Worked Multicenter Treatment Comparison

Why the Same Treatment Can Have Two Different Effects

In a multicenter study, treatment effects can be described for patients at the same underlying center risk or averaged across a population of centers. These are different questions. A conditional effect compares treatment groups while holding the center effect fixed; a population-averaged effect compares response probabilities after averaging over center heterogeneity. For a binary outcome with a logit link, those effects need not have the same numerical value, even when the model has one common treatment coefficient.

The calculations below use hypothetical treatment studies. The introductory material on conditional and marginal estimands provides the broader distinction; here the emphasis is on working through a multicenter comparison and tracing how variation among centers affects its interpretation.

Definition: Let $Y_{ij}$ be a binary outcome for patient $i$ in center $j$, and let $x_{ij}$ equal 1 for treatment and 0 for control. A random-intercept logistic GLMM specifies $$ \operatorname{logit}\{P(Y_{ij}=1\mid x_{ij},b_j)\} =\beta_0+\beta_1x_{ij}+b_j,\qquad b_j\sim N(0,\tau^2). $$ Here $b_j$ is the center-specific deviation from the overall intercept, $\tau^2$ is the between-center variance, and $\operatorname{logit}(p)=\log\{p/(1-p)\}$. The conditional treatment odds ratio at any fixed center effect is $\exp(\beta_1)$. The population-averaged response probability at treatment value $x$ is $E_{b}\{\operatorname{expit}(\beta_0+\beta_1x+b)\}$, where $\operatorname{expit}(u)=1/(1+e^{-u})$.

The model assumes that, conditional on the treatment and center effect, each patient’s outcome follows the stated Bernoulli mean model. It also assumes that center effects follow the specified normal distribution and are independent of treatment assignment. The common-slope form assumes the treatment coefficient does not vary across centers. In a randomized study, treatment should be represented in centers with suitable treatment overlap; otherwise, the comparison may depend heavily on model assumptions. The random-intercept model describes baseline differences among centers, not every possible source of heterogeneity.

From a Conditional Coefficient to an Averaged Effect

For a fixed center effect $b_j$, the control and treatment response probabilities are $\operatorname{expit}(\beta_0+b_j)$ and $\operatorname{expit}(\beta_0+\beta_1+b_j)$. Their odds ratio is $\exp(\beta_1)$, regardless of the value of $b_j$. To obtain population-averaged probabilities, however, one averages the probabilities over the distribution of center effects. In general, averaging the probabilities is not the same as setting the random effect to its mean, which is zero.

Worked Example: Integrating Over Center Effects

Suppose a hypothetical fitted model has $\beta_0=-1.20$, $\beta_1=0.80$, and $\tau=0.90$. Treat these estimates as fixed inputs for illustration; this example calculates effects, not their standard errors or uncertainty intervals.

Conditional effect. At any fixed center effect, the treatment odds ratio is $\exp(0.80)=2.226$. At the mean center effect $b=0$, the control response probability is $\operatorname{expit}(-1.20)=0.2315$, and the treatment response probability is $\operatorname{expit}(-0.40)=0.4013$. The conditional risk difference at this particular center effect is $0.4013-0.2315=0.1698$, or about 17.0 percentage points.

Population-averaged probabilities. For a numerical approximation to the normal integral, use the five-point standard-normal quadrature nodes $z=0,\ \pm1.3556,\ \pm2.8570$, with weights $0.53333,\ 0.22208$ for each of the two middle nodes, and $0.01126$ for each of the two outer nodes. Convert each node to a center effect using $b=0.90z$, then calculate the two probabilities at that effect:

\bar p_x \approx \sum_k w_k\operatorname{expit}(\beta_0+\beta_1x+0.90z_k).

The resulting values are:

Node $z_k$	Weight $w_k$	Control probability	Treatment probability
-2.8570	0.01126	0.0225	0.0488
-1.3556	0.22208	0.0816	0.1651
0	0.53333	0.2315	0.4013
1.3556	0.22208	0.5050	0.6942
2.8570	0.01126	0.7975	0.8977

For control, the weighted sum is approximately $0.01126(0.0225)+0.22208(0.0816)+0.53333(0.2315)+0.22208(0.5050)+0.01126(0.7975)=0.2630$. For treatment, it is approximately $0.01126(0.0488)+0.22208(0.1651)+0.53333(0.4013)+0.22208(0.6942)+0.01126(0.8977)=0.4155$. Thus the population-averaged risk difference is $0.4155-0.2630=0.1525$, or 15.3 percentage points. The corresponding marginal odds ratio is $(0.4155/0.5845)/(0.2630/0.7370)\approx1.992$. It is smaller than the conditional odds ratio of 2.226. These are effects for the model’s assumed center distribution, not effects at a single “average” center.

Why Logistic Averaging Changes the Odds Ratio

The difference is a consequence of the nonlinear logit link, not necessarily confounding or a treatment effect that varies by center. The following argument shows why, with a common positive conditional treatment effect and the same center-effect distribution in both groups, the marginal odds ratio is attenuated relative to the conditional odds ratio.

Justification: Write $R=\exp(\beta_1)>1$ for the conditional odds ratio and $p$ for the control probability at a particular center. The treatment probability at that center is $f(p)=Rp/\{1+(R-1)p\}$. Its second derivative is $$ f''(p)=\frac{-2R(R-1)}{\{1+(R-1)p\}^3}<0. $$ Thus $f$ is concave. Averaging over centers gives $E\{f(p)\}\leq f\{E(p)\}$. Since $f$ is the probability transformation associated with multiplying odds by $R$, this inequality implies that the marginal treatment odds are no more than $R$ times the marginal control odds. The marginal odds ratio is therefore no greater than the conditional odds ratio; with genuine variation in $p$, the inequality is strict. This argument relies on the common conditional odds ratio and the same center distribution in both treatment groups.

For a protective treatment, the analogous comparison is toward 1: averaging generally makes the marginal odds ratio less extreme than the conditional odds ratio. This is a feature of odds ratios, which are non-collapsible. It does not imply bias in either estimand. A conditional GLMM coefficient and a population-averaged coefficient answer different questions. In contrast, for an identity-link model, averaging a common treatment difference preserves that difference, provided the model and averaging population are appropriate.

Worked Center-Level Comparisons

Worked Example: Comparing Within-Center and Pooled Results

Imagine a hypothetical randomized study with 100 patients in each treatment group at each of three centers. The outcome is response by a specified follow-up time.

Center	Control responses	Treatment responses	Control rate	Treatment rate
Low baseline response	10/100	18/100	0.10	0.18
Middle baseline response	30/100	45/100	0.30	0.45
High baseline response	60/100	75/100	0.60	0.75

The center-specific estimated odds ratios are, respectively, $(18/82)/(10/90)=1.976$, $(45/55)/(30/70)=1.909$, and $(75/25)/(60/40)=2.000$. They are close to one another, as might be expected under a common conditional treatment effect, though sampling variation can make them differ.

Pooling the counts gives 100/300 responses under control and 138/300 under treatment. The pooled risk difference is $138/300-100/300=0.1267$, or 12.7 percentage points. The pooled odds ratio is $(138/162)/(100/200)=1.704$. It is lower than the center-specific odds ratios, despite the similar within-center contrasts. The centers have substantially different baseline response rates, and the pooled odds ratio is a marginal comparison across that mixture of risks. The pooled statistic should not be interpreted as the common conditional odds ratio.

Worked Example: Changing the Target Mix of Centers

Use the three center-specific risks in the table above to compare two target populations. First give the three centers equal weight. Then suppose a target population places 10% of patients in the low-response center, 30% in the middle-response center, and 60% in the high-response center.

With equal center weights, the averaged control response probability is $(0.10+0.30+0.60)/3=0.3333$, and the averaged treatment response probability is $(0.18+0.45+0.75)/3=0.4600$. The standardized risk difference is $0.4600-0.3333=0.1267$, or 12.7 percentage points.

For the stated patient-weighted target, the control probability is $0.10(0.10)+0.30(0.30)+0.60(0.60)=0.4600$. The treatment probability is $0.10(0.18)+0.30(0.45)+0.60(0.75)=0.6030$. The risk difference is $0.6030-0.4600=0.1430$, or 14.3 percentage points. The difference from the equal-center result arises because the target population gives more weight to the high-response center. A population-averaged effect is therefore incomplete unless its averaging population or center weights are specified.

Choosing and Checking the Comparison

A GLMM is useful when the scientific question concerns treatment effects conditional on center propensity, or when describing center-to-center variation is important. A GEE instead models a population-averaged mean directly. For a binary outcome, its treatment coefficient is marginal on the scale specified by its link. These approaches should not be compared as if they were competing estimates of one universal coefficient: the target estimands differ. If the scientific question is explicitly about marginal probabilities for a defined center population, report standardized probabilities or a marginal effect for that population, rather than interpreting the GLMM coefficient as if it were already marginal.

Before interpreting either result, examine whether the assumptions used for the comparison are plausible. In particular, consider whether:

Both treatment groups are represented within the centers that contribute to the contrast, and treatment allocation is not confounded with center.
A common conditional treatment slope is defensible. Marked differences in center-specific treatment estimates may signal treatment-effect heterogeneity, but can also reflect small center samples and sampling noise.
The normal random-intercept distribution and fitted model give a reasonable account of center-level baseline variation. Unusual centers or systematic residual patterns can make a simple random-intercept model inadequate.
The number of centers and the information within them support estimation of between-center variation. Few centers or sparse outcomes can make that estimate unstable, and convergence warnings require investigation rather than automatic acceptance.
The target population used for marginal predictions is explicit. An equal-center average and a patient-weighted average can answer different questions.

A common mistake is to say, “The GLMM treatment odds ratio is the treatment effect for the whole study population because the model includes all centers.” That reasoning confuses conditional and marginal effects. The correction is to state the conditioning or averaging explicitly: $\exp(\beta_1)$ is the conditional odds ratio at a fixed center effect under the model, while a population-averaged odds ratio must be calculated from averaged probabilities for a specified center distribution. Neither number is automatically preferable; the right one depends on the scientific question.

Key takeaway: Center heterogeneity can change a treatment effect’s interpretation in two ways: logistic averaging generally makes the marginal odds ratio closer to 1 than the conditional odds ratio, and the marginal response probabilities depend on which centers—and what weights—define the target population.

Check Your Understanding

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

A logistic random-intercept model has $\beta_1=\log(1.5)$. What is the conditional treatment odds ratio, and what does it compare?
Why is $\operatorname{expit}(\beta_0+\beta_1)$ generally not the population-averaged treatment probability?
In a hypothetical study, the equal-center response probabilities are 0.20 under control and 0.32 under treatment. What is the equal-center risk difference in percentage points?
Does a smaller marginal odds ratio than conditional odds ratio, by itself, show that treatment was confounded with center? Explain.

Solutions. (1) The conditional odds ratio is $\exp\{\log(1.5)\}=1.5$. It compares treatment with control at the same fixed center effect under the common-slope model. (2) That expression sets $b_j=0$; the population-averaged probability instead averages $\operatorname{expit}(\beta_0+\beta_1+b)$ over the center-effect distribution. Nonlinearity means the two calculations generally differ. (3) The risk difference is $0.32-0.20=0.12$, or 12 percentage points. (4) No. With a logistic link, non-collapsibility can make the marginal odds ratio closer to 1 even without confounding, provided the stated common-effect and common-center-distribution assumptions hold. Confounding is a separate design or modeling concern.