Highlights

1. 1. What is already known

   When using non-collapsible measures of treatment effects, such as odds ratios or hazard ratios, marginal and conditional estimands have different interpretations and will not generally coincide, even in the absence of effect modification. The presence of effect modification means that there may not be a single most effective treatment for all individuals or subgroups in the population.

2. 2. What is new

   We argue that population-average conditional and marginal estimands both quantify average effectiveness over a population but correspond to different decision questions, either to maximise the average effect for individuals in the population, or to minimise (or maximise) average event probabilities respectively. When effect modification is present, we show that these are no longer aligned and can result in conflicting treatment rankings. In such cases, making a single treatment decision can result in choosing an inferior treatment for the majority of individuals, or one with a worse expected number of events overall.

3. 3. Potential impact

   We provide recommendations for decision-making in the presence of effect modification, for decisions based both purely on effectiveness and on cost-effectiveness. ML-NMR is at present the only population adjustment method that can produce the necessary estimates in any target population of interest. Where allowable, making decisions by subgroups may result in patients being given a more effective treatment for them and result in greater cost-effectiveness overall.

2 Defining conditional and marginal estimands

Consider a binary outcome that occurs with event probability $\pi _{ik(P)}$ for an individual i receiving treatment $k = A, B, \dots $ in population P with covariates $\boldsymbol {x}_{ik(P)}$ , under the following model:

(1) $$ \begin{align} g(\pi_{ik(P)}) = \eta_{k(P)}(\boldsymbol{x}_{ik(P)}) = \mu_{(P)} + \boldsymbol{m}(\boldsymbol{x}_{ik(P)})^{\scriptscriptstyle\mathsf{T}}(\boldsymbol{\beta}_1 + \boldsymbol{\beta}_{2,k}) + \gamma_k \end{align} $$

where $g(\cdot )$ is a link function that transforms probabilities onto the linear predictor $\eta _{k(P)}(\cdot )$ , $\mu _{(P)}$ is the intercept (baseline risk), $\boldsymbol {m}(\cdot )$ is a function of the covariates, $\boldsymbol {\beta }_1$ are prognostic effects, $\boldsymbol {\beta }_{2,k}$ are effect-modifying interactions for treatment k, and $\gamma _k$ is the individual-level conditional treatment effect for an individual with $\boldsymbol {x} = \boldsymbol {0}$ . We set $\boldsymbol {\beta }_{2,A} = 0$ and $\gamma _A = 0$ .

Non-collapsibility depends on the choice of link function $g(\cdot )$ , and in particular on whether the function

(2)

is linear in $\pi $ .Reference Neuhaus and Jewell ^{12 ,} Reference Daniel, Zhang and Farewell ¹³ The function $h_{ab}(\cdot ,\cdot )$ , maps the event probabilities on one treatment a to event probabilities on another treatment b. Daniel et al.Reference Daniel, Zhang and Farewell ¹³ term $h_{ab}(\pi , \boldsymbol {x})$ the characteristic collapsibility function (CCF), and consider the case where there is no effect modification ( $\boldsymbol {\beta }_{2,k} = \boldsymbol {0}$ for all k) so the CCF no longer depends on the covariates $\boldsymbol {x}$ . When the CCF is not linear in $\pi $ , the corresponding effect measure is not collapsible; this is the case for example when $g(\cdot )$ is the logit or probit link function, which correspond to log odds ratios or probit differences. When the CCF is linear in $\pi $ , for example when $g(\cdot )$ is the identity or log link function, the corresponding effect measure (risk differences or log risk ratios, respectively) will be collapsible. However, non-collapsibility is a necessary consequence of probabilities being bounded between 0 and 1: modelling collapsible effect measures directly can result in predictions outside of this range, and induces purely mathematical treatment-covariate interactions to avoid impossible predictions.

There are several different potential estimands that may be of interest, and these do not typically coincide for non-collapsible effect measures such as log odds ratios, even in the absence of effect modification.

The individual-level conditional treatment effects between each pair of treatments b vs. a for an individual with covariates $\boldsymbol {x}$ are given by the difference in the linear predictors on each treatment:

(3) $$ \begin{align} \begin{aligned} \gamma_{ab}(\boldsymbol{x}) &= \eta_b(\boldsymbol{x}) - \eta_a(\boldsymbol{x}) \\ &= \gamma_b - \gamma_a + \boldsymbol{m}(\boldsymbol{x})^{\scriptscriptstyle\mathsf{T}}(\boldsymbol{\beta}_{2,b} - \boldsymbol{\beta}_{2,a}). \end{aligned} \end{align} $$

This estimand has an individual-specific interpretation, provided that relevant sources of subject-level heterogeneity are accounted for, and depends on the specific values of any effect modifiers for an individual. However, whilst these individual-level conditional treatment effects may be of interest to individual patients, these are not typically the focus for population-level decision making, which is instead concerned with population-average estimands. A related estimand is the individual-level conditional effect at the mean covariate values, $\gamma _{ab}(\bar {\boldsymbol {x}}_{(P)})$ , where $\bar {\boldsymbol {x}}_{(P)}$ is the mean of $\boldsymbol {x}$ in the population P. Whilst estimates of this estimand are sometimes reported, their interpretation is problematic, especially with discrete covariates since it is impossible for an individual to have the “average” value; the individual-level conditional effect at the mean is therefore not useful for decision-making.

Population-average conditional treatment effects between each pair of treatments a and b in population P are obtained by averaging the individual-level treatment effects over the covariate distribution in the population on the linear predictor scale:

(4) $$ \begin{align} \begin{aligned} d_{ab(P)} &= \int_{\mathfrak{X}} \gamma_{ab}(\boldsymbol{x}) \, f_{(P)}(\boldsymbol{x}) \mathop{}\!d\boldsymbol{x} \\ &= \int_{\mathfrak{X}} \bigl(\gamma_b - \gamma_a + \boldsymbol{m}(\boldsymbol{x})^{\scriptscriptstyle\mathsf{T}}(\boldsymbol{\beta}_{2,b} - \boldsymbol{\beta}_{2,a}) \bigr) \, f_{(P)}(\boldsymbol{x}) \mathop{}\!d\boldsymbol{x} \end{aligned} \end{align} $$

where $\mathfrak {X}$ is the support of $\boldsymbol {x}$ and $f_{(P)}(\boldsymbol {x})$ is the joint distribution of $\boldsymbol {x}$ in the population. The population-average conditional treatment effects $d_{ab(P)}$ can be interpreted as the average of the individual-level treatment effects in the population. Calculating (4) requires information on the distribution of effect-modifying covariates in the population P. In the common special case where $\boldsymbol {m}(\boldsymbol {x}) = \boldsymbol {x}$ , the covariate means $\bar {\boldsymbol {x}}_{(P)}$ are sufficient to calculate (4) since the linear predictor is linear in the covariates and the integral simplifies to $d_{ab(P)} = \gamma _b - \gamma _a + \bar {\boldsymbol {x}}^{\scriptscriptstyle \mathsf {T}}_{(P)} (\boldsymbol {\beta }_{2,b} - \boldsymbol {\beta }_{2,a})$ , where $\bar {\boldsymbol {x}}_{(P)}$ is the mean of $\boldsymbol {x}$ in the population P.

The individual-level conditional event probabilities on each treatment, for an individual with covariates $\boldsymbol {x}$ , are given by back-transforming the linear predictor onto the probability scale:

(5)

Again, as with the individual-level conditional treatment effects $\gamma _{ab(P)}$ , the individual-level conditional event probabilities are relevant to specific individuals and are not typically the focus for population-level decision making.

Population-average marginal treatment effects are obtained as a summary of the average event probabilities on each treatment:

(6) $$ \begin{align} \Delta_{ab(P)} = g(\bar{\pi}_{b(P)}) - g(\bar{\pi}_{a(P)}) \end{align} $$

where

(7) $$ \begin{align} \begin{aligned} \bar{\pi}_{k(P)} &= \int_{\mathfrak{X}} \pi_{k(P)}(\boldsymbol{x}) \, f_{(P)}(\boldsymbol{x}) \mathop{}\!d\boldsymbol{x} \\ &= \int_{\mathfrak{X}} g^{-1}(\eta_{k(P)}(\boldsymbol{x})) \, f_{(P)}(\boldsymbol{x}) \mathop{}\!d\boldsymbol{x} \end{aligned} \end{align} $$

is the average event probability on each treatment. The population-average marginal treatment effects $\Delta _{ab(P)}$ can be interpreted in terms of the effect of treatment on the average event probabilities in the population. Calculating (5), (6), and (7) requires information on the baseline risk $\mu _{(P)}$ in the population P (see Section 7 for practical considerations); (6) and (7) also require information on the joint covariate distribution $f_{(P)}(\boldsymbol {x})$ in the population P.

Here we have defined the estimands in terms of a generative outcome model. These estimands can also be defined in terms of potential outcomes without reference to any model; we give definitions in the potential outcomes framework in Appendix A. We note that, although the estimands can be defined in a “model-free” manner, all current population adjustment methods with limited IPD will impose an assumed outcome model of the form (1) in order to form an indirect comparison, either explicitly (e.g., STC, ML-NMR) or implicitly (e.g., MAIC).Reference Phillippo, Ades, Dias, Palmer, Abrams and Welton ¹⁶

The terms “population-average” and “marginal” are often used interchangeably, but here we make a conceptual distinction. Population-average refers to a quantity that has been averaged over the population, which may be conditional (like $d_{ab(P)}$ ) or marginal (like $\Delta _{ab(P)}$ ), whereas marginal refers to the scale on which this averaging has taken place (i.e., the probability scale, rather than the linear predictor scale for conditional quantities). Similarly, it is sometimes said that the population-average marginal effect is “the effect” of moving a population from one treatment to another. However, we clearly see that this intervention effect can be defined in multiple ways, depending on the scale on which the average is taken. The population-average marginal estimand $\Delta _{ab(P)}$ averages the counterfactual event probabilities on each treatment and compares them, whereas the population-average conditional estimand $d_{ab(P)}$ averages the individual counterfactual treatment effects.

For population-level decision-making, often the decision is made based on cost-effectiveness rather than purely effectiveness, such as in health technology assessment. In this case, the estimands above are not of direct interest, but are instead considered inputs to a cost-effectiveness model that evaluates the expected net benefit on each treatment. For now, we consider only effectiveness decisions based on $d_{ab(P)}$ or $\Delta _{ab(P)}$ , and we revisit cost-effectiveness decisions in Section 5.

3 Population adjustment methods

In an ideal scenario, IPD would be available from every study, in which case the “gold standard” approach is an IPD network meta-regression which accounts for differences in effect modifiers between studies and can produce population-average conditional or marginal treatment effect estimates in any population of interest (including external target populations) via equations (4) and (6).Reference Berlin, Santanna, Schmid, Szczech and Feldman ^{17 –} Reference Dias, Sutton, Welton and Ades ²⁰ However, this scenario is uncommon in many practical applications, for example in health technology assessment where a company making a submission to an agency such as the National Institute for Health and Care Excellence in England has IPD from their own study or studies, but only published aggregate data from their competitors’. Population adjustment methods are designed with this limited-IPD scenario in mind, and aim to use IPD available from a subset of studies to account for differences in the distribution of effect modifiers between studies.Reference Phillippo, Ades, Dias, Palmer, Abrams and Welton ^{16 ,} Reference Phillippo, Ades, Dias, Palmer, Abrams and Welton ²¹ Different population adjustment methods target different estimands, and produce estimates that are relevant to different target populations; these are summarised in Table 1.

Table 1

Evidence synthesis and population adjustment methods, and the estimands and target populations targeted by each

Note: Abbreviations: IPD, individual patient data; ML-NMR, multilevel network meta-regression; MAIC, matching-adjusted indirect comparison; STC, simulated treatment comparision; NMI, network meta-interpolation.

ML-NMR is a generalisation of the IPD network meta-regression framework to incorporate aggregate data, by integrating the individual-level model over each aggregate study population (as in equation (7)).Reference Phillippo, Dias and Ades ⁵ This approach avoids aggregation bias, unlike approaches to combining IPD and aggregate data in network meta-regression that simply “plug in” mean covariate values for the aggregate studies into the individual-level model.Reference Sutton, Kendrick and Coupland ^{22 –} Reference Donegan, Williamson, D’Alessandro, Garner and Tudur ²⁴ ML-NMR combines evidence at the level of the individual conditional treatment effects, and can be used to produce estimates of both conditional and marginal estimands following equations (4) and (6), in any target population of interest.Reference Phillippo, Dias and Ades ^{5 ,} Reference Phillippo, Dias, Ades and Welton ¹⁴ The marginalisation integrals (7) for each aggregate study are typically calculated using efficient quasi-Monte Carlo numerical integration, which can also be used to produce estimates for external target populations with a given covariate distribution.Reference Phillippo, Dias and Ades ⁵ The multinma R package implements ML-NMR models for a range of outcome types, as well as full-IPD and aggregate data only network meta-analysis as special cases, and provides functionality to produce estimates of all of the different estimands defined in Section 2.Reference Phillippo ²⁵

MAIC is a weighting approach, where the method of moments is used to estimate weights that match covariate means or higher order moments in an IPD study to those reported in an aggregate study.Reference Signorovitch, Wu and Yu ⁶ MAIC targets the population-average marginal estimand (6), but since no conditional regression model is fitted this bypasses the need to evaluate any marginalisation integral (equation (7)) entirely. MAIC can only produce estimates relevant to the aggregate study population in a two-study indirect comparison.Reference Phillippo, Ades, Dias, Palmer, Abrams and Welton ¹⁶

STC is a regression adjustment approach that fits a regression model in an IPD study and uses this to predict outcomes in an aggregate study population.Reference Caro and Ishak ⁷ The most common form of STC typically combines a conditional estimate from the IPD study, obtained by plugging-in mean covariate values to equation (3), with a marginal estimate as reported by the aggregate study, which are incompatible and is thus biased against both the population-average conditional (4) and marginal (6) estimands.Reference Phillippo, Dias, Ades and Welton ^{14 ,} Reference Remiro-Azócar, Heath and Baio ¹⁵ Plug-in means STC should therefore be avoided. Other forms of STC are available that avoid this problem and target the population-average marginal estimand via simulation (to evaluate equations (6) and (7)), however these incur additional sampling variation by trying to simulate a limited number of participants in the aggregate trial.Reference Caro and Ishak ⁷ All forms of STC can only produce estimates relevant to the aggregate study population in a two-study indirect comparison.Reference Phillippo, Ades, Dias, Palmer, Abrams and Welton ¹⁶

A more sophisticated form of STC based on G-computation addresses several of the issues with other forms of STC.Reference Remiro-Azócar, Heath and Baio ⁹ G-computation STC targets the population-average marginal estimand (6), fitting a regression model in an IPD study, and evaluating the marginalisation integral (7) over an aggregate study population using simulation (parametric G-computation). Uncertainty is fully quantified by implementing the approach in a Bayesian framework.Reference Remiro-Azócar, Heath and Baio ⁹ However, like other forms of STC, this approach can only produce estimates relevant to the aggregate study population in a two-study indirect comparison.Reference Remiro-Azócar, Heath and Baio ⁹ Similar simulation-based STC approaches have recently been published.Reference Zhang, Bujkiewicz and Jackson ^{26 ,} Reference Ren, Ren, Welton and Strong ²⁷

The key assumption made by all population adjustment methods in a connected network of comparisons (so-called anchored population-adjusted indirect comparisons) is conditional constancy of relative effects.Reference Phillippo, Ades, Dias, Palmer, Abrams and Welton ¹⁶ This requires all effect modifiers to be known and appropriately adjusted for, such that $\boldsymbol {x}$ includes all effect modifiers and their functional form $\boldsymbol {m}(\boldsymbol {x})$ in the outcome model (1) is correctly specified. For regression-based approaches like ML-NMR and STC, this model specification is explicit. For MAIC, the choice of covariate moments to use for matching implies the form of the underlying outcome model (1); for example, matching only on covariate means leads to $\boldsymbol {m}(\boldsymbol {x})$ being linear in $\boldsymbol {x}$ . In a disconnected network or with single-arm studies, unanchored population-adjusted indirect comparisons may be created, relying on the assumption of conditional constancy of absolute effects. This is a much stronger assumption, which requires producing accurate predictions of absolute outcomes accross populations, and is widely considered very difficult to meet.Reference Phillippo, Ades, Dias, Palmer, Abrams and Welton ¹⁶

Population adjustment analyses are often required to make simplifying assumptions for identifiability, most commonly invoking the shared effect modifier assumption, which states that effect modifier interactions are equal for a set of treatments; that is $\boldsymbol {\beta }_{2,k} = \boldsymbol {\beta }_{2,\mathcal {T}}$ for all treatments k in the set $\mathcal {T}$ . This assumption may be reasonable if treatments in $\mathcal {T}$ are from the same class and share a mode of action, otherwise this assumption is unlikely to hold; this needs to be considered on a case-by-case basis.Reference Phillippo, Ades, Dias, Palmer, Abrams and Welton ¹⁶ For MAIC and STC, with a continuous outcome and a linear outcome model, making the shared effect modifier assumption for treatments B and C means that the estimated relative treatment effect $\Delta _{BC(AC)}=\Delta _{BC}$ is constant across populations, and can be applied in any target population and not just the $AC$ trial population. For other outcomes and outcome models, however, this assumption is not sufficient to make the marginal effect $\Delta _{BC(AC)}$ transportable, as this marginal effect is specific to the distribution of covariates and baseline risk in the $AC$ population. For ML-NMR, the shared effect modifier assumption may be used in smaller networks to identify the model in the absence of sufficient data, for example in a two-study indirect comparison.Reference Phillippo, Dias and Ades ⁵ ML-NMR can use this assumption regardless of the outcome type or outcome model, since it is applied to the individual-level conditional outcome model. If this assumption does not hold and the model cannot be identified via other means (e.g., external information to inform prior distributions for the interactions or other structural assumptions about interactions), then ML-NMR is limited to producing estimates in the aggregate study population, like MAIC and STC. In even moderately-sized networks, ML-NMR may allow the shared effect modifier assumption to be assessed or removed entirely.Reference Phillippo, Dias and Ades ²⁸ We examine the shared effect modifier assumption further in Section 4.3.

Network meta-interpolation (NMI) is different to other regression-based approaches; whilst an outcome regression model is defined, this model is not estimated directly. Instead, published subgroup analyses from each trial are “interpolated” to a specific target population by solving equations based on the best linear unbiased predictor, and then these adjusted estimates are combined in a standard NMA.Reference Harari, Soltanifar and Cappelleri ²⁹ The motivation of NMI is to produce population-adjusted estimates without making the shared effect modifier assumption, by using additional information in the form of subgroup analyses from all studies.Reference Phillippo, Dias and Ades ^{5 ,} Reference Phillippo, Ades, Dias, Palmer, Abrams and Welton ¹⁶ However, NMI incurs similar biases to plug-in means STC, as the reported study-level treatment effect estimates in trial publications are unlikely to be compatible with the conditional treatment effect estimates required by NMI to perform interpolation. NMI therefore typically mixes incompatible estimates within the model (within studies, as opposed to across studies for plug-in means STC), and is thus biased against both the conditional and marginal estimands. Furthermore, since the outcome regression model is not fully estimated, it is not possible for NMI to produce population-average marginal treatment effects or absolute predictions (e.g., average event probabilities). In certain specific scenarios (i.e., binary covariates and linear outcome models) NMI does appear promising; however, the properties and performance of NMI are not yet fully understood, and simulation studies have not directly investigated bias or adequacy of variance estimation.

Standard network meta-analysis, pairwise meta-analysis, and indirect comparison methods combine aggregate data from each study without adjusting for covariates, targeting a population-average marginal estimand. The constancy of relative effects assumption is required, meaning that there is (on average) no imbalance in effect modifiers between populations. Balanace in baseline risk and prognostic factors is also required unless the outcome is continuous and the outcome generating model is linear in the covariates, as these also modify the marginal effect estimates. The target population is an average of the included study populations, which are typically assumed to all be representative of a single common population.

4 Example

To illustrate the issues that arise with non-collapsible effect measures when there is effect modification, we consider a simple example with a single covariate x that is both prognostic of outcome and effect modifying, uniformly distributed in the population between $-1$ and $1$ , and binary outcomes on three treatments $k=A,B,C$ that are modelled on the log odds ratio scale using Equation (1) with the logit link function $g(\pi ) = \mathop {\mathrm {logit}}(\pi )$ and $m(x) = x$ .

4.1 Effect modification can result in conflicting rankings

It is well-understood that the magnitudes of population-average conditional and marginal effects $d_{ab(P)}$ and $\Delta _{ab(P)}$ will typically differ, even in the absence of effect modification.Reference Gail, Wieand and Piantadosi ^{10 –} Reference Daniel, Zhang and Farewell ¹³ However, when there is effect modification the direction of effect can also differ between $d_{ab(P)}$ and $\Delta _{ab(P)}$ , resulting in conflicting treatment rankings. It is straightforward to find values of the coefficients in Equation (1) where this occurs, for example $\mu _{(P)} = 0$ , $\beta _1 = -1$ , $\beta _{2,B} = -3$ , $\beta _{2,C} = -1$ , $\gamma _B = -4$ , $\gamma _C = -3$ . This results in population-average conditional treatment effects $d_{AB(P)} = -4$ and $d_{AC(P)} = -3$ (i.e., B better than C for reducing a harmful outcome), but population-average marginal treatment effects $\Delta _{AB(P)} = -2.36$ and $\Delta _{AC(P)} = -2.49$ (i.e., C better than B). This is because treatment C results in a lower average event probability overall: $\bar {\pi }_{B(P)} = 0.087$ , $\bar {\pi }_{C(P)} = 0.077$ (and $\bar {\pi }_{A(P)} = 0.5$ ).

Basing a decision on these population-average marginal treatment effects would result in choosing treatment C, but 75% of the population are given an inferior treatment and are expected to do better on treatment B (Figure 1). On the other hand, basing a decision on these population-average conditional treatment effects would result in choosing treatment B, but a lower expected number of events $N\bar {\pi }_{k(P)}$ (in a population of size N) would be achieved by treatment C.

Figure 1

Individual-level log odds ratios $\gamma _{AB}(x)$ and $\gamma _{AC}(x)$ for treatments B and C compared to A, over the range of the covariate x in the population.

4.2 Marginal ranks depend on baseline risk and prognostic factors

Furthermore, the population-average marginal treatment effects and rankings are dependent on the baseline risk $\mu _{(P)}$ , as well as the distribution of the covariate x (even if it is only prognostic), but the population-average conditional treatment effects and rankings only change when the distribution of effect-modifying covariates changes. Figure 2 shows the population-average conditional and marginal treatment effects for a range of values of the baseline risk $\mu _{(P)}$ . The population-average conditional treatment effects $d_{ab(P)}$ are constant over all values of the baseline risk, but the population-average marginal treatment effects $\Delta _{ab(P)}$ change depending on the baseline risk and switch ranks. Also note that the widely-known result that conditional effects are always further from the null than marginal effectsReference Neuhaus and Jewell ^{12 ,} Reference Daniel, Zhang and Farewell ¹³ does not hold when there is effect modification; here the marginal log odds ratio for treatment C is further from the null for $\mu _{(P)} \ge 4.4$ . Consequently, the corollary result that power is always greater for conditional effects also does not hold when there is effect modification.

Figure 2

Population-average marginal ( $\Delta _{AB(P)}$ and $\Delta _{AC(P)}$ ) and conditional ( $d_{AB(P)}$ and $d_{AC(P)}$ ) log odds ratios for treatments B and C compared to A, for a range of values of the baseline risk $\mu _{(P)}$ .

The population-average marginal effects change because changing the baseline risk changes the individual event probabilities (Figure 3), which are averaged over the population to obtain $\bar {\pi }_{k(P)}$ and the marginal effects.

Figure 3

Individual event probabilities $\pi _{k(P)}(x)$ on each treatment over the range of covariate values x in the population, for a range of values of the baseline risk $\mu _{(P)}$ .

4.3 The shared effect modifier assumption

We see in Figure 3 that the curves of individual-level event probabilities $\pi _{k(P)}(x)$ by covariate x on each treatment intersect. This crossing of event probability curves due to effect modification is the reason that the conditional and marginal treatment effects can give different rankings. If there is no effect modification then the individual-level log odds ratios between all treatments are constant (Figure 4a), the event probability curves cannot cross (Figure 4b), and the conditional and marginal rankings and decision questions are aligned.

Figure 4

Individual-level log odds ratios $\gamma _{Ak}(x)$ (a) and event probabilities $\pi _{k(P)}(x)$ (b) when there is no effect modification ( $\beta _{2,B} = \beta _{2,C} = 0$ ), over the range of the covariate x in the population.

Moreover, the individual event probability curves of two treatments cannot cross if the effect modifier interaction coefficients are the same for these two treatments; for example, if $\beta _{2,B} = \beta _{2,C}$ (Figure 5b). In this situation the individual-level odds ratio between these two treatments is again constant (the curves in Figure 5a are parallel). The assumption that effect modifier interaction coefficients are the same for a set of treatments is called the shared effect modifier assumption (see Section 3).Reference Phillippo, Ades, Dias, Palmer, Abrams and Welton ^{16 ,} Reference Phillippo, Ades, Dias, Palmer, Abrams and Welton ²¹ The shared effect modifier assumption may sometimes be used for ML-NMR when there are insufficient data to estimate separate interaction terms for each treatment, for example in a two-study indirect comparison, in order to produce estimates for populations other than the aggregate study population.Reference Phillippo, Dias and Ades ⁵ Pairs of treatments between which the shared effect modifier assumption is not made can have individual odds ratios and event probability curves that cross; in Figure 5a the individual odds ratio for C (compared to A) intersects the line of no effect, and as a result the event probability curves for A and C cross. This means that, even if the shared effect modifier assumption is made for treatments B and C, the marginal ranking of treatment A can still change. With a continuous outcome and linear outcome model, the shared effect modifier assumption can also be used by MAIC and STC in order to produce relative effect estimates that are relevant to a target population other than that of the aggregate study in the indirect comparison, since then $\Delta _{BC(P)} = d_{BC(P)}$ is constant across all populations.Reference Phillippo, Ades, Dias, Palmer, Abrams and Welton ¹⁶ In all other cases, the population-average marginal treatment effects $\Delta _{BC(AC)}$ produced by MAIC and STC (with simulation) are specific to the distributions of covariates and baseline risk in the aggregate $AC$ study population, and cannot be transported to another population with different distributions of covariates or baseline risk.

Figure 5

Individual-level log odds ratios $\gamma _{Ak}(x)$ (a) and event probabilities $\pi _{k(P)}(x)$ (b) when the shared effect modifier assumption is made for treatments B and C ( $\beta _{2,B} = \beta _{2,C} = -4$ ), over the range of the covariate x in the population.

4.4 An example with a binary covariate in a contingency table

To help further solidify these ideas, we now consider an example with a binary outcome and a single binary covariate through a contingency table. Consider a trial of four treatments A, B, C, and D, randomised equally, in a population where the prevalence of a biomarker x is 25%. This biomarker is prognostic and effect-modifying, and we are interested in reducing occurrence of some harmful event. Table 2 shows the numbers of individuals who did and did not experience the event, within subgroups defined by the biomarker x and over the whole population.

Table 2

Contingency table for an illustrative example of four treatments, stratified by a biomarker x

Note: Population-average marginal odds ratios, subgroup-specific conditional odds ratios, and population-average conditional odds ratios are calculated vs. treatment A.

In Table 2, we then calculate the population-average marginal odds ratios, subgroup-specific conditional odds ratios, and population-average conditional odds ratios for each treatment compared to A. Here, since we have a binary covariate, calculating the population-average conditional odds ratios via the integral in equation (4) simplifies to taking the weighted average (on the log odds ratio scale) of the subgroup-specific conditional odds ratios according to the prevalence of the biomarker in the population. Based on the population-average marginal odds ratios, treatment B is the best, as it results in the lowest number of events overall. However, the population-average conditional odds ratios give a different ranking: treatments C and D are both ranked better than B, with treatment C being the best.

Treatment C is the most effective treatment for most individuals in this population, i.e., in the biomarker-negative subgroup ( $x=0$ ) which makes up 75% of the population. This leads to C having the best population-average conditional odds ratio. However, C is less effective than B for the smaller biomarker-positive subgroup ( $x=1$ ); the increased number of events in this subgroup result in a higher number of events overall on treatment C than B, and hence a worse population-average marginal odds ratio.

Treatment D is less effective than B for most of this population—the biomarker-negative subgroup ( $x=0$ )—and has a higher event rate overall. The population-average marginal odds ratio for treatment D is therefore worse than for treatment B. However, D is highly effective for the biomarker-positive subgroup ( $x=1$ ), to an extent that is sufficient to give a better population-average conditional odds ratio than treatment B.

We see here how the population-average conditional and marginal effects weigh up effectiveness over the population differently. The population-average marginal effects weigh up the expected number of events overall, i.e., the average is taken on the probability scale. The population-average conditional effects weigh up the expected individual or subgroup effectiveness over the population, i.e., the average is taken on the additive linear predictor scale.

As shown earlier in Section 4.1, we again see here that the treatment with the best population-average marginal effect (B) is not always the best treatment for the majority of individuals when effect modification is present. Furthermore, when covariates are discrete or non-symmetrically distributed, or treatment or covariate effects are non-linear, the treatment with the best population-average conditional effect is not always the best treatment for the majority of individuals. In this example with a binary covariate, treatment D has a better population-average conditional effect than treatment B, but D is less effective than B for most individuals.

Selecting the single treatment B with the best population-average marginal effect results in a decision that minimises the number of events overall. However, the rank conflict with the population-average conditional effects indicates that there is substantive differential effectiveness within the population, and a decision stratified by subgroup may be closer to optimal. In this case, treatment B is inferior for every individual in the population: treating biomarker-negative individuals with C and biomarker-positive individuals with D is the optimal decision. This stratified treatment decision would result in the least number of events overall, a population-average marginal odds ratio of $(19/981)/(358/642) = 0.03$ , and a population-average conditional odds ratio of $\exp (0.75 \times \log (0.05) + 0.25 \times \log (0.01)) = 0.04$ .

5 Interpretation

Rank-switching between the population-average conditional and marginal effects can only occur in the presence of effect modification, between treatments that have different interaction terms (i.e., no shared effect modifier assumption), and when this causes treatment ranks to change across individuals/subgroups in the population. We give a formal proof of this statement in Appendix B. However to see this intuitively, consider that rankings based on the population-average marginal treatment effects $\Delta _{ab(P)}$ can only change compared to the population-average conditional treatment effects $d_{ab(P)}$ if the individual event probabilities on each treatment $\pi _{k(P)}(\boldsymbol {x})$ change ranks within the population (i.e., if the event probability curves cross as in Figure 3). This can happen if and only if the individual-level treatment effects $\gamma _{Ak}(\boldsymbol {x})$ change ranks within the population (i.e., if the individual treatment effect curves cross as in Figure 1), which can happen if and only if there is effect modification between the two treatments.

It is well-understood that population-average marginal treatment effects are population-specific, and depend on the distributions of baseline risk and prognostic factors, as well as any effect modifiers. However, when there is effect modification we have seen that the marginal treatment rankings can also change—even if the only factors that change are those that do not affect individual treatment effects (baseline risk, prognostic factors). Conditional population-average treatment effects and rankings will only change depending on the distribution of effect modifiers, and do not depend on baseline risk or prognostic factors.

The population-average conditional and marginal effects have different interpretations, and correspond to different decision questions regarding effectiveness. The population-average conditional treatment effects represent the average of the individual treatment effects experienced in the population. They answer the question “Which treatment has the greatest effect for individuals, on average, in this population?” The population-average marginal treatment effects (whether odds ratios, risk ratios, or risk differences) quantify effectiveness in terms of the average event probabilities on each treatment, and for non-collapsible effect measures, by definition, do not represent any average of the individual or subgroup treatment effects experienced in the population, even without effect modification. They answer the question “Which treatment minimises (or maximises) the marginal average event probability in this population?”

These two decision questions are equivalent except when there is effect modification. When there is effect modification and the individual treatment effects cross within the population, the two can give different rankings. As a result, a decision based on population-average marginal treatment effects to obtain the minimum (or maximum) average event probability can result in choosing a treatment that is inferior for the majority of individuals in the population. Conversely, a decision based on population-average conditional treatment effects to obtain the most effective treatment on average for each individual in the population can result in choosing a treatment that gives a higher (or lower) average event probability than another treatment option.

Either of these decision questions and estimands might be justified. Basing a decision on the population-average conditional effects means that decision-makers want to maximise the benefit, on average, for each individual in the population. Basing a decision on the population-average marginal effects means that decision-makers want to minimise (or maximise) the expected number of events over the population, for example if (non-)events have a high associated cost or disutility. Indeed, in many cases decision-makers may wish to satisfy both decision questions; however, when there is effect modification there may not be a single treatment that achieves this over the entire population.

For cost-effectiveness decisions, neither of the above decision questions or estimands are of direct interest. Instead, the decision question is “Which treatment maximises the expected net benefit in this population?”, and the relevant quantity is the expected net benefit (NB) on each treatment $\mathbb {E}(\mathrm {NB}_{k(P)})$ . The net benefit is typically a function $\varphi _{k(P)}(\cdot , \cdot )$ of the average event probabilities $\bar {\pi }_{k(P)}$ , or the individual event probabilities $\pi _{k(P)}(\boldsymbol {x})$ for the distribution of the covariates in the population, as well as other parameters $\boldsymbol {\theta }_{k(P)}(\boldsymbol {x})$ such as resource use costs and adverse events which may also vary over the population. When there is patient heterogeneity, such as that caused by effect modification, a cost-effectiveness analysis should handle this appropriately by averaging (integrating) net benefit over the population, which necessitates constructing the net benefit as a function of the individual event probabilities $\pi _{k(P)}(\boldsymbol {x})$ Reference Welton, Soares and Palmer ³⁰ :

(8)

In simple cases this integral may be evaluated directly, however discrete event simulation is often used instead to construct and evaluate such cost-effectiveness models.Reference Karnon, Stahl, Brennan, Caro, Mar and Möller ^{31 ,} Reference Karnon and Afzali ³² Comparing equation (8) with equations (4) and (7), we see that the population-average conditional estimand corresponds to a net benefit function that is linear in individual treatment effects $\varphi _{k(P)}(\boldsymbol {x}) = \gamma _{Ak}(\boldsymbol {x})$ and the population-average marginal estimand corresponds to a net benefit function that is linear in individual event probabilities $\varphi _{k(P)}(\boldsymbol {x}) = \pi _{k(P)}(\boldsymbol {x})$ , when effectiveness is the only consideration.

6 Considerations for survival outcomes

Similar arguments can be applied to survival or time-to-event outcomes analysed using proportional hazards models, since hazard ratios are also non-collapsible. Consider a general proportional hazards model where the hazard function for an individual i receiving treatment k in population P with covariates $\boldsymbol {x}_{ik(P)}$ at time t is

(9a)

(9b) $$ \begin{align}\hspace{21pt} \eta_{k(P)}(\boldsymbol{x}_{ik(P)}) &= \mu_{(P)} + \boldsymbol{x}^{\scriptscriptstyle\mathsf{T}}_{ik(P)} (\boldsymbol{\beta}_1 + \boldsymbol{\beta}_{2,k}) + \gamma_k \end{align} $$

for some baseline hazard function $h_{0(P)}(t)$ , with corresponding survival function $S_{k(P)}(t\mid \boldsymbol {x}_{ik(P)}) = \exp ( -\int ^t_0 h(u) \mathop {}\!du )$ .

The population-average marginal hazard ratio $\Delta _{ab(P)}(t)$ is

(10) $$ \begin{align} \Delta_{ab(P)}(t) = \frac{ \bar{h}_{b(P)}(t) }{ \bar{h}_{a(P)}(t) }, \end{align} $$

the ratio of the marginal hazard functions

(11) $$ \begin{align} \bar{h}_{k(P)}(t) = \frac{ \int_{\mathfrak{X}} S_{k(P)}(t \mid \boldsymbol{x}) \, h_{k(P)}(t \mid \boldsymbol{x}) \, f_{(P)}(\boldsymbol{x}) \mathop{}\!d\boldsymbol{x} }{ \bar{S}_{k(P)}(t) }, \end{align} $$

where $\bar {S}_{k(P)}(t)$ is the population-average marginal survival function (or standardised survival function)

(12) $$ \begin{align} \bar{S}_{k(P)}(t) = \int_{\mathfrak{X}} S_{k(P)}(t \mid \boldsymbol{x}) \, f_{(P)}(\boldsymbol{x}) \mathop{}\!d\boldsymbol{x}. \end{align} $$

The marginal hazard function $\bar {h}_{k(P)}(t)$ can be considered a weighted average of the individual-level hazard functions, weighted by the probability of surviving to time t. Notably, the population-average marginal hazard ratio $\Delta _{ab(P)}(t)$ is time-varying, as well as depending on the shape of the baseline hazard function $h_{0(P)}(t)$ , and the distributions of baseline hazard and all prognostic and effect modifying covariates. This means that, if covariates are present, proportional hazards mathematically cannot hold at the marginal level. This is the case whether the covariates are prognostic or effect modifying; however, an argument analogous to that in Appendix B shows that effect modification is necessary for the marginal hazard functions to cross (assuming that covariates are balanced between arms at baseline, and that the same baseline hazard function $h_{0(P)}(t)$ applies in both arms).

The population-average conditional log hazard ratio $d_{ab(P)}$ under this model is again given by equation (4)—the average treatment effect over the population, taken on the log hazard scale. Again, $d_{ab(P)}$ depends only on the distribution of effect modifiers, not on the shape of the baseline hazard function, or the distributions of baseline hazard or prognostic factors, and is not time-varying.

As an example, consider a Weibull proportional hazards model for three treatments, A, B, and C, with a single covariate x that is uniformly distributed in the population between $-1$ and $1$ , with survival and hazard functions

(13a)

(13b) $$ \begin{align} h_{k(P)}(t \mid x) &= \nu_{(P)} t^{\nu_{(P)} - 1} \exp(\eta_{k(P)}(x)). \end{align} $$

For simplicity, we use a common shape parameter $\nu _{(P)} = 2$ for all three treatments, and set $\mu _{(P)} = -1$ , $\gamma _B = \log (0.6)$ , and $\gamma _C = \log (0.5)$ . The covariate x we set to be prognostic of survival with $\beta _1 = \log (0.25)$ . We then consider two scenarios, one where x is only prognostic so $\beta _{2,B} = \beta _{2_C} = 0$ , and the other where x is moderately effect modifying, $\beta _{2,B} = \log (0.7)$ and $\beta _{2,C} = \log (0.9)$ . The population-average marginal survival curves under this set-up are shown in Figure 6.

Figure 6

Population-average marginal survival curves with a single uniformly-distributed covariate that is (a) prognostic only, (b) prognostic and effect modifying.

The corresponding population-average conditional and marginal hazard ratios are shown in Figure 7. The presence of a prognostic covariate means that the population-average marginal hazard ratios are no longer constant over time; proportional hazards does not hold at the marginal level. When this covariate is also effect modifying, the population-average marginal hazard ratios can also change ranks over time. Figure 8 demonstrates that the population-average marginal hazard ratios depend on the shape of the baseline hazard function and the distribution of baseline hazard, whereas the population-average conditional hazard ratios do not.

Figure 7

Population-average conditional and marginal hazard ratios vs. treatment A over time with a single uniformly-distributed covariate that is (a) prognostic only, (b) prognostic and effect modifying.

Figure 8

Population-average conditional and marginal hazard ratios vs. treatment A over time, varying (a) the shape of the baseline hazard function $\nu _{(P)}$ , and (b) the distribution of baseline log hazard $\mu _{(P)}$ .

With survival outcomes, the key quantities for decision-making are typically the population-average marginal survival functions $\bar {S}_{k(P)}(t)$ on each treatment and summaries thereof, such as survival probabilities at clinically relevant time points, median survival times, or (restricted) mean survival times. $\bar {S}_{k(P)}(t)$ is also the typically the primary input to an economic model for decisions based on cost-effectiveness.

7 Recommendations for decision-making in the presence of effect modification

Decision-makers should specify a priori the target population and decision question that are of interest, and analysts should ensure that the corresponding estimand is appropriately targeted—be that population-average conditional or marginal estimates for effectiveness decisions, or the necessary inputs to an economic model for cost-effectiveness decisions. In a health technology assessment context, guidance for submissions to the National Institute of Health and Care Excellence (NICE) in England states that the choice of effect modifiers must be pre-specified prior to analysis, clinically plausible, and justified through empirical evidence, expert opinion, or systematic review.Reference Phillippo, Ades, Dias, Palmer, Abrams and Welton ^{16 , 33} Similarly, section 4.9 of the NICE health technology evaluation manual expresses a preference for pre-specified identification of subgroups with biological plausibility, and warns against post-hoc “dredging” for subgroup effects. ³³ This applies to subgroups based both on effectiveness (i.e., effect modifiers), and on other factors such as costs, baseline risk, or adverse events. In line with methodological guidance,Reference Phillippo, Ades, Dias, Palmer, Abrams and Welton ¹⁶ it is advisable that: analyses with and without adjustment for effect modifiers be presented, for example standard network meta-analysis alongside a suitable population adjustment method; any modelling assumptions such as choice of covariates to include be explored in sensitivity analyses; and the extent of extrapolation should be considered and estimates treated with caution where extrapolation is required beyond the range of the data.

7.1 Decisions based on effectiveness only

For population decision-making based purely on effectiveness, the relevant estimands are population-average conditional or marginal treatment effects in the decision target population, for example after population adjustment. ML-NMR can produce relevant estimates for any decision target populationReference Phillippo, Dias and Ades ⁵ ; other approaches like MAIC and STC are limited to producing estimates in the population of the aggregate study in an indirect comparison, which may not represent the decision target population.Reference Phillippo, Ades, Dias, Palmer, Abrams and Welton ¹⁶ In small networks like a two-study indirect comparison, ML-NMR may make use of an identifying assumption such as the shared effect modifier assumption to produce estimates for populations other than the aggregate study population (Section 3). MAIC and STC cannot make use of the shared effect modifier assumption except for continuous outcomes with a linear outcome model, and are restricted to producing estimates for the aggregate study population. Only ML-NMR at present can produce estimates of both conditional and marginal estimands. The population-average conditional and marginal estimands can result in conflicting rankings when there is effect modification, and in some cases a decision-maker may be forced to choose between a treatment that is inferior for the majority of the population or one that results in a worse expected number of events overall. If this is a concern, then the only way to resolve this conflict and realign the two effectiveness decision questions and estimands is to allow different decisions within subgroups based on covariate values.

With just one effect modifier, it is straightforward to visualise the impact on decisions by plotting the individual treatment effects against the covariate (as in Figure 1), but with multiple effect modifiers this quickly becomes infeasible. It is possible to mathematically determine the boundaries between different optimal treatment choices in the L-dimensional covariate space, where L is the number of effect-modifying covariates. However, this is likely to result in complex decisions that may be difficult to implement and hard to justify.

We propose a more pragmatic approach, where decision-makers consider both the population-average conditional and marginal treatment effects. If the respective rankings agree, then there is no substantial conflict to resolve and a single decision might be justified for the entire population. This does not rule out the possibility that smaller subgroups might obtain greater treatment benefit or lower (or higher) average event probabilities from a different decision, but it does mean that the overall decision is both the most effective on average for each individual in the population and results in the lowest (or highest) average event probability overall. If the rankings are in conflict, then decision-makers could attempt to resolve this by considering making decisions for subgroups formed by the combinations of a small number of effect modifiers (whilst still adjusting for the full set of effect modifiers at the modelling stage). These chosen effect modifiers should be those that have the most impact on treatment effects in the population, based both on the strength of the interaction and on the range of covariate values in the population. The precise cut-points could be based on examining plots of the individual treatment effects against the effect modifiers in question one covariate at a time, holding the other covariates at the population means, or they could be guided by clinical reasoning or practice (say if there are established thresholds for normal vs. high values). Careful selection of subgroups in this manner is likely to be sufficient to resolve the conflict between decision questions, whilst keeping the resulting subgroup decisions simple enough for decision-makers to justify and implement.

For example, consider a scenario with two treatments B and C compared to reference A, and three potential effect modifiers which are distributed in the target population as $x_1 \sim \mathop {\mathrm {N}}(0, 1^2)$ , $x_2 \sim \mathop {\mathrm {N}}(0, 0.25^2)$ , $x_3 \sim \mathop {\mathrm {Bern}}(0.8)$ . There is a harmful binary outcome, modelled on the logit scale following model (1), with coefficients $\mu = 0$ , $\boldsymbol {\beta }_1 = (-2, 1, -0.25)^{\scriptscriptstyle \mathsf {T}}$ , $\boldsymbol {\beta }_{2,B} = (-3, -4, -0.5)^{\scriptscriptstyle \mathsf {T}}$ , $\boldsymbol {\beta }_{2,C} = (-1, -2, 0)^{\scriptscriptstyle \mathsf {T}}$ , $\gamma _B = -4$ , and $\gamma _C = -3$ . The resulting population-average marginal and conditional log odds ratios are in conflict, $d_{BC} = 1.4$ and $d_{BC} = -0.1$ , so we proceed to consider a decision for subgroups. Plots of the individual-level treatment effects over each of the covariates in turn, holding the other covariates at their population means, are shown in Figure 9. Whilst $x_2$ has the strongest interaction estimate, $x_1$ leads to greater variation in individual-level treatment effects over the population due to the greater variation in this covariate, and the individual-level treatment effects change ranks within the range of $x_1$ in the population. $x_1$ is therefore a good candidate for forming subgroups. The individual-level treatment effects cross at $x_1 = -0.7$ , and making separate decisions for subgroups at this cut-point (treatment C for $x_1 < -0.7$ and B for $x_1 \ge -0.7$ ) does resolve the conflict between population-average marginal and conditional effects ( $d_{BC(x_1 < -0.7)} = -1.2$ and $\Delta _{BC(x_1 < -0.7)} = -0.5$ in the first subgroup, $d_{BC(x_1 \ge -0.7)} = 2.2$ and $\Delta _{BC(x_1 \ge -0.7)} = 0.4$ in the second). In many cases it may be more justifiable to base the precise cut-point(s) on clinical reasoning or practice. For example, in this case if $x_1=0$ represents a clinically-meaningful threshold, forming subgroups around this cut-point also resolves the conflict ( $d_{BC(x_1<0)} = -0.2$ and $\Delta _{BC(x_1<0)} = -0.2$ in the first subgroup, $d_{BC(x_1 \ge 0)} = 3.0$ and $\Delta _{BC(x_1 \ge 0)} = 1.6$ in the second).

Figure 9

Example of considering a subgroup decision with three effect-modifying covariates. Each plot shows the individual-level treatment effects vs. treatment A (solid lines) over each of the covariate distributions in the population in turn (histograms), holding the other covariates at their population means.

When effect modification is not present, the two decision questions are aligned and both estimands will give the same ranking of treatments. In a Bayesian setting Bayesian p-values and the precision of the ranks will be identical between the population-average conditional and marginal effects, since the action of marginalisation is a monotonic transformation of the posterior about the origin.