Centering And Scaling Covariates In Basket Models

Why the Covariate Scale Matters

In a basket trial, patient-level covariates can improve adjustment for baseline risk, but their units determine what model coefficients mean. If age is entered in years, a coefficient describes a one-year change in log odds. If age is centered at a clinically meaningful value and scaled in decades, the coefficient instead describes a ten-year change, and the intercept refers to a patient at the chosen age. These changes can make parameters easier to interpret and often put them on more manageable numerical scales.

The transformation does not, by itself, change the information in the data or the fitted probabilities. It does change the parameterization. In a Bayesian model, that distinction matters: using the same-looking priors after changing scales will generally specify a different prior model.

Model and Definitions

Let \(Y_{ij}\) be a binary response for patient \(i\) in basket \(j\), \(T_{ij}\) indicate treatment assignment, and \(x_{ij}\) be a continuous patient-level covariate. For illustration, consider the logistic model

Here \(\alpha_j\) is basket \(j\)'s control log odds at \(x_{ij}=0\), \(\theta_j\) is its treatment log odds ratio, and \(\gamma\) is the common covariate coefficient. A hierarchical specification might model \(\theta_j\) as arising from a distribution shared across baskets; the covariate transformation discussed here does not change that hierarchy in this no-interaction model.

This formulation assumes conditionally independent Bernoulli outcomes given the model parameters and covariates, a suitable logistic mean structure, and a common covariate slope across baskets. It also assumes no treatment-by-covariate interaction. The last condition is important: without an interaction, the treatment log odds ratio \(\theta_j\) is constant across values of \(x\). The model is an example, not a universal prescription; a continuous endpoint or a different link would require the corresponding likelihood and interpretation.

How Coefficients Change

Substitute \(x_{ij}=c+s z_{ij}\) into the raw-scale model. Collecting the constant and covariate terms gives

Thus the transformed model has intercept \(a_j=\alpha_j+\gamma c\) and covariate coefficient \(b=\gamma s\). The inverse transformation is \(\alpha_j=a_j-bc/s\) and \(\gamma=b/s\). These relations establish that the linear predictor, and hence each fitted probability, can remain exactly the same under the change of coordinates.

The transformed intercept \(a_j\) is the control log odds for a patient with \(x=c\), not for a patient with \(x=0\). The coefficient \(b\) is the change in control log odds for an increase of \(s\) units in the original covariate. If \(c\) is the pooled mean age and \(s\) is 10 years, for example, \(a_j\) refers to a patient at the pooled mean age and \(b\) describes a 10-year increase.

Centering and Treatment Effects

With no treatment-by-covariate interaction, centering a covariate changes the interpretation of the intercept but not the treatment contrast. In the model above, the treatment log odds ratio in basket \(j\) is \(\theta_j\) at every covariate value. Adjusting for a prognostic covariate may change its posterior estimate through the likelihood, but re-expressing that covariate alone does not make \(\theta_j\) conditional on a particular age.

If effect modification is scientifically relevant, include an interaction explicitly. For example,

Here \(\theta_j\) is the treatment log odds ratio at \(x=c\), and \(\delta\) is the change in that log odds ratio for each \(s\)-unit increase in \(x\). Centering therefore gives the treatment effect a reference value that can be stated and defended. The interaction is not created by centering; it must be part of the model if treatment effects are allowed to vary with the covariate.

Choosing the Center and Scale

A common center across baskets is usually useful when intercepts or treatment effects are to be compared. A pooled mean, a clinically meaningful threshold, or a prespecified reference value can serve, provided the choice is documented. If each basket is centered at its own mean, each intercept refers to a different covariate profile. Those intercepts are not directly comparable as baseline risks without translating them to a common reference.

Centering at the pooled mean is not automatically best. If the mean lies outside a clinically relevant range for a basket, its intercept may describe a patient unlike those in that basket. A meaningful reference value can be preferable. Scaling by a standard deviation makes coefficient magnitudes comparable across covariates, while scaling by a clinically meaningful increment may yield more direct interpretation. Scaling does not require the covariate to be normally distributed.

For a binary covariate, centering at zero or one can make the intercept refer to a clearly defined group. Standardizing a binary variable is possible, but may make the coefficient less clinically transparent. The choice should follow the intended interpretation and numerical needs rather than a rule that every covariate must be standardized.

Justification, Checks, and Limitations

The algebraic substitution shows that centering and scaling preserve the likelihood when the parameter transformation is one-to-one and the coefficients are transformed consistently. In Bayesian analysis, preserving the full model also requires transforming the prior distribution consistently. Equivalently, one may define priors directly on the new parameters, but then those priors express the intended beliefs on the new scale rather than automatically inheriting the old ones. If transforming a density explicitly, its change-of-variables Jacobian must also be included.

Before fitting, check that the chosen \(c\) is a meaningful reference and that \(s>0\) is stated in the original units. Confirm that the resulting \(z\) values are of a numerically reasonable magnitude and that the model's implied probabilities are plausible over the observed covariate range. Prior predictive simulation is useful for detecting priors that imply near-certain response probabilities or implausibly steep covariate effects. After fitting, inspect whether the covariate has enough variation within baskets to support the intended adjustment; a common slope cannot be learned well from little within-basket variation.

Centering does not correct a misspecified functional form, omitted interactions, measurement error, or lack of covariate overlap between treatment groups. Scaling does not make a coefficient clinically important, and it cannot create information where covariate values are sparse. If the model includes nonlinear terms or interactions, transform and interpret those terms explicitly; changing the main-effect reference can alter their coefficient meanings.

A common mistake is to say that centering changes only notation, then retain all raw-scale priors unchanged as independent priors on the new coefficients. The likelihood may be algebraically equivalent, but the prior distribution is not, as the prior example demonstrates. Another mistake is to interpret a centered intercept as the risk for an average patient regardless of the model. In a logistic model, a prediction at mean covariates is not generally the same as the average predicted risk across patients. State the reference covariate value and treatment condition precisely.

Using the raw covariate scale keeps coefficients in familiar units but can leave an intercept at an irrelevant value and create awkward coefficient magnitudes. Centering alone improves the intercept reference while retaining the original slope units. Scaling alone changes the slope units but leaves the intercept tied to \(x=0\). Centering and scaling together can improve both interpretability and computation, but only when the chosen reference, scale, and prior implications are made explicit.