Conditional Balance Within History Strata

Why Look Within History Strata?

In longitudinal treatment data, the treatment assigned at time \(t\) may depend on earlier treatment and covariate history. A pooled comparison of the covariates between people who do and do not receive \(A_t=1\) can obscure that dependence: imbalances in different histories may point in opposite directions and cancel in the aggregate. Conditional balance asks a more targeted question: within a selected prior-history stratum, do the weighted treatment groups have comparable distributions of measured pre-treatment covariates?

This is a diagnostic for the treatment weighting at a particular time, not by itself evidence that a causal effect is identified. The probability-calibration material concerns whether fitted treatment probabilities agree with observed frequencies; conditional balance examines the covariate distributions induced by the resulting weights. Neither diagnostic can establish that all relevant confounding has been measured.

Notation and Definition

Let \(A_t\) denote treatment at time \(t\), and let \(H_t\) denote the measured history available immediately before that treatment. It includes prior treatments and the measured covariates observed by time \(t\). Write \(g_t(a\mid H_t)=P(A_t=a\mid H_t)\) for the treatment probability. For a binary treatment, the unstabilized cumulative treatment weight through time \(t\) is

For a selected stratum, let \(S_t\) be a function of prior treatments and selected measured covariate history, defined without using the current treatment \(A_t\). The stratum may be exact or coarsened; for example, it may fix a prior-treatment pattern and place baseline age into a specified interval. A balance comparison within \(S_t=s\) considers the weighted distributions of pre-\(A_t\) covariates across current treatment groups.

For a continuous covariate \(X_t\), a mean comparison is only one summary. Distributional comparisons can also use quantiles, empirical cumulative distribution functions, or plots. A small difference in the mean does not guarantee that the distributions are alike.

What the Ideal Weights Do Within a History

The population justification for this diagnostic is an algebraic property of inverse-probability weighting. Suppose treatment takes one of \(K\) possible values, every \(g_t(a\mid h)\) is positive on histories under consideration, and the probabilities used in the weights equal the true conditional probabilities. Conditional on a fixed history \(H_t=h\), earlier factors in \(W_t\) are fixed: they depend on history already observed. The factor for the current treatment cancels its conditional assignment probability.

Proof. Conditional on \(H_t=h\), let \(C(h)\) denote the product of the weight factors for times \(s<t\). It is constant across the possible values of \(A_t\) at this fixed history. The original conditional probability of \(A_t=a\) is \(g_t(a\mid h)\), so the unnormalized weighted probability is

This quantity is identical for all \(K\) treatment values. Dividing by the sum over those values, \(K C(h)\), gives weighted conditional probability \(1/K\) for each. Since \(f(H_t)\) is fixed at \(h\), its distribution conditional on that exact history cannot differ across current-treatment groups. More generally, averaging over histories within a stratum yields balance only when the weighting and the conditioning used for the comparison support that conclusion; a coarse stratum can mix histories with different covariate distributions. Thus practical diagnostics compare the covariates across treatment groups within the chosen strata rather than treating this exact-history identity as proof of balance in every coarsened stratum. \(\square\)

Stabilized weights require a related but slightly different interpretation. If their numerator at time \(t\) is \(q_t(A_t\mid A_0,\ldots,A_{t-1})\), then after weighting the conditional treatment distribution is \(q_t(a\mid A_0,\ldots,A_{t-1})\), not necessarily uniform. Because this numerator depends on prior treatment but not on the measured covariate history, the intended diagnostic is often to compare current treatment groups within prior-treatment strata. The expected balance is conditional on that prior treatment history, not necessarily marginal over it.

Marginal Balance Can Conceal Conditional Imbalance

Let \(S\) be a history stratum, \(X\) a covariate, and \(\mu_{as}=E[X\mid A_t=a,S=s]\). The marginal mean difference can be written as

This expression combines both the within-stratum means and the stratum proportions in each treatment group. Even if the stratum proportions are equal between groups, positive and negative conditional differences can cancel. If the proportions differ, the pooled contrast also reflects that difference in composition.

Coarsened Strata and What Is Actually Fixed

Stratification makes conditional comparisons interpretable, but the exact information fixed by a stratum must be stated carefully. If a stratum is defined by the exact value of a categorical covariate, that covariate has the same value for everyone in the stratum, so its balance cannot be assessed there. By contrast, if a continuous covariate is divided into ranges, only the range is fixed. The values and distribution within that range may still differ between treatment groups.

Practical Comparisons and Failure Modes

Conditional balance complements, rather than replaces, pooled balance summaries. A pooled standardized mean difference is concise and useful for an overall comparison, but it can conceal stratum-specific discrepancies. Conversely, exact stratification can become unstable when many strata contain few observations. Probability calibration checks whether fitted probabilities match observed treatment frequencies; a calibrated model can still leave covariate imbalance in the weighted sample, and apparent balance does not by itself demonstrate calibration.

For a categorical covariate whose exact value defines a stratum, do not report its within-stratum balance as evidence: the comparison has fixed it by construction. For a categorized continuous covariate, report that the category is fixed, but inspect within-category values if relevant. More generally, choose strata before comparing current treatment groups, use the same clearly defined stratum rules across groups, and report how many observations and how much total weight each cell contains.

Sparse cells are a warning. A stratum with no observations in one treatment group cannot support a within-stratum comparison; a cell with very few observations may yield highly variable summaries. Large differences in weights can also make nominally large cells effectively depend on only a few observations. Estimated treatment probabilities, model misspecification, and limited overlap can all leave residual conditional imbalance. These diagnostics describe measured covariates and observed histories; they cannot detect imbalance in unmeasured variables.

The most useful report makes the conditioning explicit: name the treatment time, define each history stratum, identify the covariates compared, state whether weights are stabilized, and present both cell-level information and a suitable balance summary. Conditional balance is a diagnostic for the treatment assignment process represented by the measured history and fitted weights; it is not a guarantee of causal validity or of balance on every possible feature.