4  Adaptive Designs and Graphs

These notes were automatically generated by AI and lightly reviewed by the lecturer. They may still contain mistakes and factual errors.

Let’s finish our discussion of experimental designs with enrichment designs. In these trials, early results guide later recruitment. Alongside other examples, this motivates the use of causal diagrams. We will then introduce directed mixed graphs, Gaussian linear systems and an algebra for working with walks on a graph.

4.1 Enrichment designs and selection

4.1.1 Why concentrate recruitment?

The FOURIER trial compared evolocumab, a drug that inhibits PCSK9, with placebo in patients with cardiovascular disease. The main trial reported a primary-endpoint hazard ratio of 0.85, corresponding to a 15% reduction in the hazard. A later post-hoc analysis used a genetic risk score to divide patients into low-, intermediate- and high-risk groups. The figure below shows the estimated treatment effects in these groups.¹

Treatment effects by genetic risk in a post-hoc analysis of FOURIER. The endpoint here is major vascular events. Estimates and 95% confidence intervals are redrawn from Marston et al. (2020), with the 2024 correction.

On a small screen, scroll the figure horizontally to read the labels.

There is little evidence of benefit in the low-genetic-risk group, at least with the sample size available, whereas the high-risk group appears to benefit more. More generally, post-hoc subgroup analyses are common: after investing heavily in a trial, investigators want to know whether a treatment works particularly well for some patients, especially when the overall result is disappointing.

This motivates enrichment: recruit broadly, use the first-stage trial data to select a promising subgroup, then concentrate subsequent recruitment in that subgroup. The statistical difficulty is that the subgroup itself is selected using the data.

4.1.2 A two-stage simulation example

Let’s use a hypothetical trial as a simulation example to see how selecting a hypothesis from the data affects error rates. A simple normal model lets us calculate the repeated-trial probabilities by numerical integration.

Consider one treatment and five prespecified, disjoint subgroups, with treatment effects \(\beta_j\), \(j=1,\ldots,5\). Outcomes are independent normal random variables with variance 1, and larger outcomes are better. In stage 1, randomize 100 participants to treatment and 100 to control within each subgroup. Select the subgroup with the largest observed treatment–control difference, then recruit 100 new participants per arm in that subgroup alone for stage 2.

The two-stage enrichment design. Selection determines which subgroup continues; reusing the first-stage observations in the final test requires care.

On a small screen, scroll the figure horizontally to read the labels.

Write \(\bar Y_{ijk}\) for the sample mean, where \(i\in\{1,2\}\) indexes the stage, \(j\in\{1,\ldots,5\}\) the subgroup, and \(k\in\{0,1\}\) the treatment arm, with \(k=0\) denoting control. Each observed mean has variance \(1/100\). Define the standardized treatment–control differences and the selected subgroup by

\[T_{ij}=\frac{\bar Y_{ij1}-\bar Y_{ij0}}{\sqrt{2/100}}, \qquad S=\underset{j\in\{1,\ldots,5\}}{\operatorname{argmax}}T_{1j}.\]

The denominator follows because the two arm means are independent, so their variances add. In stage 2 we observe the new comparison only for \(j=S\). Our final hypothesis is \(H_S:\beta_S\le0\). This is a randomly selected hypothesis: which subgroup we test depends on the first-stage observations.

4.1.3 Marginal and conditional error

We must distinguish two error probabilities.

Definition 4.1.

Write \(H_j=0\) when the null hypothesis \(\beta_j\le0\) for subgroup \(j\) is true. The marginal type I error is

\[\text{Marginal error}=\operatorname{\mathsf P}(H_S=0\text{ and }H_S\text{ is rejected}).\]

This is the probability of selecting a true null and rejecting it. If we select a subgroup that really benefits from treatment, rejecting its null is not a type I error.

Definition 4.2.

For each \(j\) with a true null, the conditional type I error is

\[\text{Conditional error}(j) =\operatorname{\mathsf P}(H_S\text{ is rejected}\mid S=j),\qquad H_j=0.\]

For a subgroup with a positive effect, the corresponding rejection probability is conditional power instead. By the law of total probability,

\[\text{Marginal error} =\sum_{j:H_j=0}\text{Conditional error}(j)\operatorname{\mathsf P}(S=j).\]

Thus controlling each conditional error at level \(\alpha\) also controls the marginal error at \(\alpha\). The converse need not hold. A procedure can have a low overall false-rejection probability but a much higher error probability conditional on selecting a particular null subgroup.

4.1.4 Four ways to analyse the selected subgroup

Take a nominal one-sided level of 5%, and let \(\Phi\) denote the standard normal distribution function.

1. Pooled \(z\)

The simplest approach combines the two stages as though the subgroup had been fixed in advance:

\[T_j=\frac{T_{1j}+T_{2j}}{\sqrt2}, \qquad \text{reject }H_S\text{ if }T_S>\Phi^{-1}(0.95)\approx1.645.\]

Why does this fail? The selected first-stage statistic \(T_{1S}\) is the largest of five statistics. Even when all effects are zero, it tends to be larger than an ordinary standard normal observation. Pooling it with fresh data does not remove that selection effect.

2. Bonferroni–Bonferroni

Apply one Bonferroni correction across the five first-stage comparisons and another across the two stages. Reject if

\[T_{1S}>\Phi^{-1}(0.995)\approx2.576 \quad\text{or}\quad T_{2S}>\Phi^{-1}(0.975)\approx1.960.\]

The first-stage tail probability is \(0.05/(5\times2)=0.005\); the second-stage tail probability is \(0.05/2=0.025\). By the same union-bound argument used in the proof of Proposition 3.5, this procedure controls marginal error, with some conservatism. It is not designed to control error conditional on the selected subgroup.

3. Stage 2 only

Use just the fresh observations:

\[\text{Reject }H_S\text{ if }T_{2S}>\Phi^{-1}(0.95)\approx1.645.\]

For a fixed subgroup, the new observations are independent of the first-stage data used for selection. Conditional on \(S=j\), the second-stage statistic is still standard normal when \(\beta_j=0\). This gives conditional error exactly 5% at that boundary, and at most 5% when \(\beta_j\le0\).

4. Selective \(z\)

Can we retain some first-stage information while controlling conditional error? To account for selection, write the selection event as

\[\max_{j\ne S}T_{1j}\le T_{1S} =\frac1{\sqrt2}\left\{T_S+\frac{T_{1S}-T_{2S}}{\sqrt2}\right\}.\]

For a fixed subgroup \(s\), the sum \(T_s=(T_{1s}+T_{2s})/\sqrt2\) and the difference \((T_{1s}-T_{2s})/\sqrt2\) are independent under this normal model. To find the reference distribution after selection, hold that difference and the other first-stage statistics fixed. The event \(S=s\) then restricts the pooled statistic to

\[E=\{T_s\ge L_s\},\qquad L_s=\sqrt2\max_{j\ne s}T_{1j}-\frac{T_{1s}-T_{2s}}{\sqrt2}.\]

At \(\beta_s=0\), \(T_s\) is standard normal before selection. Under this conditioning, selecting subgroup \(s\) restricts it to values at or above \(L_s\). For \(t\ge L_s\), its upper-tail probability is

\[\operatorname{\mathsf P}_{\beta_s=0}\!\left(T_s>t\mid S=s, \frac{T_{1s}-T_{2s}}{\sqrt2},(T_{1j})_{j\ne s}\right) =\frac{1-\Phi(t)}{1-\Phi(L_s)}.\]

The denominator accounts for restricting the normal distribution to the values compatible with selection. Setting this tail probability to 0.05 gives the selective \(z\)-test rejection region:

\[\boxed{\text{Reject }H_S\text{ if } T_S>\Phi^{-1}\!\left(1-0.05\{1-\Phi(L_S)\}\right).}\]

Here \(L_S=\sqrt2\max_{j\ne S}T_{1j}-(T_{1S}-T_{2S})/\sqrt2\) is evaluated at the observed data. The cutoff depends on the selection event, so there is no single numerical threshold like 1.645.

At \(\beta_s=0\), this tail probability is 5% for each set of conditioning values. Averaging over them therefore gives 5% error conditional on \(S=s\). When \(\beta_s<0\), the truncated normal puts more weight on smaller values, so the error is at most 5%.

The first-stage data still contain information after selection. Accounting for selection lets us use some of that information. The power gain over using stage 2 alone can be modest or substantial, depending on the setting.

4.1.5 What the error rates show

The following probabilities come from numerical integration in the stated normal model. All entries are percentages. Bold rust-coloured error probabilities exceed the nominal 5% level.

Rejection probability (%)	Pooled \(z\)	Bonferroni–Bonferroni	Stage 2 only	Selective \(z\)
Effects \((0,0,0,0,0)\)
Marginal false rejection	16.6	4.9	5.0	5.0
Effects \((0.3,0,0,0,0)\)
Marginal false rejection	5.2	2.0	0.9	0.9
False rejection given \(S\in\{2,3,4,5\}\)	28.2	10.6	5.0	5.0
Power given \(S=1\)	95.3	73.7	68.3	69.7
Effects \((0.3,-0.3,-0.3,-0.3,-0.3)\)
Power given \(S=1\)	91.5	70.6	68.3	88.5

Under the global null, when all five effects are zero, pooling gives a 16.6% false-rejection probability rather than 5%. Bonferroni–Bonferroni brings this down to 4.9%. When only subgroup 1 benefits, however, Bonferroni–Bonferroni has conditional error 10.6% given that we select one of the other four subgroups, despite a marginal error of only 2.0%.

Why does this distinction matter? In clinical development, Phase II trials are usually smaller studies that assess whether a treatment looks promising and help choose the dose, population and hypothesis for further testing. Phase III trials are larger confirmatory studies of efficacy and safety. In the usual separation between them, Phase II helps establish the primary hypothesis that Phase III will test using new data.

If we combine the phases and control only their overall error, we may have a much larger conditional error when the earlier selection was unlucky, as the 10.6% error above illustrates. Which error should be controlled is an important question for adaptive designs.

The stage-2-only and selective \(z\)-tests both control conditional error. Selective \(z\) uses some first-stage information as well. For subgroup 1, conditional on its selection, this raises power from 68.3% to 69.7% when the other four subgroups have zero effects, and from 68.3% to 88.5% when they have negative effects.

4.2 Why use graphs?

4.2.1 Adaptive experiments

In an adaptive design such as the enrichment design, later decisions depend on earlier observations. A flowchart describes the procedure; a graphical model expresses these dependencies mathematically. The following figure illustrates a two-stage design, from work on selective randomization inference for adaptive experiments.²

A graphical representation of a two-stage adaptive experiment. Reproduced from Freidling, Zhao and Gao (2026), Figure 3, p. 9, under CC BY 4.0; cropped from the original.

On a small screen, scroll the figure horizontally to read the labels.

4.2.2 SMART designs and treatment–confounder feedback

A sequential, multiple assignment, randomized trial, or SMART, allows participants to be randomized at more than one stage. Consider the following educational trial, which involved families in 108 US elementary schools and aimed to reduce pupil absence.³

1. In October, families were randomized to messages emphasizing the benefits of attending school, messages emphasizing the consequences of missing school, or usual communications.
2. Attendance was measured during the autumn. Families in either messaging group were randomized again in January to added school-staff outreach or goal-commitment texts if any child had missed at least 8% of school days. Otherwise, they continued with the basic messages. The usual-communications group continued as before.
3. Spring absence was then measured.

The attendance SMART: families can receive successive randomized interventions. Design redrawn from Heppen, Kurki and Brown (2020), pp. 3–4 and Exhibit 5.

On a small screen, scroll the figure horizontally to read the labels.

To represent this design graphically, let \(A_1\) denote the autumn messaging assignment, \(Y_1\) autumn absence, \(A_2\) the spring support assignment and \(Y_2\) spring absence. A possible causal structure is:

Treatment–confounder feedback in the attendance example. In this teaching schematic, directed edges are blue; the red bidirected edge represents possible unmeasured common causes of autumn and spring absence.

On a small screen, scroll the figure horizontally to read the labels.

Crucially, we randomize the messages and support, not the absences. A family’s reasons for absence may persist across autumn and spring without being well measured. The bidirected arrow between \(Y_1\) and \(Y_2\) represents this possible shared, unmeasured influence.

The intermediate variable \(Y_1\) is post-treatment for \(A_1\) but pre-treatment for \(A_2\): autumn absence may be affected by the earlier messages, yet it also determines eligibility for spring support. This dual role is called treatment–confounder feedback. Simply applying ordinary regression to such data need not give the causal effect we want; the sequence of treatments and outcomes matters for the analysis.⁴

4.2.3 Genetics

Sewall Wright in 1965. UW–Madison Archives; © UW System Board of Regents, educational use permitted.

Causal diagrams have a long history. The geneticist Sewall Wright (1889–1988) developed path analysis on causal diagrams in a linear-model setting. His 1920 study of the piebald pattern of guinea pigs examined the relative importance of genetic inheritance and environment.⁵

Notice two features of the diagrams below. First, the guinea-pig diagram explicitly includes “Chance”, with arrows into the germ cells. The random transmission of genetic material connects this example to the role of randomization in causal interpretation. We will return to that connection with Mendelian randomization. Second, the diagram of two effects already has a bidirected edge between \(B\) and \(C\), representing dependence between two causes.

Wright’s guinea-pig diagram, including “Chance” in the transmission of heredity. Original Figure 5, p. 328; public domain.

Wright’s diagram of two effects, X and Y, with shared, correlated causes B and C. Original Figure 6, p. 329; public domain.

Graphical models are often introduced through directed acyclic graphs, with bidirected edges added later. Wright’s early use of a bidirected edge motivates starting with both kinds of edge here.

4.3 Directed mixed graphs and Gaussian linear systems

4.3.1 Directed mixed graphs

We now formalize the two types of edge used in these examples.

Definition 4.3.

A directed mixed graph (DMG) is a triple \(G=(V,D,B)\), where \(V\) is a finite vertex set and \(D,B\subseteq V\times V\) are its directed and bidirected edge sets. The set \(B\) is symmetric, and a bidirected edge requires bidirected loops at its endpoints:

\[(V_j,V_k)\in B\ \Longrightarrow\ (V_k,V_j),(V_j,V_j),(V_k,V_k)\in B.\]

We write \(\mathbb G(V)\) for the collection of DMGs on vertex set \(V\). Most of the graphs we consider will include a bidirected loop at every vertex.

Our running directed mixed graph: directed edges are blue, and bidirected edges, including all five self-loops, are red.

On a small screen, scroll the figure horizontally to read the labels.

Under a causal interpretation:

• \(V_j\to V_k\) represents a direct causal effect.
• \(V_j\leftrightarrow V_k\) represents exogenous dependence, for example through an unmeasured common cause.

The word direct matters. In the running graph, \(V_2\) can affect \(V_5\) indirectly through \(V_3\), as well as through the direct edge \(V_2\to V_5\). The bidirected edges describe randomness entering the system: a loop represents a variable’s own noise, while an edge between distinct vertices allows them to share some randomness.

If causality involves doing something and observing an outcome, why represent both as vertices? The vertices represent variables, whether they describe a treatment or an outcome. To give the graph a causal meaning, we must connect these variables to the potential outcomes used to define causal effects.

We will develop this connection gradually. Over the next couple of lectures, we will study graphs and the statistical models they represent. We will then introduce potential outcomes in these graphs to make the causal interpretation explicit. For now, the causal reading of the arrows motivates the mathematics; its formal meaning is still to come.

Why use a bidirected edge rather than add a hidden variable with arrows to both vertices? That is a natural idea: a common cause could explain the shared randomness. One motivation for bidirected edges comes from linear systems: the noise terms can themselves be dependent, and we need a graphical representation of that dependence. The Gaussian linear model below makes this concrete; further motivations will follow later.

4.3.2 Gaussian linear systems

Definition 4.4.

For a graph with \(d\) vertices, also use \(V\) for the column vector \((V_1,\ldots,V_d)^T\). A Gaussian linear system with respect to \(G\) satisfies

\[V=\beta^T V+E,\qquad E\sim N(0,\Lambda),\qquad \Lambda\succeq0,\]

where \(\beta\) is the matrix of coefficients and \(E\) is the vector of noise terms. The noise covariance matrix \(\Lambda\) is symmetric and positive semidefinite. The graph imposes two restrictions:

1. If \(V_j\to V_k\) is absent from \(G\), then \(\beta_{jk}=0\).
2. If \(V_j\leftrightarrow V_k\) is absent from \(G\), then \(\Lambda_{jk}=0\).

The second restriction begins to answer the question about bidirected edges. A linear model also needs to specify how its noise terms relate to one another. The matrix \(\Lambda\) does this algebraically; the bidirected edges do it graphically. A loop corresponds to a diagonal entry, the variance of a noise term.

Each variable is a linear combination of its parents—the vertices with arrows pointing into it—plus noise. For our running graph,

\[\begin{aligned} V_1&=E_1,\\ V_2&=E_2,\\ V_3&=\beta_{13}V_1+\beta_{23}V_2+E_3,\\ V_4&=\beta_{34}V_3+E_4,\\ V_5&=\beta_{25}V_2+\beta_{35}V_3+E_5. \end{aligned}\]

The noise covariance matrix has the form

\[\Lambda=\begin{pmatrix} \Lambda_{11}&0&0&\Lambda_{14}&\Lambda_{15}\\ 0&\Lambda_{22}&0&0&0\\ 0&0&\Lambda_{33}&0&0\\ \Lambda_{41}&0&0&\Lambda_{44}&0\\ \Lambda_{51}&0&0&0&\Lambda_{55} \end{pmatrix}.\]

The only off-diagonal covariances allowed to be nonzero are those between \(E_1\) and \(E_4\), and between \(E_1\) and \(E_5\). These correspond to the two cross-variable bidirected edges. The graph gives our algebraic assumptions about noise covariance a visible counterpart. Bidirected edges and loops therefore arise naturally, without first specifying extra hidden variables: they record the covariances and variances allowed by the model.

4.3.3 Solving the system

To solve the model equation, move the term \(\beta^T V\) to the left:

\[(I-\beta^T)V=E.\]

If \(I-\beta\) is invertible, then

\[V=(I-\beta)^{-T}E\sim N(0,\Sigma), \qquad \Sigma=(I-\beta)^{-T}\Lambda(I-\beta)^{-1}.\]

This is a linear transformation of a Gaussian vector, so it is still Gaussian. We have omitted an intercept; adding one changes the mean. Our main interest here is the covariance, which describes the joint relationships between the variables.

Our aim is to answer the following three statistical questions using the graph:

1. Marginal independence: which variables, or groups of variables, are independent? For example, is \(V_1\) independent of \((V_2,V_3)\)?
2. Marginalization: what happens when we retain only a subset of the variables? This means finding the marginal distribution of a subvector of \(V\). In the linear setting, variable elimination gives a system for the retained variables. Can we represent this marginalization by an operation on the graph?
3. Conditional independence: for example, are \(V_1\) and \(V_4\) independent given \(V_3\)?

To approach these questions, let’s connect matrix operations with operations on a graph.

4.4 An algebra of walks

4.4.1 The adjacency-matrix connection

Definition 4.5.

For a directed graph, its adjacency matrix \(A\) has \(A_{jk}=1\) if there is an edge \(j\to k\), and \(A_{jk}=0\) otherwise.

A basic result connects matrix powers to walks, where a walk follows a sequence of connected edges:

Proposition 4.6 (counting directed walks).

For \(r\ge1\),

\[(A^r)_{jk}=\#\{\text{directed walks from }j\text{ to }k\text{ of length }r\}.\]

Proof sketch.

To see why, consider the square of the adjacency matrix:

\[(A^2)_{jk}=\sum_l A_{jl}A_{lk}.\]

For each intermediate vertex \(l\), the product is one exactly when both edges \(j\to l\) and \(l\to k\) exist. Summing counts the length-two walks.

This elementary result gives us a starting point for connecting linear algebra and graphs. We now bring probability and statistics into the picture: a linear system gives a distribution, and we want to read its statistical relationships from the graph.

Our aim extends well beyond linear models. Linear algebra will help us discover useful operations on graphs, but eventually we want to express the resulting statistical rules without relying on a linear model. For that purpose, counts alone are not enough. We want to keep track of the actual relations connecting vertices. Instead of an adjacency-matrix entry that only says whether an edge is present, we will keep the edge itself; instead of a matrix power that only counts walks, we will keep the walks themselves.

4.4.2 Basic edge matrices

Definition 4.7.

Define two matrices whose entries are sets of edges:

\[(W[V\to V])_{jk}=W[V_j\to V_k] =\begin{cases} \{V_j\to V_k\},&(V_j,V_k)\in D,\\ \varnothing,&\text{otherwise}, \end{cases}\]

\[(W[V\leftrightarrow V])_{jk}=W[V_j\leftrightarrow V_k] =\begin{cases} \{V_j\leftrightarrow V_k\},&(V_j,V_k)\in B,\\ \varnothing,&\text{otherwise}. \end{cases}\]

Each entry is a set containing one edge, or the empty set. From these two basic matrices, we generate matrices whose \((j,k)\) entry is a set of walks from \(V_j\) to \(V_k\). The rows and columns still tell us where a walk starts and ends; what changes is that the entries contain graphical objects rather than numbers. The following operations let us combine them.

4.4.3 Three operations

Definition 4.8.

Let \(W\) and \(W'\) be two such matrices of walks. We define three operations.

1. Addition is set union, applied entrywise:

\[(W+W')[V_j,V_k]=W[V_j,V_k]\cup W'[V_j,V_k].\]

2. Multiplication is concatenation. Take a walk from \(V_j\) to an intermediate vertex \(V_l\) in the first matrix, then a walk from \(V_l\) to \(V_k\) in the second matrix, and put them together:

\[(W\cdot W')[V_j,V_k] =\sum_{V_l\in V}\{w\cdot w':w\in W[V_j,V_l],\ w'\in W'[V_l,V_k]\}.\]

Here the sum means set union over all intermediate vertices. As in ordinary matrix multiplication, we match the end of the first part to the start of the second, then collect the possibilities over every \(V_l\). Here we retain each resulting walk rather than multiply numbers and add up a count.

3. Transpose reverses the order in which we read a walk:

\[(W^T)[V_j,V_k]=\{w^T:w\in W[V_k,V_j]\}.\]

Writing a walk from right to left changes how its arrows appear on the page; it does not replace the edges of the original graph with different edges.

Returning to our five-vertex graph, concatenate the bidirected loop at \(V_2\) with each of the two directed routes from \(V_2\) to \(V_5\):

\[\begin{aligned} &\{V_2\leftrightarrow V_2\}\cdot \{V_2\to V_5,\ V_2\to V_3\to V_5\}\\ &\qquad=\{V_2\leftrightarrow V_2\to V_5,\ V_2\leftrightarrow V_2\to V_3\to V_5\}. \end{aligned}\]

For an example of transposition, read the following walk in reverse:

\[\{V_2\to V_3\to V_5\leftrightarrow V_1\}^T =\{V_1\leftrightarrow V_5\leftarrow V_3\leftarrow V_2\}.\]

The important idea is composition of relations, with multiplication as the key operation. Directed and bidirected edges give us two basic kinds of relation between variables; composing them gives new relations involving sequences of edges. Walk algebra describes their exact form rather than only counting them.

Why should these new relations have statistical meaning? Linear systems will be our bridge. In a Gaussian linear system we can calculate the distribution and study marginal and conditional independence directly. These calculations will guide us towards graphical rules. We will establish the connection in linear models first, then see how the rules also apply in nonlinear models. This is the purpose of introducing walk algebra.

© 2026 Qingyuan Zhao. Except where otherwise noted, these lecture notes are licensed under Creative Commons Attribution 4.0 International (CC BY 4.0). Third-party material is excluded from this licence; see the individual credits and rights notices. Public-domain material remains in the public domain.

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1. Sabatine et al. (2017), Evolocumab and Clinical Outcomes in Patients with Cardiovascular Disease; Marston et al. (2020), Predicting Benefit From Evolocumab Therapy in Patients With Atherosclerotic Disease Using a Genetic Risk Score, Figure 2, with the 2024 correction. The genetic analysis’s major-vascular-event endpoint differs from the original trial’s primary endpoint.

2. Freidling, Zhao and Gao (2026), Selective Randomization Inference for Adaptive Experiments. Figure 3 is reproduced under CC BY 4.0. Its caption notes that possible direct first-stage-data arrows into \(S_2\) are omitted; the drawing should not be read as asserting their absence.

3. Heppen, Kurki and Brown (2020), Can Texting Parents Improve Attendance in Elementary School?, pp. 3–4 and Exhibit 5.

4. Further reading: Hernán and Robins, Causal Inference: What If, chapters 19–21.

5. Wright (1920), The Relative Importance of Heredity and Environment in Determining the Piebald Pattern of Guinea-Pigs, Figures 5–6, pp. 328–329. Portrait record, UW–Madison Archives; educational use permitted.