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If the parametric score function has zero conditional expectation given a covariate $X$, it is orthogonal to the centered functions of $X$ that form the covariate-density nuisance tangent space. Its efficient score then equals its parametric score function, and the unknown covariate distribution causes no loss of Fisher information. This applies to Gaussian regression scores and more generally to regular conditional models with unrestricted covariate distribution and no additional nuisance components.

Alan M. Zaslavsky develops statistical methods for measuring health-care quality and disparities. His research includes missing data, hierarchical Bayesian models and Bayesian inference, bringing statistical adjustment to comparisons between health-care providers and populations.^[ref]

With Fan Li and Kari Lock Morgan, he developed a framework for balancing covariates through propensity score weighting. Their overlap weights emphasize people with a substantial chance of receiving either treatment, defining an overlap-weighted average treatment effect for that comparison population.^[ref]

A baseline covariate is measured before treatment assignment or the exposure being studied. A prognostic baseline covariate predicts the outcome. In a randomized controlled trial, adjusting for such a covariate can improve statistical efficiency without conditioning on a consequence of treatment. For instance, a measurement taken before treatment can explain persistent between-subject variation in a later measurement.^[ref]

Here is a linear regression example with explicit noise assumptions. Letwhere treatment $Z$ is independent of the baseline covariate $X$, all displayed variances are finite, $\mathbb{E}[\epsilon \mid X,Z]=0$ and $\mathrm{Var}(\epsilon \mid X,Z)={\sigma }^2$. ThenThe first identity uses the law of total variance. It isolates the unexplained population variation removed by adjustment for $X$. For $\gamma =2$, $\mathrm{Var}(X)=1$ and ${\sigma }^2=1$, this variation falls from $5$ to $1$. Estimating additional coefficients still uses data, so this population calculation is not a guarantee that every finite-sample reported standard error becomes smaller. Measurement before treatment also does not by itself make a covariate prognostic or establish a causal effect.

A logarithmic count-component contrast $b$ multiplies the susceptible mean by $e^b$, while a zero-logit contrast $g$ multiplies the structural-zero odds by $e^g$. If the baseline zero predictor is $\eta$, the marginal mean ratio isThus the count ratio alone is not a population-average effect when a covariate changes both components. With interaction terms, the relevant count contrast must first include the interactions for the stated reference group.

Conditional front-door adjustment applies the front-door argument within strata of observed covariates and then averages over those covariates. The covariates must remove exposure-mediator confounding, while the exposure together with the covariates must remove mediator-outcome confounding.

A logistic regression can specify each binary response conditionally on recorded past outcomes and baseline covariates. A chain rule for probabilities gives the path likelihood as the product of the conditional Bernoulli distribution masses. This permits dependence within a person's history without treating their outcomes as unconditionally independent. The model must specify the initial history and which past features enter its conditional mean.

A conditional randomization test tests $Y\perp X\mid Z$ by resampling the tested variable from its conditional law given the remaining covariates. For independent sample rows, hold the observed $Y,Z$ fixed and draw each replacement vector $X^{(b)}$ from the true product conditional law of $X\mid Z$. Under the null hypothesis, the original $X$ and these replacements are conditionally exchangeable. Applying the same prespecified test statistic and fitting rule to every dataset therefore gives the valid rank-based Monte Carlo testThe add-one convention and conservative treatment of ties give finite-sample type I error control. Other resampling constructions are possible when they preserve the same conditional exchangeability.^[ref]

Replacing the true conditional law by an estimate does not automatically preserve that guarantee. For a counterexample without $Z$, let the true $X$ be a fair Bernoulli random variable and let $Y$ be constant, so independence holds. If the resampling model incorrectly uses $P(X=1)=0.99$ and the test statistic is $T=1-X$, the model assigns upper-tail probability $0.01$ whenever the observed $X=0$. That event actually occurs with probability $0.5$, so the limiting simulation test rejects far more often than level $0.05$ permits.

Estimated conditional models require a separate calibration or approximation argument. More resamples reduce Monte Carlo error but do not correct a wrong conditional law. The method tests conditional independence; interpreting it as a causal effect test requires the appropriate additional causal inference assumptions.

A confounder is a pretreatment variable relevant to removing confounding for a specified treatment-outcome question. A common cause $C$ with arrows $A\leftarrow C\to Y$ is a basic example: it creates a backdoor path between treatment $A$ and outcome $Y$.

Association with both variables is not a sufficient definition. In an appropriate causal directed acyclic graph, conditioning on a common effect can instead introduce collider bias. A confounder is a causal adjustment role, rather than any predictor with a small p-value.

One precise definition identifies a confounder as a member of some minimal sufficient adjustment set: there is a set containing $C$ that makes the relevant potential outcomes independent of treatment, and no proper subset of that set does so. The role can therefore depend on which other covariates are available and on the effect being estimated. A variable that merely reduces residual bias need not satisfy this definition.^[ref]

An observed event counting process has conditional increment $\mathbb{E}(d{N}_i(t)\mid {\mathcal{H}}_{t-})=Y_i(t)h_i(t)dt$, where $Y_i$ is the at-risk process. The filtration records the past observed events, censoring, and covariates; the integral of $Y_i{h}_i$ is the compensator of a counting process.

A covariate is an observed explanatory variable used to model the conditional distribution of a response. A vector of covariates can enter a linear predictor, a treatment-effect adjustment or a hazard multiplier. Conditioning on a covariate does not itself establish that its effect is causal.

Covariate balance means that the compared treatment groups have similar distributions of specified pretreatment covariates. Exact distributional balance in a weighted population isThis is a property of the represented covariates and chosen weights. Matching and inverse probability weighting can aim to approximate it.

Equal means are weaker than equal probability distributions. For example, a covariate taking values $-1,1$ equally often in one group and $-2,2$ equally often in the other has mean zero in both, but variances one and four. A zero standardized mean difference would miss that discrepancy. Inspect relevant spreads, tails, categories and combinations as well as means.

A nonsignificant difference test does not demonstrate covariate balance: small samples may have low statistical power, while large samples can detect negligible discrepancies. Covariate balance on measured pretreatment variables also cannot establish conditional exchangeability for unmeasured potential outcomes.

For each fixed observation pattern, the probability of that pattern may depend on observed response components and fully observed covariates, but not on the response components missing under that pattern. Conditional missing completely at random is the special case with no dependence on observed response components either.

Missingness depends only on fully observed covariates $X$, not on the response after conditioning on them. A responder-only marginal estimate may still be biased when response rates and outcome probabilities both vary across covariate strata. MAR standardization over a fully observed covariate weights within-stratum respondent estimates using covariate frequencies from the whole target cohort.

In a Cox proportional-hazards model, $h_i(t)=h_0(t)\exp \{{\beta }^T{z}_i(t)\}$. At distinct event times $t_j$, let $i_j$ identify the event subject and let $R_j$ be the risk set immediately before $t_j$. With covariates evaluated just before the event, the untied Cox partial likelihood isEach ratio is the conditional probability that the next single failure belongs to the observed subject, given its event time and current risk set. Substituting the hazards into this ratio cancels their common baseline hazard $h_0(t_j)$. With three subjects whose scalar covariates are $0,0,1$, the probability that the third subject fails next is $e^{\beta }/(2+e^{\beta })$.

A risk set includes subjects who have entered observation and remain event-free and uncensored at that instant; delayed entry and time-dependent covariates must be represented accordingly. The usual inference requires appropriate independent censoring conditional on the modeled information. Tied recorded event times require a specified treatment, such as the Breslow approximation for tied event times, the Efron approximation for tied event times, or an exact tied-set conditional likelihood for its discrete-time model. These need not give the same objective as an arbitrarily ordered sequence of failures.^[ref]

A likelihood-based analogue of the coefficient of determination compares the fitted and intercept-only log-likelihoods ${\ell }_1$ and ${\ell }_0$:For a Cox proportional-hazards model, these are maximized partial likelihood logarithms with and without the covariates. The expression measures improvement in fit, rather than the fraction of variation in event times explained. Its attainable maximum can be below one. It does not test the proportional hazards assumption or establish predictive calibration.

Doubly robust estimation combines two nuisance functions so that either one can be correctly estimated while the other is misspecified. For example, let $A\in \{0,1\}$ indicate treatment, $X$ covariates, $m_a(X)=\mathbb{E}[Y\mid A=a,X]$, and $e_a(X)=\mathbb{P}(A=a\mid X)$. Under conditional exchangeability, positivity and the usual consistency of observed and potential outcomes, ${\mu }_a=\mathbb{E}[m_a(X)]$ is the mean potential outcome under treatment $a$.

For candidate functions ${\tilde{m}}_a,{\tilde{e}}_a$ with denominators bounded away from zero, the augmented signal isConditioning on $X$ shows thatThe bias vanishes if either nuisance function is correct. This algebra explains Double robustness and the second-order product bias. It does not protect against both models being wrong, failed Causal identification, or nearly zero denominators. cross-fitting helps when flexible functions are estimated from the same sample; valid confidence intervals require further rate and moment assumptions. The average treatment effect is ${\mu }_1-{\mu }_0$.^[ref]

With signed active covariate columns $X_A^s=(s_j{x}_j{)}_{j\in A}$ and their positive-definite Gram matrix $G_A=(X_A^s{)}^T{X}_A^s$, the displayed direction has unit norm and $(X_A^s{)}^{T}u=\alpha 1$. Thus it makes equal angles with unit-length signed active covariates, and movement along it reduces their signed residual correlations at the common rate $\alpha$. The regression coefficient direction is $d_A=\mathrm{diag}(s_A)w$, with inactive components zero.

For a regression function $g_{\theta }$, let $h_{\theta }={\partial }_{\theta }{g}_{\theta }$. With independent centered normal distribution error of known variance ${\sigma }^2$, the parametric score function is $h_{\theta }(X)\epsilon /{\sigma }^2$. It has zero conditional expectation given the covariate, so unknown covariate-density nuisance directions do not remove any of its Fisher information.

G-computation calculates the outcome distribution under a specified intervention from conditional laws identified under a causal model. For a single binary treatment, pretreatment covariates $X$, and outcome $Y$, the G-formula isIt follows by averaging over $X$, using conditional exchangeability to replace the conditional potential outcome mean by the mean among recipients of $a$, and using consistency in causal inference to replace $Y(a)$ by observed $Y$. Positivity ensures the required conditional outcome laws are represented in the target strata.^[ref]

For an illustrative binary outcome, suppose target weights are $0.8$ and $0.2$ in two baseline strata. Untreated risks $0.1,0.4$ give intervention risk $0.8(0.1)+0.2(0.4)=0.16$. Treated risks $0.05,0.2$ give $0.08$, so the standardized causal risk ratio is $0.5$ and the risk difference is $-0.08$. Both strategies use the same target weights.

With repeated treatments and time-varying covariates, the longitudinal G-formula integrates products of the covariate transition laws while setting each treatment according to the intervention strategy. This requires sequential exchangeability, longitudinal positivity and consistency in causal inference. Simply conditioning on every observed intermediate variable can block part of the intended effect; the sequential intervention calculation specifies which laws are averaged.

When event times are observed only in intervals and the covariate vector is constant within an interval, a proportional hazards model gives $p_{ij}=1-\exp [-\Delta H_{0j}\exp ({\beta }^T{z}_i)]$. Thus the interval event probability has a complementary log-log link, with ${\alpha }_j=\log \Delta H_{0j}$ an interval-specific intercept. The person-period Bernoulli likelihood models several events in one interval directly, without inventing their order. It needs appropriate independent censoring assumptions and a consistent account of interval entry and observation.