Factorizing the Longitudinal Data Distribution

Why Factor the Longitudinal Distribution?

A longitudinal record contains variables that occur in a particular order: baseline covariates, treatment decisions, later covariates, subsequent treatment decisions, and eventually an outcome. A factorization writes the joint distribution of that entire record as a sequence of conditional distributions that respect this order. It makes clear which part of the observed-data law describes each transition, without asserting that any one part has a causal interpretation by itself.

The factorization is useful because a complicated joint distribution can be represented through conditional distributions of familiar components. It also gives a precise accounting of what has to be described when specifying a statistical model for the observed data. The factorization itself is a probability identity: it does not require sequential exchangeability, positivity for a causal contrast, or a parametric model.

Time Order and Notation

Suppose there are treatment times \(t=0,\ldots,K\). Let \(L_0\) be the baseline covariate vector, \(A_t\) the treatment at time \(t\), and \(L_{t+1}\) the covariate vector measured after \(A_t\) and before \(A_{t+1}\), for \(t=0,\ldots,K-1\). Let \(Y\) be the final outcome, measured after \(A_K\). Define \(\bar L_t=(L_0,\ldots,L_t)\) and \(\bar A_t=(A_0,\ldots,A_t)\), with \(\bar A_{-1}\) denoting an empty history.

This convention fixes the ordering: at each time \(t\), the observed history \((\bar L_t,\bar A_{t-1})\) precedes \(A_t\); \(L_{t+1}\), when present, follows \(A_t\). Changing the measurement schedule requires changing the factorization accordingly. Assume the variables take values in standard Borel spaces, so regular conditional distributions exist. For density notation, also assume the joint law is dominated by a product of suitable reference measures. A factor may be a probability mass function, a density, or a conditional probability kernel.

The factors have distinct descriptive roles. The baseline factor \(p(l_0)\) describes the distribution of baseline covariates. Each \(p(a_t\mid\bar l_t,\bar a_{t-1})\) describes treatment assignment among records with that observed history. Each \(p(l_{t+1}\mid\bar l_t,\bar a_t)\) describes the next covariate transition given the history through the current treatment. The final factor \(p(y\mid\bar l_K,\bar a_K)\) describes the outcome conditional on the complete observed history. These descriptions do not mean the factors are independent; their conditioning histories are precisely how their dependence is represented.

Deriving the Factorization

The identity follows by repeatedly applying the chain rule for a joint distribution in the stated order. It is not a modeling assumption that the successive variables are independent.

Proof. For a finite sequence of random variables \(X_1,\ldots,X_m\) on standard Borel spaces, the chain rule expresses their joint law as the initial marginal law followed by the conditional law of each \(X_j\) given its predecessors. In density notation, where a dominated joint law is available, this is the product \(p(x_1)\prod_{j=2}^{m}p(x_j\mid x_1,\ldots,x_{j-1})\). Apply this rule to the ordered sequence \[ L_0,A_0,L_1,A_1,\ldots,L_K,A_K,Y, \] omitting \(L_{t+1}\) after the final treatment as specified. The predecessors of \(A_t\) are \((\bar L_t,\bar A_{t-1})\); those of \(L_{t+1}\) are \((\bar L_t,\bar A_t)\); and the predecessors of \(Y\) are \((\bar L_K,\bar A_K)\). Grouping the resulting factors by type gives the stated expression.

Conditional distributions are defined as regular conditional probability kernels. If two versions of the conditional distribution of a variable given a history both satisfy the defining integral identity, they agree for almost every history with respect to the marginal distribution of that history. One way to see this is to fix a measurable set \(B\) of values of the variable: both versions integrate over every measurable history set \(C\) to the same joint probability \(P(X\in B,H\in C)\). Uniqueness of Radon–Nikodym derivatives then gives equality for \(P_H\)-almost every history for that \(B\); the standard Borel setting permits a common null set using a countable determining class. Thus the law does not determine a unique conditional value at every individual history. \(\square\)

For continuous variables, a density value at a particular history is not a probability that the history occurred. In particular, a positive marginal density at \(h\) does not make the event \(H=h\) have positive probability. When joint and marginal densities exist, the conditional-density ratio formula applies \(P_H\)-almost everywhere, not as a claim that the conditional distribution is uniquely fixed at every point where a chosen density happens to be positive. Values assigned on a set of histories with marginal probability zero can be changed without changing the observed joint law.

Likelihood and Normalization

The factorization also organizes the likelihood when independent subjects are observed. Suppose there are \(n\) independent records, each observed in full, and the conditional distributions are dominated by reference measures. Write \(q_0\), \(g_t\), \(r_t\), and \(m\) for densities or mass functions corresponding respectively to the baseline, treatment, covariate-transition, and outcome factors. For a single record \(i\), its joint contribution is

Independence across subjects makes the sample likelihood the product of these contributions. Taking logarithms turns that product into a sum.

Derivation. Let \(p_\theta(o_i)\) be the joint density or mass for record \(i\), with parameter \(\theta\), and let the displayed factors be its chain-rule components. Independence gives \(L(\theta)=\prod_{i=1}^n p_\theta(o_i)\). Substituting the factorization and applying \(\log\prod_j x_j=\sum_j\log x_j\) for positive contributions yields

If a contribution is zero, the log-likelihood is \(-\infty\), consistent with the product likelihood being zero. The algebraic decomposition does not, by itself, imply that the parameters of the factors are unrelated or that separately chosen models are correctly specified. Such conclusions require additional modeling choices.

How to Read It, and What It Does Not Say

A useful way to interpret the factorization is to read from left to right and ask what information is available immediately before each variable is recorded. For example, \(p(l_{t+1}\mid\bar l_t,\bar a_t)\) is a conditional description of the next covariate given the observed history through treatment \(A_t\). It is not an assertion that the treatment caused the covariate to take its observed value. Causal interpretation requires causal assumptions and a specified intervention; the chain rule alone supplies neither.

The factorization differs from an unrestricted joint model mainly in how it organizes the distribution. An unrestricted joint model could specify the joint law of the whole record directly. The chain-rule form instead names conditional components in temporal order. It also differs from a model that assumes independence across time: no such independence is needed, because relevant past variables appear in the conditioning histories.

Diagnostics should therefore check whether each proposed conditional model is compatible with its factor's role and history. Warning signs include omitted variables in a conditioning history, reversing the order of a treatment and a covariate measured after it, or using a transition model that fails to represent observed covariate distributions across important histories. A model can fit one factor while fitting another poorly. In addition, an apparent discrepancy at one exact value of a continuous history is not, by itself, evidence of a different observed-data law: pointwise conditional versions at null histories are not identified by that law.

The factorization is limited in another important way: it is descriptive, not automatically a prescription for estimation or causal inference. It does not guarantee that a finite-dimensional parametric model for each factor is adequate. It does not address missing records, censoring, or measurement error unless those processes are explicitly included in the observed-data representation. Nor does it say that treatment probabilities should be modeled for every downstream purpose. It only states the exact conditional decomposition of the joint law under the chosen ordering.

Common Mistakes

• Calling the factors independent. The product is a chain-rule identity with conditional factors. Dependence on prior variables is carried in the conditioning histories.
• Treating a factorization as a causal result. The observed-data law can be factorized without identifying a counterfactual quantity. Causal interpretation needs additional assumptions.
• Using the wrong temporal history. A covariate measured after \(A_t\) belongs after \(A_t\) in this convention, not in the history preceding that treatment.
• Claiming pointwise identification at every continuous history. Conditional distributions are determined only up to histories with zero probability under the history distribution. A positive density value at a single point does not make that history event have positive probability.
• Assuming a correct chain rule guarantees correct models. The factorization is exact, but fitted parametric approximations to its factors can still be misspecified.