keep interpretation within the observed exposure range. If the hypothetical sample includes exposures from \(8\) to \(40\) micrograms per cubic meter, estimates within that range are supported by observed exposure values, though support may be sparse near the endpoints. Beyond \(40\), the upper segment continues as a straight line by mathematical construction; the data do not show that the same slope continues there. Report such extrapolations as unsupported by the observed range, rather than as confident dose-response conclusions. The same caution applies when the age or other adjustment values in a prediction are far from those represented in the sample.

Common Mistakes and Reporting

• Calling the knot a jump. The hinge-term spline is continuous at the knot; it changes slope, not fitted level.
• Interpreting \(\beta_2\) as the upper slope. It is the change in slope. The upper slope is \(\beta_1+\beta_2\).
• Treating a nonsignificant slope-change test as proof of a straight line. The test may have limited precision and only assesses a slope change at the specified knot.
• Turning an adjusted association into a causal claim. Adjustment for age and smoking does not rule out all confounding in an observational analysis.
• Extending the fitted upper segment confidently beyond the data. A model can calculate an extrapolated value, but calculation is not empirical support.

A clear report identifies the outcome and exposure units, the knot, the adjustment variables, the estimated segment slopes, and uncertainty relevant to the question. It also states how the spline compares with the linear fit, summarizes important residual findings, and names the observed exposure range. This makes both the curve and its limitations interpretable.