Pediatric Exposure-Response Models for Safety

Why Pediatric Safety Models Need Two Age-Related Questions

A safety event may occur at different background rates across developmental stages even when drug exposure is the same. At the same time, a given increase in exposure might have a different association with event risk in infants, children, and adolescents. A useful analysis therefore separates two questions: does baseline risk vary by age group, and does the exposure effect vary by age group? Treating these as the same question can conceal an important developmental pattern.

This tutorial considers a binary safety outcome over a defined follow-up interval, such as whether a laboratory abnormality occurs during a treatment period. The exposure measure might be an individual’s average concentration or peak concentration over a specified window. The chosen measure and time window must be clinically justified and determined so that exposure precedes, rather than follows, the event being analyzed.

Definition: An exposure-safety relationship describes how the probability of a defined safety event changes with a specified drug-exposure measure in a specified population and time interval. An age-specific baseline risk is the model-estimated event probability at the reference exposure in a particular developmental-stage group. An age-specific exposure effect allows the exposure-risk relationship to differ among those groups.

A Logistic Model With Developmental-Stage Effects

Let \(Y_{ij}=1\) if participant \(i\) in developmental-stage group \(g\) experiences the event during the defined interval, and \(Y_{ij}=0\) otherwise. Let \(x_{ij}\) be exposure, expressed in a stated unit and centered at a meaningful reference value. For example, \(x=0\) could denote the exposure value chosen as the reference, rather than no drug exposure. A logistic model with group-specific baselines and effects is

Y_{ij}\mid x_{ij},g \sim \operatorname{Bernoulli}(p_{ij}), \qquad \operatorname{logit}(p_{ij})=\alpha_g+\beta_g x_{ij}, \qquad \operatorname{logit}(p)=\log\!\left(\frac{p}{1-p}\right).

Here \(\alpha_g\) is the log odds of an event at \(x=0\) in group \(g\), and \(\beta_g\) is the change in log odds for a one-unit increase in exposure in that group. The baseline risk is \(p_g(0)=\exp(\alpha_g)/(1+\exp(\alpha_g))\). For a one-unit increase, the odds ratio is \(\exp(\beta_g)\). The event probability at exposure \(x\) is obtained by applying the inverse-logit transformation:

p_g(x)=\frac{\exp(\alpha_g+\beta_gx)} {1+\exp(\alpha_g+\beta_gx)}.

A common-effect model retains a separate baseline for each stage but constrains the exposure coefficient to be shared: \(\beta_g=\beta\) for every \(g\). This distinction matters. A shared \(\beta\) does not require equal event probabilities across ages, because the \(\alpha_g\) values can differ. Conversely, different observed event rates by age do not, on their own, demonstrate that exposure effects differ.

Model assumptions: The outcome is defined consistently across groups and is binary for the chosen interval; observations are independent conditional on the modeled covariates (or dependence is handled by an appropriate model); the log odds are linear in exposure unless a nonlinear exposure function is specified; and the exposure distributions overlap sufficiently across groups to support comparisons. Interpretation as a causal drug effect additionally requires adequate control of confounding and appropriate exposure timing. Without those conditions, the model estimates an exposure-safety association, not necessarily a causal effect.

Why Odds Ratios Must Be Translated Into Risks

The odds ratio from a logistic model is not a risk ratio and does not specify an absolute change in probability by itself. The absolute risk change depends on the starting risk. For baseline risk \(p_0\) and odds ratio \(OR\), the baseline odds are \(p_0/(1-p_0)\); multiplying by \(OR\) gives exposed odds. Converting those odds back to risk gives

p_1=\frac{OR\,p_0}{1-p_0+OR\,p_0}.

This follows because the exposed odds are \(OR\,p_0/(1-p_0)\), and a set of odds \(o\) corresponds to risk \(o/(1+o)\). Substitution and simplification yield the displayed expression. Thus, the same odds ratio can imply different absolute risk increases in developmental groups with different baseline risks.

Worked Example: The Same Odds Ratio at Two Baseline Risks

Suppose a hypothetical common-effect model estimates an odds ratio of 2.01 for a one-unit increase in exposure. At the reference exposure, the estimated baseline safety-event risk is 0.0759 in one age group and 0.0293 in another. Compute the risk after a one-unit increase and the corresponding absolute changes.

Group with baseline risk 0.0759. The baseline odds are \(0.0759/(1-0.0759)=0.0759/0.9241\approx0.08213\). The exposed odds are \(0.08213\times2.01\approx0.1651\). Therefore the exposed risk is \(0.1651/(1+0.1651)\approx0.1417\), or 14.17%. The increase is \(0.1417-0.0759=0.0658\), about 6.58 percentage points.

Group with baseline risk 0.0293. The baseline odds are \(0.0293/0.9707\approx0.03018\). The exposed odds are \(0.03018\times2.01\approx0.06067\). The exposed risk is \(0.06067/(1+0.06067)\approx0.05720\), or 5.72%. The increase is \(0.05720-0.0293=0.02790\), about 2.79 percentage points.

The model gives the same multiplicative change in odds in both groups, but not the same absolute risk increase. These are model-based hypothetical estimates, not observations from a real study. Their interpretation also depends on the validity of the model and the exposure comparison.

Allowing the Exposure Effect to Differ by Age

With \(G\) developmental-stage groups, an interaction model estimates \(G\) exposure slopes, while the common-effect model estimates one. In the interaction model, the age-by-exposure interaction is represented by the differences among the \(\beta_g\) values. The intercepts remain group-specific in both models, so the comparison addresses effect heterogeneity rather than baseline-risk heterogeneity.

Worked Example: Interpreting Age-Specific Exposure Effects

Consider a hypothetical analysis with exposure measured in mg/L above a clinically selected reference. The estimated exposure coefficients are \(\hat{\beta}_{\text{infant}}=0.55\), \(\hat{\beta}_{\text{child}}=0.70\), and \(\hat{\beta}_{\text{adolescent}}=0.68\). Interpret the one-unit odds ratios and compare the estimated effects.

For infants, the odds ratio is \(\exp(0.55)\approx1.73\). For children, it is \(\exp(0.70)\approx2.01\), and for adolescents it is \(\exp(0.68)\approx1.97\). Under this model, a 1 mg/L increase in exposure is associated with estimated event odds about 1.73, 2.01, and 1.97 times as large, respectively, within the three groups. These are odds comparisons, not statements that risk increases by 73%, 101%, and 97%.

The estimated difference between the child and infant log-odds slopes is \(0.70-0.55=0.15\). On the odds-ratio scale, the ratio of their one-unit odds ratios is \(\exp(0.15)\approx1.16\). This is a point-estimate contrast; by itself it does not establish that the true effects differ. A confidence interval or formal model comparison is needed to quantify uncertainty. Nor does the contrast establish causality unless the assumptions needed for a causal interpretation are supported.

To obtain an age-specific probability, the group’s intercept must also be used. For instance, if a group’s baseline risk at \(x=0\) is 0.0759 and its one-unit odds ratio is 2.01, the preceding calculation gives a risk of about 0.1417 at \(x=1\). The slope alone cannot determine that absolute risk.

Testing Whether a Common Exposure Effect Is Adequate

A likelihood-ratio test compares the common-effect model with the more flexible interaction model. Let \(\ell_C\) and \(\ell_I\) be their maximized log likelihoods, respectively. Because the common-effect model is nested within the interaction model, the likelihood-ratio statistic is \(2(\ell_I-\ell_C)\). Under the null hypothesis of a common exposure effect, and under regularity conditions such as adequate information and identifiable parameters, this statistic is approximately chi-squared with \(G-1\) degrees of freedom. The degrees of freedom count the additional slope parameters in the interaction model.

The test addresses \(H_0:\beta_1=\cdots=\beta_G\) against \(H_a:\) at least one slope differs. It does not test whether all age-specific baseline risks are equal. A small p-value is evidence against the common-slope constraint, conditional on the model and data. A p-value above the chosen significance level is not proof that the slopes are identical; limited events or exposure overlap may make meaningful differences hard to detect.

Worked Example: Comparing Common and Age-Specific Slopes

In a hypothetical study with three age groups, the fitted common-effect model has maximized log likelihood \(\ell_C=-312.4\), and the age-specific-slope model has \(\ell_I=-309.8\). Test the common-effect assumption at \(\alpha=0.05\).

The statistic is \(2(\ell_I-\ell_C)=2[-309.8-(-312.4)]=2(2.6)=5.20\). The interaction model adds \(3-1=2\) slope parameters, so \(df=2\). For a chi-squared distribution with two degrees of freedom, \(P(\chi^2_2\geq5.20)=\exp(-5.20/2)\approx0.0743\).

Since 0.0743 is greater than 0.05, this test does not reject the common-effect model at the 5% level. It does not demonstrate that the effects are equal: the data may not distinguish moderate differences from sampling variation. The fitted group-specific slopes, their uncertainty, the amount of exposure overlap, and the clinical consequences of possible heterogeneity should also be considered before using a common effect for prediction.

Diagnostics, Alternatives, and Limits

A model comparison is useful only if the models are credible descriptions of the data. Examine event counts and exposure distributions separately by developmental stage. Sparse events can produce unstable estimates, very wide intervals, or complete separation, where a covariate pattern predicts the outcome perfectly and ordinary maximum-likelihood estimates may diverge. Little exposure overlap between groups also weakens the comparison: an apparent age difference may rely on extrapolating one group’s fitted curve into exposure values scarcely observed in that group.

Check whether the log-odds relationship is plausible across the observed exposure range. A curved exposure-response pattern can be mistaken for age interaction if the groups occupy different parts of the exposure range. Residual and calibration checks should be examined by age and exposure, not only in the pooled data. In a repeated-measures or recurrent-event setting, treating all records as independent Bernoulli observations understates uncertainty unless within-participant dependence is modeled. Event definitions, observation time, and ascertainment should also be comparable across ages.

The main alternative specifications answer different questions. A common-slope logistic model with age-specific intercepts is parsimonious and allows baseline risks to differ. An interaction model allows slope differences but uses more parameters. A prespecified nonlinear exposure function can represent curvature; combining that function with age interactions requires sufficient data. A model of absolute risk differences may be easier to communicate for some decisions, but it imposes a different scale of effect and can predict probabilities outside the 0-to-1 range unless constrained. Model choice should reflect the scientific question, data support, and intended prediction or decision—not merely which model gives a preferred p-value.

Finally, age categories simplify communication but can hide variation within a category, and arbitrary cut points can make results sensitive to grouping choices. The logistic model’s odds ratios are conditional on included covariates; they are not generally equal to marginal population risk ratios. Unmeasured differences in disease severity, concomitant treatment, monitoring, or dose selection can confound an observational exposure-safety relationship. A model can describe data well while still failing to identify a causal effect.

Common pitfall: “The exposure odds ratio is 2.01 in every age group, so the exposure doubles risk equally for every child.” This is incorrect twice: an odds ratio is not a risk ratio, and a common odds ratio does not imply a common absolute risk increase. Convert the odds ratio using each age group’s baseline risk, and state clearly whether the model estimates a shared odds effect or age-specific effects.

Check Your Understanding

Answer each question, then compare with the solutions below.

In a model with separate age-group intercepts and one shared exposure slope, what aspect of the safety relationship is shared, and what may still differ?
A group has baseline event risk 0.10 and a one-unit exposure odds ratio of 1.50. Calculate the event risk after the exposure increase and the absolute risk increase.
For four age groups, how many degrees of freedom does the likelihood-ratio comparison add when moving from a common slope to separate slopes?
A test of slope heterogeneity gives \(p=0.18\) at \(\alpha=0.05\). What conclusion is justified, and what conclusion is not?
Give two reasons an apparent age-specific exposure effect might be unreliable even if an interaction model fits better.

Solutions. (1) The log-odds exposure slope, and therefore the per-unit odds ratio, is shared; baseline event risks can differ because intercepts are age-specific. (2) Baseline odds are \(0.10/0.90=0.1111\). Exposed odds are \(0.1111\times1.50=0.1667\), giving risk \(0.1667/(1+0.1667)=0.1429\). The increase is \(0.1429-0.10=0.0429\), or 4.29 percentage points. (3) The interaction model adds \(4-1=3\) slopes, so the comparison has 3 degrees of freedom. (4) Do not reject the common-slope hypothesis at the 5% level; do not claim that slopes are proven equal. (5) Examples include sparse events, poor exposure overlap, different exposure ranges coupled with a nonlinear relationship, or unmodeled confounding and outcome-ascertainment differences.