Separating Baseline Risk And Treatment Effect Variation

Why Separate Baseline Risk From Treatment Effect?

Basket trials study a treatment across subgroups that may differ before treatment begins. A subgroup with a high response rate under control can also have a high response rate under treatment, even if its treatment effect is modest. Another subgroup can start with a low control response rate but have a larger treatment effect. A model that treats treated-arm outcomes as if they directly measured treatment effects can confuse these patterns and borrow information in the wrong direction.

The key modeling distinction is between each basket’s baseline risk, defined here as its response probability under control, and its treatment effect, defined on the log-odds scale as the change from control to treatment. The logistic random-intercept and random-slope models introduced earlier provide the ingredients; the essential task is to give the two components distinct subgroup-level distributions.

Definition: Let $j=1,\ldots,J$ index baskets and $i$ index patients. Let $A_{ij}=0$ denote control and $A_{ij}=1$ denote treatment, and let $Y_{ij}$ be a binary response. A separate baseline-and-treatment model is $$ Y_{ij}\mid A_{ij},\alpha_j,\beta_j \sim \operatorname{Bernoulli}(p_{ij}), \qquad \operatorname{logit}(p_{ij})=\alpha_j+\beta_j A_{ij}. $$ Here $\alpha_j$ is basket $j$'s control-arm log odds of response, while $\beta_j$ is its treatment log odds ratio. Thus $p_{0j}=\operatorname{logit}^{-1}(\alpha_j)$, and the treated-arm log odds are $\alpha_j+\beta_j$.

A flexible hierarchical specification allows each component to vary across baskets:

\begin{pmatrix}\alpha_j\\ \beta_j\end{pmatrix} \mid \mu_\alpha,\mu_\beta,\tau_\alpha,\tau_\beta,\rho \sim N_2\!\left[ \begin{pmatrix}\mu_\alpha\\ \mu_\beta\end{pmatrix}, \begin{pmatrix} \tau_\alpha^2 & \rho\tau_\alpha\tau_\beta\\ \rho\tau_\alpha\tau_\beta & \tau_\beta^2 \end{pmatrix} \right].

The parameter $\tau_\alpha$ describes between-basket variation in baseline log odds, and $\tau_\beta$ describes between-basket variation in treatment log odds ratios. The correlation $\rho$ allows those two basket-level quantities to be associated. Setting $\rho=0$ is a modeling assumption of zero correlation, not a requirement for separating baseline and treatment effects. Hyperpriors must be specified for the means, scales, and any correlation parameter; their choices should give finite, clinically plausible ranges for the subgroup quantities.

This formulation assumes a suitable binary-outcome likelihood, a common treatment coding and outcome definition across baskets, and a hierarchical distribution that adequately represents the baskets as exchangeable conditional on its hyperparameters. For a causal interpretation of $\beta_j$, treatment assignment must support a valid within-basket comparison, such as randomization, and the treatment and control groups must be defined consistently. If treatment is confounded with patient characteristics, this simple model does not by itself remove that confounding.

What the Model Separates—and What It Does Not

For a given basket, control-arm data primarily inform $\alpha_j$, while treatment-arm data inform the sum $\alpha_j+\beta_j$. Treatment-versus-control comparisons therefore inform $\beta_j$, but uncertainty in the baseline and effect can remain dependent. A hierarchical model does not make the two estimates statistically independent; it gives them distinct meanings and separate between-basket variation parameters.

A short variance calculation shows why that distinction matters. The treated-arm logit is the sum of the basket’s baseline log odds and its treatment effect. Therefore, across baskets, its variance includes both sources of variation and their covariance:

\begin{aligned} \operatorname{Var}(\alpha_j+\beta_j) &=\operatorname{Var}(\alpha_j)+\operatorname{Var}(\beta_j) +2\operatorname{Cov}(\alpha_j,\beta_j)\\ &=\tau_\alpha^2+\tau_\beta^2+2\rho\tau_\alpha\tau_\beta. \end{aligned}

Thus, the spread of treated-arm logits is not generally the spread of treatment effects. If baseline risk varies substantially, a model that attributes all treated-arm differences to treatment effects can exaggerate effect heterogeneity. If baseline and treatment effects are negatively associated, their contributions can partly cancel; the treated-arm logits may then look similar even when each component varies appreciably. This identity is a statement about the model’s logit-scale random quantities, not a claim that raw observed proportions have exactly this variance.

Worked Example: Two Baskets With Different Baselines

Suppose a hypothetical randomized basket trial observes 100 control and 100 treated patients in each of two baskets. Basket A has 10 responses among controls and 25 among treated patients; Basket B has 50 control responses and 65 treated responses. Compute the plug-in baseline logits and treatment log odds ratios from the observed proportions, then compare those effects with the treated-arm logits.

For Basket A, the observed control and treated response proportions are $0.10$ and $0.25$. The estimated baseline log odds and treatment log odds ratio are $$ \hat{\alpha}_A=\log\!\left(\frac{0.10}{0.90}\right)=-2.197,\qquad \hat{\beta}_A=\log\!\left(\frac{0.25}{0.75}\right)-\hat{\alpha}_A =-1.099-(-2.197)=1.099. $$

For Basket B, the corresponding proportions are $0.50$ and $0.65$: $$ \hat{\alpha}_B=\log\!\left(\frac{0.50}{0.50}\right)=0,\qquad \hat{\beta}_B=\log\!\left(\frac{0.65}{0.35}\right)-0 =0.619. $$

The estimated treatment effects differ by $1.099-0.619=0.480$ log-odds units. But the treated-arm logits are $-1.099$ in Basket A and $0.619$ in Basket B. Their difference, A minus B, is $-1.099-0.619=-1.718$, not $0.480$. The identity makes the reason explicit: $$ (\hat{\alpha}_A+\hat{\beta}_A)-(\hat{\alpha}_B+\hat{\beta}_B) =(-2.197225-0)+(1.098612-0.619039)=-1.717652\approx -1.718. $$

The treatment-arm contrast combines a baseline-logit difference of $-2.197$ with an estimated treatment-effect difference of $0.480$. Its direction and magnitude are consequently not a measure of which basket has the larger treatment effect. These are plug-in estimates from hypothetical data; their sampling uncertainty would also need to be represented in inference.

Baseline Risk Also Changes Absolute Benefit

Separating the parameters is important even when the treatment effect is constant on the log-odds scale. A common $\beta$ does not imply the same absolute increase in response probability in every basket. If the control risk is $p_{0j}$ and the treatment odds ratio is $\exp(\beta)$, the treated risk is

p_{1j}= \frac{\exp(\beta)p_{0j}} {1-p_{0j}+\exp(\beta)p_{0j}}.

This follows by multiplying the control odds $p_{0j}/(1-p_{0j})$ by $\exp(\beta)$, then converting the resulting odds back to a probability. It is not a contradiction for baskets to share a log-odds effect but differ in risk differences: the risk difference is $p_{1j}-p_{0j}$, which depends on the baseline risk.

Worked Example: One Odds Ratio, Two Risk Differences

Suppose the treatment odds ratio is 2 in two hypothetical baskets. In Basket C, the control response risk is $0.10$; in Basket D, it is $0.50$. Calculate the treated risks and risk differences.

The common log-odds effect is $\beta=\log(2)=0.6931$. For Basket C, doubling control odds $0.10/0.90=0.1111$ gives treated odds $0.2222$, hence $$ p_{1C}=\frac{2(0.10)}{0.90+2(0.10)} =\frac{0.20}{1.10}=0.1818,\qquad p_{1C}-p_{0C}=0.1818-0.10=0.0818. $$

For Basket D, control odds are $0.50/0.50=1$, so treated odds are 2: $$ p_{1D}=\frac{2(0.50)}{0.50+2(0.50)} =\frac{1}{1.50}=0.6667,\qquad p_{1D}-p_{0D}=0.6667-0.50=0.1667. $$

Both baskets have the same treatment odds ratio, but the estimated absolute increase is about 8.18 percentage points in Basket C and 16.67 percentage points in Basket D. The log-odds treatment effect and the risk difference answer different questions; neither should be substituted for the other without specifying the target measure.

How Variation and Dependence Affect Borrowing

In a separate hierarchy, evidence about a basket’s control response contributes information about baseline risk, and contrasts between its treatment and control arms contribute information about its treatment effect. The shared hyperparameters allow borrowing among baskets in each component. A basket with a weakly estimated effect can be pulled toward the distribution of effects, while its baseline is informed by the distribution of baselines. The amount of borrowing depends on the data and the estimated between-basket variation; it is not automatically the same for both components.

The components need not be independent. If $\rho$ is not zero, evidence about a basket’s baseline can also be relevant to its effect through the modeled association, and the joint posterior can connect parameters across baskets. That association should be interpreted as a population-level modeling feature, not as proof that changing baseline risk causes a change in treatment effect.

Worked Example: Quantifying the Treated-Logit Variation

Consider a hypothetical hierarchical model with baseline-logit standard deviation $\tau_\alpha=0.8$, treatment-effect standard deviation $\tau_\beta=0.4$, and correlation $\rho=0.5$. Calculate the between-basket variance of the treated-arm logit $\alpha_j+\beta_j$, and compare it with the treatment-effect variance.

The covariance is $$ \operatorname{Cov}(\alpha_j,\beta_j) =\rho\tau_\alpha\tau_\beta =0.5(0.8)(0.4)=0.16. $$

Consequently, $$ \operatorname{Var}(\alpha_j+\beta_j) =0.8^2+0.4^2+2(0.16) =0.64+0.16+0.32=1.12. $$ The treatment-effect variance alone is $\tau_\beta^2=0.16$. The treated-arm logit standard deviation is $\sqrt{1.12}=1.058$, whereas the treatment-effect standard deviation is $0.4$. Assigning the treated-logit spread to treatment effects alone would therefore substantially overstate treatment-effect variation in this example. The calculation concerns the hierarchical variation, not a standard error for any observed basket proportion.

Choosing and Checking a Model

Three broad strategies clarify what is being assumed. A common-effect model sets $\beta_j=\beta$, allowing baseline risks to differ but assuming no treatment-effect variation. A separate random-effects model gives baselines and effects their own distributions, with independence or an estimated association as specified. An independent-basket analysis avoids hierarchical borrowing by fitting each basket separately, often at the cost of less precise basket-specific estimates. These approaches are not interchangeable: they encode different assumptions about which quantities may be shared.

Approach	Baseline risks	Treatment effects	Main implication
Common effect	May vary by basket	One shared effect	Borrowing is strong for effects because heterogeneity is excluded
Separate hierarchical components	Vary with $\tau_\alpha$	Vary with $\tau_\beta$; association may be modeled	Borrowing can differ for baseline and effect
Independent basket analyses	Estimated separately	Estimated separately	No cross-basket borrowing; estimates may be imprecise

A practical check is to ask whether the model’s reported subgroup effect is actually the treatment-versus-control contrast, rather than a treated-arm response or logit. Examine the implied baseline and treatment ranges for clinical plausibility, and check whether the data contain within-basket treatment and control information sufficient to distinguish the two. Sparse data can leave $\alpha_j$ and $\beta_j$ strongly dependent; a stable-looking treated response does not establish that both components are well determined. If a basket has no relevant control information, the baseline and treatment effect may be especially difficult to separate without substantive assumptions.

The hierarchical distribution is also a limitation. A normal distribution on the logit scale may be unsuitable if baskets form distinct groups or if effects have a shape poorly represented by that distribution. Assuming $\rho=0$ can be restrictive when baseline and treatment effects plausibly co-vary, while estimating $\rho$ can be weakly supported when there are few baskets. These uncertainties should be reflected in modeling choices and sensitivity analysis; a precise posterior under one specification does not establish that its borrowing assumptions are correct.

Common Mistake

Common mistake: “Basket B has a larger treated response rate, so it must have a larger treatment effect.” This compares outcomes under treatment without accounting for the control risk. The correction is to define the basket-specific effect as the treatment-versus-control contrast on the chosen scale. In the logistic model here, it is $\beta_j=\operatorname{logit}(p_{1j})-\operatorname{logit}(p_{0j})$, not $\operatorname{logit}(p_{1j})$ alone.

A related mistake is to use a single between-basket variance for every quantity. Baseline log odds, treatment log odds ratios, and treated-arm logits are distinct random quantities. Their variation is related by the model, but the variance identity shows that it cannot be assigned to one component without justification.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions below.

A basket has control response proportion $0.20$ and treated response proportion $0.40$. Calculate its plug-in baseline log odds and treatment log odds ratio.
If the treatment odds ratio is $1.5$ and the control risk is $0.20$, calculate the treated risk and risk difference.
Suppose $\tau_\alpha^2=0.36$, $\tau_\beta^2=0.25$, and $\rho=-0.4$. Find $\operatorname{Var}(\alpha_j+\beta_j)$.
Why is the difference between two baskets’ treated-arm logits not generally their treatment-effect difference?

Solutions. For question 1, $\hat{\alpha}=\log(0.20/0.80)=\log(0.25)=-1.386$. The treated logit is $\log(0.40/0.60)=-0.4055$, so $\hat{\beta}=-0.4055-(-1.386)=0.9808$. For question 2, treated odds are $1.5(0.20/0.80)=0.375$, giving $p_1=0.375/(1+0.375)=0.2727$. The risk difference is $0.2727-0.20=0.0727$, or 7.27 percentage points. For question 3, $\rho\tau_\alpha\tau_\beta=-0.4\sqrt{0.36}\sqrt{0.25}=-0.4(0.6)(0.5)=-0.12$, so the variance is $0.36+0.25+2(-0.12)=0.37$. For question 4, each treated-arm logit is $\alpha_j+\beta_j$, so a contrast between baskets combines their baseline-logit contrast and their treatment-effect contrast. It equals the treatment-effect contrast only if the baseline logits are equal.