Modeling Nonlinear Exposure-Response Interaction

Why Model Nonlinear Exposure-Response Interaction?

An exposure may relate to a quantitative outcome differently across genotypes, and that relation need not be a straight line. For example, an outcome might change rapidly at low exposure, then level off; another genotype might show a gradual increase throughout the observed range. A model with one exposure term and one genotype-by-exposure product term can represent differing straight-line slopes, but it cannot generally represent these curved patterns.

Spline regression addresses this limitation by expressing exposure through several fixed basis functions. Genotype-by-basis terms then allow the exposure-response curve to vary across genotype groups. The joint test of those terms asks whether the exposure-related parts of the curves differ, without requiring a linear exposure effect.

A Model for Genotype-Specific Curves

Let \(Y_i\) be a quantitative outcome for person \(i\), \(E_i\) a quantitative exposure, and \(G_i\) a categorical genotype with \(K\) groups. Choose genotype group 0 as the reference. Let \(C_{i1},\ldots,C_{ir}\) denote covariates whose effects are modeled as shared across genotype groups. With \(q\) exposure-basis functions, one linear regression model is

Here \(I(G_i=g)\) equals 1 when person \(i\) is in genotype group \(g\), and 0 otherwise. The coefficients \(\delta_g\) allow genotype groups to differ in level; the \(\theta_j\) describe the reference group's exposure curve; and \(\phi_{gj}\) allow group \(g\)'s exposure curve to differ from the reference. The model specifies \(E(\varepsilon_i\mid E_i,G_i,C_i)=0\). For the usual partial F test in a finite sample, we additionally assume independent normally distributed errors with a common variance. The regression form also assumes that the chosen basis and covariate terms adequately represent the conditional mean over the analyzed data.

Holding the covariates fixed, the fitted mean for group \(g\) is a function of exposure. For the reference group it is \(\beta_0+\sum_j\theta_jB_j(E)\), plus the shared covariate contribution. For group \(g>0\), it additionally includes \(\delta_g+\sum_j\phi_{gj}B_j(E)\). Thus group-specific curves can have different levels as well as different exposure patterns.

The basis must be chosen before interpreting the model. For example, a truncated linear spline with a knot at \(c\) can use \(B_1(E)=E\) and \(B_2(E)=(E-c)_+\), where \((E-c)_+=\max(E-c,0)\). This gives a continuous piecewise-linear curve with a possible change in slope at \(c\). More flexible cubic spline bases can represent smoother curvature. Knot locations, basis dimension, and any boundary constraints determine which curves the model can represent; they are not learned merely by adding interaction terms.

What Does “The Curves Differ” Mean?

There are two related but distinct comparisons. The exposure-interaction null is

Under this null, genotype groups share the same exposure-related basis coefficients. Their fitted curves can still have different constant levels because the \(\delta_g\) terms remain in the model. This test therefore assesses whether the exposure-response patterns differ, not whether the full mean curves coincide at every exposure.

If the scientific question instead concerns equality of the entire curves, the null must also set every genotype-level difference to zero: \(H_0:\delta_g=0\) and \(\phi_{gj}=0\) for all relevant \(g,j\). That broader test mixes baseline differences with exposure-related differences. Stating which null is being tested prevents a genotype main effect from being mistaken for evidence of exposure effect modification.

With a reference-group coding and a basis that excludes a constant function, the exposure-interaction null imposes \(m=(K-1)q\) restrictions. It is a joint test: individual coefficient tests do not replace it, because the basis coefficients jointly determine the curve and may be correlated.

Joint Wald Test

Let \(\widehat{\boldsymbol{\phi}}\) collect the \(m\) estimated genotype-by-basis coefficients, and let \(\widehat V\) be their estimated covariance matrix from the fitted model. The Wald statistic for the interaction null is

When the null is true and the usual large-sample approximation is appropriate, \(W\) has an approximate chi-square distribution with \(m\) degrees of freedom. The p-value is the probability, under that reference distribution and the null, of a statistic at least as large as the observed \(W\). This test accounts for the estimated covariance among the interaction coefficients.

Nested-Model Partial F Test

For a linear model with independent, normally distributed errors of common variance, the same null can be tested by comparing two nested fits. The unrestricted model includes all genotype-by-basis terms. The reduced model removes those terms but retains genotype main effects, the shared exposure basis, and the same covariates. Let \(RSS_R\) and \(RSS_F\) be their residual sums of squares, and let \(p_F\) be the number of fitted parameters in the unrestricted model. The partial F statistic is

Under the null and the stated model assumptions, this statistic has an \(F\) distribution with \(m\) and \(n-p_F\) degrees of freedom. The numerator measures the reduction in residual variation per restriction; the denominator estimates the error variance in the unrestricted model. The comparison is valid only when the two fits use the same observations and differ only by the tested interaction terms.

Choosing and Interpreting the Basis

A spline basis offers flexibility, not freedom from modeling decisions. The basis dimension controls how many distinct exposure patterns can be represented. Too few basis functions may miss important curvature; too many may produce imprecise estimates, especially when some genotype groups have little data across the exposure range. Knots and boundary behavior should be selected using subject-matter knowledge and a prespecified analysis plan where possible.

Interpretation is usually clearer from fitted curves or contrasts at meaningful exposure values than from a single basis coefficient. A coefficient on \(B_j(E)\) describes a change associated with that particular basis function while the other model terms are held fixed; it is not generally a per-unit exposure effect. Predictions should be restricted to exposure values with adequate data support in the genotype groups being compared. A fitted curve beyond that support is extrapolation and may be driven mainly by the chosen basis.

As in other regression models, adjustment for measured covariates does not automatically make an association causal. A causal interpretation requires an appropriate design and assumptions about confounding, selection, and measurement, in addition to a correctly specified outcome model. The interaction test alone addresses a statistical comparison of modeled exposure patterns.

Common Mistake

A frequent error is to test only the coefficient of \(G\times E\) in a linear model and conclude that there is no genotype-by-exposure interaction when that coefficient is not significant. That conclusion is limited to the chosen straight-line interaction. If the true difference is curvature or a change in slope over part of the exposure range, a single product term may not represent it. The correction is to use a prespecified spline basis suited to the scientific question and test all genotype-by-basis coefficients jointly. Conversely, a significant joint test indicates differences within the fitted basis model, not proof of a particular biological mechanism.