The Mathematics of Indirect Treatment Comparisons
Population Adjustment, Transportability, and Comparative Effect Estimation
Preface
This book is about the mathematics of comparing treatments that have never been studied head to head.
When two treatments have been compared against a common comparator in separate randomized trials, but never against each other, we can still learn something about how they compare. When the populations enrolled in those trials differ in ways that change the size of a treatment effect, the naive indirect comparison becomes biased, and we need population adjustment. When we want the answer not for a trial population but for a decision population that neither trial sampled, we need transportability. The methods that do this work, network meta-analysis, matching-adjusted indirect comparison, simulated treatment comparison, multilevel network meta-regression, and multilevel unanchored meta-regression, are the subject of this book. Our goal is to develop all of them as mathematics: from definitions, through lemmas, to theorems with proofs.
Who this book is for
The book is written for the reader who wants to understand indirect treatment comparison from the ground up and emerge able to derive, criticize, and extend the methods rather than only run them. We assume no more than comfort with calculus and the willingness to follow a proof. Everything else is built here. A reader who has asked “why do we minimize the sum of squared residuals, and what makes that the right thing to do,” or “what exactly is being assumed when an indirect comparison is called valid,” is the reader we are writing for.
We begin with linear algebra and probability, build ordinary least squares and the generalized linear model from first principles, develop the causal inference machinery of potential outcomes and transportability, and only then turn to evidence synthesis and the population-adjustment methods that are the heart of the book. By the final part the same reader who started at the definition of a vector space will be reading the aggregation integral of multilevel network meta-regression and the identification theory of unanchored comparisons.
The contract between us
This is a pure mathematics text. Every substantive claim is either proved in full or explicitly marked as cited or assumed. We hold ourselves to three rules.
Foundations are stated rigorously and the load-bearing ones are proved. From linear regression onward, results are proved in full. For the prerequisite linear algebra and real analysis, we state every result precisely and prove the ones the later theory rests upon, such as the orthogonal projection theorem and the spectral theorem, while citing standard references for routine facts.
Assumptions are never hidden. Each theorem displays its hypotheses. Where a result holds only on a particular scale, or only for a collapsible effect measure, or only under a shared effect modifier assumption, the statement says so. A recurring lesson of this book is that in indirect comparison the validity of a method is inseparable from the scale and the estimand on which it is asserted.
Where a result has been verified by machine, we say so. A companion formal development in the Coq proof assistant checks many of the central theorems of this book with no admitted steps. Results with a machine-checked counterpart carry a tag pointing to the corresponding formal lemma, and the final chapter explains the formalization and exactly what it assumes.
How the book is organized
The book has six parts.
Part I, Mathematical Foundations, builds linear algebra, probability, statistical inference, ordinary least squares, the generalized linear model, and hierarchical and Bayesian modeling. This part is self-contained and proved from the basics.
Part II, Causal Inference and Estimands, develops potential outcomes, identification, the adjustment estimators, effect modification, collapsibility, and the estimand-based theory of transportability that frames the rest of the book.
Part III, Meta-Analysis and Network Meta-Analysis, covers pairwise meta-analysis, the Bucher indirect comparison, the network meta-analysis model and its identifiability, and the precise way standard network meta-analysis fails under effect modification.
Part IV, Population Adjustment, is the theory of matching-adjusted indirect comparison and simulated treatment comparison, including the derivation of the matching weights, the effective sample size bound, and the non-collapsibility pitfall.
Part V, Multilevel Network Meta-Regression, derives multilevel network meta-regression from the individual-level model by integration, develops the approximation and numerical-integration theory it requires, and extends it to general likelihoods and to the unanchored setting of multilevel unanchored meta-regression.
Part VI, Bias, Comparator Selection, and Rigor, treats covariate-distance matching, quantitative bias analysis for indirect comparison, simulation-study theory, and the machine-checked formalization.
Appendices collect the notation glossary, matrix calculus, a distributions reference, the special functions used in numerical integration, the mapping from the mathematics to the reference software, and solutions to selected exercises.
How to read it
A first reading can follow the conceptual spine: the preface, then the notation chapter, then Chapters 6, 9, 11, 12, 14, 16, 17, and 20. These carry the main ideas. A reader who wants the full development should read in order. A reader who already knows the foundations may start at Part II and treat Part I as reference. Each method chapter is written to be readable once its prerequisites in Parts I and II are in hand.
A note on the notation chapter that follows: it is the single source of truth for symbols used throughout the book. We recommend reading it once and returning to it as needed.